Monthly Notices of the Royal Astronomical Society

Huang,, Shuiyao;Katz,, Neal;Davé,, Romeel;Oppenheimer, Benjamin, D;Weinberg, David, H;Fardal,, Mark;Kollmeier, Juna, A;Peeples, Molly, S

doi: 10.1093/mnras/staa135pmid: N/A

ABSTRACT Many phenomenologically successful cosmological simulations employ kinetic winds to model galactic outflows. Yet systematic studies of how variations in kinetic wind scalings might alter observable galaxy properties are rare. Here we employ gadget-3 simulations to study how the baryon cycle, stellar mass function, and other galaxy and CGM predictions vary as a function of the assumed outflow speed and the scaling of the mass-loading factor with velocity dispersion. We design our fiducial model to reproduce the measured wind properties at 25 per cent of the virial radius from the Feedback In Realistic Environments simulations. We find that a strong dependence of η ∼ σ5 in low-mass haloes with |$\sigma \lt 106\mathrm{\, km\, s^{-1}}$| is required to match the faint end of the stellar mass functions at |$z$| > 1. In addition, faster winds significantly reduce wind recycling and heat more halo gas. Both effects result in less stellar mass growth in massive haloes and impact high ionization absorption in halo gas. We cannot simultaneously match the stellar content at |$z$| = 2 and 0 within a single model, suggesting that an additional feedback source such as active galactic nucleus might be required in massive galaxies at lower redshifts, but the amount needed depends strongly on assumptions regarding the outflow properties. We run a 50 |$\mathrm{Mpc}\, h^{-1}$|⁠, 2 × 5763 simulation with our fiducial parameters and show that it matches a range of star-forming galaxy properties at |$z$| ∼ 0–2. methods: numerical, galaxies: evolution, galaxies: general 1 INTRODUCTION Galactic scale outflows (galactic winds) driven by star formation processes have been recognized as a critical ingredient in galaxy evolution. Galactic winds are observed ubiquitously among star-forming galaxies in both the local and distant Universe, and their properties are often found to correlate with the properties of the central galaxy such as the star formation rate (SFR) and the circular velocity (Martin 2005; Rupke, Veilleux & Sanders 2005; Heckman & Borthakur 2016). The short-lived, massive stars formed in star-forming galaxies release a considerable amount of energy and momentum during their short lifetimes through radiation, stellar winds, and supernova (SN) explosions. Collectively, these effects could efficiently drive the large-scale outflow of dense, metal-enriched gas from the interstellar medium (ISM) to large distances from the galaxy, making a strong impact on galaxy growth and also on the properties of the circumgalactic medium (CGM). Galactic winds have been implemented as a subgrid model in cosmological simulations, in which they play a critical role in explaining the suppressed star formation in dwarf galaxies and the metal content in the CGM (e.g. Oppenheimer & Davé 2008; Oppenheimer et al. 2012; Ford et al. 2013). However, implementing galactic winds in cosmological simulations remains a challenge because of our limited knowledge of the wind driving mechanism, and the limited resolution of large-volume simulations. Self-consistently generating galactic winds by explicitly modelling the key wind driving mechanisms is still a challenging problem that is under active study (Zhang 2018). More importantly, the physical processes that are critical to driving winds occur on scales that are so small that they remain unresolvable in even the highest resolution zoom-in simulations of today (e.g. Scannapieco & Brüggen 2015; Schneider & Robertson 2017; Hopkins et al. 2018). As a consequence, modern cosmological simulations adopt a variety of subgrid prescriptions that describe how to launch galactic winds from simulated galaxies (Springel & Hernquist 2003; Oppenheimer & Davé 2006; Stinson et al. 2006; Agertz et al. 2013; Schaye et al. 2015; Hopkins et al. 2018; Pillepich et al. 2018a). This diversity of numerical recipes for galactic winds leads to many different predictions from these simulations (Scannapieco et al. 2012; Sadoun et al. 2016; Sembolini et al. 2016b; Valentini et al. 2017). The kinetic feedback models (Springel & Hernquist 2003; Oppenheimer & Davé 2006), like those that we employ, rely on scaling relations that connect the macroscopic properties of galactic winds, such as the wind velocity |$v$||$\mathrm{ w}$| and the mass-loading factor η, defined as the ratio between the outflow rate (⁠|$\dot{M}_\mathrm{ w}$|⁠) and the SFR to the resolved properties of their host galaxies such as the halo mass Mh, or some characteristic velocity (e.g. the velocity dispersion σ). Though the properties of galactic winds and the physical mechanisms that generate them are still poorly understood, there have been many constraints on these scaling relations from observations, analytic calculations, and simulations (Murray, Quataert & Thompson 2005; Rupke et al. 2005; Murray, Ménard & Thompson 2011). The fiducial wind prescription that we have used in many of our previous papers (e.g. Oppenheimer & Davé 2006; Davé et al. 2013; Ford et al. 2016) was motivated by the analytic momentum-driven and energy-driven wind models developed by Murray et al. (2005). In the momentum-driven scenario, the outflow is driven in a momentum-conserving manner by the radiation pressure from massive stars and SNe acting on the dust particles that is coupled to the cool gas. The momentum flux overwhelms the gravitational potential of the dark matter halo in early phases and accelerates the cool gas from within the star-forming region to an asymptotic velocity at the virial radius of the dark matter halo. Assuming an isothermal potential and ignoring hydrodynamic forces, Murray et al. (2005) derived the evolution of the wind speed as a function of radius as $$\begin{eqnarray*} v_\mathrm{ w}(r) = 2\sigma _\mathrm{1D}\sqrt{(f_\mathrm{L}- 1)\ln \left(\frac{r}{R_0}\right)} , \end{eqnarray*}$$ (1) where σ1D is the one-dimensional velocity dispersion measured for an isothermal sphere, fL = L/LM is the ratio between the luminosity of the galaxy and the critical Eddington luminosity, and R0 is the radius from which the wind is launched. They also derived the scalings between η and σ as η ∝ σ−1 based on the conservation of momentum. In our more recent simulations, we actually assume that η ∝ σ−2 for small galaxies, which is the scaling one would expect for energy-driven winds by an SN. Even if this is not the correct physics behind real galactic winds, this modified momentum-driven model predicts scaling relations between global quantities such as mass loading, wind velocity, and the stellar mass that, when included in cosmological hydrodynamic simulations, are broadly consistent with many observational constraints (Oppenheimer & Davé 2006, 2008; Davé et al. 2010; Oppenheimer et al. 2010; Davé, Finlator & Oppenheimer 2011b; Davé, Oppenheimer & Finlator 2011a; Davé et al. 2013; Ford et al. 2016). However, implementing the wind model into our simulations is more complicated than suggested by the above equations. Instead of launching a wind from any radius R0 as in equation (1), we eject wind particles with an initial velocity |$v$||$\mathrm{ w}$| from star-forming regions that inhabit the centre of the galactic potential and let them propagate out under the combined gravitational and hydrodynamical forces (we ignore hydrodynamic interactions for a short period after wind launch; see below for details). Furthermore, the dynamical evolution of wind particles in our simulation is very different from the analytical solution of Murray et al. (2005) for several reasons. First, the gravitational potentials in our simulated haloes are steeper than the isothermal sphere assumed in Murray et al. (2005), especially in the central region where baryonic matter dominates. Secondly, our simulations do not explicitly include radiation pressure, which in their calculation accelerates the outflow all the way out to the virial radius. Thirdly, wind particles in our simulation are further slowed down by hydrodynamic interactions with the gas in the CGM or the intergalactic medium (IGM). Finally, these interactions are probably not accurately evolved owing to resolution and other numerical issues. Recent zoom-in simulations of individual galaxies provide further insights into the scaling relations between the launched winds and their host galaxies (Muratov et al. 2015; Christensen et al. 2016). Capable of resolving GMC scales and the turbulent nature of the ISM, these simulations drive winds by explicitly modelling physical processes that depend on the local ISM properties and analyse how the wind behaviours depend on the global properties of their host galaxies, therefore better bridging the gap between the governing physics on small scales and the impact of the winds in the broader context of galaxy formation [but still not resolving all the scales critical for driving winds (e.g. Schneider & Robertson 2017)]. Using a series of simulations that span four decades in halo mass up to |$10^{12}\, \mathrm{M}_\odot$| and covering a redshift range from |$z$| = 0 to 4, the Feedback In Realistic Environments (FIRE) project (Muratov et al. 2015, hereafter M15) derives how mass-loading factors and wind speeds depend on the circular velocity, the halo mass, and the stellar mass of the host galaxies. They report faster wind speeds in massive haloes than in our previous simulations using the fiducial wind model described above (and in more detail below). They also report a stronger scaling between the mass-loading factor and the circular velocity with |$\eta \propto v_\mathrm{c}^{-3.3}$|⁠, steeper than the energy-driven wind scaling, |$\eta \propto v_\mathrm{c}^{-2}$|⁠, which we assume in our simulations for low-mass galaxies. Christensen et al. (2016) simulate and analyse over 20 spiral and dwarf galaxies covering halo masses from |$10^{9.5}$| to |$10^{12}\, \mathrm{M}_\odot$|⁠. Despite using a very different feedback model, they obtain a similar scaling for the mass-loading factor, |$\eta \propto v_\mathrm{c}^{-2.2}$|⁠. One key issue is that M15 report their results at R25, one quarter of the virial radius, while by necessity we impose our wind scalings at wind launch, which occurs inside star-forming regions within the galaxy at much smaller radii. Clearly, it makes more sense to talk about galactic wind properties outside the galaxy and R25 is a reasonable radius to choose. As we discuss below, M15 motivated us to look at our wind scaling properties at R25, and we find that they are very different from those at launch. Motivated by this recognition, in this paper we revisit the basic assumptions made in our subgrid wind model. In particular, we recalibrate our prescriptions for launching winds from galaxies using the scaling relations found in the FIRE simulations as constraints. We will examine how the new prescription, now capable of qualitatively reproducing the wind behaviours seen in the FIRE simulations, will affect some of the basic predictions of our cosmological simulations, such as the galactic stellar mass functions (GSMFs) and the galactic mass–metallicity relation (MZR) at various redshifts. Furthermore, we also experimented with a range of wind parameters, all allowed by current observational and theoretical constraints on galactic winds, to examine the robustness of these predictions to small changes in the wind implementation and were surprised to find that these basic observational quantities were actually very sensitive to small changes in the wind scalings, changes that are much smaller than the differences between wind models employed by different simulation groups (e.g. Agertz et al. 2013; Schaye et al. 2015; Davé, Thompson & Hopkins 2016; Pillepich et al. 2018a). Recent cosmological simulations also often adopt a subgrid active galactic nucleus (AGN) feedback model and show that it is crucial to reproduce the number density of massive galaxies and the fraction of red galaxies at low redshifts to match observations. The simulations presented in this paper, like our past published work, do not include any such subgrid model for AGN feedback, or any other mechanism that has been proposed in the literature to specifically quench star formation in massive galaxies (e.g. Crain et al. 2015; Davé et al. 2016; Weinberger et al. 2016). Simulations without AGN feedback tend to produce too many blue massive galaxies, indicating that the stellar feedback alone is insufficient to keep these galaxies quenched. However, the implementation of AGN feedback often has little effect at higher redshift or in small galaxies, where stellar feedback dominates galaxy growth. The implementation of any AGN feedback model that includes free parameters that are tuned to match observations will thus inevitably be affected by the stellar feedback models. Therefore, in this paper we will not focus on reproducing the observed Universe in every detail, but we will rather explore how sensitively galaxy evolution depends on the star formation-driven wind model. This knowledge will also help characterize the limits of what stellar feedback alone can accomplish and thus provide further constraints on any additional required feedback mechanism. The paper is organized as follows. Section 2 reviews our original subgrid model for launching winds in our simulations and also introduces the new wind algorithm. Section 3 describes our cosmological simulations and introduces the test simulations that we use in this paper. Section 4 studies how sensitively our simulations depend on the parameters of the wind algorithm by comparing results from the test simulations. Section 5 studies in detail how stellar mass grows within galaxies in our simulations by focusing on their accretion and merger histories, and how it is affected by certain wind parameters. It also discusses the challenge of matching observations relying only on stellar-driven winds and the requirements for any additional feedback mechanism. Section 6 presents results from our high-resolution simulation using the new wind algorithm with a fiducial choice of wind parameters and compares them to results from earlier published versions of our cosmological simulations. Section 7 summarizes our results. 2 THE KINETIC FEEDBACK MODEL 2.1 Our published wind model We based our original subgrid wind model (ezw; Davé et al. 2013; Ford et al. 2016) on the analytic formulation of energy-driven and momentum-driven winds from Murray et al. (2005). Here we summarize our numerical algorithm for including it in our simulations. For any SPH particle i in a galaxy that is above a critical density threshold ρSF for star formation, we determine whether or not to turn it from a normal SPH particle into a wind particle according to a probability pi that scales with the local SFR: $$\begin{eqnarray*} p_i \propto \eta \times \mathrm{SFR}_i. \end{eqnarray*}$$ We choose the critical density threshold as ρSF = 0.13 μmH (Springel & Hernquist 2003), where μ is the mean atomic weight and mH is the mass of the hydrogen atom. Once an SPH particle becomes a wind particle, we add a velocity boost of |$v$||$\mathrm{ w}$| to the particle in the direction of |$\boldsymbol {v_i}\times \boldsymbol {a_i}$|⁠, where |$\boldsymbol {v_i}$| and |$\boldsymbol {a_i}$| are the velocity and acceleration of the particle, respectively, before launch, as outflows are often seen perpendicular to the disc where the resistance from the cold dense ISM is minimized. All hydrodynamical interactions relating to the newly created wind particle are ignored for an interval of tdelay = 20 kpc/|$v$||$\mathrm{ w}$| or until the particle has reached a density threshold of ρth = 0.1ρSF. This decoupling from hydrodynamical forces ensures that wind particles are able to escape the disc where the resolution is insufficient to correctly model the hydrodynamical interactions (Dalla Vecchia & Schaye 2008). The two free parameters, η and |$v$||$\mathrm{ w}$|, are crucial to the wind model, whose values are constrained from the analytical scalings that correlate them with the galaxy velocity dispersion σ. For our preferred published model (Davé et al. 2013), which we refer to as the ezw model, those scalings are as follows: $$\begin{eqnarray*} v_{\mathrm{ w}; ezw} = 4.29\sigma \sqrt{f_\mathrm{L}- 1} + 2.9\sigma \end{eqnarray*}$$ (2) $$\begin{eqnarray*} \eta =\left\lbrace \begin{split} &\frac{150\mathrm{\, km\, s^{-1}}}{\sigma }\frac{75\mathrm{\, km\, s^{-1}}}{\sigma } &(\sigma \leqslant 75\mathrm{\, km\, s^{-1}})\\ &\frac{150\mathrm{\, km\, s^{-1}}}{\sigma } &(\sigma \geqslant 75\mathrm{\, km\, s^{-1}}) \end{split} ,\right. \end{eqnarray*}$$ (3) where fL depends on the metallicity of the SPH particle as constrained by observations (Rupke et al. 2005), and we adjust the normalization factor σ0 = 150 km s−1 to match the enrichment level of the high-redshift IGM (Oppenheimer & Davé 2008). This original wind velocity formula (equation 2) rescales the launch wind velocity by adding 2.9σ in an attempt to get the correct asymptotic velocity at the virial radius, to account for the dynamical evolution of the winds inside the halo. However, as we will show in Section 3.1, this does not work very well. The formula for η introduces a characteristic velocity |$\sigma _{ezw} = 75\ \mathrm{\, km\, s^{-1}}$| that separates the momentum-driven wind scaling regime from the energy-driven one. The momentum-driven mass-loading factor, which scales with σ−1, applies to relatively large systems where outflows could be driven primarily by the momentum flux from young stars and SNe while the thermal energy from SNe would be dissipated too quickly to become dynamically important. However, in dwarf galaxies with σ below this limit, we assume that energy feedback from SNe starts to dominate, based on analytical and numerical models by Murray, Quataert & Thompson (2010) and Hopkins, Quataert & Murray (2012). In this energy conserving regime, we assume η ∝ σ−2. Whether or not the physical models behind these scaling relations are correct, Davé et al. (2013) show that this hybrid scaling leads to better agreement with both the low mass stellar mass and H i mass functions at |$z$| = 0. We determine the velocity dispersion σ of the host halo on the fly. We identify galaxies using a friends-of-friends (FoF) algorithm that binds star-forming particles to their closest neighbours. We estimate the velocity dispersion using the total mass of the galaxy Mgal: $$\begin{eqnarray*} \sigma _\mathrm{FoF}= 200\left(\frac{M_\mathrm{gal}}{5\times 10^{12}\, \mathrm{M}_\odot }\frac{H(z)}{H_0}\right)^{1/3} \mathrm{\, km\, s^{-1}} , \end{eqnarray*}$$ (4) where Mgal is the total mass of the FoF group, and H(⁠|$z$|⁠) and H0 are the Hubble constants at redshifts |$z$| and 0, respectively. Mgal includes dark matter, gas, and stars and we choose the FoF linking length to be smaller than one that would make Mgal equal to the virial mass (see Section 2.3 for details). In principle, we could measure the velocity dispersion σFoF for each galaxy directly. However, uncertainties arise owing to poor resolution particularly in the inner regions of each galaxy (Oppenheimer & Davé 2006). Moreover, the numerical noise would in some cases yield an unrealistic σFoF that would lead to unphysical results. Finally, satellite galaxies often have their σFoF overestimated owing to contamination by particles that actually belong to the central galaxy but that are impossible to separate. Therefore, we use the above empirical relation given that any error that arises from using this relation is subdominant to the uncertainties that come from our assumptions about the wind model itself. 2.2 A new algorithm for launching winds The wind speed formula above (equation 2) derives from the analytic calculations of Murray et al. (2005) (equation 1 in this paper) for radiation-driven winds. However, as discussed in the introduction, the actual propagation of wind particles in SPH simulations is very different from that assumed in this analytic model. In the simulations, a wind particle is initially decoupled from the hydrodynamics until it reaches the critical density ρth, typically several kiloparsecs from the galactic centre. After that, the particle can interact hydrodynamically with the surrounding gas and will slow down and heat. How this occurs depends on the numerical details of the hydrodynamic solver, since the interaction is poorly resolved. In dwarf galaxies, the winds are typically much faster than the escape velocity and are able to escape their host haloes, but most winds in massive galaxies only travel to a certain distance from the galaxy and eventually fall back within a recycling time-scale trec (Oppenheimer et al. 2010). We will show in Section 4 that both the distances to which wind particles travel and their recycling time-scales are highly sensitive to the initial wind speed and numerical resolution. This leads to large uncertainties in the evolution of galaxies and their CGM properties because the behaviour of the ejected wind particles has a crucial impact on the gas supply for star formation inside galaxies, and the density, temperature, and metal distribution in the surrounding halo gas. Here we present an improved algorithm to determine how winds are ejected from their host galaxies. In this new method, we keep the velocity of a wind particle relative to its host galaxy constant until it reaches the density threshold ρth where the particle recouples hydrodynamically to the other gas. We choose the density threshold to be 0.1 ρSF, where ρSF is the physical density threshold above which star formation occurs in the simulation. Therefore, before recoupling, the wind particle effectively also decouples from gravity so that its kinetic evolution remains temporarily independent of the central potential dominated by baryons. The wind particle still contributes to the overall gravitational field as it leaves the disc, preventing galaxies with strong outflows from having large artificial dynamical disturbances in the disc. We also adopt a new formula for the initial wind speed that is parametrized differently from before. As we will see in Section 3.1, this results in |$v$||$\mathrm{ w}$| ∝ |$v$|c at R25 as found in M15 and Murray et al. (2005). We keep the tangential velocity relative to the galaxy fixed so that the wind particle conserves its angular momentum as it is launched. We determine the radial component of the velocity by $$\begin{eqnarray*} v_\mathrm{ w} = \alpha _v\sigma \sqrt{f_\mathrm{L}}\left(\frac{\sigma }{50\ \mathrm{km\ s^{-1}}}\right)^{\beta _v} , \end{eqnarray*}$$ (5) where α|$v$| and β|$v$| are free parameters that will be discussed later, and fL is the metallicity-dependent ratio between the galaxy luminosity and the Eddington luminosity. We adopt the Oppenheimer & Davé (2006) formula for fL: $$\begin{eqnarray*} f_\mathrm{L} = f_{\mathrm{L};\odot }\times 10^{-0.0029(\log Z_\mathrm{gal} + 9)^{2.5} + 0.417\,694} , \end{eqnarray*}$$ (6) where fL; ⊙ varies randomly between 1.05 and 2. Here we now use the mass-weighted average metallicity Zgal of the host galaxy to compute fL, instead of directly using the metallicity of the wind particle as in their paper and in our past work. This is more appropriate since it is the global properties of the galaxy that will determine the wind properties at R25 and the metallicities may have large variances inside a galaxy. In most galaxies, fL is only slightly above 1. Note that the |$\sqrt{f_\mathrm{L}-1}$| term in the original ezw velocity formula (equation 2) becomes |$\sqrt{f_\mathrm{L}}$| after adding in the kinetic energy lost to the gravitational field as the wind particle climbs up the potential, so that our winds can match the asymptotic velocity predicted by equation (1). In the original formula, this correction is included as the second term 2.9σ, which is normalized at a radius of Resc = 0.1Rvir. Note that this is different from the R0 that appears in the analytic formula (equation 1). Oppenheimer & Davé (2006) chose R0 = 0.01Rvir to obtain the constant factor 4.29 in the first term of equation (2). In our new formula, we use the same normalization radius Resc = R0 for the two terms. We incorporate the freedom of choosing R0 into the parameter α|$v$|. The parameters α|$v$| and β|$v$| determine the overall wind speed and its scaling with the depth of the halo potential. Since σ scales with |$M_\mathrm{gal}^{1/3}$|⁠, the wind speed scales with the halo mass by a power-law index (1 + β|$v$|)/3. The parameter β|$v$|, therefore, characterizes how much momentum the wind particles need to overcome the central, baryon-dominated gravitational potential and reflects how the central potential varies with halo mass. The parameter α|$v$| reflects the uncertainties in choosing the normalization radius R0 in equation (1) and in measuring the σ from simulations. We calibrate our parameters to make our wind scalings at R25 consistent with the results from M15 (see Section 3.1 for details). Note that this wind speed should not be directly compared with observations because this wind speed formula only applies to winds that are close to the disc (R0), where they are launched, while observationally the location of the winds is usually unknown. In addition to the wind speed, we also explore different choices for the mass-loading factor scalings. Instead of the original formula (equation 3), we now parametrize η as $$\begin{eqnarray*} \eta =\left\lbrace \begin{split} &\alpha _\eta \left(\frac{150\mathrm{\, km\, s^{-1}}}{\sigma }\right)\left(\frac{\sigma _{ezw}}{\sigma }\right)^{\beta _\eta }(1 +z)^{1.3} &(\sigma \leqslant \sigma _{ezw})\\ &\alpha _\eta \left(\frac{150\mathrm{\, km\, s^{-1}}}{\sigma }\right)(1 +z)^{1.3} &(\sigma \geqslant \sigma _{ezw}) \end{split} ,\right. \end{eqnarray*}$$ (7) where αη is a normalization factor and βη is the power-law index. Therefore, the mass-loading factor η still follows a momentum-driven wind scaling η ∝ σ−1 in massive systems above a certain threshold σezw. Below that threshold, |$\eta \propto \sigma ^{-(1+\beta _\eta)}$|⁠. The original energy-driven scaling in small galaxies corresponds to βη = 1. However, we will show in later sections that to match the observed number densities of small galaxies at high redshifts requires a higher βη value. We also adopt a redshift-dependent factor (1 + |$z$|⁠)1.3 motivated by the results from the FIRE simulations (M15). To avoid unphysically large η at high redshifts, we only allow η to change with |$z$| at |$z$| < 4 and use a constant factor of 51.3 at |$z$| > 4. 2.3 The FoF group finder Both the mass-loading factor and the wind speed in our wind algorithm depend on the properties of the galaxy from which the winds are launched. To identify galaxies and their host haloes on the fly, we use an FoF group finder with a linking length of 0.04 times the mean interparticle spacing, and multiply the resulting mass by a constant factor determined via a calibration against the results from using so, a more accurate but more computationally costly spherical overdensity halo finder (Kereš et al. 2005; Huang et al. 2019). Above the mass resolution limit, the on-the-fly FoF group finder typically underestimates the total mass (including both baryons and dark matter) by a factor of 2–3 with a scatter of ∼0.1 dex. The discrepancies between the estimated halo masses are nearly scale independent, introducing only a small additional factor of σ0.05 to the wind scalings (equations 5 and 7). Furthermore, we now identify the group centre and velocity centroid by including all cold baryons (including star-forming gas and stars) within the FoF group instead of just using stars as in our previous work. This is especially important for dwarf galaxies that have only begun assembling, which can be almost devoid of stars. 2.4 Wind energy In this section, we calculate the total kinetic energy flux of the star formation-driven winds according to our new wind speed formula (equation 5). For a total mass M* of stars formed, the total kinetic energy added to the winds that are generated as a consequence is $$\begin{eqnarray*} E_\mathrm{kin}(M_*) = \frac{1}{2}M_\mathrm{ w}\bar{v}_\mathrm{ w}^2 , \end{eqnarray*}$$ (8) where the amount of winds launched over a given time M|$\mathrm{ w}$| relates to the amount of star formation by M|$\mathrm{ w}$| ≡ ηM*, and |$\bar{v}_\mathrm{ w}$| is the average wind speed. Combining this equation with the definition of η and |$v$||$\mathrm{ w}$| results in $$\begin{eqnarray*} E_\mathrm{kin} = 0.7\bar{f_\mathrm{L}}\alpha _v^2\sigma ^2\left(\frac{106}{\sigma } \right)^{\beta _\eta -2\beta _v+1}(1+z)^{1.3}M_* , \end{eqnarray*}$$ (9) where |$\bar{f_\mathrm{L}} \sim 1.0$| is the average value for the luminosity factor that appears in equations (1) and (5). Therefore, the wind energy generated per solar mass of star formation is $$\begin{eqnarray*} E_\mathrm{kin}(\mathrm{M}_\odot) \, = \, 1.6\times 10^{47}\nonumber \\ && \times \alpha _v^2 (2.1)^{2\beta _v} \left(\frac{106}{\sigma }\right)^{\beta _\eta -2\beta _v-1}\nonumber \\ && \times (1+z)^{1.3}\ \mathrm{erg} . \end{eqnarray*}$$ (10) Using the parameters for our fiducial simulation: α|$v$| = 4.0, β|$v$| = 0.6, and βη = 4.0, the wind energy for massive galaxies with |$\sigma \gt 106\mathrm{\, km\, s^{-1}}$| is $$\begin{eqnarray*} E_\mathrm{kin}(\mathrm{M}_\odot) = 6.3\times 10^{48}\left(\frac{106}{\sigma } \right)^{-2.2}(1+z)^{1.3}\ \mathrm{erg}. \end{eqnarray*}$$ (11) One can compare this value to the amount of energy available from Type II SNe. According to a Chabrier IMF, the average number of type II SNe per one solar mass is ηSN = 8.3 × 10−3. Assuming each SN produces ESN = 1051 erg of energy, the total energy produced by SN per solar mass of stars formed is ϵSN = 8.3 × 1048 erg. In our new wind model, this equals the wind energy from a galaxy at |$z$| = 0 with |$\sigma _\mathrm{FoF}\sim 120\mathrm{\, km\, s^{-1}}$|⁠. After taking into account the factor of ∼31/3 underestimate of the real σ by the on-the-fly FoF finder and using |$v_\mathrm{c}= \sqrt{2}\sigma$|⁠, this corresponds to a circular velocity of |$v_\mathrm{c}\sim 240\mathrm{\, km\, s^{-1}}$|⁠. For lower mass galaxies, the wind energy is a fraction (<1) of the energy available from SNe, but for massive galaxies it is larger. Hence, in addition to equation (5), we adopt an upper limit for the wind speed that requires the total kinetic energy of winds to be less than 5 times the total available energy from type II SNe. Of course, in these more massive systems one expects the winds to be dominated by photon momentum if one takes the model seriously. Even so, we think it prudent to limit the total wind energy. The calculation above shows that the upper limit only matters at high redshifts or in the most massive systems. 3 SIMULATIONS We implemented the new wind algorithm into our SPH code based on gadget-3 [see Springel (2005) for reference]. The code includes several recent numerical improvements in the SPH technique (Huang et al. 2019). To summarize, we use the pressure–entropy formulation (Hopkins 2013) of SPH to integrate the fluid equations and a quintic spline kernel to measure fluid quantities over 128 neighbouring particles. We also use the Cullen & Dehnen (2010) viscosity algorithm and artificial conduction as in Read & Hayfield (2012) to capture shocks more accurately and to reduce numerical noise. Both the artificial viscosity and the conduction are turned on only in converging flows with |$\nabla \cdot \boldsymbol {v} \lt 0$| to minimize unwanted numerical dissipation. We also include the Hubble flow while calculating the velocity divergence. Our fiducial code leads to considerable improvements in resolving the instabilities at fluid interfaces in subsonic flows and produces consistent results with other state-of-art hydrodynamic codes in various numerical tests (Sembolini et al. 2016a, b; Huang et al. 2019). In the current version, we also add metal-line cooling including photoionization effects for 11 elements as in Wiersma, Schaye & Smith (2009), and we recalculate cooling rates according to an updated ionizing background (Haardt & Madau 2012). The star formation processes are modelled as in Springel & Hernquist (2003), which includes a subgrid model for the multiphase ISM in dense regions with nH > 0.13 cm−3, and a star formation recipe that is scaled to match the Kennicutt–Schmidt relation. In this paper, we will distinguish SPH particles as star forming or non-star forming based on whether or not their densities are higher than this density threshold. We specifically trace the enrichment of four metal species C, O, Si, and Fe that are produced from type II SNe, type Ia SNe, and AGB stars as in Oppenheimer & Davé (2008). These processes also generate energy that we put in the simulations with subgrid models. However, the input energy from these feedback processes only has subdominant effects to galaxy formation compared to the wind feedback (Oppenheimer & Davé 2008). We evolve the fiducial simulation (RefHres) for this paper in a comoving periodic box with L = 50 h−1 Mpc on each side that initially contains 2 × 5763 gas and dark matter particles. The initial masses of each gas particle and dark matter are |$m_\mathrm{gas}=1.1\times 10^7\, \mathrm{M}_\odot$| and |$m_\mathrm{dark}=6.6 \times 10^7\, \mathrm{M}_\odot$|⁠, respectively. Our Plummer equivalent gravitational softening of 1.2 kpc determines our spatial resolution. In addition, we run several simulations with the same initial conditions in L = 50 h−1 Mpc boxes but at a lower resolution with 2 × 2883 particles (with eight times worse mass resolution and two times worse spatial resolution). Most of these simulations use the same wind algorithm as in the fiducial simulation with only differences in the wind parameters. We use these simulations to test the numerical convergence of the wind algorithm and also to determine the sensitivity of the simulations to the wind parameters. We choose the 50 h−1 Mpc size to balance between a decent numerical resolution and a representative volume, given the computational resources that are feasible. Furthermore, many of our previous results are based on simulations of similar volume and resolution. We are therefore able to verify the robustness of these previous results with the volume chosen. Zoom-in simulations (e.g. Sadoun et al. 2016; Valentini et al. 2017) provide a wide range of observables and diagnostics different from those probed by cosmological simulations and are therefore complementary to this study. In Table 1, we summarize and classify all the simulations into three categories. The first category simulations differ from the fiducial simulation only in terms of the mass-loading factor. The second category simulations differ only in terms of wind speed. In Section 4, we will focus on comparing simulations of each category to demonstrate the sensitivity to the wind parameters. The rest of the simulations differ in both the mass-loading factor and the wind speed or use different wind algorithms such as our original hybrid ezw wind model. In addition, the ezwDESPH simulation is the only simulation that uses the traditional SPH technique and, therefore, is the simulation that is closest to the original numerical model used in our previously published simulations (e.g. Davé et al. 2013; Ford et al. 2016). In Section 6, we will focus on results from the fiducial simulation and compare them to observations as well as the original ezw model to show how much the new fiducial wind algorithm changes some of our basic results from our previous simulations. Table 1. Simulations and their wind parameters. Simulation . αη . βη . |$\sigma _{ezw}\, ^a$| . α|$v$| . |$\beta _v\, ^b$| . Colourc . Remark . RefHres 0.1 4 106 4.0 0.6 black Fiducial wind model, high resolution with Ngas = 5763 Ref 0.1 4 106 3.5 0.6 magenta Fiducial wind model, Ngas = 2883 ezw 4.29 1 75 – 0.0 dark red The ezw model from Davé et al. (2013) ezwFast 4.29 1 75 3.5 0.6 teal ezw mass loading, but wind speeds of Ref ezwDESPH 4.29 1 75 – 0.0 blue The ezw wind model, traditional SPH methods StrongFB 0.2 4 106 3.5 0.6 red αη = 0.2 instead of the fiducial value 0.1 WeakFB 0.05 4 106 3.5 0.6 orange αη = 0.05 instead of the fiducial value 0.1 Sigma75 0.4 4 75 3.5 0.6 brown |$\sigma _{ezw} = 75\mathrm{\, km\, s^{-1}}$| instead of the fiducial |$106\mathrm{\, km\, s^{-1}}$| Refσ4 0.1 3 106 3.5 0.6 steel blue η scales with σ−4, not η ∝ σ−5 as in Ref Refσ3 0.1 2 106 3.5 0.6 plum η scales with σ−3 not η ∝ σ−5 as in Ref RefSlow 0.1 4 106 3.0 0.3 green Same as the Ref simulation but slower winds Simulation . αη . βη . |$\sigma _{ezw}\, ^a$| . α|$v$| . |$\beta _v\, ^b$| . Colourc . Remark . RefHres 0.1 4 106 4.0 0.6 black Fiducial wind model, high resolution with Ngas = 5763 Ref 0.1 4 106 3.5 0.6 magenta Fiducial wind model, Ngas = 2883 ezw 4.29 1 75 – 0.0 dark red The ezw model from Davé et al. (2013) ezwFast 4.29 1 75 3.5 0.6 teal ezw mass loading, but wind speeds of Ref ezwDESPH 4.29 1 75 – 0.0 blue The ezw wind model, traditional SPH methods StrongFB 0.2 4 106 3.5 0.6 red αη = 0.2 instead of the fiducial value 0.1 WeakFB 0.05 4 106 3.5 0.6 orange αη = 0.05 instead of the fiducial value 0.1 Sigma75 0.4 4 75 3.5 0.6 brown |$\sigma _{ezw} = 75\mathrm{\, km\, s^{-1}}$| instead of the fiducial |$106\mathrm{\, km\, s^{-1}}$| Refσ4 0.1 3 106 3.5 0.6 steel blue η scales with σ−4, not η ∝ σ−5 as in Ref Refσ3 0.1 2 106 3.5 0.6 plum η scales with σ−3 not η ∝ σ−5 as in Ref RefSlow 0.1 4 106 3.0 0.3 green Same as the Ref simulation but slower winds Notes. aThe first three parameters, αη, βη, and σezw determine the mass-loading factor according to equation (7). The ezw wind model uses a slightly different formula (equation 3) where these parameters have a similar effect. bThe next two parameters, α|$v$| and β|$v$|, determine the initial wind speed according to equation (5). The wind speed in the ezw model is formulated in a quite different way so that the parameters do not apply. cWe use a consistent colour scheme for the entire paper to distinguish simulations from each other. This column indicates the unique colour that is used to represent the corresponding simulation. Open in new tab Table 1. Simulations and their wind parameters. Simulation . αη . βη . |$\sigma _{ezw}\, ^a$| . α|$v$| . |$\beta _v\, ^b$| . Colourc . Remark . RefHres 0.1 4 106 4.0 0.6 black Fiducial wind model, high resolution with Ngas = 5763 Ref 0.1 4 106 3.5 0.6 magenta Fiducial wind model, Ngas = 2883 ezw 4.29 1 75 – 0.0 dark red The ezw model from Davé et al. (2013) ezwFast 4.29 1 75 3.5 0.6 teal ezw mass loading, but wind speeds of Ref ezwDESPH 4.29 1 75 – 0.0 blue The ezw wind model, traditional SPH methods StrongFB 0.2 4 106 3.5 0.6 red αη = 0.2 instead of the fiducial value 0.1 WeakFB 0.05 4 106 3.5 0.6 orange αη = 0.05 instead of the fiducial value 0.1 Sigma75 0.4 4 75 3.5 0.6 brown |$\sigma _{ezw} = 75\mathrm{\, km\, s^{-1}}$| instead of the fiducial |$106\mathrm{\, km\, s^{-1}}$| Refσ4 0.1 3 106 3.5 0.6 steel blue η scales with σ−4, not η ∝ σ−5 as in Ref Refσ3 0.1 2 106 3.5 0.6 plum η scales with σ−3 not η ∝ σ−5 as in Ref RefSlow 0.1 4 106 3.0 0.3 green Same as the Ref simulation but slower winds Simulation . αη . βη . |$\sigma _{ezw}\, ^a$| . α|$v$| . |$\beta _v\, ^b$| . Colourc . Remark . RefHres 0.1 4 106 4.0 0.6 black Fiducial wind model, high resolution with Ngas = 5763 Ref 0.1 4 106 3.5 0.6 magenta Fiducial wind model, Ngas = 2883 ezw 4.29 1 75 – 0.0 dark red The ezw model from Davé et al. (2013) ezwFast 4.29 1 75 3.5 0.6 teal ezw mass loading, but wind speeds of Ref ezwDESPH 4.29 1 75 – 0.0 blue The ezw wind model, traditional SPH methods StrongFB 0.2 4 106 3.5 0.6 red αη = 0.2 instead of the fiducial value 0.1 WeakFB 0.05 4 106 3.5 0.6 orange αη = 0.05 instead of the fiducial value 0.1 Sigma75 0.4 4 75 3.5 0.6 brown |$\sigma _{ezw} = 75\mathrm{\, km\, s^{-1}}$| instead of the fiducial |$106\mathrm{\, km\, s^{-1}}$| Refσ4 0.1 3 106 3.5 0.6 steel blue η scales with σ−4, not η ∝ σ−5 as in Ref Refσ3 0.1 2 106 3.5 0.6 plum η scales with σ−3 not η ∝ σ−5 as in Ref RefSlow 0.1 4 106 3.0 0.3 green Same as the Ref simulation but slower winds Notes. aThe first three parameters, αη, βη, and σezw determine the mass-loading factor according to equation (7). The ezw wind model uses a slightly different formula (equation 3) where these parameters have a similar effect. bThe next two parameters, α|$v$| and β|$v$|, determine the initial wind speed according to equation (5). The wind speed in the ezw model is formulated in a quite different way so that the parameters do not apply. cWe use a consistent colour scheme for the entire paper to distinguish simulations from each other. This column indicates the unique colour that is used to represent the corresponding simulation. Open in new tab To illustrate how differences in the mass-loading factors affect the simulations, we show in the upper panel of Fig. 1 the input scaling laws (equation 7) between the mass-loading factor η and the velocity dispersion σ measured from the on-the-fly FoF group finder. In the lower panel, we show the actual mass-loading factor of individual galaxies in each simulation as a function of their stellar mass at |$z$| = 2. For comparison, we also show the empirical fit from the FIRE simulations (M15), and a formula used in the Somerville et al. (2012) semi-analytic model.1 When making these comparisons, remember, however, that the η referred to in the simulations is at wind launch inside the galaxy while the η values in the FIRE simulations are measured well outside the galaxy (R25). Figure 1. Open in new tabDownload slide Upper panels: The mass-loading factor η, as a function of the halo velocity dispersion σgal at |$z$| = 2. In the simulations, we calculate σgal from the mass of each halo identified by the on-the-fly FoF group finder. The scalings follow equation (3) for the ezw winds and equation (7) for the new wind algorithm. Lower panels: η as a function of stellar mass M*. The shaded area traces the median and includes 68 per cent of galaxies within each mass bin for the Ref simulation. Different simulations are colour coded according to Table 1. We also show the analytic formula from M15 (black solid line) and Somerville et al. (2012) (black dashed line) for comparison. Figure 1. Open in new tabDownload slide Upper panels: The mass-loading factor η, as a function of the halo velocity dispersion σgal at |$z$| = 2. In the simulations, we calculate σgal from the mass of each halo identified by the on-the-fly FoF group finder. The scalings follow equation (3) for the ezw winds and equation (7) for the new wind algorithm. Lower panels: η as a function of stellar mass M*. The shaded area traces the median and includes 68 per cent of galaxies within each mass bin for the Ref simulation. Different simulations are colour coded according to Table 1. We also show the analytic formula from M15 (black solid line) and Somerville et al. (2012) (black dashed line) for comparison. 3.1 Wind speed at R25 A major update to our fiducial simulation from our original ezw model (Davé et al. 2013) is the readjustment of the initial wind velocity so that we approximately have |$v$||$\mathrm{ w}$| ∝ σ outside the galaxy as opposed to at wind launch, in better correspondence with the Murray et al. (2005) model. In this section, we will characterize how this modification changes the behaviour of winds as they propagate into the halo after they have been launched. We will compare the wind behaviour to that predicted from the FIRE simulations, which follow the evolution of wind particles with more detailed physics and at higher resolution. Using these zoom-in simulations, M15 derived an empirical relation between the wind speed at R25 ≡ 0.25 Rvir and the circular velocity |$v$|c of the host halo from which the winds are launched. They find that the median wind speed could be fitted with $$\begin{eqnarray*} v_{\mathrm{ w},50} = 0.85 v_\mathrm{c}^{1.1} \end{eqnarray*}$$ (12) and that the upper 95th percentile wind speed is fitted with $$\begin{eqnarray*} v_{\mathrm{ w},95} = 1.9 v_\mathrm{c}^{1.1} . \end{eqnarray*}$$ (13) They obtain these relations from their data at high and medium redshifts but do not find apparent redshift evolution of these relations. This is very close to the |$v_\mathrm{c}^{1.0}$| in the Murray et al. (2005) model and our scaling at wind launch. However, we want to compare with our winds not at wind launch but at R25, where they are measured in M15. To measure the wind speed at R25 in our simulations at a given redshift, we track the evolution of all wind particles that are ejected within a small redshift bin centred at that redshift. We identify the host haloes of all these wind particles from the halo catalogue generated with the post-processing halo finder so (Kereš et al. 2005). Then for each wind particle, we define the wind velocity at R25, |$v$|25, as the radial velocity of the particle when it first crosses the R25 of its host halo. In the left-hand panel of Fig. 2, we compare the speed of wind particles in our original ezw model (ezw) to the empirical relations derived from the FIRE simulations (M15) measured at R25. The ezw winds (cyan line) are much slower than in M15 in massive galaxies. In fact, most wind particles launched in galaxies above a certain |$v$|c in the ezw simulation do not have sufficient initial momentum to ever reach R25 at all. This is more clearly illustrated in the bottom panels, which show as solid lines the fraction of wind particles that reach R25. In the massive galaxies in the ezw simulation, the winds fall back on to their host galaxies within a very short time-scale and, therefore, play little role in regulating the star formation of their host galaxies. Even though the winds were launched with |$v$||$\mathrm{ w}$| ∝ |$v$|c (red line), their velocities are almost independent of Rvir at R25 (cyan line). It was this realization that originally motivated us to re-examine our wind model. Figure 2. Open in new tabDownload slide Upper panels: The relation between the velocity of wind particles and the circular velocity |$v$|c of their host galaxy. These wind particles are launched within a small redshift window at |$z$| = 2. In each panel, the red line shows the running medians of the initial launch velocities and the cyan line shows |$v$|25 – the velocities of the wind particles when they reach 0.25 the virial radius (R25). The green segment indicates the |$v$||$\mathrm{ w}$| ∝ σ1.6 scaling imposed at launch for the fiducial simulation. The colour map shows the distribution of wind particles according to their |$v$|25 and |$v$|c. The colour scale indicates the number of wind particles in each cell. We also include the empirical fit from the FIRE simulations (M15). The black solid and dashed lines in each panel correspond to their 50th and 95th percentiles, respectively. Lower panels: In each panel, the black line shows the fraction of wind particles that reach R25 in their host halo. The grey histogram shows the unweighted distribution of |$v$|c for all wind particles. The three panels from left to right are plotted for the ezw, Ref, and RefSlow simulations, respectively. Agreement between the cyan and black solid lines thus represents agreement between our wind launch prescription and the median FIRE results, but agreement between the cyan solid and black dashed lines may be a better measure for reasons described in the text. The new wind algorithm in our fiducial simulation is able to reproduce the trend seen in the FIRE simulations. However, winds in the original ezw simulations are in general too slow, particularly in massive galaxies, where only a small fraction of wind particles is able to reach R25 before falling back. Figure 2. Open in new tabDownload slide Upper panels: The relation between the velocity of wind particles and the circular velocity |$v$|c of their host galaxy. These wind particles are launched within a small redshift window at |$z$| = 2. In each panel, the red line shows the running medians of the initial launch velocities and the cyan line shows |$v$|25 – the velocities of the wind particles when they reach 0.25 the virial radius (R25). The green segment indicates the |$v$||$\mathrm{ w}$| ∝ σ1.6 scaling imposed at launch for the fiducial simulation. The colour map shows the distribution of wind particles according to their |$v$|25 and |$v$|c. The colour scale indicates the number of wind particles in each cell. We also include the empirical fit from the FIRE simulations (M15). The black solid and dashed lines in each panel correspond to their 50th and 95th percentiles, respectively. Lower panels: In each panel, the black line shows the fraction of wind particles that reach R25 in their host halo. The grey histogram shows the unweighted distribution of |$v$|c for all wind particles. The three panels from left to right are plotted for the ezw, Ref, and RefSlow simulations, respectively. Agreement between the cyan and black solid lines thus represents agreement between our wind launch prescription and the median FIRE results, but agreement between the cyan solid and black dashed lines may be a better measure for reasons described in the text. The new wind algorithm in our fiducial simulation is able to reproduce the trend seen in the FIRE simulations. However, winds in the original ezw simulations are in general too slow, particularly in massive galaxies, where only a small fraction of wind particles is able to reach R25 before falling back. In the middle panel of Fig. 2, we make the same plot for our fiducial simulation (Ref). Now the distribution of wind particles from our simulation roughly agrees with the empirical scaling relations from the FIRE simulations (M15). In detail, the median velocities of our winds (cyan lines) are higher than their medians but are typically lower than their upper 95th percentile values. Most of the wind particles launched using the new algorithm are now able to reach R25, even in the most massive systems. There are caveats when directly comparing the median velocities between our simulations and FIRE. First, the nature of our winds differs from theirs. In M15, the winds are explicitly accelerated by the physical processes that they adopt in their simulations (Hopkins et al. 2012). Their winds are multiphase by nature but they do not distinguish the cold and warm phases when calculating the wind speed. In contrast, our winds are imposed on the galaxies and represent only the cold, mass loaded outflow, which cannot be self-consistently generated from the physics included in our simulations. Before reaching R25 where the wind speeds are measured and compared, the wind particles in our simulations have been slowing down owing to hydrodynamic interactions and gravity, while in their simulations the wind particles keep being accelerated by radiation pressure and it is not clear whether or not they have started to slow down at that radius. Therefore, the kinematic structure and the evolution of winds in our simulation are likely very different from theirs. Hence, we do not know if the comparison to FIRE would be substantially different at a different radius, e.g. R10 or R40. Second and more importantly, we measure the wind speeds in fundamentally different ways. M15 measure the wind speed using the flux-weighted average of all outflowing particles over a given epoch. This measurement preferentially selects particles where the outflowing flux is maximal, i.e. when the winds are strongest. Furthermore, their definition of outflowing particles includes all gas particles in the halo at that radius as long as their radial velocities are above the halo velocity dispersion, while in our simulations we only include the actual ejected wind particles in the measurement. Since in lower mass haloes the typical wind speed is much larger than the random motions of the halo component, their measurement likely underestimates the speed of the outflowing material that is actually accelerated from the galaxies. In contrast, in larger mass haloes their method could measure winds even for gas that is roughly in hydrostatic equilibrium within the halo and hence may overestimate the wind velocities (and η). Our measurement reflects the speed of the fastest outflowing particles within each halo and, therefore, should be more comparable to their 95th percentile values. In fact, if we try to measure our winds in a way more similar to that in M15, it lowers our median wind velocities to agree better with their median values. Unlike M15’s finding that the |$v$|25–|$v$|c relation is independent of redshift, in our simulations the |$v$|25 slightly decreases for a given |$v$|c at lower redshifts, especially in massive haloes, even though we launch our winds with a redshift-independent initial velocity (equation 5). The winds lose more momentum as they move from the launching radius to R25 at low redshifts, likely owing to the combined effects of a deeper potential, enhanced hydrodynamic forces, and an underestimate of σ for massive haloes using the on-the-fly FoF group finder. The significant differences between the wind behaviours in our simulations are particularly interesting considering that the initial wind velocities are not very different from each other. A relatively small difference in the initial wind speed at launch has a significant impact on the wind behaviour at larger radii. This indicates that this kinetic wind algorithm, which is adopted in many cosmological simulations (e.g. Springel & Hernquist 2003; Oppenheimer & Davé 2006; Agertz et al. 2013; Schaye et al. 2015; Davé et al. 2016; Hopkins et al. 2018; Pillepich et al. 2018a) as a subgrid model for stellar feedback, is very sensitive to the details of its implementation and the choice of wind parameters. We will discuss this sensitivity more in Section 4. 4 SENSITIVITY TO THE WIND MODEL In this section, we explore the scaling laws that determine the mass-loading factor and the wind launch speed in our new wind algorithm, and study the sensitivity of our simulations to these parameters. All the test simulations we use in this section are listed in Table 1 and all have the same numerical resolution. We will explore the effects of numerical resolution in Section 6. We identify galaxies using skid and so as in Huang et al. (2019) and measure the stellar mass and halo mass for every galaxy that we identify. First, we will focus on comparing the GSMFs at four different redshifts and discuss the effects of changing the mass-loading factor and the wind speed separately. In addition, we will also examine the growth of individual galaxies and how they differ among the simulations, since their different star formation histories are an immediate consequence of the wind algorithm, which controls their gas supply. To make direct comparisons between individual galaxies, we cross-match galaxies from different simulations to those in the Ref simulation based on their phase-space information. We also require matched galaxies to have a stellar mass difference smaller than 1 dex to avoid matching satellite galaxies to centrals. We define the differences of stellar masses between matched galaxies as Δlog (M*) ≡ log (M*) − log (M*, f), where M* is the stellar mass of a galaxy in a simulation and M*, f is the stellar mass of its matched galaxy in the fiducial simulation. Finally, we will look at how the cosmic mean stellar density evolves with time in each test simulation. As an integrated quantity, the cosmic stellar density at different redshifts indicates whether or not a simulation produces the right amount of stars in total. 4.1 The GSMFs The GSMFs are one of the most robust measurements from observations and have been used as an important constraint for calibrating subgrid models in cosmological simulations. To compare our simulated GSMFs with observations, we use results from multiple large galactic surveys. All of these observations assume a Chabrier (2003) IMF for their stellar mass estimate as in our simulation. Different measurements of the |$z$| = 0 GSMFs (e.g. Li & White 2009; Baldry et al. 2012; Bernardi et al. 2013; Moustakas et al. 2013) generally agree at the faint end up to log (M*) ∼ 10.5 and start to deviate slightly from each other at higher masses. Both the choice of the aperture (Bernardi et al. 2013) and the choice of the stellar population synthesis template (e.g. Mitchell et al. 2013; Tomczak et al. 2014) contribute to relatively large uncertainties in stellar masses at the massive end. Conroy, Gunn & White (2009) estimated that the systematic error on stellar masses ranges from 0.3 dex at |$z$| ∼ 0 to 0.6 dex at |$z$| ∼ 2. Our choices of the observed GSMFs at |$z$| = 0 reflect these uncertainties. The Baldry et al. (2012) result is based on single-Sersic fits to the light profiles of galaxies at |$z$| < 0.06 from the Galaxy And Mass Assembly (GAMA) survey, while Bernardi et al. (2013) use a Sersic-bulge + exponential disc model that results in larger stellar masses at the bright end. For |$z$| = 1 and 2 GSMFs, we use the data from Tomczak et al. (2014), who compiled GSMFs over a broad redshift range 0.2 < |$z$| < 3 using deep observations from the FourStar Galaxy Evolution Survey (ZFOURGE) and the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS), obtaining a completeness limit of log (M*) ∼ 9.5. We use the Song et al. (2016) results from the CANDELS survey for comparison at |$z$| = 4. First, we will show how the GSMFs are sensitive to the mass-loading factor. The mass-loading factor η is controlled by three parameters (equation 7): a normalization factor αη, a power-law index βη that determines how it scales with σFoF, which is the velocity dispersion of the host halo identified from the on-the-fly group finder, and the characteristic velocity σezw above which the scaling with σFoF changes from |$\eta \propto \sigma ^{-(1+\beta _\eta)}$| to η ∝ σ−1. All the simulations that we use for this comparison are listed in Table 1 and use the same parameters for the wind speed as the fiducial simulation, but have different parameters for η. Here we describe the features of each simulation using the low-resolution fiducial simulation Ref as a reference. StrongFB increases the overall mass loading by a factor of 2 and WeakFB reduces it by a factor of 2. Sigma75 uses a smaller σezw of 75 |$\mathrm{\, km\, s^{-1}}$| than the fiducial 106 |$\mathrm{\, km\, s^{-1}}$|⁠, but also renormalizes αη so that it has the same scaling with σFoF for small haloes below σezw. Refσ4 and Refσ3 use shallower scalings with σFoF for haloes smaller than the characteristic σezw, with a power index parameter βη equal to 3 and 2, respectively, instead of the fiducial value of 4. ezwFast has the same mass loading as ezw but has the wind launch speed scalings of Ref, which produce faster winds. The bottom panel of Fig. 1 shows the relation between η and σFoF for these simulations. 4.1.1 Dependence on βη and σezw Fig. 3 shows how changing the power index βη and the characteristic velocity σezw affects GSMFs at different times. Not surprisingly, the faint end of the GSMFs is particularly sensitive to βη. Since all these simulations except for ezwFast have the same η|$\mathrm{ w}$|–σFoF relation above σezw, a higher βη means more mass in outflows from smaller haloes and less star formation. Ref and Sigma75 are the only simulations that successfully reproduce the observed faint end at all redshifts, and both have a strong scaling with |$\eta \propto \sigma _\mathrm{FoF}^{-5}$| (βη = 4). Simulations with a shallower dependence on σFoF tend to produce more low-mass galaxies at |$z$| > 1, creating a stronger tension with the observational data. For example, the ezwFast simulation shows that the shallow scaling |$\eta _\mathrm{ w} \propto \sigma _\mathrm{FoF}^{-2}$| predicted from the analytic momentum/energy-driven model results in too many faint galaxies formed at high redshifts. Figure 3. Open in new tabDownload slide The GSMF at different redshifts. At each redshift, we compare the GSMFs from a set of test simulations, which are shown in different colours according to the colour scheme defined in Table 1. The dotted vertical line in each panel indicates the resolution limit for galaxies corresponding to a total mass of 128 SPH particles in these low-resolution simulations. The observational data for these redshifts are described in the text. Figure 3. Open in new tabDownload slide The GSMF at different redshifts. At each redshift, we compare the GSMFs from a set of test simulations, which are shown in different colours according to the colour scheme defined in Table 1. The dotted vertical line in each panel indicates the resolution limit for galaxies corresponding to a total mass of 128 SPH particles in these low-resolution simulations. The observational data for these redshifts are described in the text. The above result shows that a steep scaling between the mass-loading factor and the circular velocity, or equivalently the halo mass, is essential to explain the flat faint end of GSMFs at |$z$| = 1 and 2 when one uses a kinetic feedback model such as ours. This requirement for a strong dependence between η and σ has also been recently found in other work.2 In the IllustrisTNG simulations, Pillepich et al. (2018b) reported a scaling with |$\eta \propto M_\mathrm{h}^{-1} \propto \sigma ^{-3}$| (see their fig. 7) as their fiducial choice to fit a wide range of observables. The FIRE simulations generate galactic outflows self-consistently instead of using a simple scaling law and found η ∝ σ−3.3 (M15). Semi-analytic studies (SAM) also require steep scalings to fit the GSMFs at different redshifts. Lu et al. (2014) reported a rather steep scaling with η ∝ σ−6. Somerville et al. (2012) parametrize their mass loading as |$\eta \propto \sigma ^{-\beta _{LD}}[1+\sigma ^{\beta _{EJ}}]^{-1}$| with fiducial parameters βLD = 2.25 and βEJ = 6. This scaling, also shown in Fig. 1, is similar in form to our scalings, though the normalization is lower. Peeples & Shankar (2011) also find with their chemical evolution model that very steep mass loading scalings (⁠|$\eta \propto v_\mathrm{c}^{-3}$| or steeper) are required to explain the steep slope of the observed MZR at |$z$| = 0. Even though the specific wind models used in these other works have important differences, such as whether or not they add additional heating, it is clear that the efficient suppression of star formation in less massive galaxies requires stronger winds than those predicted from the energy-driven model (η ∝ σ−2) or the momentum-driven model (η ∝ σ−1), which were previously assumed in many cosmological simulations. Note that the GSMF at |$z$| = 0 alone is insufficient to differentiate between these different scalings. Accurate measurement of the stellar content at higher redshifts could, therefore, place important constraints on the wind models. For more massive galaxies, βη has a less significant effect than the parameter σezw, which determines the mass scale where the steep scaling |$\eta \propto \sigma ^{-(1+\beta _\eta)}$| changes to the momentum-driven wind scaling η ∝ σ−1. The Ref, Refσ4, and Refσ3 simulations have indistinguishable GSMFs above log (M*) = 10.5 in spite of their different βη values. In contrast, the GSMFs from the ezwFast and Sigma75 simulations start showing clear differences at the massive end from the other three simulations as early as |$z$| = 2, indicating that the growth of massive galaxies is very sensitive to the choice of σezw. For example, Fig. 1 shows that the mass-loading factor in the Sigma75 simulation is the same as that in the fiducial simulation at σFoF < 75 |$\mathrm{\, km\, s^{-1}}$|⁠, but it becomes larger by a steadily increasing factor for |$\sigma _{ezw} \gt 75\mathrm{\, km\, s^{-1}}$| and is a factor of ∼4 higher in all haloes with |$\sigma _{ezw} \ge 106\mathrm{\, km\, s^{-1}}$|⁠, our fiducial value of σezw. As a result, the growth of massive galaxies in the Sigma75 simulation is significantly suppressed at all redshifts. Interestingly, the ezwFast and Sigma75 simulations have the best overall agreement with the observed |$z$| = 0 GSMF among all these test simulations. However, they significantly underproduce the number of massive galaxies at higher redshifts. Some of the other simulations, including the fiducial simulation, agree with observations better at higher redshifts at the cost of a slight excess of the most massive galaxies at |$z$| = 0. We find matching the massive end of the GSMFs at both |$z$| = 0 and 2 simultaneously to be very challenging within our current framework for feedback. A successful feedback model must allow a rapid build-up of massive galaxies at |$z$| = 2 but also must account for the slow evolution of the massive end from |$z$| = 0 to 2 as suggested by observations. The success of our Ref model at |$z$| ≥ 1 but failure at |$z$| = 0 suggests that an additional mechanism such as AGN feedback suppresses the growth of massive galaxies at low redshifts, or else that the galaxy scalings of stellar feedback change sharply at |$z$| < 1. 4.1.2 Dependence on the normalization αη Fig. 4 shows the effects of changing the linear normalization factor αη by comparing three simulations with αη incrementally varying by a factor of 2. In general, a higher mass loading normalization results in less stars being formed because more cold gas is ejected in galactic winds. To a rough approximation, the GSMFs of the WeakFB and StrongFB simulations are offset horizontally by a factor of 2–4 at all redshifts, with the Ref simulation mid-way between them. Figure 4. Open in new tabDownload slide Same as Fig. 3, except that here we focus on the effect of the linear factor αη of the mass-loading factor. The η in these simulations are different by a factor of 2 so that for the same galaxy, the StrongFB model (red) produces 2× more massive winds than the fiducial Ref model (magenta), while the StrongFB model (orange) produces 2× less massive winds. Figure 4. Open in new tabDownload slide Same as Fig. 3, except that here we focus on the effect of the linear factor αη of the mass-loading factor. The η in these simulations are different by a factor of 2 so that for the same galaxy, the StrongFB model (red) produces 2× more massive winds than the fiducial Ref model (magenta), while the StrongFB model (orange) produces 2× less massive winds. Fig. 5 compares the stellar mass differences between individual, matched galaxies from these simulations. We use the fiducial simulation as the reference so that all galaxies from that simulation lie on the black horizontal line. For each galaxy in the fiducial simulation, we find its matched galaxy in the other two simulations and calculate the stellar mass differences. The medians are shown as coloured lines. Figure 5. Open in new tabDownload slide The stellar mass differences between galaxies that are cross-matched between the different simulations and the Ref simulation at the given redshifts. The shaded region indicates the 1σ scatter in each M* bin. In each panel, the dotted vertical lines indicate the resolution limit of galaxies corresponding to a total mass of 128 SPH particles. The dashed horizontal lines indicate the offset in stellar masses (±0.3 dex) predicted by equation (17). Figure 5. Open in new tabDownload slide The stellar mass differences between galaxies that are cross-matched between the different simulations and the Ref simulation at the given redshifts. The shaded region indicates the 1σ scatter in each M* bin. In each panel, the dotted vertical lines indicate the resolution limit of galaxies corresponding to a total mass of 128 SPH particles. The dashed horizontal lines indicate the offset in stellar masses (±0.3 dex) predicted by equation (17). Using a simple analytic model, we could predict the stellar mass of an isolated galaxy whose growth is governed by gas outflow, star formation, and the inflow of both pristine gas and recycled winds. The equilibrium condition is $$\begin{eqnarray*} \dot{M}_{\mathrm{ in}} + \dot{M}_{\mathrm{ rec}} = \dot{M}_{\mathrm{ out}} + \dot{M}_{*} . \end{eqnarray*}$$ (14) This equilibrium equation is typically a good approximation in hydrodynamic simulations (Finlator & Davé 2008). Assuming that a fraction frec of the outflow recycles, so |$\dot{M}_{\mathrm{ rec}} \approx f_\mathrm{\mathrm{ rec}}\dot{M}_{\mathrm{ out}}$|⁠, the final stellar mass of the galaxy would be $$\begin{eqnarray*} M_* = M_{\mathrm{ in}}\left(\frac{1}{1+\eta } + f_\mathrm{rec}\frac{\eta }{(1+\eta)^2}\right), \end{eqnarray*}$$ (15) where Min is the time-integrated mass of unrecycled gas that accretes on to the galaxy. This derivation assumes a constant η and frec for the galaxy. However, since both of these are functions of halo mass, this assumption breaks down for galaxies that undergo a strong evolution in their halo mass such as in a major merger. Since two matched galaxies in different simulations have similar assembly histories, gravitational potential, and outflow speeds, their Min and frec should remain approximately the same, provided that the outflows do not themselves disrupt accretion or recycling. The ratio of final stellar masses between two simulations is thus determined by the different η: $$\begin{eqnarray*} \frac{M_{*,1}}{M_{*,2}} = \left[ \frac{1+(1+f_\mathrm{rec})\eta _2}{1+(1+f_\mathrm{rec})\eta _1} \right]. \end{eqnarray*}$$ (16) For low-mass galaxies, we can assume η >> 1 and frec ∼ 0 because winds can easily escape from their shallow gravitational potential. Hence, the ratio above will asymptotically approach $$\begin{eqnarray*} \left(\frac{M_{*,1}}{M_{*,2}}\right)_{\eta \gg 1} = \frac{\eta _2}{\eta _1}. \end{eqnarray*}$$ (17) Therefore, this simple model predicts that the stellar mass of small galaxies in the WeakFB and StrongFB simulations should differ from their corresponding galaxies in the fiducial simulation also by a factor of 2 (0.3 dex). Fig. 5 shows that this prediction agrees with our simulations reasonably well. It works almost perfectly when we increase η by a factor of 2. When we decrease η by the same factor, it does not work as well and the error increases with galaxy mass. This is because the approximation that η ≫ 1 is less robust with decreasing η. For example, Fig. 1 shows that η ∼ 1 at log (M*) ∼ 10 in the WeakFB simulation. In fact, in the opposite limit, i.e. η ≪ 1, equation (16) predicts that M*, 1 = M*, 2, consistent qualitatively with the convergence of curves at high mass in Fig. 5. The degree to which this toy model works is perhaps surprising, since it makes many oversimplified assumptions. First, the equilibrium equation (equation 14) assumes that the total amount of cold gas in galaxies does not change with time while in the simulations this is not guaranteed. Secondly, it treats galaxies in isolation, neglecting any interactions with other galaxies, but in reality a significant fraction of gas accretion may have come from previous outflows from other galaxies (e.g. Ford et al. 2014; Anglés-Alcázar et al. 2017). Thirdly, it assumes that any outflow will have no subsequent effect on the galaxy except through a nearly constant fraction of immediate wind recycling, but the outflowing gas may interact with the surrounding gas and thus affect further gas accretion. The success of the toy model suggests that these concerns are likely not dominant in our cosmological simulations. 4.1.3 Wind speed In this section, we will show how the GSMFs are sensitive to the initial wind speed. The initial wind speed at launch depends on two parameters (equation 5): a normalization factor α|$v$| and a power-law index β|$v$| that determines how wind speed scales with σFoF. The fiducial simulation adopts α|$v$| = 3.5 and β|$v$| = 0.6. These values are tuned to match the |$v$|25–|$v$|c scaling from the FIRE simulation (M15) at |$z$| = 2. The ezw simulation uses the original ezw formula for the wind speed (equation 2). The ezwFast simulation uses the fiducial wind speed scaling but the mass loading of the ezw simulation. The RefSlow simulation uses a slightly shallower slope β|$v$| = 0.5 than the fiducial Ref simulation. Therefore, we will demonstrate the effects of wind speed by comparing the ezw and ezwFast simulations under the original η formula, and comparing the Ref and RefSlow simulations under the fiducial η formula. Fig. 6 shows that the massive end of the GSMFs is very sensitive to the initial wind speed. First, we compare the two ezw simulations. At |$z$| = 4, the GSMFs are still very similar, because even the slower winds are above the escape velocities of haloes at this redshift. But the massive ends of the GSMFs start to show clear differences after |$z$| = 2. As Fig. 2 has shown, with the original ezw wind speed a significant fraction of wind particles fall back towards the galaxy before reaching R25 and become star forming again very soon after being launched. The new wind speed in the ezwFast allows wind particles to travel much further, and they return much later, if at all. This reduces the amount of stars formed in intermediate-mass haloes and at least delays stellar growth in massive haloes. As a result, galaxies in the ezwFast simulation are less massive, with the mass difference increasing towards more massive systems. Figure 6. Open in new tabDownload slide Same as Fig. 3, except that here we focus on the effect of different initial wind speeds on the GSMFs. The fiducial Ref (magenta) and the RefSlow (green) simulations use the new wind launch algorithm and are only different in |$v$||$\mathrm{ w}$|. The other two simulations use the original ezw wind model. However, the ezwFast (teal) simulation has fast winds as in the fiducial simulation while the ezw (dark red) simulation uses the ezw wind speed. Figure 6. Open in new tabDownload slide Same as Fig. 3, except that here we focus on the effect of different initial wind speeds on the GSMFs. The fiducial Ref (magenta) and the RefSlow (green) simulations use the new wind launch algorithm and are only different in |$v$||$\mathrm{ w}$|. The other two simulations use the original ezw wind model. However, the ezwFast (teal) simulation has fast winds as in the fiducial simulation while the ezw (dark red) simulation uses the ezw wind speed. Now compare the fiducial simulation to the RefSlow simulation. The only difference between them is that the wind speed in the fiducial simulation increases slightly faster with halo mass. Even in the most massive haloes, the difference in wind speed is only a factor of 2. However, the massive galaxies in the RefSlow simulations are much more massive than those in the Ref simulation. Even more strikingly, the massive ends of the Ref and ezwFast simulations are quite close, even though their mass-loading factors at |$\sigma \gt 106\mathrm{\, km\, s^{-1}}$| differ by a factor of ∼10. This similarity shows that the wind speed (matched between these simulations) is a crucial governor of high-mass galaxy growth, probably because of its impact on recycling rates. 4.2 Stellar density evolution (SDE) In Fig. 7 we show the SDE as a function of redshift. This has been measured observationally in many studies (Li & White 2009; González et al. 2011; Baldry et al. 2012; Ilbert et al. 2013; Moustakas et al. 2013; Muzzin et al. 2013; Tomczak et al. 2014). Most of these measurements agree well within 0.1 dex at redshifts below |$z$| = 2, but the differences become larger at higher redshifts. Here we use the data from Muzzin et al. (2013) for redshifts 0–3 and González et al. (2011) for higher redshifts. The Muzzin et al. (2013) sample has a mass completeness limit of log (M*) = 10.76 at |$z$| = 2.5–3.0 and log (M*) = 10.94 at |$z$| = 3.0–4.0 and hence the data have to be extrapolated to estimate the stellar mass densities at these redshifts. The González et al. (2011) data can be interpreted as upper limits since they did not correct for nebular emission when using the UV data to derive the stellar densities (Smit et al. 2014). Figure 7. Open in new tabDownload slide The evolution of the comoving stellar mass density with redshift from the test simulations. We use results from Muzzin et al. (2013) and González et al. (2011) as observational constraints, though other measurements in the literature agree with each other in general. The colour scheme for the different lines is defined in Table 1. Only the fiducial Ref and Refσ4 simulations agree with the observations. The other simulations either overproduce stars at high redshifts or fail to match the evolution at low redshifts. The upper panel compares simulations with different mass-loading factors. The lower panel compares simulations with different wind speeds. Figure 7. Open in new tabDownload slide The evolution of the comoving stellar mass density with redshift from the test simulations. We use results from Muzzin et al. (2013) and González et al. (2011) as observational constraints, though other measurements in the literature agree with each other in general. The colour scheme for the different lines is defined in Table 1. Only the fiducial Ref and Refσ4 simulations agree with the observations. The other simulations either overproduce stars at high redshifts or fail to match the evolution at low redshifts. The upper panel compares simulations with different mass-loading factors. The lower panel compares simulations with different wind speeds. Only the Refσ4 simulation and the fiducial simulation are consistent with the observations at all redshifts. These two simulations differ only in βη, with the Refσ4 simulation having a shallower η − σ slope βη = 3 that launches less winds in low-mass galaxies. The slight excess of low-mass galaxies in the Refσ4 simulation (Fig. 3) explains its overall higher stellar densities in Fig. 7. The simulations with a lower βη, i.e. the ezw, ezwFast, and Refσ3 simulations, overproduce low-mass galaxies at high redshifts, leading to much higher stellar densities at |$z$| > 2. This discrepancy supports the claim in Section 4.1.1 that one requires a strong dependence of the mass-loading factor on the halo velocity dispersion. The lower panel of Fig. 7 shows that the wind speed has a strong effect on the evolution at lower redshifts. The wind dynamics is more sensitive to the initial wind speed in massive haloes as shown in Fig. 2. The fast wind in the Ref and ezwFast simulations significantly limits the growth of massive galaxies compared to the RefSlow and ezw simulations, resulting in a slower growth of ρ* at |$z$| < 2. However, ezwFast still overpredicts the total stellar mass at |$z$| = 0 because of too much star formation at earlier redshifts owing to a relatively smaller mass-loading factor in low-mass galaxies. Here we have shown how the stellar content of galaxies in our simulations is sensitive to wind parameters, namely η and |$v$||$\mathrm{ w}$|. To make more robust comparisons to observations, we will need to further transform our simulation data into mock observations and take into account various observational effects, such as corrections for aperture (Schaye et al. 2015; Pillepich et al. 2018b) and completeness (Furlong et al. 2015). We did not make these corrections in this work as these comparisons are not meant to be interpreted as rigorous tests of our galaxy formation model, but rather to demonstrate the sensitivity of the numerical results to the wind implementations and their associated parameters. 5 DISCUSSION In the previous section, we demonstrated that the properties of galaxies in our simulations are sensitive to the subgrid model for galactic winds. To summarize, first, the low-mass end slopes of the GSMFs are sensitive to the mass-loading factor η, especially to the power-law index βη that determines how strongly η scales with σ in low-mass galaxies; secondly, the stellar masses of massive galaxies are sensitive to the initial wind launch speed |$v$||$\mathrm{ w}$|. In this section, we study in detail how galaxies build up their stellar masses in our simulations and how they are affected by the wind algorithm. The stellar content in a given halo at any redshift is closely related to the baryon cycles (inflow/outflow) that it has experienced over cosmic time. The subgrid wind algorithm controls outflow in our simulations while the amount of cosmic accretion through filaments (cold accretion) or cooling flows from the shocked halo gas (hot accretion) governs the inflow. In addition, galactic winds that were launched in the past can also fall back on to the galaxies after they lost their initial momentum. This wind recycling often dominates at low redshifts (Oppenheimer et al. 2010; Anglés-Alcázar et al. 2017). In this section, we will study the accretion history of the gas that ultimately forms stars. The particle nature of our SPH simulations makes it convenient to track the evolution of individual gas particles. In Section 5.1, we will describe how we differentiate between cold and hot primordial accretion and wind recycling through cosmic time. We will focus on analysing and comparing three simulations: the fiducial Ref, RefSlow, and Refσ3 simulations. These simulations show clear differences in the resultant GSMFs, SDEs, and stellar mass–halo mass functions (SMHMs) at different redshifts, even though they use the same wind algorithm, albeit with different wind parameters. Using the fiducial simulation as a baseline for comparison, the RefSlow simulation represents test simulations that explore the effects of the wind speed, which we will show affects not only wind recycling but also pristine gas accretion. On the other hand, the Refσ3 simulation represents a test simulation with varying parameters for the mass-loading factor, which directly controls the amount of outflows from haloes of different masses. In Section 5.2, we will first show how the SMHMs evolve from |$z$| = 4 to 0 in these simulations. In Section 5.3, we will analyse how the wind algorithms shape the SMHMs at |$z$| = 2 and address the differences between the simulations. Galaxy evolution at higher redshifts (⁠|$z$| > 2) is much less complicated than later evolution for lower mass galaxies, since it involves various important additional processes, such as mergers, cold halo–hot halo dichotomy, and group and cluster formation. They play less significant roles at higher redshifts. However, galaxies at |$z$| = 2 are the building blocks for those at lower redshifts and must also agree with the observational constraints. In Section 5.4, we will focus on the late evolution after |$z$| = 2 and the formation histories of galaxies at |$z$| = 0. In Section 5.5, we will discuss the importance of mergers to the assembly of stellar mass, and in Section 5.6 we will study in detail the properties and the effects of wind recycling. Finally, in Section 5.7, we will discuss what might be missing from our feedback prescriptions and what might be needed to remove the remaining discrepancies between the simulations and the observations. 5.1 Tracking the accretion history To understand how galaxies acquire the gas that ultimately forms their stars, we track the evolution of individual SPH particles that at some point become star forming.3 At each time-step, we track all the accretion events, i.e. whenever a gas particle changes from non-star forming to star forming at that time-step, and output the properties of the accreted particle and the galaxy on to which it accretes. To distinguish these accretion events, we introduce a parameter Tmax to characterize the thermodynamic history of the accreted particle as in Kereš et al. (2005) and Oppenheimer et al. (2010). We define Tmax as the maximum temperature the particle ever reaches before becoming star forming. We define an accretion event as hot mode accretion if log (Tmax) > 5.5 or cold mode accretion otherwise. Both of these accretion modes are also referred to as pristine gas accretion. On the other hand, if a particle is launched as a wind and subsequently re-accretes into a galaxy, we define this accretion as wind recycling. Unlike in our previous work, we reset Tmax to 0 once a particle is launched as a wind so that at the time it recycles, it will have a different Tmax. In this way, we can further divide wind recycling events into hot wind accretion and cold wind accretion based on the same temperature criteria. In addition, once a gas particle spawns or turns into a star particle, we associate this star-forming event with the last accretion event of that SPH particle. Therefore, for each star particle in the simulation we can tell when, where, and in which mode its progenitor gas particle accretes. In the following sections, we will study the formation history of galaxies by looking at their star particles. 5.2 The stellar mass–halo mass functions The stellar mass–halo mass function (SMHM) complements the GSMFs by showing how efficiently baryons turn into stars in haloes of different masses. Instead of directly plotting the ratio of stellar mass to halo mass, in Fig. 8 we instead plot the baryon conversion efficiency, i.e. ϵb ≡ M*/Mh(Ωb/Ωm)−1 to visually capture the small differences between the models more easily. Observationally, one determines this relation using empirical models that connect observed galaxies to dark matter haloes from N-body simulations. The empirically constrained SMHMs depend on the method used, but overall they agree with each other fairly well [see Moster, Naab & White (2018), for a recent compilation]. In Fig. 15, we compare the |$z$| = 0 SMHMs for the central galaxies from our simulations to the SMHM that is obtained in Behroozi et al. (2013) using subhalo abundance matching. Figure 8. Open in new tabDownload slide The stellar mass–halo mass functions (SMHMs) at |$z$| = 0, 1, 2, and 4. We compare the SMHMs from the same set of simulations as in Fig. 3. The solid lines are the running medians of the relation. We also show the scatter of the relation for the RefHres simulation as shaded regions that enclose 68 per cent of all galaxies within each Mh bin. The upturn in the SMHMs below |$M_\mathrm{ h} \lt 10^{11}\, \mathrm{M}_\odot$| at |$z$| = 0 and 1 is a selection effect owing to an artificial stellar mass cut for underresolved galaxies. The blue lines show the empirical best-fitting models from Behroozi et al. (2013) as observational constraints. Figure 8. Open in new tabDownload slide The stellar mass–halo mass functions (SMHMs) at |$z$| = 0, 1, 2, and 4. We compare the SMHMs from the same set of simulations as in Fig. 3. The solid lines are the running medians of the relation. We also show the scatter of the relation for the RefHres simulation as shaded regions that enclose 68 per cent of all galaxies within each Mh bin. The upturn in the SMHMs below |$M_\mathrm{ h} \lt 10^{11}\, \mathrm{M}_\odot$| at |$z$| = 0 and 1 is a selection effect owing to an artificial stellar mass cut for underresolved galaxies. The blue lines show the empirical best-fitting models from Behroozi et al. (2013) as observational constraints. The Ref and RefHres simulations agree reasonably well with the observationally inferred (Behroozi et al. 2013) SMHM up to the peak at log (Mvir) ∼ 12, with the largest difference at |$z$| = 1. The 2883 and 5763 simulations of this model predict similar results for log (Mvir) > 11.2, but the lower resolution simulation artificially boosts M*/Mh at lower masses. The turnover of the SMHM is much sharper in the observations than in any of the simulations, and all models drastically overpredict M*/Mh for log (Mvir) > 13 at |$z$| = 0. The Refσ3 simulation, with weaker outflows in low-mass haloes, overpredicts the observed M*/Mh in low-mass haloes with log (Mvir) < 11.5 at all redshifts and converges to the Ref model at high masses. The RefSlow simulation, with lower wind velocities, makes similar predictions to the Ref model at |$z$| = 4, but at |$z$| = 2, and increasingly at lower redshifts, it predicts higher M*/Mh in haloes near or above the SMHM turnover. The agreements and disagreements in Fig. 8 closely track those seen previously in the GSMF (Fig. 3). We now examine the contributions to galaxy stellar masses in more detail, focusing first on |$z$| = 2 and then on |$z$| = 0. 5.3 Galaxies at redshift z = 2 Fig. 9 shows the contribution of cold, hot, cold wind, and hot wind accretion to the stellar mass content of galaxies at |$z$| = 2 in the Ref, RefSlow, and Refσ3 simulations. The qualitative trends are similar between the simulations. Cold mode accretion dominates, contributing to nearly 100 per cent of all star formation in small haloes with log (Mvir) < 11.0 and over half of all stars in the most massive haloes. The hot mode fraction grows with halo mass and becomes comparable to the cold mode in the most massive haloes. Wind recycling is not yet important at |$z$| = 2, especially in less massive haloes where most winds are able to escape the halo potential and not return. Figure 9. Open in new tabDownload slide A closer look into the SMHM at |$z$| = 2. Upper panel: The black lines show the total mass of stars, averaged over all central galaxies from each halo mass bin, as a function of the halo mass, i.e. SMHM. The stellar masses are further divided into four categories, based on the accretion histories of their progenitor gas particle. The blue, red, cyan, and magenta lines show stellar mass formed from cold, hot, cold wind, and hot wind accretion, respectively. We also plot the empirically determined relation between stellar mass and halo masses from Behroozi et al. (2013) as the green line.Lower panel: The fraction of stellar mass that falls into each category. In each panel, the line styles indicate the three simulations used in this comparison. Figure 9. Open in new tabDownload slide A closer look into the SMHM at |$z$| = 2. Upper panel: The black lines show the total mass of stars, averaged over all central galaxies from each halo mass bin, as a function of the halo mass, i.e. SMHM. The stellar masses are further divided into four categories, based on the accretion histories of their progenitor gas particle. The blue, red, cyan, and magenta lines show stellar mass formed from cold, hot, cold wind, and hot wind accretion, respectively. We also plot the empirically determined relation between stellar mass and halo masses from Behroozi et al. (2013) as the green line.Lower panel: The fraction of stellar mass that falls into each category. In each panel, the line styles indicate the three simulations used in this comparison. Comparing the dotted lines and the solid lines shows the effect of changing the mass-loading factor. The Refσ3 simulation has a smaller η in low-mass galaxies compared to the fiducial simulation and, therefore, allows more gas to turn into stars. As a result, there is much more stellar mass formed from cold accretion in these galaxies, while in the other two simulations this gas is more likely to be launched as a wind before forming stars. These simulations have the same η values in massive haloes, and the stellar mass production converges at log (Mvir) > 12. This convergence implies that the winds from low-mass galaxies are not affecting the pristine gas accretion into high-mass galaxies. In principle, one expects the Ref simulation to have more wind recycling than the Refσ3 simulation owing to the larger amount of wind ejection, but this is not seen because the ejected particles have not yet had enough time to recycle. Hence, wind recycling remains a small fraction up to |$z$| = 2 in both models. Comparing the dashed lines and the solid lines shows the effect of changing the wind speed. The most significant effect is that the slower winds in the RefSlow simulation result in much more wind recycling, especially the cold wind accretion in haloes of all sizes at |$z$| = 2. This is a direct consequence of the shorter recycling time. Another clear effect is that the fiducial simulation has less cold and hot accretion than the RefSlow simulation, indicating that the fast wind speed not only suppresses wind recycling but also plays a role in preventing pristine accretion through hydrodynamic interactions with the fresh, in-falling gas. Since these two simulations have the same outflow rate for a given halo mass, the higher stellar mass in the RefSlow galaxies can be explained by the enhanced accretion rate owing to the slow wind speed. In summary, two major factors contribute to the different |$z$| = 2 SMHMs from our test simulations. The mass-loading factor controls the amount of outflow for a given halo but has little effect on the total amount of cold or hot accretion, which dominates at that redshift and above. The wind speed affects the amount of inflow. Faster winds reduce cold and hot accretion and also reduce the wind recycling by a similar amount. 5.4 Galaxies at redshift z = 0 The observed SMHMs at |$z$| = 0 show a characteristic Λ-shape with the intermediate-mass haloes [log (Mvir) ∼ 12] having the peak baryonic conversion efficiency. The efficiency declines in more massive haloes as well as in less massive ones, although the reasons are likely very different: theoretical models of galaxy formation suggest that the formation of massive galaxies is characterized by the late assembly of smaller systems that formed at early times and by having little in situ star formation at low redshifts. On the other hand, the low-mass haloes in the local Universe followed more linear growth histories and formed many of their stars more recently. As shown in Fig. 8, our simulations in general fail to match the observed SMHMs at |$z$| = 0 over the entire mass range. The discrepancies are most prominent in the most massive haloes, motivating us to perform separate analyses on galaxies that form in haloes of different masses. Here we select haloes from three different mass bins and study the formation histories of their central galaxies. The low-mass bin consists of ∼2000 haloes with 11.0 < log (Mvir) < 11.5. The intermediate-mass bin consists of ∼400 haloes with 11.85 < log (Mvir) < 12.15 and the massive bin consists of ∼60 haloes with 12.85 < log (Mvir) < 13.15. The exact number of haloes within each mass bin varies slightly among the simulations, but we focus on comparing the average baryonic conversion efficiency, which is normalized by the total virial mass of all haloes within each mass bin. In Figs 10 and 11, we show how stellar mass grows with time in haloes selected from the three mass regimes and divide the stellar mass at any time into categories based on their accretion histories as in the previous section. Figure 10. Open in new tabDownload slide We select and divide |$z$| = 0 central galaxies into three groups based on their halo virial mass. Columns from left to right show how stellar mass on average grows with time in low-mass [11.0 < log (Mvir) < 11.5], intermediate-mass [11.85 < log (Mvir) < 12.15], and massive [12.85 < log (Mvir) < 13.15] haloes. Similar to Fig. 9, at each redshift, we divide star particles in these galaxies into four channels based on their accretion history: The blue, red, cyan, and magenta lines indicate cold, hot, cold wind, and hot wind accretion, respectively. Thetop row shows the cumulative mass growth history. The middle row shows differential stellar mass growth within a constant redshift interval Δ|$z$|. In the upper and middle rows, we have normalized the stellar mass by the halo mass to indicate the baryon conversion factor. The green stars are the empirical results from Moster et al. (2018) for comparison. The bottom row shows the fraction of stars formed within each subcategory, with the grey line showing the fraction of total stellar mass at |$z$| = 0 that has already formed at a certain redshift. In each panel, we compare two simulations: the Ref and RefSlow simulations, indicated by the solid and dashed lines, respectively. Figure 10. Open in new tabDownload slide We select and divide |$z$| = 0 central galaxies into three groups based on their halo virial mass. Columns from left to right show how stellar mass on average grows with time in low-mass [11.0 < log (Mvir) < 11.5], intermediate-mass [11.85 < log (Mvir) < 12.15], and massive [12.85 < log (Mvir) < 13.15] haloes. Similar to Fig. 9, at each redshift, we divide star particles in these galaxies into four channels based on their accretion history: The blue, red, cyan, and magenta lines indicate cold, hot, cold wind, and hot wind accretion, respectively. Thetop row shows the cumulative mass growth history. The middle row shows differential stellar mass growth within a constant redshift interval Δ|$z$|. In the upper and middle rows, we have normalized the stellar mass by the halo mass to indicate the baryon conversion factor. The green stars are the empirical results from Moster et al. (2018) for comparison. The bottom row shows the fraction of stars formed within each subcategory, with the grey line showing the fraction of total stellar mass at |$z$| = 0 that has already formed at a certain redshift. In each panel, we compare two simulations: the Ref and RefSlow simulations, indicated by the solid and dashed lines, respectively. Figure 11. Open in new tabDownload slide Same as Fig. 10, except that here we compare the Ref and Refσ3 simulations, indicated by the solid and dotted lines, respectively. Figure 11. Open in new tabDownload slide Same as Fig. 10, except that here we compare the Ref and Refσ3 simulations, indicated by the solid and dotted lines, respectively. Low-mass haloes: The stellar mass from the Ref and RefSlow simulations matches the observed value at |$z$| = 0, but the Refσ3 simulation overproduces stellar mass by 0.6 dex. At |$z$| = 0, cold accretion and cold wind recycling each contributes roughly half of the total stars formed, while hot mode accretion contributes |$\text{$\sim$} 10 \!-\! 15{{ per\ cent}}$| of star formation. Cold accretion dominates the supply of star-forming gas in all three simulations until |$z$| = 2.0, after which cold wind recycling and hot mode accretion start to be important. Compared to the other simulations, the haloes in Refσ3 form many more stars from both cold accretion and wind recycling, not because of more inflow but because they have less outflow as a result of the smaller mass-loading factors. The slower wind speed in the RefSlow model increases cold wind recycling by a large amount compared to the fiducial simulation. As a result, the galaxies at |$z$| = 0 are slightly more massive as wind recycling gains importance after |$z$| = 2, but their stellar masses are still consistent with the observations. Intermediate-mass haloes: The |$z$| = 0 stellar mass from all three simulations is consistent with the observations within a small factor. The evolution of stellar content in these haloes is qualitatively similar to the small-mass haloes but with several major differences. First, stars from all accretion channels formed earlier in these more massive haloes, as is expected from the hierarchical assembly of galaxies. Secondly, both hot accretion and hot wind recycling, though still subdominant over most of the time, become more important at low redshifts, and together contribute |$\text{$\sim$} 30{{ per\ cent}}$| of the total stars formed at |$z$| = 0. Thirdly, cold accretion still dominates star formation at |$z$| > 2 but nearly stops after |$z$| = 2 when the shock-heated gas starts to develop a hot corona in these haloes. In the end, cold accretion only accounts for |$\text{$\sim$} 20{{ per\ cent}}$| of the total stars formed. Cold wind recycling still plays a critical role in determining the final stellar mass of the galaxies, and largely accounts for the differences among the three simulations. Its contribution is more prominent in the RefSlow simulation. Massive haloes: At the massive end, galaxies from the Ref and Refσ3 simulations evolve very similarly and overproduce stars by a factor of 3 at |$z$| = 0 (Fig. 11). The Ref simulation has larger η in small haloes, but the differences in η decrease with σ and become the same when |$\sigma \gt 106\mathrm{\, km\, s^{-1}}$|⁠. Therefore, the larger mass-loading factor in the Ref simulation only affects the progenitor galaxies during the earliest stages of their assembly when they were still small. Since these haloes assembled fast at high redshifts, the different scalings of η and σ in the low-mass haloes have little effect on the massive galaxies in our simulations. Compared to the intermediate-mass haloes, they have even earlier star formation and a higher fraction of hot accretion and hot wind recycling. Except for the RefSlow simulation, where cold wind recycling is clearly more important for stellar growth than the other channels, all four accretion channels contribute comparable amounts in the other simulations, with cold accretion + cold wind recycling and hot accretion + hot wind recycling each responsible for half of the stars formed. The RefSlow simulation overproduces stellar mass by a factor of 5, more than the other simulations. Fig. 10 shows that this owes not only to more cold wind recycling because of the slower wind speed, but also because of a significantly higher amount of hot accretion and hot wind recycling than the other simulations. Furthermore, hot accretion and hot wind recycling are also higher in the low-mass and intermediate-mass regimes, but unlike in the massive haloes, they are always subdominant to the total mass budget in less massive haloes. Naturally, any feedback mechanism designed to suppress star formation at these masses would strongly impact these trends. 5.5 The importance of mergers In the above discussions, we focused on studying the histories of star particles that end up in certain galaxies at a certain redshift. However, galaxies in a hierarchical Universe are often the result of many merging events. In particular, massive galaxies are often assembled from many smaller galaxies that formed in a wide range of haloes and environments. Since the in situ star formation efficiency, which is regulated by feedback, strongly depends on the halo mass, the final mass of a galaxy could be sensitive to feedback in those haloes where star formation was most efficient. Therefore, to understand how the feedback algorithms affect the final stellar mass of massive galaxies at |$z$| = 0, it is necessary to study when and where their progenitors formed. To evaluate the importance of mergers, we need to trace the evolution of galaxies in our simulations over time. At each output, if most stars within a galaxy are found in some galaxy at the next output, we define the first galaxy as a progenitor of the second galaxy. A galaxy could have more than one progenitor at any time, and we define its main progenitor as the most massive progenitor. We consider any other progenitor of this galaxy as a merger into this galaxy between the two outputs. Therefore, we can define the main evolutionary path of a galaxy at |$z$| = 0 by sequentially tracking its main progenitors over time. At any time when a merger occurs, we calculate the mass ratio between the two galaxies. In this work, we define major mergers as those that involve two galaxies with a mass ratio over 1/5. One caveat is that some galaxies take longer than a few outputs to completely merge with their host galaxies. In some situations, they were first grouped with the host galaxy during the first pass-by but left and became a separate galaxy later on, before they finally merged again. To avoid counting these galaxies as individual mergers multiple times, we consider only the first merging event by requiring that the mass of the host galaxy be at its maximum up to the merging event. Therefore, if the merging galaxy later left, the mass of the host galaxy would decrease and any subsequent pass-by will not be counted until the merger is complete. This criterion effectively removes most of the spurious mergers without missing any real mergers. To evaluate the importance of mergers, we look at galaxies at |$z$| = 0 and determine what fraction of stars each galaxy accreted through major mergers and where and when the stars present at |$z$| = 0 form. Fig. 12 shows that for most galaxies, the fraction of stars acquired through major mergers is less than 10 per cent. In general, more massive galaxies have a higher fraction of their stars formed in other galaxies and merged with it at later times, but even in the most massive bins, only 30 per cent of galaxies have more than half of their stars added through major mergers. The major merger fractions are also similar between the Ref and RefSlow simulations. Galaxies in the RefSlow simulation in general have a higher fraction of stars formed in situ because of more wind recycling on to the main progenitors. These results do not change very much if we include mergers with a mass ratio less than 1/5. Figure 12. Open in new tabDownload slide The dotted, dashed, and solid lines indicate the fraction of galaxies that have more than 10, 20, and 50 per cent of their |$z$| = 0 stellar mass gained by major mergers. The Ref and RefSlow simulations are shown in magenta and green, respectively. In general, the importance of mergers increases with M*, but even in the most massive galaxies, the fraction of stars from mergers are less than those formed in situ. Figure 12. Open in new tabDownload slide The dotted, dashed, and solid lines indicate the fraction of galaxies that have more than 10, 20, and 50 per cent of their |$z$| = 0 stellar mass gained by major mergers. The Ref and RefSlow simulations are shown in magenta and green, respectively. In general, the importance of mergers increases with M*, but even in the most massive galaxies, the fraction of stars from mergers are less than those formed in situ. In summary, the stellar growth of galaxies in our simulations is dominated by in situ star formation, with major mergers contributing a small fraction, except in the most massive galaxies. The final stellar mass of a galaxy is in most cases determined by the growth of its most massive progenitor, which is in turn regulated by how efficiently feedback suppresses star formation during the entire time of the evolution of the progenitor and its host halo. However, the relative importance of mergers in galaxy growth could increase if one added additional feedback to remove all the late-time star formation in massive galaxies, as required to match observations. 5.6 Wind recycling Wind recycling dominates the supply of star-forming gas at lower redshifts and is responsible for a considerable fraction of the total stellar mass in most haloes. In this section, we will show that wind speed strongly affects the recycling time-scale trec of the launched winds. The amount of winds that re-accrete after being launched is closely related to the recycling time-scale trec (Oppenheimer et al. 2010), defined as the time between a particle being launched as wind and it becoming star forming again. Fig. 13 compares the trec of wind particles from the three simulations with different wind speeds (see Fig. 2). The wind particles from the ezw simulation have trec that strongly depends on the halo mass. The deep gravitational potential of massive haloes causes wind particles to fall back shortly after being launched, creating a galactic fountain that is categorically different from the galactic scale winds in smaller galaxies. Oppenheimer et al. (2010) use the same wind algorithm in their simulations and find a similar trend. They refer to it as ‘differential recycling’, which is key to regulating star formation as a function of halo mass and thereby shaping the GSMFs. However, the recycling times from Oppenheimer et al. (2010) are greatly affected by the fact that their wind speeds do not scale as |$v$||$\mathrm{ w}$| ∝ σ after they leave the galaxy (Fig. 2) as they were originally intended, which was the motivation for our new wind model. Figure 13. Open in new tabDownload slide Upper panels: The median recycling time trec of winds that have recycled by |$z$| = 0 as a function of the virial mass Mvir of the halo from which the winds were launched. We only include winds launched from central galaxies. The dotted lines show trec only for winds that become hot. The left- and right-hand panels show results for winds launched at |$z$| = 2 and 1, respectively. We include all wind particles that are launched during a small redshift window with Δ|$z$| = 0.002 at these redshifts. The dotted horizontal line in each panel indicates the lookback time at that redshift and is the upper limit of trec for those winds. The shaded area shows the 1σ scatter in each Mh bin. Middle panels: The solid lines indicate the fraction of all winds that have ever re-accreted on to any galaxy at least once by |$z$| = 0. The dotted lines include only those winds that become hot. In general, the fraction of hot winds that recycle is lower. Bottom panels: The fraction of all winds that become hot, regardless of whether or not they have recycled by |$z$| = 0. Figure 13. Open in new tabDownload slide Upper panels: The median recycling time trec of winds that have recycled by |$z$| = 0 as a function of the virial mass Mvir of the halo from which the winds were launched. We only include winds launched from central galaxies. The dotted lines show trec only for winds that become hot. The left- and right-hand panels show results for winds launched at |$z$| = 2 and 1, respectively. We include all wind particles that are launched during a small redshift window with Δ|$z$| = 0.002 at these redshifts. The dotted horizontal line in each panel indicates the lookback time at that redshift and is the upper limit of trec for those winds. The shaded area shows the 1σ scatter in each Mh bin. Middle panels: The solid lines indicate the fraction of all winds that have ever re-accreted on to any galaxy at least once by |$z$| = 0. The dotted lines include only those winds that become hot. In general, the fraction of hot winds that recycle is lower. Bottom panels: The fraction of all winds that become hot, regardless of whether or not they have recycled by |$z$| = 0. The other two simulations have a mass-dependent enhancement to the wind speed. Increasing wind speed with mass has a direct effect on trec, with a stronger enhancement (Ref) leading to a longer trec in massive haloes. Similar to what Fig. 2 indicates, the wind dynamics inside these haloes are sensitive to their initial speed. For example, in the most massive galaxies at |$z$| = 2, the average wind speed in the Ref simulation is ∼2 times faster than in the RefSlow simulation while the recycling time is ∼3 times longer. We have further divided wind recycling into cold wind and hot wind recycling based on whether or not the wind particle heats up to 105.5 K. In the middle row of Fig. 13, the dotted lines show that the hot winds are less likely to re-accrete into galaxies by |$z$| = 0 than cold winds. For the winds that did recycle, the upper rows of Fig. 13 show that the recycling time-scales are significantly longer for the hot winds (dotted) on average. Most cold winds that formed stars at |$z$| ∼ 0 were launched well below |$z$| = 1, while most hot winds were launched around |$z$| = 2. Moreover, Figs 10 and 11 show that even though hot wind recycling is nearly negligible in low-mass and intermediate-mass haloes, it is important to the late star formation in massive galaxies. Because of the long trec of hot winds, it also indicates that a considerable fraction of stars in massive galaxies formed from outflow material launched long ago, at least in our simulations. The wind particles in our simulations are shock heated immediately after they hydrodynamically recouple to the ambient SPH particles. The initial wind speed plays a critical role in determining the post-shock temperature of the wind particles. Once they heat up to a temperature where cooling becomes inefficient, they will likely stay hot and become indistinguishable from a normal gas particle of the hot corona gas. The evolution of the hot wind particles thus depends more on the cooling physics than the dynamics that governs the recycling of cold wind particles. The bottom row of Fig. 13 shows that the fraction of winds that became hot is very sensitive to wind speed. The hot wind fraction is significantly higher in the Ref simulation, where wind heating is more efficient owing to the faster wind speeds. Note that with a sufficiently fast wind speed, our wind algorithm naturally results in a multiphase outflow, without the need to artificially add a hot component to the winds at launch as in MUFASA (Davé et al. 2016). We also find that the hot wind fraction is negligible in even the most massive galaxies in the ezw simulation, where the wind speeds are even lower. This was the wind model used in Davé et al. (2013). However, we must caution that the interactions between the winds and the halo gas likely involve processes such as hydrodynamic instabilities and thermal conduction that are unresolved in our simulations and likely even in galaxy zoom-in simulations with the highest resolution today (e.g. Schneider & Robertson 2017). The evolution of winds inside and outside galactic haloes in galaxy simulations, therefore, likely depends as much on numerics as on the true underlying physics. Hence, the behaviour and the effects of wind recycling must be re-examined with future simulations that have higher resolution or accurate and numerically robust subgrid models that incorporate necessary physics that has been neglected or incorrectly modelled in simulations up until now. Simulations that concentrate resolution in gaseous haloes (Hummels et al. 2019; Peeples et al. 2019; van de Voort et al. 2019) can improve modelling of physics in the CGM, though even with this approach the resolution may not be sufficient to accurately model interactions within the multiphase CGM (Scannapieco & Brüggen 2015; Schneider & Robertson 2017) and in addition it may be difficult to quantify recycling for ensembles of galaxies with a range of properties. 5.7 Implications for additional feedback Feedback processes are essential in cosmological simulations to successfully reproduce the observed stellar content of the Universe. Stellar feedback such as galactic winds generated from the brightest stars and SNe has been widely applied to explain the growth of small galaxies, but these processes alone are usually insufficient to suppress the growth of massive galaxies. The stellar feedback models in simulations are usually tuned to match observational constraints at the low-mass end, while one often invokes additional feedback such as AGN feedback to produce more realistic massive galaxies. It is also unclear what exact role AGN feedback plays in suppressing star formation. It could work as preventative feedback that limits the amount of inflow, or as kinetic feedback that drives additional outflows from galaxies. In the previous sections, we have shown that changing the parameters of our particular stellar feedback model within our explored range could significantly affect galaxy growth, even in the most massive haloes. This hints at the possibility that a combination of carefully tuned galactic wind parameters might be able to simultaneously reproduce the stellar content on all mass scales at any redshift. Even if the wind model is unable to meet all the constraints, it is important to understand the shortcomings of the current model that have to be solved with additional feedback processes. A successful galaxy formation model that reproduces |$z$| = 0 results must also be able to match observations from higher redshifts. At |$z$| = 2, we have shown that a strong halo mass dependence of the mass-loading factor is key to matching the faint-end slope of the SMHM, while different wind speeds are responsible for variations of the stellar mass in massive haloes. The Ref and RefSlow simulations both reasonably match the observed relation at |$z$| = 2. The RefSlow simulation produces more stars in intermediate to massive haloes owing to slower winds and a short recycling time, and agrees better with observations at the ‘knee’ but worse at the massive end. At |$z$| = 0, the discrepancy at the massive end grows, resulting in a factor of 3 times more stars in the massive bin and even larger discrepancies in more massive haloes. The grey lines in the bottom panels of Figs 10 and 11 show the build-up of stars that end up in the massive galaxies. They are consistent among the simulations, with more than |$80{{ per\ cent}}$| of stars formed after |$z$| = 2 and |$60{{ per\ cent}}$| of stars formed after |$z$| = 1. Therefore, in our simulations it is the late star formation in massive galaxies that must be greatly reduced to match the |$z$| = 0 observations. Section 5.5 confirms that most of these stars formed in situ instead of through merging. A successful model must maintain the level of star formation up to |$z$| = 2 as in the RefSlow simulation but significantly reduce the amount of stars formed afterwards. In fact, the top right panels of Figs 10 and 11 show that galaxies in the massive bin have already formed by |$z$| = 2 as many stars as required to match the |$z$| = 0 observations. Hence, to make these galaxies agree with the |$z$| = 0 constraint, nearly all in situ and ex situ star formation after |$z$| = 2 needs to be suppressed. Observationally, this phenomenon is known as downsizing, i.e., massive galaxies at |$z$| = 0 formed earlier but have little late-time star formation. However, it is challenging to reproduce this effect in our simulations without additional feedback. Stars from cold accretion mostly formed at high redshifts in small haloes that later assembled into the massive galaxies. The most efficient way to remove them from our simulations is to have stronger winds, i.e. stronger mass-loading factors in those haloes. However, having too much winds early will unavoidably fail to match observations at higher redshifts. The right-hand panels of Figs 10 and 11 show that cold accretion has nearly stopped after |$z$| = 2 for these galaxies but hot accretion and wind recycling continue growing rapidly at low redshifts, and is responsible for most of the excess stars formed. After |$z$| = 2, the stars in massive galaxies that must be prevented from forming come from almost equal parts: hot accretion, hot wind re-accretion, and cold wind re-accretion. Hence, preventing hot gas from cooling and forming stars at these times will eliminate both the hot mode accretion and hot wind re-accretion and will lessen the tension in massive haloes. There are several potential mechanisms, such as AGN and cosmic ray heating, that could reduce the amount of this cooling gas but they are not yet included in our simulations. The hot wind recycling would be harder to affect by extra heating, because the higher metallicity of this gas makes it cool faster. In hydrodynamic simulations, mixing between ejected wind elements and the surrounding gas may have a large impact on the amounts of hot and hot wind accretion. It is, however, unclear whether this type of feedback could prevent the ∼1/3 of stars formed through cold wind re-accretion after |$z$| = 2 from forming, which is also necessary to match the observations. It is possible that a more accurate treatment of the cloud-CGM interaction would allow a larger fraction of these winds to become hot, alleviating this problem. 6 THE HIGH-RESOLUTION SIMULATION OF THE REFERENCE MODEL In this section, we present key results from the high-resolution RefHres simulation. It adopts the new wind launch algorithm as described in the previous sections using our fiducial set of wind parameters listed in Table 1. It is also implemented with the numerical improvements to the SPH hydrodynamics introduced by Huang et al. (2019). We will focus on those predictions that have changed significantly from Huang et al. (2019) and from our previously published work with our new wind algorithm. 6.1 The stellar content Fig. 14 shows that the GSMFs from our fiducial simulation, shown as black lines in each panel, are mostly consistent with observations at all redshifts from |$z$| = 0 to 4. The agreement is particularly good at the faint end except for |$z$| = 0, where our simulation slightly underproduces the number of these low-mass galaxies. At the massive end, our GSMFs agree with observations at |$z$| = 1 and 4. However, our fiducial simulation produces too many massive galaxies at |$z$| = 0 and too few massive galaxies at |$z$| = 2, even after taking account for systematic uncertainties in the stellar mass measurements at these redshifts. Figure 14. Open in new tabDownload slide Same as Fig. 3, except that here we show a different set of simulations, including the fiducial high-resolution simulation RefHres. See the text and Table 1 for descriptions of these models. The vertical dotted lines correspond to the mass of 1024 SPH particles in the RefHres simulation and 128 SPH particles in the other simulations. Figure 14. Open in new tabDownload slide Same as Fig. 3, except that here we show a different set of simulations, including the fiducial high-resolution simulation RefHres. See the text and Table 1 for descriptions of these models. The vertical dotted lines correspond to the mass of 1024 SPH particles in the RefHres simulation and 128 SPH particles in the other simulations. The level of our agreement is comparable to other cosmological hydrodynamic simulations such as EAGLE (Furlong et al. 2015), MUFASA (Davé et al. 2016), and illustrisTNG (Pillepich et al. 2018b), except for massive galaxies at |$z$| = 0 where AGN feedback incorporated in these other simulations more strongly suppresses stellar mass growth. The ezw model as implemented by Davé et al. (2013) includes an ad hoc quenching scheme in massive galaxies to reproduce the |$z$| = 0 GSMF. Without this quenching scheme, however, the ezw wind formulation produces worse agreement with observations: too many small galaxies at higher redshifts (⁠|$z$| = 2 and 4) and too many massive galaxies at lower redshifts (⁠|$z$| = 1 and 0). We have shown in Section 4 that the success of reproducing the faint end of GSMFs at |$z$| > 1 relies on a steeper scaling between the mass-loading factor η and the halo mass. On the other hand, suppressing the growth of massive galaxies relies on a higher wind velocity to effectively remove cold gas from the galaxies. Fig. 14 also shows that different hydrodynamic algorithms (comparing ezwDESPH and ezw) have noticeable effects on the GSMFs, principally at the massive end, although these are much less significant than the changes driven by the wind algorithm. Comparing the Ref simulation and the RefHres simulation shows that the results are also robust to numerical resolution, but note that in the higher resolution simulation we have increased the overall wind speed by a small factor to obtain a similar |$v$|25–|$v$|c relation. Fig. 15 shows that the baryon conversion efficiency from our fiducial simulation agrees well with observations (Behroozi et al. 2013; Moster et al. 2018) in small haloes and reaches a similar peak value, but it becomes too high in more massive haloes. This is a more clear illustration of the excess of stars in massive haloes than that seen in the |$z$| = 0 GSMF. Comparing to the ezw simulation, which uses the same SPH method but the ezw wind model, the new wind algorithm significantly reduces the stellar content in these massive galaxies but it is still not enough to match the observations. Also, increasing the numerical resolution has little effect on the SMHM. Figure 15. Open in new tabDownload slide The stellar mass–halo mass functions at |$z$| = 0. We compare the SMHMs from the same set of simulations as in Fig. 14. The solid lines are the running medians of the relation. We show the scatter of the relation for the RefHres and ezwDESPH simulations as shaded regions that enclose 68 per cent of all galaxies within each Mh bin. The green lines show the empirical best-fitting model from Behroozi et al. (2013) and Moster et al. (2018) as observational constraints. Figure 15. Open in new tabDownload slide The stellar mass–halo mass functions at |$z$| = 0. We compare the SMHMs from the same set of simulations as in Fig. 14. The solid lines are the running medians of the relation. We show the scatter of the relation for the RefHres and ezwDESPH simulations as shaded regions that enclose 68 per cent of all galaxies within each Mh bin. The green lines show the empirical best-fitting model from Behroozi et al. (2013) and Moster et al. (2018) as observational constraints. 6.2 SDE Fig. 16 shows the SDE of a few simulations. Our fiducial simulation, shown as the thick, black line in Fig. 16, agrees with the observational data to within 0.1 dex below redshift |$z$| = 3. At higher redshifts, it falls in between the Muzzin et al. (2013) data and the upper limits from González et al. (2011). Our simulation is capable of capturing the general trend of the cosmic SDE. Since the stellar density at any epoch is equivalent to the integration of the GSMFs at that redshift, the success of matching the GSMFs at various redshifts is key to matching the observed SDE. Figure 16. Open in new tabDownload slide Same as Fig. 7, except that here we show results from the fiducial high-resolution simulation compared with a few test simulations. Our fiducial wind model (in the RefHres and Ref simulations) reproduces the observations well, but the original ezw wind model starts overproducing stars at very early times. Figure 16. Open in new tabDownload slide Same as Fig. 7, except that here we show results from the fiducial high-resolution simulation compared with a few test simulations. Our fiducial wind model (in the RefHres and Ref simulations) reproduces the observations well, but the original ezw wind model starts overproducing stars at very early times. The original ezw model not only produces too many stars at |$z$| ∼ 0, owing mostly to the excess of stellar mass in massive galaxies, but also has started overproducing stars since |$z$| = 5 owing to an insufficiently large η. Changing the numerics from the ezwDESPH simulation to the ezw simulation allows more star formation at lower redshifts, but the effects are less significant than the effects of changing the wind algorithm. 6.3 Gas fractions and metallicity Fig. 17 shows the cold gas fractions fgas at |$z$| = 0 and 2 in the left-hand panels. In the simulations, we define fgas as $$\begin{eqnarray*} f_\mathrm{gas}\equiv \frac{M_\mathrm{gas}}{M_\mathrm{gas}+ M_{*}} , \end{eqnarray*}$$ (18) where M* is the total stellar mass of the galaxy, and Mgas is the total mass of the ISM gas in that galaxy. To determine which SPH particles are treated as multiphase ISM gas in our simulations, we assume a physical density threshold of nH > 0.13 cm−3 and a temperature threshold of log T/K < 4.5. Any SPH particles within a galaxy that meets these criteria are included when calculating Mgas. These sharp thresholds are somewhat arbitrary, so comparisons to observations should be interpreted with caution (Davé et al. 2011b). Figure 17. Open in new tabDownload slide Left-hand panels: cold gas fractions (defined in the text) as a function of stellar mass at |$z$| = 2 (upper panel) and |$z$| = 0 (lower panel). The observational data in the lower panel are compiled by Peeples et al. (2014), with error bars denoting the 16–84 per cent range. Right-hand panels: gas-phase mass–metallicity relations at |$z$| = 2 (upper panel) and |$z$| = 0 (lower panel). The |$z$| = 2 data are from Sanders et al. (2015) and the |$z$| = 0 data are from Tremonti et al. (2004). The shaded area in each panel shows the 16–84 per cent range of the results from the fiducial high-resolution RefHres simulation. We also show the medians from the lower resolution Ref as magenta lines. Figure 17. Open in new tabDownload slide Left-hand panels: cold gas fractions (defined in the text) as a function of stellar mass at |$z$| = 2 (upper panel) and |$z$| = 0 (lower panel). The observational data in the lower panel are compiled by Peeples et al. (2014), with error bars denoting the 16–84 per cent range. Right-hand panels: gas-phase mass–metallicity relations at |$z$| = 2 (upper panel) and |$z$| = 0 (lower panel). The |$z$| = 2 data are from Sanders et al. (2015) and the |$z$| = 0 data are from Tremonti et al. (2004). The shaded area in each panel shows the 16–84 per cent range of the results from the fiducial high-resolution RefHres simulation. We also show the medians from the lower resolution Ref as magenta lines. At |$z$| = 0, we add the data from Peeples et al. (2014), which are compiled from various data sets (McGaugh 2005, 2012; Leroy et al. 2008; Saintonge et al. 2011). The data points show the averaged atomic + molecular gas fractions in each stellar mass bin, with error bars indicating the 16th and 84th percentiles, which is the same range chosen for the simulated data. Our fiducial simulation reproduces the observed trend very well, though it slightly overpredicts the cold fractions in massive galaxies. Compared to the ezwDESPH simulation, small galaxies with |$\log (M_*/\mathrm{M}_\odot) \lt 10$| in the fiducial simulation are more gas rich, making the scaling relation at the faint end agree with the observations from Peeples et al. (2014). Comparing the Ref and RefHres in the figure shows that this result is resolution independent. At the massive end, the gas fractions in the fiducial simulation are close to those from the ezwDESPH simulation. Galaxies in the ezwDESPH simulation are in general more massive than their counterparts in the fiducial simulation. Therefore, at a fixed halo mass galaxies in the fiducial simulation actually contain a higher gas mass. In future work, we will track gas accretion through cosmic time in detail to understand the origin of cold gas in galaxies. At |$z$| = 2, a detailed comparison with observations is unavailable owing to a lack of direct measurements of the cold gas content at high redshifts. Nevertheless, it is consistent with the indirect observations (e.g. Popping et al. 2015) that the cold gas fractions are generally higher than at |$z$| = 0 at a fixed stellar mass. The differences between the fiducial and ezwDESPH simulations are much larger at this redshift, with much lower gas fractions in massive galaxies than in the fiducial simulation. In the right-hand panels of Fig. 17, we compare the gas-phase MZRs from our fiducial simulation with observations. We calculate the gas-phase metallicity for each galaxy by averaging over all ISM particles within the galaxy, weighted by their SFR. We use oxygen as the metallicity tracer and adopt a solar value of [O/H]⊙ + 12 = 8.69 (Asplund et al. 2009). We use the Sanders et al. (2015) (⁠|$z$| ∼ 2.3) and Tremonti et al. (2004) data for comparisons. Since they measure metallicity using different calibrations, we convert the Sanders et al. (2015) data to the Tremonti et al. (2004) calibration using the fitting formula from Kewley & Ellison (2008). This increases the overall normalization of the Sanders et al. (2015) data by 0.1–0.3 dex. At face value, the comparison in Fig. 17 shows a slight overproduction of gas-phase metallicity with the right overall trend at |$z$| = 2, but a more severe discrepancy at |$z$| = 0 where the simulations underpredict the metallicity of low-mass galaxies and overpredict the metallicity of high-mass galaxies. The caveat is that calibration and measurement uncertainties have a large impact on the observed mass–metallicity relation (Kewley & Ellison 2008). Furthermore, the initial mass function (IMF) averaged oxygen yield is uncertain, and it could change with galaxy mass if the IMF itself changes. In principle, the mass–metallicity relation is a strong diagnostic of outflow efficiency (Finlator & Davé 2008), and it should also be sensitive to the amount of metal recycling in winds. 6.4 Intergalactic and circumgalactic medium Galactic winds are not only important as a feedback mechanism that suppresses galaxy growth, but are also essential to explain the enrichment of the IGM and the CGM as they carry the metals that are produced inside the galaxy into the outer halo and beyond. Measurements of the metal content in the IGM/CGM using quasar absorption spectroscopy [see Tumlinson, Peeples & Werk (2017), for a review] provide crucial constraints for cosmological simulations (Oppenheimer & Davé 2006; Oppenheimer et al. 2012; Ford et al. 2014, 2016). In this section, we show how the new wind algorithm in our fiducial simulation affects the metal distributions in the IGM/CGM. To mimic the observational measurements, we create mock quasar absorption spectra using specexbin as in Huang et al. (2019). A more detailed description of the technique can be found in Oppenheimer & Davé (2006). In short, we generate random sightlines covering a redshift range from |$z$| = 0 to 0.5 through the simulation volume. On each of these long sightlines, we calculate the optical depth of multiple ions in redshift space based on the properties of the surrounding gas, such as the density, temperature, velocity, and metallicity. We use a uniform ultraviolet background (Haardt & Madau 2012) to calculate the ionization level of each ion. We normalize the strength of the background to match the Lyman α decrement measurements (Huang et al. 2019). From these mock spectra, we further obtain observational quantities such as column densities and equivalent widths for each ion by fitting their line profiles using the Voigt profile fitting software autovp (Davé et al. 1997). In this paper, we generate 71 sightlines for each of the low-resolution simulations and 400 sightlines for the RefHres simulation. Fig. 18 compares the column density distributions (CDDs) of O vi and Ne viii from the four simulations. Comparison of ezw to ezwDESPH shows that numerics have a strong effect on the CDDs of these ions, as shown by Huang et al. (2019). The new wind model (Ref) slightly increases the number of high-column density absorbers compared to the ezw wind (ezw) but does not strongly affect the low-column density absorbers. The CDDs are also sensitive to numerical resolution, as the higher resolution simulation RefHres has fewer absorbers than the lower resolution simulation. Figure 18. Open in new tabDownload slide The CDDs of O vi (upper panel) and Ne viii (lower panel). We obtain the statistics from random sightlines that span from |$z$| = 0.0 to 0.5 as described in the text. Results from the four simulations are colour coded according to Table 1. Figure 18. Open in new tabDownload slide The CDDs of O vi (upper panel) and Ne viii (lower panel). We obtain the statistics from random sightlines that span from |$z$| = 0.0 to 0.5 as described in the text. Results from the four simulations are colour coded according to Table 1. The contours in Fig. 19 show how metals are distributed in the temperature–density phase space at |$z$| = 0 in the three simulations. Comparing the ezw (middle panels) and the ezwDESPH (left-hand panels) shows the effects of changing numerics and cooling physics on the metal distributions and the high-ion absorbers. We have studied those effects in greater detail in previous work (Huang et al. 2019). The main effect is that there are more metals in the warm–hot gas (WHIM; upper left quadrants) owing to better resolved shocks around filaments. Figure 19. Open in new tabDownload slide The metal distributions at |$z$| = 0 in phase space from the ezwDESPH (left), ezw (middle), and RefHres (right) simulations. The purple background colour scale indicates the mass-weighted average metallicity in each cell. In each panel, we show the O vi absorbers or the Ne viii absorbers on random sightlines that are generated using the technique described in the text. The absorbers are colour coded according to their column densities. The two dotted lines in each panel divide the phase space into four regions: the warm–hot IGM (WHIM, upper left), the diffuse IGM (lower left), hot halo gas (upper right), and cold dense galactic gas (lower right). Several contours lines are stressed for better visualization. Figure 19. Open in new tabDownload slide The metal distributions at |$z$| = 0 in phase space from the ezwDESPH (left), ezw (middle), and RefHres (right) simulations. The purple background colour scale indicates the mass-weighted average metallicity in each cell. In each panel, we show the O vi absorbers or the Ne viii absorbers on random sightlines that are generated using the technique described in the text. The absorbers are colour coded according to their column densities. The two dotted lines in each panel divide the phase space into four regions: the warm–hot IGM (WHIM, upper left), the diffuse IGM (lower left), hot halo gas (upper right), and cold dense galactic gas (lower right). Several contours lines are stressed for better visualization. Comparing the RefHres simulation (right-hand panels) to the ezw simulation (middle panels) shows that the new wind algorithm spreads a considerable amount of metals into the warm–hot IGM gas and the hot, dense gas as a result of both the stronger mass loading in low-mass galaxies and the faster wind speed. Since we do not allow metal mixing between the enriched wind particles and the pristine IGM gas, the enhanced metallicity at below cosmic mean density comes directly from wind particles that escape into the IGM. The higher metallicity in the hot gas is likely because of wind particles being able to remain longer in hot haloes before re-accreting on to the galaxies. One numerical caveat is that when the original specexbin calculates the local gas properties such as the temperature at a given location in a sightline, it averages over all neighbouring particles close to the sightline without distinguishing wind particles from normal SPH particles. This potentially leads to errors in a multiphase gas, such as when cold, metal-rich wind particles are among hot CGM particles. Therefore, we modified specexbin to take into account the contribution of each surrounding particle to the spectra on a particle-by-particle basis. However, we do not find any significant differences in the results for the high ions from using these two different methods. 7 SUMMARY Galactic winds are crucial to galaxy formation. At present, hydrodynamic simulations that model cosmological volumes (i.e. many Mpc on a side) lack the resolution to generate winds from physical processes in the ISM. Such simulations, therefore, employ subgrid prescriptions that are designed to capture the phenomenological behaviours of these processes. In this paper, we revisit a wind implementation that is based on a numerical algorithm proposed and developed by Springel (2005), Oppenheimer & Davé (2006), and Davé et al. (2013). We take into account new constraints from high-resolution zoom-in simulations (M15) and statistical properties of galaxies at high redshifts, such as their stellar mass functions, and make several changes to our wind algorithm. We examine the ability of the new algorithm to reproduce a wide range of observations and study the sensitivity of these predictions to variations in model parameters. The basic design of the wind algorithm is that in star-forming galaxies, cold and dense SPH particles are stochastically ejected from their host galaxies with an initial momentum kick to model large-scale star formation-driven winds. The mass-loading factor η determines the rate at which particles are ejected and the wind speed |$v$||$\mathrm{ w}$| determines the initial velocity given to the ejected particles. Observations and analytic calculations have shown that both of these parameters correlate with properties of their host galaxy or host halo such as the SFR and the characteristic velocity σ (Rupke et al. 2005), but an accurate determination of these scalings is unknown. Previous wind algorithms often parametrize them as η ∝ σ−1 or σ−2, and |$v$||$\mathrm{ w}$| ∝ σ, following the analytic formulation for momentum-driven or energy-driven winds (Murray et al. 2005). However, it becomes clear in cosmological simulations that artificial numerical treatments as well as fine-tuning of the model parameters are required to successfully reproduce key observables, such as the galaxy stellar mass function, owing to limitations in the numerical resolution of simulations and the simplicity of the analytic models. Furthermore, recent zoom-in galaxy simulations (e.g. M15) suggest different wind scalings than the analytic models. Most importantly, simulations necessarily impose these scalings at wind launch, while they are supposed to hold for gas that has escaped the dense ISM. When we measure the resultant wind scalings outside of galaxies, the original scalings no longer hold. We have therefore altered our wind launch algorithm to reproduce, approximately, the wind properties measured by M15 at 25 per cent of the halo virial radius. Major updates from our previous wind algorithm include the following: (1) We allow more freedom when assigning η and |$v$||$\mathrm{ w}$|. In particular, we allow a stronger dependence of η on σ, or equivalently, the halo mass. (2) We allow newly ejected wind particles to temporarily decouple from their host galaxies dynamically before they reach a density threshold of 0.1ρSF. The new algorithm may appear to be less deterministic than the original one by having a few more tunable parameters, but it is an unavoidable compromise to the uncertainties and limitations of our current knowledge of the nature of galactic winds. The primary focus of this paper is, therefore, not to extensively search for a set of parameters that best reproduce the observed Universe but rather to explore and characterize how some of the well-established observational results on galaxy formation could be affected by a physically plausible range of wind model parameters. Naturally, we perform this exploration within the narrow confines of our wind model. Differences between the methods used by different simulation groups in the literature could be larger. We find that the faint-end slopes of the GSMFs at |$z$| > 1 in our simulations are most sensitive to the power-law index βη, which determines how strongly the mass-loading factor η depends on σ in low-mass galaxies (Section 4.1.1). The energy-driven scaling η ∝ σ−2 that was used in our previous simulations (e.g. Davé et al. 2013) produces a faint-end slope that is too steep compared to observations. We find that to match the observed flatter slope, we need a scaling as steep as η ∝ σ−5 for |$\sigma \lt 106\mathrm{\, km\, s^{-1}}$| in our fiducial simulation. All of our simulations adopt η ∝ σ−1 at high masses. The need for such a strong scaling at low σ has also been found in the FIRE simulations, which predict an intermediate scaling of η ∝ σ−3.3 (M15), as well as in semi-analytic works (Peeples & Shankar 2011; Somerville et al. 2012; Lu et al. 2014) and other cosmological simulations (Pillepich et al. 2018a) that include kinetic feedback. Even though βη critically affects sub-L* galaxies at |$z$| = 1, the different scalings adopted in our test simulations produce similar faint-end slopes of the GSMFs at |$z$| = 0 and also have little effect on the final masses of massive galaxies. This emphasizes that robust statistical properties of dwarf galaxies at high redshifts are essential to distinguish between different feedback models and to understand how stellar feedback regulates galaxy growth. Such observations will have to await the launch of JWST. Changing the overall strength of outflows by changing the normalization factor αη also has a clear effect on galaxy growth, with a higher mass-loading factor leading to less star formation, especially in dwarf galaxies at high redshifts (Fig. 4). This dependence of M* on αη can be qualitatively explained by a simple analytic model that assumes isolated galaxy growth and negligible wind recycling (Fig. 5). The evolution of wind particles in a halo is very sensitive to the initial wind speed and the gravitational potential near the centre, which is usually dominated by baryons and is numerically poorly resolved. The winds launched with our new method have wind velocities that agree with the FIRE simulations (M15) at R25, while those launched with the original velocity formula often lose most of their momentum at small radii and even fail to reach R25 in massive haloes (Fig. 2). As a consequence, the initial wind speed has a strong impact on the growth of massive galaxies. Contrary to some previous findings that the stellar feedback is only efficient enough to suppress star formation in sub-L* galaxies, in some of our simulations, including the fiducial simulation, the fast winds do significantly reduce star formation in massive galaxies and bring the massive end of the predicted GSMF at |$z$| = 0 much closer to observations as long as they are capable of escaping their host galaxies instead of almost instantly falling back as in our original algorithm. Note, however, that the FIRE simulations only explore haloes as massive as 1013 M⊙. Below this mass scale, the FIRE simulations are able to reproduce the stellar mass–halo mass relation without any AGN feedback (Feldmann et al. 2017), supporting our results that stellar feedback alone might be sufficient to suppress star formation up to this mass scale. However, in our wind algorithm, we extrapolate the empirical relation between |$v$|25 and |$v$|c to even more massive systems by adjusting the initial wind velocities. Therefore, our results at the massive end of the GSMFs should not be interpreted as a consequence derived from physical assumptions but they rather show that the wind speed, as well as how winds propagate and stay in the halo, has strong effects on galaxy evolution. We further study how the initial wind speed could affect our simulation results by comparing our fiducial simulation with the RefSlow simulation, a simulation with slower wind speeds (Figs 6–8). Changing the wind speed significantly affects star formation in massive galaxies but has little effect in low-mass galaxies. In the most massive galaxies of the two simulations, the average wind speed differs by a factor of ∼2, and the stellar masses differ by ∼0.2–0.4 dex at different redshifts. This leads to clear differences at the massive end of the GSMFs, where the statistical variance is large. The faster wind speeds in the fiducial simulation relative to our older ezw algorithm drive wind particles further from their host galaxy. It also heats more wind particles to the temperature of the hot corona, making them have to cool before re-accreting and hence reducing their re-accretion rate (Figs 9–11). Both effects lengthen the recycling time of the wind particles and make wind recycling less efficient than simulations with slower wind speeds (Fig. 13). However, wind recycling still dominates accretion on to the massive galaxies at low redshifts, fuelling too much late star formation. Hot accretion is also responsible for 25 per cent of the total mass of stars formed in the massive galaxies at |$z$| = 0 and also needs to be significantly suppressed to have these galaxy stellar masses match observations. Mergers play only a limited role in the growth of massive galaxies and are nearly negligible for low-mass and intermediate-mass galaxies (Fig. 12). However, if one removes all the late-time star formation in massive galaxies required to match observations, the merger growth could become much more important. This sensitivity to the initial launch speed also implies that the simulations are sensitive to numerical resolution that affects the accuracy of force calculations near the centre of the haloes and the physical assumptions that govern the propagation of winds in the haloes. We empirically find that in our fiducial simulation RefHres, which has twice the spatial resolution and eight times the mass resolution as the other simulations, we need to enhance the wind speed by an overall factor of ∼1.14 to match the constraints at R25. After this correction, the galaxy properties of the fiducial simulation are similar to those of the corresponding lower resolution simulation. It implies that recalibration of the initial wind speed at different resolutions is necessary in subgrid wind implementations that are similar to ours. Instead of matching observational constraints such as the stellar mass functions, it is likely sufficient to tune the parameters to reproduce the same wind speed at a certain radius, after which wind propagation becomes largely independent of resolution. In this work, we choose R25 because of the constraints from the FIRE simulations (M15). With the new wind model and a fiducial set of wind parameters, we run a simulation (RefHres) with higher numerical resolution than these test simulations. This simulation results in GSMFs, SMHMs, and SDEs that are in much better agreement with observations at all redshifts than the original ezw wind. However, it still produces too many stars in massive galaxies at |$z$| = 0. The cold gas fractions agree well with observations and are not significantly affected by the new wind algorithm. The fiducial simulation produces slightly more high column density absorbers for high ions such as O vi and Ne viii, but this result is sensitive to numerical resolution. Despite many changes in both numerical algorithms and wind implementations, our new simulations confirm three key conclusions of our previous work: cold accretion produces most of the gas that forms stars in low-mass haloes, hot accretion takes over from cold accretion in high-mass haloes, and wind recycling is an essential component of galaxy growth at redshifts |$z$| < 1 (Kereš et al. 2005, 2009a, b; Oppenheimer et al. 2010). However, the details of the wind implementation have a large impact on the amount and mass dependence of wind recycling. Reproducing the observed stellar masses in high-mass haloes likely requires an additional mechanism that suppresses hot gas accretion, and AGN feedback is a natural candidate for this mechanism (Benson et al. 2003; Bower et al. 2006; Croton et al. 2006). However, we should be cautious in drawing lessons about AGN feedback scaling because the amount of feedback required is sensitive to still uncertain aspects of galactic winds driven by stellar feedback. In this paper, we have focused on the effects of wind launch algorithms, but our simulations also suffer from underresolving the physics of ejected wind gas after it has entered the CGM. This is probably true of all current cosmological simulations, even zoom-in simulations that attempt to resolve parsec-level structure on the ISM. Forcing high resolution in the CGM is one approach to this problem (Hummels et al. 2019; Peeples et al. 2019; van de Voort et al. 2019), though even so it may be difficult to resolve the relevant scales of instabilities and fluid mixing (Scannapieco & Brüggen 2015; Schneider & Robertson 2017). Another approach is to develop an explicit subgrid model for evolving wind particles after they leave the galaxy, so that wind propagation and recycling, which we have shown to critically affect many simulation results, are controlled by physical parameters instead of unresolved numerics. We will present such a model in future work. ACKNOWLEDGEMENTS We acknowledge support by NSF grant AST-1517503, NASA ATP grant 80NSSC18K1016, and HST Theory grant HST-AR-14299. DW acknowledges support of NSF grant AST-1909841. Footnotes 1 Since Somerville et al. (2012) parametrizes η as a function of halo mass, we used the SMHM relation at |$z$| = 2 from Behroozi, Wechsler & Conroy (2013) to obtain the halo mass from the stellar mass for any galaxy. 2 In other work, the mass-loading factor is often correlated with either the halo mass Mh or a characteristic velocity that scales with |$M_\mathrm{h}^{1/3}$|⁠, though the specific definitions for the mass and the velocity are slightly different. 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Das,, Susmita;Kanbur, Shashi, M;Bellinger, Earl, P;Bhardwaj,, Anupam;Singh, Harinder, P;Meerdink,, Brett;Proietti,, Nicholas;Chalmers,, Anthony;Jordan,, Ryan

doi: 10.1093/mnras/staa182pmid: N/A

Koen,, Chris

doi: 10.1093/mnras/staa190pmid: N/A

ABSTRACT Large monitoring campaigns, particularly those using multiple filters, have produced replicated time series of observations for literally millions of stars. The search for periodicities in such replicated data can be facilitated by comparing the periodograms of the various time series. In particular, frequency spectra can be searched for common peaks. The sensitivity of this procedure to various parameters (e.g. the time base of the data, length of the frequency interval searched, number of replicate series, etc.) is explored. Two additional statistics that could sharpen results are also discussed: the closeness (in frequency) of peaks identified as common to all data sets, and the sum of the ranks of the peaks. Analytical expressions for the distributions of these two statistics are presented. The method is illustrated by showing that a ‘dubious’ periodicity in an 'Asteroid Terrestrial-impact Last Alert System' data set is highly significant. methods: data analysis, methods: statistical 1 INTRODUCTION The type of data considered in this paper is a low-amplitude sinusoidal signal embedded in white noise: $$\begin{eqnarray*} y(t_{kj})\, = \,A_k\cos (2\pi \nu _0 t_{kj}+\phi _k)+e(t_{kj});\nonumber \\ j\, = \,1,2,\ldots ,N_k; \quad k=1,2,\ldots ,K , \end{eqnarray*}$$ (1) where the Ak are amplitudes, ν0 is the (common) frequency, ϕk are the phases of the signal, and e is uncorrelated noise. The K time series in equation (1) are each observed in Nk time points tkj, which will in general be irregularly spaced. In the simulations presented below, the noise is conveniently assumed zero-mean Gaussian with standard deviation σk, $$\begin{eqnarray*} e(t_{kj}) \sim N(0,\sigma _k^2), \end{eqnarray*}$$ but the specific distribution is not too important. Note also that the noise standard deviations, amplitudes, and number of observations per time series may differ, although it will be assumed in the theoretical treatment below that $$\begin{eqnarray*} A_1=A_2=\cdots A_K \quad \,\, \sigma _1=\sigma _2=\cdots \sigma _K \quad \,\, N_1=N_2=\cdots N_K . \end{eqnarray*}$$ This assumption is made for the sake of expositional clarity, not out of necessity. In fact, the units of the different time series need not be the same; for example, intensity measurements from different parts of the electromagnetic spectrum (radio, optical, X-ray, etc.) and/or quantities derived from spectra (radial velocities, equivalent widths, etc.) can all be included. The analyses below will be performed in the frequency domain. A central role will therefore be played by spectral transformations of the data. The amplitude spectrum S of data { y(tj)} is defined in terms of the periodogram $$\begin{eqnarray*} I(\nu)=\frac{1}{N} \left| \sum _{j=1}^N [y(t_j)-\overline{y}] \exp (-2\pi i\nu t_j) \right|^2 \end{eqnarray*}$$ (2) as $$\begin{eqnarray*} H(\nu)=\frac{2}{N}\sqrt{I(\nu)} \,\, . \end{eqnarray*}$$ (3) In equations (2) and (3), ν is the frequency, |$\overline{y}$| is the mean of the time series, and |$i=\sqrt{-1}$|⁠. Searching for sinusoidal signals embedded in a noisy time series usually involves plotting I or H against ν, to see whether there are power or amplitude excesses at any frequencies. The hypothesis test $$\begin{eqnarray*} &&\textrm{H0:~The~time~series~is~pure~noise}\nonumber \\ &&\textrm{H1:~There~is~a~signal~in~the~time~series} \end{eqnarray*}$$ (4) is usually performed by comparing, in some way, the height of the largest spectral peak to the level of the rest of the spectrum – see e.g. Frescura, Engelbrecht & Frank (2008). A somewhat different scenario is considered in this paper, namely establishing significance levels for the presence of sinusoidal signals when more than one independent realization of the time series is available, as in equation (1). Fig. 1 shows the amplitude spectra of three independent simulated data sets of the form equation (1). In each data set, ν0 = 0.05 d−1, A = 0.007 mag, σ = 0.02 mag, and N = 150. In order to obtain realistic irregular data spacings, observation times of a few stars were taken from the 'Asteroid Terrestrial-impact Last Alert System’ (ATLAS) variable star catalogue (see Heinze et al. 2018). The time spans covered by the three data sets are 506, 658, and 501 d. The red vertical line in the Figure marks the position of the signal frequency ν0. The highest peak in the bottom spectrum is indeed at ν0, but in both the other two spectra, the peak at ν0 is ranked fourth. The most likely conclusion drawn from Fig. 1 is that there is no evidence for a periodicity in any of the three spectra. Figure 1. Open in new tabDownload slide Amplitude spectra of three simulated data sets, each consisting of a sinusoid with amplitude 7 mmag with superposed white noise with σ = 20 mmag. The (red) vertical line shows the position of the sinusoid frequency ν0 = 0.05 d−1. Figure 1. Open in new tabDownload slide Amplitude spectra of three simulated data sets, each consisting of a sinusoid with amplitude 7 mmag with superposed white noise with σ = 20 mmag. The (red) vertical line shows the position of the sinusoid frequency ν0 = 0.05 d−1. Fig. 2 demonstrates that coadding the spectra does not greatly improve the situation. Although the tallest peak is at the correct frequency, its height (7.2 mmag) is not markedly in excess of all other peaks, and only 2.4 times the mean noise level (3.06 mmag). Figure 2. Open in new tabDownload slide The average of the three spectra in Fig. 1. The (red) vertical line shows the position of the sinusoid frequency ν0 = 0.05 d−1. Figure 2. Open in new tabDownload slide The average of the three spectra in Fig. 1. The (red) vertical line shows the position of the sinusoid frequency ν0 = 0.05 d−1. Nonetheless, as will be demonstrated below, it is possible to correctly identify ν0 from the three spectra in Fig. 1, and with high significance. This is essentially done by comparing the positions of peaks extracted from each of the spectra. The first step in the analysis is therefore to identify the positions (frequencies) of all peaks, and then to search for peak positions which are closely similar across all spectra. It is well known that the frequency resolution of the periodogram is ∼1/ΔT, where ΔT is the time base covered by the observations (e.g. Kovács 1981). Experimentation gave good results with peak widths taken to be 0.8/ΔT. If there are K data sets, peaks are considered coincident if there is some overlap of each peak with all others. In set-theoretic notation, $$\begin{eqnarray*} P_1 \cap P_2 \cap \cdots \cap P_K \ne \emptyset \end{eqnarray*}$$ (5) is required, where for each peak Pk = [P0k − 0.4/ΔTk, P0k + 0.4/ΔTk], P0k being the frequency of maximum power in spectrum k. The next section of the paper presents the results of simulations for the noise only case (A = 0 in equation 1). This paves the way for the significance level determinations demonstrated in Section 3. 2 NULL HYPOTHESIS SIMULATIONS It is, of course, possible to obtain chance coincidences of spectral peak positions. In this section of the paper, the impact of various properties of the time series on such false alarm probabilities is studied. Aside from the number K of independent time series and the number of observations N, included in the study are the following: the mean time baseline |$\overline{\Delta T}$|⁠, the frequency interval (0,νE) searched, the number of peaks L from each spectrum, which is taken into consideration (starting with the highest), and the spacing of the measurements. As far as the latter is concerned, the ATLAS spacings are supplemented by random exponentially distributed intervals between measurements. Table 1 summarizes the results of an extensive study – each line is the outcome of at least 10 000 simulations. Briefly: Comparing results for different K, the probability of finding peaks coincidences decreases rapidly with K. This is no surprise – finding four peaks aligned by chance is clearly more rare than finding two or three. The larger νE, the smaller p0. The reason is that if the top ranked L peaks are spread over a wider frequency interval (0, νE), the probability of a chance peak coincidence is reduced. Not surprisingly, the probability of finding peak coincidences increases with L, the number of peaks from each data set, which is taken into account. The probability p0 of a spurious peak alignment decreases with increasing |$\overline{\Delta T}$|⁠. This follows because the spectral resolution improves, spectral peaks are narrower, and the probability of a chance coincidence is therefore reduced. The number of observations N in each data set has little influence on p0. The same is true of the two types of data spacing considered. Table 1. Null hypothesis simulation results. Meanings of the symbols are: K – number of spectra compared; L – the peak rank to which each spectrum is searched; N – number of observations in the simulated time series; νE – the upper limit of the frequency range over which the spectrum is calculated; |$\overline{\Delta T}$| – mean (over the K time series) time span of the observations; A, E – observation spacing, either resembling ATLAS or with exponentially distributed times between observations; p0 – the probability of finding at least one spectral peak alignment. K . L . N . νE . |$\overline{\Delta T}$| . Spacing . p0 . 2 5 150 0.1 582 A 0.52 2 5 150 0.1 575 E 0.55 2 5 150 0.2 582 A 0.29 2 5 150 0.1 1202 A 0.28 2 5 150 0.1 2930 A 0.13 2 5 150 0.5 582 A 0.13 2 5 250 0.1 624 A 0.51 2 10 150 0.1 582 A 0.96 2 10 150 0.1 572 E 0.97 3 5 150 0.1 555 A 0.071 3 5 150 0.1 561 E 0.075 3 5 150 0.2 555 A 0.020 3 5 150 0.1 1135 A 0.017 3 5 150 0.5 555 A 0.0030 3 5 250 0.1 605 A 0.067 3 10 150 0.1 555 A 0.46 3 10 150 0.1 568 E 0.47 4 5 150 0.1 546 A 0.0081 4 5 150 0.1 586 E 0.0070 4 5 150 0.2 561 A 0.0011 4 5 150 0.1 1117 A 7.9E-4 4 5 150 0.5 561 A 1.0E-4 4 5 250 0.1 587 A 0.0063 4 10 150 0.1 546 A 0.12 4 10 150 0.1 553 E 0.12 K . L . N . νE . |$\overline{\Delta T}$| . Spacing . p0 . 2 5 150 0.1 582 A 0.52 2 5 150 0.1 575 E 0.55 2 5 150 0.2 582 A 0.29 2 5 150 0.1 1202 A 0.28 2 5 150 0.1 2930 A 0.13 2 5 150 0.5 582 A 0.13 2 5 250 0.1 624 A 0.51 2 10 150 0.1 582 A 0.96 2 10 150 0.1 572 E 0.97 3 5 150 0.1 555 A 0.071 3 5 150 0.1 561 E 0.075 3 5 150 0.2 555 A 0.020 3 5 150 0.1 1135 A 0.017 3 5 150 0.5 555 A 0.0030 3 5 250 0.1 605 A 0.067 3 10 150 0.1 555 A 0.46 3 10 150 0.1 568 E 0.47 4 5 150 0.1 546 A 0.0081 4 5 150 0.1 586 E 0.0070 4 5 150 0.2 561 A 0.0011 4 5 150 0.1 1117 A 7.9E-4 4 5 150 0.5 561 A 1.0E-4 4 5 250 0.1 587 A 0.0063 4 10 150 0.1 546 A 0.12 4 10 150 0.1 553 E 0.12 Open in new tab Table 1. Null hypothesis simulation results. Meanings of the symbols are: K – number of spectra compared; L – the peak rank to which each spectrum is searched; N – number of observations in the simulated time series; νE – the upper limit of the frequency range over which the spectrum is calculated; |$\overline{\Delta T}$| – mean (over the K time series) time span of the observations; A, E – observation spacing, either resembling ATLAS or with exponentially distributed times between observations; p0 – the probability of finding at least one spectral peak alignment. K . L . N . νE . |$\overline{\Delta T}$| . Spacing . p0 . 2 5 150 0.1 582 A 0.52 2 5 150 0.1 575 E 0.55 2 5 150 0.2 582 A 0.29 2 5 150 0.1 1202 A 0.28 2 5 150 0.1 2930 A 0.13 2 5 150 0.5 582 A 0.13 2 5 250 0.1 624 A 0.51 2 10 150 0.1 582 A 0.96 2 10 150 0.1 572 E 0.97 3 5 150 0.1 555 A 0.071 3 5 150 0.1 561 E 0.075 3 5 150 0.2 555 A 0.020 3 5 150 0.1 1135 A 0.017 3 5 150 0.5 555 A 0.0030 3 5 250 0.1 605 A 0.067 3 10 150 0.1 555 A 0.46 3 10 150 0.1 568 E 0.47 4 5 150 0.1 546 A 0.0081 4 5 150 0.1 586 E 0.0070 4 5 150 0.2 561 A 0.0011 4 5 150 0.1 1117 A 7.9E-4 4 5 150 0.5 561 A 1.0E-4 4 5 250 0.1 587 A 0.0063 4 10 150 0.1 546 A 0.12 4 10 150 0.1 553 E 0.12 K . L . N . νE . |$\overline{\Delta T}$| . Spacing . p0 . 2 5 150 0.1 582 A 0.52 2 5 150 0.1 575 E 0.55 2 5 150 0.2 582 A 0.29 2 5 150 0.1 1202 A 0.28 2 5 150 0.1 2930 A 0.13 2 5 150 0.5 582 A 0.13 2 5 250 0.1 624 A 0.51 2 10 150 0.1 582 A 0.96 2 10 150 0.1 572 E 0.97 3 5 150 0.1 555 A 0.071 3 5 150 0.1 561 E 0.075 3 5 150 0.2 555 A 0.020 3 5 150 0.1 1135 A 0.017 3 5 150 0.5 555 A 0.0030 3 5 250 0.1 605 A 0.067 3 10 150 0.1 555 A 0.46 3 10 150 0.1 568 E 0.47 4 5 150 0.1 546 A 0.0081 4 5 150 0.1 586 E 0.0070 4 5 150 0.2 561 A 0.0011 4 5 150 0.1 1117 A 7.9E-4 4 5 150 0.5 561 A 1.0E-4 4 5 250 0.1 587 A 0.0063 4 10 150 0.1 546 A 0.12 4 10 150 0.1 553 E 0.12 Open in new tab 3 CALCULATING p-VALUES The probabilities in the last column of Table 1 provide a first step towards obtaining significance levels for peak coincidences. Two further independent statistics are useful for improving the estimation of peak-coincidence p-values. The first is $$\begin{eqnarray*} S=\sum _{k=1}^K r_\mathrm{ k}, \end{eqnarray*}$$ where rk is the rank of the peak amongst peaks in spectrum k. For example, there is a single peak coincidence in Fig. 1, near ν = 0.05; the ranks of the three peaks are r1 = 4, r2 = 4, and r3 = 1, giving S = 9. In the absence of any signal, the ranks rj will be completely random between unity and L, whereas, in general, the rk will tend to be smaller in the presence of signals, hence S will also be small. An expression for the probability function of S is available, on recognizing that its genesis is similar to a so-called ‘urn’ problem. Imagine an urn, with L balls inside, numbered from one to L. Now draw, with replacement, K balls from the urn, and note the sum S of the K numbers drawn. The probability function PS of S is given by $$\begin{eqnarray*} P_\mathrm{ S}(x)=L^{-K} \sum _{r=0}^M (-1)^r {{K}\choose {r}} {{x-rL-1}\choose {K-1}} \quad \,\, K \le x \le KL, \end{eqnarray*}$$ (6) where $$\begin{eqnarray*} M=\min \left(K,\frac{x-K}{L} \right) \end{eqnarray*}$$ (Charalambides 2005). The cumulative probability function is $$\begin{eqnarray*} F_\mathrm{ S}(x)= L^{-K} \sum _{r=0}^M (-1)^r {{K}\choose {r}} {{x-rL}\choose {K}}. \end{eqnarray*}$$ (7) Fig. 3 compares equation (6) with the results of one of the simulation experiments reported in Table 1. Figure 3. Open in new tabDownload slide Distribution of the sum of ranks for one of the simulations reported in Table 1 (K = 3, L = 10, νE = 0.1, N = 150, and |$\overline{\Delta T}=568$|⁠). The bars show the result of 10 000 simulations, while the filled circles denote the probability function (equation 6). Figure 3. Open in new tabDownload slide Distribution of the sum of ranks for one of the simulations reported in Table 1 (K = 3, L = 10, νE = 0.1, N = 150, and |$\overline{\Delta T}=568$|⁠). The bars show the result of 10 000 simulations, while the filled circles denote the probability function (equation 6). The second statistic is the width |$w$| of the interval covered by the K coincident-peak frequencies. Again, this is expected to be smaller if peaks are due to a sinusoidal signal rather than a chance near-alignment of peaks due to noise. Analytical expressions for the probability density function (PDF) f|$w$| and the cumulative distribution function (CDF) F|$w$| of |$w$| are derived in Appendix A, for the special case where all ΔT are equal. The situation is considerably more complicated if the time baselines of the different data sets are different, and it is unclear if simple expressions for f|$w$| and F|$w$| exist for this general case. Results for one of the simulations in Table 1 are displayed in Fig. 4. The dotted line in the bottom panel is F|$w$| derived in the Appendix, calculated using the mean of the three values of ℓ = 0.4/ΔT. It evidently provides a good approximation of the empirical CDF in the lower tail, which is of greatest interest. Figure 4. Open in new tabDownload slide The distribution of the frequency interval widths defined by the positions of three near-coincident peaks, for one of the simulations reported in Table 1 (K = 3, L = 10, νE = 0.1, N = 150, and |$\overline{\Delta T}=568$|⁠). The top panel shows a histogram estimate of the PDF. In the bottom panel, the solid (black) line is the empirical CDF based on the data in the top panel, while the (red) dotted line shows the theoretical result (equation A3). Figure 4. Open in new tabDownload slide The distribution of the frequency interval widths defined by the positions of three near-coincident peaks, for one of the simulations reported in Table 1 (K = 3, L = 10, νE = 0.1, N = 150, and |$\overline{\Delta T}=568$|⁠). The top panel shows a histogram estimate of the PDF. In the bottom panel, the solid (black) line is the empirical CDF based on the data in the top panel, while the (red) dotted line shows the theoretical result (equation A3). Returning to Fig. 1, there is a single coincidence of peaks amongst the three spectra, close to the true signal frequency of 0.05 d−1. According to the relevant entry in Table 1 (K = 3, L = 5, N = 150, νE = 0.2 d−1, |$\overline{\Delta T}=555$| d), the probability of obtaining a peak coincidence by chance is p0 = 0.02. The sum of ranks is S = 9. Using equation (5), the probability of obtaining S ≤ 9, given that there is a peak coincidence, is p1 = 0.58. Finally, |$w$| = 6.86 × 10−4 d−1, with p2 = F|$w$|(0.00069) = 0.23. It may be concluded that observing this particular configuration of spectral peaks, or any more convincing, is p0p1p2 = 0.0026. It is also possible to compare the spectra two at a time. The results of doing so are summarized in Table 2. The only comparison that is significant at the conventional 5 per cent level is that of spectra 1 and 3 (p = 0.036). Table 2. Parameters extracted from the amplitude spectra in Fig. 1. Spectra are numbered (1–3) from top to bottom. Spectra compared . Mean frequency . Range |$w$| . Sum of ranks . p-value . 1,2 0.0500 3.8E-4 8 0.096 1,3 0.0504 3.1E-4 5 0.036 2,3 0.0502 6.9E-4 5 0.091 1,2,3 0.0502 6.9E-4 9 0.0026 Spectra compared . Mean frequency . Range |$w$| . Sum of ranks . p-value . 1,2 0.0500 3.8E-4 8 0.096 1,3 0.0504 3.1E-4 5 0.036 2,3 0.0502 6.9E-4 5 0.091 1,2,3 0.0502 6.9E-4 9 0.0026 Open in new tab Table 2. Parameters extracted from the amplitude spectra in Fig. 1. Spectra are numbered (1–3) from top to bottom. Spectra compared . Mean frequency . Range |$w$| . Sum of ranks . p-value . 1,2 0.0500 3.8E-4 8 0.096 1,3 0.0504 3.1E-4 5 0.036 2,3 0.0502 6.9E-4 5 0.091 1,2,3 0.0502 6.9E-4 9 0.0026 Spectra compared . Mean frequency . Range |$w$| . Sum of ranks . p-value . 1,2 0.0500 3.8E-4 8 0.096 1,3 0.0504 3.1E-4 5 0.036 2,3 0.0502 6.9E-4 5 0.091 1,2,3 0.0502 6.9E-4 9 0.0026 Open in new tab If the search had been to weaker peaks, specifically L = 10, then for K = 3, p0 = 0.14, and Pr(S = 9) = 0.084, and hence the significance level would be p0p1p2 = 0.0027. If only two spectra are compared, the probability of obtaining a spurious peak match rises to p0 = 0.76. 4 POWER FUNCTIONS The power of a statistical test is defined as the probability of rejecting the null hypothesis when it is indeed false. In the present context, the alternative hypothesis in equation (4) is not precise enough, and ‘power’ here will rather be the probability of correctly identifying ν0, or an alias of it. Fig. 5 illustrates that why it is necessary to allow the possibility that an alias of ν0 is identified. Each panel of the diagram is based on 10 000 simulations of K data sets with irregular time spacings, with each simulated data set consisting of a sinusoid (frequency ν0 = 0.02 d−1 and amplitude A = 0.01 mag) plus white noise with σ = 0.02 mag. The spectra of each set of K data sets is searched for peak coincidences. Typically, more than one such coincidence is found. Only that with the smallest p-value is retained. In Fig. 5, the further requirement p < 0.01 is imposed, i.e. effectively the hypothesis test is carried out at the 1 per cent level; this leaves 6609 (K = 2), 9075 (K = 3), and 8917 (K = 4) estimates of ν0. Of these, 56 (K = 2), 4 (K = 3), and 0 (K = 4) are neither close to ν0 nor to any nearby alias: Put another way, the percentages of incorrect frequency identifications amongst those significant at the 1 per cent level are 0.85 per cent, 0.04 per cent, and 0 per cent for K = 2, 3, and 4, respectively. Only ‘correct’ frequency estimates are shown in Fig. 5. Figure 5. Open in new tabDownload slide Estimates of ν0 from periodogram peak correspondences for K = 4 (top), K = 3 (middle), and K = 2 (bottom). The results in each of the panel are based on 10 000 simulations. Note the obvious alias near 0.017 d−1 of ν0 = 0.02 d−1. Given the uneven data spacings, aliasing is to be expected. Figure 5. Open in new tabDownload slide Estimates of ν0 from periodogram peak correspondences for K = 4 (top), K = 3 (middle), and K = 2 (bottom). The results in each of the panel are based on 10 000 simulations. Note the obvious alias near 0.017 d−1 of ν0 = 0.02 d−1. Given the uneven data spacings, aliasing is to be expected. The probability of correctly extracting ν0 (or a close alias) from time series with σ = 0.02 mag white noise is plotted in Fig. 6. Time intervals between observations were taken to be exponentially distributed, with |$\overline{\Delta T} \approx 550$| d. The highest L = 5 peaks over the range (0, νE = 0.1 d−1] in each spectrum were taken into account. Figure 6. Open in new tabDownload slide The probability of correctly identifying ν0 (or a close alias), as a function of the signal amplitude. The open circles, the dots, and the broken line respectively indicate results for K = 2, K = 3, and K = 4. Top panel: frequencies tested at the 5 per cent level. Bottom panel: frequencies tested at the 1 per cent level. Other relevant parameter values are N = 150 and L = 5. Figure 6. Open in new tabDownload slide The probability of correctly identifying ν0 (or a close alias), as a function of the signal amplitude. The open circles, the dots, and the broken line respectively indicate results for K = 2, K = 3, and K = 4. Top panel: frequencies tested at the 5 per cent level. Bottom panel: frequencies tested at the 1 per cent level. Other relevant parameter values are N = 150 and L = 5. Perhaps the biggest surprise in Fig. 6 is the good performance of the two-peak comparison when testing at the 5 per cent level. The smaller probability of identifying the correct frequency when comparing four spectra is likewise a surprise. These results are put into proper context by Fig. 7, which shows rather high probabilities of an incorrect frequency determination when testing at the 5 per cent level for K = 2. For K = 4, on the other hand, frequencies found significant are quite unlikely to be wrong. Figure 7. Open in new tabDownload slide The probability that an incorrect frequency is found to be significant. The open circles, the dots, and the broken line respectively indicate results for K = 2, K = 3, and K = 4. Top panel: frequencies tested at the 5 per cent level. Bottom panel: frequencies tested at the 1 per cent level. Other relevant parameter values are N = 150 and L = 5. Figure 7. Open in new tabDownload slide The probability that an incorrect frequency is found to be significant. The open circles, the dots, and the broken line respectively indicate results for K = 2, K = 3, and K = 4. Top panel: frequencies tested at the 5 per cent level. Bottom panel: frequencies tested at the 1 per cent level. Other relevant parameter values are N = 150 and L = 5. When testing the frequency at the 1 per cent level, ‘powers’ of K = 3 and K = 4 are very similar. The probabilities of an incorrect determination are also comparable for amplitudes larger than about 9 mmag. This suggests that there are situations where it is sufficient to search for alignments of three peaks, with little to be gained by attempting to match four peaks. (Note though, that Fig. 5 shows that the correct alias is more likely to be selected if K = 4). For K = 2, when testing at the 1 per cent level, the ‘power’ is excellent for amplitudes 12 mmag or so, with the probability of a spurious frequency identification being at the 0.5 per cent level or lower. 5 TWO APPLICATIONS TO REAL DATA The first data sets analysed in this section, observations of the star ATO 129.0947−26.1809, were extracted from the ATLAS variable star catalogue (Heinze et al. 2018). The brightness of the star was measured 84 times through the c (cyan) filter and 151 times through the o (orange) filter. Single outliers were removed from each of the two data sets. The respective time baselines covered were ΔT1 = 477 (c) and ΔT2 = 558 (o) d. Amplitude spectra of the two data sets can be seen in Fig. 8. The ATLAS period is given as 0.054255 d (ν = 18.4315 d−1), but it is classified as a ‘dubious’ (‘probably not real’) variable. The highest peak in the o data spectrum is indeed at 18.431 25 d−1. Figure 8. Open in new tabDownload slide Amplitude spectra of ATLAS observations of ATO 129.0947−26.1809. Top panel: c filter. Bottom panel: o filter. Figure 8. Open in new tabDownload slide Amplitude spectra of ATLAS observations of ATO 129.0947−26.1809. Top panel: c filter. Bottom panel: o filter. There are no peak correspondences between the two spectra for L < 7, so the two limits L = 10 and L = 20 are investigated. Results are in Table 3. For L = 10, there is a single correspondence, at ν = 18.4312 d−1. In order to evaluate the significance level, 12 000 permutations of each data set were performed, and the peak correspondences were searched for. At least one peak correspondence was found in 62 of the 12 000 synthetic data sets, i.e. p0 = 0.0052. The probability p2 of the observed frequency range was evaluated both from the simulation results and from equation (A3), while the probability p1 of the sum of ranks follows from equation (7). The overall p-value is p = p0p1p2 = 0.00012, where the larger of the two values of p2 was used. It follows that the periodicity in the ATO 129.0947−26.1809 data is highly significant. Table 3. An analysis of frequency spectra of ATLAS observations of ATO 129.0947−26.1809. Two probabilities are given for the differences in the frequencies (i.e. range) obtained, respectively, from the c and o data sets. The first was calculated from spectra of permuted data, while the second follows from equation (A3). L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 10 18.4312 9 0.36 9.99E-5 0.011, 0.065 1.2E-4 20 18.4312 9 0.090 9.99E-5 0.049, 0.065 8.8E-5 – 19.4338 17 0.34 1.50E-4 0.11, 0.097 5.6E-4 – 17.4286 19 0.43 1.40E-4 0.098, 0.090 6.3E-4 L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 10 18.4312 9 0.36 9.99E-5 0.011, 0.065 1.2E-4 20 18.4312 9 0.090 9.99E-5 0.049, 0.065 8.8E-5 – 19.4338 17 0.34 1.50E-4 0.11, 0.097 5.6E-4 – 17.4286 19 0.43 1.40E-4 0.098, 0.090 6.3E-4 Open in new tab Table 3. An analysis of frequency spectra of ATLAS observations of ATO 129.0947−26.1809. Two probabilities are given for the differences in the frequencies (i.e. range) obtained, respectively, from the c and o data sets. The first was calculated from spectra of permuted data, while the second follows from equation (A3). L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 10 18.4312 9 0.36 9.99E-5 0.011, 0.065 1.2E-4 20 18.4312 9 0.090 9.99E-5 0.049, 0.065 8.8E-5 – 19.4338 17 0.34 1.50E-4 0.11, 0.097 5.6E-4 – 17.4286 19 0.43 1.40E-4 0.098, 0.090 6.3E-4 L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 10 18.4312 9 0.36 9.99E-5 0.011, 0.065 1.2E-4 20 18.4312 9 0.090 9.99E-5 0.049, 0.065 8.8E-5 – 19.4338 17 0.34 1.50E-4 0.11, 0.097 5.6E-4 – 17.4286 19 0.43 1.40E-4 0.098, 0.090 6.3E-4 Open in new tab Table 4. An analysis of frequency spectra of two sets of radial velocity measurements of ϵ Eri. L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 5 3.954E-4 2 0.13 1.81E-5 0.094 0.006 L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 5 3.954E-4 2 0.13 1.81E-5 0.094 0.006 Open in new tab Table 4. An analysis of frequency spectra of two sets of radial velocity measurements of ϵ Eri. L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 5 3.954E-4 2 0.13 1.81E-5 0.094 0.006 L . Mean frequency . Sum ranks . Prob. (ranksum) . Frequency range . Prob. (range) . Overall p . 5 3.954E-4 2 0.13 1.81E-5 0.094 0.006 Open in new tab For L = 20, there are three peak correspondences. The first is, of course, the same as was found for L = 10. The other two, less significant correspondences, appear to be at ∼1 d−1 aliases of ν = 18.312 d−1 (Table 3). For L = 20, p0 = 0.015 was obtained from 5000 permutations of the data. Overall p-values are listed in Table 3. The second example analysis is of radial velocity measurements of ϵ Eridani (HD 22049). Mawet et al. (2019) extensively discuss the evidence for an exoplanet associated with the star and provide two new sets of radial velocities (their tables 5 and 6). Since the second of these data sets spans a time period of only 3.3 yr, whereas the periodicity of interest is ∼7.4 yr, it is replaced by the considerably more extensive radial velocities of the star collected by Howard & Fulton (2016). Amplitude spectra of the two data sets (Mawet et al. 2019, table 5: N = 91, ΔT = 7.45 yr and Howard & Fulton 2016 : N = 176, ΔT = 24.11 yr) are plotted in Fig. 9. Fig. 10 shows low frequency details of the spectra. Figure 9. Open in new tabDownload slide Amplitude spectra of two sets of radial velocity measurements of ϵ Eri. Top panel: recent Keck/HIRES data (Mawet et al. 2019). Bottom panel: Lick/Hamilton Spectrograph data (Howard & Fulton 2016). Figure 9. Open in new tabDownload slide Amplitude spectra of two sets of radial velocity measurements of ϵ Eri. Top panel: recent Keck/HIRES data (Mawet et al. 2019). Bottom panel: Lick/Hamilton Spectrograph data (Howard & Fulton 2016). Figure 10. Open in new tabDownload slide The low frequency sections of the spectra in Fig. 9. Figure 10. Open in new tabDownload slide The low frequency sections of the spectra in Fig. 9. Inspection of Figs 9 and 10 reveals that both spectra are dominated by low frequency features. For νE = 0.5 d−1 (i.e. a short period limit of 2 d), and L = 5, there are two peak coincidences between the two spectra, both at very low frequencies – 3.95 × 10−4 d−1 (P = 6.94 yr) and 3.06 × 10−3 d−1 (P = 327 d). The two frequencies are not independent: If the two data sets are prewhitened by the lower (more significant) frequency, the spectra of the residuals show no coincidences whatsoever for L ≤ 10. We therefore proceed only with f = 3.95 × 10−4 d−1 – see Table 4. In order to find the peak correspondence probability p0, the same recipe as in the case of ATO 129.0947−26.1809 could be followed. At least one peak coincidence is obtained in only 247 out of the 12 000 permutations, i.e. p0 = 0.002. However, there is an implicit assumption in the use of the permutation method, namely that the time dependence of the data can be adequately mimicked by the random rearrangement of the observations amongst the times of observation. In the frequency domain, this is tantamount to assuming that the overall shape of the spectrum is flat – something which clearly does not apply in the case of the radial velocity measurements. Fig. 11 shows the results of smoothing the two spectra; the plot demonstrates the substantial power excess, and hence inflated likelihood of substantial peaks, at low frequencies. Figure 11. Open in new tabDownload slide Smoothed versions of the two spectra in Fig. 9. The thick blue line: Keck/HIRES data. The thin black line: Lick/Hamilton Spectrograph data. Figure 11. Open in new tabDownload slide Smoothed versions of the two spectra in Fig. 9. The thick blue line: Keck/HIRES data. The thin black line: Lick/Hamilton Spectrograph data. Fortunately, a modification of the premutation method, which delivers more realistc spectra, is fairly easy: The flat spectra of permuted data are simply multiplied by the smooth functions in Fig. 11, so that the simulated have the same overall shapes as the observed spectra. The importance of this correction is manifested by the increase of p0 to 0.51. The reason for the large change is not difficult to find: Effectively, the largest peaks will overwhelmingly be found in a narrow range at very low frequencies so that the probability of peak coincidences is greatly inflated. The spectral shape also affects the distribution of the sum of peak ranks, increasing the probabilities of small sums relative to larger values. Thus p1 = 0.13 instead of 0.04 for the flat-spectrum case. The statistic |$w$| is uniformly distributed over (0, 1.925E − 4), giving p2 = 0.094 for the observed value of 1.808 × 10−5 d−1. The overall significance level for the peak coincidence is 0.006, i.e. highly significant. 6 DISCUSSION An intriguing possibility raised by the theory of this paper is applications to single data sets, by separating these into multiple sets. This should be particularly useful in cases where there might be multiple weak periodicities present, as such data present problems for the usual approaches based on contrasting peak heights with noise levels. The results above suggest that separating time series into K = 3 subsets may be particularly fruitful. ACKNOWLEDGEMENTS The scientific editor made a useful suggestion which led to the inclusion of the second analysis in Section 5. REFERENCES Charalambides C. A. , 2005 , Combinatorial Methods in Discrete Distributions . John Wiley & Sons Inc ., Hoboken, New Jersey, USA Google Scholar Crossref Search ADS Google Scholar Google Preview WorldCat COPAC Frescura F. A. M. , Engelbrecht C. A. , Frank B. S. , 2008 , MNRAS , 388 , 1693 10.1111/j.1365-2966.2008.13499.x Crossref Search ADS Crossref Heinze A. N. et al. . , 2018 , AJ , 156 , 241 10.3847/1538-3881/aae47f Crossref Search ADS Crossref Howard A. W. , Fulton B. J. , 2016 , PASP , 128 , 114401 10.1088/1538-3873/128/969/114401 Crossref Search ADS Crossref Kovács G. , 1981 , Ap&SS , 78 , 175 10.1007/BF00654032 Crossref Search ADS Crossref Mawet D. et al. . , 2019 , AJ , 157 , 33 10.3847/1538-3881/aaef8a Crossref Search ADS Crossref APPENDIX A: THE PROBABILITY DENSITY FUNCTION OF THE FREQUENCY INTERVAL SPANNED BY K NEAR-COINCIDENT SPECTRAL PEAKS It is given that the K peaks are coincident, i.e. equation (4) holds, where Pj = [P0j − ℓ, P0j + ℓ] = [aj, bj] (j = 1, 2, ⋅⋅⋅, K). It is required to find, in the first instance, the cumulative distribution function (CDF) of the interval spanned by the P0j, i.e. $$\begin{eqnarray*} w=\max _j{P_{0j}}-\min _j{P_{0j}} \,\, . \end{eqnarray*}$$ For convenience, assign the index j = 1 to the smallest P0j, i.e. a1 < aj (j = 2, 3, …, K) also. Since all intervals Pj (j ≥ 2) must overlap P1, it follows that all aj (j > 2) lie in the interval P1. In fact, each of the aj (j > 2) is uniformly randomly distributed in [a1, b1]. The maximum of the aj, i.e. the upper order statistic a(K), then has CDF $$\begin{eqnarray*} F_y(y)=[F_a(y)]^{K-1}, \end{eqnarray*}$$ (A1) where Fa is the CDF of a uniform distribution on [a1, b1]. The latter is easily shown to be $$\begin{eqnarray*} F_a(y) \left\lbrace \begin{array}{ll}0 & y\lt a_1\\ (y-a_1)(b_1-a_1) & a_1 \le y \le b_1. \\ 1 & y\gt b_1 \end{array} \right. \end{eqnarray*}$$ (A2) From the definition of |$w$|⁠, $$\begin{eqnarray*} w=a_{(K)}-a_1 \end{eqnarray*}$$ and hence, from equations (A1) and (A2), $$\begin{eqnarray*} F_w(w)=F_y(w+a_1)\, = \, \left\lbrace \begin{array}{ll}0 & w \lt 0 \\ \left[ w/(b_1-a_1)\right]^{K-1} & 0 \le w \le b_1-a_1 \\ 1 & w\gt b_1-a_1 \end{array} \right. \nonumber \\ = \, \left\lbrace \begin{array}{ll}0 & w \lt 0 \\ (w/2\ell)^{K-1} & 0 \le w \le 2\ell. \\ 1 & w\gt 2\ell \end{array} \right. \end{eqnarray*}$$ (A3) The PDF of |$w$| follows immediately as $$\begin{eqnarray*} f_w(w)= \left\lbrace \begin{array}{ll}0 & w\lt 0 \\ (K-1)w^{K-2}/(2\ell)^{K-1} & 0 \le w \le 2\ell. \\ 0 & w\gt 2\ell \end{array} \right. \end{eqnarray*}$$ (A4) © 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Young, J, E;Kuzio de Naray,, Rachel;Wang, Sharon, X

doi: 10.1093/mnras/staa191pmid: N/A

Carbognani,, Albino;Buzzoni,, Alberto

doi: 10.1093/mnras/staa208pmid: N/A

ABSTRACT We report on accurate BVRc observations of (6478) Gault, a 5–6 km diameter inner main-belt asteroid in the Phocaea family, notable for its sporadic, comet-like ejection of dust. This curious behaviour has been mainly interpreted as reconfigurations after YORP spin-up, although merging of a contact binary system cannot be fully excluded. We collected optical observations along the 2019 March–April period, at orbital phase angles between 12° and 21°, to search for direct evidence of asteroid quick spinning rotation. A prevailing period value of 3.34 ± 0.02 h is supported by our and other photometric observations. In the YORP spin-up hypothesis, this period points to a bulk density ρ ≈ 1 |$\textrm{g}\, \textrm{cm}^{-3}$|⁠. The mean colours are B − V = +0.82±0.3, V − Rc = +0.28±0.06, and B − Rc = +1.11±0.4, but we have observed a trend towards bluer colour during the April session, with about Δ(B − V) ∼ 0.35 ± 0.09 mag. This colour change can be due to asteroid rotation and support the hypothesis that there is a bluer surface under the Gault’s dust. minor planets, asteroids: individual: (6478) Gault 1 INTRODUCTION Main belt asteroid (6478) Gault (hereafter ‘Gault’), recently surged to very special attention (Smith & Denneau 2019) as an outstanding member of the active asteroids class, sporting typical morphological features of comets, such as a coma and tail (see Fig. 1). Pre-covery research in the NOAO image data base allowed to trace Gault’s outbursts back to year 2013 (Chandler et al. 2019). As the outbursts appeared along the full heliocentric orbit, even about the aphelion distance of 2.75 au, this feature tends to exclude the sublimation of volatile material as a cause of the activity. Furthermore, spectroscopic observations showed a prevailing presence of dust, rather than gas, both in the coma and in the asteroid tails (Jewitt et al. 2019). Figure 1. Open in new tabDownload slide A 14 × 11 arcmin picture of (6478) Gault with its tail taken from OAVdA on 2019 March 23, about 20:20 ut (α = 10h 04m 25.2s, δ = −01° 08′ 06|${^{\prime \prime}_{.}}$|7; J2000.0). North is up, east to the left. The main tail length is 5 arcmin 30 arcsec at position angle PA ≈ 272°. Its also visible a fainter 12 arcsec-extended antitail at PA ≈ 91°. The image is a stack of 38 frames, each with 180 s exposure time. Figure 1. Open in new tabDownload slide A 14 × 11 arcmin picture of (6478) Gault with its tail taken from OAVdA on 2019 March 23, about 20:20 ut (α = 10h 04m 25.2s, δ = −01° 08′ 06|${^{\prime \prime}_{.}}$|7; J2000.0). North is up, east to the left. The main tail length is 5 arcmin 30 arcsec at position angle PA ≈ 272°. Its also visible a fainter 12 arcsec-extended antitail at PA ≈ 91°. The image is a stack of 38 frames, each with 180 s exposure time. A re-iterate sequence of outbursts in the last year, namely on 2018 October 28 ± 5, December 31 ± 5, 2019 February 10 ± 7 (Jewitt et al. 2019), also including part of the present observations, may rule out as well the unlikely case of multiple impacts with smaller bodies as the triggering physical mechanism of Gault’s activity. Rather, this may definitely restrain the focus to an intervening dynamical instability of the asteroid’s structure, where a nearly spin-barrier rotation could strongly ease the on-going disintegration of a ‘fluffy’ body (Kleyna et al. 2019). Alternative to any rotation-driven process, however, also binary-system merging could be invoked as the main responsible of Gault’s outbursts (Ye et al. 2019). The presence of the spin-barrier in the ‘realm of asteroids’ can be explained by the cohesionless ‘rubble-pile’ structure model, assuming asteroids to consist in fact of collisional break-up fragments mainly bunching together under mutual gravitation, but in some case perturbed by centrifugal forces according to body’s rotation speed (Pravec et al. 2002). Simple physical arguments lead to estimate, for the critical rotation period (PS) of a spherical object of bulk density ρ (expressed in g cm−3) $$\begin{eqnarray*} P_\mathrm{ S} = \sqrt{\frac{3\pi }{G\rho }} \approx \frac{3.3 \textrm{h}}{\sqrt{\rho }}. \end{eqnarray*}$$ (1) Asteroid’s bulk density is a crucial but difficult parameter to obtain, as we need to know both mass and volume of the body. In general, S-type asteroids are denser than C-type ones, the latter likely displaying a larger macroporosity. Reference figures indicate ρS = 2.72±0.54 g cm−3 for S-type and ρC = 1.33±0.58 g cm−3 for C-type objects (Carbognani 2017). Things get slightly more entangled in case of a non-spherical geometry. If we deal in particular with the relevant case of a ‘cigar-shaped’ triaxial ellipsoid (spinning around the ‘c’ axis and with three axis constrain: a ≥ b = c, according to Richardson, Elankumaran & Sanderson (2005), then the spin-barrier critical period (PE) exceeds the spherical case of equation (1) as |$P_E = {\cal {F}}\ P_S$|⁠, with the shape factor |${\cal F}(\epsilon)$| fully depending on body’s (equatorial) eccentricity1 in the form: $$\begin{eqnarray*} {\cal {F}} = \sqrt{\frac{2 \epsilon ^3}{3\left(\epsilon ^2-1 \right)\left(2 \epsilon +\ln \frac{1-\epsilon }{1+\epsilon }\right)}}. \end{eqnarray*}$$ (2) By combining equations (1) and (2), a straight PE versus ρ relationship can be derived, as displayed in Fig. 2 for different values of eccentricity. According to previous bulk-density figures, one sees from the plot that centrifugal break-up may be reached by C-type asteroids for a spin-barrier critical period PE ∼ 2.5–4.0 h, while a shorter period, always well less than 2.5 h, might be required for a denser S-type object. Figure 2. Open in new tabDownload slide The expected PE versus ρ relationship according to Richardson et al. (2005). Spin-barrier critical period PE in case of a ‘cigar-shaped’ triaxial ellipsoid is obtained from the spherical case modulated by the shape factor |$\cal F$| of equation (2), fully depending on the body’s eccentricity. In addition to the spherical geometry (e = 0) two cases are displayed in the plot, respectively, with e = 0.6 and 0.4, with a slower critical period increasing with body’s eccentricity, at fixed bulk density ρ, as labelled on the plot. The reference bulk-density figures for C- and S-type asteroids, according to Carbognani (2017) are reported as yellow and red bands, respectively. The prevailing estimate of Gault’s rotation period of P = 3.34 h is marked in the plot, together with the implied range for asteroid’s density (arrows) ρ ∼ 1 g cm−3. See text for a discussion. Figure 2. Open in new tabDownload slide The expected PE versus ρ relationship according to Richardson et al. (2005). Spin-barrier critical period PE in case of a ‘cigar-shaped’ triaxial ellipsoid is obtained from the spherical case modulated by the shape factor |$\cal F$| of equation (2), fully depending on the body’s eccentricity. In addition to the spherical geometry (e = 0) two cases are displayed in the plot, respectively, with e = 0.6 and 0.4, with a slower critical period increasing with body’s eccentricity, at fixed bulk density ρ, as labelled on the plot. The reference bulk-density figures for C- and S-type asteroids, according to Carbognani (2017) are reported as yellow and red bands, respectively. The prevailing estimate of Gault’s rotation period of P = 3.34 h is marked in the plot, together with the implied range for asteroid’s density (arrows) ρ ∼ 1 g cm−3. See text for a discussion. No firm estimate of Gault’s rotation period was available until cometary activity was first discovered on 2019 January 5 (Smith & Denneau 2019). Subsequent photometric follow-up to obtain an accurate light curve of the object did not lead to any conclusive result, likely due to the masking effect of dust in the coma (Jewitt et al. 2019; Kleyna et al. 2019; Man-To, Yoonyoung & Xing 2019; Sanchez et al. 2019; Ye et al. 2019). Based on a Lomb–Scargle and ANOVA light-curve analysis, Kleyna et al. (2019) recently proposed for Gault a rotation period about 2 h, which implied a density of about 2.7 g cm−3 as for a typical S-type asteroid. However, a slower period, about 3 h, more suitable for a C-type object, has later been claimed by Ferrin (2019). Until now, no phased light curve can be reported to explicitly support any of these values. 2 OBSERVATIONS AND DATA REDUCTION Thanks to asteroid’s closer distance, near opposition with Earth, and taking advantage of the declining trend of dust activity, we surveyed Gault along the 2019 March–April trajectory arc, with the purpose of determining the asteroid’s rotation period from its optical light curve. A first observing batch was carried out with the OAVdA Ritchey–Chrétien 0.81-m f/4.75 telescope at Saint-Barthelemy (Aosta, Italy, MPC ID code B04) along the three nights of 2019 March 23, 26, and 27. The sky was with some sporadic veils the first night, while in the following two nights it was clear and transparent. The telescope was equipped with an FLI 1001E CCD array of 1024 × 1024 pixels with 24 |$\, \mu$|m pixel size used in 2 × 2 binning mode such as to provide a platescale of 2.54 arcsec px−1 across a 21.9 × 21.9 arcmin field of view. Gault’s imaging was performed with C filter (i.e. ‘white light’), in order to maximize target detection (estimated about V ∼ 17). The frames were dark subtracted and then flat-fielded according to the standard procedure. The SNR for the three sessions was near 50, the mean uncertainty is, respectively, 0.020, 0.022, and 0.023 mag. Although fully successful ones, these observations caught the asteroid still in full activity, with a detectable coma and an extended dust tail visible over 5.5 arcmin away at PA ∼ 272°, as well evident from Fig. 1. A further observing run was then attempted one month later, along the night of 2019 April 15, with the asteroid now definitely ‘turned off’ in its quiescent state (see Fig. 3). However, as Gault was becoming about one magnitude fainter with increasing its orbital phase angle, we had to rely on the larger ‘G.B. Cassini’ 152 cm f/4.6 Ritchey–Chrétien telescope of the Loiano Observatory (Bologna, Italy, MPC ID code 598) for these new observations. The BFOSC camera was attached the telescope, equipped with a Princeton Instruments EEV 1340 × 1300 pixel back-illuminated CCD with 20 |$\, \mu$|m pixel size. Platescale was 0.58 arcsec px−1 leading to a field of view of 13.0 × 12.6 arcmin. Broad-band Johnson/Cousins B, V, Rc filters were used to measure asteroid’s colours. The telescope was tracked at non-sidereal rates to follow Gault’s motion and increase S/N of detection. Figure 3. Open in new tabDownload slide Rc-band isophotal contour plot of a Gault’s illustrative image from the Loiano data set, along the night of 2019 April 15. Esposure time is 240 s with telescope tracked at non-sidereal rates to follow Gault’s motion. The displayed field of view is about 60 × 45 arcsec across, with North up and East to the left. Coordinate axes are labelled in pixel scale (1 px = 0.58 arcsec). Gault is the ‘rounded’ object about (x, y) = (362, 591) coordinates. The vertically elongated object to the left of the image is a saturated star distorted by on-target tracking. Seeing on the image is about 2.2 arcsec FWHM. A bright full Moon, only 12° apart was strongly affecting the sky background, here estimated in μR ∼ 16.8 mag arcsec−2. From the image, we can however rule out at an S/N ≥ 3 confidence level any activity signature around the asteroid, brighter than 21.9 mag arcsec−2. Figure 3. Open in new tabDownload slide Rc-band isophotal contour plot of a Gault’s illustrative image from the Loiano data set, along the night of 2019 April 15. Esposure time is 240 s with telescope tracked at non-sidereal rates to follow Gault’s motion. The displayed field of view is about 60 × 45 arcsec across, with North up and East to the left. Coordinate axes are labelled in pixel scale (1 px = 0.58 arcsec). Gault is the ‘rounded’ object about (x, y) = (362, 591) coordinates. The vertically elongated object to the left of the image is a saturated star distorted by on-target tracking. Seeing on the image is about 2.2 arcsec FWHM. A bright full Moon, only 12° apart was strongly affecting the sky background, here estimated in μR ∼ 16.8 mag arcsec−2. From the image, we can however rule out at an S/N ≥ 3 confidence level any activity signature around the asteroid, brighter than 21.9 mag arcsec−2. The Loiano observations were carried out under clear but partly scattered sky, with seeing about 2.2 arcsec [full width at half-maximum (FWHM)] and a bright full Moon about 12° apart from the target. Nevertheless, a good sequence of Rc images each with 240 s integration (mean uncertainty about 0.09 mag), was obtained spanning about 4 h in total, interleaved by three B, V series to sample asteroid’s colours. The Landolt (1992) PG1047+003 calibration field was taken at similar airmass of Gault images in the three B, V, Rc bands, providing to avoid cloud interference. Image processing included bias subtraction and flat-fielding procedure, as usual. Due to scattered clouds, however, special care has been devoted for photometric reduction of the entire data set, as discussed in more detail in the next sections. Along the total of four OAVdA and Loiano observing runs, we collected about 12 h of observation on the target, as summarized in Table 1. Table 1. Summary of the 2019 OAVdA (B04) and Loiano (598) observing sessions. Date . No. of . Band . Exposure . Timespan . MPC . . frames . . (s) . (h) . ID . March 23 47 C 180 2.0 B04 March 26 61 C 180 3.0 B04 March 27 65 C 180 3.0 B04 April 15 49 Rc 240 4.0 598 3 V 300 598 3 B 480 598 Date . No. of . Band . Exposure . Timespan . MPC . . frames . . (s) . (h) . ID . March 23 47 C 180 2.0 B04 March 26 61 C 180 3.0 B04 March 27 65 C 180 3.0 B04 April 15 49 Rc 240 4.0 598 3 V 300 598 3 B 480 598 Open in new tab Table 1. Summary of the 2019 OAVdA (B04) and Loiano (598) observing sessions. Date . No. of . Band . Exposure . Timespan . MPC . . frames . . (s) . (h) . ID . March 23 47 C 180 2.0 B04 March 26 61 C 180 3.0 B04 March 27 65 C 180 3.0 B04 April 15 49 Rc 240 4.0 598 3 V 300 598 3 B 480 598 Date . No. of . Band . Exposure . Timespan . MPC . . frames . . (s) . (h) . ID . March 23 47 C 180 2.0 B04 March 26 61 C 180 3.0 B04 March 27 65 C 180 3.0 B04 April 15 49 Rc 240 4.0 598 3 V 300 598 3 B 480 598 Open in new tab 2.1 On-frame photometry MPO Canopus package (Warner 2009) was used for differential aperture photometry of our data. We especially relied on the Comp Star Selector (CSS) and Derivedmags software feature to pick up a reference grid of (whenever possible) solar-type standards, from the CMC15 star catalogue (Muiños & Montojo 2014), and therefrom lead to an accurate calibration (to within a few hundredths of magnitude internal uncertainty) of Gault’ magnitude directly on the observed field. This is very useful because it allows the different photometric sessions to be linked together. Gault’s aperture photometry has been carried out through an ∼3 FWHM circular aperture, throughout, to account for seeing effects. Only the best frames, with the target clearly unaffected by star crowding, were retained. According to the CMC15/UCAC4/APASS photometric characterization (Carbognani 2016), we can confidently match the Johnson–Cousins Rc system with our observations with the equation: $$\begin{eqnarray*} R_c = r^{\prime } - 0.112 - 0.128\left(B - V \right)\quad {\rm mag}. \end{eqnarray*}$$ (3) In equation (3) r′ is the apparent red mag of the star in the Sloan system adopted by CMC15/UCAC4/APASS catalogues, while B and V are the mag in the Johnson system. The RMS, when using equation (3), is about 0.05 mag. For a solar-type star, as our comparisons, B − V ≃ 0.656 ± 0.005, so: $$\begin{eqnarray*} R_c \simeq r^{\prime } - 0.2 \end{eqnarray*}$$ (4) This and the previous correction was applied throughout in the reported Rc magnitudes of this paper. A subset of three-to-five comparison stars across the full frame sequence for each observing run were measured in order to assess sky transparency conditions along the night. In particular, for the Loiano observations, this procedure allowed us to track in some detail the temporal behaviour of thin cloud absorption affecting Gault’s imaging and recover colours to fiducially cloud-free conditions. This correction is of paramount importance in order to derive the asteroid’s colours variation. 3 GAULT’S COLOURS Three series of deeper B, V images (referred to hereafter as Batch #1, 2, and 3, with exposure time of 8 min in B and 5 min in V) have been accompanying the Rc-band sequence along the Loiano session of 2019 April 15. As marked in the lower panel of Fig. 5, the B, V luminosity was sampled around 20:03-20:17 ut (Batch #1), 21:06-21:20 ut (Batch #2), and 22:10-22:24 ut (Batch #3), in order to assess Gault’s apparent colours at different light-curve phase. The photometric reduction has been carried out according to the usual standard calibration procedure (Harris, Fitzgerald & Reed 1981; Landolt 1992). In addition, special care has been devoted to take the Landolt field at similar airmass than Gault’s frames in order to minimize differential corrections. If we look at the photometric trend of the comparison stars present in Gault’s field of view we see that cloud absorption did not affect Batch #1, while a thinner coverage was in place, on the contrary, at Batch #2 and Batch #3. To estimate the effect of cloud absorption on colours we chose three stars, from the UCAC4 (Zacharias et al. 2013) stars catalogue,2 placed near Gault and computed the colours with the same photometric parameters used for Gault. The results are shown in Table 2. From this we can see how the average colours value and the one from the UCAC4 catalogue are compatible within a few hundredths of magnitude. Thus, despite the presence of veils and the full Moon, the observed colours are reliable. If we look at the individual Batches, we can see how the stars colours tend to become redder, going from Batch #1 to Batch #3 as expected, which appears to be the most conditioned by cloud veils. Taking as reference the Batch #1, we can compute a set of mean correction terms defined as (colours Batch #1)-(colours Batch #2) or (colours Batch #1)-(colours Batch #3). We can use these additive terms to ‘delete’ the veils effect on Gault’s colours (see Table 3). Table 2. UCAC4 stars BVRc colours along the observing night of 2019 April 15. The last two columns provide, respectively, the average colour value on three Batch and the catalogue value. 473-044752 . Batch #1 . Batch #2 . Batch #3 . Average . Cat. . B − V 0.33±0.03 0.34±0.03 0.52±0.03 0.40±0.06 0.43 V − Rc 0.23±0.02 0.27±0.02 0.27±0.02 0.26±0.01 0.25 B − Rc 0.56±0.02 0.61±0.02 0.79±0.02 0.66±0.07 0.68 473-044753 B − V 0.75±0.04 0.76±0.04 0.96±0.04 0.82±0.07 0.90 V − Rc 0.39±0.02 0.46±0.02 0.44±0.02 0.43±0.02 0.41 B − Rc 1.14±0.03 1.22±0.03 1.40±0.03 1.25±0.08 1.32 473-044754 B − V 1.42±0.06 1.37±0.06 1.49±0.06 1.43±0.04 1.49 V − Rc 0.90±0.04 0.96±0.04 0.95±0.04 0.94±0.02 0.93 B − Rc 2.32±0.05 2.33±0.05 2.45±0.05 2.37±0.04 2.42 473-044752 . Batch #1 . Batch #2 . Batch #3 . Average . Cat. . B − V 0.33±0.03 0.34±0.03 0.52±0.03 0.40±0.06 0.43 V − Rc 0.23±0.02 0.27±0.02 0.27±0.02 0.26±0.01 0.25 B − Rc 0.56±0.02 0.61±0.02 0.79±0.02 0.66±0.07 0.68 473-044753 B − V 0.75±0.04 0.76±0.04 0.96±0.04 0.82±0.07 0.90 V − Rc 0.39±0.02 0.46±0.02 0.44±0.02 0.43±0.02 0.41 B − Rc 1.14±0.03 1.22±0.03 1.40±0.03 1.25±0.08 1.32 473-044754 B − V 1.42±0.06 1.37±0.06 1.49±0.06 1.43±0.04 1.49 V − Rc 0.90±0.04 0.96±0.04 0.95±0.04 0.94±0.02 0.93 B − Rc 2.32±0.05 2.33±0.05 2.45±0.05 2.37±0.04 2.42 Open in new tab Table 2. UCAC4 stars BVRc colours along the observing night of 2019 April 15. The last two columns provide, respectively, the average colour value on three Batch and the catalogue value. 473-044752 . Batch #1 . Batch #2 . Batch #3 . Average . Cat. . B − V 0.33±0.03 0.34±0.03 0.52±0.03 0.40±0.06 0.43 V − Rc 0.23±0.02 0.27±0.02 0.27±0.02 0.26±0.01 0.25 B − Rc 0.56±0.02 0.61±0.02 0.79±0.02 0.66±0.07 0.68 473-044753 B − V 0.75±0.04 0.76±0.04 0.96±0.04 0.82±0.07 0.90 V − Rc 0.39±0.02 0.46±0.02 0.44±0.02 0.43±0.02 0.41 B − Rc 1.14±0.03 1.22±0.03 1.40±0.03 1.25±0.08 1.32 473-044754 B − V 1.42±0.06 1.37±0.06 1.49±0.06 1.43±0.04 1.49 V − Rc 0.90±0.04 0.96±0.04 0.95±0.04 0.94±0.02 0.93 B − Rc 2.32±0.05 2.33±0.05 2.45±0.05 2.37±0.04 2.42 473-044752 . Batch #1 . Batch #2 . Batch #3 . Average . Cat. . B − V 0.33±0.03 0.34±0.03 0.52±0.03 0.40±0.06 0.43 V − Rc 0.23±0.02 0.27±0.02 0.27±0.02 0.26±0.01 0.25 B − Rc 0.56±0.02 0.61±0.02 0.79±0.02 0.66±0.07 0.68 473-044753 B − V 0.75±0.04 0.76±0.04 0.96±0.04 0.82±0.07 0.90 V − Rc 0.39±0.02 0.46±0.02 0.44±0.02 0.43±0.02 0.41 B − Rc 1.14±0.03 1.22±0.03 1.40±0.03 1.25±0.08 1.32 473-044754 B − V 1.42±0.06 1.37±0.06 1.49±0.06 1.43±0.04 1.49 V − Rc 0.90±0.04 0.96±0.04 0.95±0.04 0.94±0.02 0.93 B − Rc 2.32±0.05 2.33±0.05 2.45±0.05 2.37±0.04 2.42 Open in new tab Table 3. The mean colours correction terms for Batch #1, Batch #2, and Batch #3 derived from colours of Table 2. . Batch #1 . Batch #2 . Batch #3 . B − V 0 +0.01±0.03 −0.16±0.07 V − Rc 0 −0.06±0.01 −0.05±0.006 B − Rc 0 −0.05±0.03 −0.21±0.07 . Batch #1 . Batch #2 . Batch #3 . B − V 0 +0.01±0.03 −0.16±0.07 V − Rc 0 −0.06±0.01 −0.05±0.006 B − Rc 0 −0.05±0.03 −0.21±0.07 Open in new tab Table 3. The mean colours correction terms for Batch #1, Batch #2, and Batch #3 derived from colours of Table 2. . Batch #1 . Batch #2 . Batch #3 . B − V 0 +0.01±0.03 −0.16±0.07 V − Rc 0 −0.06±0.01 −0.05±0.006 B − Rc 0 −0.05±0.03 −0.21±0.07 . Batch #1 . Batch #2 . Batch #3 . B − V 0 +0.01±0.03 −0.16±0.07 V − Rc 0 −0.06±0.01 −0.05±0.006 B − Rc 0 −0.05±0.03 −0.21±0.07 Open in new tab The apparent Gault colours along the three observing windows are displayed in Table 4, together with their average values. As far as the latter ones are considered, our colours are fully consistent with Man-To et al. (2019), after correcting the latter ones to the Johnson–Cousins system, according to Bessel (1979). One has to report, however, the evident trend towards ‘bluer colour’ along the Table 4 observations (see Fig. 4 for a plot), with the asteroid becoming increasingly bluer of about Δ(B − V) ∼ 0.35 ± 0.09 mag, near the luminosity minimum (see Fig. 5). This trend was not present in the UCAC4 stars of Table 2, where, on the contrary, there is a little redshift due to veils, as expected. We consider this trend towards bluer colour a real effect. Probably there is a bluer region on the Gault surface that we had observed during the session, thanks to asteroid rotation. Indeed, the temporal difference between Batch #1 and Batch #3 is about 2 h, more than half of the best rotation period we estimated for Gault (see Section 4.1). These colours variation towards blue is consistent with what was found by Marsset et al. (2019), that in NIR spectroscopic observations of 2019 March 31 found Gault bluer than similar observations of 2019 April 8. In this last observation the Gault spectrum appears an S-type asteroid, compatible with Phocaea’s spectrum. To our knowledge, no one had observed such a marked change in Gault’s colour during the same session. It is assumed as a rule that the surface of asteroids is uniform so finding these changes, as for NEA (297274) 1996 SK (Lin et al. 2014), is very interesting. Figure 4. Open in new tabDownload slide Gault’s colours variations over time. Figure 4. Open in new tabDownload slide Gault’s colours variations over time. Figure 5. Open in new tabDownload slide Gault’s observing sessions of 2019 March 23–27 from OAVdA (the mean uncertainty for the three sessions are, respectively, 0.020, 0.022, and 0.023 mag) and 2019 April 15 from Loiano (mean uncertainty 0.09 mag), are summarized in the upper and lower panels, respectively. The Rc magnitude scale is reproduced, throughout, from the local CMC-15 calibration, according to equation (4). Along the OAVdA observations, the asteroid was about its Earth opposition, at orbital phase angle ϕ ∼ 12.9°, a figure that increased to ϕ ∼ 21.4° for the Loiano data. The B, V magnitude sampling from Batch #1–3 observations is marked on the plot. Note a substantial difference in light-curve amplitude and shape between the two observing sessions. See text for a discussion. Figure 5. Open in new tabDownload slide Gault’s observing sessions of 2019 March 23–27 from OAVdA (the mean uncertainty for the three sessions are, respectively, 0.020, 0.022, and 0.023 mag) and 2019 April 15 from Loiano (mean uncertainty 0.09 mag), are summarized in the upper and lower panels, respectively. The Rc magnitude scale is reproduced, throughout, from the local CMC-15 calibration, according to equation (4). Along the OAVdA observations, the asteroid was about its Earth opposition, at orbital phase angle ϕ ∼ 12.9°, a figure that increased to ϕ ∼ 21.4° for the Loiano data. The B, V magnitude sampling from Batch #1–3 observations is marked on the plot. Note a substantial difference in light-curve amplitude and shape between the two observing sessions. See text for a discussion. Table 4. Gault’s BVRc colours along the observing night of 2019 April 15, and comparison with Man-To et al. (2019). The trend towards bluer colour from Batch #1 to Batch #3 is evident (see also Fig. 4). . Batch |$\#1^{a}$| . Batch |$\#2^{a}$| . Batch |$\#3^{a}$| . Average . MT19b . B − V 0.84±0.04 1.15±0.06 0.49±0.08 0.82±0.3 0.79±0.06 V − Rc 0.34±0.02 0.29±0.03 0.22±0.02 0.28±0.06 0.31±0.02 B − Rc 1.19±0.03 1.44±0.05 0.71±0.08 1.11±0.4 1.10±0.06 . Batch |$\#1^{a}$| . Batch |$\#2^{a}$| . Batch |$\#3^{a}$| . Average . MT19b . B − V 0.84±0.04 1.15±0.06 0.49±0.08 0.82±0.3 0.79±0.06 V − Rc 0.34±0.02 0.29±0.03 0.22±0.02 0.28±0.06 0.31±0.02 B − Rc 1.19±0.03 1.44±0.05 0.71±0.08 1.11±0.4 1.10±0.06 aAfter cloud veils correction, as discussed in the text. bAs from Man-To et al. (2019). Open in new tab Table 4. Gault’s BVRc colours along the observing night of 2019 April 15, and comparison with Man-To et al. (2019). The trend towards bluer colour from Batch #1 to Batch #3 is evident (see also Fig. 4). . Batch |$\#1^{a}$| . Batch |$\#2^{a}$| . Batch |$\#3^{a}$| . Average . MT19b . B − V 0.84±0.04 1.15±0.06 0.49±0.08 0.82±0.3 0.79±0.06 V − Rc 0.34±0.02 0.29±0.03 0.22±0.02 0.28±0.06 0.31±0.02 B − Rc 1.19±0.03 1.44±0.05 0.71±0.08 1.11±0.4 1.10±0.06 . Batch |$\#1^{a}$| . Batch |$\#2^{a}$| . Batch |$\#3^{a}$| . Average . MT19b . B − V 0.84±0.04 1.15±0.06 0.49±0.08 0.82±0.3 0.79±0.06 V − Rc 0.34±0.02 0.29±0.03 0.22±0.02 0.28±0.06 0.31±0.02 B − Rc 1.19±0.03 1.44±0.05 0.71±0.08 1.11±0.4 1.10±0.06 aAfter cloud veils correction, as discussed in the text. bAs from Man-To et al. (2019). Open in new tab 4 DERIVED LIGHT CURVE A general summary of the OAVdA and Loiano observations is summarized in the two panels of Fig. 5. As far as the OAVdA data set is concerned, a first outstanding feature of Gault’s observed light curve along all the three nights of 2019 March 23, 26, and 27 is a quite regular trend with the object almost steady at a ‘flat’ maximum interspersed with ‘spiky’ minima, where magnitude gets some 0.1–0.15 mag fainter. This feature, strongly reminiscent of the photometric behaviour of eclipsing binary stars, closely recalls a similar trend seen weeks before by the Indian HCT and ESA OGS telescopes, as reported by Kleyna et al. (2019, see their fig. 3). A change of status occurs, however, in the April observations from Loiano (lower panel of the figure) where, on the contrary, the asteroid variation shows a smoother ‘sinusoidal’ light curve and much larger amplitude (i.e. ARc ∼ 0.5 mag). Such a strong change in the light-curve amplitude and shape prevented us from put all observing sessions together in a coherent period analysis. However, one may argue that this photometric behaviour is typical of an elongated body as the orbital phase angle (ϕ) increases. In fact, the observations from the HCT and OGS telescopes, and from OAVdA as well were taken close to asteroid’s opposition, at the mean orbital phase angle ϕ ∼ 10° ± 3° and 12.9°, respectively, while from Loiano we observed at a phase angle ϕ = 21.4°. A distinctive relationship is recognized for asteroids of different taxonomic type between amplitude and orbital phase angle (Zappalá et al. 1990) in the form: $$\begin{eqnarray*} A(\phi) = A(0)\, (1+ m\phi). \end{eqnarray*}$$ (5) In the equation, A(0) is the light-curve amplitude (in mag) at the opposition (namely at ϕ = 0o). If we express ϕ in degrees, then the scaling coefficient m depends on the taxonomic type and can be empirically calibrated (Zappalá et al. 1990) as m = 0.030, 0.015, and 0.013, respectively for S-, C-, and M-type asteroids. If we enter l.h. term of equation (5) with the amplitude observed from Loiano, that is A(21.4°) ≈ 0.5 mag, which is our best value as observed when Gault’s activity was decreasing, then an opposition value of A(0) = 0.35±0.05 mag is inferred, accounting for the full range of m along the taxonomic class. If we assume the magnitude variation to be fully induced by a change of reflective surface in a ‘cigar-shaped’ ellipsoid (with fixed albedo), then the (b/a) ratio can be constrained as |$(b/a) \approx 10^{-0.4\, A(0)} = 0.73_{\pm 0.03}$|⁠. According to footnote 1 definition, this leads to a plausible range for body’s (sagittal) eccentricity of ϵ ≈ 0.68±0.03. This estimate implies that the light-curve amplitude is due entirely to the asteroid shape. If there are albedo patches on surface, as discussed above, the elongation will be smaller. 4.1 Rotation and spin-barrier critical period Given our sparse data set, a more pondered statistical approach was pursued to constrain the possible periodicity in Gault’s light curve. The OAVdA data are better suited for this exercise because they span a larger timeline, between March 23 and 27. On the contrary, the Loiano observations restrain to a 4 h interval only, although they more firmly appear to constrain the allowed range of possible period values, yet hardly shorter than 3 h (see Fig. 5). To further extend the temporal coverage of the OAVdA’s photometry, data taken from Sanchez et al. (2019) on March 26 and 30, 2019 were also used (see Fig. 6) in our analysis. We did a Lomb–Scargle analysis, between 0.5 and 10 h, of the Gault light curve using all the OAVdA and Sanchez sessions. If all sessions are used, the dominant period is 1.3 h completely given by the March 26 Sanchez session. This peak is a fake, due to two interruptions in the observations at a distance of 0.05 d, or about 1.2 h which gives a false periodicity in a substantially flat light curve. Another problem with the March 26 Sanchez session is that the average error on the mag is 0.05, while for the OAVdA and Sanchez sessions on March 30 it goes from 0.015 to 0.023. So Sanchez’s March 26th session is twice as noisy as the others and it makes sense to remove it from the analysis. Removing this session the best period is 3.34 ± 0.02 h (see Fig. 7). In the periodogram remains a widened peak between 7 and 8 h, which could correspond to the period of a hypothetical binary system, see Section 4.2 for a more detailed discussion. This value also confirms the Ferrin (2019) preliminary estimate from his own photometry, and the period is also compatible with the Loiano observations, as evident from the plot of Fig. 5. Indeed, a Lomb–Scargle analysis of the Loiano data set also show a peak around a period of about 3.4 h. Figure 6. Open in new tabDownload slide Gault’s observing sessions of 2019 March 26 and 30 from Sanchez et al. (2019) are summarized in the upper and lower panels, respectively. Figure 6. Open in new tabDownload slide Gault’s observing sessions of 2019 March 26 and 30 from Sanchez et al. (2019) are summarized in the upper and lower panels, respectively. Figure 7. Open in new tabDownload slide The Lomb–Scargle periodogram of spectral power versus period (between 0.5 and 10 h) for OAVdA’s sessions and Sanchez 30 March. The best period is 3.34 ± 0.02 h. Figure 7. Open in new tabDownload slide The Lomb–Scargle periodogram of spectral power versus period (between 0.5 and 10 h) for OAVdA’s sessions and Sanchez 30 March. The best period is 3.34 ± 0.02 h. Our results were also corroborated by independently cross-checking the OAVdA and Sanchez data with the falc Fourier analysis algorithm by Harris et al. (1989), implemented in the MPO canopus package. The resulting MPO Canopus phased light curves from OAVdA and Sanchez are shown in Fig. 8. The best period is 3.36 h very close to the 3.34 h period that we had found using Lomb–Scargle. One major concern deals with the lack of any evident ‘secondary’ minimum, about mid-way from two ‘primary’ minima (i.e. ‘double-peaked’ light curve), as usual for an asteroids. Probably Gault’s reflectance have been heavily affected by dust activity which may have partially erased the light-curve characteristics. Figure 8. Open in new tabDownload slide The phased light curves of the March 23, 26, 27 OAVdA, and March 30 Sanchez according to the Falc algorithm implemented in MPO Canopus. In this case, the best period is 3.36 h, very near to the 3.34 ± 0.02 h best period of Fig. 7, found with the Lomb–Scargle periodogram. Figure 8. Open in new tabDownload slide The phased light curves of the March 23, 26, 27 OAVdA, and March 30 Sanchez according to the Falc algorithm implemented in MPO Canopus. In this case, the best period is 3.36 h, very near to the 3.34 ± 0.02 h best period of Fig. 7, found with the Lomb–Scargle periodogram. In a disrupting ‘rubble-pile’ structure model, a glance to Fig. 2 clearly points to an asteroid bulk density |$\rho \lesssim 1.2 ~\textrm{g}~\textrm{cm}^{-3}$|⁠, a value compatible with a internally fragmented S-type asteroid, i.e. with large macroporosity. As a main conclusion, our analysis definitely rules out the spin-barrier classical value of about 2 h, as claimed by Kleyna et al. (2019). 4.2 A merging binary system? Patching absorption by Gault’s surrounding dust layers could naturally give reason of the the lack of any ‘secondary’ minimum in the phased light curves of Fig. 83 and the so erratic luminosity trend discussed in the Kleyna et al. (2019) paper, as well. Alternatively, we can match the expected ‘double-peaked’ photometric trend by moving on the ∼2 × period pattern. The resulting phased light curve of the OAVdA and Sanchez data given by MPO Canopus, is shown in Fig. 9. Such new physical scenario, with a best period of about 7 h, could explain Gault’s activity in terms of a near-contact binary that merge itself in a contact binary through the loss of angular momentum due to BYORP effect (Ye et al. 2019). Indeed a careful analysis of Fig. 9 may recall a contact binary system of two elongated bodies of similar size whose orbital plane is tilted enough with respect to our point of view such as to avoid full occultation between the two components (see e.g. Descamps 2008, for illustrative examples). Note that the second minimum in Fig. 9 does not fall exactly at the 0.75 phase as expected for a contact binary system, probably the light curve is ‘dirty’ as a result of Gault’s activity (with OAVdA’s session only, the second minimum fall in 0.75 phase). In this case, if we assume the same bulk density (ρG) and size (RG) for the two Gault’s components, orbiting at a distance |$n\, R_G$| apart,4 then the Kepler third law provides: $$\begin{eqnarray*} \frac{4\, {\pi} ^2}{P^2} = \rho _G\, G\frac{(8/3)\, {\pi} R_G^3}{(n\, R_G)^3}, \end{eqnarray*}$$ (6) or $$\begin{eqnarray*} P = \sqrt{\frac{3{\pi} \, n^3}{2\, G}\frac{1}{\rho }} \sim \frac{2.33\, n^{3/2}}{\sqrt{\rho }} \quad {\rm (hr)} \end{eqnarray*}$$ (7) Figure 9. Open in new tabDownload slide The phased light curves of the March 23, 26, 27 OAVdA, and March 30 Sanchez according to the Falc algorithm implemented in MPO Canopus but plotted with the ∼2 × period values. This light curve, with a best period of about 7 h, is compatible with a contact binary system with equal components (Descamps 2008). Figure 9. Open in new tabDownload slide The phased light curves of the March 23, 26, 27 OAVdA, and March 30 Sanchez according to the Falc algorithm implemented in MPO Canopus but plotted with the ∼2 × period values. This light curve, with a best period of about 7 h, is compatible with a contact binary system with equal components (Descamps 2008). Fig. 10 summarizes our results for the full range of possible configurations. In case of a preferred fiducial period of P ≈ 7 h or larger, a contact double asteroid could be admitted with an implied bulk density |$\rho \lesssim 1.0$| g cm−3, as marked in the figure. A much larger value for ρ would however allowed in case of a close but semidetached system. Figure 10. Open in new tabDownload slide The expected P versus ρ relationship for a close binary system with the asteroid consisting of two components of similar size and mass, according to equation (7). The component distance is parametrized in terms of multiple ‘n’ of the body’s reference radius, RG, as in equation (6). Accordingly, a contact system is obtained for n = 2, while for n = 3 the two asteroid components are orbiting one RG apart. The nominal periodicity of case P ≈ 7 h is singled out in the plot, with an implied density for Gault of |$\rho \lesssim 1.0$| g cm−3, in case of a contact binary system. The 6.7-h double periodicity, that occurred in MPO Canopus without using the March 30 Sanchez session, is also indicated. This gives an idea of the ‘weight’ that even a single session can have in these critical measures. Figure 10. Open in new tabDownload slide The expected P versus ρ relationship for a close binary system with the asteroid consisting of two components of similar size and mass, according to equation (7). The component distance is parametrized in terms of multiple ‘n’ of the body’s reference radius, RG, as in equation (6). Accordingly, a contact system is obtained for n = 2, while for n = 3 the two asteroid components are orbiting one RG apart. The nominal periodicity of case P ≈ 7 h is singled out in the plot, with an implied density for Gault of |$\rho \lesssim 1.0$| g cm−3, in case of a contact binary system. The 6.7-h double periodicity, that occurred in MPO Canopus without using the March 30 Sanchez session, is also indicated. This gives an idea of the ‘weight’ that even a single session can have in these critical measures. 5 SUMMARY AND CONCLUSIONS In this paper, we comprehensively reviewed the observations made in early 2019 on the new active asteroid (6478) Gault. The most likely cause is that the asteroid activity was due to reconfigurations after YORP spin-up. However, also binary-system merging could be invoked as the main responsible of Gault’s outbursts. For this reason an accurate estimate of the inherent photometric periodicity could actually discriminate between the different scenarios. Until very recently, in their 2019 observations, Kleyna et al. (2019) proposed a spinning value about 2 h, which implied a density of some 2.7 g cm−3, as for a typical S-type asteroid (see Fig. 2). This result was consistent with Gault’s asteroid family: Phocaea. Two NIR spectra taken by Marsset et al. (2019) show deep absorption band near 1 and 2 |$\, \mu$|m consistent with an S-type asteroid, this support the link between Phocaea collisional family and Gault. To better clarify the situation about the rotation period, we added fresh photometric observations from OAVdA, in the second half of 2019 March (see Fig. 5). To extend the temporal coverage of the OAVdA’s photometry, data taken from Sanchez et al. (2019) on 2019 March 26 and 30, were also used (see Fig. 6). Finally we did a Lomb–Scargle analysis, between 0.5 and 10 h, of the Gault light curves using OAVdA and Sanchez sessions. From the periodogram (see Fig. 7), a best period is identified, namely 3.34±0.02 h, with no evident sign of any ∼2 h periodicity. The 3.34 h period also confirms the Ferrin (2019) preliminary estimate and it may be taken as the most probable, although the other near values cannot be firmly excluded at the current state of observations. If this is the real context, then by invoking the spin-barrier limit, Fig. 2 shows that Gault’s bulk density should not exceed ρ ∼ 1.2 g cm−3, compatible with a fragmented S-type asteroid. By forcing twice a photometric period in order to fit with a ‘double-peaked’ light curve (Fig. 9), we challenged the possibility for Gault to be a merging contact (or semidetached) binary system consisting in fact of similar twin bodies. A realistic solution in this case points to a best period of about 7 h, leading to quite a ‘fluffy’ bulk density |$\rho \lesssim 1.0 ~\textrm{g}~\textrm{cm}^{-3}$|⁠, in force of equation (7). The mid-April light curve from Loiano, sampling Gault’s more quiescent status compared to March (see Fig. 3 and compare with Fig. 1), shows a greater amplitude and a more sinusoidal shape compared to the OAVdA observations. Also this data set show a peak around a period of about 3.4 h. In case of constant albedo, this may be suggestive of an elongated (roughly cigar-like) shape for the body, with an implied (sagittal) eccentricity ϵ ≈ 0.68±0.03. Gault colours were also assessed along the Loiano observing run, leading to the average figures summarized in Table 4, namely (B − V) = 0.82±0.3, (V − Rc) = 0.28±0.06 and (B − Rc) = 1.11±0.4, in quite a good agreement with Man-To et al. (2019) but with a remarkable trend towards bluer colour, with the asteroid becoming much bluer near the minimum light-curve luminosity (see Fig. 5). The change in (B − V) colour was about Δ(B − V) ∼ 0.35 ± 0.09 mag. This strange behaviour is supported by the aforementioned spectroscopic observations made on March 31 and April 8 by Marsset et al. (2019). The first spectrum was bluer than the second one and this indicates a macroscopic difference of albedo in different Gault’s areas. It is possible that this difference is due to an active area that has exposed new fresh material not been reddened by solar radiation. Further photometric and spectroscopic observations are needed to fully characterize this very interesting minor body. ACKNOWLEDGEMENTS The authors wish to thank Sanchez J. A. for granting the use of Gault photometric data and the Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), managed by the Fondazione Clément Fillietroz-ONLUS, for granting the use of the Main Telescope. Many thanks to the referee for the useful suggestions that have greatly improved the quality of the manuscript. Footnotes 1 As usual, we define ϵ = [1 − (b/a)2]1/2, in terms of minor-to-major axial ratio (b/a) of the body. 2 In the UCAC4 catalogue the B and V mag are in the Johnson system, while the red mag are in the Sloan r′ system. To transform from r′ to Rc, we use equation (3). 3 Actually, in a dust-free ‘cigar-shaped’ ellipsoid of fixed albedo, spinning around the principal momentum axis, one must expect ‘secondary’ minimum to be of equal amplitude than the ‘primary’ one, both being generated by the opposite end-to-end extrema of the spinning ‘cigar’. 4 In our notation, we have a contact binary if n = 2, that is if the two asteroid components are separated by twice their reference radius RG. REFERENCES Bessell M. S. , 1979 , PASP , 91 , 589 10.1086/130542 Crossref Search ADS Crossref Carbognani A. , 2016 , Minor Planet Bull. , 43 , 290 Carbognani A. , 2017 , Planet. Space Sci. , 147 , 1 10.1016/j.pss.2017.07.019 Crossref Search ADS Crossref Chandler C. 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Astashenok, Artyom, V;Odintsov, Sergey, D

doi: 10.1093/mnras/staa214pmid: N/A

ABSTRACT We investigated realistic neutron stars in axion R2 gravity. The coupling between curvature and axion field ϕ is assumed in the simple form ∼R2ϕ. For the axion mass in the range ma ∼ 10−11–10−10 eV the solitonic core within neutron star and corresponding halo with size ∼100 km can exist. Therefore the effective contribution of R2 term grows inside the star and it leads to change of star parameters (namely, mass, and radius). We obtained the increase of star mass independent from central density for wide range of masses. Therefore, maximal possible mass for given equation of state grows. At the same time, the star radius increases not so considerably in comparison with GR. Hence, our model may predict possible existence of supermassive compact stars with masses |$M\sim 2.2\!-\!2.3\, \mathrm{M}_\odot$| and radii Rs ∼ 11 km for realistic equation of state (we considered APR equation of state). In general relativity one can obtain neutron stars with such characteristics only for unrealistic, extremely stiff equations of state. Note that this increase of mass occurs due to change of solution for scalar curvature outside the star. In GR curvature drops to zero on star surface where ρ = p = 0. In the model underconsideration the scalar curvature dumps more slowly in comparison with vacuum R2 gravity due to axion ‘galo’ around the star. stars: neutron, dark matter 1 INTRODUCTION There are still unresolved fundamental puzzles in modern cosmology and relativistic astrophysics. One of them is so-called dark energy which governs the observed accelerated Universe expansion (Riess et al. 1998; Perlmutter et al. 1999; Riess et al. 2004). According to the well-known lambda cold dark matter (ΛCDM) model, dark energy is simply cosmological constant and its density is 72 per cent of the global energy budget of the Universe. The remaining 28 per cent, clustered in galaxies and clusters of galaxies, consist of baryons (only 4 per cent) and CDM the nature of which is unclear. Alternative approach to description of late-time cosmological dynamics of Universe is proposed by theories of modified gravity (Capozziello & Fang 2002; Nojiri & Odintsov 2003b; Carroll et al. 2004). Furthermore, the unified description of observed late-time acceleration and early Universe expansion is also possible in the context of f(R) theory (Nojiri & Odintsov 2003a). Matter and radiation dominance eras can be also described in frames of this approach (for review, see refs. Capozziello & de Laurentis 2011; Nojiri & Odintsov 2011; Olmo 2011; de la Cruz-Dombriz & Sáez-Gómez 2012; Nojiri, Odintsov & Oikonomou 2017). Another problem of current cosmology and high-energy physics is dark matter. There are many evidences in favour for particle nature of dark matter [for example, see data about collision of galaxies in the Bullet Cluster and cluster MACSJ0025 (Markevitch et al. 2003; Clowe et al. 2006; Bradač et al. 2008; Robertson, Massey & Eke 2017)]. As the candidates on the role of dark matter the weakly interacted massive particles (WIMPs) have been considered usually. Many approaches to direct observation of such particles were proposed but so far the direct experiments for detection of such particles have not been successful (see CDMS II Collaboration 2010; Davis, McCabe & Bœhm 2014; Davis 2015; Roszkowski, Sessolo & Trojanowski 2018; Schumann 2019). Another possibility is that dark matter is nothing else than axions (Sakharov & Khlopov 1994; Sakharov, Sokoloff & Khlopov 1996; Khlopov, Sakharov & Sokoloff 1999; Marsh 2016; Marsh et al. 2017; Caputo 2019; Cicoli, Guidetti & Pedro 2019; Fukunaga, Kitajima & Urakawa 2019; Odintsov & Oikonomou 2019). Some recent experiments indicate in favour for its existence (Avignone, Creswick & Vergados 2018; Caputo, Garay & Witte 2018; Du et al. 2018; Henning et al. 2018; Caputo et al. 2019; Lawson et al. 2019; Ouellet et al. 2019; Rozner et al. 2019; Safdi, Sun & Chen 2019). Mass of axions can be very low (theoretical estimations give value in the wide range ∼10−12–10−3 eV). The possibility of axions detection is based on axion–photon interaction in the presence of magnetic fields (Balakin & Ni 2010; Balakin, Bochkarev & Tarasova 2012; Balakin, Muharlyamov & Zayats 2014). Recently, axion F(R) gravity was discussed in ref. Odintsov & Oikonomou (2019) in which the unification of dark energy with axion dark matter (and eventually, with inflation) was proposed. For the description of axion in this model misalignment model was used (Anisimov & Dine 2005). According to misagliment model the primordial U(1) Peccei–Quinn symmetry is broken during inflation. For f(R) it was chosen simple R2 model and non-minimal coupling with axion field in the form h(ϕ)Rγ. It is interesting to address the question about possible manifestations of modified gravity with various scalar fields (including axions) on astrophysical level. First of all, one should pay attention to relativistic stars for which the energy density in the centre is around 1015–1016 g cm−3 and therefore gravitational field is extremely high. In principle, for such strong gravity regime the possible deviations from general relativity can be visible somehow. Unfortunately, we have no well-established data about mass–radius diagram for neutron stars from astronomical observations. Mass, radius, and other parameters of relativistic stars also depend from the equation of state chosen for dense matter. There is now clear understanding how nuclear matter behaves at such extreme densities and tens of EoS were proposed over the years (for some introduction, see ref. Rezzolla et al. 2018). This paper is devoted to the study of neutron stars in frames of R2 gravity coupled with axion field ϕ in the form βR2ϕ where β is some constant. It is interesting to note that Compton wavelength λa for particle with mass ma ∼ 10−11–10−10 eV is ∼10–102 km. This scale is comparable with characteristic size of neutron stars. Therefore, one can consider the possibility of existence of solitonic core containing dark matter in the centre of the star. The contribution to energy density from such core is negligible itself and therefore could not influence on the parameters of star. However, the assumption of coupling ∼R2ϕ can lead to non-trivial deviations from general relativity. F(R) gravity was considered as viable alternative to GR for description of stellar structure in some papers. The question of hydrostatic equilibrium was studied in ref. Capozziello et al. (2011). Dynamics and collapse of collisionless self-gravitating systems in R2 gravity was firstly investigated by Capozziello et al. (2012). It is interesting to note also iterative procedure for the solution of modified Lane–Emden equation considered in Farinelli et al. (2014). For neutron stars initially the perturbative approach was used. The scalar curvature R is defined by Einstein equations at zeroth order on the small parameter, i.e. R ∼ T, where T is the trace of energy–momentum tensor. This approach is applied to construction of neutron star models in f(R) = R + αR2 + βR3 and f(R) = R + αR2(1 + γln R) gravity also in Arapoǧlu, Deliduman & Eksi (2011), Alavirad & Weller (2013), and Astashenok, Capozziello & Odintsov (2013). In modified f(R) gravity model with cubic and quadratic terms, it is possible to obtain neutron stars with |$M\sim 2\, \mathrm{M}_{\odot }$| for simple hyperon equations of state (EoS) although the soft hyperon equation of state is usually treated as non-realistic in the standard general relativity (Astashenok, Capozziello & Odintsov 2014). The possible signatures of modified gravity in neutron star astrophysics also can include existence of neutron stars with extremely high magnetic fields (Cheoun et al. 2013; Astashenok, Capozziello & Odintsov 2015). Interesting results were obtained for f(R) = R1 + ϵ gravity in Capozziello et al. (2016). Detailed analysis of neutron stars structure in R2 gravity was given by authors in ref. Astashenok, Odintsov & de la Cruz-Dombriz (2017). It is shown that so-called gravitational sphere with non-zero curvature appears around the star. Recently, the mass–radius relation in both metric and torsional R2 gravity were investigated in Feola et al. (2019). For recent review of compact star models in modified theories of gravity see Olmo, Rubiera-Garcia & Wojnar (2019) and references therein. The main problem of simple R2 gravity is that possible observable consequences appear only if the contribution of R2-term is sufficiently large. The motivation for the consideration of non-minimal coupling in the form ∼R2ϕ is that axion field can strengthen the contribution of R2-term only inside star due to solitonic core. Outside the star, the scalar field and scalar curvature quickly drop in fact to zero value (of course, in comparison with corresponding values inside the star). In the next section, we start from the equations for stellar configurations in f(R) gravity in quasi-isotropic coordinates. For their derivation the so-called 3+1 formalism is used. as a result we get the system of equations of elliptical type for unknown metric functions and scalar curvature R. The account of scalar field requires one more equation. The results of the calculation including mass profile, mass–radius diagram, scalar curvature, and axion field are given in the Section 3. For neutron star matter the well-known APR equation of state (EoS) is used. Our calculations show that qualitative behaviour of solutions does not depend from chosen EoS. 2 3+1 formalism in f(R) gravity We start from the well-known Einstein equations in frames of general relativity $$\begin{eqnarray*} R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R={8\pi } T_{\mu \nu }. \end{eqnarray*}$$ (1) Here, Rμν is the Ricci tensor associated with the Levi–Civita connection ∇ in four-dimensional space–time, R = gμνRμν is the scalar curvature and Tμν is the energy–momentum tensor of matter. The system of units in which G = c = 1 is used. First, we consider simple f(R) gravity with the action $$\begin{eqnarray*} S=\frac{1}{2}\int f(R) \sqrt{-g} d^{4} x \end{eqnarray*}$$ (2) from which it follows that Einstein equations are $$\begin{eqnarray*} f_{R}(R) R_{\mu \nu }-\frac{f(R)}{2} \, g_{\mu \nu } -\left(\nabla _{\mu } \nabla _{\nu }- g_{\mu \nu }\Box \right) f_{R}(R))=8\pi T_{\nu \mu }. \end{eqnarray*}$$ (3) Here |$\Box =\nabla ^{\mu }\nabla _{\mu }$| is covariant D’Alamber operator and fR(R) ≡ df/dR. Then, we drop arguments of f(R). One can rewrite (2) in equivalent form $$\begin{eqnarray*} {f_{R}R_{\mu \nu }-\frac{1}{2}(f_{R}R-f)g_{\mu \nu }-\left(\frac{1}{2}\Box +\nabla _{\mu } \nabla _{\nu }\right)f_{R}}\nonumber\\ {\quad=8\pi \left(T_{\mu \nu }-\frac{1}{2}g_{\mu \nu }T\right), } \end{eqnarray*}$$ (4) where T is the trace of energy–momentum tensor. For the study of compact stars in general relativity the 3+1 formalism is often used (Gourgoulhon 2007; Alcubierre 2008; Baumgarte & Shapiro 2010; Gourgoulhon 2010). One can adopt this method for f(R) gravity without any significant changes. The key moment is foliation of space–time by space-like hypersurfaces Σt. Parameter t can be associated with coordinate time. The next step is the definition of metric γij induced by metric gμν on hypersurface Σ. The components of induced metric can be given via the components of unit time-like normal vector n and metric gμν: $$\begin{eqnarray*} \gamma _{\alpha \beta }=g_{\alpha \beta }+n_{\alpha }n_{\beta }. \end{eqnarray*}$$ (5) We also define the components of orthogonal projector on to hypersurface Σt by raising of the first index: $$\begin{eqnarray*} \gamma ^{\alpha }_{\cdot \beta }=\delta ^{\alpha }_{\cdot \beta }+n^{\alpha } n_{\beta }. \end{eqnarray*}$$ (6) We use metric of special form $$\begin{eqnarray*} \mathrm{ d}s^{2}=-N^{2}\mathrm{ d}\mathrm{ }t^{2}+\gamma _{ij}(\mathrm{ d}x^{i}+\xi ^{i}\mathrm{ d}t)(\mathrm{ d}\mathrm{ }x_{j}+\xi _{j}\mathrm{ d}t), \end{eqnarray*}$$ (7) where N is so-called lapse function and |$\vec{\boldsymbol {\xi }}$| is shift vector. The three-dimensional metric γij is the metric induced on surface. Then projecting the Einstein equation (4) twice on to Σt, (ii) twice along to vector |$\vec{\boldsymbol {n}}$| normal to Σt, and (iii) once on Σt and once along |$\vec{\boldsymbol {n}}$|⁠, one gets the following three equations: $$\begin{eqnarray*} {f_{R}\left(\frac{\partial K_{ij} }{\partial t} - {\mathcal {L}}_{\vec{\boldsymbol {\xi }}}\, K_{ij}\right) +f_{R}\big(D_i D_j N - N}\nonumber\\ {\qquad\times\big\lbrace {}^3 R_{ij} + K K_{ij} -2 K_{ik} K^k_{\ \, j}\big\rbrace \big)}\nonumber \\ {\quad=4\pi N \left[ (\sigma -\epsilon) \gamma _{ij} - 2 \sigma _{ij} \right]-\frac{1}{2}(f_{R}R-f)N\gamma _{ij}}\nonumber\\ {\qquad-N\left(\frac{1}{2}\gamma _{ij}\Box +D_{i}D_{j}\right)f_{R}, } \end{eqnarray*}$$ (8) $$\begin{eqnarray*} f_{R}({}^3 R + K^2 - K_{ij} K^{ij}) = 16\pi E+f_{R}R-f+2D^{i} D_{i} f_{R}, \end{eqnarray*}$$ (9) $$\begin{eqnarray*} f_{R}\left(D_j K^j_{\ \, i} - D_i K\right) = 8\pi p_i -n^{\mu }\nabla _{\mu }(D_{i}f_{R}), \end{eqnarray*}$$ (10) where Kij is tensor of extrinsic curvature and |$K=K^{i}_{i}$|⁠. One also introduces the components of Lie derivative of tensor and Kij along the vector |$\vec{\boldsymbol {\xi }}$|⁠: $$\begin{eqnarray*} {\mathcal {L}}_{\vec{\boldsymbol {\xi }}}\, K_{ij} = \xi ^k \frac{\partial K_{ij}}{\partial x^k} + K_{kj} \frac{\partial \xi ^k}{\partial x^i} + K_{ik} \frac{\partial \xi ^k}{\partial x^j}. \end{eqnarray*}$$ (11) The covariant derivatives Di can be expressed in terms of partial derivatives with respect to the spatial coordinates (xi) by means of the Christoffel symbols |${}^3 \Gamma ^i_{\ \, jk}$| of |$\boldsymbol {D}$| associated with (xi): $$\begin{eqnarray*} D_i D_j N = \frac{\partial ^2 N}{\partial x^i\partial x^j} - {}^3 \Gamma ^k_{\ \, ij} \frac{\partial N}{\partial x^k} , \end{eqnarray*}$$ (12) $$\begin{eqnarray*} D_j K^j_{\ \, i} = \frac{\partial K^j_{\ \, i}}{\partial x^j} + {}^3 \Gamma ^j_{\ \, jk} K^k_{\ \, i} - {}^3 \Gamma ^k_{\ \, ji} K^j_{\ \, k} , \end{eqnarray*}$$ (13) $$\begin{eqnarray*} D_i K = \frac{\partial K}{\partial x^i} . \end{eqnarray*}$$ (14) For three-dimensional Ricci tensor 3Rij and scalar curvature 3R, we have following relations: $$\begin{eqnarray*} {}^3 R_{ij} = \frac{\partial \, {}^3 \Gamma ^k_{\ \, ij}}{\partial x^k} - \frac{\partial \, {}^3 \Gamma ^k_{\ \, ik}}{\partial x^j} + {}^3 \Gamma ^k_{\ \, ij} {}^3 \Gamma ^l_{\ \, kl} - {}^3 \Gamma ^l_{\ \, ik} {}^3 \Gamma ^k_{\ \, lj} \end{eqnarray*}$$ (15) $$\begin{eqnarray*} {}^3 R = \gamma ^{ij} \, {}^3 R_{ij}. \end{eqnarray*}$$ (16) The quantities ϵ, Sij, and pi are energy density, components of stress tensor, and vector of energy flux density correspondingly. These values can be obtained from stress–energy tensor Tμν: $$\begin{eqnarray*} \epsilon \, = \, n^{\mu } n^{\nu } T_{\mu \nu },\nonumber \\ \sigma _{ij}\, = \,\gamma ^{\mu }_{i}\gamma ^{\nu }_{j} T_{\mu \nu },\quad \sigma =\sigma ^{i}_{i}. \\ p_{i}\, = \,-n^{\mu }\gamma ^{\nu }_{i}T_{\mu \nu }. \nonumber \end{eqnarray*}$$ (17) Then we take the trace of first equation: $$\begin{eqnarray*} f_{R}D_{i}D^{i}N\, = \,Nf_{R}({}^{3}R+K^{2}-2K_{ik}K^{ik})+4\pi N (S-3E)\nonumber \\ &&+\,\frac{3}{2}N(f-f_{R}R)-\frac{3}{2}N\Box f_{R}-ND_{i}D^{i}f_{R}. \end{eqnarray*}$$ (18) From equation (9) it follows that $$\begin{eqnarray*} f_{R}({}^{3}R+K^{2})=f_{R}K_{ij}K^{ij}+16\pi E+f_{R}R-f+2D_{i}D^{i}f_{R} \end{eqnarray*}$$ and therefore one can rewrite the previous equation as $$\begin{eqnarray*} f_{R}D_{i}D^{i}N \, = \, Nf_{R}K_{ij}K^{ij}+4\pi N(\epsilon +\sigma)-\frac{1}{2}N(f_{R}R-f)\nonumber \\ && -\,\frac{3}{2}N\Box f_{R}+ND_{i}D^{i}f_{R}. \end{eqnarray*}$$ (19) Let us consider the non-rotating stellar configurations. In this case, all metric functions depend only from radial coordinate. For convenience, we used isotropic spatial coordinates with metric in the form $$\begin{eqnarray*} \mathrm{ d}s^{2}=-N^{2}(r)\mathrm{ d}t^{2}+A^{2}(r)(\mathrm{ d}r^{2}+r^{2}\mathrm{ d}\Omega ^{2}). \end{eqnarray*}$$ (20) For this metric one can obtain that $$\begin{eqnarray*} K=0,\quad K_{i j} K^{i j}=0. \end{eqnarray*}$$ We also have the following expression for three-dimensional scalar curvature: $$\begin{eqnarray*} {}^3 R = -\frac{4}{A^3} \left(\frac{\mathrm{ d}\mathrm{ }^{2}A}{\mathrm{ d}r^{2}} + \frac{2}{r} \frac{\mathrm{ d}A}{\mathrm{ d}r} - \frac{1}{2A} \left(\frac{\mathrm{ d}A}{\mathrm{ d}r} \right) ^2 \right) .\nonumber \end{eqnarray*}$$ Taking into account that $$\begin{eqnarray*} \frac{1}{A}\frac{\mathrm{ d}^{2}A}{\mathrm{ d}r^{2}}=\frac{\mathrm{ d}\mathrm{ }^{2}\ln A}{\mathrm{ d}r^{2}}+\left(\frac{\mathrm{ d}\ln A}{\mathrm{ d}r}\right)^{2}, \end{eqnarray*}$$ one can rewrite the equation for (3)R in the following form $$\begin{eqnarray*} {}^3 R = -\frac{4}{A^2}\left(\triangle ^{r}_{(3)}\ln A+\frac{1}{2}\left(\frac{\mathrm{ d}\ln A}{\mathrm{ d}r}\right)^{2}\right). \end{eqnarray*}$$ (21) In this relation, |$\triangle ^{r}_{(3)}$| is nothing else than radial part of three-dimensional Laplace operator in Euclidean space. The energy–momentum tensor in the case of spherical symmetry can be presented in diagonal form |$T_{\mu }^{nu}=\mbox{diag}(-\epsilon , p, p, p)$|⁠, where p is pressure of matter and therefore $$\begin{eqnarray*} \sigma ^{r}_{r}=\sigma ^{\theta }_{\theta }=\sigma ^{\phi }_{\phi }=p, \quad \sigma =3p. \end{eqnarray*}$$ The action of three-dimensional covariant D’Alamber operator for any scalar function Φ(r) depending only from radial coordinate is reduced to $$\begin{eqnarray*} D^{i}D_{i}\Phi (r)=\frac{1}{A^{2}}\left(\triangle ^{r}_{(3)}\Phi +\frac{\mathrm{ d}\ln A}{\mathrm{ d}r}\frac{\mathrm{ d}\Phi }{\mathrm{ d}\mathrm{ }r}\right). \end{eqnarray*}$$ (22) For four-dimensional d’Alambertian one obtains the simple relation: $$\begin{eqnarray*} \Box \Phi (r)=\frac{1}{A^{2}}\left(\triangle ^{r}_{(3)}\Phi +\frac{\mathrm{ d}\ln (NA)}{\mathrm{ d}r}\frac{\mathrm{ d}\Phi }{\mathrm{ d}r}\right). \end{eqnarray*}$$ (23) Finally, equation (19) for our task can be presented in the form: $$\begin{eqnarray*} f_{R}\triangle ^{r}_{(3)}\nu +\frac{1}{2}\triangle ^{r}_{(3)}f_{R}\, = \,4\pi A^{2}(\epsilon +3p)-\frac{A^{2}}{2}(f_{R}R-f)\nonumber \\ &&-\,f_{R}\frac{\mathrm{ d}\eta }{\mathrm{ d}\mathrm{ }r}\frac{\mathrm{ d}\nu }{\mathrm{ d}r} - \frac{1}{2}\frac{\mathrm{ d}\eta }{\mathrm{ d}r}\frac{\mathrm{ d}f_{R}}{\mathrm{ d}\mathrm{ }r}-\frac{\mathrm{ d}\nu }{\mathrm{ d}r}\frac{\mathrm{ d}f_{R}}{\mathrm{ d}r}. \end{eqnarray*}$$ (24) Here, η = ln (AN) and ν = ln N. Let us consider equation (9) for metric (20). Using relations for 3R and covariant d’Alambertian one gets $$\begin{eqnarray*} 2f_{R}\triangle ^{r}_{(3)}\ln A+\triangle ^{r}_{(3)}f_{R}\, = \,-8\pi A^{2} \epsilon -\frac{A^2}{2}(f_{R}R-f)\nonumber \\ &&-\,f_{R}\left(\frac{\mathrm{ d}(\ln A)}{\mathrm{ d}r}\right)^{2} - \frac{\mathrm{ d}(\ln A)}{\mathrm{ d}\mathrm{ }r}\frac{\mathrm{ d}f_{R}}{\mathrm{ d}r}. \end{eqnarray*}$$ (25) Next, one considers the ϕϕ-component of (8). One need to use the following representations for DϕϕN and 3Rϕϕ: $$\begin{eqnarray*} D_{\phi }D_{\phi }N\, = \,r^{2}\sin ^{2}\theta \left(\frac{1}{A}\frac{\mathrm{ d}A}{\mathrm{ d}r}+\frac{1}{r}\right)\frac{\mathrm{ d}N}{\mathrm{ d}r},\\ ^{3}R_{\phi \phi }\, = \,-r^{2}\sin ^{2}\theta \frac{1}{A}\left(\frac{\mathrm{ d}\mathrm{ }^{2}A}{\mathrm{ d}r^{2}}+\frac{3}{r}\frac{\mathrm{ d}A}{\mathrm{ d}r}\right). \end{eqnarray*}$$ One obtains after simple calculations $$\begin{eqnarray*} f_{R}\triangle ^{r}_{(4)}\ln A+\frac{1}{2}\triangle ^{r}_{(3)}f_{R}\, = \,4\pi A^{2}(p-\epsilon)-\frac{A^{2}}{2}(f_{R}R-f)\nonumber \\ &&-\,f_{R}\left(\frac{\mathrm{ d}\ln A}{\mathrm{ d}r}\right)^{2}-f_{R}\frac{\mathrm{ d}\ln (Ar)}{\mathrm{ d}r}\frac{\mathrm{ d}\nu }{\mathrm{ d}r}\nonumber \\ &&-\,\frac{1}{2}\frac{\mathrm{ d}\eta }{\mathrm{ d}r}\frac{\mathrm{ d}f_{R}}{\mathrm{ d}r}-\frac{\mathrm{ d}\ln (Ar)}{\mathrm{ d}r}\frac{\mathrm{ d}f_{R}}{\mathrm{ d}r}. \end{eqnarray*}$$ (26) Adding (24) and (26) gives the following equation for η: $$\begin{eqnarray*} f_{R}\triangle ^{r}_{(4)}\eta +\triangle ^{r}_{(4)}f_{R}\, = \,16\pi A^{2}p-A^{2}(f_{R}R-f)\nonumber \\ &&-\,f_{R}\left(\frac{\mathrm{ d} \eta }{\mathrm{ d}r}\right)^{2}-2\frac{\mathrm{ d} \eta }{\mathrm{ d}r}\frac{\mathrm{ d} f_{R}}{\mathrm{ d}r} . \end{eqnarray*}$$ (27) Then adding (24) to (25) and subtracting (26) we obtain: $$\begin{eqnarray*} f_{R}\triangle ^{r}_{(2)}\eta +\triangle ^{r}_{(2)}f_{R}\, = \,8\pi A^{2}p-\frac{1}{2}A^{2}(f_{R}R-f)\nonumber \\ &&-\,f_{R}\left(\frac{\mathrm{ d} \nu }{\mathrm{ d}r}\right)^{2}-\frac{\mathrm{ d} \nu }{\mathrm{ d}r}\frac{\mathrm{ d }f_{R}}{\mathrm{ d}\mathrm{ }r} . \end{eqnarray*}$$ (28) In general relativity scalar curvature R is simply −8πT and therefore it approaches to zero on the surface of star where ϵ = p = 0. For f(R) gravity one needs additional equation for scalar curvature. This equation can be obtained from trace of Einstein equations and for our case of radial dependence of scalar curvature it takes the form: $$\begin{eqnarray*} \triangle ^{r}_{(3)}f_{R}=\frac{8\pi }{3} A^{2}(3p-\epsilon)-\frac{A^{2}}{3}(f_{R}R-2f)-\frac{\mathrm{ d} \eta }{\mathrm{ d}r}\frac{\mathrm{ d} f_{R}}{\mathrm{ d}\mathrm{ }r}. \end{eqnarray*}$$ (29) For the case of function fR = F(R, ϕ) depending also from scalar field ϕ these equations are valid. For radial derivatives of function F(R, ϕ) one should remember that $$\begin{eqnarray*} \frac{\mathrm{ d}F}{\mathrm{ d}r}\, = \,F_{R}\frac{\mathrm{ d}R}{\mathrm{ d}r}+F_{\phi }\frac{\mathrm{ d}\phi }{\mathrm{ d}r},\\ \frac{\mathrm{ d}^{2}F}{\mathrm{ d}r^2}\, = \,F_{R}\frac{\mathrm{ d}^{2}R}{\mathrm{ d}r^2}+F_{RR}\left(\frac{\mathrm{ d}R}{\mathrm{ d}r}\right)^{2}+F_{\phi }\frac{\mathrm{ d}^{2}\phi }{\mathrm{ d}r^2}\nonumber \\ &&+\,F_{\phi \phi }\left(\frac{\mathrm{ d}\phi }{\mathrm{ d}r}\right)^{2}+ 2F_{R\phi }\frac{\mathrm{ d}R}{\mathrm{ d}r}\frac{\mathrm{ d}\phi }{\mathrm{ d}r}. \end{eqnarray*}$$ Assuming the action for axion field in the following form $$\begin{eqnarray*} S_{\phi }=\int \mathrm{ d}^{4}x\sqrt{-g}\left(-\frac{1}{2}\partial ^{\mu }\phi \partial _{\mu }\phi -V(\phi)\right). \end{eqnarray*}$$ (30) One can obtain the equation for scalar field ϕ = ϕ(r): $$\begin{eqnarray*} \triangle ^{r}_{(3)}\phi =A^{2}\frac{\mathrm{ d}\mathrm{ }V}{\mathrm{ d}\mathrm{ }\phi }-\frac{A^{2}}{8\pi }\frac{\mathrm{ d}f}{\mathrm{ d}\phi }-\frac{\mathrm{ d}\phi }{\mathrm{ d}r}\frac{\mathrm{ d}\eta }{\mathrm{ d}r}. \end{eqnarray*}$$ (31) The system of equations (24), (27), (29), (31) should be supplemented by a set of boundary conditions for η, ν, R, and ϕ. Those are provided by the asymptotic flatness assumption. On spatial infinity the metric tensor tends towards Minkowski metric and therefore $$\begin{eqnarray*} \nu \rightarrow 0, \quad \eta \rightarrow 0, \quad R\rightarrow 0\quad \mbox{for}\quad r\rightarrow \infty . \end{eqnarray*}$$ For scalar field, we also assume that ϕ → 0 when r → +∞ because the density of dark matter in the space (∼10−29 g cm−3) is extremely low in comparison with densities inside relativistic stars. Asymptotical behaviour of A(r) at r → ∞ defines gravitational mass M of star for distant observer. In general relativity the solution of Einstein equations outside the star has the form: $$\begin{eqnarray*} A(r)=\left(1+\frac{M}{2r}\right)^{2}, \quad N(r)=\left(1-\frac{M}{2r}\right)\left(1+\frac{M}{2r}\right)^{-1}. \end{eqnarray*}$$ (32) Therefore, the gravitational mass of star can be found as an asymptotical limit $$\begin{eqnarray*} M=2\lim _{r\rightarrow \infty }r(\sqrt{A}-1). \end{eqnarray*}$$ One should also account that physical radial coordinate |$\tilde{r}$| is $$\begin{eqnarray*} \tilde{r}=Ar. \end{eqnarray*}$$ Note that in the following the symbol ‘r’ on figures means physical distance. Tildes are omitted for simplicity. Table 1. Parameters of compact stars (mass and radius) in general relativity, pure R2 gravity (α = 0.25), and for several values of β (α = 0.25) in the model with axion field for three values of energy density in the centre ϵ0. The corresponding values of curvature and axion field in the centre of star are given also. For massive stars one can see that increase of mass for β = 103 consists of |$\sim 0.2\, \mathrm{M}_{\odot }$| (in comparison with general relativity) for the same density in the centre of star. ϵ0, MeV fm−3 . β, |$r_{g}^{2}$| . M/M⊙, . Rs, km . R0, |$r_{g}^{-2}$| . ϕ0 . GR 1.64 11.25 0.0674 – 0 1.68 11.40 0.0674 – 650 100 1.65 11.38 0.0273 0.0097 250 1.67 11.42 0.0183 0.0094 1000 1.73 11.48 0.0077 0.0076 GR 1.92 11.03 0.0146 – 0 1.98 11.15 0.0137 – 800 100 2.00 11.26 0.0238 0.0097 250 2.04 11.33 0.0162 0.0096 1000 2.12 11.44 0.0071 0.0077 GR 2.10 10.68 −0.0889 – 0 2.14 10.80 −0.0899 – 1000 100 2.19 10.89 0.0142 0.0077 250 2.22 10.95 0.0108 0.0080 1000 2.30 11.07 0.0051 0.0068 ϵ0, MeV fm−3 . β, |$r_{g}^{2}$| . M/M⊙, . Rs, km . R0, |$r_{g}^{-2}$| . ϕ0 . GR 1.64 11.25 0.0674 – 0 1.68 11.40 0.0674 – 650 100 1.65 11.38 0.0273 0.0097 250 1.67 11.42 0.0183 0.0094 1000 1.73 11.48 0.0077 0.0076 GR 1.92 11.03 0.0146 – 0 1.98 11.15 0.0137 – 800 100 2.00 11.26 0.0238 0.0097 250 2.04 11.33 0.0162 0.0096 1000 2.12 11.44 0.0071 0.0077 GR 2.10 10.68 −0.0889 – 0 2.14 10.80 −0.0899 – 1000 100 2.19 10.89 0.0142 0.0077 250 2.22 10.95 0.0108 0.0080 1000 2.30 11.07 0.0051 0.0068 Open in new tab Table 1. Parameters of compact stars (mass and radius) in general relativity, pure R2 gravity (α = 0.25), and for several values of β (α = 0.25) in the model with axion field for three values of energy density in the centre ϵ0. The corresponding values of curvature and axion field in the centre of star are given also. For massive stars one can see that increase of mass for β = 103 consists of |$\sim 0.2\, \mathrm{M}_{\odot }$| (in comparison with general relativity) for the same density in the centre of star. ϵ0, MeV fm−3 . β, |$r_{g}^{2}$| . M/M⊙, . Rs, km . R0, |$r_{g}^{-2}$| . ϕ0 . GR 1.64 11.25 0.0674 – 0 1.68 11.40 0.0674 – 650 100 1.65 11.38 0.0273 0.0097 250 1.67 11.42 0.0183 0.0094 1000 1.73 11.48 0.0077 0.0076 GR 1.92 11.03 0.0146 – 0 1.98 11.15 0.0137 – 800 100 2.00 11.26 0.0238 0.0097 250 2.04 11.33 0.0162 0.0096 1000 2.12 11.44 0.0071 0.0077 GR 2.10 10.68 −0.0889 – 0 2.14 10.80 −0.0899 – 1000 100 2.19 10.89 0.0142 0.0077 250 2.22 10.95 0.0108 0.0080 1000 2.30 11.07 0.0051 0.0068 ϵ0, MeV fm−3 . β, |$r_{g}^{2}$| . M/M⊙, . Rs, km . R0, |$r_{g}^{-2}$| . ϕ0 . GR 1.64 11.25 0.0674 – 0 1.68 11.40 0.0674 – 650 100 1.65 11.38 0.0273 0.0097 250 1.67 11.42 0.0183 0.0094 1000 1.73 11.48 0.0077 0.0076 GR 1.92 11.03 0.0146 – 0 1.98 11.15 0.0137 – 800 100 2.00 11.26 0.0238 0.0097 250 2.04 11.33 0.0162 0.0096 1000 2.12 11.44 0.0071 0.0077 GR 2.10 10.68 −0.0889 – 0 2.14 10.80 −0.0899 – 1000 100 2.19 10.89 0.0142 0.0077 250 2.22 10.95 0.0108 0.0080 1000 2.30 11.07 0.0051 0.0068 Open in new tab 3 RESULTS We considered in detail the model with the action $$\begin{eqnarray*} \mathcal {S}=\int \mathrm{ d}^{4}x\sqrt{-g}\left(\frac{R+\alpha R^2}{16\pi }+\beta \frac{R^{2}\phi }{16{\pi}}-\frac{1}{2}\partial _{\mu }{\phi}\partial ^{\mu }\phi -V(\phi)\right).\nonumber\\ \end{eqnarray*}$$ (33) Following to Marsh (2016), we consider only small deviations from the potential minimum. Therefore, the leading term is $$\begin{eqnarray*} V(\phi)\simeq \frac{1}{2}{m^{2}_{\mathrm{ a}}}\phi ^{2}. \end{eqnarray*}$$ The behaviour of axion scalar is mainly governed by |$m_{\mathrm{ a}}^{2}\phi ^{2}$| term. Another key moment of this model is the influence of scalar field on the dependence of scalar curvature from radial coordinate. For axion mass, we take the value corresponding to Compton wavelength 10rg where rg is gravitational radius of Sun (2.95 km). For the parameter β the range 100 < β < 1000 in units of |$r_{g}^2$| is explored. First of all, let us consider the case when the parameter α is relatively small (for example α = 0.25). The mass–radius diagram can be seen on Fig. 1 with the dependence of stellar mass from the energy density in the centre of star. For massive stars one can see (Table 1) that increase of mass for β = 103 consists of |$\sim 0.2\, \mathrm{M}_{\odot }$| or ∼10 per cent (in comparison with general relativity) for the same density in the centre of star. The radius of star also increases and therefore for given radius the increase of mass looks even bigger. Therefore, one can expect that in such modified gravity the fraction of supermassive neutron stars (with |$M\gt 2.0\, \mathrm{M}_{\odot }$|⁠) should increase. Another interesting point is the maximal possible mass for given EoS. For APR equation, we have that |$M_{\mathrm{ max}\mathrm{ }}=2.31\, \mathrm{M}_{\odot }$| at β = 103 in comparison with |$M_{\mathrm{ max}}=2.16\, \mathrm{M}_{\odot }$|⁠) in general relativity. Note that mass increases with β non-linearly and for very large β this increase does not exceed significantly the result obtained for β = 1000. In general relativity one can obtain compact stars with |$M\sim 2.3 \, \mathrm{M}_{\odot }$| and Rs ∼ 11 km only for very stiff equations of state usually treated as unrealistic. Figure 1. Open in new tabDownload slide Mass–radius diagram for |$M\gt 1.3\, \mathrm{M}_{\odot }$| and dependence mass from central energy density (in Mev fm−3) for neutron stars in general relativity and model (33) for various values of β and α = 0.25. Figure 1. Open in new tabDownload slide Mass–radius diagram for |$M\gt 1.3\, \mathrm{M}_{\odot }$| and dependence mass from central energy density (in Mev fm−3) for neutron stars in general relativity and model (33) for various values of β and α = 0.25. It is interesting to consider the mass profile m(r) for various parameters of model. On Fig. 2, the m(r) for three values of ϵ0 (650, 800, and 1000 MeV fm−3) is depicted. The increase of gravitational mass of star for distant observer occurs due to non-trivial behaviour of scalar curvature outside the star (Fig. 3). For GR R = 0 outside the star and for simple R2-model of gravity scalar curvature decreases quickly outside the star. Due to axion field (see Fig. 4) the damping of scalar curvature became smoother. Figure 2. Open in new tabDownload slide The radial profile neutron star mass for case α = 0.25 at ϵ0 = 650, 800, and 1000 Mev fm−3 for β = 0 (thick lines), β = 100 (dotted lines), β = 250 (solid lines), and β = 1000 (dashed lines). Figure 2. Open in new tabDownload slide The radial profile neutron star mass for case α = 0.25 at ϵ0 = 650, 800, and 1000 Mev fm−3 for β = 0 (thick lines), β = 100 (dotted lines), β = 250 (solid lines), and β = 1000 (dashed lines). Figure 3. Open in new tabDownload slide The radial profile of curvature R (in units of |$r_{g}^{-2}$|⁠) for case α = 0.25 at ϵ0 = 650 (upper panel), 800 (middle panel), and 1000 Mev fm−3 (down panel) correspondingly. Black lines correspond to R2 gravity without axion field. Note that for the case of ϵ0 = 1000 Mev fm−3 curvature goes strongly negative in a range r < 4 km. This effect is only artefact of the choice of EoS. In simple R2 gravity for our interval of α the solution for scalar curvature R does not significantly differ from the solution in GR where scalar curvature is simply 8π(ρ − 3p). For APR EoS for large densities 3p > ρ and therefore R < 0. Figure 3. Open in new tabDownload slide The radial profile of curvature R (in units of |$r_{g}^{-2}$|⁠) for case α = 0.25 at ϵ0 = 650 (upper panel), 800 (middle panel), and 1000 Mev fm−3 (down panel) correspondingly. Black lines correspond to R2 gravity without axion field. Note that for the case of ϵ0 = 1000 Mev fm−3 curvature goes strongly negative in a range r < 4 km. This effect is only artefact of the choice of EoS. In simple R2 gravity for our interval of α the solution for scalar curvature R does not significantly differ from the solution in GR where scalar curvature is simply 8π(ρ − 3p). For APR EoS for large densities 3p > ρ and therefore R < 0. Figure 4. Open in new tabDownload slide The radial profile of axion field ϕ for case α = 0.25 at ϵ0 = 650 (upper panel), 800 (middle panel), and 1000 (down panel) Mev fm−3 correspondingly. Figure 4. Open in new tabDownload slide The radial profile of axion field ϕ for case α = 0.25 at ϵ0 = 650 (upper panel), 800 (middle panel), and 1000 (down panel) Mev fm−3 correspondingly. The increase of parameter α in considered area (0.25 < α < 2.5) does not lead to significant consequences for stellar masses. For example on Fig. 5, the radial profiles of scalar curvature, axion field, and mass are depicted for fixed value of β = 250 and two values of α: 0.25 and 2.5. One can see that increase of α leads to some decrease of axion field and curvature (therefore contribution of term βR2ϕ decreases). Note that we do not consider the case of very large α (∼O(100)) by the following reason. One can see that value of curvature within star is ∼0.01–0.015. Therefore term αR2 for α ∼ O(100) will be larger in comparison with R. Therefore from physical viewpoint such values of α seems unrealistic. Figure 5. Open in new tabDownload slide Scalar curvature, axion field, and mass profiles for some densities in the centre of star in a case of β = 250 for two values of α: 0.25 (solid lines) and 2.5 (dashed lines). Figure 5. Open in new tabDownload slide Scalar curvature, axion field, and mass profiles for some densities in the centre of star in a case of β = 250 for two values of α: 0.25 (solid lines) and 2.5 (dashed lines). One can also mention about the influence of axion mass on the solution of gravitational equations. We considered ma = 0.01 (corresponding to Compton wavelength 100rg and found that in this case only solution of scalar field outside the star changes considerably (see Fig. 6) but scalar curvature for r > 50 km is very close to zero and therefore contribution of term Rϕ2 is negligible. Figure 6. Open in new tabDownload slide Scalar curvature, axion field, and mass profiles for some densities in the centre of star in a case of β = 250 for two values of ma: 0.1 (solid lines) and 0.01 (dashed lines). Figure 6. Open in new tabDownload slide Scalar curvature, axion field, and mass profiles for some densities in the centre of star in a case of β = 250 for two values of ma: 0.1 (solid lines) and 0.01 (dashed lines). We also considered in our preliminary calculations the potential with ∼ϕ4 term but in this case deviations from simple model are very negligible. 4 CONCLUSION We investigated neutron stars in axion R2 gravity with non-minimal curvature–axion coupling in form ∼R2ϕ. Our main result is possible increase of stellar mass due to axion presence. We obtained the increase of mass |$\sim 0.2\, \mathrm{M}_{\odot }$| for massive stars in the case of β = 1000. This value is sufficient for possible observational indication of such model. The star radius increases not so considerably (∼400 m for β = 1000). Therefore, axion F(R) gravity underconsideration may explain the possible existence of supermassive neutron stars (⁠|$M\gt 2.2\, \mathrm{M}_\odot$|⁠) compact as in general relativity at the same time. There are some indications in favour of the existence of such neutron stars [for example the possible masses of B1957+20 (van Kerkwijk, Breton & Kulkarni 2011) and 4U 1700−377 (Clark et al. 2002) are |$M\sim 2.4 \, \mathrm{M}_{\odot }$|]. Axion field affects on behaviour of scalar curvature inside and outside star in comparison with general relativity and vacuum R2 gravity. Increase of mass for distant observer occurs due to ‘gravitational sphere’ outside the star with non-zero curvature. The star radius increases and mass confined inside the stellar surface decreases in comparison with general relativity but the contribution of gravitational sphere overcompensates this decrease. In contrast to simple square gravity increase of gravitational mass is relatively equal for various values of density in the centre of star up to the masses close to maximum. Therefore, the increase of maximal mass takes place. For instance, we obtained that in the case of APR equation of state maximal mass is 2.31|$\, \mathrm{M}_{\odot }$| instead of 2.15 |$\, \mathrm{M}_{\odot }$| in general relativity for some choice of parameters. Increase of radius also takes place but it is not so significant and hardly observable. Our calculations show also interesting effect of some ‘compensation’ between two terms, i.e. αR2 and βϕR2. If α increases the contribution of second term decreases due to damping mean value of curvature and axion field. As the consequence, we have no possibility to discriminate between various solutions corresponding to various parameters. Characteristic scale of curvature damping in pure R2 gravity is ∼α1/2. We considered the case when this value is smaller in comparison with Compton wavelength of axion field (10rg). For very large Compton wavelength (∼100rg, for example) we have no observable consequences on masses and radii of stellar configurations. Only radius of axion ‘galo’ around the star grows. Although, for illustration we used the star models based on well-known APR EoS our calculations lead to qualitatively similar results for other realistic choices of EoS for dense matter in neutron stars. Finally, one can expect that combined effect of axion dark matter and modified gravity maybe quite significant in stellar astrophysics at strong gravitational regime. One should mention recent paper (Riley et al. 2019) in which authors found promising mass–radius posteriors for mass and radius of pulsar PSR J0030+0451. We think that these results on current stage can exclude some equations of state for which radius of star with mass |$M\sim 1.3\!-\!1.5\, \mathrm{M}_\odot$| differs significantly from narrow range. In the light of these data APR equation of state is under question in general relativity and as consequence in our model because for |$M=1.3\!-\!1.5\, \mathrm{M}_{\odot }$| possible value of radius differs from GR value negligibly [this difference is ∼100 m for model with axions and this value is less in comparison with error of measurements (∼1 km)]. For EoS describing these data the picture is the same. In this case, if GR fits these data well then our theory does the same. One can ask how would one differentiate between considered model of simple f(R) gravity with axions and simply for example the case with different Eos for dense matter? 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Trujillo,, Ignacio;Chamba,, Nushkia;Knapen, Johan, H

doi: 10.1093/mnras/staa236pmid: N/A

ABSTRACT Present-day multiwavelength deep imaging surveys allow to characterize the outskirts of galaxies with unprecedented precision. Taking advantage of this situation, we define a new physically motivated measurement of size for galaxies based on the expected location of the gas density threshold for star formation. Employing both theoretical and observational arguments, we use the stellar mass density contour at 1 M⊙ pc−2 as a proxy for this density threshold for star formation. This choice makes our size definition operative. With this new size measure, the intrinsic scatter of the global stellar mass (M⋆)–size relation (explored over five orders of magnitude in stellar mass) decreases to ∼0.06 dex. This value is 2.5 times smaller than the scatter measured using the effective radius (∼0.15 dex) and between 1.5 and 1.8 times smaller than those using other traditional size indicators such as R23.5, i (∼0.09 dex), the Holmberg radius RH (∼0.09 dex), and the half-mass radius |$R_{\rm e,M_{\star }}$| (∼0.11 dex). Moreover, galaxies with 107 M⊙ < M⋆ < 1011 M⊙ increase monotonically in size following a power law with a slope very close to 1/3, equivalent to an average stellar mass 3D density of ∼4.5 × 10−3 M⊙ pc−3 for galaxies within this mass range. Galaxies with M⋆ > 1011 M⊙ show a different slope with stellar mass, which is suggestive of a larger gas density threshold for star formation at the epoch when their star formation peaks. methods: data analysis, methods: observational, techniques: photometric, galaxies: formation, galaxies: fundamental parameters, galaxies: photometry00 1 INTRODUCTION The sizes of galaxies play a pivotal role in our understanding of how they form and evolve. While the size of an everyday object is quite an intuitive concept, in the case of galaxies where there are no clear edges, measuring their extent is a non-trivial task. The absence of a clear border leads to two different ways of measuring the size of galaxies in the astronomical literature. The first and today’s most popular approach is identifying the size of a galaxy as the radial distance containing half of its light (i.e. its effective radius Re). A second and fairly common approach to indicate galaxy size is the location of a fixed surface brightness isophote. The effective radius has been used to characterize the size of galaxies since at least the publication by de Vaucouleurs (1948). Obviously, using half of the light of a galaxy to indicate its size is an arbitrary definition. Other fractions of light could be and in fact have been used as well for this task, for example, the radial distance containing 90 per cent of the light of the galaxy (R90) or the Petrosian and Kron radii (Petrosian 1976; Kron 1980, for a review on how the Petrosian and Kron radii relate to the commonly used surface brightness distribution provided by the Sérsic 1968 model and how such size definitions are affected by the depth of images see Graham & Driver 2005). One of the reasons for the popularity of Re is its robustness against many observational issues. In particular, as the surface brightness profiles of the vast majority of galaxies decline very rapidly (with a steepness equal to or larger than an exponential), the effective radius is barely affected by the depth of the images (Trujillo, Graham & Caon 2001). This robustness makes Re quite appealing as a measurement for galaxy size as different authors using different data sets can reach an agreement on the size. However, despite its undeniable value, Re is incapable of describing the global (luminous) size of galaxies (see an in-depth discussion in Graham 2019). This limits our use of Re as a direct measurement of galaxy size because Re measures light concentration and strongly depends on the shape of the light profile. Consider, for example, two disc galaxies with similar appearance but with bulges of very different brightness. The global Re of the galaxy with a prominent bulge will be significantly smaller than that of the one with a faint bulge. For this reason, and as we will show in this work, galaxies with the same extension can have very different effective radii. This is not a minor issue and has serious consequences when one wants to address or infer the nature of galaxies (Chamba, Trujillo & Knapen 2020). In addition, the Re of a galaxy can vary significantly with wavelength (see e.g. Kennedy et al. 2015). The second approach for measuring galaxy size is based on the radial location of a given isophote. The two most common size definitions are R25 (also known as the de Vaucouleurs radius) based on the radial location of the isophote at μB = 25 mag arcsec−2 and the Holmberg radius (RH) defined as the radial distance of the isophote at μB = 26.5 mag arcsec−2 (Holmberg 1958). R25 was popularized in the famous Second Reference Catalogue of Bright Galaxies by de Vaucouleurs, de Vaucouleurs & Corwin (1976). The authors of the catalogue refer to Redman (1936) as the first to propose R25 and to Liller (1960) as the first to adopt it. These two surface brightness values correspond roughly to 10 per cent and 3 per cent (respectively) of the brightness of the (darkest) night sky in the B-band in ground-based observatories. R25 and RH were motivated by the typical depth of optical images 60 yr ago, and created to measure the maximum extension of galaxies visible at that time (de Vaucouleurs et al. 1976). In this sense, measuring galaxy size using such a definition was not motivated by any particular physical reason and both R25 and RH simply reflect the technological limitation in the 1960s. Such isophotal size definitions are not limited to the optical bands only. For example, Muñoz-Mateos et al. (2015) characterized the global extensions of the galaxies in the infrared Spitzer Survey of Stellar Structure in Galaxies (S4G) survey (Sheth et al. 2010) using μ3.6 = 25.5 mag arcsec−2 as a size indicator. In the context of exploring the scaling relations between size, luminosity, and velocity of late-type galaxies, Saintonge & Spekkens (2011) and Hall et al. (2012) found that the use of R23.5, i (i.e. the radial location of the isophote μi = 23.5 mag arcsec−2) yields the smallest scatter in the size–luminosity relation. In contrast to Re and the isophotal size measures, there has also been some effort to characterize galaxy size using physically motivated parameters. An example of such a size parameter is the exponential scale length rd which is connected with the angular momentum of dynamically stable discs (see e.g. Mo, Mao & White 1998; Mo, van den Bosch & White 2010). However, in practice, due to the complexity of galactic discs (which include bars, spiral arms, etc) the use of rd has been shown to be complicated to reproduce by different authors. In fact, for the same galaxies, rd has been measured with a scatter of ∼25 per cent (see e.g. Knapen & van der Kruit 1991; Möllenhoff 2004). All the above size measures were introduced using relatively shallow imaging surveys. More recently, however, a revolution in the limiting depth of new astronomical imaging surveys has happened. As we will propose in this paper, image depth is no longer the limitation it once was to find a more representative and physically motivated definition for galaxy size. While the most commonly used astronomical survey, the Sloan Digital Sky Survey (SDSS; Abazajian et al. 2003), reaches a comparable depth that obtained in photographic plates (i.e. ∼26.5 mag arcsec−2 in the g-band, which is equivalent to a 3σ fluctuation with respect to the background of the image measured in an area of 10 × 10 arcsec2; Kniazev et al. 2004; Pohlen & Trujillo 2006), surveys conducted a decade later (i.e. Martínez-Delgado et al. 2010; Ferrarese et al. 2012; Merritt, van Dokkum & Abraham 2014; Capaccioli et al. 2015; Duc et al. 2015; Koda et al. 2015; Fliri & Trujillo 2016; Mihos et al. 2017) are regularly observing 2–3 mag deeper than SDSS. The current observational limit taken from ground-based telescopes is 31.5 mag arcsec−2 in the r-band, (equivalent to a 3σ fluctuation with respect to the background of the image in an area of 10 × 10 arcsec2; Trujillo & Fliri 2016) and a similar depth is expected to be achieved with ultradeep surveys that are currently in operation such as the Hyper Suprime Cam Survey (Aihara et al. 2018) and the future Large Synoptic Survey Telescope (LSST; Ivezic et al. 2008) survey. Going beyond this depth has been only possible with ultradeep imaging taken from space (see e.g. Borlaff et al. 2019). In this paper we propose a physically motivated definition to measure the size of galaxies. We suggest using the location of the gas density threshold for star formation in galaxies as a natural size indicator, where by natural we mean a size indicator that is connected with the intuitive concept of an edge. In other words, a size indicator that can be linked to a sharp contrast or change in the properties of the objects we explore. In practical terms, we will show that using the radial location of the contour at a stellar mass density of 1 M⊙ pc−2 (R1) corresponds roughly to the location of the gas density threshold for star formation. This definition is innately linked to the separation of the majority of stars that were born in-situ from stars that were mostly accreted throughout a galaxy’s history, potentially extending its use to define the stellar halo (N. Chamba et al. in preparation). In addition, as we will show below, R1 provides a more direct association to what an observer recognizes as the total extension of a galaxy than Re. While measuring R1 would have been difficult using past imaging surveys due to the required level (μr>26 mag arcsec−2) to identify isophotes with a low mass density around 1 M⊙ pc−2, we will see that current surveys are able to reach such depth without much difficulty. Finally, using a physically motivated definition for measuring the size of galaxies is not just another way of measuring the extensions of these objects such as RH, R25, or their variants. In fact, the use of R1 substantially modifies the scaling relations where galaxy size is an important parameter. This is particularly the case compared to Re. We will show that the use of R1 significantly decreases the scatter of the stellar mass–size relation by a factor of 2.5. Moreover, using R1, galaxies with stellar masses from 107 to 1011 M⊙ share the same stellar mass–size trend. The overall decrease in scatter essentially tightens the observed correlation between galaxy size and stellar mass, thus allowing us to gain insight about the size of an object if its stellar mass is known or vice versa. We will discuss whether these findings indicate a more fundamental meaning of the new size estimator R1 compared to the more arbitrary effective radius. We will also explore how R1 compares with other size indicators such as the radius enclosing half of the stellar mass (⁠|$R_{\rm e,M_{\star }}$|⁠), the Holmberg radius, and R23.5, i. This paper is structured as follows. In Section 2, we motivate the new size definition based on the location of the gas density threshold for star formation in galaxies. In Sections 3 and 4, we describe the data used and the selection of targets. The methodology is described in Section 5 and our results presented in Section 6. Section 7 discusses the results obtained and they are summarized in Section 8. Through the paper we assume a standard ΛCDM cosmology with Ωm = 0.3, |$\Omega _\Lambda$|= 0.7, and H0 = 70 km s−1 Mpc−1. 2 TOWARDS A PHYSICALLY MOTIVATED DEFINITION FOR THE SIZE OF GALAXIES When defining a new way to measure the size of galaxies, it is important to select a physical criterion intimately linked to the way galaxies increase in extension. Galaxies are expected to grow both in stellar mass and size by two different phenomena. The first is based on the transformation of gas into stars and the second is due to the accretion of new stars by merging and tidal interactions with other galaxies. While the merging process is stochastic and difficult to model, the transformation of gas into stars is strongly connected with the gas density of these systems. Above a given gas density threshold, gas is transformed into stars. Consequently, the position of these newborn stars is encircled by the location of such a critical gas density (Spitzer 1968; Quirk 1972; Fall & Efstathiou 1980; Kennicutt 1989). The radial location of this gas density threshold is thus suggestive of a natural way to define the size of galaxies. This is the expectation for the vast majority of galaxies, i.e. those whose main channel of stellar mass growth is the transformation of gas into stars. This includes almost all the dwarf galaxies and the majority of disc galaxies where growth by merging activity with other minor objects is expected to be small (see e.g. Toth & Ostriker 1992). The critical gas surface density for star formation is theoretically estimated to be Σc ∼3–10 M⊙ pc−2 (see e.g. Schaye 2004). If the efficiency of transforming gas into stars is not 100 per cent, a reasonable way of defining the size of a galaxy would be to locate a stellar mass isocontour at Σ⋆ ∼1–3 M⊙ pc−2. Such a range in surface density corresponds to an efficiency of gas-to-star transformation between ∼10 and 30 per cent. A way to test whether such a definition is reliable and a better proxy for the global luminous extension of galaxies compared to other size indicators such as Re is to explore the stellar mass density at which the edges of disc galaxies appear (i.e. the location of their truncations). To the best our knowledge, such work has not been conducted exhaustively yet. However, we have some examples where this has been done in detail. For instance, in UGC00180, a galaxy with similar properties to M31, the truncation is located at ∼2.5 M⊙ pc−2 (this is an upper limit as the projection effect has not been taken into account; Trujillo & Fliri 2016). For another two edge-on nearby galaxies (NGC 4565 and NGC 5907), the stellar mass density at their truncation radii is between 1 and 2 M⊙ pc−2 (Martínez-Lombilla, Trujillo & Knapen 2019). The fact that the fraction of stars beyond the truncation of NGC 4565 and NGC 5907 declines to 0.1–0.2 per cent reinforces the idea that such a stellar mass density is a good proxy for defining the luminous size of a galaxy. This number is compatible with a tiny fraction of stars that migrated from a region within the truncation radius to the outskirts. Unlike Re, an added value to the physically motivated size definition we are proposing is that the measurement corresponds to what the human eye identifies as the border of an object. In what follows, we propose the radial location of the gas density threshold for star formation as our size definition. Based on theoretical arguments and observational evidence of Milky Way-like galaxies, we suggest an operative way to estimate this density threshold for star formation by using a stellar mass density isocontour at Σ⋆ ∼1 M⊙ pc−2. We refer to the radial position of such an isomass contour as R1. Obviously, the choice of 1 M⊙ pc−2, instead of, for example, 0.5, 2, or 3 M⊙ pc−2, depends on the exact efficiency of star formation among different galaxies. Therefore, depending on the galaxies’ characteristics other values could perhaps better enclose the location of in-situ star formation. In this paper, we have preferred to adopt a relatively low efficiency in transforming gas into stars to be as inclusive as possible. In this regard, if anything, our measure for the size of galaxies could lead to slightly larger sizes where the efficiency of forming stars is higher than what we assume in this work (see Appendix A). On the contrary, if the star formation were very inefficient (as may well be the case for dwarf galaxies), our measure of size will be biased towards smaller sizes (see Chamba et al. 2020). In Appendix B, we discuss the use of alternative proxies to locate the radial location of the gas density threshold for star formation. In the previous paragraphs, we have motivated the use of the radial location of the stellar mass isocontour at Σ⋆ ∼ 1 M⊙ pc−2 as an operative method to locate the gas density threshold for star formation and thus characterize a physical size for galaxies. This size measure should work particularly well for galaxies whose main growth channel is the transformation of gas into stars. What would be the plight of such a definition for spheroidal galaxies? Those galaxies are thought to form a significant fraction of their stars in an early-on intense starburst and later on add new stars (mostly to their periphery) through merging with other (satellite) galaxies. Most of this secondary growth is produced by dry minor mergers (see e.g. Trujillo, Ferreras & de La Rosa 2011). As a matter of fact, we will show in this paper that the proposed size definition is useful to separate the core of spheroidal galaxies (predominantly formed by an intense star formation burst) from the material that is later on accreted by minor merging. A discussion of the limits of the new size measure is given in Appendix C. 3 IMAGING DATA: THE IAC STRIPE82 LEGACY PROJECT To estimate the location of the density threshold for star formation through the position of the 1 M⊙ pc−2 isomass contour, a survey with multiwavelength colour information is necessary. As we will explain in Section 5, the stellar mass density profiles of the objects can be estimated using different combinations of optical bands. In this work, we have used the IAC Stripe82 Legacy Project (hereafter IAC Stripe82) data set (Fliri & Trujillo 2016; Román & Trujillo 2018) as our deep imaging survey. This data set is a co-addition of the SDSS Stripe82 data (Frieman et al. 2008) with the goal of retaining the faintest surface brightness structures. The average seeing is 1 arcsec and the pixel scale is 0.396 arcsec. The total area of the survey is 275 deg2. To conduct this work we have used publicly available rectified images from this data set (http://research.iac.es/proyecto/stripe82/). In addition to the imaging data, the public release also includes photometric catalogues (Fliri & Trujillo 2018). The mean limiting surface brightness of the survey are μg = 29.1, μr = 28.6, and μi = 28.1 mag arcsec2 (equivalent to a 3σ fluctuation with respect to the background of the image in an area of 10 × 10 arcsec2). 4 TARGET SELECTION Having introduced a new size definition based on the radial location of the gas density threshold for star formation, we will now explore its use across a galaxy mass range as large as possible and how it performs for different morphological types. This is relevant as the star formation history could be very different depending on the galaxy’s characteristics. The cosmological volume covered by the Stripe82 data, together with its depth, thus allows us to collect a relatively large sample of galaxies with a wide range of stellar masses and morphologies. We have selected 1005 galaxies with z < 0.09 spanning five orders of magnitude in stellar mass (107  M⊙ < M⋆ < 1012  M⊙). This collection of galaxies extends from the dwarf galaxies regime up to giant spirals and ellipticals. All the galaxies with M⋆ > 109 M⊙ were selected from the Nair & Abraham (2010) catalogue, which includes a detailed visual classification of about 14 000 galaxies in the SDSS footprint. We have selected all the galaxies listed in this catalogue that are within the Stripe82 area (i.e. 1010 objects). Unfortunately, the Nair & Abraham (2010) catalogue lacks objects with stellar masses below 109 M⊙. For this reason, to increase our sample towards less massive galaxies (107 M⊙ < M⋆ < 3 × 109 M⊙), we retrieve them directly from the Maraston et al. (2013) catalogue. For the dwarf sample we lack morphological information. In order to have enough spatial resolution for our size analysis, we select only nearby dwarf galaxies, i.e. with 0.002 < z < 0.018. Within such a redshift range and the Stripe82 area we find 323 galaxies from the Maraston et al. (2013) catalogue. Of the total 1333 initially selected galaxies, 1005 were used for the final analysis, and 328 galaxies removed for multiple reasons. In some cases, the galaxies are located very close to a bright star or galaxy (152 objects), making the retrieval of their surface brightness profile unreliable. In other cases (103 objects) the galaxies are dramatically affected by dust contamination from Galactic cirri or several neighbouring objects that crowd the outskirts of the galaxy. Forty-eight galaxies that have an axial ratio smaller than 0.3 were also removed (See 5.2). Twenty-two galaxies for which the TType was classified as ‘unknown’ in the Nair & Abraham (2010) catalogue were discarded. And finally, three galaxies were removed as they appeared at the edge of the Stripe82 footprint and only part of the galaxy was visible in the images. The sample of massive galaxies was separated into two morphological groups depending on the TType classification by Nair & Abraham (2010). Those galaxies with TType>−1 (in total 464 objects) were called ‘spiral galaxies’ and contain morphologies from S0/a to Im, while those with TType<−1 (in total 279 objects) are dubbed ‘ellipticals’ and contain the morphological classes E0 to S0+. After the cleaning process, the remaining ‘dwarfs’ comprise 262 galaxies in our final sample. For completeness, in the table where we provide the properties of these galaxies, the TType for dwarf galaxies is indicated as −99 (see Section 6). 5 METHODOLOGY As explained in the Introduction, in this paper we explore the stellar mass–size relation of galaxies using our new size definition. In addition, we compare this mass–size relation with those resulting from the use of traditional size measurements such as the effective radius, the half-mass radius, and the Holmberg and μi = 23.5 mag arcsec−2 isophotal radii. In order to estimate the structural parameters necessary for this analysis, we need to conduct a number of steps that are explained in the following subsections. For all galaxies, we create images in the g and r filters of 600 kpc × 600 kpc in size in the rest-frame of each galaxy and centred on the object of interest. The pipeline developed for this work is written using PYTHON v. 3.6.5.1 5.1 Removal of scattered light from point sources and masking The scattered light from bright stars was modelled and subtracted using the procedure of Trujillo & Fliri (2016). This is a key step that is necessary to explore low-surface brightness features with confidence (see e.g. Uson, Boughn & Kuhn 1991; Slater, Harding & Mihos 2009). All stars brighter than 17 mag were identified using the G-band reported in the GAIA DR 1 catalogue (Gaia Collaboration 2016). To produce the scattered light field, we use the extended (radial size of ∼8 arcmin) point spread function (PSF) models in all the SDSS bands created by Infante-Sainz, Trujillo & Román (2020). The pipeline to remove the scattered light from point sources in the IAC Stripe82 fields will be fully described in a future publication and applied to the full IAC Stripe82 survey (N. Chamba et al. in preparation). Once the scattered light is removed from the images, it is necessary to mask all remaining sources that are affecting the light distribution of the galaxy we are exploring. To conduct this task, we used a python implementation of MTOBjects (Teeninga et al. 2016), a tree-based detection scheme which is robust against false positives, especially important for the identification of extended low signal-to-noise structures in deep imaging (C. Haigh et al. submitted). For this work, the algorithm parameter move_up = 0.3 and the α parameter for statistical testing was set to its default value. 5.2 The effect of inclination The surface brightness of galaxies (particularly those following a disc-like configuration) are strongly affected by the inclination of the object. The larger the inclination of a galaxy, the brighter it appears to an observer as the number of stars along the line-of-sight increases. As our proposed size definition requires a proper estimation of the flux in the outer regions of galaxies, we correct the brightness of galaxies by the effect of its inclination. This is not straightforward and has been investigated in-depth in multiple papers (see e.g. Holmberg 1958, 1975; Tully & Fouque 1985; Giovanelli et al. 1994). To estimate the correction we need to apply to the data, we build a 3D disc model assuming an exponential decline for the radial light distribution (de Vaucouleurs 1959; Freeman 1970) and a sech2 in the vertical |$z$| direction. This is expected for an isothermal population in a plane–parallel system (Spitzer 1942; Camm 1950). The luminosity distribution of the model is: $$\begin{eqnarray*} L(R,z) = L_0\exp \bigg (\frac{-R}{h}\bigg)\mathrm{ sech}^2\bigg (\frac{z}{2z_0}\bigg), \end{eqnarray*}$$ (1) where L0 is the central luminosity density, h is the scale length, and |$z$|0 is the scale height. The model was created using IMFIT (we used the model ExponentialDisk3D with n = 1; Erwin 2015). We probe three different models with |$z$|0/h = 0.08, 0.12, and 0.17. The ratio of these parameters covers the values measured for the thin disc of our own Milky Way and its uncertainties (Bland-Hawthorn & Gerhard 2016). This model is an idealized version of discs. Real discs are much more complex, containing clumps, dust, warps, etc. In addition, we have not considered possible corrections due to internal dust. Therefore, any dependence of the model on wavelength is neglected (for a detailed analysis of this issue see Kourkchi et al. 2019). Since we are mostly interested in the effect of inclination on the brightness of the intermediate-outer regions of galaxies, we calculate the difference in surface brightness (Δμ) at a given inclination i (μinc) compared to the face-on orientation (μface-on) at a radial distance of R = 5h (i.e. Δμ = μface-on–μinc). The difference Δμ was estimated at all inclinations along the semimajor axis of the model galaxy (see Fig. 1). Figure 1. Open in new tabDownload slide Difference in surface brightness Δμ = μface-on − μinc between the face-on orientation and a given inclination for a disc-like galaxy. The figure shows Δμ along the semimajor axis of a galaxy model at a radial distance R = 5h and is explored for three different disc thicknesses (shown in the legend) parametrized using |$z$|0/h. Figure 1. Open in new tabDownload slide Difference in surface brightness Δμ = μface-on − μinc between the face-on orientation and a given inclination for a disc-like galaxy. The figure shows Δμ along the semimajor axis of a galaxy model at a radial distance R = 5h and is explored for three different disc thicknesses (shown in the legend) parametrized using |$z$|0/h. Fig. 1 shows that the difference in brightness Δμ produced by different disc thicknesses (as parametrized by |$z$|0/h) is only noticeable at very large inclinations (i.e. i >70 deg). For this reason, we remove any galaxy with an axial ratio smaller than 0.3 from the sample (Section 4). To facilitate the reader with the application of this inclination correction, we provide in Table 1 the values of the coefficients of a polynomial fit to the different models shown in Fig. 1: $$\begin{eqnarray*} \Delta \mu = \sum _{j = 0}^{4}\alpha _j(b/a)^j \end{eqnarray*}$$ (2) with b/a = cos(i), the ratio of the semiminor to the semimajor axis of the isophote used to measured the inclination of galaxies. The polynomial fit we provide is very accurate, with an error in Δμ < 0.01 mag. In the next Section, we explain how to apply this correction to real data. In this work, we have used the correction corresponding to the ratio |$z$|0/h = 0.12. Table 1. Values of the polynomial coefficients to correct the surface brightness profiles of disc-like galaxies for the inclination effect. The coefficients have been calculated for three different disc thicknesses which are parametrized by the ratio |$z$|0/h. As we are interested in the effect of inclination on the intermediate and outer regions of galaxies, this correction has been estimated at a distance R = 5h (see the text for details). |$z$|0/h . α0 . α1 . α2 . α3 . α4 . 0.08 3.195 −10.396 17.584 −16.033 5.657 0.12 2.845 −7.833 10.792 −8.482 2.679 0.17 2.440 −5.273 4.577 −1.932 0.185 |$z$|0/h . α0 . α1 . α2 . α3 . α4 . 0.08 3.195 −10.396 17.584 −16.033 5.657 0.12 2.845 −7.833 10.792 −8.482 2.679 0.17 2.440 −5.273 4.577 −1.932 0.185 Open in new tab Table 1. Values of the polynomial coefficients to correct the surface brightness profiles of disc-like galaxies for the inclination effect. The coefficients have been calculated for three different disc thicknesses which are parametrized by the ratio |$z$|0/h. As we are interested in the effect of inclination on the intermediate and outer regions of galaxies, this correction has been estimated at a distance R = 5h (see the text for details). |$z$|0/h . α0 . α1 . α2 . α3 . α4 . 0.08 3.195 −10.396 17.584 −16.033 5.657 0.12 2.845 −7.833 10.792 −8.482 2.679 0.17 2.440 −5.273 4.577 −1.932 0.185 |$z$|0/h . α0 . α1 . α2 . α3 . α4 . 0.08 3.195 −10.396 17.584 −16.033 5.657 0.12 2.845 −7.833 10.792 −8.482 2.679 0.17 2.440 −5.273 4.577 −1.932 0.185 Open in new tab 5.3 Stellar mass density profiles After the removal of scattered light and masking the images, we extract the surface brightness profiles of the galaxies in the g and r bands to obtain their stellar mass density profiles. The surface brightness profiles are obtained using elliptical apertures with a fixed centre, axial ratio q, and position angle (PA). As a first guess for the centre of the galaxies, we use the RA and Dec information provided by the SDSS catalogues. To determine the axial ratio and PA, for each galaxy we use those pixels where the surface brightness is between 25 and 26 mag arcsec−2 in the g-band. The spatial distribution of these pixels were fit to an ellipse. The PA (in degrees) is the angle between the semimajor axis and the horizontal axis, measured in the counter-clockwise direction from the horizontal axis. The fit parameters (centre, axial ratio, and PA) of the ellipses were then visually checked to ensure the outermost parts of the galaxies were characterized properly. If not, they are corrected accordingly. Once fixed, surface brightness profiles of the galaxies are extracted by averaging their flux over annuli parametrized by the fit ellipse. These profiles are extracted up to a radial distance of 200 arcsec which is well beyond the visual extension of our galaxies. This is crucial in order to retrieve a sensible characterization of the outer part of galaxies, particularly when the criterion we are proposing in this work is based on the location of a low stellar mass density contour such as 1 M⊙ pc−2. Another important effect that must be accounted for when obtaining surface brightness profiles is defining the level of the background. Although the IAC Stripe82 images we have used are background subtracted, in some occasions the subtraction was not precise enough to be a reliable representation of the (local) surrounding background value of the galaxies (i.e. a slight under- or overestimation). For this reason, in order to have the most accurate background subtraction as possible, we followed the procedure developed by Pohlen & Trujillo (2006). The radial distance up to which the profiles have been extracted (i.e. 200 arcsec) is about two times the location of the isophote at 26.5 mag arcsec−2 (r-band) in the case of ellipticals and three times for the spiral and dwarf galaxies. This allows us to determine the background brightness in regions very close to the galaxies by identifying the asymptotic value in the number of counts around the object. We fit that value, subtract/add it to the images, and obtain the profile once more. We then correct the surface brightness profiles for Galactic extinction. The extinction corrections Ag and Ar are obtained from NED taking into account the location of each galaxy on the sky (https://ned.ipac.caltech.edu/forms/calculator.html). Following this, the effect of the inclination (see Section 5.2) is corrected for spiral and dwarf galaxies as follows. For each galaxy we measure its inclination based on the axial ratio we have determined before. The inclination correction, Δμ, is then directly applied to the derived surface brightness profiles. The same inclination correction is applied for both g and r profiles, therefore the colour radial profile of galaxies remains unaffected. Due to our limited photometric information, we do not attempt any correction for internal dust. The final step is to obtain the stellar mass-to-light ratio (M/L) profile. Once the M/L is known, the following equation (see e.g. Bakos, Trujillo & Pohlen 2008): $$\begin{eqnarray*} \log \Sigma _\star = \log (M/L)_\lambda - 0.4(\mu _\lambda -\mu _{\rm abs,\odot ,\lambda })+8.629 , \end{eqnarray*}$$ (3) where μabs, ⊙, λ is the absolute magnitude of the Sun at wavelength λ, is used to obtain the stellar mass density (in M⊙ pc−2) as a function of the surface brightness. To compute M/L, we followed the procedure described by Roediger & Courteau (2015). As a basis for our estimation, we used the g−r colour and the surface brightness in the g-band. We use the parameters provided by Roediger & Courteau (2015) that correspond to the Bruzual & Charlot (2003, BC03) models and a Chabrier IMF (Chabrier 2003). Despite the obvious advantage of decreasing the effect of galactic dust by using the i-band instead of the bluer g and r bands, we prefer to use the latter filters to estimate our size indicator for two main reasons. First, the sky brightness in the i-band is around two magnitudes brighter than in the g-band (see e.g. fig. 1 in Hall et al. 2012). This effect is not compensated by the brighter emission of the stellar populations towards the red (which is typically between g–i = 0.5 to 1 mag for spiral galaxies). As a result, the redder SDSS bands are noisier at a given surface brightness because all the SDSS bands have the same integration time. Secondly, as our size indicator is estimated through a colour combination, the effect of the PSF on the surface brightness profiles should not be very different from band to band. This applies for g and r, but in the case of the i-band, the SDSS PSF is significantly different for those in g and r, as can be seen in de Jong (2008, fig. 2) and Infante-Sainz et al. (2020, fig. 8). 5.4 Estimating the structural parameters of galaxies Once the stellar mass density profiles of the galaxies are created, it is straightforward to obtain the total stellar mass and the location of R1, the proxy for the location of the gas density threshold for star formation we have adopted as a measure of size in this work. This procedure has been performed for all the galaxies in our sample. In order to get a homogeneous determination of the total stellar mass of all our galaxies, we have integrated their mass density profiles. The integration takes into account the axial ratio of the galaxy and therefore assumes an elliptical symmetry for the distribution of light, from the central position of the object up to the radial location provided by the 29 mag arcsec−2 isophote (g-band). This estimate of the total stellar mass is a lower limit to the total mass of the object. However, the limiting isophote we are using is extremely faint, therefore the amount of stellar mass beyond such an isophote is expected to be very low (<3 per cent; Trujillo et al. 2001). We prefer to use this approach for estimating the total stellar mass instead of assuming a shape for the light distribution (i.e. exponential, de Vaucouleurs, etc) and extrapolating the stellar mass density profiles to infinity. In Appendix D, we compare our stellar mass determination with that of the Portsmouth Spectro-Photometric Stellar Mass computation (Maraston et al. 2013) and find that both mass determinations are similar. Finally, we determine the location of R1 directly using the stellar mass density profiles. Estimates of the half-mass radii are done using the cumulative mass density profiles. The effective radii of galaxies are determined from the g-band images (our deepest data).2 We use the growth curve in g to obtain the radial location within which half of the total light of the galaxy is contained. As we have done for the total stellar mass, the total light of the galaxy is measured as the light enclosed by the observed 29 mag arcsec−2 isophote (g-band). By definition, Re is not affected by Galactic extinction nor the inclination correction of the profiles except indirectly for the location of the 29 mag arcsec−2 isophote (g-band). In addition, we also estimate the Holmberg Radius (RH) for all our galaxies. Lacking the B-band in our survey, the location of the observed isophote at μg = 26 mag arcsec−2 was considered as a proxy for RH. Using the i-band profiles, we also determine the radial location of the isophote corresponding to 23.5 mag arcsec−2 (i.e. R23.5, i). Both isophotal sizes were estimated after correcting the profiles for Galactic extinction and cosmological dimming. All these structural parameters of our galaxies are provided in Table 2. Table 2. Characteristics of the galaxies used in this work. Unless explicitly stated otherwise, the quantities provided in this table have been derived in this work. This table includes the name of the galaxies (Abolfathi et al. 2018), their spatial location (Abolfathi et al. 2018), their axial ratio (q), and the position angle of the ellipses (PA) used to extract the surface brightness profiles (measured counter-clockwise starting from the horizontal axis), Galactic extinctions in the g, r, and i bands (from NED), spectroscopic redshift (⁠|$z$|⁠) (Abolfathi et al. 2018), morphological type (Nair & Abraham 2010; TType = −99 corresponds to dwarf galaxies), effective radius Re (measured in the g-band), the half-mass radius (⁠|$R_{\rm e,M_{\star }}$|⁠), the radius corresponding to the location of the isophote with μi = 23.5 mag arcsec−2 (i.e. R23.5, i), the Holmberg Radius (RH), the radial location R1 of the isomass contour at 1 M⊙ pc −2 and the stellar mass of galaxies (assuming a Chabrier IMF). The quantities are given showing only the significant figures up to which the values can be regarded reliable. The table shows those galaxies in Figs 2 and 3, in order of appearance. The complete table is available in the online version of the paper. JID . RA . Dec . q . PA . Ag . Ar . Ai . |$z$| . TType . Re . |$R_{\rm e,M_{\star }}$| . R23.5, i . RH . R1 . log(M⋆/M⊙) . . (deg) . (deg) . . (deg) . (mag) . (mag) . (mag) . . . (kpc) . (kpc) . (kpc) . (kpc) . (kpc) . . J010301.72-010639.46 15.75723 −1.11113 0.88 95.0 0.134 0.093 0.069 0.0175 −2 2.45 3.67 7.94 11.67 18.56 10.38 J005753.69-004852.90 14.47382 −0.81479 0.96 120.0 0.094 0.065 0.048 0.0419 −2 2.78 4.47 12.41 17.27 25.08 11.03 J000150.32+010155.24 0.45973 1.03172 0.49 179.0 0.085 0.059 0.044 0.0862 −5 24.99 29.13 46.05 71.98 148.12 11.82 J003934.82+005135.83 9.89529 0.85979 0.71 61.0 0.066 0.046 0.034 0.0146 5 8.64 7.70 16.39 20.87 23.01 10.37 J012223.77-005230.73 20.59913 −0.87523 0.44 42.0 0.169 0.117 0.087 0.0271 4 16.17 9.65 32.27 44.35 44.04 10.96 J021219.69-004841.46 33.08210 −0.81153 0.89 37.0 0.097 0.067 0.050 0.0408 0 9.18 5.90 23.68 32.61 37.17 11.24 J021808.12+004529.8 34.53385 0.75830 0.64 133.0 0.130 0.090 0.067 0.0092 −99 1.00 1.49 1.79 2.86 2.97 8.05 J233646.86+003724.2 354.19526 0.62341 0.86 88.0 0.113 0.078 0.058 0.0088 −99 2.13 2.38 2.58 4.44 5.44 8.65 J010607.19+004633.5 16.52997 0.77599 0.84 48.0 0.084 0.058 0.043 0.0174 −99 5.43 5.21 4.14 10.87 10.91 9.10 J235618.80-001820.17 359.07860 −0.30583 0.71 99.0 0.124 0.086 0.064 0.0241 −3 4.11 5.08 16.70 26.56 48.39 11.15 J024331.30+001824.49 40.88040 0.30676 0.36 108.0 0.124 0.086 0.064 0.0267 3 8.82 5.72 22.89 30.75 29.43 10.57 JID . RA . Dec . q . PA . Ag . Ar . Ai . |$z$| . TType . Re . |$R_{\rm e,M_{\star }}$| . R23.5, i . RH . R1 . log(M⋆/M⊙) . . (deg) . (deg) . . (deg) . (mag) . (mag) . (mag) . . . (kpc) . (kpc) . (kpc) . (kpc) . (kpc) . . J010301.72-010639.46 15.75723 −1.11113 0.88 95.0 0.134 0.093 0.069 0.0175 −2 2.45 3.67 7.94 11.67 18.56 10.38 J005753.69-004852.90 14.47382 −0.81479 0.96 120.0 0.094 0.065 0.048 0.0419 −2 2.78 4.47 12.41 17.27 25.08 11.03 J000150.32+010155.24 0.45973 1.03172 0.49 179.0 0.085 0.059 0.044 0.0862 −5 24.99 29.13 46.05 71.98 148.12 11.82 J003934.82+005135.83 9.89529 0.85979 0.71 61.0 0.066 0.046 0.034 0.0146 5 8.64 7.70 16.39 20.87 23.01 10.37 J012223.77-005230.73 20.59913 −0.87523 0.44 42.0 0.169 0.117 0.087 0.0271 4 16.17 9.65 32.27 44.35 44.04 10.96 J021219.69-004841.46 33.08210 −0.81153 0.89 37.0 0.097 0.067 0.050 0.0408 0 9.18 5.90 23.68 32.61 37.17 11.24 J021808.12+004529.8 34.53385 0.75830 0.64 133.0 0.130 0.090 0.067 0.0092 −99 1.00 1.49 1.79 2.86 2.97 8.05 J233646.86+003724.2 354.19526 0.62341 0.86 88.0 0.113 0.078 0.058 0.0088 −99 2.13 2.38 2.58 4.44 5.44 8.65 J010607.19+004633.5 16.52997 0.77599 0.84 48.0 0.084 0.058 0.043 0.0174 −99 5.43 5.21 4.14 10.87 10.91 9.10 J235618.80-001820.17 359.07860 −0.30583 0.71 99.0 0.124 0.086 0.064 0.0241 −3 4.11 5.08 16.70 26.56 48.39 11.15 J024331.30+001824.49 40.88040 0.30676 0.36 108.0 0.124 0.086 0.064 0.0267 3 8.82 5.72 22.89 30.75 29.43 10.57 Open in new tab Table 2. Characteristics of the galaxies used in this work. Unless explicitly stated otherwise, the quantities provided in this table have been derived in this work. This table includes the name of the galaxies (Abolfathi et al. 2018), their spatial location (Abolfathi et al. 2018), their axial ratio (q), and the position angle of the ellipses (PA) used to extract the surface brightness profiles (measured counter-clockwise starting from the horizontal axis), Galactic extinctions in the g, r, and i bands (from NED), spectroscopic redshift (⁠|$z$|⁠) (Abolfathi et al. 2018), morphological type (Nair & Abraham 2010; TType = −99 corresponds to dwarf galaxies), effective radius Re (measured in the g-band), the half-mass radius (⁠|$R_{\rm e,M_{\star }}$|⁠), the radius corresponding to the location of the isophote with μi = 23.5 mag arcsec−2 (i.e. R23.5, i), the Holmberg Radius (RH), the radial location R1 of the isomass contour at 1 M⊙ pc −2 and the stellar mass of galaxies (assuming a Chabrier IMF). The quantities are given showing only the significant figures up to which the values can be regarded reliable. The table shows those galaxies in Figs 2 and 3, in order of appearance. The complete table is available in the online version of the paper. JID . RA . Dec . q . PA . Ag . Ar . Ai . |$z$| . TType . Re . |$R_{\rm e,M_{\star }}$| . R23.5, i . RH . R1 . log(M⋆/M⊙) . . (deg) . (deg) . . (deg) . (mag) . (mag) . (mag) . . . (kpc) . (kpc) . (kpc) . (kpc) . (kpc) . . J010301.72-010639.46 15.75723 −1.11113 0.88 95.0 0.134 0.093 0.069 0.0175 −2 2.45 3.67 7.94 11.67 18.56 10.38 J005753.69-004852.90 14.47382 −0.81479 0.96 120.0 0.094 0.065 0.048 0.0419 −2 2.78 4.47 12.41 17.27 25.08 11.03 J000150.32+010155.24 0.45973 1.03172 0.49 179.0 0.085 0.059 0.044 0.0862 −5 24.99 29.13 46.05 71.98 148.12 11.82 J003934.82+005135.83 9.89529 0.85979 0.71 61.0 0.066 0.046 0.034 0.0146 5 8.64 7.70 16.39 20.87 23.01 10.37 J012223.77-005230.73 20.59913 −0.87523 0.44 42.0 0.169 0.117 0.087 0.0271 4 16.17 9.65 32.27 44.35 44.04 10.96 J021219.69-004841.46 33.08210 −0.81153 0.89 37.0 0.097 0.067 0.050 0.0408 0 9.18 5.90 23.68 32.61 37.17 11.24 J021808.12+004529.8 34.53385 0.75830 0.64 133.0 0.130 0.090 0.067 0.0092 −99 1.00 1.49 1.79 2.86 2.97 8.05 J233646.86+003724.2 354.19526 0.62341 0.86 88.0 0.113 0.078 0.058 0.0088 −99 2.13 2.38 2.58 4.44 5.44 8.65 J010607.19+004633.5 16.52997 0.77599 0.84 48.0 0.084 0.058 0.043 0.0174 −99 5.43 5.21 4.14 10.87 10.91 9.10 J235618.80-001820.17 359.07860 −0.30583 0.71 99.0 0.124 0.086 0.064 0.0241 −3 4.11 5.08 16.70 26.56 48.39 11.15 J024331.30+001824.49 40.88040 0.30676 0.36 108.0 0.124 0.086 0.064 0.0267 3 8.82 5.72 22.89 30.75 29.43 10.57 JID . RA . Dec . q . PA . Ag . Ar . Ai . |$z$| . TType . Re . |$R_{\rm e,M_{\star }}$| . R23.5, i . RH . R1 . log(M⋆/M⊙) . . (deg) . (deg) . . (deg) . (mag) . (mag) . (mag) . . . (kpc) . (kpc) . (kpc) . (kpc) . (kpc) . . J010301.72-010639.46 15.75723 −1.11113 0.88 95.0 0.134 0.093 0.069 0.0175 −2 2.45 3.67 7.94 11.67 18.56 10.38 J005753.69-004852.90 14.47382 −0.81479 0.96 120.0 0.094 0.065 0.048 0.0419 −2 2.78 4.47 12.41 17.27 25.08 11.03 J000150.32+010155.24 0.45973 1.03172 0.49 179.0 0.085 0.059 0.044 0.0862 −5 24.99 29.13 46.05 71.98 148.12 11.82 J003934.82+005135.83 9.89529 0.85979 0.71 61.0 0.066 0.046 0.034 0.0146 5 8.64 7.70 16.39 20.87 23.01 10.37 J012223.77-005230.73 20.59913 −0.87523 0.44 42.0 0.169 0.117 0.087 0.0271 4 16.17 9.65 32.27 44.35 44.04 10.96 J021219.69-004841.46 33.08210 −0.81153 0.89 37.0 0.097 0.067 0.050 0.0408 0 9.18 5.90 23.68 32.61 37.17 11.24 J021808.12+004529.8 34.53385 0.75830 0.64 133.0 0.130 0.090 0.067 0.0092 −99 1.00 1.49 1.79 2.86 2.97 8.05 J233646.86+003724.2 354.19526 0.62341 0.86 88.0 0.113 0.078 0.058 0.0088 −99 2.13 2.38 2.58 4.44 5.44 8.65 J010607.19+004633.5 16.52997 0.77599 0.84 48.0 0.084 0.058 0.043 0.0174 −99 5.43 5.21 4.14 10.87 10.91 9.10 J235618.80-001820.17 359.07860 −0.30583 0.71 99.0 0.124 0.086 0.064 0.0241 −3 4.11 5.08 16.70 26.56 48.39 11.15 J024331.30+001824.49 40.88040 0.30676 0.36 108.0 0.124 0.086 0.064 0.0267 3 8.82 5.72 22.89 30.75 29.43 10.57 Open in new tab 6 RESULTS Fig. 2 shows a few representative galaxies in our sample to illustrate the difference between the location of their R1 and Re contours. The size based on the location of the gas density threshold for star formation much better represents the intuitive concept of the size of galaxies, such as its edge or boundary compared to Re. Expanding on this point, Fig. 3 shows the location of R1 and Re for two galaxies with clear signatures of on-going stellar accretion. In these examples, the location of R1 may serve as a marker to separate the stellar material which is in the form of streams (formed ex-situ) from those stars which are located in the bulk (in-situ) of the main galaxy. An in-depth analysis of the use of R1 (and its variants) for this purpose will be presented in a future publication (N. Chamba et al. in preparation). Figure 2. Open in new tabDownload slide Collection of galaxy images showing the location of their Re and R1, the isomass contour at 1 M⊙ pc−2. The top row shows galaxies which have been classified as ellipticals (E0-S0+), the middle row shows spiral (S0/a-Sm) galaxies, and the lower row shows dwarf galaxies. The galaxies are displayed with increasing stellar mass from left to right. This figure clearly illustrates how the proxy for the gas density threshold for star formation (R1) nicely encloses the bulk of the stellar mass of galaxies. The coloured regions of the images are the IAC Stripe82 g, r, and i band composite, while the black and white background is the sum of the three bands. The background level of these images is ∼29.1 mag arcsec−2 (3σ 10 × 10 arcsec2; r-band). Table 2 lists details of the galaxies shown. Figure 2. Open in new tabDownload slide Collection of galaxy images showing the location of their Re and R1, the isomass contour at 1 M⊙ pc−2. The top row shows galaxies which have been classified as ellipticals (E0-S0+), the middle row shows spiral (S0/a-Sm) galaxies, and the lower row shows dwarf galaxies. The galaxies are displayed with increasing stellar mass from left to right. This figure clearly illustrates how the proxy for the gas density threshold for star formation (R1) nicely encloses the bulk of the stellar mass of galaxies. The coloured regions of the images are the IAC Stripe82 g, r, and i band composite, while the black and white background is the sum of the three bands. The background level of these images is ∼29.1 mag arcsec−2 (3σ 10 × 10 arcsec2; r-band). Table 2 lists details of the galaxies shown. Figure 3. Open in new tabDownload slide Separating in-situ star formation from ex-situ stellar accretion using our size measure (see N. Chamba et al. in preparation for further details). The two images show how the isomass contour at 1 M⊙ pc−2 (i.e. R1 or the proxy for the location of the gas density threshold for star formation) nicely divides the structure of galaxies into two different parts: the inner region where the bulk of the stars is contained and where in-situ star formation is or has been taking place and the external region where streams of on-going accretion are clearly visible. The left-hand panel shows a massive elliptical and the right-hand panel corresponds to a spiral galaxy with similar stellar mass to the Milky Way. See Table 2 for details on the galaxies shown. Figure 3. Open in new tabDownload slide Separating in-situ star formation from ex-situ stellar accretion using our size measure (see N. Chamba et al. in preparation for further details). The two images show how the isomass contour at 1 M⊙ pc−2 (i.e. R1 or the proxy for the location of the gas density threshold for star formation) nicely divides the structure of galaxies into two different parts: the inner region where the bulk of the stars is contained and where in-situ star formation is or has been taking place and the external region where streams of on-going accretion are clearly visible. The left-hand panel shows a massive elliptical and the right-hand panel corresponds to a spiral galaxy with similar stellar mass to the Milky Way. See Table 2 for details on the galaxies shown. 6.1 The properties of the R1–mass relation The main result of this paper is shown in Fig. 4: the mass–size relation spanning over five orders of magnitude in stellar mass (107–1012 M⊙). The figure illustrates how the mass–size relation changes when using R1 instead of Re as a size measurement of galaxies. To extract both the slope and dispersion of the relations, we used a Huber Regressor (Huber 1964), which is a linear regression model that is robust to outliers. We list a number of enlightening results: R1 is a factor of 5 to 10 larger than Re in all galaxies. The observed scatter of the stellar mass–size relation is significantly lower by a factor of ∼2 (from |$\sigma _{\rm R_e}$|  ∼ 0.17 dex to |$\sigma _{R_1}$| ∼ 0.09 dex) compared to the scatter using Re as a size indicator. As we will show in the next section, once the observational and methodological uncertainties are accounted for, the scatter of the stellar mass–size relation drops even more to a tiny 0.06 dex (i.e. a factor of ∼2.5 smaller than the intrinsic scatter using Re). The observed global scatter of the R1–mass size relation is also lower than the observed one found using other popular size estimators (⁠|$\sigma _{R_{e,M_{\star }}}$| ∼ 0.12 dex, |$\sigma _{R_H}$| ∼ 0.11 dex, and |$\sigma _{R_{23.5,i}}$| ∼ 0.11 dex; see Table 3). The average 2D stellar density (as measured within R1) changes from ∼10 M⊙ pc−2 for the less massive galaxies to ∼100 M⊙ pc−2 for the most massive spiral galaxies. Above 1011 M⊙, the average 2D stellar density of the galaxies decreases again. From 107 to 1011 M⊙ all galaxies are located on the same mass–size relation following a power law, |$R_{\rm 1}\propto M_\star \, ^{\beta }$|⁠, with β = 0.35 ± 0.01. This value is compatible with the one found by Hall et al. (2012) who compared the disc scale lengths of spiral galaxies with their luminosities and found β = 0.377 ± 0.007. Interestingly, β ∼1/3 would correspond to almost the same 3D stellar mass density (∼4.5 × 10−3 M⊙ pc−3) if all the stars were distributed in a sphere of radius R1. Above 1011 M⊙, the slope of the relation rises to β = 0.58 ± 0.02. This likely indicates that the most massive galaxies have formed or gained their stars very differently compared to galaxies with lower masses. Figure 4. Open in new tabDownload slide Stellar mass–size relation for the galaxies in our sample. Upper panel: The observed R1–mass and Re–mass relations, where Re has been measured using the g-band. The scatter of the relation using R1 is significantly smaller compared to that with Re. Lower panel: The same R1–mass relation after splitting our sample into three categories: ellipticals (E0-S0+), spirals (S0/a-Im), and dwarfs as labelled in the legend. Spiral and dwarf galaxies follow the same trend, while massive ellipticals with M⋆>1011 M⊙ show a tilt with respect to less massive galaxies. The grey dashed lines correspond to locations in the plane with constant (projected) stellar mass density. Figure 4. Open in new tabDownload slide Stellar mass–size relation for the galaxies in our sample. Upper panel: The observed R1–mass and Re–mass relations, where Re has been measured using the g-band. The scatter of the relation using R1 is significantly smaller compared to that with Re. Lower panel: The same R1–mass relation after splitting our sample into three categories: ellipticals (E0-S0+), spirals (S0/a-Im), and dwarfs as labelled in the legend. Spiral and dwarf galaxies follow the same trend, while massive ellipticals with M⋆>1011 M⊙ show a tilt with respect to less massive galaxies. The grey dashed lines correspond to locations in the plane with constant (projected) stellar mass density. In Table 3 we show the best-fitting parameters to a power law of the form R ∝ M|$_{\star }^{\beta }$| using all the galaxies in the size–stellar mass relation as well as separate fits using each subsample (i.e. dwarfs, S0/a-Sm, and E0-S0+). We have performed our analysis using the new size indicator R1 as well as other popular size indicators: Re, |$R_{\rm e,M_{\star }}$|⁠, RH, and R23.5, i. The uncertainties in the best-fitting slope (β) and dispersion of the observed relations (⁠|$\sigma _{\rm R_{obs}}$|⁠) are computed using a simple bootstrap method. One-third of the measured points on the relation were randomly selected and fit at each iteration, for 1000 iterations. The spread in the distribution of the fits from this exercise is what is reported as the uncertainty in β and σ. As we mentioned above, the observed global scatter of the R1–mass relation is significantly smaller than the one observed using Re and lower than the observed scatter with all other size estimators, i.e. |$R_{\rm e,M_{\star }}$|⁠, RH, and R23.5, i. The values of the scatter in the size–mass relations are, however, affected by uncertainties in estimating the stellar mass and the background around the galaxies. In order to quantify how these uncertainties affect the observed scatter of the different relations and therefore compare the intrinsic scatter of the relation using R1 with the other size indicators, we have conducted a number of tests which we describe in the next subsection. Table 3. Best-fitting power-law slope β and the observed dispersion of different size–mass relations. We provide the values for the entire sample as well as for the different families of galaxies. The third column corresponds to the Pearson r correlation coefficient. We also calculate the contribution to the observed scatter produced by the uncertainty in the background of the images (⁠|$\sigma _{\rm R_{back}}$|⁠) and in our stellar mass estimation (⁠|$\sigma _{\rm R_{mass}}$|⁠). The last column shows the intrinsic scatter of the size–mass relation (⁠|$\sigma _{\rm R_{int}}$|⁠) after accounting for the uncertainty in the background and the stellar mass of the objects. Galaxy type . β . |$\sigma _{\rm R_{obs}}$| . r . |$\sigma _{\rm R_{back}}$| . |$\sigma _{\rm R_{mass}}$| . |$\sigma _{\rm R_{int}}$| . R1–stellar mass All 0.365 ± 0.005 0.089 ± 0.005 0.971 0.045 ± 0.003 0.047 ± 0.003 0.061 ± 0.005 E0-S0+ 0.580 ± 0.022 0.090 ± 0.006 0.936 0.060 ± 0.004 0.054 ± 0.004 0.040 ± 0.006 S0/a-Sm 0.332 ± 0.014 0.089 ± 0.005 0.881 0.045 ± 0.003 0.035 ± 0.002 0.068 ± 0.005 Dwarfs 0.362 ± 0.016 0.088 ± 0.006 0.931 0.020 ± 0.003 0.056 ± 0.006 0.065 ± 0.006 Re–stellar mass All 0.247 ± 0.011 0.168 ± 0.009 0.811 ∼0.001 0.067 ± 0.005 0.154 ± 0.009 E0-S0+ 0.553 ± 0.032 0.108 ± 0.009 0.894 ∼0.001 0.092 ± 0.006 0.057 ± 0.009 S0/a-Sm 0.225 ± 0.026 0.162 ± 0.009 0.556 ∼0.001 0.047 ± 0.002 0.155 ± 0.009 Dwarfs 0.283 ± 0.040 0.221 ± 0.012 0.621 ∼0.001 0.064 ± 0.004 0.212 ± 0.012 |$R_{\rm e,M_{\star }}$|–stellar mass All 0.204 ± 0.006 0.117 ± 0.008 0.854 ∼0.001 0.052 ± 0.003 0.105 ± 0.008 E0-S0+ 0.509 ± 0.021 0.086 ± 0.007 0.918 ∼0.001 0.075 ± 0.006 0.042 ± 0.007 S0/a-Sm 0.196 ± 0.022 0.120 ± 0.009 0.595 ∼0.001 0.040 ± 0.002 0.113 ± 0.009 Dwarfs 0.175 ± 0.027 0.139 ± 0.008 0.638 ∼0.001 0.038 ± 0.003 0.134 ± 0.008 RH–stellar mass All 0.304 ± 0.007 0.109 ± 0.005 0.940 0.003 ± 0.001 0.056 ± 0.004 0.094 ± 0.005 E0-S0+ 0.478 ± 0.017 0.065 ± 0.005 0.950 0.004 ± 0.001 0.026 ± 0.003 0.059 ± 0.005 S0/a-Sm 0.266 ± 0.014 0.105 ± 0.005 0.783 0.003 ± 0.001 0.035 ± 0.004 0.099 ± 0.005 Dwarfs 0.307 ± 0.028 0.148 ± 0.008 0.790 0.002 ± 0.001 0.095 ± 0.014 0.113 ± 0.008 R23.5, i–stellar mass All 0.330 ± 0.006 0.106 ± 0.006 0.944 0.002 ± 0.001 0.062 ± 0.005 0.086 ± 0.005 E0-S0+ 0.443 ± 0.013 0.056 ± 0.005 0.956 0.002 ± 0.001 0.021 ± 0.002 0.052 ± 0.005 S0/a-Sm 0.285 ± 0.016 0.104 ± 0.006 0.804 0.003 ± 0.001 0.052 ± 0.005 0.090 ± 0.006 Dwarfs 0.354 ± 0.026 0.143 ± 0.010 0.836 0.001 ± 0.001 0.097 ± 0.012 0.105 ± 0.010 Galaxy type . β . |$\sigma _{\rm R_{obs}}$| . r . |$\sigma _{\rm R_{back}}$| . |$\sigma _{\rm R_{mass}}$| . |$\sigma _{\rm R_{int}}$| . R1–stellar mass All 0.365 ± 0.005 0.089 ± 0.005 0.971 0.045 ± 0.003 0.047 ± 0.003 0.061 ± 0.005 E0-S0+ 0.580 ± 0.022 0.090 ± 0.006 0.936 0.060 ± 0.004 0.054 ± 0.004 0.040 ± 0.006 S0/a-Sm 0.332 ± 0.014 0.089 ± 0.005 0.881 0.045 ± 0.003 0.035 ± 0.002 0.068 ± 0.005 Dwarfs 0.362 ± 0.016 0.088 ± 0.006 0.931 0.020 ± 0.003 0.056 ± 0.006 0.065 ± 0.006 Re–stellar mass All 0.247 ± 0.011 0.168 ± 0.009 0.811 ∼0.001 0.067 ± 0.005 0.154 ± 0.009 E0-S0+ 0.553 ± 0.032 0.108 ± 0.009 0.894 ∼0.001 0.092 ± 0.006 0.057 ± 0.009 S0/a-Sm 0.225 ± 0.026 0.162 ± 0.009 0.556 ∼0.001 0.047 ± 0.002 0.155 ± 0.009 Dwarfs 0.283 ± 0.040 0.221 ± 0.012 0.621 ∼0.001 0.064 ± 0.004 0.212 ± 0.012 |$R_{\rm e,M_{\star }}$|–stellar mass All 0.204 ± 0.006 0.117 ± 0.008 0.854 ∼0.001 0.052 ± 0.003 0.105 ± 0.008 E0-S0+ 0.509 ± 0.021 0.086 ± 0.007 0.918 ∼0.001 0.075 ± 0.006 0.042 ± 0.007 S0/a-Sm 0.196 ± 0.022 0.120 ± 0.009 0.595 ∼0.001 0.040 ± 0.002 0.113 ± 0.009 Dwarfs 0.175 ± 0.027 0.139 ± 0.008 0.638 ∼0.001 0.038 ± 0.003 0.134 ± 0.008 RH–stellar mass All 0.304 ± 0.007 0.109 ± 0.005 0.940 0.003 ± 0.001 0.056 ± 0.004 0.094 ± 0.005 E0-S0+ 0.478 ± 0.017 0.065 ± 0.005 0.950 0.004 ± 0.001 0.026 ± 0.003 0.059 ± 0.005 S0/a-Sm 0.266 ± 0.014 0.105 ± 0.005 0.783 0.003 ± 0.001 0.035 ± 0.004 0.099 ± 0.005 Dwarfs 0.307 ± 0.028 0.148 ± 0.008 0.790 0.002 ± 0.001 0.095 ± 0.014 0.113 ± 0.008 R23.5, i–stellar mass All 0.330 ± 0.006 0.106 ± 0.006 0.944 0.002 ± 0.001 0.062 ± 0.005 0.086 ± 0.005 E0-S0+ 0.443 ± 0.013 0.056 ± 0.005 0.956 0.002 ± 0.001 0.021 ± 0.002 0.052 ± 0.005 S0/a-Sm 0.285 ± 0.016 0.104 ± 0.006 0.804 0.003 ± 0.001 0.052 ± 0.005 0.090 ± 0.006 Dwarfs 0.354 ± 0.026 0.143 ± 0.010 0.836 0.001 ± 0.001 0.097 ± 0.012 0.105 ± 0.010 Open in new tab Table 3. Best-fitting power-law slope β and the observed dispersion of different size–mass relations. We provide the values for the entire sample as well as for the different families of galaxies. The third column corresponds to the Pearson r correlation coefficient. We also calculate the contribution to the observed scatter produced by the uncertainty in the background of the images (⁠|$\sigma _{\rm R_{back}}$|⁠) and in our stellar mass estimation (⁠|$\sigma _{\rm R_{mass}}$|⁠). The last column shows the intrinsic scatter of the size–mass relation (⁠|$\sigma _{\rm R_{int}}$|⁠) after accounting for the uncertainty in the background and the stellar mass of the objects. Galaxy type . β . |$\sigma _{\rm R_{obs}}$| . r . |$\sigma _{\rm R_{back}}$| . |$\sigma _{\rm R_{mass}}$| . |$\sigma _{\rm R_{int}}$| . R1–stellar mass All 0.365 ± 0.005 0.089 ± 0.005 0.971 0.045 ± 0.003 0.047 ± 0.003 0.061 ± 0.005 E0-S0+ 0.580 ± 0.022 0.090 ± 0.006 0.936 0.060 ± 0.004 0.054 ± 0.004 0.040 ± 0.006 S0/a-Sm 0.332 ± 0.014 0.089 ± 0.005 0.881 0.045 ± 0.003 0.035 ± 0.002 0.068 ± 0.005 Dwarfs 0.362 ± 0.016 0.088 ± 0.006 0.931 0.020 ± 0.003 0.056 ± 0.006 0.065 ± 0.006 Re–stellar mass All 0.247 ± 0.011 0.168 ± 0.009 0.811 ∼0.001 0.067 ± 0.005 0.154 ± 0.009 E0-S0+ 0.553 ± 0.032 0.108 ± 0.009 0.894 ∼0.001 0.092 ± 0.006 0.057 ± 0.009 S0/a-Sm 0.225 ± 0.026 0.162 ± 0.009 0.556 ∼0.001 0.047 ± 0.002 0.155 ± 0.009 Dwarfs 0.283 ± 0.040 0.221 ± 0.012 0.621 ∼0.001 0.064 ± 0.004 0.212 ± 0.012 |$R_{\rm e,M_{\star }}$|–stellar mass All 0.204 ± 0.006 0.117 ± 0.008 0.854 ∼0.001 0.052 ± 0.003 0.105 ± 0.008 E0-S0+ 0.509 ± 0.021 0.086 ± 0.007 0.918 ∼0.001 0.075 ± 0.006 0.042 ± 0.007 S0/a-Sm 0.196 ± 0.022 0.120 ± 0.009 0.595 ∼0.001 0.040 ± 0.002 0.113 ± 0.009 Dwarfs 0.175 ± 0.027 0.139 ± 0.008 0.638 ∼0.001 0.038 ± 0.003 0.134 ± 0.008 RH–stellar mass All 0.304 ± 0.007 0.109 ± 0.005 0.940 0.003 ± 0.001 0.056 ± 0.004 0.094 ± 0.005 E0-S0+ 0.478 ± 0.017 0.065 ± 0.005 0.950 0.004 ± 0.001 0.026 ± 0.003 0.059 ± 0.005 S0/a-Sm 0.266 ± 0.014 0.105 ± 0.005 0.783 0.003 ± 0.001 0.035 ± 0.004 0.099 ± 0.005 Dwarfs 0.307 ± 0.028 0.148 ± 0.008 0.790 0.002 ± 0.001 0.095 ± 0.014 0.113 ± 0.008 R23.5, i–stellar mass All 0.330 ± 0.006 0.106 ± 0.006 0.944 0.002 ± 0.001 0.062 ± 0.005 0.086 ± 0.005 E0-S0+ 0.443 ± 0.013 0.056 ± 0.005 0.956 0.002 ± 0.001 0.021 ± 0.002 0.052 ± 0.005 S0/a-Sm 0.285 ± 0.016 0.104 ± 0.006 0.804 0.003 ± 0.001 0.052 ± 0.005 0.090 ± 0.006 Dwarfs 0.354 ± 0.026 0.143 ± 0.010 0.836 0.001 ± 0.001 0.097 ± 0.012 0.105 ± 0.010 Galaxy type . β . |$\sigma _{\rm R_{obs}}$| . r . |$\sigma _{\rm R_{back}}$| . |$\sigma _{\rm R_{mass}}$| . |$\sigma _{\rm R_{int}}$| . R1–stellar mass All 0.365 ± 0.005 0.089 ± 0.005 0.971 0.045 ± 0.003 0.047 ± 0.003 0.061 ± 0.005 E0-S0+ 0.580 ± 0.022 0.090 ± 0.006 0.936 0.060 ± 0.004 0.054 ± 0.004 0.040 ± 0.006 S0/a-Sm 0.332 ± 0.014 0.089 ± 0.005 0.881 0.045 ± 0.003 0.035 ± 0.002 0.068 ± 0.005 Dwarfs 0.362 ± 0.016 0.088 ± 0.006 0.931 0.020 ± 0.003 0.056 ± 0.006 0.065 ± 0.006 Re–stellar mass All 0.247 ± 0.011 0.168 ± 0.009 0.811 ∼0.001 0.067 ± 0.005 0.154 ± 0.009 E0-S0+ 0.553 ± 0.032 0.108 ± 0.009 0.894 ∼0.001 0.092 ± 0.006 0.057 ± 0.009 S0/a-Sm 0.225 ± 0.026 0.162 ± 0.009 0.556 ∼0.001 0.047 ± 0.002 0.155 ± 0.009 Dwarfs 0.283 ± 0.040 0.221 ± 0.012 0.621 ∼0.001 0.064 ± 0.004 0.212 ± 0.012 |$R_{\rm e,M_{\star }}$|–stellar mass All 0.204 ± 0.006 0.117 ± 0.008 0.854 ∼0.001 0.052 ± 0.003 0.105 ± 0.008 E0-S0+ 0.509 ± 0.021 0.086 ± 0.007 0.918 ∼0.001 0.075 ± 0.006 0.042 ± 0.007 S0/a-Sm 0.196 ± 0.022 0.120 ± 0.009 0.595 ∼0.001 0.040 ± 0.002 0.113 ± 0.009 Dwarfs 0.175 ± 0.027 0.139 ± 0.008 0.638 ∼0.001 0.038 ± 0.003 0.134 ± 0.008 RH–stellar mass All 0.304 ± 0.007 0.109 ± 0.005 0.940 0.003 ± 0.001 0.056 ± 0.004 0.094 ± 0.005 E0-S0+ 0.478 ± 0.017 0.065 ± 0.005 0.950 0.004 ± 0.001 0.026 ± 0.003 0.059 ± 0.005 S0/a-Sm 0.266 ± 0.014 0.105 ± 0.005 0.783 0.003 ± 0.001 0.035 ± 0.004 0.099 ± 0.005 Dwarfs 0.307 ± 0.028 0.148 ± 0.008 0.790 0.002 ± 0.001 0.095 ± 0.014 0.113 ± 0.008 R23.5, i–stellar mass All 0.330 ± 0.006 0.106 ± 0.006 0.944 0.002 ± 0.001 0.062 ± 0.005 0.086 ± 0.005 E0-S0+ 0.443 ± 0.013 0.056 ± 0.005 0.956 0.002 ± 0.001 0.021 ± 0.002 0.052 ± 0.005 S0/a-Sm 0.285 ± 0.016 0.104 ± 0.006 0.804 0.003 ± 0.001 0.052 ± 0.005 0.090 ± 0.006 Dwarfs 0.354 ± 0.026 0.143 ± 0.010 0.836 0.001 ± 0.001 0.097 ± 0.012 0.105 ± 0.010 Open in new tab 6.2 The intrinsic scatter of the R1–mass relation There are two main sources of uncertainty which affect the observed scatter in the size–mass relations. The first is the accuracy in measuring the background level around the galaxies. For some galaxies (particularly the massive ellipticals or those with red stellar populations) the surface brightness at which R1 is measured is very faint (μg ∼28 mag arcsec−2) and therefore, a slight under- or oversubtraction of the background would bend the surface brightness profiles of these objects and move the location of R1. To quantify how this can affect the position of R1 and the rest of the size indicators, we have taken all our observed surface brightness profiles and randomly subtracted/added a number of counts compatible with the uncertainty in the background level around each galaxy. This variation of the background allows us to measure the variation in size for each galaxy which can then be used to estimate its contribution to the total observed dispersion in the size–mass plane. This contribution (⁠|$\sigma _{\rm R_{back}}$|⁠) is shown in Table 3. The background determination affects the size determination for massive ellipticals more than for spirals and/or small dwarfs. This is because the latter are mainly star-forming objects and therefore the surface brightness at which R1 is located is brighter (μg ∼ 26–27 mag arcsec−2). This explanation also applies for the isophotal sizes RH and R23.5, i. However, for Re and |$R_{\rm e,M_{\star }}$|⁠, the scatter due to the background correction is negligible. The other significant source of scatter in the size–mass plane is the uncertainty in measuring the total stellar mass of galaxies from the integrated stellar mass density profile of the objects. As explained in Section 5.4, we measure our total stellar mass by integrating the stellar mass density profiles. To quantify how the uncertainty in the total stellar mass affects our results, we have assumed the following uncertainties in measuring the stellar mass: δmass = 0.24 ± 0.01 dex (for the entire sample), δmass = 0.19 ± 0.01 dex (for the E0-S0 + subsample), δmass = 0.24 ± 0.01 dex (for the S0/a-Sm subsample) and δmass = 0.25 ± 0.03 dex (for the Dwarfs subsample). These values were computed by an analysis of the differences between the Portsmouth stellar masses of our galaxies (Maraston et al. 2013) and those we measured using the g–r colour profile (Roediger & Courteau 2015, see Appendix D for further details). To model the effect of the mass uncertainty (⁠|$\sigma _{\rm R_{mass}}$|⁠) on the scatter of the scaling relationship, all the observed stellar mass profiles were either scaled up or down in mass to place the galaxies on the best-fitting line through the observed stellar mass plane. This has been performed self-consistently, i.e. taking into account the change in the location of R1 due to the scaling of the profile. Once all the galaxies are located exactly on top of the best fitting stellar mass–size relation (i.e. with zero scatter), we randomly scale the stellar mass density profiles up or down again, this time by a quantity compatible with a Gaussian distribution whose standard deviation is given by the above δmass values. We repeat this procedure 1000 times and on each occasion we measure the scatter of the stellar mass–size plane produced by the uncertainty in measuring the stellar mass. We show an illustration of the scatter of the stellar mass–size relation caused by the uncertainty in stellar mass in Fig. D2. The scatter in the stellar mass–size plane generated by the uncertainty in mass is shown in Table 3. Interestingly, for R1, RH, and R23.5, i, we find that the dwarfs are the most affected by the uncertainty due to our mass determination. This is once again expected as the star formation activity of dwarf galaxies is, on average, more stochastic (Kauffmann 2014) and complicated to model than that of massive spirals and ellipticals. Therefore, a single colour is not a good proxy for the M/L ratio of dwarfs as it is in the case for more gentle star formation histories. Once the scatter produced by both the uncertainty in the background and the stellar mass determination have been characterized, we can calculate the intrinsic scatter of the stellar mass–size relations. To do this, we have taken the observed scatter and removed in quadrature the two scatters generated by the background level and stellar mass uncertainty. Obviously, the exact intrinsic scatter of the mass–size relation is difficult to measure as there is some ambiguity in choosing the uncertainty in stellar mass. Here we have used the above uncertainty values in the stellar mass motivated by what we find comparing the Portsmouth stellar masses (Maraston et al. 2013) with the ones we retrieve using the g–r colour (Roediger & Courteau 2015). We acknowledge that our intrinsic scatter values are an approximation, but do estimate that the intrinsic scatter of the R1–mass relation is about a factor of 1.5 smaller than the observed one (i.e. ∼0.06 dex), as a crude evaluation. This implies that the intrinsic stellar mass–R1 relation is, indeed, very tight. In future work, we will address this issue in much more detail. This will be possible as a result of deeper data and therefore a decrease in the uncertainty in measuring the image background level. In addition, we plan to explore the stellar mass–R1 relation using 3.6 |$\mu$|m images (S. Díaz-García et al, in preparation) from Spitzer, where the uncertainty in measuring the stellar mass is smaller. In particular, we will use the S4G survey (Sheth et al. 2010) analysis where images have been corrected by the contamination from young stars (Querejeta et al. 2015) and the depth is enough to reach 1 M⊙ pc−2 (Muñoz-Mateos et al. 2015). Finally, it is worth mentioning how the intrinsic scatter of the new mass–size relation compares with the intrinsic scatter of the other popular size–mass relations. In the case of Re and |$R_{\rm e,M_{\star }}$|⁠, the uncertainty produced by an incorrect background determination is almost negligible. We have found for these cases that |$\sigma _{R_{back}}\sim$|0.001 dex. This is because our images are very deep and therefore the effect of the uncertainty in our background estimation on the surface brightness profiles barely affects the location of Re and |$R_{\rm e,M_{\star }}$| which are found at relatively high surface brightness values. Therefore, we do not expect a large contribution to the observed scatter of these size–mass relations from incorrect background level measurements. In the case of the uncertainty in stellar mass, the first thing to note is given that Re (⁠|$R_{\rm e,M_{\star }}$|⁠) is defined as the location where half of the total light (stellar mass) is enclosed, its measurement is not affected by an incorrect mass determination of the object. This is because the only effect any uncertainty in mass could transfer to the shape of the profile is a scaling factor towards higher or lower stellar density. However, although the scatter in the size axis is negligible, the uncertainty in the mass axis will play a role in the total scatter of the size–mass plane. Nonetheless, there are two reasons why such an uncertainty will play a minor role in the observed scatter of these relations. First, the slope of the Re (⁠|$R_{\rm e,M_{\star }}$|⁠)–mass relation is rather flat in the region 109 to 1011 M⊙, therefore the contribution from an uncertainty in the mass to enlarge the scatter of the relationships in this region would be close to zero (for example, in spirals and Re, the observed scatter is |$\sigma _{\rm R_e}$| = 0.162 ± 0.009 while the intrinsic scatter is almost the same |$\sigma _{\rm R_{e,int}}$| = 0.155 ± 0.009). Secondly, as the observed scatters of the Re (⁠|$R_{\rm e,M_{\star }}$|⁠)–mass relations are already larger than in the case of R1 (⁠|$\sigma _{\rm R_{obs}}$|⁠), the contribution of a similar uncertainty in mass to the intrinsic scatter is very small. In the case of the Re–mass relation, as can be seen in Table 3, the global observed scatter is only reduced by 9 per cent after accounting for the mass uncertainty, giving an intrinsic scatter of |$\sigma _{\rm R_{e,int}}$| = 0.154 ± 0.009. Therefore, it is reasonable to compare the scatter of the observedRe–mass relation with the intrinsicR1–mass relation. As shown in Table 3, the decrease in scatter from R1 to Re ranges from a factor of 2.5 (comparing both intrinsic scatters) to 2.75 (comparing the intrinsic scatter using R1 with the observed scatter using Re). We illustrate in Fig. 5 how the R1–mass relation would be observed without the scatter produced by the background level and the stellar mass determination. Finally, for the isophotal sizes RH and R23.5, i, the main contributor to the observed scatter is also the uncertainty in measuring the global mass of galaxies. In these cases, the intrinsic scatter for the global size–mass relation decreases by 15–20 per cent compared to the observed values. Compared to the R1–mass relation, the intrinsic scatter of the global size–mass relations using RH and R23.5, i is a factor of 1.5 and 1.4 larger, respectively. In Section 7.1, we expand on these results by comparing the scatter of the size–mass relations as a function of galaxy morphology. Figure 5. Open in new tabDownload slide Similar to Fig. 4 (bottom panel), this figure shows the R1–stellar mass relation as it would be observed without any uncertainty in measuring the background level of the images and the stellar mass of galaxies. The intrinsic scatter of the relation (0.06 dex) is a factor of 2.5 smaller than the scatter of the Re–mass relation. Figure 5. Open in new tabDownload slide Similar to Fig. 4 (bottom panel), this figure shows the R1–stellar mass relation as it would be observed without any uncertainty in measuring the background level of the images and the stellar mass of galaxies. The intrinsic scatter of the relation (0.06 dex) is a factor of 2.5 smaller than the scatter of the Re–mass relation. 7 DISCUSSION The results of this paper show that the use of a physically motivated definition for the size of galaxies based on the location of the gas density threshold for star formation produces a global stellar mass–size relation with a very narrow intrinsic scatter (0.06 dex). In the following subsections, we compare the characteristics of the new size parameter R1 as well as the R1–mass relation with other popular size measurements. 7.1 R1 compared to other popular size definitions In this paper we have used R1 as a proxy for the location of the gas density threshold for star formation in galaxies. Nonetheless, the use of R1 as a size indicator is reminiscent of definitions based on the B-band isophote at 25 mag arcsec−2, at 26.5 mag arcsec−2 (i.e. the Holmberg radius) or in the i-band such as R23.5, i. Although our size definition is not based on the depth of current surveys (as was the case for the size parameters that were defined using photographic plates), it is worth exploring the stellar mass–size plane with popular isophotal size definitions. In this work we lack the B-band filter. As a compromise, we thus decided to use g-band imaging (the closest filter to the B-band with enough depth) available to us to show the stellar mass–size plane when using the position of the 26 mag arcsec−2 (g-band) isophote as a size indicator. It is this isophote to which we refer to as the Holmberg radius (RH). We also include a comparison with size based on the location of the i-band isophote 23.5 mag arcsec−2 (R23.5, i). The results of this exercise are shown in Fig. 6. Figure 6. Open in new tabDownload slide Comparing the R1–mass relation with other size–mass relations using: Holmberg Radius, RH defined in this work as the 26 mag arcsec−2 isophote in SDSS g-band (upper panel), R23.5, i, the radial location of the μi = 23.5 mag arcsec−2 isophote (middle panel) and |$R_{\rm e,M_{\star }}$|⁠, the half-mass radius (lower panel). Figure 6. Open in new tabDownload slide Comparing the R1–mass relation with other size–mass relations using: Holmberg Radius, RH defined in this work as the 26 mag arcsec−2 isophote in SDSS g-band (upper panel), R23.5, i, the radial location of the μi = 23.5 mag arcsec−2 isophote (middle panel) and |$R_{\rm e,M_{\star }}$|⁠, the half-mass radius (lower panel). The observed scatter of the global stellar mass–RH relation is 0.109 ± 0.005. This value is larger than the one observed for R1 (0.089 ± 0.005). Interestingly, the scatter is particularly larger for the dwarfs and spirals than for the massive ellipticals. This is understandable as the variability in star formation activity among the less massive galaxies is larger than for the most massive ellipticals. Different star formation levels produce different g-band luminosities for the same stellar mass density, and therefore the scatter is larger when using size indicators based on blue bands (as is the case of RH). A potential way to decrease the scatter using a single photometric band would be to use a redder band (i.e. one less affected by recent star formation activity). For instance, one would expect the use of the i-band to decrease the scatter of the stellar mass–size relation. This is in fact the case. The observed scatter of the global stellar mass–R23.5, i relation is a bit lower (0.106 ± 0.006 dex) than in the case of RH using the g-band. While the observed scatter of the R1–mass relation is predominantly affected by the uncertainty in background and mass estimation, in the case of the RH–mass and R23.5, i–mass relations, the main contributor to the scatter is the mass uncertainty. This is because the 26 mag arcsec−2 isophote in the g-band and the 23.5 mag arcsec−2 in the i-band are brighter than the typical brightness of the isomass contour 1 M⊙ pc−2 (see Section 6.2). Consequently, the contribution of the uncertainty in the background to the estimation of the location of RH (and R23.5, i) is not very important. Therefore, while the intrinsic scatter of the R1–mass relation is around 0.06 dex, for RH-mass (and R23.5, i–mass) the global intrinsic scatter decreases to ∼0.09 dex. We discuss our findings in the context of the H i–mass relation of galaxies in Appendix E. Although the global size–mass relation using R1 produces the smallest scatter, it is worth checking whether this is also the case for different galaxy families. The family of galaxies that consistently shows both the lowest observed and intrinsic scatter values in the stellar mass–size plane is the E0-S0+ group. This applies to all the size indicators explored, including the effective and half-mass radii. It is particularly remarkable that the observed scatter using R23.5, i (0.056 ± 0.005 dex) is almost comparable to the lowest intrinsic scatter values obtained for this galaxy type using R1 and the half-mass radius (∼0.04 dex). The small scatter of the elliptical galaxies is a direct consequence of their low level of internal structure compared to other galaxies. This fact makes the members of this family almost homologous. Therefore, if one is interested in a relative comparison between the size of galaxies within such a family, any size indicator already suggested in the literature is useful. In the case of the S0/a-Sm family, i.e. those galaxies with very complex internal structure consisting of bars, rings, spiral arms, etc., the difference in scatter among the size indicators is much larger than for the ellipticals. As expected, the size indicators showing the larger scatter for this galaxy type are those which better reflect the light concentration of the objects: i.e. the effective and the half-mass radii. However, those size measurements that are closer to a characterization of the boundaries of the galaxies (e.g. R1, RH, and R23.5, i) are the ones with lower scatter. A similar result is found for the dwarf galaxies. In addition to the above results, we also quantitatively compare the scatters found here for spiral galaxies with those measured in the literature. Similar to Saintonge & Spekkens (2011) and Hall et al. (2012), we divide our spiral galaxy sample into three categories: Sa-Sab, Sb-Sbc, and Sc-Sd. In Fig. 7 we show the stellar mass–size relations for these types using R1 and R23.5i as size indicators. We find a similar stratification as the one reported by Saintonge & Spekkens (2011), Hall et al. (2012), and Muñoz-Mateos et al. (2015), i.e. at fixed stellar mass (or luminosity), those galaxies having later types are the largest. This is especially manifested at the low mass end. Saintonge & Spekkens (2011) has a sample mostly composed of Sc galaxies. Using R23.5i and the luminosity in the i-band, they found an observed scatter of 0.05 dex. For the same morphological type, we find here (this time using the stellar mass) an observed scatter of 0.101 ± 0.007 dex. The larger scatter is connected to the fact that we use the stellar mass instead of the luminosity. The observed scatter using R1 for Sc-Sd galaxies is 0.082 ± 0.007 dex. As our main source of the scatter is the determination of the stellar mass, it is worth giving the intrinsic scatter values: 0.096 ± 0.007 dex (R23.5i) and 0.066 ± 0.007 dex (R1). Within the common mass range 109.5–1011 M⊙ for all the spiral galaxy types, the observed scatters for the Sc-Sd galaxies are: 0.095 ± 0.008 dex (R23.5i) and 0.077 ± 0.008 dex (R1). The scatter reported by Saintonge & Spekkens (2011) is extraordinarily tight. Using a similar sample, Hall et al. (2012) found an observed scatter value for the R23.5i–mass relation which ranges from 0.070 dex (for their higher quality sample) to 0.096 dex (their entire sample), which is in closer agreement with our observed value. Figure 7. Open in new tabDownload slide Stellar mass–size relation for three morphological groups within our spiral galaxy sample: Sa-Sab (orange squares), Sb-Sbc (green triangles), and Sc-Sd (blue dots). The top panel shows the relation using R23.5, i while the bottom panel shows the same relation using R1 as the size indicator. Similar to Saintonge & Spekkens (2011), Hall et al. (2012), and Muñoz-Mateos et al. (2015), we find that spiral galaxies are stratified, with the largest ones at fixed stellar mass being those of later morphological types. Figure 7. Open in new tabDownload slide Stellar mass–size relation for three morphological groups within our spiral galaxy sample: Sa-Sab (orange squares), Sb-Sbc (green triangles), and Sc-Sd (blue dots). The top panel shows the relation using R23.5, i while the bottom panel shows the same relation using R1 as the size indicator. Similar to Saintonge & Spekkens (2011), Hall et al. (2012), and Muñoz-Mateos et al. (2015), we find that spiral galaxies are stratified, with the largest ones at fixed stellar mass being those of later morphological types. Another aspect to highlight is the change in the global slope of the stellar mass–size relation as a function of the size indicators we have explored. Using R1, we find a slope a bit above 1/3 between 107 and 1011 M⊙. This value is in line with the one found using isophotal radii (RH abd R23.5, i) as a size measure. The slopes, however, decrease significantly when using the effective and half-mass radii. We expand on the potential meaning of the slope we measure using R1 in subsection 7.2. 7.2 The slope of the stellar mass–size relation The slope of the stellar mass–size relation we report for galaxies within the mass range 107 to 1011 M⊙ is very close to 1/3. A straightforward calculation shows that if all the stars within R1 were located within a sphere of such radius, the stellar mass density (in 3D) of all the galaxies in this mass range will be equivalent to ∼4.5 × 10−3 M⊙ pc−3. Obviously, the spatial configuration of both dwarf and spiral galaxies is not spherical but disc-like. Nonetheless, it is suggestive to think that the gas that originally formed all these objects was in a spherical-like configuration at an early galaxy phase before its collapse to form the disc configuration. In other words, it is worth speculating whether the currently observed 3D stellar density for all the galaxies in our sample is a reflection of a common 3D gas density at an early phase (before collapsing) of our objects. In fact, this constant 3D stellar density could be linked with the expected constant density of dark matter haloes which formed at a given age of the Universe (see e.g. Mo et al. 1998). It is also worth indicating that while we see a monotonic increase in the size of galaxies with stellar mass using R1 (as well as for RH and R23.5, i), the same is not true for Re or |$R_{\rm e,M_{\star }}$|⁠. This is particularly manifested in the interval 109 to 1011 M⊙ in stellar mass where the increase in effective (or in the half-mass) radius of the (mostly) spiral galaxies is very modest. This mass range is where the bulges of spiral galaxies appear. What we are witnessing here is the enormous impact of using Re (or |$R_{\rm e,M_{\star }}$|⁠) for measuring sizes when a significant amount of the light (or stellar mass) of galaxies can be concentrated in the inner parts of galaxies with a bulge. This small increase in Re (or |$R_{\rm e,M_{\star }}$|⁠) between 109 and 1011 M⊙ is not a minor issue. As the vast majority of works aiming to understand the connection between the galaxy size and the dark matter halo properties use Re as a size indicator (see e.g. Kravtsov 2013; Jiang et al. 2019; Zanisi et al. 2020), the small increase in the effective radius in this mass range can hide a potential connection between the dark and the luminous component of the galaxies. In a companion paper (C. Dalla Vecchia et al. in preparation), we show that the use of R1 permits the connection of both galaxy components directly, ultimately facilitating our understanding about how these objects form. 7.3 The tilt of the stellar mass–size relation at 1011 M⊙ A notable feature of the new stellar mass–size relation is the change in slope observed at ∼1011 M⊙. The slope changes from ∼1/3 to ∼3/5 (see Table 3) for the most massive galaxies. The abrupt change in slope is found in all the size indicators probed in this work. This ∼1011 M⊙ stellar mass value marks the shift between objects with disc-like configuration to objects with a spherical symmetry. In addition, this is the stellar mass where the transition from rotationally to pressure-supported systems has been reported (see e.g. Emsellem et al. 2011). As mentioned in Appendix A, we speculate that this change in slope is a manifestation of different gas density threshold values for star formation in galaxies that formed at high-|$z$| could have had. Observational evidence has shown that the most massive galaxies underwent a huge burst of star formation at high-z with star formation rates reaching values ≳1000 M⊙ yr−1 (see e.g. Riechers et al. 2013). The high star formation rates these galaxies have undergone could have injected a lot of energy into the gas, thereby preventing the star formation at low mass densities and consequently increasing the gas density threshold for star formation. A remnant of this huge star formation burst is the core of these massive galaxies that later undergo important merger activity, creating their envelopes (see e.g. Trujillo et al. 2011; Ferreras et al. 2014; Buitrago et al. 2017). In short, we speculate that the tilt we observe at ∼1011 M⊙ in the new stellar mass–size plane is a reflection of a change in the gas density threshold for star formation when the bulk of the most massive galaxies originated. 8 CONCLUSIONS We introduce a new approach to define the luminous size of galaxies, aiming to link size with the region where galaxies form stars. In order to make such a physically motivated size definition operative, we propose using the average radial location of the gas density threshold for star formation to measure this quantity. We suggest the use of the radial position of the isomass contour at 1 M⊙ pc−2 (here referred to as R1) as a proxy for measuring this threshold. This value is motivated by both theoretical and observational arguments. In particular, the density value found at the location of the truncation in galaxies similar to our own Milky Way. When using R1 as a size indicator for galaxies, the global scatter of the stellar mass–size relation explored over five orders of magnitude in stellar mass drops significantly, reaching a value of ∼0.06 dex. This value is 2.5 times smaller than the scatter measured using the effective radius (∼0.15 dex) and 1.5 to 1.8 times smaller than those using other traditional sizes indicators such as R23.5, i (∼0.09 dex), RH (∼0.09 dex), and |$R_{\rm e,M_{\star }}$| (∼0.11 dex). Between 107 and 1011 M⊙, the slope of the stellar mass–size relation is very close to 1/3. In a 3D spherical distribution, this corresponds to a constant stellar density of ∼4.5 × 10−3 M⊙ pc−3, which could be a reflection of a common gas density when the primordial gas collapsed to form stars. Beyond 1011 M⊙, the stellar mass–size relation gets steeper, reaching a slope of ∼3/5. We speculate that this drastic increase in size of the most massive galaxies could be linked to its different star formation histories, reflecting that the gas density threshold for star formation was higher at the epoch of their main formation burst. SUPPORTING INFORMATION Table 2. Characteristics of the galaxies used in this work. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. ACKNOWLEDGEMENTS We acknowledge the referee for a careful reading of this paper and a for a large number of suggestions to improve its presentation. We thank Raúl Infante-Sainz and Javier Román for providing the extended SDSS Point Spread Functions (PSFs) of all filters for use in this work and Stéphane Courteau for interesting comments. NC thanks Caroline Haigh for providing the latest version of MTObjects. We acknowledge financial support from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement No 721463 to the SUNDIAL ITN network, from the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation and Universities (MCIU) and the European Regional Development Fund (FEDER) under the grants with reference AYA2016-76219-P and AYA2016-77237-C3-1-P, from IAC projects P/300624 and P/300724, financed by the Ministry of Science, Innovation and Universities, through the State Budget and by the Canary Islands Department of Economy, Knowledge and Employment, through the Regional Budget of the Autonomous Community, and from the Fundación BBVA under its 2017 programme of assistance to scientific research groups, for the project ‘Using machine-learning techniques to drag galaxies from the noise in deep imaging’. This work has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Software: Astropy,3 a community-developed core Python package for Astronomy (Robitaille et al. 2013; Price-Whelan et al. 2018); SciPy (Jones et al. 2001); NumPy (Oliphant 2006; Walt, Colbert & Varoquaux 2011); Scikit-learn (Pedregosa et al. 2011); Matplotlib (Hunter 2007); Jupyter Notebooks (Kluyver et al. 2016); TOPCAT (Taylor 2005); IMFIT (Erwin 2015); MTObjects (Teeninga et al. 2016); SWarp (Bertin 2010); and SAO Image DS9 (Smithsonian Astrophysical Observatory 2000). Footnotes 1 https://www.python.org/ 2 To check the robustness of our estimation of Re using the g-band, we also estimated the same quantity using the i-band. We found a very tight correlation between both effective radii (Pearson correlation coefficient r = 0.996). 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R. , 1991 , ApJ , 369 , 46 10.1086/169737 Crossref Search ADS Crossref Walt S. v. d. , Colbert S. C. , Varoquaux G. , 2011 , Comput. Sci. Eng. , 13 , 22 10.1109/MCSE.2011.37 Crossref Search ADS Crossref Wang J. , Koribalski B. S. , Serra P. , van der Hulst T. , Roychowdhury S. , Kamphuis P. , Chengalur J. N. , 2016 , MNRAS , 460 , 2143 10.1093/mnras/stw1099 Crossref Search ADS Crossref Zanisi L. et al. . , 2020 , MNRAS , 492 , 1671 10.1093/mnras/stz3516 Crossref Search ADS Crossref APPENDIX A: IS 1 M⊙ PC−2 A GOOD PROXY FOR THE LOCATION OF THE GAS DENSITY THRESHOLD FOR STAR FORMATION ALONG THE 107 TO 1012 M⊙ MASS RANGE? In this paper we propose the radial position of the isomass contour at 1 M⊙ pc−2 as a size indicator motivated by its proximity to the location of the gas density threshold for star formation in galaxies found theoretically (see e.g. Schaye 2004) and because this value is representative of the location of the disc truncation in galaxies similar to the Milky Way (Martínez-Lombilla et al. 2019). However, this value is not guaranteed to be representative of the gas density threshold for star formation in galaxies with very different stellar mass (like dwarfs) or those who were formed during an enormous burst of star formation at high-|$z$| (as is the case of the most massive ellipticals). In fact, there are some hints in our data indicating that a running gas density threshold for star formation could be more representative of the size of galaxies. Looking at Fig. 2, we can appreciate that while 1 M⊙ pc−2 very well represents the size of spiral galaxies, the use of this value does not fully enclose the extension of dwarf galaxies. A potential explanation for this is that in the case of dwarf galaxies the gas density threshold for star formation is lower than in the case of more massive spirals, and therefore using, for instance, 0.3–0.5 M⊙ pc−2 instead of 1 M⊙ pc−2 could be a better proxy for measuring the size of these small systems. This idea of a lower gas density threshold for star formation for dwarf galaxies is observationally supported as on average their amount of H2 compared to H i (and therefore their star-forming efficiency) is lower than for larger mass systems (see e.g. Leroy et al. 2008; Huang et al. 2012). On the other hand, the situation would be reversed in the case of massive elliptical galaxies where R1 is a bit larger than the visual extent of these galaxies. The gas density threshold for star formation during its high-z formation could be larger than what it is currently for the spiral galaxies. In the local Universe, for instance, it has been found that starburst galaxies have significantly enhanced star formation (Jaskot et al. 2015). In this case, using an isomass contour with a larger density value (i.e. >1 M⊙ pc−2) as size could be a better option for these very massive galaxies. In fact, one of the most notable features of the stellar mass–size relation we have analysed is the change in slope of the relation for galaxies above 1011 M⊙ in stellar mass (Section 7.3). We believe that this could be an indication that the proxy we are using to calculate the location of the gas density threshold for star formation for galaxies (i.e. the isomass contour at 1 M⊙ pc−2) can be slightly different for galaxies where the bulk of star formation occurred at high-|$z$|⁠. For this reason, we have explored how the relation changes when we use an isomass contour at 3 M⊙ pc−2 for the size of the most massive objects. We show the result of doing such an exercise in Fig. A1. Figure A1. Open in new tabDownload slide Stellar mass–size relation using different stellar mass densities as proxies for indicating the size of the most massive galaxies: R1 and R3. As can be seen, the size of the most massive objects changes depending on the isomass contour used to measure its size. The figure shows that most of the galaxies from 107 to 1012 could be allocated in the same stellar mass–size relation (i.e. with a similar slope) if R3 were used for measuring the size of the galaxies for objects with stellar mass larger than 1011 M⊙. Figure A1. Open in new tabDownload slide Stellar mass–size relation using different stellar mass densities as proxies for indicating the size of the most massive galaxies: R1 and R3. As can be seen, the size of the most massive objects changes depending on the isomass contour used to measure its size. The figure shows that most of the galaxies from 107 to 1012 could be allocated in the same stellar mass–size relation (i.e. with a similar slope) if R3 were used for measuring the size of the galaxies for objects with stellar mass larger than 1011 M⊙. Using R3 instead of R1 for the most massive galaxies decreases the size of these objects and thus follows the trend observed in R1 for the less massive galaxies (M⋆ < 1011 M⊙) as marked by the best-fit dashed line in Fig. A1. Beyond 1011.4 M⊙, however, even with R3 the galaxies are above this trend in R1. This is suggestive of an even larger gas density threshold for star formation in these ultramassive objects. In Chamba et al. (2020), we describe the opposite exercise using the dwarf galaxies. APPENDIX B: OTHER POTENTIAL SIZE INDICATORS BASED ON THE LOCATION OF THE GAS DENSITY THRESHOLD FOR STAR FORMATION Intimately linked to the gas density threshold for star formation is the drop-off of galaxy profiles, both in the optical and in H |${\small {\alpha}}$|⁠. The position of a sharp decline in the optical profile, often referred to in the literature as breaks or truncations (see a discussion about this in Martín-Navarro et al. 2012), has been used to explore the cosmic size evolution of galaxies (Trujillo & Pohlen 2005; Azzollini, Trujillo & Beckman 2008). Unfortunately, not all galaxies show a clear deviation from the exponential decline in their profiles that can be used to trace their sizes (Pohlen & Trujillo 2006). In this paper, we have used the fact that for Milky Way-like galaxies, the truncation is found near a stellar mass density of ∼1 M⊙ pc−2. However, we do not know whether this result also holds for galaxies of different stellar mass (see Appendix A). In a future paper, we will explore this issue with our sample by identifying the location of the truncations in the profiles of our galaxies and determining the values of the stellar mass densities at those positions. In addition to the drop-off in the optical profile, many disc galaxies show a sharp decline in their H |${\small {\alpha}}$| profiles. The location of this truncation in H |${\small {\alpha}}$| is also considered a good indicator of the radial position of the gas density threshold for star formation in actively star forming galaxies (see e.g. Martin & Kennicutt 2001). The H |${\small {\alpha}}$| emission is directly linked to the presence of massive O stars and therefore the presence of recent star formation. In this sense, the location of the truncation radius in H |${\small {\alpha}}$| could be a direct indicator of the position of the gas density threshold for star formation and consequently, a very good proxy for galaxy size. Despite the obvious advantages of using H |${\small {\alpha}}$| profiles to derive the size of galaxies, there are a number of practical limitations to a massive use of this technique. First, to the best of our knowledge, there are currently no deep H |${\small {\alpha}}$| surveys covering a large area of the sky that allow us to make this analysis on all kinds of galaxies. Secondly, contrary to the use of a stellar mass isocontour as we have done in this paper, the use of H |${\small {\alpha}}$| will be restricted to only those galaxies that are currently forming stars. In addition, as is also the case in the optical, not all the disc galaxies show a clear feature in their profiles that can be associated to a sharp decline in on-going star formation (see e.g. Koopmann, Haynes & Catinella 2006). Nonetheless, in those cases where the break radii of H |${\small {\alpha}}$| surface brightness profiles have been measured, a bimodal distribution has been found, with the majority of the galaxies showing breaks at 0.7 or 1.1 R25 (Christlein, Zaritsky & Bland-Hawthorn 2010). It remains to be explored whether the breaks in H |${\small {\alpha}}$| are at the same location as that of the truncations in the optical for large samples of galaxies. Finally, though the presence of H |${\small {\alpha}}$| is a clear indication of recent star formation, the H |${\small {\alpha}}$| line is connected with the presence of stars massive enough to generate a Stroemgren sphere. However, it is easy to imagine (particularly in the outer parts of galaxies where the density is very low) star formation occurring at low levels without the formation of very massive stars. In such a case, where H |${\small {\alpha}}$| is no longer present, the UV emission could be used as an indicator of recent star formation. As mentioned before, it would be necessary to search for a clear feature that can be associated to a gas density threshold for star formation in a large number of galaxies and thus approach the issue of galaxy size and star formation activity from multiple perspectives. APPENDIX C: THE LIMITS OF THE NEW SIZE DEFINITION Our size definition is motivated by the location of the gas density threshold for star formation in galaxies. This definition is rather general and there is no reason why this could not be applied to any galaxy where in-situ star formation is present. However, in order to make our size definition operative, we have used the position of the stellar mass density contour at 1 M⊙ pc−2. By taking this specific value, we are limiting ourselves to characterize only the size of objects whose maximum stellar mass density is ≳1 M⊙ pc−2. Therefore, we quantify the brightness of objects that cannot be characterized with such a specific prescription. Some old and metal-poor galaxies with central surface brightness μg(0) >27 mag arcsec−2 would have a maximum stellar mass density <1 M⊙ pc−2 and therefore, they would be unable to have a size estimation based on our proposed isomass contour. As the surface brightness of galaxies is not usually reported by their central value but as the brightness averaged within their effective radius (i.e. <μ > e), we compute which <μ > e limits our size measure R1. This is done by using the relation between central surface brightness and the average surface brightness according to a Sérsic profile: $$\begin{eqnarray*} \lt \mu \gt _e = \mu (0)-2.5\log \frac{n\Gamma (2n)}{b_n^{2n}} , \end{eqnarray*}$$ (C1) where n is the Sérsic index of the model. For 0.5<n <2 (a typical range of variation of the Sérsic index for galaxies with very low-surface brightness; see e.g. Román & Trujillo 2017), this would imply that the use of R1 is limited to galaxies with <μ > e ≲ 28–29 mag arcsec−2. This surface brightness is extremely low, none the less the current faintest galaxies detected in our Local Group (see e.g. Fig. 7 in McConnachie 2012) have such extreme brightness. This again reinforces the idea we mentioned in Appendix A that low mass systems have very low star formation efficiency and therefore a lower gas density threshold for star formation. Future ultradeep imaging surveys would allow us to better calibrate the gas density threshold for star formation values for low dense systems. Nevertheless, our size definition could be readily adjusted to these extreme objects by using a lower mass density contour for characterizing their sizes. APPENDIX D: STELLAR MASS DETERMINATION As explained in the main body of the paper, stellar masses of all galaxies were calculated using their stellar mass density profiles which were derived using the g–r colour profiles (Roediger & Courteau 2015). In order to evaluate the reliability of this mass estimate, we have compared our stellar mass estimations with those obtained by the Portsmouth group (Maraston et al. 2013). To do that we have retrieved the stellar masses from the Portsmouth Spectro-Photometric Model Fitting catalogue available on the SDSS webpage.4 The Portsmouth stellar masses are calculated using the BOSS spectroscopic redshift, ZNOQSO and u, g, r, i, z photometry by means of broad-band SED fitting of stellar population models. Two separate stellar mass calculations were conducted: one assuming a passive template and the other a star-forming template. The version we have used here is the star-forming template with a Kroupa IMF. In Fig. D1, we show the comparison between the stellar masses estimated in this work and those obtained by the Portsmouth group (when available). The ratio of these two mass estimates is approximately described by a Gaussian distribution. We find a systematic offset in our mass estimates by a factor of 1.6 with respect to the Portsmouth masses. This offset could be due to multiple reasons – we are not using the same IMF nor the same aperture to estimate the total flux of the objects. However, for our paper, the most relevant aspect is not the offset but the scatter in the difference between the two mass measurements. We calculate this scatter by measuring the rms of the ratio between the two masses. For the total sample, the rms of this distribution is 0.24 dex. In addition to showing the distribution of the ratio between the two masses of the entire sample, we also overplot the distribution of the same ratio for the different galaxy subsamples defined in this work. We find that the rms of the subsample distributions are: 0.19 dex (E0-S0+), 0.24 dex (S0/a-Sm), and 0.25 (Dwarfs). Figure D1. Open in new tabDownload slide Uncertainty in measuring the stellar masses of galaxies used in our sample. MP corresponds to the stellar mass of the objects reported by the Portsmouth group (Maraston et al. 2013) and M⋆ is the mass we have estimated in this work using the g–r colour profile (Roediger & Courteau 2015). We show the ratio of these two estimates for the full sample (yellow) and the different galaxy subsamples labelled in the legend. Figure D1. Open in new tabDownload slide Uncertainty in measuring the stellar masses of galaxies used in our sample. MP corresponds to the stellar mass of the objects reported by the Portsmouth group (Maraston et al. 2013) and M⋆ is the mass we have estimated in this work using the g–r colour profile (Roediger & Courteau 2015). We show the ratio of these two estimates for the full sample (yellow) and the different galaxy subsamples labelled in the legend. In our analysis, we have used the above rms of the subsample distributions as a way of estimating the uncertainty in the stellar mass–size relation. In Fig. D2, we illustrate how the R1–mass relation with an intrinsic scatter equal to zero would look with only the uncertainty contribution from the stellar mass estimate. In other words, this plot shows the contribution of the mass uncertainty to the observed scatter of the mass–size relation. The scatter of the R1–mass size relation plotted in Fig. D2, which is the scatter simply produced by the uncertainty in measuring the stellar mass, is |$\sigma _{R_{1,mass}}$| = 0.047 dex. That is the contribution of the mass uncertainty to the total observed scatter of the R1–mass size relation (i.e. |$\sigma _{R_1}$| = 0.089 dex). Figure D2. Open in new tabDownload slide Scatter in the R1–mass relation produced by the uncertainty in stellar mass. The figure shows how galaxies spread over the size– mass plane simply due to effect of the uncertainty in measuring the stellar mass of each galaxy. To build this plot we have assumed that the intrinsic scatter of the size–mass relation is exactly zero and follow how the uncertainty in measuring the mass will spread around the best-fitting lines to the observed measurements. The contour shows the 3σ distribution of the final sample. Figure D2. Open in new tabDownload slide Scatter in the R1–mass relation produced by the uncertainty in stellar mass. The figure shows how galaxies spread over the size– mass plane simply due to effect of the uncertainty in measuring the stellar mass of each galaxy. To build this plot we have assumed that the intrinsic scatter of the size–mass relation is exactly zero and follow how the uncertainty in measuring the mass will spread around the best-fitting lines to the observed measurements. The contour shows the 3σ distribution of the final sample. APPENDIX E: COMPARING THE R1–STELLAR MASS RELATION WITH THE H i SIZE–MASS RELATION OF GALAXIES An extremely tight scaling relation for galaxies is the one which links their total H i mass with the diameter of their H i discs defined by the location of the gas surface density at 1 M⊙ pc−2 (Broeils & Rhee 1997). This relation spans over five orders of magnitude in H  i mass. A subsequent analysis of this relation has shown that its scatter is extremely low: 0.06 dex and the slope almost exactly 1/2, i.e. 0.506 ± 0.003 (Wang et al. 2016). These values are suggestive of a uniform characteristic H i surface density of ∼5 M⊙ pc−2. In the case of the H i size –mass relation of galaxies, the use of 1 M⊙ pc−2 is a subjective choice and, contrary to this work, there is no a priory physical motivation to select this value. Interestingly though, the scatter of the H i size–mass relation is minimized when the H i size is measured at surface densities between 1 and 2 M⊙ pc−2. The reason we cite this relation here is because of its potential connection to the size–mass relation we are exploring in this paper. In galaxy formation models (see e.g. Lagos et al. 2011), the H  i surface density is regulated by the conversion process from atomic to molecular hydrogen, i.e. H i-to-H2. Observationally, it is found that the H i surface density saturates at ∼9 M⊙ pc−2, and gas at higher surface densities are converted to molecular gas (Bigiel et al. 2008). As H2 and star formation are closely linked (Krumholz, Leroy & McKee 2011), the H i-to-H2 process should be reflected in the star formation activity and, therefore, in the location of the gas density threshold for star formation in galaxies. As galaxies observationally have a very similar average H i surface density of ∼5 M⊙ pc−2 (as reflected by the tight H  i size–mass relation of galaxies), it seems reasonable to assume that the gas density threshold for star formation is also similar among very different galaxies. For this reason, we speculate that the small intrinsic scatter of the stellar mass–size relation we have explored in this work (based on the location of the gas density threshold for star formation) could be connected with or be a reflection of the tight H i size–mass relation of galaxies. © 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Irwin, Patrick G, J;Parmentier,, Vivien;Taylor,, Jake;Barstow,, Jo;Aigrain,, Suzanne;Lee, Graham K, H;Garland,, Ryan

doi: 10.1093/mnras/staa238pmid: N/A

McCormac, James; Gillen, Edward; Jackman, James A G; Brown, David J A; Bayliss, Daniel; Wheatley, Peter J; Anderson, David R; Armstrong, David J; Bouchy, François; Briegal, Joshua T; Burleigh, Matthew R; Cabrera, Juan; Casewell, Sarah L; Chaushev, Alexander; Chazelas, Bruno; Chote, Paul; Cooke, Benjamin F; Costes, Jean C; Csizmadia, Szilárd;

Santos,, S;Sobral,, D;Matthee,, J;Calhau,, J;da Cunha,, E;Ribeiro,, B;Paulino-Afonso,, A;Arrabal Haro,, P;Butterworth,, J

doi: 10.1093/mnras/staa093pmid: N/A

ABSTRACT We explore deep rest-frame UV to FIR data in the COSMOS field to measure the individual spectral energy distributions (SED) of the ∼4000 SC4K (Sobral et al.) Lyman α (Ly α) emitters (LAEs) at z ∼ 2–6. We find typical stellar masses of 109.3 ± 0.6 M⊙ and star formation rates (SFR) of SFR|$_{\rm SED}=4.4^{+10.5}_{-2.4}$| M⊙ yr−1 and SFR|$_{\rm Ly\,\alpha }=5.9^{+6.3}_{-2.6}$| M⊙ yr−1, combined with very blue UV slopes of |$\beta =-2.1^{+0.5}_{-0.4}$|⁠, but with significant variations within the population. MUV and β are correlated in a similar way to UV-selected sources, but LAEs are consistently bluer. This suggests that LAEs are the youngest and/or most dust-poor subset of the UV-selected population. We also study the Ly α rest-frame equivalent width (EW0) and find 45 ‘extreme’ LAEs with EW0 > 240 Å (3σ), implying a low number density of (7 ± 1) × 10−7 Mpc−3. Overall, we measure little to no evolution of the Ly α EW0 and scale length parameter (w0), which are consistently high (EW|$_0=140^{+280}_{-70}$| Å, |$w_0=129^{+11}_{-11}$| Å) from z ∼ 6 to z ∼ 2 and below. However, w0 is anticorrelated with MUV and stellar mass. Our results imply that sources selected as LAEs have a high Ly α escape fraction (fesc,Ly α) irrespective of cosmic time, but fesc,Ly α is still higher for UV-fainter and lower mass LAEs. The least massive LAEs (<109.5 M⊙) are typically located above the star formation ‘main sequence’ (MS), but the offset from the MS decreases towards z ∼ 6 and towards 1010 M⊙. Our results imply a lack of evolution in the properties of LAEs across time and reveals the increasing overlap in properties of LAEs and UV-continuum selected galaxies as typical star-forming galaxies at high redshift effectively become LAEs. galaxies: evolution, galaxies: formation, galaxies: high-redshift, galaxies: star formation 1 INTRODUCTION The Lyman α (Ly α; λ0,vacuum = 1215.67 Å) emission line has been predicted to be associated with young star-forming galaxies (SFGs; e.g. Partridge & Peebles 1967) but it can also be emitted by active galactic nuclei (AGNs; e.g. Miley & De Breuck 2008; Sobral et al. 2018b). Typical Ly α emitters (LAEs) selected with deep surveys have been found to have low stellar mass (M⋆ ≲ 109 M⊙), low dust content, and high specific star formation rates (SFRs; e.g. Gawiser et al. 2006, 2007), but LAEs can span a wide range in different properties (e.g. Hagen et al. 2016; Matthee et al. 2016). Observationally, the transition between the dominant powering source in LAEs seems to occur at ∼1043 erg s−1, roughly two times the characteristic Ly α luminosity (L|$^\star _{\rm Ly\,\alpha }$|⁠) at z ∼ 2–3 (see Sobral et al. 2018b). Searches using the Ly α emission line have been extremely successful at selecting young SFGs through narrow-band searches (e.g. Hu et al. 2004; Ouchi et al. 2008; Matthee et al. 2015; Santos, Sobral & Matthee 2016; Sobral et al. 2017; Arrabal Haro et al. 2018; Harikane et al. 2018) and spectroscopically confirming bright LAEs (e.g. Sobral et al. 2015, 2018b; Hu et al. 2016; Matthee et al. 2017b; Shibuya et al. 2018) due to the bright Ly α feature. Other studies have successfully selected samples of LAEs using integral field spectroscopy observations (e.g. van Breukelen, Jarvis & Venemans 2005; Blanc et al. 2011; Bacon et al. 2015; Drake et al. 2017) and blind spectroscopy (e.g. Martin & Sawicki 2004; Rauch et al. 2008; Cassata et al. 2011; Le Fèvre et al. 2015). LAEs typically have faint continua, and thus the study of properties of individual sources has typically only been done for extreme LAEs with L ≳ L|$^\star _{\rm Ly\,\alpha }$| (e.g. Ouchi et al. 2013; Sobral et al. 2015). For ≲L|$^\star _{\rm Ly\,\alpha }$| LAEs, studies have typically resorted to stacking of sources (e.g. Momose et al. 2014; Kusakabe et al. 2018). More commonly, large samples of high-redshift SFGs have been selected by searching for the presence of a Lyman break (e.g. Madau et al. 1996; Steidel et al. 1996, 1999). Currently, there are |$\gt 10\, 000$|s of known galaxies at z ∼ 2–10 (see e.g. Bouwens et al. 2014a, 2015), mostly consisting of faint sub-L|$^\star _{\rm UV}$| galaxies found through deep small area searches, typically too faint to follow-up with current spectroscopic instrumentation. While Ly α surveys are efficient at selecting galaxies, inferring intrinsic properties of a galaxy directly from its Ly α emission is challenging due to the complex nature of Ly α radiative transfer. Ly α photons suffer resonant scattering from gas in the interstellar/circumgalactic medium (ISM/CGM) and get easily absorbed by dust (for a review on the process of Ly α radiative transfer, see Dijkstra 2017), which can suppress Ly α emission even in young SFGs. The complex physics of Ly α radiative transfer means that the Ly α escape fraction (fesc,Ly α – the ratio between observed and intrinsic Ly α luminosity) is difficult to predict. Multiple studies have taken different approaches to this problem. Observationally, fesc,Ly α has been measured by comparing Ly α to dust-corrected H α luminosities (Oteo et al. 2015; Matthee et al. 2016; Sobral et al. 2017). Some studies estimate fesc,Ly α by computing the ratio between SFR derived from Ly α (assuming case B recombination) and SFR derived from alternative methods such as from spectral energy distributions (SEDs; Cassata et al. 2015) or the far-infrared (FIR; Wardlow et al. 2014). Others measure the ratio between the observed Ly α luminosity density and the dust-corrected H α luminosity density (Sobral et al. 2017). Alternatively, studies have measured the ratio between Ly α SFR density (SFRD) and UV SFRD by integrating the respective luminosity functions (Sobral et al. 2018a). Typical SFGs at z ∼ 2–3 are found to have very low fesc,Ly α (⁠|$\lt 5{{\ \rm per\ cent}}$|⁠, e.g. Hayes et al. 2010; Cassata et al. 2015; Oteo et al. 2015; Matthee et al. 2016). However, sources selected due to their Ly α emission have much higher fesc,Ly α (as high as |$\sim 40{{\ \rm per\ cent}}$| at z = 2.2, Sobral et al. 2017). Despite the complexity of the Ly α radiative transfer, properties of the Ly α line such as its equivalent width (EW) have been shown to hold important information. Sources selected by their Ly α emission typically have high EWs, with rest-frame Ly α EW (EW0) ∼ 50–150 Å at z ∼ 0.3–6 (see e.g. Gronwall et al. 2007; Hashimoto et al. 2017; Wold et al. 2017), which can be explained by young stellar ages, low metallicities, and/or top-heavy initial mass functions (IMFs; Schaerer 2003; Raiter, Schaerer & Fosbury 2010) or complex radiative transfer effects (Neufeld 1991). The high Ly α EW0 measured for LAEs even at low redshift (z ∼ 0.3, Wold et al. 2017) contrasts with rest-frame EW measurements from other emission lines for galaxies at similar redshifts (e.g. H α, [O ii] and H β + [O iii] EW0), which are measured to be ≤25 Å at z ∼ 0.3 (e.g. SDSS: Thomas et al. 2013; HETDEX: Adams et al. 2011). It should be noted, however, that LAEs with very low EW0 (down to 5 Å) have been detected in some studies (e.g. Sobral et al. 2017; Arrabal Haro et al. 2018), highlighting the diversity of LAE populations. Sobral & Matthee (2019) derived a simple empirical relation that estimates fesc,Ly α from EW0: fesc,Ly α = 0.0048 × EW0. This relation implies a connection between the intrinsic EW and the dust attenuation. A non-evolution of typical EW0 with redshift could thus imply a non-evolution of fesc,Ly α in Ly α-selected samples. A constant typical EW0 = 80 Å across redshift would result in a typical f|$_{\rm esc, Ly\alpha }\sim 40{{\ \rm per\ cent}}$| for LAEs. With the measurement of fesc, Ly α from EW0, it is possible to derive the SFR of LAEs by translating Ly α flux into dust-corrected H α flux with simple assumptions. This provides an SFR computation that is independent of SED fitting and provides a comparison with SED-derived SFRs for LAEs even before observations with James Webb Space Telescope. Exploring how LAEs, which are typically low stellar mass galaxies, fit in the star formation ‘main sequence’ (MS; Brinchmann et al. 2004; Daddi et al. 2007; Noeske et al. 2007; Schreiber et al. 2015) can shed light in a stellar mass range of the SFR–M⋆ relation, which is still widely unconstrained at z > 2. Previous studies have found that LAEs occupy the low stellar mass end of the MS at z = 2.5 (e.g. Shimakawa et al. 2017) but also measured to be significantly above the MS extrapolation (Whitaker et al. 2014) for low stellar masses at z ∼ 2 (e.g. Hagen et al. 2016; Kusakabe et al. 2018) and even at z = 4.9 (Harikane et al. 2018). This suggests that LAEs are experiencing more intense star formation than the general population of galaxies of similar mass at similar redshifts, which may be explained by a burstier nature of star formation. We intend to expand these studies using a large sample of LAEs at z ∼ 2–6. In this work, we use a uniformly selected sample of ∼4000 LAEs (SC4K, Sobral et al. 2018a) to measure rest-frame UV properties and their evolution from the end of reionization at z ∼ 6 until the peak of star formation history at z ∼ 2. For our sample of galaxies, we measure EW0, SFR, M⋆, UV luminosity (MUV), and UV continuum slope (β) for individual LAEs, using photometry measurements that we conduct ourselves, including data from UltraVISTA DR4, and by modelling SEDs using magphys (da Cunha, Charlot & Elbaz 2008; da Cunha et al. 2015). Additionally, we discuss different approaches to measure SFR and how they influence our findings and we provide all our measurements in a public catalogue. This paper is structured as follows: in Section 2.1, we present the SC4K sample of LAEs and detail how we conduct point spread function (PSF) aperture photometry and obtain SEDs and SED fits for each individual LAE. We present the properties of LAEs in Section 3, where we show the methodology we use to derive EW0, SFR, MUV, and β. We present our results in Section 4, looking into the MUV–β and SFR–M⋆ relations and the potential evolution of EW0 with redshift, along with physical interpretations. Finally, we present our conclusions in Section 5. Throughout this work, we use a ΛCDM cosmology with H0 = 70 km s−1 Mpc−1, ΩM = 0.3, and |$\Omega _\Lambda = 0.7$|⁠. All magnitudes in this paper are presented in the AB system (Oke & Gunn 1983) and we use a Chabrier (2003) IMF. 2 SAMPLE, PHOTOMETRY, AND SED FITTING 2.1 The sample: SC4K We use the public SC4K sample of LAEs (Slicing COSMOS with 4k LAEs, Sobral et al. 2018a), which contains 3908 sources selected due to their high Ly α EW at z ∼ 2–6. These LAEs were selected with wide-field surveys conducted with Subaru and the Isaac Newton telescopes, using 16 (12 + 4) medium + narrow-bands (MB + NB) over 2 deg2 in the COSMOS field (Capak et al. 2007; Scoville et al. 2007; Taniguchi et al. 2015), covering a full comoving volume of ∼108 Mpc3. For full details on the selection of the sample see Sobral et al. (2018a). Briefly, the selection criteria applied were (i) EW0 cut of 50 Å for MBs, 25 Å for NBs, and 5 Å for the NB at z = 2.23 (see Sobral et al. 2017); (ii) significant excess emission in the selection medium/narrow-band, Σ > 3 (see Bunker et al. 1995; Sobral et al. 2013); (iii) colour break blueward of the detected Ly α emission, due to the expected presence of a Lyman break; (iv) removal of sources with strong red colours which are typically lower redshift contaminants where the Balmer break mimics a Lyman break; (v) visual inspection of all candidates to remove spurious sources and star artefacts. We show an overview of the properties of the SC4K LAEs, split by selection bands, in Table 1. For each selection band, we provide the median of each property and the 16th (84th) percentiles of its distribution as lower (upper) uncertainties. Additionally, in Fig. 1 we show a histogram distribution of Ly α luminosity (LLy α), EW0 (see Section 3.1) and SFR using the Sobral & Matthee (2019) calibration (see Section 3.6). The differences in the lower end distribution of LLy α are driven by an increasing luminosity distance and a roughly similar flux limit. The evolution of the Ly α luminosity function is presented in Sobral et al. (2018a). Figure 1. Open in new tabDownload slide Distributions of parameters derived directly from photometry. Ly α luminosity (left-hand panel), EW0 (middle panel), and SFR derived directly from LLy α and EW0 (Sobral & Matthee 2019, see Section 3.6.1; right-hand panel). MB (NB) data are shown as filled (dashed) lines. For each parameter, top panels show the z ≤ 3.1 sample and bottom panels show the higher redshift LAEs. The EW0 peak at z = 3.1 (NB) is artificial and it is the upper limit of the EW0, obtained from the flux upper limit. AGNs have been removed. Figure 1. Open in new tabDownload slide Distributions of parameters derived directly from photometry. Ly α luminosity (left-hand panel), EW0 (middle panel), and SFR derived directly from LLy α and EW0 (Sobral & Matthee 2019, see Section 3.6.1; right-hand panel). MB (NB) data are shown as filled (dashed) lines. For each parameter, top panels show the z ≤ 3.1 sample and bottom panels show the higher redshift LAEs. The EW0 peak at z = 3.1 (NB) is artificial and it is the upper limit of the EW0, obtained from the flux upper limit. AGNs have been removed. Table 1. Overview of the SC4K sample of LAEs. We present the median of all measurements for each galaxy property, with the errors being the 16th and 84th percentile of the distribution. (1) LAE selection filter (Sobral et al. 2018a); (2) Mean redshift of the sample based on Ly α within the filter FWHM; (3) Number of LAEs (Number of LAEs after removing sources with AGN signatures, see Section 2.1.1); (4) Number of non-AGN LAEs with SEDs (percentage, see Section 2.5.1); (5) Ly α luminosity; (6) Ly α rest-frame EW; (7) SFR derived directly from LLy α and EW0 (Sobral & Matthee 2019, see Section 3.6.1); (8) Best likelihood SFR parameter from SED fitting; (9) Best likelihood stellar mass parameter from SED fitting; (10) UV magnitude computed by integrating the SED at λ0 = 1500 Å, see Section 3.3; (11) slope of the UV continuum measured from the SED fits, see Section 3.4. (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . (11) . Filter . Ly |$\alpha \, z$| . # LAEs . # SEDs . log|$_{10}\,$|LLy α . EW0 . SFRLy α . SFRSED . M⋆ . MUV . β . . . (no AGNs) . . (erg s−1) . ( Å) . (M⊙ yr−1) . (M⊙ yr−1) . (log|$_{10}\,$|(M⋆/M⊙)) . (AB) . . NB392 2.2 159 (137) 129 (94 per cent) |$42.55^{+0.15}_{-0.15}$| 79|$^{+52}_{-44}$| |$4.7^{+4.9}_{-2.2}$| |$5.5^{+20.5}_{-3.6}$| |$9.5^{+0.5}_{-0.6}$| |$-19.6^{+1.0}_{-0.6}$| |$-1.8^{+0.9}_{-0.5}$| IA427 2.5 741 (686) 673 (98 per cent) |$42.64^{+0.22}_{-0.14}$| 128|$^{+220}_{-62}$| |$4.0^{+3.1}_{-1.8}$| |$2.9^{+6.9}_{-1.5}$| |$9.2^{+0.5}_{-0.5}$| |$-19.7^{+0.6}_{-0.6}$| |$-2.0^{+0.3}_{-0.4}$| IA464 2.8 311 (284) 283 (100 per cent) |$42.88^{+0.22}_{-0.15}$| 121|$^{+152}_{-52}$| |$6.8^{+4.6}_{-2.4}$| |$4.0^{+9.1}_{-1.6}$| |$9.1^{+0.6}_{-0.3}$| |$-20.2^{+0.5}_{-0.5}$| |$-2.1^{+0.5}_{-0.3}$| IA484 3.0 711 (636) 625 (98 per cent) |$42.83^{+0.18}_{-0.11}$| 176|$^{+340}_{-95}$| |$5.0^{+4.5}_{-2.0}$| |$3.1^{+5.8}_{-1.4}$| |$9.0^{+0.7}_{-0.3}$| |$-20.0^{+0.6}_{-0.7}$| |$-2.4^{+0.6}_{-0.0}$| NB501 3.1 45 (38) 31 (82 per cent) |$42.92^{+0.19}_{-0.13}$| 170|$^{+2259}_{-99}$| |$6.6^{+7.5}_{-3.2}$| |$6.2^{+15.2}_{-3.1}$| |$9.6^{+0.4}_{-0.5}$| |$-20.4^{+1.1}_{-0.8}$| |$-2.3^{+1.1}_{-0.2}$| IA505 3.2 483 (437) 433 (99 per cent) |$42.89^{+0.19}_{-0.13}$| 142|$^{+351}_{-71}$| |$6.3^{+4.9}_{-2.5}$| |$4.5^{+6.5}_{-2.0}$| |$9.4^{+0.5}_{-0.5}$| |$-20.2^{+0.6}_{-0.6}$| |$-2.1^{+0.4}_{-0.4}$| IA527 3.3 641 (593) 573 (97 per cent) |$42.84^{+0.19}_{-0.10}$| 149|$^{+245}_{-74}$| |$5.7^{+5.1}_{-2.3}$| |$4.1^{+5.7}_{-1.9}$| |$9.4^{+0.6}_{-0.6}$| |$-20.2^{+0.5}_{-0.6}$| |$-2.0^{+0.3}_{-0.5}$| IA574 3.7 98 (88) 87 (99 per cent) |$42.98^{+0.14}_{-0.13}$| 97|$^{+72}_{-39}$| |$10.9^{+6.4}_{-4.9}$| |$6.7^{+6.9}_{-2.7}$| |$9.3^{+0.7}_{-0.2}$| |$-20.8^{+0.5}_{-0.4}$| |$-2.4^{+0.8}_{-0.0}$| IA624 4.1 142 (139) 116 (83 per cent) |$43.02^{+0.18}_{-0.06}$| 186|$^{+666}_{-99}$| |$6.7^{+8.2}_{-1.8}$| |$6.1^{+9.1}_{-2.8}$| |$9.2^{+0.5}_{-0.5}$| |$-20.5^{+0.5}_{-0.6}$| |$-1.9^{+0.3}_{-0.5}$| IA679 4.6 79 (75) 69 (92 per cent) |$43.25^{+0.15}_{-0.05}$| 186|$^{+267}_{-89}$| |$11.6^{+12.2}_{-2.8}$| |$9.3^{+18.6}_{-4.0}$| |$9.5^{+0.8}_{-0.3}$| |$-21.2^{+0.6}_{-0.5}$| |$-2.4^{+0.8}_{-0.0}$| IA709 4.8 81 (77) 73 (95 per cent) |$43.16^{+0.13}_{-0.10}$| 124|$^{+200}_{-56}$| |$13.2^{+9.9}_{-5.5}$| |$9.1^{+15.8}_{-3.8}$| |$9.4^{+0.5}_{-0.3}$| |$-21.1^{+0.5}_{-0.4}$| |$-2.0^{+0.3}_{-0.5}$| NB711 4.8 78 (74) 56 (76 per cent) |$42.74^{+0.28}_{-0.16}$| 80|$^{+64}_{-42}$| |$7.8^{+11.2}_{-3.6}$| |$14.4^{+61.0}_{-9.5}$| |$9.7^{+0.6}_{-0.6}$| |$-20.9^{+0.5}_{-0.8}$| |$-1.9^{+0.8}_{-0.5}$| IA738 5.1 79 (75) 65 (87 per cent) |$43.25^{+0.17}_{-0.14}$| 120|$^{+222}_{-47}$| |$15.7^{+15.5}_{-7.6}$| |$16.0^{+32.4}_{-9.2}$| |$9.6^{+0.7}_{-0.3}$| |$-21.3^{+0.4}_{-0.7}$| |$-1.8^{+0.2}_{-0.6}$| IA767 5.3 33 (30) 29 (97 per cent) |$43.37^{+0.20}_{-0.07}$| 134|$^{+169}_{-48}$| |$18.7^{+15.0}_{-7.4}$| |$20.6^{+50.5}_{-10.8}$| |$9.7^{+0.3}_{-0.4}$| |$-21.6^{+0.4}_{-0.5}$| |$-2.0^{+0.3}_{-0.4}$| NB816 5.7 192 (186) 108 (58 per cent) |$42.82^{+0.27}_{-0.11}$| 235|$^{+547}_{-169}$| |$5.2^{+6.4}_{-2.4}$| |$28.5^{+83.7}_{-20.8}$| |$9.9^{+0.4}_{-0.5}$| |$-21.4^{+0.6}_{-0.6}$| |$-1.8^{+0.7}_{-0.6}$| IA827 5.8 35 (35) 27 (77 per cent) |$43.44^{+0.19}_{-0.11}$| 325|$^{+963}_{-266}$| |$22.0^{+47.5}_{-8.4}$| |$25.3^{+80.1}_{-16.1}$| |$9.9^{+0.6}_{-0.4}$| |$-22.0^{+0.8}_{-1.0}$| |$-1.8^{+0.7}_{-0.6}$| GLOBAL 4.1 3908 (3590) 3377 (94 per cent) |$42.84^{+0.27}_{-0.17}$| 138|$^{+281}_{-70}$| |$5.9^{+6.3}_{-2.6}$| |$4.4^{+10.5}_{-2.4}$| |$9.3^{+0.6}_{-0.5}$| |$-20.2^{+0.7}_{-0.8}$| |$-2.1^{+0.5}_{-0.4}$| (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . (11) . Filter . Ly |$\alpha \, z$| . # LAEs . # SEDs . log|$_{10}\,$|LLy α . EW0 . SFRLy α . SFRSED . M⋆ . MUV . β . . . (no AGNs) . . (erg s−1) . ( Å) . (M⊙ yr−1) . (M⊙ yr−1) . (log|$_{10}\,$|(M⋆/M⊙)) . (AB) . . NB392 2.2 159 (137) 129 (94 per cent) |$42.55^{+0.15}_{-0.15}$| 79|$^{+52}_{-44}$| |$4.7^{+4.9}_{-2.2}$| |$5.5^{+20.5}_{-3.6}$| |$9.5^{+0.5}_{-0.6}$| |$-19.6^{+1.0}_{-0.6}$| |$-1.8^{+0.9}_{-0.5}$| IA427 2.5 741 (686) 673 (98 per cent) |$42.64^{+0.22}_{-0.14}$| 128|$^{+220}_{-62}$| |$4.0^{+3.1}_{-1.8}$| |$2.9^{+6.9}_{-1.5}$| |$9.2^{+0.5}_{-0.5}$| |$-19.7^{+0.6}_{-0.6}$| |$-2.0^{+0.3}_{-0.4}$| IA464 2.8 311 (284) 283 (100 per cent) |$42.88^{+0.22}_{-0.15}$| 121|$^{+152}_{-52}$| |$6.8^{+4.6}_{-2.4}$| |$4.0^{+9.1}_{-1.6}$| |$9.1^{+0.6}_{-0.3}$| |$-20.2^{+0.5}_{-0.5}$| |$-2.1^{+0.5}_{-0.3}$| IA484 3.0 711 (636) 625 (98 per cent) |$42.83^{+0.18}_{-0.11}$| 176|$^{+340}_{-95}$| |$5.0^{+4.5}_{-2.0}$| |$3.1^{+5.8}_{-1.4}$| |$9.0^{+0.7}_{-0.3}$| |$-20.0^{+0.6}_{-0.7}$| |$-2.4^{+0.6}_{-0.0}$| NB501 3.1 45 (38) 31 (82 per cent) |$42.92^{+0.19}_{-0.13}$| 170|$^{+2259}_{-99}$| |$6.6^{+7.5}_{-3.2}$| |$6.2^{+15.2}_{-3.1}$| |$9.6^{+0.4}_{-0.5}$| |$-20.4^{+1.1}_{-0.8}$| |$-2.3^{+1.1}_{-0.2}$| IA505 3.2 483 (437) 433 (99 per cent) |$42.89^{+0.19}_{-0.13}$| 142|$^{+351}_{-71}$| |$6.3^{+4.9}_{-2.5}$| |$4.5^{+6.5}_{-2.0}$| |$9.4^{+0.5}_{-0.5}$| |$-20.2^{+0.6}_{-0.6}$| |$-2.1^{+0.4}_{-0.4}$| IA527 3.3 641 (593) 573 (97 per cent) |$42.84^{+0.19}_{-0.10}$| 149|$^{+245}_{-74}$| |$5.7^{+5.1}_{-2.3}$| |$4.1^{+5.7}_{-1.9}$| |$9.4^{+0.6}_{-0.6}$| |$-20.2^{+0.5}_{-0.6}$| |$-2.0^{+0.3}_{-0.5}$| IA574 3.7 98 (88) 87 (99 per cent) |$42.98^{+0.14}_{-0.13}$| 97|$^{+72}_{-39}$| |$10.9^{+6.4}_{-4.9}$| |$6.7^{+6.9}_{-2.7}$| |$9.3^{+0.7}_{-0.2}$| |$-20.8^{+0.5}_{-0.4}$| |$-2.4^{+0.8}_{-0.0}$| IA624 4.1 142 (139) 116 (83 per cent) |$43.02^{+0.18}_{-0.06}$| 186|$^{+666}_{-99}$| |$6.7^{+8.2}_{-1.8}$| |$6.1^{+9.1}_{-2.8}$| |$9.2^{+0.5}_{-0.5}$| |$-20.5^{+0.5}_{-0.6}$| |$-1.9^{+0.3}_{-0.5}$| IA679 4.6 79 (75) 69 (92 per cent) |$43.25^{+0.15}_{-0.05}$| 186|$^{+267}_{-89}$| |$11.6^{+12.2}_{-2.8}$| |$9.3^{+18.6}_{-4.0}$| |$9.5^{+0.8}_{-0.3}$| |$-21.2^{+0.6}_{-0.5}$| |$-2.4^{+0.8}_{-0.0}$| IA709 4.8 81 (77) 73 (95 per cent) |$43.16^{+0.13}_{-0.10}$| 124|$^{+200}_{-56}$| |$13.2^{+9.9}_{-5.5}$| |$9.1^{+15.8}_{-3.8}$| |$9.4^{+0.5}_{-0.3}$| |$-21.1^{+0.5}_{-0.4}$| |$-2.0^{+0.3}_{-0.5}$| NB711 4.8 78 (74) 56 (76 per cent) |$42.74^{+0.28}_{-0.16}$| 80|$^{+64}_{-42}$| |$7.8^{+11.2}_{-3.6}$| |$14.4^{+61.0}_{-9.5}$| |$9.7^{+0.6}_{-0.6}$| |$-20.9^{+0.5}_{-0.8}$| |$-1.9^{+0.8}_{-0.5}$| IA738 5.1 79 (75) 65 (87 per cent) |$43.25^{+0.17}_{-0.14}$| 120|$^{+222}_{-47}$| |$15.7^{+15.5}_{-7.6}$| |$16.0^{+32.4}_{-9.2}$| |$9.6^{+0.7}_{-0.3}$| |$-21.3^{+0.4}_{-0.7}$| |$-1.8^{+0.2}_{-0.6}$| IA767 5.3 33 (30) 29 (97 per cent) |$43.37^{+0.20}_{-0.07}$| 134|$^{+169}_{-48}$| |$18.7^{+15.0}_{-7.4}$| |$20.6^{+50.5}_{-10.8}$| |$9.7^{+0.3}_{-0.4}$| |$-21.6^{+0.4}_{-0.5}$| |$-2.0^{+0.3}_{-0.4}$| NB816 5.7 192 (186) 108 (58 per cent) |$42.82^{+0.27}_{-0.11}$| 235|$^{+547}_{-169}$| |$5.2^{+6.4}_{-2.4}$| |$28.5^{+83.7}_{-20.8}$| |$9.9^{+0.4}_{-0.5}$| |$-21.4^{+0.6}_{-0.6}$| |$-1.8^{+0.7}_{-0.6}$| IA827 5.8 35 (35) 27 (77 per cent) |$43.44^{+0.19}_{-0.11}$| 325|$^{+963}_{-266}$| |$22.0^{+47.5}_{-8.4}$| |$25.3^{+80.1}_{-16.1}$| |$9.9^{+0.6}_{-0.4}$| |$-22.0^{+0.8}_{-1.0}$| |$-1.8^{+0.7}_{-0.6}$| GLOBAL 4.1 3908 (3590) 3377 (94 per cent) |$42.84^{+0.27}_{-0.17}$| 138|$^{+281}_{-70}$| |$5.9^{+6.3}_{-2.6}$| |$4.4^{+10.5}_{-2.4}$| |$9.3^{+0.6}_{-0.5}$| |$-20.2^{+0.7}_{-0.8}$| |$-2.1^{+0.5}_{-0.4}$| Open in new tab Table 1. Overview of the SC4K sample of LAEs. We present the median of all measurements for each galaxy property, with the errors being the 16th and 84th percentile of the distribution. (1) LAE selection filter (Sobral et al. 2018a); (2) Mean redshift of the sample based on Ly α within the filter FWHM; (3) Number of LAEs (Number of LAEs after removing sources with AGN signatures, see Section 2.1.1); (4) Number of non-AGN LAEs with SEDs (percentage, see Section 2.5.1); (5) Ly α luminosity; (6) Ly α rest-frame EW; (7) SFR derived directly from LLy α and EW0 (Sobral & Matthee 2019, see Section 3.6.1); (8) Best likelihood SFR parameter from SED fitting; (9) Best likelihood stellar mass parameter from SED fitting; (10) UV magnitude computed by integrating the SED at λ0 = 1500 Å, see Section 3.3; (11) slope of the UV continuum measured from the SED fits, see Section 3.4. (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . (11) . Filter . Ly |$\alpha \, z$| . # LAEs . # SEDs . log|$_{10}\,$|LLy α . EW0 . SFRLy α . SFRSED . M⋆ . MUV . β . . . (no AGNs) . . (erg s−1) . ( Å) . (M⊙ yr−1) . (M⊙ yr−1) . (log|$_{10}\,$|(M⋆/M⊙)) . (AB) . . NB392 2.2 159 (137) 129 (94 per cent) |$42.55^{+0.15}_{-0.15}$| 79|$^{+52}_{-44}$| |$4.7^{+4.9}_{-2.2}$| |$5.5^{+20.5}_{-3.6}$| |$9.5^{+0.5}_{-0.6}$| |$-19.6^{+1.0}_{-0.6}$| |$-1.8^{+0.9}_{-0.5}$| IA427 2.5 741 (686) 673 (98 per cent) |$42.64^{+0.22}_{-0.14}$| 128|$^{+220}_{-62}$| |$4.0^{+3.1}_{-1.8}$| |$2.9^{+6.9}_{-1.5}$| |$9.2^{+0.5}_{-0.5}$| |$-19.7^{+0.6}_{-0.6}$| |$-2.0^{+0.3}_{-0.4}$| IA464 2.8 311 (284) 283 (100 per cent) |$42.88^{+0.22}_{-0.15}$| 121|$^{+152}_{-52}$| |$6.8^{+4.6}_{-2.4}$| |$4.0^{+9.1}_{-1.6}$| |$9.1^{+0.6}_{-0.3}$| |$-20.2^{+0.5}_{-0.5}$| |$-2.1^{+0.5}_{-0.3}$| IA484 3.0 711 (636) 625 (98 per cent) |$42.83^{+0.18}_{-0.11}$| 176|$^{+340}_{-95}$| |$5.0^{+4.5}_{-2.0}$| |$3.1^{+5.8}_{-1.4}$| |$9.0^{+0.7}_{-0.3}$| |$-20.0^{+0.6}_{-0.7}$| |$-2.4^{+0.6}_{-0.0}$| NB501 3.1 45 (38) 31 (82 per cent) |$42.92^{+0.19}_{-0.13}$| 170|$^{+2259}_{-99}$| |$6.6^{+7.5}_{-3.2}$| |$6.2^{+15.2}_{-3.1}$| |$9.6^{+0.4}_{-0.5}$| |$-20.4^{+1.1}_{-0.8}$| |$-2.3^{+1.1}_{-0.2}$| IA505 3.2 483 (437) 433 (99 per cent) |$42.89^{+0.19}_{-0.13}$| 142|$^{+351}_{-71}$| |$6.3^{+4.9}_{-2.5}$| |$4.5^{+6.5}_{-2.0}$| |$9.4^{+0.5}_{-0.5}$| |$-20.2^{+0.6}_{-0.6}$| |$-2.1^{+0.4}_{-0.4}$| IA527 3.3 641 (593) 573 (97 per cent) |$42.84^{+0.19}_{-0.10}$| 149|$^{+245}_{-74}$| |$5.7^{+5.1}_{-2.3}$| |$4.1^{+5.7}_{-1.9}$| |$9.4^{+0.6}_{-0.6}$| |$-20.2^{+0.5}_{-0.6}$| |$-2.0^{+0.3}_{-0.5}$| IA574 3.7 98 (88) 87 (99 per cent) |$42.98^{+0.14}_{-0.13}$| 97|$^{+72}_{-39}$| |$10.9^{+6.4}_{-4.9}$| |$6.7^{+6.9}_{-2.7}$| |$9.3^{+0.7}_{-0.2}$| |$-20.8^{+0.5}_{-0.4}$| |$-2.4^{+0.8}_{-0.0}$| IA624 4.1 142 (139) 116 (83 per cent) |$43.02^{+0.18}_{-0.06}$| 186|$^{+666}_{-99}$| |$6.7^{+8.2}_{-1.8}$| |$6.1^{+9.1}_{-2.8}$| |$9.2^{+0.5}_{-0.5}$| |$-20.5^{+0.5}_{-0.6}$| |$-1.9^{+0.3}_{-0.5}$| IA679 4.6 79 (75) 69 (92 per cent) |$43.25^{+0.15}_{-0.05}$| 186|$^{+267}_{-89}$| |$11.6^{+12.2}_{-2.8}$| |$9.3^{+18.6}_{-4.0}$| |$9.5^{+0.8}_{-0.3}$| |$-21.2^{+0.6}_{-0.5}$| |$-2.4^{+0.8}_{-0.0}$| IA709 4.8 81 (77) 73 (95 per cent) |$43.16^{+0.13}_{-0.10}$| 124|$^{+200}_{-56}$| |$13.2^{+9.9}_{-5.5}$| |$9.1^{+15.8}_{-3.8}$| |$9.4^{+0.5}_{-0.3}$| |$-21.1^{+0.5}_{-0.4}$| |$-2.0^{+0.3}_{-0.5}$| NB711 4.8 78 (74) 56 (76 per cent) |$42.74^{+0.28}_{-0.16}$| 80|$^{+64}_{-42}$| |$7.8^{+11.2}_{-3.6}$| |$14.4^{+61.0}_{-9.5}$| |$9.7^{+0.6}_{-0.6}$| |$-20.9^{+0.5}_{-0.8}$| |$-1.9^{+0.8}_{-0.5}$| IA738 5.1 79 (75) 65 (87 per cent) |$43.25^{+0.17}_{-0.14}$| 120|$^{+222}_{-47}$| |$15.7^{+15.5}_{-7.6}$| |$16.0^{+32.4}_{-9.2}$| |$9.6^{+0.7}_{-0.3}$| |$-21.3^{+0.4}_{-0.7}$| |$-1.8^{+0.2}_{-0.6}$| IA767 5.3 33 (30) 29 (97 per cent) |$43.37^{+0.20}_{-0.07}$| 134|$^{+169}_{-48}$| |$18.7^{+15.0}_{-7.4}$| |$20.6^{+50.5}_{-10.8}$| |$9.7^{+0.3}_{-0.4}$| |$-21.6^{+0.4}_{-0.5}$| |$-2.0^{+0.3}_{-0.4}$| NB816 5.7 192 (186) 108 (58 per cent) |$42.82^{+0.27}_{-0.11}$| 235|$^{+547}_{-169}$| |$5.2^{+6.4}_{-2.4}$| |$28.5^{+83.7}_{-20.8}$| |$9.9^{+0.4}_{-0.5}$| |$-21.4^{+0.6}_{-0.6}$| |$-1.8^{+0.7}_{-0.6}$| IA827 5.8 35 (35) 27 (77 per cent) |$43.44^{+0.19}_{-0.11}$| 325|$^{+963}_{-266}$| |$22.0^{+47.5}_{-8.4}$| |$25.3^{+80.1}_{-16.1}$| |$9.9^{+0.6}_{-0.4}$| |$-22.0^{+0.8}_{-1.0}$| |$-1.8^{+0.7}_{-0.6}$| GLOBAL 4.1 3908 (3590) 3377 (94 per cent) |$42.84^{+0.27}_{-0.17}$| 138|$^{+281}_{-70}$| |$5.9^{+6.3}_{-2.6}$| |$4.4^{+10.5}_{-2.4}$| |$9.3^{+0.6}_{-0.5}$| |$-20.2^{+0.7}_{-0.8}$| |$-2.1^{+0.5}_{-0.4}$| (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . (11) . Filter . Ly |$\alpha \, z$| . # LAEs . # SEDs . log|$_{10}\,$|LLy α . EW0 . SFRLy α . SFRSED . M⋆ . MUV . β . . . (no AGNs) . . (erg s−1) . ( Å) . (M⊙ yr−1) . (M⊙ yr−1) . (log|$_{10}\,$|(M⋆/M⊙)) . (AB) . . NB392 2.2 159 (137) 129 (94 per cent) |$42.55^{+0.15}_{-0.15}$| 79|$^{+52}_{-44}$| |$4.7^{+4.9}_{-2.2}$| |$5.5^{+20.5}_{-3.6}$| |$9.5^{+0.5}_{-0.6}$| |$-19.6^{+1.0}_{-0.6}$| |$-1.8^{+0.9}_{-0.5}$| IA427 2.5 741 (686) 673 (98 per cent) |$42.64^{+0.22}_{-0.14}$| 128|$^{+220}_{-62}$| |$4.0^{+3.1}_{-1.8}$| |$2.9^{+6.9}_{-1.5}$| |$9.2^{+0.5}_{-0.5}$| |$-19.7^{+0.6}_{-0.6}$| |$-2.0^{+0.3}_{-0.4}$| IA464 2.8 311 (284) 283 (100 per cent) |$42.88^{+0.22}_{-0.15}$| 121|$^{+152}_{-52}$| |$6.8^{+4.6}_{-2.4}$| |$4.0^{+9.1}_{-1.6}$| |$9.1^{+0.6}_{-0.3}$| |$-20.2^{+0.5}_{-0.5}$| |$-2.1^{+0.5}_{-0.3}$| IA484 3.0 711 (636) 625 (98 per cent) |$42.83^{+0.18}_{-0.11}$| 176|$^{+340}_{-95}$| |$5.0^{+4.5}_{-2.0}$| |$3.1^{+5.8}_{-1.4}$| |$9.0^{+0.7}_{-0.3}$| |$-20.0^{+0.6}_{-0.7}$| |$-2.4^{+0.6}_{-0.0}$| NB501 3.1 45 (38) 31 (82 per cent) |$42.92^{+0.19}_{-0.13}$| 170|$^{+2259}_{-99}$| |$6.6^{+7.5}_{-3.2}$| |$6.2^{+15.2}_{-3.1}$| |$9.6^{+0.4}_{-0.5}$| |$-20.4^{+1.1}_{-0.8}$| |$-2.3^{+1.1}_{-0.2}$| IA505 3.2 483 (437) 433 (99 per cent) |$42.89^{+0.19}_{-0.13}$| 142|$^{+351}_{-71}$| |$6.3^{+4.9}_{-2.5}$| |$4.5^{+6.5}_{-2.0}$| |$9.4^{+0.5}_{-0.5}$| |$-20.2^{+0.6}_{-0.6}$| |$-2.1^{+0.4}_{-0.4}$| IA527 3.3 641 (593) 573 (97 per cent) |$42.84^{+0.19}_{-0.10}$| 149|$^{+245}_{-74}$| |$5.7^{+5.1}_{-2.3}$| |$4.1^{+5.7}_{-1.9}$| |$9.4^{+0.6}_{-0.6}$| |$-20.2^{+0.5}_{-0.6}$| |$-2.0^{+0.3}_{-0.5}$| IA574 3.7 98 (88) 87 (99 per cent) |$42.98^{+0.14}_{-0.13}$| 97|$^{+72}_{-39}$| |$10.9^{+6.4}_{-4.9}$| |$6.7^{+6.9}_{-2.7}$| |$9.3^{+0.7}_{-0.2}$| |$-20.8^{+0.5}_{-0.4}$| |$-2.4^{+0.8}_{-0.0}$| IA624 4.1 142 (139) 116 (83 per cent) |$43.02^{+0.18}_{-0.06}$| 186|$^{+666}_{-99}$| |$6.7^{+8.2}_{-1.8}$| |$6.1^{+9.1}_{-2.8}$| |$9.2^{+0.5}_{-0.5}$| |$-20.5^{+0.5}_{-0.6}$| |$-1.9^{+0.3}_{-0.5}$| IA679 4.6 79 (75) 69 (92 per cent) |$43.25^{+0.15}_{-0.05}$| 186|$^{+267}_{-89}$| |$11.6^{+12.2}_{-2.8}$| |$9.3^{+18.6}_{-4.0}$| |$9.5^{+0.8}_{-0.3}$| |$-21.2^{+0.6}_{-0.5}$| |$-2.4^{+0.8}_{-0.0}$| IA709 4.8 81 (77) 73 (95 per cent) |$43.16^{+0.13}_{-0.10}$| 124|$^{+200}_{-56}$| |$13.2^{+9.9}_{-5.5}$| |$9.1^{+15.8}_{-3.8}$| |$9.4^{+0.5}_{-0.3}$| |$-21.1^{+0.5}_{-0.4}$| |$-2.0^{+0.3}_{-0.5}$| NB711 4.8 78 (74) 56 (76 per cent) |$42.74^{+0.28}_{-0.16}$| 80|$^{+64}_{-42}$| |$7.8^{+11.2}_{-3.6}$| |$14.4^{+61.0}_{-9.5}$| |$9.7^{+0.6}_{-0.6}$| |$-20.9^{+0.5}_{-0.8}$| |$-1.9^{+0.8}_{-0.5}$| IA738 5.1 79 (75) 65 (87 per cent) |$43.25^{+0.17}_{-0.14}$| 120|$^{+222}_{-47}$| |$15.7^{+15.5}_{-7.6}$| |$16.0^{+32.4}_{-9.2}$| |$9.6^{+0.7}_{-0.3}$| |$-21.3^{+0.4}_{-0.7}$| |$-1.8^{+0.2}_{-0.6}$| IA767 5.3 33 (30) 29 (97 per cent) |$43.37^{+0.20}_{-0.07}$| 134|$^{+169}_{-48}$| |$18.7^{+15.0}_{-7.4}$| |$20.6^{+50.5}_{-10.8}$| |$9.7^{+0.3}_{-0.4}$| |$-21.6^{+0.4}_{-0.5}$| |$-2.0^{+0.3}_{-0.4}$| NB816 5.7 192 (186) 108 (58 per cent) |$42.82^{+0.27}_{-0.11}$| 235|$^{+547}_{-169}$| |$5.2^{+6.4}_{-2.4}$| |$28.5^{+83.7}_{-20.8}$| |$9.9^{+0.4}_{-0.5}$| |$-21.4^{+0.6}_{-0.6}$| |$-1.8^{+0.7}_{-0.6}$| IA827 5.8 35 (35) 27 (77 per cent) |$43.44^{+0.19}_{-0.11}$| 325|$^{+963}_{-266}$| |$22.0^{+47.5}_{-8.4}$| |$25.3^{+80.1}_{-16.1}$| |$9.9^{+0.6}_{-0.4}$| |$-22.0^{+0.8}_{-1.0}$| |$-1.8^{+0.7}_{-0.6}$| GLOBAL 4.1 3908 (3590) 3377 (94 per cent) |$42.84^{+0.27}_{-0.17}$| 138|$^{+281}_{-70}$| |$5.9^{+6.3}_{-2.6}$| |$4.4^{+10.5}_{-2.4}$| |$9.3^{+0.6}_{-0.5}$| |$-20.2^{+0.7}_{-0.8}$| |$-2.1^{+0.5}_{-0.4}$| Open in new tab We note that extensive analysis of the SC4K public sample have already been conducted in previous works. For example, Paulino-Afonso et al. (2018) studied the UV morphologies of the sample and found that UV sizes of LAEs are constant from z ∼ 2 to z ∼ 6 with effective radii sizes of re ∼ 1.0 ± 0.1 kpc. Shibuya et al. (2019) analysed the radial surface brightness profiles of ∼9000 LAEs (including SC4K) and found that LAEs typically have small sizes, similar to those presented by Paulino-Afonso et al. (2018). This means SC4K LAEs are unresolved in the continuum in ground-based data. Khostovan et al. (2019) derived clustering properties of the sample and measured typical halo masses of ∼1011 M⊙ in NB-selected LAEs and ∼1011–1012 M⊙ in MB-selected LAEs, showing the clustering and typical dark matter halo masses that host LAEs is strongly dependent on LLy α. They find more luminous LAEs reside in more massive dark matter haloes. Calhau et al. (2019) study the X-ray and radio properties of the SC4K sample, estimating black hole accretion rates which can reach ∼3 M⊙ yr−1 in the most extreme sources. They also find that the overall AGN fraction of LAEs is low (⁠|$\lt 10{{\ \rm per\ cent}}$|⁠) but dependent on LLy α, significantly increasing with increasing luminosity and approaching |$100{{\ \rm per\ cent}}$| at LLy α > 1044 erg s−1. 2.1.1 X-ray and radio AGNs in SC4K In total we have 3908 LAEs in our sample, with 254 detected in X-ray and 120 detected in radio (56 in both), resulting in 318 AGN candidates (Calhau et al. 2019). LAEs which are detected in the X-ray and/or radio are classified as AGNs as star-forming processes would require SFR ≳ 1000 M⊙ yr−1 to be detected above the flux limit at such wavelengths and redshifts (see discussion in Calhau et al. 2019). The number of AGNs reported in this paper constitutes an extra 177 sources compared to the ones originally reported in Sobral et al. (2018a), with the additional sources being identified by reaching lower S/N with deep Chandra data (COSMOS Chandra Legacy, Civano et al. 2016) and VLA radio data at 1.4 GHz (VLA-COSMOS Survey, Schinnerer et al. 2004, 2007; Bondi et al. 2008; Schinnerer et al. 2010) and by including 3 GHz radio data (Smolčić et al. 2017). We note, however, that due to available coverage, Calhau et al. (2019) only probe 3705 SC4K LAEs with X-ray and radio data. Throughout this work, SC4K AGNs may be shown in figures (clearly highlighted as such) but are removed from any fitting/binning and median values in tables unless stated otherwise as we focus on the properties of the star-forming population. The catalogue that is provided in this paper has a flag for sources detected in X-ray and radio (see Section 3.7). 2.1.2 Redshift binning To improve the S/N in certain redshift ranges and for clearer visualization of results, we frequently group multiple MB filters in specific redshift bins throughout this paper, following the same grouping scheme as Sobral et al. (2018a): z = 2.5 ± 0.1 (IA427), z = 3.1 ± 0.4 (IA464, IA484, IA505, IA527); z = 3.9 ± 0.3 (IA574, IA624); z = 4.7 ± 0.2 (IA679, IA709); z = 5.4 ± 0.5 (IA738, IA767, IA827). We generally study the NBs separately as there are some relevant distinctions between MBs and NBs, most significantly the flux limit and EW0 cut. Additionally, analysing the two separately provides independent results and allows checks for systematics. 2.2 Multiwavelength data We use the extensive archive of publicly available multiwavelength data in the COSMOS field to conduct accurate photometric measurements in the UV, optical, near-infrared (NIR), mid-infrared (MIR), and FIR wavelengths for each SC4K LAE, individually. A summary of the filters used, effective wavelength, width, and limiting magnitude is provided in Table 2. We use optical broad-band (B, V, g+, r+, i+, z++), medium-band (IA427, IA464, IA484, IA505, IA527, IA574, IA624, IA679, IA709, IA738, IA767, IA827), and narrow-band (NB711, NB816) data taken with the Subaru/SuprimeCam (Capak et al. 2007; Taniguchi et al. 2007), retrieved from the COSMOS Archive.1 Additionally, we use the u band from CFHT/MegaCam. We use deep NIR data (Y, J, H, Ks) from UltraVISTA DR4 (McCracken et al. 2012), taken with VISTA/VIRCAM (Sutherland et al. 2015). Data used have a 0.15 arcsec pix−1 pixel scale and are calibrated to a zero-point of 31.4 mag (30 mag for UltraVISTA and u images). For MIR coverage, we use data from Spitzer/IRAC, channels 1 (3.6 |$\rm{\mu m}$|⁠) and 2 (4.5 |$\rm{\mu m}$|⁠) from SPLASH (Steinhardt et al. 2014) and channels 3 (5.6 |$\rm{\mu m}$|⁠) and 4 (8.0 |$\rm{\mu m}$|⁠) from S-COSMOS (Sanders et al. 2007). IRAC data have a zero-point of 21.5814 mag and a pixel scale of 0.6 arcsec pix−1. Table 2. Overview of the photometric filters used in this work ranked from the lowest to highest wavelengths. (1) Photometric filter; (2) Effective wavelength; (3) Filter FWHM; (4) 3σ magnitude depth measured in a fixed 2 arcsec aperture; (5) Correction term summed to the measured magnitudes to correct for systematic offsets; (6) Filter dependent dust correction that is subtracted from the measured magnitudes; (7) Instrument and telescope used for the observations; (8) Source of the data. Filter . λeff . FWHM . Depth . sf . Aλ . Instrument, telescope . Source . . (Å) . (Å) . (⁠|$3\sigma$|⁠, 2 arcsec) . . . . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . u 3911.0 538.0 27.8 0.054 0.0878 MegaCam, CFHT Capak et al. (2007) IA427 4256.3 206.5 27.0 0.037 0.0816 Suprime-Cam, Subaru Capak et al. (2007) B 4439.6 806.7 28.3 −0.242 0.0784 Suprime-Cam, Subaru Capak et al. (2007) IA464 4633.3 218.0 26.9 0.013 0.0750 Suprime-Cam, Subaru Capak et al. (2007) g+ 4728.3 1162.9 27.6 0.024 0.0733 Suprime-Cam, Subaru Capak et al. (2007) IA484 4845.9 228.5 27.0 0.000 0.0713 Suprime-Cam, Subaru Capak et al. (2007) IA505 5060.7 230.5 26.8 −0.002 0.0678 Suprime-Cam, Subaru Capak et al. (2007) IA527 5258.9 242.0 27.1 0.026 0.0646 Suprime-Cam, Subaru Capak et al. (2007) V 5448.9 934.8 27.6 0.046a 0.0616 Suprime-Cam, Subaru Capak et al. (2007) IA574 5762.1 271.5 26.8 0.078 0.0570 Suprime-Cam, Subaru Capak et al. (2007) IA624 6230.0 300.5 26.8 0.002 0.0506 Suprime-Cam, Subaru Capak et al. (2007) r+ 6231.8 1348.8 27.7 0.003 0.0506 Suprime-Cam, Subaru Capak et al. (2007) IA679 6778.8 336.0 26.7 0.039a 0.0442 Suprime-Cam, Subaru Capak et al. (2007) IA709 7070.7 315.5 26.8 −0.024 0.0411 Suprime-Cam, Subaru Capak et al. (2007) NB711 7119.6 72.5 25.9 0.014 0.0406 Suprime-Cam, Subaru Capak et al. (2007) IA738 7358.7 323.5 26.5 0.017 0.0383 Suprime-Cam, Subaru Capak et al. (2007) i+ 7629.1 1489.4 27.2 0.019 0.0360 Suprime-Cam, Subaru Capak et al. (2007) IA767 7681.2 364.0 26.5 0.041 0.0356 Suprime-Cam, Subaru Capak et al. (2007) NB816 8149.0 119.5 26.6 0.068 0.0320 Suprime-Cam, Subaru Capak et al. (2007) IA827 8240.9 343.5 26.5 −0.019 0.0313 Suprime-Cam, Subaru Capak et al. (2007) z++ 9086.6 955.3 26.8 −0.037 0.0265 Suprime-Cam, Subaru Capak et al. (2007) Y 10 211.2 930.0 26.2 0.0 0.0211 VIRCAM, VISTA McCracken et al. (2012) (DR4) J 12 540.9 172.0 25.8 0.0 0.0144 VIRCAM, VISTA McCracken et al. (2012) (DR4) H 16 463.7 2910 26.1 0.0 0.0088 VIRCAM, VISTA McCracken et al. (2012) (DR4) Ks 21 487.7 3090 25.8 0.0 0.0053 VIRCAM, VISTA McCracken et al. (2012) (DR4) IRAC1 35 262.5 7412 25.6 0.002 0.0021 IRAC, Spitzer Steinhardt et al. (2014) IRAC2 44 606.7 10 113 25.5 0.000 0.0014 IRAC, Spitzer Steinhardt et al. (2014) IRAC3 56 764.4 13 499 22.6 0.013 0.0010 IRAC, Spitzer Sanders et al. (2007) IRAC4 77 030.1 28 397 22.5 −0.171 0.0007 IRAC, Spitzer Sanders et al. (2007) 100 |$\rm{\mu m}$| 97 9036.1 356 866 15.4 0.20b 0.0000 PACS, Herschel Lutz et al. (2011) 160 |$\rm{\mu m}$| 153 9451.3 749 540 14.3 −0.06b 0.0000 PACS, Herschel Lutz et al. (2011) 250 |$\rm{\mu m}$| 247 1245.1 658 930 10.9 −0.49b 0.0000 SPIRE, Herschel Oliver et al. (2012) 350 |$\rm{\mu m}$| 346 7180.4 937 200 10.6 −0.15b 0.0000 SPIRE, Herschel Oliver et al. (2012) 500 |$\rm{\mu m}$| 496 1067.7 1848 042 10.6 0.03b 0.0000 SPIRE, Herschel Oliver et al. (2012) Filter . λeff . FWHM . Depth . sf . Aλ . Instrument, telescope . Source . . (Å) . (Å) . (⁠|$3\sigma$|⁠, 2 arcsec) . . . . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . u 3911.0 538.0 27.8 0.054 0.0878 MegaCam, CFHT Capak et al. (2007) IA427 4256.3 206.5 27.0 0.037 0.0816 Suprime-Cam, Subaru Capak et al. (2007) B 4439.6 806.7 28.3 −0.242 0.0784 Suprime-Cam, Subaru Capak et al. (2007) IA464 4633.3 218.0 26.9 0.013 0.0750 Suprime-Cam, Subaru Capak et al. (2007) g+ 4728.3 1162.9 27.6 0.024 0.0733 Suprime-Cam, Subaru Capak et al. (2007) IA484 4845.9 228.5 27.0 0.000 0.0713 Suprime-Cam, Subaru Capak et al. (2007) IA505 5060.7 230.5 26.8 −0.002 0.0678 Suprime-Cam, Subaru Capak et al. (2007) IA527 5258.9 242.0 27.1 0.026 0.0646 Suprime-Cam, Subaru Capak et al. (2007) V 5448.9 934.8 27.6 0.046a 0.0616 Suprime-Cam, Subaru Capak et al. (2007) IA574 5762.1 271.5 26.8 0.078 0.0570 Suprime-Cam, Subaru Capak et al. (2007) IA624 6230.0 300.5 26.8 0.002 0.0506 Suprime-Cam, Subaru Capak et al. (2007) r+ 6231.8 1348.8 27.7 0.003 0.0506 Suprime-Cam, Subaru Capak et al. (2007) IA679 6778.8 336.0 26.7 0.039a 0.0442 Suprime-Cam, Subaru Capak et al. (2007) IA709 7070.7 315.5 26.8 −0.024 0.0411 Suprime-Cam, Subaru Capak et al. (2007) NB711 7119.6 72.5 25.9 0.014 0.0406 Suprime-Cam, Subaru Capak et al. (2007) IA738 7358.7 323.5 26.5 0.017 0.0383 Suprime-Cam, Subaru Capak et al. (2007) i+ 7629.1 1489.4 27.2 0.019 0.0360 Suprime-Cam, Subaru Capak et al. (2007) IA767 7681.2 364.0 26.5 0.041 0.0356 Suprime-Cam, Subaru Capak et al. (2007) NB816 8149.0 119.5 26.6 0.068 0.0320 Suprime-Cam, Subaru Capak et al. (2007) IA827 8240.9 343.5 26.5 −0.019 0.0313 Suprime-Cam, Subaru Capak et al. (2007) z++ 9086.6 955.3 26.8 −0.037 0.0265 Suprime-Cam, Subaru Capak et al. (2007) Y 10 211.2 930.0 26.2 0.0 0.0211 VIRCAM, VISTA McCracken et al. (2012) (DR4) J 12 540.9 172.0 25.8 0.0 0.0144 VIRCAM, VISTA McCracken et al. (2012) (DR4) H 16 463.7 2910 26.1 0.0 0.0088 VIRCAM, VISTA McCracken et al. (2012) (DR4) Ks 21 487.7 3090 25.8 0.0 0.0053 VIRCAM, VISTA McCracken et al. (2012) (DR4) IRAC1 35 262.5 7412 25.6 0.002 0.0021 IRAC, Spitzer Steinhardt et al. (2014) IRAC2 44 606.7 10 113 25.5 0.000 0.0014 IRAC, Spitzer Steinhardt et al. (2014) IRAC3 56 764.4 13 499 22.6 0.013 0.0010 IRAC, Spitzer Sanders et al. (2007) IRAC4 77 030.1 28 397 22.5 −0.171 0.0007 IRAC, Spitzer Sanders et al. (2007) 100 |$\rm{\mu m}$| 97 9036.1 356 866 15.4 0.20b 0.0000 PACS, Herschel Lutz et al. (2011) 160 |$\rm{\mu m}$| 153 9451.3 749 540 14.3 −0.06b 0.0000 PACS, Herschel Lutz et al. (2011) 250 |$\rm{\mu m}$| 247 1245.1 658 930 10.9 −0.49b 0.0000 SPIRE, Herschel Oliver et al. (2012) 350 |$\rm{\mu m}$| 346 7180.4 937 200 10.6 −0.15b 0.0000 SPIRE, Herschel Oliver et al. (2012) 500 |$\rm{\mu m}$| 496 1067.7 1848 042 10.6 0.03b 0.0000 SPIRE, Herschel Oliver et al. (2012) Notes. aincludes an additional offset to correct the systematic uncertainties, Section 2.4.6. bdenotes values obtained from the deblended FIR catalogue presented by Jin et al. 2018. Open in new tab Table 2. Overview of the photometric filters used in this work ranked from the lowest to highest wavelengths. (1) Photometric filter; (2) Effective wavelength; (3) Filter FWHM; (4) 3σ magnitude depth measured in a fixed 2 arcsec aperture; (5) Correction term summed to the measured magnitudes to correct for systematic offsets; (6) Filter dependent dust correction that is subtracted from the measured magnitudes; (7) Instrument and telescope used for the observations; (8) Source of the data. Filter . λeff . FWHM . Depth . sf . Aλ . Instrument, telescope . Source . . (Å) . (Å) . (⁠|$3\sigma$|⁠, 2 arcsec) . . . . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . u 3911.0 538.0 27.8 0.054 0.0878 MegaCam, CFHT Capak et al. (2007) IA427 4256.3 206.5 27.0 0.037 0.0816 Suprime-Cam, Subaru Capak et al. (2007) B 4439.6 806.7 28.3 −0.242 0.0784 Suprime-Cam, Subaru Capak et al. (2007) IA464 4633.3 218.0 26.9 0.013 0.0750 Suprime-Cam, Subaru Capak et al. (2007) g+ 4728.3 1162.9 27.6 0.024 0.0733 Suprime-Cam, Subaru Capak et al. (2007) IA484 4845.9 228.5 27.0 0.000 0.0713 Suprime-Cam, Subaru Capak et al. (2007) IA505 5060.7 230.5 26.8 −0.002 0.0678 Suprime-Cam, Subaru Capak et al. (2007) IA527 5258.9 242.0 27.1 0.026 0.0646 Suprime-Cam, Subaru Capak et al. (2007) V 5448.9 934.8 27.6 0.046a 0.0616 Suprime-Cam, Subaru Capak et al. (2007) IA574 5762.1 271.5 26.8 0.078 0.0570 Suprime-Cam, Subaru Capak et al. (2007) IA624 6230.0 300.5 26.8 0.002 0.0506 Suprime-Cam, Subaru Capak et al. (2007) r+ 6231.8 1348.8 27.7 0.003 0.0506 Suprime-Cam, Subaru Capak et al. (2007) IA679 6778.8 336.0 26.7 0.039a 0.0442 Suprime-Cam, Subaru Capak et al. (2007) IA709 7070.7 315.5 26.8 −0.024 0.0411 Suprime-Cam, Subaru Capak et al. (2007) NB711 7119.6 72.5 25.9 0.014 0.0406 Suprime-Cam, Subaru Capak et al. (2007) IA738 7358.7 323.5 26.5 0.017 0.0383 Suprime-Cam, Subaru Capak et al. (2007) i+ 7629.1 1489.4 27.2 0.019 0.0360 Suprime-Cam, Subaru Capak et al. (2007) IA767 7681.2 364.0 26.5 0.041 0.0356 Suprime-Cam, Subaru Capak et al. (2007) NB816 8149.0 119.5 26.6 0.068 0.0320 Suprime-Cam, Subaru Capak et al. (2007) IA827 8240.9 343.5 26.5 −0.019 0.0313 Suprime-Cam, Subaru Capak et al. (2007) z++ 9086.6 955.3 26.8 −0.037 0.0265 Suprime-Cam, Subaru Capak et al. (2007) Y 10 211.2 930.0 26.2 0.0 0.0211 VIRCAM, VISTA McCracken et al. (2012) (DR4) J 12 540.9 172.0 25.8 0.0 0.0144 VIRCAM, VISTA McCracken et al. (2012) (DR4) H 16 463.7 2910 26.1 0.0 0.0088 VIRCAM, VISTA McCracken et al. (2012) (DR4) Ks 21 487.7 3090 25.8 0.0 0.0053 VIRCAM, VISTA McCracken et al. (2012) (DR4) IRAC1 35 262.5 7412 25.6 0.002 0.0021 IRAC, Spitzer Steinhardt et al. (2014) IRAC2 44 606.7 10 113 25.5 0.000 0.0014 IRAC, Spitzer Steinhardt et al. (2014) IRAC3 56 764.4 13 499 22.6 0.013 0.0010 IRAC, Spitzer Sanders et al. (2007) IRAC4 77 030.1 28 397 22.5 −0.171 0.0007 IRAC, Spitzer Sanders et al. (2007) 100 |$\rm{\mu m}$| 97 9036.1 356 866 15.4 0.20b 0.0000 PACS, Herschel Lutz et al. (2011) 160 |$\rm{\mu m}$| 153 9451.3 749 540 14.3 −0.06b 0.0000 PACS, Herschel Lutz et al. (2011) 250 |$\rm{\mu m}$| 247 1245.1 658 930 10.9 −0.49b 0.0000 SPIRE, Herschel Oliver et al. (2012) 350 |$\rm{\mu m}$| 346 7180.4 937 200 10.6 −0.15b 0.0000 SPIRE, Herschel Oliver et al. (2012) 500 |$\rm{\mu m}$| 496 1067.7 1848 042 10.6 0.03b 0.0000 SPIRE, Herschel Oliver et al. (2012) Filter . λeff . FWHM . Depth . sf . Aλ . Instrument, telescope . Source . . (Å) . (Å) . (⁠|$3\sigma$|⁠, 2 arcsec) . . . . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . u 3911.0 538.0 27.8 0.054 0.0878 MegaCam, CFHT Capak et al. (2007) IA427 4256.3 206.5 27.0 0.037 0.0816 Suprime-Cam, Subaru Capak et al. (2007) B 4439.6 806.7 28.3 −0.242 0.0784 Suprime-Cam, Subaru Capak et al. (2007) IA464 4633.3 218.0 26.9 0.013 0.0750 Suprime-Cam, Subaru Capak et al. (2007) g+ 4728.3 1162.9 27.6 0.024 0.0733 Suprime-Cam, Subaru Capak et al. (2007) IA484 4845.9 228.5 27.0 0.000 0.0713 Suprime-Cam, Subaru Capak et al. (2007) IA505 5060.7 230.5 26.8 −0.002 0.0678 Suprime-Cam, Subaru Capak et al. (2007) IA527 5258.9 242.0 27.1 0.026 0.0646 Suprime-Cam, Subaru Capak et al. (2007) V 5448.9 934.8 27.6 0.046a 0.0616 Suprime-Cam, Subaru Capak et al. (2007) IA574 5762.1 271.5 26.8 0.078 0.0570 Suprime-Cam, Subaru Capak et al. (2007) IA624 6230.0 300.5 26.8 0.002 0.0506 Suprime-Cam, Subaru Capak et al. (2007) r+ 6231.8 1348.8 27.7 0.003 0.0506 Suprime-Cam, Subaru Capak et al. (2007) IA679 6778.8 336.0 26.7 0.039a 0.0442 Suprime-Cam, Subaru Capak et al. (2007) IA709 7070.7 315.5 26.8 −0.024 0.0411 Suprime-Cam, Subaru Capak et al. (2007) NB711 7119.6 72.5 25.9 0.014 0.0406 Suprime-Cam, Subaru Capak et al. (2007) IA738 7358.7 323.5 26.5 0.017 0.0383 Suprime-Cam, Subaru Capak et al. (2007) i+ 7629.1 1489.4 27.2 0.019 0.0360 Suprime-Cam, Subaru Capak et al. (2007) IA767 7681.2 364.0 26.5 0.041 0.0356 Suprime-Cam, Subaru Capak et al. (2007) NB816 8149.0 119.5 26.6 0.068 0.0320 Suprime-Cam, Subaru Capak et al. (2007) IA827 8240.9 343.5 26.5 −0.019 0.0313 Suprime-Cam, Subaru Capak et al. (2007) z++ 9086.6 955.3 26.8 −0.037 0.0265 Suprime-Cam, Subaru Capak et al. (2007) Y 10 211.2 930.0 26.2 0.0 0.0211 VIRCAM, VISTA McCracken et al. (2012) (DR4) J 12 540.9 172.0 25.8 0.0 0.0144 VIRCAM, VISTA McCracken et al. (2012) (DR4) H 16 463.7 2910 26.1 0.0 0.0088 VIRCAM, VISTA McCracken et al. (2012) (DR4) Ks 21 487.7 3090 25.8 0.0 0.0053 VIRCAM, VISTA McCracken et al. (2012) (DR4) IRAC1 35 262.5 7412 25.6 0.002 0.0021 IRAC, Spitzer Steinhardt et al. (2014) IRAC2 44 606.7 10 113 25.5 0.000 0.0014 IRAC, Spitzer Steinhardt et al. (2014) IRAC3 56 764.4 13 499 22.6 0.013 0.0010 IRAC, Spitzer Sanders et al. (2007) IRAC4 77 030.1 28 397 22.5 −0.171 0.0007 IRAC, Spitzer Sanders et al. (2007) 100 |$\rm{\mu m}$| 97 9036.1 356 866 15.4 0.20b 0.0000 PACS, Herschel Lutz et al. (2011) 160 |$\rm{\mu m}$| 153 9451.3 749 540 14.3 −0.06b 0.0000 PACS, Herschel Lutz et al. (2011) 250 |$\rm{\mu m}$| 247 1245.1 658 930 10.9 −0.49b 0.0000 SPIRE, Herschel Oliver et al. (2012) 350 |$\rm{\mu m}$| 346 7180.4 937 200 10.6 −0.15b 0.0000 SPIRE, Herschel Oliver et al. (2012) 500 |$\rm{\mu m}$| 496 1067.7 1848 042 10.6 0.03b 0.0000 SPIRE, Herschel Oliver et al. (2012) Notes. aincludes an additional offset to correct the systematic uncertainties, Section 2.4.6. bdenotes values obtained from the deblended FIR catalogue presented by Jin et al. 2018. Open in new tab For the FIR coverage, we use 100 and 160|$\rm{\mu m}$| data (PEP, Lutz et al. 2011) taken with Herschel/PACS (Pilbratt et al. 2010) and 250, 350, and 500|$\rm{\mu m}$| data (HerMES, Griffin et al. 2010; Oliver et al. 2012) taken with Herschel/SPIRE. The five listed FIR images have a pixel scale of 1.2, 2.4, 6, 8.3, and 12 arcsec pix−1, respectively. FIR images are calibrated to provide fluxes in Jansky and thus have a zero-point of 8.9 mag. 2.3 Multiwavelength photometry Accurate photometric measurements are essential to obtain robust SEDs and derive accurate galaxy properties, particularly for sources that are faint in the continuum. While there is a plethora of publicly available catalogues for the COSMOS field (e.g. Ilbert et al. 2009; Laigle et al. 2015), such catalogues are typically broad-band selected and thus miss a significant number of line emitters, especially faint, high EW sources. For example, 9 per cent of our LAEs are not detected in the i band-selected catalogue from Ilbert et al. (2009) with 1 arcsec radius matching and 29 per cent of SC4K LAEs are not detected in the NIR-selected catalogue from Laigle et al. (2015). Continuum faint sources with very blue UV continuum slopes have low fluxes in the observed optical and will fall below the detection thresholds of NIR selected catalogues (e.g. Laigle et al. 2015), particularly if they have low stellar masses. Therefore, to obtain consistent, controllable and uniform measurements for the entire sample of LAEs, we conduct our own aperture photometry and estimate errors locally using empty apertures. We also compare our photometry with measurements from the COSMOS catalogues and find a very good agreement. Furthermore, because we have measured the sizes in the rest-frame UV and found SC4K LAEs to be very compact (point-like for the data we use; re = 1.0 kpc corresponds to 0.13 arcsec at z = 3), we opt to conduct PSF photometry, as fully explained in Section 2.4. 2.4 Aperture photometry of SC4K LAEs 2.4.1 Overview of our aperture photometry In order to obtain accurate PSF aperture photometry for individual LAEs, for each band, we estimate the total magnitude by following the steps: conducting photometry in fixed apertures (Section 2.4.2); applying aperture corrections based on PSF stars around each LAE (Section 2.4.3); applying reddening corrections (Section 2.4.4); introducing systematic offset corrections based on known offsets and COSMOS catalogues (Section 2.4.6); Magnitudes per source and per band are computed as $$\begin{eqnarray} {\rm mag=mag_0+aper_{cor}}+s_\mathrm{ f}-A_\lambda , \end{eqnarray}$$ (1) where mag0 is the magnitude calculated by converting the flux obtained in fixed apertures (typically 2 arcsec diameter for most of the data) to the AB magnitude system before any correction is applied, apercor is the aperture correction derived per band and per source, based on PSF stars around each LAE, sf the systematic offset correction for the filter, and Aλ the reddening correction computed for the effective wavelength of the filter. The error in the final magnitude is obtained by propagating the error in flux, scaling the error with the correction that was applied to the flux, and then adding 30 per cent of the total correction to the error in flux.2 Aperture photometry in the FIR is discussed separately in Section 2.4.5. 2.4.2 Aperture photometry in fixed apertures We conduct aperture photometry centred on the position of each SC4K LAE (Sobral et al. 2018a) over all the filters listed in Table 2. We do this by creating 200 × 200 pixel (30 arcsec × 30 arcsec for a 0.15 arcsec pix−1 pixel scale) cutouts, where we conduct the photometry.3 For optical to MIR images, we use 2 arcsec diameter apertures. We estimate the background noise by placing 2000 2 arcsec apertures in random positions of the field where there are no detections above 2σ (given by the segmentation maps per filter produced by sextractor; Bertin & Arnouts 1996) and subtract it from the counts of the aperture placed on the LAE. Upper and lower errors are measured as the 84th and 16th percentiles of all random apertures. We repeat this procedure per band per source. 2.4.3 Aperture correction The original PSF was kept across all images as we have opted for correcting the photometry with PSF stars, instead of PSF matching the data, in order to avoid modifying the data and confuse nearby sources. Fixed aperture photometry in non-PSF matched images requires correction of the PSF effect on photometry so we can obtain total fluxes and total magnitudes for point-like sources. To do this, we measure the magnitude of stars4 in 2 arcsec apertures and with mag_auto (Bertin & Arnouts 1996).5 We define the correction factor (apercor in equation 1) as the difference between mag_auto and magnitudes measured in 2 arcsec apertures. This correction is valid for point-like sources, an assumption that should be valid for our LAEs given the rest-frame UV sizes as measured by Paulino-Afonso et al. (2018) using high-resolution HST/ACS images. The correction term is measured for each filter, and it is the median correction of stars within a 0.3 deg radius around each LAE, accounting for spatial variations of the PSF per band. 2.4.4 Galactic extinction correction We correct for dust attenuation along the line of sight due to our Galaxy. For the COSMOS field, the median galactic extinction is measured to be E(B − V) = 0.0195 ± 0.006 (Capak et al. 2007). The slope of the extinction curve with wavelength is parametrized by the factor R(V) $$\begin{eqnarray} R(V)\equiv \frac{A(V)}{E(B-V)}, \end{eqnarray}$$ (2) where A(V) is the total extinction at the V band. For the diffuse interstellar medium, the median value of R(V) is estimated to be 3.1 (e.g. Fitzpatrick 1999) and it is the value used in this paper. We use the model from Fitzpatrick & Massa (2007), where the attenuation at a wavelength (λ) becomes $$\begin{eqnarray} A_\lambda =A(V)\left(1+\frac{k}{R(V)}\right), \end{eqnarray}$$ (3) where k is a polynomial expansion of λ−1 (equation 2 from Fitzpatrick & Massa 2007) with a linear component for UV wavelengths, a curvature term for the far-UV and a Lorentzian-like bump at 2175 Å. We determine Aλ for the effective wavelength of each filter and show its value for each filter in Table 2. 2.4.5 FIR photometry For FIR data, due to the large PSF of 7.2, 12, 18.15, 25.15, and 36.30 arcsec (100, 160, 250, 350, and 500 |$\rm{\mu m}$|⁠, respectively), the usage of 2 arcsec diameter aperture photometry is not viable. We conduct PSF aperture photometry using apertures which are the size of the PSF: radius of 6, 5, 3, 3, and 3 pixels, respectively (retrieving 67 per cent of the total flux), with the same random empty aperture procedure to estimate background. This allows us to then apply aperture corrections of 1/0.67 to get full fluxes for point-like sources. For 100 |$\rm{\mu m}$| (160 |$\rm{\mu m}$|⁠), we multiply the flux by the filter correction factor 1.1 (1.2) as described in the PEP public data release notes (see Lutz et al. 2011). However, the blending of sources is still a serious issue, as the large pixel scale makes it difficult to establish if a detection is produced by one of our LAEs or by a neighbouring source. To solve this, we use the FIR measurements from the publicly available deblended COSMOS catalogue (Jin et al. 2018), where FIR emission is deblended to match optical–NIR coordinates. With a 1 arcsec match to the deblended catalogue, there are 14, 11, 29, 19, and 12 SC4K LAEs with 3σ detections in 100, 160, 250, 350, and 500 |$\rm{\mu m}$|⁠, respectively. Whenever a source is undetected in the FIR, we assign the local estimate of the background as an upper limit, which we measure with 2000 empty apertures the size of the PSF. We ensure our own flux measurements are consistent with Jin et al. (2018, see Section 2.4.6). 2.4.6 Systematic offsets We correct for systematic offsets (sf) in the photometry by applying the corrections derived by Ilbert et al. (2009) (we present these values in Table 2). After applying the systematic offsets and all previous correction terms, we compare our total magnitudes with measurements from Ilbert et al. (2009) and Laigle et al. (2015). We find no statistically significant difference with our measurements except for two filters (IA679, V) which have systematic offsets of ∼0.5 mag. We apply a further correction (included in the sf, Table 2) to our magnitudes, so the median of the magnitude difference becomes zero. For FIR magnitudes, we estimate the systematic correction term from the FIR deblended catalogue (Jin et al. 2018), also presented in Table 2. 2.5 Spectral energy distributions of SC4K LAEs Having conducted photometry in the 34 filters listed in Table 2, we can now explore the SED of each individual LAE, observed from UV to FIR. We use the publicly available SED-fitting code magphys6 (da Cunha et al. 2008, 2012) with the high-redshift extension (see da Cunha et al. 2015), to obtain SED fits for each individual galaxy, using our rest-frame UV, optical, and NIR–FIR photometric measurements. magphys is based on dust attenuation models from Charlot & Fall (2000) and uses the stellar population synthesis model from Bruzual & Charlot (2003) with a Chabrier (2003) IMF (range 0.1–100 M⊙) to compute the emission of simple stellar populations (SSPs, populations of coeval stars with similar properties). We use the prescription of Madau (1995) to model the intergalactic medium (IGM). The software generates a library of model SEDs for galaxies at the mean redshift of the NB/MB filter (see Table 1) and for the given photometric bands. The modelled SED of a galaxy is composed by the weighted sum of SSPs, with the star formation history (SFH) being a continuously delayed exponential function with an early rise followed by a decay. Instantaneous bursts of star formation of random duration (lasting 30–300 Myr) and amplitude (forming mass between 0.1 and 100 times the mass formed by the continuous SFH) are superimposed. A Bayesian approach is then used to compare model SEDs with observed photometry, creating a parameter likelihood distribution for several galaxy properties such as stellar mass, SFR, and dust attenuation. As the models are purely stellar (no nebular line fitting), we do not fit photometry from filters where we expect strong nebular emission, namely Ly α at the selection NB or MB filters, as it is by definition significant in our Ly α-selected sample. While we do not remove photometry from filters which may have contribution from other emission lines such as H α (IRAC filters at z ∼ 4–6) or [O iii] (H − K bands at z ∼ 2–3), by removing the Ly α-contaminated filter, combined with the large number of filters used, we do not expect an overestimation of masses due to nebular line contamination. We explore this by rerunning magphys for the entire z = 2.5 sample (IA427) after removing the H and K bands, which may be contaminated from [O iii] and H α emission, respectively, and compare the difference of estimated stellar masses. We find that when removing both H and K, the median difference of stellar masses is 0, with no dependence on mass, and the average difference -0.07 (⁠|$M_{\rm \star , no\, HK}$| being slightly smaller). Removing H and K makes the estimation of stellar mass more uncertain as the rest-frame optical becomes more poorly constrained. Additionally, we test the effect of only removing H, with the IR still being constrained by the other bands. We also find no significant difference in stellar masses, with the median of the difference being 0 and the average −0.08 (⁠|$M_{\rm \star , no\, H}$| being slightly smaller). Overall, we find that not removing photometric bands outside Ly α does not lead to a significant overestimation of stellar masses for our sources. However, including nebular lines may still be important, particularly if we look at other parameters (e.g. ages), as there may be some systematics, particularly for the faintest sources with the highest EWs. This will be addressed in a forthcoming paper with an SED-fitting code that models nebular emission (cigale, Noll et al. 2009; Boquien et al. 2019). For our z ∼ 2–6 LAEs, the optical bands are essential to fit the rest-frame UV continuum, IRAC filters can constrain fluxes redward of D4000 and the FIR measurements provide upper constraints in the dust emission, which can improve the SFR estimates. We note that, as explained in Section 2.1.1, while we exclude sources with evidence of AGN activity when computing median properties of the sample, we still obtain SED fits (without using any AGN SED model) for those sources. In Fig. 2, we show observed and intrinsic SED fits and photometric measurements/upper limits for two LAEs. The SEDs were purposely chosen to show two very distinct galaxies within the SC4K sample: one with a very blue and steep UV continuum slope, with low stellar mass that dominates the sample and one with a more red continuum, more massive and with higher dust extinction which is much more rare in the sample of LAEs. While the latter is not well representative of a typical LAE, it is still important to show that LAEs can span a large variety of physical properties. This LAE is detected in two Herschel bands, which shows that FIR can be important to constrain the SED fits and derive properties of high-redshift LAEs. Figure 2. Open in new tabDownload slide Left: SED of SC4K-IA427-134461 (at z = 2.5), for observed UV–IR wavelengths as we only obtain upper limits in the FIR. Red circles show the luminosity (in solar units) measured at the corresponding observed wavelength and green arrows show the upper limits for non-detections, where the flux is |$\lt 3\sigma$|⁠. Unfilled circles are the luminosity at the NB/MB where the LAE was selected, and we note that this filter was not used to derive the SED fit. The black line is the best-fitting SED to the observed photometry and the blue dashed line the intrinsic (dust-free) SED. This is an example of a very blue (β = −2.0) and low stellar mass (M⋆ = 109.2 M⊙) LAE. Right: Same as left-hand panel but for SC4K-IA427-10601 (at z = 2.5) and at a wider wavelength range, showing FIR wavelengths as this LAE is detected in 250 and 350 |$\rm{\mu m}$| due to the presence of dust. This LAE is redder (β = −0.3) and more massive (M⋆ = 1010.5 M⊙). Note that this LAE is not representative of the SC4K sample as only |$\sim 3{{\ \rm per\ cent}}$| (2 per cent) non-AGN LAEs are as massive (as red). Figure 2. Open in new tabDownload slide Left: SED of SC4K-IA427-134461 (at z = 2.5), for observed UV–IR wavelengths as we only obtain upper limits in the FIR. Red circles show the luminosity (in solar units) measured at the corresponding observed wavelength and green arrows show the upper limits for non-detections, where the flux is |$\lt 3\sigma$|⁠. Unfilled circles are the luminosity at the NB/MB where the LAE was selected, and we note that this filter was not used to derive the SED fit. The black line is the best-fitting SED to the observed photometry and the blue dashed line the intrinsic (dust-free) SED. This is an example of a very blue (β = −2.0) and low stellar mass (M⋆ = 109.2 M⊙) LAE. Right: Same as left-hand panel but for SC4K-IA427-10601 (at z = 2.5) and at a wider wavelength range, showing FIR wavelengths as this LAE is detected in 250 and 350 |$\rm{\mu m}$| due to the presence of dust. This LAE is redder (β = −0.3) and more massive (M⋆ = 1010.5 M⊙). Note that this LAE is not representative of the SC4K sample as only |$\sim 3{{\ \rm per\ cent}}$| (2 per cent) non-AGN LAEs are as massive (as red). 2.5.1 Number of derived SEDs Although all LAEs are by definition detected in the MB/NB where they were selected (Sobral et al. 2018a), a small fraction of our LAEs have few to no detections in other photometric bands. For such cases, SED fitting may fail. Out of the 3590 non-AGN LAEs, we obtain reliable SEDs for 3377 (94 per cent, see Table 1). The catalogue that we release with this paper (see Section 3.7) has an SED flag which marks unreliable SEDs. The AGN flag indicates AGN LAEs (Section 2.1.1, Calhau et al. 2019), and we reiterate that while we compute parameters for these sources, the SED-derived parameters are not reliable and are not included in any median property estimation done in this work. 3 THE PROPERTIES OF LAEs In this section, we present our methodology and computations to derive galaxy properties for individual LAEs, using our full photometric measurements and SED fits from magphys. EW0 and LLy α of all LAEs in the SC4K sample have been derived and published in Sobral et al. (2018a). 3.1 Ly α luminosity (LLy α) LLy α is calculated from the Ly α line flux (fLy α) $$\begin{eqnarray} {\rm \mathit{ L}}_{\rm Ly\,\alpha } \left[\rm erg\, s^{-1}\right]= 4 pi\times \mathit{ f}_{\rm Ly\,\alpha }{\rm \mathit{ D}_L^2}(\it z) , \end{eqnarray}$$ (4) where DL(z) is the luminosity distance at the redshift of each source, computed from the redshifted Ly α at the effective wavelength of the detection NB/MB. In Fig. 1 (left), we show the LLy α distribution of our LAEs, spanning a wide range of luminosities LLy α = 1042–44 erg s−1. 3.2 Ly α rest-frame equivalent width (EW0) The observed EW (EWobs) of an emission line is the ratio between the flux of the line and the continuum flux density and can be calculated as $$\begin{eqnarray} \rm EW_{\rm obs} [\mathring{\rm A}]= \Delta \lambda _{1}\frac{\it f_{1}-f_{2}}{\it f_{2}-f_{2}(\Delta \lambda _{1}/\Delta \lambda _{2})}, \end{eqnarray}$$ (5) where Δλ1 is the FWHM of the NB or MB, Δλ2 the excess broad-band filter (Sobral et al. 2018a), f1 is the flux density measured in the NB or MB, and f2 is the flux density computed from two adjacent BB filters, which avoids assumptions of the slope of the continuum (for details see Sobral et al. 2018a). The rest-frame EW (EW0) is calculated as $$\begin{eqnarray} {\rm EW}_{\rm 0} [\mathring{\rm A}]= \frac{\rm EW_{\rm obs}}{1+z}, \end{eqnarray}$$ (6) where z is the redshift of Ly α at the effective wavelength of the NB or MB (Sobral et al. 2018a). We provide the median EW0 for different redshifts and for the full SC4K sample in Table 1. 3.2.1 EW0 scale length (w0) An exponential fit of the form N = N0 exp(-EW0/w0) has been widely used to describe Ly α EW0 distributions (e.g. Gronwall et al. 2007; Hashimoto et al. 2017; Wold et al. 2017), with the rate of decay being determined by the scale length parameter w0. With our sample of LAEs, we analyse EW0 distributions in multiple well-defined redshift ranges between z ∼ 2 and z ∼ 6. To estimate w0, we define bins of 20 Å and fit the exponential function to the observed distribution (see Fig. 3), taking into account Poissonian errors. Bins with less than two sources are excluded from the fits. To account for bin width choice, we add 10 Å (half the bin width) in quadrature to the errors of w0. We also explore how an EW0 upper cut affects w0 as it removes sources with extreme (and more uncertain) EWs. We apply a cut of EW0 = 240 Å, the theoretical limit of EW0 powered by Population II star formation (e.g. Charlot & Fall 1993) and the value which has been extensively used in Ly α emission studies to identify ‘extreme’ EW galaxies (e.g. Cantalupo, Lilly & Haehnelt 2012; Marino et al. 2018). We compute |$\chi ^2_{\rm red}$| by comparing the best exponential fit to the histogram of observed counts and their associated Poisson errors. Figure 3. Open in new tabDownload slide Left: EW0 distribution of the full SC4K sample of LAEs. We fit an exponential function of the form N = N0 exp(-EW0/w0), and derive the parameter w0. Fit derived with the distribution of EW0 (EW0 < 240 Å) is shown in red (blue). Right: Same but for an individual filter (IA427) with LAEs at z = 2.5. Figure 3. Open in new tabDownload slide Left: EW0 distribution of the full SC4K sample of LAEs. We fit an exponential function of the form N = N0 exp(-EW0/w0), and derive the parameter w0. Fit derived with the distribution of EW0 (EW0 < 240 Å) is shown in red (blue). Right: Same but for an individual filter (IA427) with LAEs at z = 2.5. Additionally, we fully explore how the errors on EW0 influence the measurement of w0 by using an MCMC approach. For each iteration, we perturb the EW0 of each LAE in that specific sample within their asymmetric error bars (assuming a double normal probability distribution function centred at each EW0 and with FWHM equal to the errors derived from photometry; Sobral et al. 2018a). We impose a hard lower limit equal to the detection threshold (50 Å for MBs, 25 Å for NBs except for NB392 which has a lower limit of 5 Å; see Section 2.1) and an upper limit of 1000 Å, with any source outside these values not being included in a specific realization. With the perturbed EW0, we construct the histogram of the current iteration, using bins of 20 Å. We fit an exponential to the generated histogram bins, taking into account the associated Poissonian error (⁠|$\sqrt{N}$|⁠) of each bin. We iterate this process 200 times, and the final w0 is the median value of all fits with error up (down) being the 84th (16th) percentile of all fits. In addition, to account for the uncertainty introduced by the bin width choice, we also add 10 Å in quadrature to the errors of w0. We also apply the MCMC approach with a cut of EW0 = 240 Å. For the MCMC approach, where EW0 are perturbed, |$\chi ^2_{\rm red}$| is computed by comparing the best fit to the median histogram of all iterations and its Poisson errors. In Table B1 (available online), we show the inferred w0 values (including perturbed estimates) for different redshift ranges and filter combinations. Furthermore, it is important to establish how the EW0 distribution depends on MUV and M⋆. To understand this dependence, we measure w0 in three MUV and M⋆ ranges and show our measurements in Table B1 (available online). For the faintest and the lowest mass ranges, we are significantly incomplete to the low EW0 end of the EW distribution, resulting in a peak at ∼100 Å. Thus, we only fit EW0 > 100 Å to accurately estimate the exponential decay of the distribution for these two cases. 3.3 Rest-frame UV luminosity (MUV) The UV luminosity of a galaxy is associated with continuum emission from massive stars and traces SFR in the past 100 Myr (e.g. Boselli et al. 2001; Salim et al. 2009). A priori, sources selected by their strong Ly α emission could be expected to have strong MUV as both trace recent star formation (neglecting AGN contribution), although Ly α can trace slightly more recent star formation because stars dominating the ionizing photon budget have lifetimes of ∼10 Myr. However, as shown by e.g. Matthee et al. (2017b) and Sobral et al. (2018a) more factors come into play as Ly α and MUV do not necessarily correlate with each other, due to e.g. highly ISM dependent fesc,Ly α (which can result in most Ly α emission being absorbed by dust particles or scattered off neutral hydrogen) or an ionizing efficiency which is evolving with redshift. We compute MUV by integrating the best-fitting SEDs at rest-frame λ0 = 1400–1600 Å. We show the MUV histogram distribution in Fig. 4 (centre). Due to the magnitude limits, at higher redshift we are only sensitive to more luminous MUV sources. We detect SC4K LAEs as bright as MUV = −23 and as faint as MUV = −17. Figure 4. Open in new tabDownload slide Distribution of properties derived from the SED fitting (magphys, see Section 2.5). We show the stellar mass, M⋆ (left), rest-frame UV luminosity, MUV (middle), and rest-frame UV slope, β (right). Top panels show the z ≤ 3.1 sample and the bottom panels show the higher redshift LAEs. AGNs have been removed. Figure 4. Open in new tabDownload slide Distribution of properties derived from the SED fitting (magphys, see Section 2.5). We show the stellar mass, M⋆ (left), rest-frame UV luminosity, MUV (middle), and rest-frame UV slope, β (right). Top panels show the z ≤ 3.1 sample and the bottom panels show the higher redshift LAEs. AGNs have been removed. 3.4 UV continuum slope (β) The slope of the UV continuum can be parametrized in the form fλ∝λβ (e.g. Meurer, Heckman & Calzetti 1999). The slope β is sensitive to the age, metallicity, and dust content of a galaxy. Bruzual & Charlot (2003) models used by magphys have a hard limit to how negative (blue) β can be (β = −2.44), a natural consequence of an upper limit in the IMF. While β may be intrinsically even bluer for more ‘extreme’ stellar populations, in this study, we do not explore those. We measure β directly from the best fit as the slope of the continuum at rest-frame λ0 = 1300–2100 Å. We apply a conservative approach and only use β measurements from sources with at least two detections in this wavelength range. This ensures the β slope is directly constrained and not a direct consequence of assumed SED templates. As expected, due to an increasing luminosity distance, combined with rest-frame λ0 = 1300–2100 Å moving into IR wavelengths, there are fewer β measurements at higher redshift. In addition, we also compute β by fitting a power law (βpl) to the photometric measurements (similar to e.g. Bouwens et al. 2014b), with no SED fitting assumptions. We fit βpl in the range λ0 = 1400–2100 Å, which is smaller than the range used to compute β from the SED fit to avoid broad-band filters at ∼1300 Å, which can be contaminated by the Ly α break. Only sources with at least three 3σ detections in that range are considered for the power-law measurement. For the full SC4K, we measure a median |$\beta _{\rm pl}=-1.8^{+0.8}_{-0.7}$|⁠, which is redder (+0.3) than β from the SED fit (0.2 when only considering sources with βpl measurements), but still within the error bars. Overall, β is better constrained through SED fitting as it uses a prior – the SED models included in magphys. Furthermore, the SED models take into account ∼30 filters over the full UV–FIR wavelength range, preventing it from being as sensitive to individual filter measurements in the smaller λ0 = 1400–2100 Å range. Thus, throughout this paper, we use β computed from the SED fits. We show the histogram distributions of β in Fig. 4 (right). LAEs tend to be very blue across all redshift ranges (median |$\beta =-2.1^{+0.5}_{-0.4}$|⁠, Table 1). LAEs at z = 2.2 are found to have the reddest β slopes, albeit still very blue and comparable to the Lyman Break Galaxy (LBG) population (see further discussion in 4.1). We note, none the less, that the z = 2.2 sample has some key differences compared to other LAEs in SC4K sample, as it selects LAEs down to 5 Å EW0 in addition to reaching the faintest LLy α. This allows redder sources to be picked up, while the much higher EW0 LAEs tend to have much bluer β slopes. 3.5 Stellar mass (M⋆) The total mass of stars in a galaxy (stellar mass, M⋆) is a fundamental galaxy property which is a reflection of its star formation history. We use M⋆ derived from the likelihood parameter distribution from magphys modelling. We show the histogram distribution of M⋆ in our sample in Fig. 4 (left). Most LAEs (88 per cent) have stellar masses <1010 M⊙, although it is important to stress there are some more massive galaxies, which shows a significant diversity. We observe a slight shift to higher masses as we move to higher redshifts (see also Table 1) but this is a natural consequence of only being sensitive to intrinsically more luminous galaxies at higher redshift. We find that typical LAEs are low stellar mass galaxies, with the median of the SC4K sample of LAEs being M|$_\star =10^{9.3^{+0.6}_{-0.5}}$| M⊙. 3.6 Star formation rates (SFRs) 3.6.1 Emission line-based SFRs with Ly α We estimate the SFR directly from LLy α and EW0, using the recipe from Sobral & Matthee (2019) which has calibrated EW0 as a good empirical indicator of fesc,Ly α. With a measurement of fesc,Ly α, LLy α can be converted to dust-corrected H α luminosity assuming case-B recombination (Brocklehurst 1971) and transformed into SFR following Kennicutt (1998). For a Chabrier IMF (0.1–100 M⊙) and assuming fesc, LyC = 0, LLy α in erg s−1 and EW0 in Å, the SFR thus becomes (Sobral & Matthee 2019) $$\begin{eqnarray} {\rm SFR_{Ly\,\alpha }\, \left[{\rm M_{\odot }\, yr^{-1}}\right]}=\frac{{\rm \mathit{ L}}_{\rm Ly\,\alpha }\times 4.4\times 10^{-42}}{\rm 0.042\, {\rm EW}_{0}}. \end{eqnarray}$$ (7) For EW0 > 210 Å, following Sobral & Matthee (2019), we set fesc, Ly α = 1 which corresponds to SFR |${}[{\rm M_{\odot }\, yr^{-1}}]=4.98\times 10^{-43}\times {\rm \mathit{ L}}_{\rm Ly\,\alpha }$|⁠, with LLyα in erg s−1. This SFR is calibrated with dust-corrected H α luminosities and thus should be interpreted as dust-corrected SFR. We show the SFR distribution in Fig. 1 (right). As the SFR is derived from LLy α, it is limited by the same detection limits, which causes a shift to higher SFR with increasing redshift. We measure SFRs in the range |$\sim 1\!-\!300\, {\rm M_{\odot }\, yr^{-1}}$|⁠, and measure a median SFR|$_{\rm Ly\,\alpha }=5.9^{+6.3}_{-2.6}$| for SC4K LAEs (see Table 1). 3.6.2 SED-derived SFRs As previously stated, magphys uses a Bayesian approach to estimate the best likelihood SFR, comparing model SEDs (generated using some assumptions, see Section 2.5) with observed photometry. Due to our FIR measurements being mostly upper limits for |$\gt 99{{\ \rm per\ cent}}$| of SC4K LAEs, it is not possible to directly measure the amount of SFR that is obscured by dust and the optical thickness of dust from IR to FIR. As such, the amount of dust and SFR is inferred from the UV–optical slope. We measure SFRs in the range |$\sim 0.1\!-\!3000\, {\rm M_{\odot }\, yr^{-1}}$|⁠, and measure a median SFR|$_{\rm SED}=4.4^{+10.5}_{-2.4}$| M⊙ yr−1 for SC4K LAEs (Table 1). 3.6.3 SFRLy α versus SFRSED In this work, we estimate SFRs of individual LAEs using two approaches: emission line-based with Ly α (SFRLy α, Section 3.6.1) and from SED fitting (SFRSED, Section 3.6.2). These two approaches are independent as SFRLy α is derived directly from two properties of the Ly α emission line (luminosity and EW0), while SFRSED is obtained with magphys by removing the filter contaminated by Ly α and using up to ≈30 photometric data points from the rest-frame UV to the rest-frame FIR. In Fig. 5, we show a comparison between SFRLy α and SFRSED at different mass ranges. We measure a small systematic offset at M⋆ = 108–9 M⊙ and SFRSED = 1–10 M⊙ yr−1 for all stellar mass ranges, with the emission line-based approach predicting slightly higher SFRs. As Ly α traces more recent star formation than the UV-continuum, the higher predicted SFRs could be explained by ongoing bursts of star formation, which lead to slightly higher SFRLy α. Only for SFRs which are measured to be high from SED (SFRSED > 10 M⊙ yr−1) there is a significant difference, with SFRLy α being lower and its median maxing at ≈10 M⊙ yr−1. Such SFR ranges are typically only seen in more massive ranges (M⋆ = 109–11 M⊙), which are thus more susceptible to have underestimated SFRs from Ly α. This is in line with what could be expected for very massive galaxies as Ly α will only be able to measure the contribution in regions of the galaxy which are actively star forming and unobscured, leading to underestimated SFRs in these regimes. Nevertheless, it is remarkable that two largely independent methods obtain such similar results. For the global populations of SC4K LAEs, these two methods also retrieve very similar SFRs of |$5.9^{+6.3}_{-2.6}$| and |$4.4^{+10.5}_{-2.4}$| M⊙ yr−1 for the emission line based and SED based, respectively. Additionally, in Appendix (Fig. B1), we show SFRLy α versus SFRSED at different redshift ranges. Both approaches predict very similar SFRs at all redshifts, outside the differences at aforementioned ranges as the emission line-based approach cannot reach such SFR ranges. Figure 5. Open in new tabDownload slide Emission line-based SFR versus SED-fitting SFR for the full sample of LAEs at different stellar masses. Blue circles are the median bin and individual points are plotted as scatter in the background. The black line is the one-to-one ratio. There is a small systematic offset at M⋆ = 108–9 M⊙ yr−1 and for SFRSED = 1–10 M⊙ yr−1 for all stellar mass ranges. For higher stellar masses and SFRs, there is a more significant difference between the two methods, with the emission line-based approach predicting lower SFRs. This is a likely consequence of Ly α not being sensitive to obscured regions in very massive galaxies, thus not being sensitive to their full contribution. Additionally, we plot AGN LAEs with black stars (purely stellar + dust SED fitting with no AGN models) to show they are typically measured as having high stellar masses when blindly running SED codes with no AGN models in AGN samples. Figure 5. Open in new tabDownload slide Emission line-based SFR versus SED-fitting SFR for the full sample of LAEs at different stellar masses. Blue circles are the median bin and individual points are plotted as scatter in the background. The black line is the one-to-one ratio. There is a small systematic offset at M⋆ = 108–9 M⊙ yr−1 and for SFRSED = 1–10 M⊙ yr−1 for all stellar mass ranges. For higher stellar masses and SFRs, there is a more significant difference between the two methods, with the emission line-based approach predicting lower SFRs. This is a likely consequence of Ly α not being sensitive to obscured regions in very massive galaxies, thus not being sensitive to their full contribution. Additionally, we plot AGN LAEs with black stars (purely stellar + dust SED fitting with no AGN models) to show they are typically measured as having high stellar masses when blindly running SED codes with no AGN models in AGN samples. Furthermore, in a recent study by Calhau et al. (2019), the SFR of the SC4K sample is derived through the stacking of radio imaging in the 3 GHz band. For the stacking procedure, individual sources with direct detections are removed as these are likely AGNs. They find median SFR|$_{\rm radio}=5.1^{+1.3}_{-1.2}$| M⊙ yr−1 from the z ∼ 2–6 stack, which is in very good agreement with emission line-based and SED-based SFR estimates of the sample. 3.7 Catalogue of SC4K LAE properties With this paper, we make public a catalogue with multiple measurements for individual LAEs in the SC4K sample. For each LAE, we provide RA, Dec., LLy α, EW0, X-ray and radio flags (as given by Sobral et al. 2018a) and updated X-ray and radio flags (as given by Calhau et al. 2019), M⋆, β, MUV, SFRLy α, and SFRSED, with associated errors. We also provide our photometric measurements in Jansky for the 34 filters used in this work and a boolean SED flag which indicates unreliable SEDs. For LAEs with true SED flag, we set all SED-derived properties to −99. We provide the catalogue of SC4K LAEs in electronic format in Appendix A. 4 RESULTS AND DISCUSSION 4.1 MUV–β relation for LAEs and its evolution The UV rest-frame luminosity (MUV) and the UV β slope follow a tight correlation in UV-continuum selected samples (e.g. Bouwens et al. 2014b), with faint MUV galaxies being typically bluer (more negative β). We measure how these two parameters are correlated for LAEs, whether they follow a similar MUV–β relation as UV-continuum selected samples, and whether the relation evolves. In Fig. 6, we show the relation between MUV (Section 3.3) and β (Section 3.4) for six redshift intervals (z = 2.2, 2.5, 3.1, 3.9, 4.7, 5.4). We note that at very faint MUV we are biased towards redder sources. This is a consequence of redder sources being easier to detect in the optical filters, while sources with a very steep continuum slope will fall below our detection limits, particularly faint MUV sources. As such, in Fig. 6, we show the faintest MUV bin as unfilled. Figure 6. Open in new tabDownload slide UV-continuum slope β (measured from SED fitting, see Section 3.4) versus UV luminosity MUV (derived by integrating the SED fits at ∼1500 Å, see Section 3.3). Each panel contains LAEs from different redshift intervals (from left to right z = 2.2, 2.5, 3.1, 3.9, 4.7, 5.4). The median β of each MUV bin of LAEs selected through medium (narrow) band filters is shown as filled coloured circles (squares) with the individual points being plotted as scatter in the background. Unfilled markers are likely biased bins, as discussed in Section 4.1. The clustering of points at β = −2.44 is a physically imposed model limitation as β cannot become bluer without increasing the upper mass of the IMF to unreasonable values. For comparison we add measurements from LAEs at z ∼ 2–3 (Sobral et al. 2018b) and UV-continuum selected samples at z ∼ 2–2.5 (Hathi et al. 2016) and z ∼ 4, z ∼ 5, and z ∼ 6 (Bouwens et al. 2014b). The black arrow is the size in β of AUV = 0.5 (AUV = 4.43 + 1.99β, Meurer et al. 1999). We find the median β in LAEs to be as blue or bluer than UV-selected samples at the same MUV for all redshifts. Figure 6. Open in new tabDownload slide UV-continuum slope β (measured from SED fitting, see Section 3.4) versus UV luminosity MUV (derived by integrating the SED fits at ∼1500 Å, see Section 3.3). Each panel contains LAEs from different redshift intervals (from left to right z = 2.2, 2.5, 3.1, 3.9, 4.7, 5.4). The median β of each MUV bin of LAEs selected through medium (narrow) band filters is shown as filled coloured circles (squares) with the individual points being plotted as scatter in the background. Unfilled markers are likely biased bins, as discussed in Section 4.1. The clustering of points at β = −2.44 is a physically imposed model limitation as β cannot become bluer without increasing the upper mass of the IMF to unreasonable values. For comparison we add measurements from LAEs at z ∼ 2–3 (Sobral et al. 2018b) and UV-continuum selected samples at z ∼ 2–2.5 (Hathi et al. 2016) and z ∼ 4, z ∼ 5, and z ∼ 6 (Bouwens et al. 2014b). The black arrow is the size in β of AUV = 0.5 (AUV = 4.43 + 1.99β, Meurer et al. 1999). We find the median β in LAEs to be as blue or bluer than UV-selected samples at the same MUV for all redshifts. LAEs are found to be consistently bluer than UV-selected samples (Bouwens et al. 2014b; Hathi et al. 2016) at similar redshifts (up to ∼1 dex bluer), regardless of being NB or MB selected, at all redshifts studied (see also Hashimoto et al. 2017). Our results are consistent with z ∼ 2–3 LAEs measurements from Sobral et al. (2018b). Additionally, we measure an increase of β with MUV (∼0.5 dex per ΔMUV = 2), indicating that brighter MUV LAEs are redder at all redshift ranges, even though LAEs are typically bluer. This tight correlation between MUV and β is very similar to the one observed in LBG populations, implying an important overlap between the populations and also an important diversity within the LAE population. In Fig. 7, we show the 1σ contours for the MUV versus β distribution. We compute the 1σ contours by bootstrapping our individual data points. We choose a random subset of 50 per cent of the data points, determine the best fit, iterate the process 1000 times and define the 1σ contours as the 16th and 84th percentiles of all fits. As previously mentioned, faint MUV bins will be biased towards redder sources, which are easier to detect in the continuum. As such, we apply a MUV cut to our fits, equal to the lower limit of the faintest filled MUV bin (Fig. 6). Figure 7. Open in new tabDownload slide The evolution of the MUV–β relation for LAEs. Shaded regions are the 1σ intervals obtained by bootstrapping the individual measurements for which we are not significantly biased (see Section 4.1). β increases with MUV and this relation shifts down to smaller β as we move to higher redshifts. Most of this trend seems to be captured by a decrease in the normalization of the relation, but we also find some evidence of the relation steepening. Figure 7. Open in new tabDownload slide The evolution of the MUV–β relation for LAEs. Shaded regions are the 1σ intervals obtained by bootstrapping the individual measurements for which we are not significantly biased (see Section 4.1). β increases with MUV and this relation shifts down to smaller β as we move to higher redshifts. Most of this trend seems to be captured by a decrease in the normalization of the relation, but we also find some evidence of the relation steepening. Overall we find a MUV–β relation for LAEs, which is qualitatively very similar to the one observed in UV-selected samples. As it can be seen in Fig. 7, the normalization of the MUV–β relation slowly moves to bluer β with increasing redshift for LAEs, and there is also tentative evidence for the relation to become steeper at higher redshift. This can also be seen in Fig. 6, where the lowest redshift LAEs show a much flatter relation, while at higher redshift the relation seems to be steeper. These results might be explained due to a consistent average decrease in dust content and metallicity even within LAEs from low to high redshift. 4.2 Implications of MUV–β relation for LAEs The UV continuum β slope can be an indicator of the dust attenuation of a galaxy as well as the age and metallicity of its stellar population, but because it is sensitive to all these effects, it can also be very complicated to interpret (see e.g. Popping, Puglisi & Norman 2017). As shown by Bouwens et al. (2012, see fig. 13 therein), a negative offset of ∼0.5–1 dex in β should be dominated by a change in dust, albeit age and metallicity can also significantly steepen β, with a hotter population of stars. This suggests that LAEs are a subset of the SFG population which is very young and likely more metal-poor, with significant contribution from O and B stars which make the UV continuum steeper. In LBGs, β has been shown to depend on the UV luminosity, with a similar slope independent of redshift (e.g. Bouwens et al. 2012, 2014b). The normalization of the relation is shifted to bluer β as we move to higher redshifts which can be explained by a lower dust content/lower dust extinction in galaxies at higher redshift (e.g. Finkelstein et al. 2012). As shown in Fig. 7, LAEs have a very similar behaviour to LBG galaxies: β is tightly correlated with MUV, with brighter MUV galaxies being redder and the normalization of this slope shifting to lower β with increasing redshift, which can be explained by a lower dust content at higher redshift even for LAEs. Similar observations of the MUV–β trend and the β evolution with redshift have been shown by Hashimoto et al. (2017). The work presented by Hashimoto et al. (2017) reaches fainter MUV than the work presented here and thus provides a consistent view of UV properties in LAEs from a complementary work using a different selection method (integral field spectroscopy with muse). 4.3 Ly α EW0 and w0: evolution for LAEs? EW0 is an indicator of the strength of an emission line relatively to the continuum. As such, it holds important information about a galaxy, with high EW0 being associated with young stellar ages, low metallicities, and top-heavy IMFs (Schaerer 2003; Raiter et al. 2010). We use our sample of LAEs at well-defined redshift ranges to probe for redshift evolution of EW0. We find the median Ly α EW0 of SC4K LAEs to remain constant at ∼140 Å with redshift, both in MB and NB-selected samples (median EW|$_0=138^{+284}_{-70}$| Å). We show the little to no evolution of median EW0 in Fig. B2. For individual filters, we detect a tentative higher than average EW0 at z ∼ 5.7–5.8, which could be caused by the small sample size or higher contamination fraction, although we highlight the large error bars. The calculated median Ly α EW0 can be very sensitive to selection effects, and it is possible that the non-evolution we measure is a consequence of the relatively high EW0 > 50 Å cut applied in SC4K. In order to further tackle this, we also investigate the evolution of the scale parameter w0 (Section 3.2.1). w0 has been extensively probed in the literature (see e.g. Ciardullo et al. 2012; Hashimoto et al. 2017), particularly because the exponential decay of the EW0 distribution should be less affected by observational EW0 cuts. Our results are presented in Fig. 8. We find no statistically significant evolution of the Ly α w0 with redshift. Generally, w0 is slightly higher when determining it without any upper constraints on the Ly α EW0, and lower if we restrict its calculation to LAEs with EW0, but no significant evolution is seen when using a single self-consistent method. We therefore conclude that both the observed median Ly α EW0 and the distributions of Ly α w0 for LAEs are not changing significantly from z ∼ 2 to z ∼ 6. A non-evolution of w0 suggests there is no significant evolution in the typical or average properties of sources selected as LAEs across cosmic time. These include their typical metallicities and dust properties, but also perhaps more importantly their Ly α escape fraction, fesc, Ly α. As shown by Sobral & Matthee (2019), the observed Ly α EW0 can be used to estimate fesc, Ly α. The non-evolution of Ly α EW0 and w0 across time implies non-evolving fesc, Ly α for LAEs. For SC4K LAEs, we infer a constant fesc, Ly α of ≈0.6–0.7 across cosmic time (≈0.5–0.6 when applying the EW0 > 240 Å cut). These median fesc, Ly α values are consistent with those derived using radio SFRs for SC4K Ly α emitters (0.7 ± 0.2, see Calhau et al. 2019). Figure 8. Open in new tabDownload slide Global Ly α w0 evolution with redshift. Best w0 estimates are shown as blue circles (squares) for the full range of EW0 (EW0 < 240 Å). Blue contours are estimated by perturbing the w0 bins within error bars (see Section 3.2.1 for details). We find evidence for little to no evolution of w0. The white points show Ly α w0 of the full SC4K sample. We present a compilation of Ly α w0 from z = 0.3 to z ∼ 6 (Gronwall et al. 2007; Nilsson et al. 2009; Guaita et al. 2010; Blanc et al. 2011; Ciardullo et al. 2012; Wold, Barger & Cowie 2014; Hashimoto et al. 2017; Wold et al. 2017). In addition, we show the [O ii] (H β + [O iii]) rest-frame EWs of emitters selected by these lines (Khostovan et al. 2016) as purple (green) fits and H α EW0 (Faisst et al. 2016; Matthee et al. 2017a) as red. Overall, the consensus of all data points is that there is no significant Ly α w0 evolution with redshift despite the strong increase in the typical EW0 of non-resonant lines for a wider population of SFGs. Figure 8. Open in new tabDownload slide Global Ly α w0 evolution with redshift. Best w0 estimates are shown as blue circles (squares) for the full range of EW0 (EW0 < 240 Å). Blue contours are estimated by perturbing the w0 bins within error bars (see Section 3.2.1 for details). We find evidence for little to no evolution of w0. The white points show Ly α w0 of the full SC4K sample. We present a compilation of Ly α w0 from z = 0.3 to z ∼ 6 (Gronwall et al. 2007; Nilsson et al. 2009; Guaita et al. 2010; Blanc et al. 2011; Ciardullo et al. 2012; Wold, Barger & Cowie 2014; Hashimoto et al. 2017; Wold et al. 2017). In addition, we show the [O ii] (H β + [O iii]) rest-frame EWs of emitters selected by these lines (Khostovan et al. 2016) as purple (green) fits and H α EW0 (Faisst et al. 2016; Matthee et al. 2017a) as red. Overall, the consensus of all data points is that there is no significant Ly α w0 evolution with redshift despite the strong increase in the typical EW0 of non-resonant lines for a wider population of SFGs. However, it should be noted that different redshifts do not necessarily probe the same MUV ranges (Fig. 4, middle panel), which should be considered when discussing w0 evolution with redshift, particularly as w0 depends on MUV (see Section 4.3.2). We attempt to explore potential bias effects by computing w0 with a consistent MUV cut. For the full SC4K sample, we compute w0 for −22 < MUV < −19 which is a MUV range probed by all redshifts (see middle panel of Fig. 4). For this cut, a flat relation (non-evolution) is still observed within 0.9σ for EW0 < 240 Å and 1.8σ for the full EW0 range. The different MUV ranges probed by different selection filters/redshifts do not seem to be sufficient to explain the non-evolution of w0 with redshift, which is likely a characteristic of the LAE population itself. 4.3.1 Comparison with other studies In order to compare our results with other studies across different redshifts, in Fig. 8 we show a compilation of Ly α w0 in samples of LAEs, from z ∼ 0 to z ∼ 6 (Gronwall et al. 2007; Nilsson et al. 2009; Guaita et al. 2010; Blanc et al. 2011; Ciardullo et al. 2012; Wold et al. 2014, 2017; Hashimoto et al. 2017). Our results agree well with Hashimoto et al. (2017), Guaita et al. (2010), and Blanc et al. (2011). Furthermore, our extrapolation of w0 to low redshift is consistent with the results from Wold et al. (2014, 2017). Our measurements reveal higher values than those by Nilsson et al. (2009), Gronwall et al. (2007), and Ciardullo et al. (2012), all at intermediate redshifts (z = 2.25–3.1) and with selections that go to much lower EWs. We note however that the w0 measured by Nilsson et al. (2009) is below our MB detection threshold and that our blind selection of LAEs is not sensitive to the lowest EW0, as highlighted in Fig. 8. Our LAE selection of high EW LAEs is much more similar to blind surveys done with muse (Hashimoto et al. 2017), but SC4K allows the selection and study of much higher luminosity LAEs. Furthermore, we note that our w0 measurements shift to smaller values when the EW0 < 240 Å cut is applied, becoming even more similar to the measurements reported in the literature. While there are observed variations due to different sample selections which contribute to the scatter (Fig. 8), overall we conclude that there is no clear evolution of the Ly α EW0 and w0 for LAEs when taking into account all measurements. Such parameters remaining constant for LAEs contrasts with measurements from other non-resonant emission lines for the general star-forming population, which are found to increase significantly with redshift. In order to provide a rough comparison, in Fig. 8 we also show the redshift evolution of the rest-frame EW of line emitters, including [O ii] and H β + [O iii] emitters (Khostovan et al. 2016) and H α EW0 (Sobral et al. 2014). While at z ∼ 0 those non-resonant rest-frame optical emission lines have typical EW0 < 25 Å, by z ∼ 2 they already exceed Ly α EW0. This reveals a very significant evolution of the typical stellar populations of the general population of SFGs, while those selected to be LAEs have high Ly α EW0 at all cosmic times. Since LAEs have typically high EWs in their rest-frame optical lines, it is very likely that we are seeing SFGs becoming, on average, LAEs, towards z ∼ 6. Such possibility would easily explain the rise in the global Ly α/UV luminosity densities (see full discussion and implications in Sobral et al. 2018a). 4.3.2 The w0 and fesc, Ly α dependence on M⋆ and MUV LAEs seem to show no evolution in their typical Ly α w0 across cosmic time. However, one could expect that LAEs with different physical properties may show different w0, particularly as a consequence of different Ly α escape fractions (see e.g. Matthee et al. 2016; Oyarzún et al. 2017; Sobral & Matthee 2019). We start by investigating how Ly α w0 may depend on the stellar mass of LAEs. The results are presented on the left-hand panel of Fig. 9, where we show the results when restricting the measurements to EW0 < 240 Å and when using full samples. We find an anticorrelation between Ly α w0 and stellar mass, with the least massive LAEs having w0 ≈ 180 Å and the most massive having w0 ≈ 70 Å. By using Sobral & Matthee (2019), this could be seen as a significant difference in the typical fesc, Ly α which would decline from ≈90 per cent for |$\rm \mathit{ M}\sim 10^{8.5}$| M⊙ LAEs to f|$_{\rm esc, Ly\,\alpha }\approx 30{{\ \rm per\ cent}}$| for |$\rm \mathit{ M}\sim 10^{10.5}$| M⊙ LAEs. This trend is very similar to those found by Matthee et al. (2016) for a general population of H α emitters with much higher SFRs and lower fesc, Ly α than our LAEs and by Oyarzún et al. (2017). Figure 9. Open in new tabDownload slide The Ly α w0 dependence on M⋆ and MUV. Best w0 estimates are shown as blue circles (squares) for the full range of EW0 (EW0 < 240 Å). A label with fesc, Ly α (=0.048w0; Sobral & Matthee 2019) is added for a potential physical interpretation of results. Left: Ly α w0 is anticorrelated with stellar mass, such that the most massive LAEs have the lowest w0 and likely the lowest fesc, Ly α. Right: Ly α w0 is also anticorrelated with UV luminosity, with the faintest UV LAEs having the highest Ly α w0. Figure 9. Open in new tabDownload slide The Ly α w0 dependence on M⋆ and MUV. Best w0 estimates are shown as blue circles (squares) for the full range of EW0 (EW0 < 240 Å). A label with fesc, Ly α (=0.048w0; Sobral & Matthee 2019) is added for a potential physical interpretation of results. Left: Ly α w0 is anticorrelated with stellar mass, such that the most massive LAEs have the lowest w0 and likely the lowest fesc, Ly α. Right: Ly α w0 is also anticorrelated with UV luminosity, with the faintest UV LAEs having the highest Ly α w0. In Fig. 9 (right-hand panel), we also show how Ly α w0 is clearly anticorrelated with MUV. Our results show that UV luminous LAEs in our sample (MUV ≈ −21.5) have Ly α w0 ≈ 50 Å, which rises with declining UV luminosity to w0 ≈ 180 Å for MUV ≈ −19.5 LAEs. This implies that the UV faintest sources have the highest fesc, Ly α (Sobral & Matthee 2019) of around ≈85 per cent, while the most UV luminous LAEs have fesc, Ly α ≈ 20–30 per cent. Our results are in good agreement with Oyarzún et al. (2017) and reveal that even though LAEs have high Ly α w0 across cosmic time, the population still shows important trends with stellar mass and rest-frame UV luminosity. 4.3.3 LAEs with extreme EW0 The nature of LAEs with extremely high EW0 and the processes behind the creation of such extreme lines are still a relatively unexplored topic despite a range of discoveries (e.g. Cantalupo et al. 2012; Kashikawa et al. 2012; Hashimoto et al. 2017; Maseda et al. 2018). Typical internal star formation processes should not be enough to power EW0 > 240 Å in Ly α (Schaerer 2003; Raiter et al. 2010), but studies like Cantalupo et al. (2012) suggest that such extreme objects which have been found could be explained by fluorescent ‘illumination’ from e.g. a nearby quasar (see also Rosdahl & Blaizot 2012; Yajima et al. 2012). Additionally, an extreme z = 6.5 LAE with EW0 = 436 Å is reported in Kashikawa et al. (2012), with the authors arguing that such a high EW0 requires a very young, massive and metal-poor stellar population, or even Population III stars. The large volume covered by SC4K (∼108 Mpc3) and the sensitivity to the highest EWs provides a unique opportunity to identify and quantify the number density of extremely high EW LAEs. In order to do so in a conservative way, rather than simply selecting sources with Ly α EW0 higher than 240 Å, we take the photometric errors fully into account, and we use the 3σ errors. In practice, we look for LAEs within SC4K which satisfy EW0 > 240 Å at a 3σ level7 and for which we have no evidence of AGN activity. We find a total of 45 ‘extreme’ non-AGN LAEs in ∼61.5 × 106 Mpc3 and we investigate how these are distributed across redshift. The results are shown in Table 3, where we use Poisson errors. Most of the extreme LAEs are found at z ∼ 2–3. Furthermore, by taking into account the volumes surveyed, we find that the number density of extreme LAEs within SC4K rises, from (0.12 ± 0.08) × 10−6 Mpc−3 at z ∼ 5.4 to (1.50 ± 0.61) × 10−6 Mpc−3 at z ∼ 2.5, although such increase should be treated with caution, as the higher redshift sample does not reach very faint MUV (>−20) ranges. Overall, we find a number density of (0.73 ± 0.11) × 10−6 Mpc−3 at z ∼ 2–6, revealing that these sources are exceptionally rare. At 1σ confidence level, we find 318 LAEs with EW0 > 240 Å, resulting in a number density of (5.17 ± 0.29) × 10−6 Mpc−3. Spectroscopic follow-up observations are required to further understand their nature. We find our 45 ‘extreme’ sources to be a diverse population, as they are found at all Ly α luminosities and stellar masses, but preferentially at faint UV luminosities which is a consequence of high EW + ‘random’ Ly α luminosities. They typically have blue UV β slopes but some reach redder values (β ∼ −1.2). We do not observe a spatial correlation between ‘extreme’ LAEs and AGN, which we would expect if the high EWs in this sample of LAEs were generated by fluorescent ‘illumination’. Table 3. Number count and number density of LAEs with EW0 > 240 Å at a 3σ level, for different redshift intervals, using comoving volumes from Sobral et al. (2018a). Errors are Poissonian. We find very low number densities of extreme LAEs, but these increase with decreasing redshift. Redshift interval . N . Φ . . (# LAEs) . (10−6 Mpc−3) . MB, z = 2.5 ± 0.1 6 (±2) 1.50 ± 0.61 MB, z = 3.1 ± 0.4 15 (±4) 0.82 ± 0.21 MB, z = 3.9 ± 0.3 4 (±2) 0.40 ± 0.20 MB, z = 4.7 ± 0.2 2 (±1) 0.17 ± 0.12 MB, z = 5.4 ± 0.5 2 (±1) 0.12 ± 0.08 Full sample 45 (±7) 0.73 ± 0.11 Redshift interval . N . Φ . . (# LAEs) . (10−6 Mpc−3) . MB, z = 2.5 ± 0.1 6 (±2) 1.50 ± 0.61 MB, z = 3.1 ± 0.4 15 (±4) 0.82 ± 0.21 MB, z = 3.9 ± 0.3 4 (±2) 0.40 ± 0.20 MB, z = 4.7 ± 0.2 2 (±1) 0.17 ± 0.12 MB, z = 5.4 ± 0.5 2 (±1) 0.12 ± 0.08 Full sample 45 (±7) 0.73 ± 0.11 Open in new tab Table 3. Number count and number density of LAEs with EW0 > 240 Å at a 3σ level, for different redshift intervals, using comoving volumes from Sobral et al. (2018a). Errors are Poissonian. We find very low number densities of extreme LAEs, but these increase with decreasing redshift. Redshift interval . N . Φ . . (# LAEs) . (10−6 Mpc−3) . MB, z = 2.5 ± 0.1 6 (±2) 1.50 ± 0.61 MB, z = 3.1 ± 0.4 15 (±4) 0.82 ± 0.21 MB, z = 3.9 ± 0.3 4 (±2) 0.40 ± 0.20 MB, z = 4.7 ± 0.2 2 (±1) 0.17 ± 0.12 MB, z = 5.4 ± 0.5 2 (±1) 0.12 ± 0.08 Full sample 45 (±7) 0.73 ± 0.11 Redshift interval . N . Φ . . (# LAEs) . (10−6 Mpc−3) . MB, z = 2.5 ± 0.1 6 (±2) 1.50 ± 0.61 MB, z = 3.1 ± 0.4 15 (±4) 0.82 ± 0.21 MB, z = 3.9 ± 0.3 4 (±2) 0.40 ± 0.20 MB, z = 4.7 ± 0.2 2 (±1) 0.17 ± 0.12 MB, z = 5.4 ± 0.5 2 (±1) 0.12 ± 0.08 Full sample 45 (±7) 0.73 ± 0.11 Open in new tab Through a narrow-band filter search, Cantalupo et al. (2012) targeted a field centred in a hyper luminous quasar and identified 18 LAEs at z = 2.4 in a comoving volume of 5500 Mpc3. Stacking of these sources results in EW0 > 800 Å (1σ), which cannot be explained by typical star formation processes. This implies a higher number density of extreme LAEs than the conservative number density we report in this paper, although this can be easily explained by Cantalupo et al. (2012) specifically targeting a quasar field. In a more comparable blank search, using deep MUSE data, Hashimoto et al. (2017) selected six LAEs with EW0 > 240 Å at a 1σ level (zero at 3σ) in 9.31 × 104 Mpc3 (Drake et al. 2017) at z ∼ 2–6. This results in a number density of ∼6 × 10−5 Mpc−3, suggesting these ‘extreme’ LAEs may be even more common at fainter luminosities than those in the SC4K sample. 4.4 SFR–M⋆ relation and evolution We test the dependence of SFR on M⋆ in our sample of LAEs and its potential evolution with redshift. In Fig. 10, we show SFR derived from Ly α and EW0 (see Section 3.6) versus M⋆ (derived from SEDs, Section 3.5) for our sample of LAEs and compare with SFRs derived from SED fitting. We compare our measurements with the MS relation as derived in Schreiber et al. (2015) (converted from Salpeter to Chabrier IMF, extrapolated to low-mass ranges when required) and a few studies at different redshifts. Figure 10. Open in new tabDownload slide SFR (derived from Ly α and EW0 and derived from SED fits, see Section 3.6) versus M⋆ (derived from SED fits, see Section 2.5). Each panel contains LAEs from different redshift intervals (from left to right z = 2.2, 2.5, 3.1, 3.9, 4.7, 5.4). The median SFRLy α of each M⋆ bin for LAE selected through medium (narrow) band filters is shown as filled coloured circles (squares) with the individual points being plotted as scatter in the background. The median SFRSED of each M⋆ bin for LAE selected through medium (narrow) band filters is shown as open circles (squares). The dotted horizontal line is the average SFR depth, computed from the flux depth and average EW0 of the sample. The continuous black lines are the best-fitting relations from Schreiber et al. (2015) computed for the redshift of each panel and converted from Salpeter to Chabrier. These relations are shown as dashed lines for the mass ranges where they were extrapolated. Figure 10. Open in new tabDownload slide SFR (derived from Ly α and EW0 and derived from SED fits, see Section 3.6) versus M⋆ (derived from SED fits, see Section 2.5). Each panel contains LAEs from different redshift intervals (from left to right z = 2.2, 2.5, 3.1, 3.9, 4.7, 5.4). The median SFRLy α of each M⋆ bin for LAE selected through medium (narrow) band filters is shown as filled coloured circles (squares) with the individual points being plotted as scatter in the background. The median SFRSED of each M⋆ bin for LAE selected through medium (narrow) band filters is shown as open circles (squares). The dotted horizontal line is the average SFR depth, computed from the flux depth and average EW0 of the sample. The continuous black lines are the best-fitting relations from Schreiber et al. (2015) computed for the redshift of each panel and converted from Salpeter to Chabrier. These relations are shown as dashed lines for the mass ranges where they were extrapolated. We find that in general there is a relation between SFR and M⋆ at all redshifts for LAEs. The relation is relatively shallow when using Ly α SFRs and steeper when using SED SFRs, as can be seen in Fig. 10. The relation between SFR and M⋆ seems to steepen with increasing redshift for LAEs when using SED SFRs, as can also be seen in Fig. 11 (right-hand panel). This steepening with increasing redshift also seems to make the SFR–M⋆ relation much more in line with the extrapolated relations found for UV-continuum selected sources (e.g. Schreiber et al. 2015). Figure 11. Open in new tabDownload slide Left: Running average of SFR (derived from Ly α and EW0, see Section 3.6) versus M⋆ (derived from SED fits, see Section 2.5). Right: Same but with SFR derived from magphys (see Section 3.6). The SFR–M⋆ slopes derived from the two methods are different, with the SED-derived slope being steeper. The difference is likely a consequence of SFRLy α not being able to reach very low (<1 M⊙ yr−1) and very high SFRs (>20–30 M⊙ yr−1), but we provide further discussion in Section 3.6.3. For comparison, we show the MS line for UV-continuum selected sources from Schreiber et al. (2015), where the dashed lines show the extrapolated values. Figure 11. Open in new tabDownload slide Left: Running average of SFR (derived from Ly α and EW0, see Section 3.6) versus M⋆ (derived from SED fits, see Section 2.5). Right: Same but with SFR derived from magphys (see Section 3.6). The SFR–M⋆ slopes derived from the two methods are different, with the SED-derived slope being steeper. The difference is likely a consequence of SFRLy α not being able to reach very low (<1 M⊙ yr−1) and very high SFRs (>20–30 M⊙ yr−1), but we provide further discussion in Section 3.6.3. For comparison, we show the MS line for UV-continuum selected sources from Schreiber et al. (2015), where the dashed lines show the extrapolated values. At z < 4, we find that LAEs are typically above the MS relation at their corresponding redshift. This is particularly evident for low stellar masses (M⋆ < 109.5 M⊙) although we find that more massive LAEs tend to be within the MS or even below it, a consequence of the slope of the relation being shallower. At higher redshifts, we find that even at low stellar masses (109.0–9.5 M⊙) LAEs are closer to the MS or that the MS becomes closer to the relation valid for LAEs, as SFGs may become more LAE-like. Our results therefore suggest that at higher redshifts there is a wider overlap between LAEs and more ‘normal’ populations of galaxies, as UV-continuum selected galaxies become LAE-like. This could explain the agreement between high-z LAEs and the results of Salmon et al. (2015). It is none the less important to point out (as shown in Fig. 10) that the flux limit in Ly α corresponds to a rough cut in SFR and therefore a bias towards higher SFRs at the lowest masses. Similar flux cuts also affect continuum-selected samples, placing them well above the MS (see e.g. Tasca et al. 2015). Our results are in good agreement with measurements from Ly α-selected samples from Kusakabe et al. (2018) at z = 2 within error bars. We also compare our results with those presented by Harikane et al. (2018). While we do not reach such low masses, our results are consistent with LAEs being above the MS at low stellar masses. With our SC4K sample of LAEs, we can now analyse the evolution of the SFR–M⋆ in wide mass ranges at different redshifts, no longer being constrained by single bins or having to stack sources to SED fit the stacked photometry, being able to probe the evolution of the relation within the same sample. As previously discussed in Section 3.6.3, there are limitations to different SFR methods, which are important to highlight when comparing the SFR–M⋆ relation. SFRLy α consistently predicts higher SFR than SFRSED for low stellar masses and lower SFR for very high stellar masses. In fact, individual measurements of SFRLy α seem to fully saturate at ∼100 M⊙ yr−1 with the medians typically not going above ∼20–30 M⊙ yr−1 (see Sobral et al. 2018b). SFRLy α also implies higher SFRs at lower masses, possibly due to tracing more recent star formation which would be higher than the one measured from the continuum if LAEs are going through bursts of star formation. This can be clearly seen with NB LAEs measurements at z = 5.7, where the low luminosity sample predicts SFRs ≈ 10 M⊙ yr−1. SFRSED may be better suited for such conditions, and as seen in Fig. 10, it points towards a relation similar to the Schreiber et al. (2015) extrapolations for the entire mass range we can probe. Nevertheless, we find that the SFRs derived from the two approaches to be consistent, with the same trends being observed from both. In Fig. 11, we show the running averages for M⋆ versus SFR. We find the normalization of the relation to increase with redshift (left-hand panel) but, as previously discussed, this is mostly driven by detection limits, as we are only capable of reaching down to SFR < 5 M⊙ yr−1 at z ∼ 2. 4.5 LAEs: are they ‘main-sequence’ galaxies? The stellar mass of a galaxy and its SFR are correlated in typical galaxies, creating a trend known as the ‘main sequence’8 of SFGs (Brinchmann et al. 2004; Noeske et al. 2007). A priori, we can naively expect this correlation to occur as the stellar mass of a galaxy is the integral of SFR across time, so the total amount of stars produced will be proportional to the current SFR, assuming a continuous SFR. This dependence can lead to ‘tracks’ in SED-fitting derived values which lead to a more stringent correlation between SFR and M⋆. Galaxies going through periods of intensive star formation, which may be a consequence of bursty star formation, will occupy a region above the MS. In typical galaxies, SFR and M⋆ are in tight correlation and the normalization of the relation increases with redshift (e.g. Schreiber et al. 2015). Understanding whether the MS trend holds for LAEs provides important insight into how star formation occurs and how it is driven in this population of predominantly early, primeval galaxies. In principle, we do not expect a Ly α-selected sample to span uniformly around the MS, because we select an emission line strength which at fixed stellar mass always gives high sSFR ≡ SFR/M⋆. We therefore do not expect to use LAEs to measure the MS in an unbiased way, but we can use the comparison to the MS to determine how LAEs fit in the general galaxy population. Several measurements at z > 2 have measured the MS relation by probing M⋆ > 1010 M⊙, with the low-mass limit typically rising to M⋆ > 1011 M⊙ at z > 3.5 (Schreiber et al. 2015), but some recent studies have measured the SFR–M⋆ slope and scatter down to M⋆ = 109 M⊙ (Salmon et al. 2015). Our sample of high redshift, typically low M⋆ SFGs reaches a region still widely uncharted at these redshift ranges. Our results point towards an intensive star formation nature for low-mass LAEs at z < 4, which places them significantly above the extrapolation of the MS to the lowest masses. A more bursty star-forming nature could explain these SFRs above the MS. However, we cannot directly infer burstiness from our measurements. More massive LAEs seem to fall within the MS. At higher redshifts, SFRSED–M⋆ measurements for LAEs start to resemble more the MS at all mass ranges. We also find SFRLy α–M⋆ to follow an MS-like relation at z > 4, except for M⋆ ≳ 1010.5, when SFRLy α seems to saturate, likely due to dust, and is not able to reach SFRs as high as SFRSED. This can easily be explained by more massive galaxies showing much higher dust extinction (see e.g. Garn & Best 2010; Sobral et al. 2012; Whitaker et al. 2017), which at some point might completely absorb Ly α and UV photons in high SFR regions (Sobral et al. 2018b), making it impossible for them to be observed. In such cases, the FIR and some visible and NIR light can still escape, leading to a large discrepancy between SFRSED and SFRLy α. We note that SFRLy α contains an empirical correction for dust extinction (see Sobral & Matthee 2019), but this was calibrated for typical LAEs where only moderate to low levels of dust extinction are present leading to Ly α and UV photons being attenuated, but not fully destroyed. At the highest masses, we are likely seeing LAEs with several star-forming regions that may be completely invisible in the UV and Ly α but where at least one region has a hole or a porous ISM (see also Popping et al. 2017). Overall, we find that the SFR–M⋆ relation for LAEs steepens with redshift and that its normalization also rises with look-back time (see Fig. 11). As a consequence, by z ∼ 5–6, LAEs and the general UV-continuum selected population essentially become indistinguishable. This increasing overlap of populations with increasing redshift is also observed in the morphologies and sizes of SFGs, which become LAE-like (compact, re ∼ 1 kpc) towards high redshift (Paulino-Afonso et al. 2018) and diverge towards lower redshift as LAEs remain compact at all redshifts. Our results are also fully consistent with the rapid rise of the cosmic average Ly α/UV luminosity density ratio with increasing redshift (Sobral et al. 2018a), which imply that a higher fraction of SFGs share the properties associated with LAEs, leading to a rise of the cosmic averaged Ly α escape fraction and the cosmic averaged ionization efficiency, ξion. Such results are also in agreement with other studies showing a rise of the LAE fraction in UV-selected sources towards z ∼ 6 (Curtis-Lake et al. 2012; Schenker et al. 2014; Stark et al. 2017), and globally imply that by z ∼ 6 LAEs become representative of the majority of the star-forming population. 5 CONCLUSIONS In this work, we determined and explored key properties of a large sample of LAEs from the publicly available SC4K survey (∼4000 LAEs at z ∼ 2–6 in the COSMOS field; Sobral et al. 2018a). We conducted PSF photometry over 34 bands from rest-frame UV to FIR and derived the best-fitting SEDs using magphys. We computed SFRs, MUV, β, and M⋆ for each individual LAE and we provide a full catalogue of SC4K LAEs with all the photometric measurements and derived properties. Our main results are SC4K LAEs are typically low stellar mass sources (median M⋆ = 10|$^{9.3^{+0.6}_{-0.5}}$| M⊙), very blue in the rest-frame UV (β = −2.1 |$^{+0.5}_{-0.4}$|⁠), and have low SFRs (SFRLy α: |$5.9^{+6.3}_{-2.6}$| M⊙ yr−1; SFRSED: |$4.4^{+10.5}_{-2.4}$| M⊙ yr−1). We observe a tight correlation between β and MUV, qualitatively similar to the one observed in UV-selected samples. The normalization of this correlation shifts to smaller β (bluer) with increasing redshift, which is consistent with a decreasing dust content with increasing redshift in galaxies even for LAEs. Our LAEs are as blue or bluer than UV-selected LBGs at similar redshifts (up to ∼1 dex in the redshift range z ∼ 2–6), suggesting they always constitute the youngest, most metal-poor and/or most dust-poor subset of the UV-selected sources. We find evidence for little to no evolution in the typical Ly α EW0 and the scale parameter w0 with redshift, suggesting the median fesc, Ly α in LAEs is always high and not evolving strongly with redshift. We find that the Ly α w0 (and thus fesc, Ly α) for LAEs declines with increasing stellar mass, implying that fesc, Ly α is highest for the lowest stellar mass LAEs and lowest for the most massive LAEs. A similar trend is found with rest-frame UV luminosity, where the faintest LAEs have the highest typical EWs and the highest fesc, Ly α. We explore extreme EW0 measurements in our large sample of LAEs and find 45 non-AGN LAEs with EW0 > 240 Å at a 3σ level, resulting in a number density (7 ± 1) × 10−7 Mpc−3. These extreme emitters are incredibly rare but can provide insight into extreme Ly α emission that is neither purely from typical star formation or AGNs. By using Ly α EW0 to infer fesc, Ly α (Sobral & Matthee 2019), we compute Ly α SFRs which are independent of SED fitting measurements and we compare both. Ly α and SED-fitting based SFRs show a remarkable agreement for M⋆ = 109–10 M⊙ and SFRSED = 1–10 M⊙ yr−1. SFRLy α predicts lower SFRs at more massive regimes, likely due to not being sensitive to heavily obscured parts of very massive galaxies. LAEs show a relation between stellar mass and SFR at all redshifts, but this is typically shallower than the relation found for the general star-forming population. We also find that the relation steepens and rises with increasing redshift for LAEs. LAEs are typically above the ‘main sequence’ at z < 4 and M⋆ < 109.5 M⊙, indicating LAEs are experiencing more intense star formation than the general population of galaxies of similar mass at similar redshifts, with one possible explanation being a bursty star formation nature of LAEs. For higher masses and redshifts, this offset decreases, implying a larger overlap between LAEs and more ‘normal’ SFGs. Overall, we find that LAEs are typically very young, low-mass galaxies, albeit they still span an important range of properties, and within the LAE population there are important trends with stellar mass and UV luminosity. Typical properties of LAEs seem to have little evolution between z = 2 and z = 6, although they still become bluer and the relation between SFR and stellar mass steepens and rises slightly. By z ≳ 4, the overlap between LAEs and the more general UV-selected population becomes significant and by z ∼ 6 they seem to become undistinguishable, as typical SFGs essentially become LAE-like. Our results reveal how galaxies selected as LAEs constitute mostly the youngest, most primeval galaxies at any redshift, and also that LAEs are ideal sources to study the dominant population of SFGs towards z ≳ 6 and therefore also likely the population that reionized the Universe. SUPPORTING INFORMATION SC4K_properties_Santos19_to_submit_141219.fits Table B1.EW0 scale length (w0) for different redshift bins, derived as fully detailed in Section 3.2.1. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. ACKNOWLEDGEMENTS We thank the anonymous referee for the valuable feedback that significantly improved the quality and clarity of this paper. SS and JC acknowledge studentships from Lancaster University. APA acknowledges support from Fundação para a Ciência e a Tecnologia through the project PTDC/FIS-AST/31546/2017. The authors would like to thank Ali Khostovan, Sara Perez Sanchez, Alex Bennett, and Tom Rose for contributions and discussions in the early stages of this work. Based on data products from observations made with European Southern Observatory (ESO) Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by Complementary and Legacy data for the Euclid Telescope (CALET) and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. Finally, the authors acknowledge the unique value of the publicly available analysis software topcat (Taylor 2005) and publicly available programming language python, including the numpy, pyfits, matplotlib, scipy, and astropy (Astropy Collaboration 2013) packages. 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Open in new tabDownload slide Emission line-based SFR versus SED-fitting SFR for the full sample of LAEs at different redshift ranges. Coloured circles (squares) are the median bin for MBs (NBs) and individual points are plotted as scatter in the background. The black line is the one-to-one ratio. While the two approaches roughly follow the one-to-one ratio, there are some key differences. Similar to what is observed in Fig. 5, median SFRLy α is slightly higher than SFRSED for SFRSED < 10 M⊙ yr−1. However, SFRLy α seems to saturate at median SFRLy α ≈ 10–30 M⊙ yr−1. Figure B1. Open in new tabDownload slide Emission line-based SFR versus SED-fitting SFR for the full sample of LAEs at different redshift ranges. Coloured circles (squares) are the median bin for MBs (NBs) and individual points are plotted as scatter in the background. The black line is the one-to-one ratio. While the two approaches roughly follow the one-to-one ratio, there are some key differences. Similar to what is observed in Fig. 5, median SFRLy α is slightly higher than SFRSED for SFRSED < 10 M⊙ yr−1. However, SFRLy α seems to saturate at median SFRLy α ≈ 10–30 M⊙ yr−1. Figure B2. Open in new tabDownload slide Global median EW0 evolution with redshift. The median EW0 values for medium (narrow) bands are shown as blue circles (green stars). Blue stars are the measurements for individual MBs. The thin (thick) error bars are the 16th and 84th percentiles of the EW0 distribution (divided by the Poissonian error |$\sqrt{N}$|⁠). The median and errors of EW0 can be found in Table 1. Blue (green) shaded region is the 1σ contour obtained by perturbing the EW0 within the thick error bars for medium (narrow) band selected LAE. We find evidence of little EW0 evolution with redshift for the global sample of LAEs, with the median EW0 remaining roughly constant at ∼140 Å, although there is a tentative higher EW0 at z = 5.7, albeit with large error bars. Figure B2. Open in new tabDownload slide Global median EW0 evolution with redshift. The median EW0 values for medium (narrow) bands are shown as blue circles (green stars). Blue stars are the measurements for individual MBs. The thin (thick) error bars are the 16th and 84th percentiles of the EW0 distribution (divided by the Poissonian error |$\sqrt{N}$|⁠). The median and errors of EW0 can be found in Table 1. Blue (green) shaded region is the 1σ contour obtained by perturbing the EW0 within the thick error bars for medium (narrow) band selected LAE. We find evidence of little EW0 evolution with redshift for the global sample of LAEs, with the median EW0 remaining roughly constant at ∼140 Å, although there is a tentative higher EW0 at z = 5.7, albeit with large error bars. Author notes Zwicky Fellow. © 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)