Accretion of satellites on to central galaxies in clusters: merger mass ratios and orbital parameters

Accretion of satellites on to central galaxies in clusters: merger mass ratios and orbital... Abstract We study the statistical properties of mergers between central and satellite galaxies in galaxy clusters in the redshift range 0 < z < 1, using a sample of dark-matter only cosmological N-body simulations from Le SBARBINE data set. Using a spherical overdensity algorithm to identify dark-matter haloes, we construct halo merger trees for different values of the overdensity Δc. While the virial overdensity definition allows us to probe the accretion of satellites at the cluster virial radius rvir, higher overdensities probe satellite mergers in the central region of the cluster, down to ≈0.06rvir, which can be considered a proxy for the accretion of satellite galaxies on to central galaxies. We find that the characteristic merger mass ratio increases for increasing values of Δc: more than 60 per cent of the mass accreted by central galaxies since z ≈ 1 comes from major mergers. The orbits of satellites accreting on to central galaxies tend to be more tangential and more bound than orbits of haloes accreting at the virial radius. The obtained distributions of merger mass ratios and orbital parameters are useful to model the evolution of the high-mass end of the galaxy scaling relations without resorting to hydrodynamic cosmological simulations. galaxies: clusters: general, galaxies: elliptical and lenticular, cD, galaxies: evolution, galaxies: formation, dark matter 1 INTRODUCTION Central galaxies (CGs) in galaxy groups and clusters are typically massive early-type galaxies with relatively old stellar populations and little ongoing star formation. CGs are believed to form in two phases (Merritt 1985; Tremaine 1990; Dubinski 1998; Ruszkowski & Springel 2009; Lauer et al. 2014). A first phase of in situ star formation at redshift z ≳ 1 is followed by a second phase of growth via the so-called galactic cannibalism process (Ostriker & Tremaine 1975; White 1976; Hausman & Ostriker 1978), that is accretion of satellite galaxies driven by dynamical friction (Chandrasekhar 1943). Quantitatively, both theoretical (De Lucia & Blaizot 2007; Feldmann et al. 2010; Tonini et al. 2012; Shankar et al. 2015) and observational (Marchesini et al. 2014; Bellstedt et al. 2016; Buchan & Shankar 2016; Vulcani et al. 2016) arguments suggest that about half of the stellar mass of CGs is assembled in situ at z ≳ 1, and the other half is assembled at relatively late times (z ≲ 1) via cannibalism processes. The effect of this cannibalism-driven growth phase on the properties of the CG (for instance size and velocity dispersion) is determined not only by the properties of the cannibalised galaxies (e.g. mass ratio between satellite and central; Naab, Johansson & Ostriker 2009), but also by the merging orbital parameters (Boylan-Kolchin, Ma & Quataert 2006; Nipoti et al. 2012). Measures of size, velocity dispersion, luminosity, and stellar mass of observed CGs lie on tight scaling relations (Bernardi et al. 2007; Liu et al. 2008; Vulcani et al. 2014). Knowing the properties of the mergers that occur during the late growth of CGs is thus important to theoretically understand the origin and evolution of their scaling relations. Nipoti (2017, hereafter N17) made the point that, given the very special location of CGs, at the bottom of the deep potential well of the host group or cluster, the distribution of the orbital parameters of the central-satellite encounters can be quite different from that of the encounters between galaxies not belonging to groups or clusters. N17 has characterized the distribution of the orbital parameters for central-satellite mergers using idealized N-body simulations in which the host system (a cluster or a group) is modelled as an isolated, spherical, collisionless N-body system and the satellite is rigid, being represented by a single massive, softened particle. In particular, the simulations of N17 are not framed within a cosmological context: the initial orbital parameters of the satellites are extracted from the host-halo distribution function, based on the assumption that violent relaxation (Lynden-Bell 1967) is rapid and the satellite population does not retain much memory of the cosmological distribution of the orbital parameters at the time of infall. The orbital parameters of the satellites then evolve due to dynamical friction (i.e. the satellites lose orbital energy and angular momentum). In this work we improve on the analysis of N17 by considering the problem in a fully cosmological setting, focusing on the growth of CGs in clusters of galaxies. For this purpose, we take advantage of the suite of cosmological simulations Le SBARBINE (Despali et al. 2016). These simulations are dark-matter only and so they do not contain a realistic galaxy population. Nevertheless, if we assume that CGs sit at the centre of dark-matter haloes and if we select the central regions of these host haloes at overdensities typical for the location of the CGs, they can be used for our purposes. Following Despali et al. (2016), we identified haloes in Le SBARBINE simulations for different overdensity threshold: Δc = Δvir, where Δvir is the redshift-dependent virial overdensity, and Δc = 200, 5000, 10 000 and 20 000, independent of redshift. For each of these halo definitions, we also built the corresponding merger history trees. When Δvir is considered, the entire virialized region of the halo is selected (in the case of a galaxy cluster, the entire cluster-sized dark-matter halo). When higher overdensities are considered, smaller regions of the halo are selected: for sufficiently high overdensity we select only the central part of the virialized halo, which we can roughly identify with the CG. Moreover, at these high overdensities, dense substructures within the main virial halo can be identified as independent structures. For each given overdensity we measure the properties of the mergers (specifically, the mass ratios and the orbital parameters). We then compare the properties of cosmological accretion (at the virial radius rvir) with those of accretion on to CGs (at some inner radius r ≪ rvir). The paper is organized as follows. Section 2 describes the simulations and the numerical methods. The orbital parameters of halo–halo encounters are defined in Section 3. The results are presented in Section 4, while in Section 5 we draw our conclusions. 2 NUMERICAL METHODS 2.1 N-body simulations and identification of dark-matter haloes In this work we make use of the results from the cosmological dark-matter only N-body simulations Le SBARBINE (Despali et al. 2016). The assumed background cosmology and initial conditions of the simulations are consistent with the results from the Planck Collaboration XVI (2014). In particular, in the simulations and throughout this paper we adopt a standard Λ cold dark matter cosmological model with the following parameters: matter density parameter Ωm = 0.307, cosmological constant density parameter $$\Omega _\Lambda = 0.693$$, linear power spectrum amplitude σ8 = 0.829, and dimensionless Hubble constant h = 0.677. Here we use only the two highest resolution runs among Le SBARBINE simulations: Ada (dark-matter particle mass m = 2.87 × 107M⊙) and Bice (m = 2.29 × 108M⊙). The two simulations have the same number of particles (N = 10243), but different box size: 92.3 Mpc for Ada and 184.6 Mpc for Bice. At each stored snapshot haloes are identified using a spherical overdensity algorithm (e.g. Tormen 1998; Tormen, Moscardini & Yoshida 2004; Giocoli, Tormen & van den Bosch 2008). In practice, haloes are defined as spherical overdensities with radius such that the average density is   \begin{eqnarray} {\overline{\rho }}=\Delta _{\rm c}\rho _{\rm crit}, \end{eqnarray} (1)where Δc is the critical overdensity and   \begin{eqnarray} \rho _{\rm crit}(z)= \frac{3H^2(z)}{8\pi G} \end{eqnarray} (2)is the critical density of the Universe, depending on redshift through the Hubble parameter H(z). Using this method, the overdensity threshold that defines the halo boundaries can be varied depending on the observational data that one wants to compare with. While the virial overdensity Δc = Δvir is commonly used in structure formation studies, other definitions can be chosen to be more similar to observational data sets: Δc = 500 is typically used in X-ray observations to define the mass of a galaxy cluster, while Δc = 200 is often used to fit weak lensing shear profiles. In our case Δc = 20 000 corresponds to the region within r ≃ 0.06rvir, which is a proxy for the size of the CG in a cluster. In this work, we consider different choices for Δc. For each value of Δc, we define the halo mass MΔ and the halo radius rΔ. First of all, we consider the virial value Δc = Δvir(z), which depends on z and on the cosmological parameters, as given by Eke, Cole & Frenk (1996). For the assumed cosmology Δvir increases with redshift: reference values are Δvir ≃ 97.9 at z = 0 and Δvir ≃ 154 at z = 1. When Δvir = Δc the halo radius and mass are, respectively, the virial radius rvir and the virial mass Mvir. For comparison with previous work we consider also the standard value Δc = 200, independent of redshift: in this case the halo mass and radius are M200c and r200c, respectively. Finally, in order to study the behaviour of mergers in central parts of the haloes, we explore the following other values of the critical overdensity, independent of redshift: Δc = 5000 (with mass M5000c and radius r5000c), Δc = 10 000 (with mass M10000c and radius r10000c), and Δc = 20 000 (with mass M20000c and radius r20000c). It must be noted that in each catalogue (i.e. for each value of the considered overdensity Δc) the haloes are identified independently. Therefore, the number of haloes is in general different in each catalogue, because a halo that, at a given redshift, is identified for a given value of Δc, at the same redshift, might be ‘incorporated’ in a bigger halo when a lower overdensity is considered (for more details see fig. 2 of Despali et al. 2016). 2.2 Sample of haloes and catalogues of halo–halo encounters We study the redshift range 0 ≲ z ≲ 1, in which Le SBARBINE simulations have 13 snapshots at the following redshifts: z1 = 1.012, z2 = 0.904, z3 = 0.796, z4 = 0.694, z5 = 0.597, z6 = 0.507, z7 = 0.421, z8 = 0.34, z9 = 0.264, z10 = 0.192, z11 = 0.124, z12 = 0.06, z13 = 0. We create a sample of galaxy clusters by selecting, in the Δvir catalogue, all haloes with Mvir ≥ 1014M⊙ at z = 0. The resulting sample consists of 101 haloes at z = 0 (12 haloes in Ada and 89 haloes in Bice). We identify these 101 haloes also in the higher-overdensity (Δc > Δvir) catalogues (Δc = 200, 5000, 10 000, and 20 000), finding that all of them have counterparts in all the considered catalogues. We note that, as the selection in mass is done on Mvir, the selected z = 0 haloes can have mass MΔ < 1014M⊙ for Δc > Δvir, because the mass of a given halo decreases for increasing Δc. In order to identify halo-halo encounters we proceed as follows. From the halo catalogues built for the 13 simulation snapshots and for each overdensity, we construct the halo merging history tree. Starting from each halo at z = 0, we define its progenitors at the previous output, z = 0.06, as all haloes that, in the time elapsed between two snapshots, have given at least 50 per cent of their particles to the considered z = 0 halo. The main progenitor at z = 0.06 is defined as the most massive progenitor of the z = 0 halo. We then repeat the same procedure, now starting from the main progenitor at z = 0.06 and considering its progenitors at z = 0.124, and we proceed backwards in time, always following the main progenitor halo. The resulting merger tree consists of a main trunk, which traces the main progenitor back in time, and of satellites; these last are all the progenitors that at any time merge directly on to the main progenitor. By construction, for given simulation and descendant halo, the definition of the main progenitor depends on the time sampling (i.e. the number of snapshots): in principle it is possible that the main branch is not identified correctly if the time sampling is insufficient1. However, Giocoli (2008), using simulations with time sampling similar to Le SBARBINE, has shown that the probability of misidentifying the main progenitor branch for galaxy cluster scale haloes is much below 10 per cent in the redshift interval 0 ≲ z ≲ 1. In addition, the consistence of our results with those obtained by Jiang et al. (2015, hereafter J15) with the DOVE simulation (Appendix A), which has a better time sampling, ensures us that the time sampling of our runs is good enough to uniquely follow the main halo progenitor branch back in time. We define the halo–halo encounters between two subsequent snapshots by selecting in the higher-redshift snapshot all the satellite-main progenitor pairs. We indicate the physical properties of the main progenitor (sometimes referred to also as host halo) with the subscript ‘host’ and those of the less massive progenitors (satellites) with the subscript ‘sat’. For each pair we then measure the masses (MΔ, host and MΔ, sat), radii (rΔ, host and rΔ, sat), and relative position (r) and velocity (v) of the centres of mass. We apply the procedure described above to all the z = 0 haloes of our sample and to all their main progenitors, back to the z = 0.904 snapshot. In this way we build our catalogue containing all the halo–halo encounters in the redshift range 0 < z < 1 that end up in cluster-sized haloes at z = 0. Of course, the number of encounters and their properties are different for different overdensities. We note that not all the halo–halo encounters must be considered rapid mergers. By construction, our catalogue of halo–halo encounters includes cases of satellite haloes that, after the encounter, escape from the main halo. In these cases, the haloes are distinct at a given snapshot, are identified as a single halo at a later snapshot, but are again distinct at an even later time step. In the terminology of this work we then distinguish halo–halo encounters and halo–halo mergers. In Section 4.2 we will define a criterion to select the subsample of mergers in the whole sample of halo–halo encounters. For each halo–halo encounter we define the mass ratio ξ = MΔ, sat/MΔ, host. In order to have a robust measure of the properties of halo–halo encounters and mergers, we limit our exploration to ξ ≥ ξmin, where ξmin is a minimum mass ratio such that the number of particles of the satellite is at least N ≈ 100. Clearly ξmin depends on the mass resolution of the simulations, on the explored redshift range, and on the considered overdensity Δc. In particular, we adopt ξmin = 0.005 for Δc = Δvir and Δc = 200, ξmin = 0.01 for Δc = 5000 and Δc = 10 000, and ξmin = 0.1 for Δc = 20 000. We verified that with these choices, for our sample of encounters in the redshift range 0 < z < 1, the number of satellites with N < 100 never exceeds 5 per cent of the entire satellite population. Using dark-matter only simulations to infer the properties of central-satellite galaxy mergers in clusters, we are implicitly assuming that the dynamical evolution of subhaloes orbiting massive host haloes is not significantly influenced by the presence of baryons. In fact, comparisons between dark-matter only and hydrodynamic cosmological simulations indicate that some properties of the subhalo mass function can be modified by the presence of baryons (Chua et al. 2017, and references therein). However, the effect is negligible for the relatively high satellite-host mass ratios considered in this work (see fig. 1 of Chua et al. 2017). 3 ORBITAL PARAMETERS OF HALO–HALO ENCOUNTERS It is useful to describe an encounter between two haloes in terms of the orbital parameters calculated in the point-mass two-body approximation. This description, though not rigorous for extended objects, is often used in the study and classification of mergers of galaxies (e.g. Boylan-Kolchin, Ma & Quataert 2006; Nipoti, Treu & Bolton 2009) and dark-matter haloes (e.g. Khochfar & Burkert 2006; Posti et al. 2014). Here, we define the point-mass two-body approximation orbital parameters of halo–halo encounters, following the formalism of N17. The orbit can be fully characterized by the pair of parameters orbital energy and angular momentum. For a halo–halo encounter we define the two-body approximation orbital energy per unit mass   \begin{eqnarray} {E_{\rm 2b}}=\frac{1}{2}v^2-\frac{G{M}_{\rm 2b}}{r}, \end{eqnarray} (3)where M2b ≡ MΔ, host + MΔ, sat, $$r\equiv |{\boldsymbol r}|$$ is the relative distance and $$v\equiv |{\boldsymbol v}|$$ is the relative speed, between their centres of mass. It is useful to decompose $${\boldsymbol v}$$ in its radial component $$v_{\rm r}={\boldsymbol v}\cdot {\boldsymbol r}/r$$ and its tangential component, with modulus $$v_{\rm tan}=\sqrt{v^2-v_{\rm r}^2}$$. The modulus of the orbital angular momentum per unit mass is L = rvtan. At fixed separation r and energy E2b, the modulus of the maximum allowed specific angular momentum is   \begin{eqnarray} {L_{\rm max}}=r\sqrt{2\left({E_{\rm 2b}}+\frac{G{M}_{\rm 2b}}{r}\right)}=rv, \end{eqnarray} (4)where v is the relative speed when the two haloes have separation r. Clearly, Lmax is such that L/Lmax = vtan/v. Another set of orbital parameters used to classify halo–halo encounters (e.g. Benson 2005; J15) is the pair (v/vcirc, |vr|/v), where, given a distance r from the centre of the host system (for instance the overdensity radius rΔ, host), v is the relative speed at r = rΔ, host, vr is the radial component of the relative velocity at r = rΔ, host and   \begin{eqnarray} v_{\rm circ}=\sqrt{\frac{G{M}_{\rm \Delta ,host}}{{r}_{\rm \Delta ,host}}} \end{eqnarray} (5)is the host circular velocity at rΔ, host. Given the finite time sampling (i.e. the finite number of snapshots), in our simulations we have information on the halo–halo relative velocity $${{\boldsymbol v}_{\rm snap}}$$ when the two haloes have a separation rsnap that is in general larger than rΔ, host. As discussed in several previous works (e.g. Benson 2005, J15), it is thus necessary to apply a correction to recover (v/vcirc, |vr|/v) measured when the satellite crosses the desired overdensity radius of the host (i.e. when the separation is rΔ, host). We correct the velocity as follows. We first compute the relative velocity $${\boldsymbol v}_{\rm 2b}$$ at rΔ, host, assuming that the point-mass two-body energy and angular momentum are conserved: $${\boldsymbol v}_{\rm 2b}$$ is such that   \begin{eqnarray} \frac{1}{2}v_{\rm snap}^2-\frac{G({M}_{\rm \Delta ,host}+{M}_{\rm \Delta ,sat})}{r_{\rm snap}} =\frac{1}{2}v_{\rm 2b}^2-\frac{G({M}_{\rm \Delta ,host}+{M}_{\rm \Delta ,sat})}{{r}_{\rm \Delta ,host}}\nonumber\\ \end{eqnarray} (6)and   \begin{eqnarray} r_{\rm snap}v_{\rm tan,snap}={r}_{\rm \Delta ,host}v_{\rm tan,2b}, \end{eqnarray} (7)where vtan, 2b is the tangential component of $${\boldsymbol v}_{\rm 2b}$$. If, as in most cases, equations (6) and (7) give vtan, 2b ≤ v2b, the modulus of the radial component of v2b is $$|v_{\rm r, 2b}|=\sqrt{v_{\rm 2b}^2-v_{\rm tan,2b}^2}$$. If, instead, equations (6) and (7) give vtan, 2b > v2b (which indicates that the point-mass two-body orbit is too crude an approximation), we simply fix vtan, 2b = v2b and vr, 2b = 0. Finally, we define the corrected velocity $${\boldsymbol v}$$ at the time of crossing (r = rΔ, host) to be such that   \begin{eqnarray} \left(\frac{v}{v_{\rm circ}}\right)_{{r}_{\rm \Delta ,host}}=\frac{1}{2}\frac{v_{\rm snap}+v_{\rm 2b}}{v_{\rm circ}} \end{eqnarray} (8)and   \begin{eqnarray} \left(\frac{v_{\rm r}}{v}\right)_{{r}_{\rm \Delta ,host}}=\frac{1}{2}\left[\left(\frac{v_{\rm r}}{v}\right)_{\rm snap}+\left(\frac{v_{\rm r}}{v}\right)_{\rm 2b}\right]. \end{eqnarray} (9)We verified that this is a reasonably good approximation by comparing the distributions of our sample of haloes with previous literature work (see Appendix A). 4 RESULTS 4.1 Halo masses and radii at different overdensities The ratios MΔ/Mvir and rΔ/rvir are decreasing functions of Δc (see Despali et al. 2017). The exact values of these ratios depend on the halo mass density distribution through the halo concentration parameter (Giocoli, Tormen & Sheth 2012). We computed MΔ/Mvir and rΔ/rvir for all haloes in our sample (i.e. the 101 z = 0 haloes and all their main progenitors in all previous snapshots back to z = 0.904; altogether 1224 haloes; Section 2.2). The means and standard deviations of the distributions of MΔ/Mvir and rΔ/rvir are reported in Table 1 for Δc = 200, 5000, 10 000, and 20 000. For the highest overdensity here considered Δc = 20 000, the average values are MΔ/Mvir ≈ 0.05 and rΔ/rvir ≈ 0.06. The distributions of MΔ/Mvir and rΔ/rvir are broader for increasing Δc, with standard deviations in the range 10-17 per cent for rΔ/rvir and 11-44 per cent for MΔ/Mvir. Table 1. Mean (μ) and standard deviation (σ) of the distributions of MΔ/Mvir and rΔ/rvir, for different values of Δc, for our sample consisting of the 101 z = 0 haloes with Mvir ≥ 1014M⊙ and all their main progenitor haloes in the previous snapshots, back to z = 0.904 (altogether 1224 haloes). Δc  μ(MΔ/Mvir)  σ(MΔ/Mvir)  μ(rΔ/rvir)  σ(rΔ/rvir)  200  0.848  0.097  0.807  0.082  5000  0.148  0.047  0.153  0.020  10 000  0.085  0.032  0.100  0.015  20 000  0.045  0.020  0.064  0.011  Δc  μ(MΔ/Mvir)  σ(MΔ/Mvir)  μ(rΔ/rvir)  σ(rΔ/rvir)  200  0.848  0.097  0.807  0.082  5000  0.148  0.047  0.153  0.020  10 000  0.085  0.032  0.100  0.015  20 000  0.045  0.020  0.064  0.011  View Large In the following sections we will compare our results with those of simulations of satellites in isolated host haloes (N17), in which the merger orbital parameters were measured at a radius rcen = 0.12rs, where rs is the halo scale radius. If we identify the truncation radius of N17 with rvir, we have rcen ≃ 0.024rvir and that the mass contained within rcen is Mcen ≃ 0.0075Mvir. Therefore, the region probed by N17 is somewhat smaller than the most central region here considered (Δc = 20 000) and roughly corresponds to an overdensity Δc = 50 000. It is useful to note that Kravtsov (2013) finds that, on average, the three-dimensional half-mass radius of the stellar distribution of observed CGs is r*, 1/2 ≈ 0.015r200c ≈ 0.012rvir (using the average ratio r200c/rvir ≃ 0.85 found for our sample of haloes; see Table 1). Therefore, in terms of r*, 1/2, we have r20000c ≈ 5r*, 1/2 and rcen ≈ 2r*, 1/2, which indicates that both r20000c and rcen probe the region of the halo occupied by the stellar distribution of the CG. 4.2 Mergers and fly-bys As mentioned in Section 2.2, we do not expect to have a rapid merger for all halo–halo encounters. Rapid mergers occur when the orbits are bound (E2b < 0), but also for unbound orbits (E2b ≥ 0), provided the orbital angular-momentum modulus L is sufficiently low (see section 7.4 of Binney & Tremaine 1987). For this reason, a convenient parameter that can be used to identify mergers is the orbit eccentricity   \begin{eqnarray} e = \sqrt{1+\frac{2{E_{\rm 2b}}L^2}{G^2{M}_{\rm 2b}^2}}, \end{eqnarray} (10)which, for E2b > 0, is an increasing function of both E2b and L. As in N17, we take as fiducial discriminating value of eccentricity ecrit = 1.5 and classify an encounter as a merger when e ≤ ecrit and as a fly-by when e > ecrit. The eccentricity distributions for our samples of halo–halo encounters with mass ratio 0.01 ≤ ξ < 0.1 are shown in Fig. 1 (upper panel) for Δc = Δvir, Δc = 5000, and Δc = 10 000 (corresponding to radii rvir, r5000c ≈ 0.15rvir and r10000c ≈ 0.1rvir, respectively; see Section 4.1). The distribution of the eccentricity for encounters with mass ratio ξ ≥ 0.1 is shown in the lower panel of Fig. 1 for Δc = Δvir and Δc = 20 000 (corresponding to radii rvir and r20000c ≈ 0.06rvir, respectively; see Section 4.1). In the same panel we plot also the distribution found by N17 for numerical models of satellites orbiting in isolated haloes with ξ ≃ 0.13 and ξ ≃ 0.67, measured at rcen ≈ 0.02rvir (see Section 4.1). From Fig. 1 it is clear that most of the encounters are indeed classified as mergers: the adopted cut in eccentricity allows us to effectively exclude the tail of high-eccentricity orbits, which are most likely fly-bys. The number of mergers and the total number of encounters for our sample are reported in Table 2 for different values of Δc and intervals of ξ. Figure 1. View largeDownload slide Upper panel: probability distribution p = dn/dx of the logarithm of the orbital eccentricity of halo–halo encounters computed in the two-body approximation (x = log10e), for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dotted histogram), and Δc = 10 000 (dashed histogram). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. The vertical dashed line (e = 1.5) discriminates mergers (e ≤ 1.5) and fly-bys (e > 1.5). Lower panel: same as the upper panel, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid histogram) and Δc = 20 000 (dot–dashed histogram). The dashed histogram represents the results obtained by N17 for encounters at rcen ≈ 0.02rvir, using simulations of satellites in isolated host haloes. Figure 1. View largeDownload slide Upper panel: probability distribution p = dn/dx of the logarithm of the orbital eccentricity of halo–halo encounters computed in the two-body approximation (x = log10e), for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dotted histogram), and Δc = 10 000 (dashed histogram). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. The vertical dashed line (e = 1.5) discriminates mergers (e ≤ 1.5) and fly-bys (e > 1.5). Lower panel: same as the upper panel, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid histogram) and Δc = 20 000 (dot–dashed histogram). The dashed histogram represents the results obtained by N17 for encounters at rcen ≈ 0.02rvir, using simulations of satellites in isolated host haloes. Table 2. Total number of encounters and number of encounters classified as mergers experienced by all the haloes in our sample (see Section 2.2). The data for rΔ = rcen refer to the results of N17. ξ  rΔ  Encounters  Mergers  0.005-0.05  r200c  1855  1733  0.01–0.1  rvir  1049  998  0.01–0.1  r5000c  456  330  0.01–0.1  r10000c  275  210  0.1–1  rvir  216  207  0.1–1  r20000c  98  69  0.1–1  rcen  82  44  ξ  rΔ  Encounters  Mergers  0.005-0.05  r200c  1855  1733  0.01–0.1  rvir  1049  998  0.01–0.1  r5000c  456  330  0.01–0.1  r10000c  275  210  0.1–1  rvir  216  207  0.1–1  r20000c  98  69  0.1–1  rcen  82  44  View Large As a quantitative test of our classification of mergers and fly-bys, we analysed the post-encounter evolution of the satellites in the Δc = 20 000 catalogue with mass ratio ξ ≥ 0.1. In practice, for each encounter occurring between the snapshots at redshifts zi − 1 and zi, we check whether the satellite and the main halo are distinct (i.e. the satellite has escaped) in the snapshot at redshift zi + 1 (clearly we exclude the case i = 13, because the snapshot at z13 = 0 is the last; see Section 2.2). We find that the satellite escapes in 80 per cent of the encounters classified as fly-bys and in 15 per cent of the encounters classified as mergers, which suggests that our classification is sufficiently accurate. We verified that the selection of mergers is not sensitive to the exact value of ecrit: the main results of the present work are essentially the same for values of ecrit in the range 1.25 ≲ ecrit ≲ 2. 4.3 Distribution of merger mass ratio There are good reasons to expect mergers on to CGs in clusters to be characterized by a distribution of mass ratios ξ different from that of cosmological halo–halo mergers measured at the virial radius. It is well known that dynamical friction, which is the main driver of galactic cannibalism, is more effective for more massive satellites, so we expect the typical mass ratio of mergers on to CGs to be higher than that of mergers at the virial radius of the host cluster. We can quantitatively explore this question by comparing the distributions of ξ in halo catalogues obtained for different values of Δc. The upper panel of Fig. 2 shows, as a function of the mass ratio ξ, the fraction Maccr( > ξ) of mass accreted in mergers with mass ratio larger than ξ, normalized to the total mass accreted in mergers with mass ratio ξ ≥ 0.01, for mergers measured at2rvir, r5000c, and r10000c. A clear trend emerges from this plot: in line with the expectations, mergers with higher mass ratios contribute more when more central halo regions are considered. The difference between the mergers measured at Δvir and those measured at higher overdensities becomes more and more important for ξ → 1 (see lower panel of Fig. 2). The median value ξmed, such that half of the mass is accreted in mergers with mass ratio larger than ξmed, is 0.26 at rvir, 0.37 at ≈0.15rvir and 0.53 at ≈0.1rvir. Another useful indicator of the characteristic mass ratio of mass accretion is the mass-weighted merger mass ratio 〈ξ〉M (see Nipoti et al. 2012), which can be written as   \begin{eqnarray} \left\langle \xi \right\rangle _{\rm M}=\frac{\left\langle \xi ^2\right\rangle _{\rm N}}{\left\langle \xi \right\rangle _{\rm N}}, \end{eqnarray} (11)where 〈⋅⋅⋅〉N is the number-weighted average. As shown in Fig. 2 (upper panel), 〈ξ〉M ≃ 0.38 at rvir, 〈ξ〉M ≃ 0.44 at ≈0.15rvir, and 〈ξ〉M ≃ 0.49 at ≈0.1rvir. For the innermost radius here probed (≈0.1rvir) the characteristic merger mass ratio is close to 1/2. Figure 2. View largeDownload slide Upper panel: fraction of mass accreted in mergers with mass ratio larger than ξ, relative to the total mass accreted in mergers with 0.01 ≤ ξ ≤ 1, for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dotted curve), and Δc = 10 000 (dashed curve). The measures are for mergers in the redshift interval 0 < z < 1 for our sample of dark-matter haloes with Mvir ≥ 1014M⊙ at z = 0. The horizontal lines indicate, for the distributions with the corresponding line styles, the fraction fmajor of mass accreted in major mergers, assuming major-merger mass-ratio threshold ξmajor = 1/3 (lower lines) or ξmajor = 1/4 (upper lines). The vertical lines indicate the mass-weighted average merger mass ratio (equation 11) for the distributions with the corresponding line styles. Lower panel: dotted curve: relative difference between the dotted and solid curves in the upper panel. Dashed curve: relative difference between the dashed and solid curve in the upper panel. Figure 2. View largeDownload slide Upper panel: fraction of mass accreted in mergers with mass ratio larger than ξ, relative to the total mass accreted in mergers with 0.01 ≤ ξ ≤ 1, for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dotted curve), and Δc = 10 000 (dashed curve). The measures are for mergers in the redshift interval 0 < z < 1 for our sample of dark-matter haloes with Mvir ≥ 1014M⊙ at z = 0. The horizontal lines indicate, for the distributions with the corresponding line styles, the fraction fmajor of mass accreted in major mergers, assuming major-merger mass-ratio threshold ξmajor = 1/3 (lower lines) or ξmajor = 1/4 (upper lines). The vertical lines indicate the mass-weighted average merger mass ratio (equation 11) for the distributions with the corresponding line styles. Lower panel: dotted curve: relative difference between the dotted and solid curves in the upper panel. Dashed curve: relative difference between the dashed and solid curve in the upper panel. Given a discriminant mass ratio ξmajor between major and minor mergers, we can define fmajor ≡ Maccr( ≥ ξmajor)/Maccr( ≥ ξmin) as the fraction of mass accreted in major mergers in the redshift range 0 < z < 1 (here the minimum mass ratio is ξmin = 0.01.). For, respectively, Δc = Δvir, 5000, and 10 000 we find fmajor = 0.48, 0.55, and 0.64 (assuming ξmajor = 1/3), and fmajor = 0.54, 0.62, and 0.69 (assuming ξmajor = 1/4). Taking the results for Δc = 10 000 as a proxy for accretion on to the CG, we can conclude that (at least in the explored mass ratio interval 0.01 ≤ ξ ≤ 1) more than 60 per cent of the mass accreted at z < 1 by CGs in clusters is due to major mergers. This conclusion is qualitatively consistent with previous observational (Lidman et al. 2013) and theoretical (Rodriguez-Gomez et al. 2016) results on the role of major mergers in the build-up of massive CGs. By definition, for given ξmajor, fmajor depends on the minimum mass ratio ξmin. Here, for the reasons explained in Section 2.2, we have fixed ξmin = 0.01, but of course also mergers with lower mass ratio contribute to the actual halo mass growth. The slopes at low ξ of the curves in the upper panel of Fig. 2 suggest that the relative contribution of mergers with ξ < 0.01 is more important at rvir than at r5000c and r10000c. To quantify this effect, we computed fmajor assuming ξmin = 0.005 (thus relaxing our requirement that the satellites have at least N ≈ 100 particles): in this case we get values of fmajor that are only slightly smaller than those obtained for ξmin = 0.01 (for instance, by ≲ 6 per cent for measures at rvir and by ≲ 3 per cent for measures at r10000c). Thus, in this respect, our conclusion about the predominance of major mergers in the z ≲ 1 build-up of cluster CGs appears robust. 4.4 Orbital parameters for mergers with mass ratio $$\boldsymbol{0.01\le \xi <0.1}$$ In this section we discuss the distribution of orbital parameters for mergers (i.e. encounters with eccentricity e ≤ 1.5) with mass ratio in the range 0.01 ≤ ξ < 0.1, comparing the results for Δc = Δvir, Δc = 5000, and Δc = 10 000. Fig. 3 shows, for these samples of mergers, the distributions of the two-body specific orbital energy E2b and of the modulus of the specific orbital angular momentum L. E2b is normalized to Ψ0 ≡ G(MΔ, host + MΔ, sat)/rΔ, host, which is the absolute value of the two-body gravitational potential of the encounter when the separation is rΔ, host. L is normalized to Lmax (equation 4), which is the modulus of the maximum angular momentum for given orbital energy E2b. We have fitted the distributions of E2b/Ψ0 with a Gaussian distribution   \begin{eqnarray} p(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp {\left[-\frac{(x-\mu )^2}{2\sigma ^2}\right]}, \end{eqnarray} (12)where μ is the mean and σ is the standard deviation, and the distributions of L/Lmax with a beta distribution   \begin{eqnarray} p(x)=\frac{x^{\alpha -1}(1-x)^{\beta -1}}{B(\alpha , \beta )}, \end{eqnarray} (13)where   \begin{eqnarray} B(\alpha , \beta )=\frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )} \end{eqnarray} (14)and Γ is the gamma function. In Fig. 4 we plot the distributions of the orbital parameters v/vcirc and |vr|/v, which, as pointed out in Section 3, are a pair of parameters, alternative to E2b and L, often used to characterize the orbits of galaxy and halo encounters. We emphasize that, in the present context, this pair of parameters does not carry exactly the same information as E2b and L: v/vcirc and |vr|/v are evaluated at a separation rΔ, host (equations 8 and 9), while E2b and L are evaluated at the snapshot before the merger. Moreover, while v is normalized to the main halo circular velocity vcirc, which is independent of the properties of the satellite, E2b is normalized to Ψ0, which depends also on the mass of the satellite (and therefore on the mass ratio ξ). The distributions of v/vcirc are fitted with a Gaussian (equation 12), while the distributions of |vr|/v are fitted with a beta distribution (equation 13). The best-fitting distributions of E2b/Ψ0, L/Lmax, v/vcirc, and |vr|/v are over-plotted in the corresponding panels of Figs 3 and 4, and their parameters are reported in Tables 3 and 4. Figure 3. View largeDownload slide Probability distribution p = dn/dx of the normalized orbital energy, computed in the two-body approximation (x = E2b/Ψ0; left-hand panel), and of the normalized angular-momentum modulus (x = L/Lmax; right-hand panel) for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dashed histogram), and Δc = 10 000 (dotted histogram). Here Ψ0 ≡ G(MΔ, host + MΔ, sat)/rΔ, host and Lmax is defined by equation (4). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Figure 3. View largeDownload slide Probability distribution p = dn/dx of the normalized orbital energy, computed in the two-body approximation (x = E2b/Ψ0; left-hand panel), and of the normalized angular-momentum modulus (x = L/Lmax; right-hand panel) for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dashed histogram), and Δc = 10 000 (dotted histogram). Here Ψ0 ≡ G(MΔ, host + MΔ, sat)/rΔ, host and Lmax is defined by equation (4). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Figure 4. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and of the radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) when the satellite crosses the virial radius of the host rΔ, host for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dashed curve), and Δc = 10 000 (dotted curve). Here vcirc is the host circular velocity at rΔ, host. v/vcirc and |vr|/v are evaluated at rΔ, host as in equations (8) and (9). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Figure 4. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and of the radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) when the satellite crosses the virial radius of the host rΔ, host for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dashed curve), and Δc = 10 000 (dotted curve). Here vcirc is the host circular velocity at rΔ, host. v/vcirc and |vr|/v are evaluated at rΔ, host as in equations (8) and (9). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Table 3. Mean (μ) and standard deviation (σ) of the best-fitting Gaussian distributions (equation 12) of the orbital parameters E2b/Ψ0 and v/vcirc, for different values of the overdensity radius rΔ and intervals of mass ratios ξ. The data for rΔ = rcen refer to the results of N17. Parameter  ξ  rΔ  μ  σ  E2b/Ψ0  0.01–0.1  rvir  −0.21  0.29  E2b/Ψ0  0.01–0.1  r5000c  −0.30  0.39  E2b/Ψ0  0.01–0.1  r10000c  −0.39  0.40  E2b/Ψ0  0.1–1  rvir  −0.24  0.23  E2b/Ψ0  0.1–1  r20000c  −0.31  0.37  E2b/Ψ0  0.1–1  rcen  −0.33  0.31  v/vcirc  0.01–0.1  rvir  1.17  0.25  v/vcirc  0.01–0.1  r5000c  0.86  0.31  v/vcirc  0.01–0.1  r10000c  0.76  0.32  v/vcirc  0.1–1  rvir  1.28  0.21  v/vcirc  0.1–1  r20000c  0.99  0.39  v/vcirc  0.1–1  rcen  1.39  0.35  Parameter  ξ  rΔ  μ  σ  E2b/Ψ0  0.01–0.1  rvir  −0.21  0.29  E2b/Ψ0  0.01–0.1  r5000c  −0.30  0.39  E2b/Ψ0  0.01–0.1  r10000c  −0.39  0.40  E2b/Ψ0  0.1–1  rvir  −0.24  0.23  E2b/Ψ0  0.1–1  r20000c  −0.31  0.37  E2b/Ψ0  0.1–1  rcen  −0.33  0.31  v/vcirc  0.01–0.1  rvir  1.17  0.25  v/vcirc  0.01–0.1  r5000c  0.86  0.31  v/vcirc  0.01–0.1  r10000c  0.76  0.32  v/vcirc  0.1–1  rvir  1.28  0.21  v/vcirc  0.1–1  r20000c  0.99  0.39  v/vcirc  0.1–1  rcen  1.39  0.35  View Large Table 4. Parameters α and β, mean (μ) and standard deviation (σ) of the best-fitting beta distributions (equation 13) of the orbital parameters L/Lmax and vr/v, for different values of the overdensity radius rΔ and intervals of mass ratios ξ. The data for rΔ = rcen refer to the results of N17. Parameter  ξ  rΔ  α  β  μ  σ  L/Lmax  0.01–0.1  rvir  1.58  1.38  0.53  0.25  L/Lmax  0.01–0.1  r5000c  1.01  0.71  0.59  0.30  L/Lmax  0.01–0.1  r10000c  0.96  0.69  0.58  0.30  L/Lmax  0.1–1  rvir  1.57  2.16  0.42  0.23  L/Lmax  0.1–1  r20000c  0.81  0.85  0.49  0.31  L/Lmax  0.1–1  rcen  1.99  0.68  0.75  0.23  vr/v  0.01–0.1  rvir  2.69  0.78  0.78  0.20  vr/v  0.01–0.1  r5000c  1.51  0.67  0.69  0.26  vr/v  0.01–0.1  r10000c  1.39  0.60  0.70  0.26  vr/v  0.1–1  rvir  3.70  0.67  0.85  0.16  vr/v  0.1–1  r20000c  1.24  0.43  0.75  0.27  vr/v  0.1–1  rcen  1.28  1.05  0.55  0.27  Parameter  ξ  rΔ  α  β  μ  σ  L/Lmax  0.01–0.1  rvir  1.58  1.38  0.53  0.25  L/Lmax  0.01–0.1  r5000c  1.01  0.71  0.59  0.30  L/Lmax  0.01–0.1  r10000c  0.96  0.69  0.58  0.30  L/Lmax  0.1–1  rvir  1.57  2.16  0.42  0.23  L/Lmax  0.1–1  r20000c  0.81  0.85  0.49  0.31  L/Lmax  0.1–1  rcen  1.99  0.68  0.75  0.23  vr/v  0.01–0.1  rvir  2.69  0.78  0.78  0.20  vr/v  0.01–0.1  r5000c  1.51  0.67  0.69  0.26  vr/v  0.01–0.1  r10000c  1.39  0.60  0.70  0.26  vr/v  0.1–1  rvir  3.70  0.67  0.85  0.16  vr/v  0.1–1  r20000c  1.24  0.43  0.75  0.27  vr/v  0.1–1  rcen  1.28  1.05  0.55  0.27  View Large The distributions of E2b/Ψ0 (left-hand panel in Fig. 3) suggest that for higher values of Δc (i.e. when more central regions of the haloes are considered) the orbits of mergers tend to be slightly more bound: the mean orbital energy for Δc = 10 000 is more negative than that for Δc = 5000, which in turn is more negative than that at Δc = Δvir (see Table 3). However, the distributions are relatively broad and there is substantial overlap. A qualitatively similar, but stronger trend can be seen in the distributions of v/vcirc (which is another measure of the binding energy of the orbit; left-hand panel in Fig. 4), which are characterized by larger offsets between the peak of the measures at rvir and those at r5000c or r10000c. The fact that the distributions are more offset in v/vcirc than in E2b/Ψ0 comes from the fact that the accretion history for higher Δc is characterized by higher merger mass ratios (see Section 4.3 and Fig. 2). As pointed out above, while vcirc ignores the properties of the satellite, the normalization potential Ψ0 accounts for the mass ratio. In this sense, E2b/Ψ0 should give a cleaner measure of the binding energy of the orbits, when samples with different mass-ratio distributions are compared. We also note that the distributions of both E2b/Ψ0 and v/vcirc have higher scatter (larger standard deviation; see Table 3) for increasing Δc. The distributions of L/Lmax (right-hand panel in Fig. 3) and |vr|/v (right-hand panel in Fig. 4) suggest that for higher values of Δc (i.e. when more central regions of the haloes are considered) the orbits of mergers tend to be significantly more tangential. Comparing the right-hand panels of Figs 3 and 4, it is apparent that, for Δc = 5000 and Δc = 10 000, the distributions of both L/Lmax = vtan/v and |vr|/v peak at ≈1. This might be counterintuitive, because |vr|/v → 0 when |vtan|/v → 1. However, the relation between |vr|/v and |vtan|/v, namely   \begin{eqnarray} \frac{|v_{\rm r}|}{v}=\sqrt{1-\left(\frac{v_{\rm tan}}{v}\right)^2}, \end{eqnarray} (15)is non-linear and such that, for instance, |vr|/v ≳ 0.9 corresponds to vtan/v ≲ 0.44. Thus, a peak at |vr|/v ≈ 1 does not necessarily imply a peak at L/Lmax ≈ 0. Moreover, we recall that while L/Lmax is evaluated at rsnap, |vr|/v is evaluated at rΔ, host, so the right-hand panels of Figs 3 and 4 do not contain exactly the same information. The distributions of L/Lmax and |vr|/v for Δc = 5000 and Δc = 10 000 are almost indistinguishable, but they are significantly different from those obtained for Δc = Δvir. For the higher overdensities, the mean values of L/Lmax and |vr|/v are, respectively, higher and lower than those obtained for Δc = Δvir (see Table 4). The difference between measures at rvir and those at ≲ 0.15rvir is best visualized in the right-hand panel of Fig. 3: the distribution of L/Lmax for measures at rvir drops for L/Lmax → 1, where instead the distributions for more central measures peak. The values of the standard deviation reported in Table 4 indicate that the distributions of L/Lmax and |vr|/v have larger scatter for Δc = 5000 and Δc = 10 000 than for Δc = Δvir. Overall, the results presented in this section lead us to conclude that, for mergers with 0.01 ≤ ξ < 0.1, the orbits of mergers at r10000c ≈ 0.1rvir are more bound and more tangential than those at rvir. 4.5 Orbital parameters for mergers with mass ratio $$\boldsymbol{\xi \ge 0.1}$$ We focus here on the distribution of orbital parameters for mergers with mass ratios in the range 0.1 ≤ ξ ≤ 1. As we limit ourselves to the mass ratios higher than ξ = 0.1, we can consider here the overdensity Δc = 20 000 (see Section 2.2), which probes the central region of the halo r20000c ≈ 0.06rvir (see Section 4.1). As done in Section 4.4, we compare the results obtained for this central halo region with those obtained at rvir. Moreover, in this case we can include in our analysis also the results of N17, who, using non-cosmological simulations, explored the distribution of orbital parameters at rcen ≈ 0.02rvir, roughly corresponding to Δc = 50 000, for merger mass ratios ξ ≃ 0.13 and ξ ≃ 0.67 (see Sections 4.1 and 4.2). Both r20000c ≈ 0.06rvir and rcen ≈ 0.02rvir can be considered proxies for the characteristic size of the CG in a cluster of galaxy (see Section 4.1). In Figs 5 and 6 we plot the distributions of E2b/Ψ0, L/Lmax, v/vcirc, and |vr|/v, together with their best fits. The parameters of the best fits (Gaussian distributions – equation 12 – for E2b/Ψ0 and v/vcirc, and beta distributions—equation 13—for L/Lmax and |vr|/v) are reported in Tables 3 and 4. The distribution of E2b/Ψ0 that we find at r20000c ≈ 0.06rvir has slightly more negative mean than the distribution measured at rvir (see the right-hand panel of Fig. 5), confirming the trend found in Section 4.4 for mergers with ξ < 0.1: the orbits of satellites accreting on to CGs tend to be slightly more bound than those of satellites accreting at the virial radius of the host cluster. The same result is visualized in Fig. 6 (left-hand panel), showing that the distribution of v/vcirc at r20000c peaks at lower values than the distribution of v/vcirc measured at rvir. Figure 5. View largeDownload slide Same as Fig. 3, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curve) and Δc = 20 000 (short-dashed curve). The long-dashed curve represents the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. Figure 5. View largeDownload slide Same as Fig. 3, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curve) and Δc = 20 000 (short-dashed curve). The long-dashed curve represents the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. Figure 6. View largeDownload slide Same as Fig. 4, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curves), and Δc = 20 000 (short-dashed curves). The long-dashed curves represent the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. Figure 6. View largeDownload slide Same as Fig. 4, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curves), and Δc = 20 000 (short-dashed curves). The long-dashed curves represent the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. It is interesting to compare the results at r20000c ≈ 0.06rvir with those obtained at rcen ≈ 0.02rvir by N17. While the distributions of E2b/Ψ0 are very similar for rcen and r20000c, the distributions of v/vcirc are significantly offset: in the experiments of N17 the values of v/vcirc measured at rcen tend to be higher than those measured at rvir, in contrast with the results obtained here for r20000c. The different behaviour between the distributions of v/vcirc and E2b/Ψ0 can be explained as follows. Though both quantities measure the binding energy of the orbit, as pointed out in Section 4.4, they are normalized quite differently: while Ψ0 accounts for the merger mass ratio ξ, vcirc is independent of ξ. The experiments of N17 have average merger mass ratio 〈ξ〉N ≃ 0.62 higher than our samples of mergers at rvir and r20000c (〈ξ〉N ≈ 0.5 in the interval 0.1 ≤ ξ ≤ 1). Moreover, we recall that in this work E2b is measured at the snapshot before merger, while v is corrected to be evaluated at rΔ, host (see Section 3): this is another source of difference between the distributions of E2b/Ψ0 and v/vcirc. Based only on measures of v/vcirc, compared to those of J15 (measured at r200c), N17 concluded that the orbits for CG–satellite mergers tend to be less bound than those of cosmological halo–halo mergers. The present analysis reveals that the higher values of v/vcirc found in the simulations of N17 at least partly reflect a bias in the merger mass ratios, which tend to be higher than cosmologically motivated values. In any case, the simulations here considered should, in general, be more realistic than the idealized simulations of N17, thus we believe that the distribution of v/vcirc here obtained for r20000c should be more representative for real CGs than the distribution found for rcen in N17. Therefore, based on the results found for E2b/Ψ0 and v/vcirc, we can conclude that the orbits of satellites accreting on to CGs in clusters tend to be more bound than the orbits of satellites accreting on to the host cluster-sized haloes. The distributions of L/Lmax (right-hand panel in Fig. 5) and |vr|/v (right-hand panel in Fig. 6) indicate that, as it happens for mergers with ξ < 0.1, also for mass ratios ξ ≥ 0.1 the orbits measured at r20000c tend to be more tangential than those measured at rvir (see also Table 4). In particular, the probability density function of L/Lmax is flatter for Δc = 20 000 than for Δc = Δvir (the latter peaks at ≈0.3 and drops above ≈0.8); the probability density function of |vr|/v measured at r20000c has a strong peak at ≈1, which is absent for measures at rvir. As far as the eccentricity of the orbits is concerned, the results obtained by N17 are consistent with those obtained here: the distributions of L/Lmax and |vr|/v measured at rcen are biased towards, respectively, high and low values, even more than those found here for Δc = 20 000. Therefore the present results confirm and strengthen the finding of N17 that the orbits of satellites accreting on to CGs tend to be more tangential than those of cosmological halo–halo accretion at the virial radius. As far as the scatter in the distributions is concerned, the trend for ξ ≥ 0.1 is the same as that for ξ < 0.1: for all the considered parameters (E2b/Ψ0, v/vcirc, L/Lmax, |vr|/v) the scatter is larger for higher values of Δc (i.e. smaller radii; see Tables 3 and 4). The effect is strongest for v/vcirc, for which the best-fitting probability density function has standard deviation almost a factor of 2 higher for measures at ≈0.06rvir than for measures at rvir. This is qualitatively in agreement with the findings of N17: the standard deviations for measures at rcen are higher than those for measures at rvir, with the only exception of L/Lmax, for which the scatter is the same in the two cases (Tables 3 and 4). 5 CONCLUSIONS In this paper we have used the results of the dark-matter only cosmological N-body simulations Le SBARBINE (Despali et al. 2016) to study the statistical properties of mergers between central and satellite galaxies in galaxy clusters. In particular we selected a sample of 101 cluster-sized haloes at z = 0 from the simulations Ada and Bice and traced their merging history in the redshift interval 0 < z < 1. We constructed merger trees for different overdensities Δc. When we use the virial overdensity [Δc = Δvir, with 100 ≲ Δvir(z) ≲ 150] we probe the accretion of satellites at the cluster virial radius rvir (Giocoli et al. 2008; Giocoli 2010). When we use higher overdensities (Δc = 5000, 10 000, and 20 000) we probe the accretion of satellites in the central region of the cluster (at radii r5000c ≈ 0.15rvir, r10000c ≈ 0.1rvir, and r20000c ≈ 0.06rvir), which can be considered as a proxy for the accretion of satellite galaxies on to CGs. We measured the distributions of merger mass ratios and orbital parameters for these merger histories. The main results of this work are the following. Though minor mergers largely outnumber major mergers, the latter contribute to the mass accreted at z < 1 at least as much as minor mergers, for all values of Δc. The mass-weighted merger mass ratio 〈ξ〉M increases for increasing Δc, so major mergers are even more important for CGs than for the accretion at the cluster virial radius. In the mass-ratio interval 0.01 ≤ ξ ≤ 1, more than 60 per cent of the mass accreted by CGs at z < 1 is due to major mergers. For higher overdensities (i.e. more central regions), the orbits of the accreting satellites tend to be less bound and more tangential. Therefore, the orbits of satellites accreting on to CGs are characterized by higher specific orbital angular momentum and lower specific orbital energy than orbits of halo accretion at the virial radius. The scatter in the orbital parameters tends to be larger for accretion on to CGs than for accretion at the halo virial radius. In this respect, the strongest effect is found for the distribution of v/vcirc, which, for ξ ≥ 0.1, has standard deviation almost a factor of 2 higher at ≈0.06rvir than at rvir. We compared the results obtained at r20000c ≈ 0.06rvir in our cosmological simulations with those obtained by N17 at rcen ≈ 0.02rvir in idealized non-cosmological simulations. We found good agreement on the distribution of orbital angular momentum, but we revised N17's conclusions on the binding energy of the orbits, which were somewhat biased by the non-cosmological setting. We provided parameters of the analytic best-fitting distributions of the pairs of orbital parameters (E2b, L) and (v/vcirc, |vr|/v) for different values of Δc (Tables 3 and 4). The distributions obtained for Δc = 20 000 (i.e. measured at r20000c ≈ 0.06rvir) can be taken as reference for modelling accretion on to CGs in clusters. In particular, the provided analytic distributions could be included in models attempting to predict the evolution of the scaling relations of cluster CGs without resorting to hydrodynamic cosmological simulations (e.g. Bernardi et al. 2011; Volonteri & Ciotti 2013; Shankar et al. 2015). Acknowledgements We would like to thank Giuseppe Tormen (and the Physics and Astronomy Department of Padova) who provided the computational resources to run the simulations. CG acknowledges support from the Italian Ministry for Education, University and Research (MIUR) through the SIR individual grant SIMCODE, project number RBSI14P4IH. Footnotes 1 For instance, consider a halo with mass M at redshift zi that at redshift zi − 1 > zi splits in two haloes of mass M1 and M2 < M1, which, in turn, at redshift zi − 2 > zi − 1 split, respectively, in two haloes of mass M1, 1 and M1, 2 < M1, 1, and in two haloes of mass M2, 1 and M2, 2 < M2, 1. 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Dynamics and Interactions of Galaxies . Springer-Verlag, Berlin, Heidelberg, p. 394 Google Scholar CrossRef Search ADS   Volonteri M., Ciotti L., 2013, ApJ , 768, 29 https://doi.org/10.1088/0004-637X/768/1/29 CrossRef Search ADS   Vulcani B., et al.  , 2014, ApJ , 797, 62 https://doi.org/10.1088/0004-637X/797/1/62 CrossRef Search ADS   Vulcani B., et al.  , 2016, ApJ , 816, 86 https://doi.org/10.3847/0004-637X/816/2/86 CrossRef Search ADS   White S. D. M., 1976, MNRAS , 174, 19 https://doi.org/10.1093/mnras/174.1.19 CrossRef Search ADS   APPENDIX A: DISTRIBUTIONS OF THE ORBITAL PARAMETERS MEASURED AT $$\boldsymbol{r_{\rm 200c}}$$ Here we compare the distributions of orbital parameters of the encounters experienced by the haloes of our sample with those found by J15 in the cosmological dark-matter only simulation DOVE. Specifically, we consider here the distributions of v/vcirc and |vr|/v found by J15 for dark-matter haloes of z = 0 mass M200c ≈ 1014M⊙ considering mergers with mass ratios in the range 0.005 ≤ ξ ≤ 0.05 in the redshift interval 0 < z < zHF, where zHF is the formation redshift of the halo (for M200c ≈ 1014M⊙ the distribution of zHF peaks between z = 0.5 and z = 1; see J15). In order to compare our results with those of J15, we built the Δc = 200 merger tree of our sample of 101 haloes with z = 0 mass Mvir ≥ 1014M⊙, taking all encounters with mass ratio 0.005 ≤ ξ ≤ 0.05. Altogether, in this way we select a sample of 1855 encounters (see Table 2). For these encounters we evaluated v/vcirc and vr/v at r200c using equations (8) and (9). The results are shown in Fig. A1: overall the agreement between our distributions and those of J15 is remarkable. The peaks and the widths of the two distributions of v/vcirc (left-hand panel of Fig. A1) almost coincide, while the tail at low values of v/vcirc is somewhat stronger in our distribution than in the distribution of J15. The two distributions of |vr|/v (right-hand panel of Fig. A1) are in excellent agreement over the entire range 0 ≤ |vr|/v ≤ 1. It must be noted that the time sampling of the DOVE simulation is significantly better than that of Le SBARBINE: for instance, in the redshift range 0 ≲ z ≲ 1 the number of available snapshots is 38 for DOVE and 13 for Le SBARBINE. Thus, the agreement in the distributions of v/vcirc and |vr|/v between J15's sample of encounters and ours suggests that the correction (Section 3) we applied to estimate the parameters at separation rΔ should be reliable. Figure A1. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) measured at r200c for halo–halo mergers in this work (solid curves) and in J15 (dashed curves). The solid curves are obtained for the cosmological simulations Ada and Bice considering mergers in the redshift range 0 < z < 1 for haloes with z = 0 mass Mvir ≥ 1014M⊙. The dashed curves are obtained for the cosmological simulation DOVE considering mergers from z = 0 up to the halo formation redshift (see text), for haloes with z = 0 mass M200c ≈ 1014M⊙. In both cases the satellite-to-host mass ratio is in the range 0.005 ≤ ξ ≤ 0.05. Figure A1. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) measured at r200c for halo–halo mergers in this work (solid curves) and in J15 (dashed curves). The solid curves are obtained for the cosmological simulations Ada and Bice considering mergers in the redshift range 0 < z < 1 for haloes with z = 0 mass Mvir ≥ 1014M⊙. The dashed curves are obtained for the cosmological simulation DOVE considering mergers from z = 0 up to the halo formation redshift (see text), for haloes with z = 0 mass M200c ≈ 1014M⊙. In both cases the satellite-to-host mass ratio is in the range 0.005 ≤ ξ ≤ 0.05. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

Accretion of satellites on to central galaxies in clusters: merger mass ratios and orbital parameters

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Abstract

Abstract We study the statistical properties of mergers between central and satellite galaxies in galaxy clusters in the redshift range 0 < z < 1, using a sample of dark-matter only cosmological N-body simulations from Le SBARBINE data set. Using a spherical overdensity algorithm to identify dark-matter haloes, we construct halo merger trees for different values of the overdensity Δc. While the virial overdensity definition allows us to probe the accretion of satellites at the cluster virial radius rvir, higher overdensities probe satellite mergers in the central region of the cluster, down to ≈0.06rvir, which can be considered a proxy for the accretion of satellite galaxies on to central galaxies. We find that the characteristic merger mass ratio increases for increasing values of Δc: more than 60 per cent of the mass accreted by central galaxies since z ≈ 1 comes from major mergers. The orbits of satellites accreting on to central galaxies tend to be more tangential and more bound than orbits of haloes accreting at the virial radius. The obtained distributions of merger mass ratios and orbital parameters are useful to model the evolution of the high-mass end of the galaxy scaling relations without resorting to hydrodynamic cosmological simulations. galaxies: clusters: general, galaxies: elliptical and lenticular, cD, galaxies: evolution, galaxies: formation, dark matter 1 INTRODUCTION Central galaxies (CGs) in galaxy groups and clusters are typically massive early-type galaxies with relatively old stellar populations and little ongoing star formation. CGs are believed to form in two phases (Merritt 1985; Tremaine 1990; Dubinski 1998; Ruszkowski & Springel 2009; Lauer et al. 2014). A first phase of in situ star formation at redshift z ≳ 1 is followed by a second phase of growth via the so-called galactic cannibalism process (Ostriker & Tremaine 1975; White 1976; Hausman & Ostriker 1978), that is accretion of satellite galaxies driven by dynamical friction (Chandrasekhar 1943). Quantitatively, both theoretical (De Lucia & Blaizot 2007; Feldmann et al. 2010; Tonini et al. 2012; Shankar et al. 2015) and observational (Marchesini et al. 2014; Bellstedt et al. 2016; Buchan & Shankar 2016; Vulcani et al. 2016) arguments suggest that about half of the stellar mass of CGs is assembled in situ at z ≳ 1, and the other half is assembled at relatively late times (z ≲ 1) via cannibalism processes. The effect of this cannibalism-driven growth phase on the properties of the CG (for instance size and velocity dispersion) is determined not only by the properties of the cannibalised galaxies (e.g. mass ratio between satellite and central; Naab, Johansson & Ostriker 2009), but also by the merging orbital parameters (Boylan-Kolchin, Ma & Quataert 2006; Nipoti et al. 2012). Measures of size, velocity dispersion, luminosity, and stellar mass of observed CGs lie on tight scaling relations (Bernardi et al. 2007; Liu et al. 2008; Vulcani et al. 2014). Knowing the properties of the mergers that occur during the late growth of CGs is thus important to theoretically understand the origin and evolution of their scaling relations. Nipoti (2017, hereafter N17) made the point that, given the very special location of CGs, at the bottom of the deep potential well of the host group or cluster, the distribution of the orbital parameters of the central-satellite encounters can be quite different from that of the encounters between galaxies not belonging to groups or clusters. N17 has characterized the distribution of the orbital parameters for central-satellite mergers using idealized N-body simulations in which the host system (a cluster or a group) is modelled as an isolated, spherical, collisionless N-body system and the satellite is rigid, being represented by a single massive, softened particle. In particular, the simulations of N17 are not framed within a cosmological context: the initial orbital parameters of the satellites are extracted from the host-halo distribution function, based on the assumption that violent relaxation (Lynden-Bell 1967) is rapid and the satellite population does not retain much memory of the cosmological distribution of the orbital parameters at the time of infall. The orbital parameters of the satellites then evolve due to dynamical friction (i.e. the satellites lose orbital energy and angular momentum). In this work we improve on the analysis of N17 by considering the problem in a fully cosmological setting, focusing on the growth of CGs in clusters of galaxies. For this purpose, we take advantage of the suite of cosmological simulations Le SBARBINE (Despali et al. 2016). These simulations are dark-matter only and so they do not contain a realistic galaxy population. Nevertheless, if we assume that CGs sit at the centre of dark-matter haloes and if we select the central regions of these host haloes at overdensities typical for the location of the CGs, they can be used for our purposes. Following Despali et al. (2016), we identified haloes in Le SBARBINE simulations for different overdensity threshold: Δc = Δvir, where Δvir is the redshift-dependent virial overdensity, and Δc = 200, 5000, 10 000 and 20 000, independent of redshift. For each of these halo definitions, we also built the corresponding merger history trees. When Δvir is considered, the entire virialized region of the halo is selected (in the case of a galaxy cluster, the entire cluster-sized dark-matter halo). When higher overdensities are considered, smaller regions of the halo are selected: for sufficiently high overdensity we select only the central part of the virialized halo, which we can roughly identify with the CG. Moreover, at these high overdensities, dense substructures within the main virial halo can be identified as independent structures. For each given overdensity we measure the properties of the mergers (specifically, the mass ratios and the orbital parameters). We then compare the properties of cosmological accretion (at the virial radius rvir) with those of accretion on to CGs (at some inner radius r ≪ rvir). The paper is organized as follows. Section 2 describes the simulations and the numerical methods. The orbital parameters of halo–halo encounters are defined in Section 3. The results are presented in Section 4, while in Section 5 we draw our conclusions. 2 NUMERICAL METHODS 2.1 N-body simulations and identification of dark-matter haloes In this work we make use of the results from the cosmological dark-matter only N-body simulations Le SBARBINE (Despali et al. 2016). The assumed background cosmology and initial conditions of the simulations are consistent with the results from the Planck Collaboration XVI (2014). In particular, in the simulations and throughout this paper we adopt a standard Λ cold dark matter cosmological model with the following parameters: matter density parameter Ωm = 0.307, cosmological constant density parameter $$\Omega _\Lambda = 0.693$$, linear power spectrum amplitude σ8 = 0.829, and dimensionless Hubble constant h = 0.677. Here we use only the two highest resolution runs among Le SBARBINE simulations: Ada (dark-matter particle mass m = 2.87 × 107M⊙) and Bice (m = 2.29 × 108M⊙). The two simulations have the same number of particles (N = 10243), but different box size: 92.3 Mpc for Ada and 184.6 Mpc for Bice. At each stored snapshot haloes are identified using a spherical overdensity algorithm (e.g. Tormen 1998; Tormen, Moscardini & Yoshida 2004; Giocoli, Tormen & van den Bosch 2008). In practice, haloes are defined as spherical overdensities with radius such that the average density is   \begin{eqnarray} {\overline{\rho }}=\Delta _{\rm c}\rho _{\rm crit}, \end{eqnarray} (1)where Δc is the critical overdensity and   \begin{eqnarray} \rho _{\rm crit}(z)= \frac{3H^2(z)}{8\pi G} \end{eqnarray} (2)is the critical density of the Universe, depending on redshift through the Hubble parameter H(z). Using this method, the overdensity threshold that defines the halo boundaries can be varied depending on the observational data that one wants to compare with. While the virial overdensity Δc = Δvir is commonly used in structure formation studies, other definitions can be chosen to be more similar to observational data sets: Δc = 500 is typically used in X-ray observations to define the mass of a galaxy cluster, while Δc = 200 is often used to fit weak lensing shear profiles. In our case Δc = 20 000 corresponds to the region within r ≃ 0.06rvir, which is a proxy for the size of the CG in a cluster. In this work, we consider different choices for Δc. For each value of Δc, we define the halo mass MΔ and the halo radius rΔ. First of all, we consider the virial value Δc = Δvir(z), which depends on z and on the cosmological parameters, as given by Eke, Cole & Frenk (1996). For the assumed cosmology Δvir increases with redshift: reference values are Δvir ≃ 97.9 at z = 0 and Δvir ≃ 154 at z = 1. When Δvir = Δc the halo radius and mass are, respectively, the virial radius rvir and the virial mass Mvir. For comparison with previous work we consider also the standard value Δc = 200, independent of redshift: in this case the halo mass and radius are M200c and r200c, respectively. Finally, in order to study the behaviour of mergers in central parts of the haloes, we explore the following other values of the critical overdensity, independent of redshift: Δc = 5000 (with mass M5000c and radius r5000c), Δc = 10 000 (with mass M10000c and radius r10000c), and Δc = 20 000 (with mass M20000c and radius r20000c). It must be noted that in each catalogue (i.e. for each value of the considered overdensity Δc) the haloes are identified independently. Therefore, the number of haloes is in general different in each catalogue, because a halo that, at a given redshift, is identified for a given value of Δc, at the same redshift, might be ‘incorporated’ in a bigger halo when a lower overdensity is considered (for more details see fig. 2 of Despali et al. 2016). 2.2 Sample of haloes and catalogues of halo–halo encounters We study the redshift range 0 ≲ z ≲ 1, in which Le SBARBINE simulations have 13 snapshots at the following redshifts: z1 = 1.012, z2 = 0.904, z3 = 0.796, z4 = 0.694, z5 = 0.597, z6 = 0.507, z7 = 0.421, z8 = 0.34, z9 = 0.264, z10 = 0.192, z11 = 0.124, z12 = 0.06, z13 = 0. We create a sample of galaxy clusters by selecting, in the Δvir catalogue, all haloes with Mvir ≥ 1014M⊙ at z = 0. The resulting sample consists of 101 haloes at z = 0 (12 haloes in Ada and 89 haloes in Bice). We identify these 101 haloes also in the higher-overdensity (Δc > Δvir) catalogues (Δc = 200, 5000, 10 000, and 20 000), finding that all of them have counterparts in all the considered catalogues. We note that, as the selection in mass is done on Mvir, the selected z = 0 haloes can have mass MΔ < 1014M⊙ for Δc > Δvir, because the mass of a given halo decreases for increasing Δc. In order to identify halo-halo encounters we proceed as follows. From the halo catalogues built for the 13 simulation snapshots and for each overdensity, we construct the halo merging history tree. Starting from each halo at z = 0, we define its progenitors at the previous output, z = 0.06, as all haloes that, in the time elapsed between two snapshots, have given at least 50 per cent of their particles to the considered z = 0 halo. The main progenitor at z = 0.06 is defined as the most massive progenitor of the z = 0 halo. We then repeat the same procedure, now starting from the main progenitor at z = 0.06 and considering its progenitors at z = 0.124, and we proceed backwards in time, always following the main progenitor halo. The resulting merger tree consists of a main trunk, which traces the main progenitor back in time, and of satellites; these last are all the progenitors that at any time merge directly on to the main progenitor. By construction, for given simulation and descendant halo, the definition of the main progenitor depends on the time sampling (i.e. the number of snapshots): in principle it is possible that the main branch is not identified correctly if the time sampling is insufficient1. However, Giocoli (2008), using simulations with time sampling similar to Le SBARBINE, has shown that the probability of misidentifying the main progenitor branch for galaxy cluster scale haloes is much below 10 per cent in the redshift interval 0 ≲ z ≲ 1. In addition, the consistence of our results with those obtained by Jiang et al. (2015, hereafter J15) with the DOVE simulation (Appendix A), which has a better time sampling, ensures us that the time sampling of our runs is good enough to uniquely follow the main halo progenitor branch back in time. We define the halo–halo encounters between two subsequent snapshots by selecting in the higher-redshift snapshot all the satellite-main progenitor pairs. We indicate the physical properties of the main progenitor (sometimes referred to also as host halo) with the subscript ‘host’ and those of the less massive progenitors (satellites) with the subscript ‘sat’. For each pair we then measure the masses (MΔ, host and MΔ, sat), radii (rΔ, host and rΔ, sat), and relative position (r) and velocity (v) of the centres of mass. We apply the procedure described above to all the z = 0 haloes of our sample and to all their main progenitors, back to the z = 0.904 snapshot. In this way we build our catalogue containing all the halo–halo encounters in the redshift range 0 < z < 1 that end up in cluster-sized haloes at z = 0. Of course, the number of encounters and their properties are different for different overdensities. We note that not all the halo–halo encounters must be considered rapid mergers. By construction, our catalogue of halo–halo encounters includes cases of satellite haloes that, after the encounter, escape from the main halo. In these cases, the haloes are distinct at a given snapshot, are identified as a single halo at a later snapshot, but are again distinct at an even later time step. In the terminology of this work we then distinguish halo–halo encounters and halo–halo mergers. In Section 4.2 we will define a criterion to select the subsample of mergers in the whole sample of halo–halo encounters. For each halo–halo encounter we define the mass ratio ξ = MΔ, sat/MΔ, host. In order to have a robust measure of the properties of halo–halo encounters and mergers, we limit our exploration to ξ ≥ ξmin, where ξmin is a minimum mass ratio such that the number of particles of the satellite is at least N ≈ 100. Clearly ξmin depends on the mass resolution of the simulations, on the explored redshift range, and on the considered overdensity Δc. In particular, we adopt ξmin = 0.005 for Δc = Δvir and Δc = 200, ξmin = 0.01 for Δc = 5000 and Δc = 10 000, and ξmin = 0.1 for Δc = 20 000. We verified that with these choices, for our sample of encounters in the redshift range 0 < z < 1, the number of satellites with N < 100 never exceeds 5 per cent of the entire satellite population. Using dark-matter only simulations to infer the properties of central-satellite galaxy mergers in clusters, we are implicitly assuming that the dynamical evolution of subhaloes orbiting massive host haloes is not significantly influenced by the presence of baryons. In fact, comparisons between dark-matter only and hydrodynamic cosmological simulations indicate that some properties of the subhalo mass function can be modified by the presence of baryons (Chua et al. 2017, and references therein). However, the effect is negligible for the relatively high satellite-host mass ratios considered in this work (see fig. 1 of Chua et al. 2017). 3 ORBITAL PARAMETERS OF HALO–HALO ENCOUNTERS It is useful to describe an encounter between two haloes in terms of the orbital parameters calculated in the point-mass two-body approximation. This description, though not rigorous for extended objects, is often used in the study and classification of mergers of galaxies (e.g. Boylan-Kolchin, Ma & Quataert 2006; Nipoti, Treu & Bolton 2009) and dark-matter haloes (e.g. Khochfar & Burkert 2006; Posti et al. 2014). Here, we define the point-mass two-body approximation orbital parameters of halo–halo encounters, following the formalism of N17. The orbit can be fully characterized by the pair of parameters orbital energy and angular momentum. For a halo–halo encounter we define the two-body approximation orbital energy per unit mass   \begin{eqnarray} {E_{\rm 2b}}=\frac{1}{2}v^2-\frac{G{M}_{\rm 2b}}{r}, \end{eqnarray} (3)where M2b ≡ MΔ, host + MΔ, sat, $$r\equiv |{\boldsymbol r}|$$ is the relative distance and $$v\equiv |{\boldsymbol v}|$$ is the relative speed, between their centres of mass. It is useful to decompose $${\boldsymbol v}$$ in its radial component $$v_{\rm r}={\boldsymbol v}\cdot {\boldsymbol r}/r$$ and its tangential component, with modulus $$v_{\rm tan}=\sqrt{v^2-v_{\rm r}^2}$$. The modulus of the orbital angular momentum per unit mass is L = rvtan. At fixed separation r and energy E2b, the modulus of the maximum allowed specific angular momentum is   \begin{eqnarray} {L_{\rm max}}=r\sqrt{2\left({E_{\rm 2b}}+\frac{G{M}_{\rm 2b}}{r}\right)}=rv, \end{eqnarray} (4)where v is the relative speed when the two haloes have separation r. Clearly, Lmax is such that L/Lmax = vtan/v. Another set of orbital parameters used to classify halo–halo encounters (e.g. Benson 2005; J15) is the pair (v/vcirc, |vr|/v), where, given a distance r from the centre of the host system (for instance the overdensity radius rΔ, host), v is the relative speed at r = rΔ, host, vr is the radial component of the relative velocity at r = rΔ, host and   \begin{eqnarray} v_{\rm circ}=\sqrt{\frac{G{M}_{\rm \Delta ,host}}{{r}_{\rm \Delta ,host}}} \end{eqnarray} (5)is the host circular velocity at rΔ, host. Given the finite time sampling (i.e. the finite number of snapshots), in our simulations we have information on the halo–halo relative velocity $${{\boldsymbol v}_{\rm snap}}$$ when the two haloes have a separation rsnap that is in general larger than rΔ, host. As discussed in several previous works (e.g. Benson 2005, J15), it is thus necessary to apply a correction to recover (v/vcirc, |vr|/v) measured when the satellite crosses the desired overdensity radius of the host (i.e. when the separation is rΔ, host). We correct the velocity as follows. We first compute the relative velocity $${\boldsymbol v}_{\rm 2b}$$ at rΔ, host, assuming that the point-mass two-body energy and angular momentum are conserved: $${\boldsymbol v}_{\rm 2b}$$ is such that   \begin{eqnarray} \frac{1}{2}v_{\rm snap}^2-\frac{G({M}_{\rm \Delta ,host}+{M}_{\rm \Delta ,sat})}{r_{\rm snap}} =\frac{1}{2}v_{\rm 2b}^2-\frac{G({M}_{\rm \Delta ,host}+{M}_{\rm \Delta ,sat})}{{r}_{\rm \Delta ,host}}\nonumber\\ \end{eqnarray} (6)and   \begin{eqnarray} r_{\rm snap}v_{\rm tan,snap}={r}_{\rm \Delta ,host}v_{\rm tan,2b}, \end{eqnarray} (7)where vtan, 2b is the tangential component of $${\boldsymbol v}_{\rm 2b}$$. If, as in most cases, equations (6) and (7) give vtan, 2b ≤ v2b, the modulus of the radial component of v2b is $$|v_{\rm r, 2b}|=\sqrt{v_{\rm 2b}^2-v_{\rm tan,2b}^2}$$. If, instead, equations (6) and (7) give vtan, 2b > v2b (which indicates that the point-mass two-body orbit is too crude an approximation), we simply fix vtan, 2b = v2b and vr, 2b = 0. Finally, we define the corrected velocity $${\boldsymbol v}$$ at the time of crossing (r = rΔ, host) to be such that   \begin{eqnarray} \left(\frac{v}{v_{\rm circ}}\right)_{{r}_{\rm \Delta ,host}}=\frac{1}{2}\frac{v_{\rm snap}+v_{\rm 2b}}{v_{\rm circ}} \end{eqnarray} (8)and   \begin{eqnarray} \left(\frac{v_{\rm r}}{v}\right)_{{r}_{\rm \Delta ,host}}=\frac{1}{2}\left[\left(\frac{v_{\rm r}}{v}\right)_{\rm snap}+\left(\frac{v_{\rm r}}{v}\right)_{\rm 2b}\right]. \end{eqnarray} (9)We verified that this is a reasonably good approximation by comparing the distributions of our sample of haloes with previous literature work (see Appendix A). 4 RESULTS 4.1 Halo masses and radii at different overdensities The ratios MΔ/Mvir and rΔ/rvir are decreasing functions of Δc (see Despali et al. 2017). The exact values of these ratios depend on the halo mass density distribution through the halo concentration parameter (Giocoli, Tormen & Sheth 2012). We computed MΔ/Mvir and rΔ/rvir for all haloes in our sample (i.e. the 101 z = 0 haloes and all their main progenitors in all previous snapshots back to z = 0.904; altogether 1224 haloes; Section 2.2). The means and standard deviations of the distributions of MΔ/Mvir and rΔ/rvir are reported in Table 1 for Δc = 200, 5000, 10 000, and 20 000. For the highest overdensity here considered Δc = 20 000, the average values are MΔ/Mvir ≈ 0.05 and rΔ/rvir ≈ 0.06. The distributions of MΔ/Mvir and rΔ/rvir are broader for increasing Δc, with standard deviations in the range 10-17 per cent for rΔ/rvir and 11-44 per cent for MΔ/Mvir. Table 1. Mean (μ) and standard deviation (σ) of the distributions of MΔ/Mvir and rΔ/rvir, for different values of Δc, for our sample consisting of the 101 z = 0 haloes with Mvir ≥ 1014M⊙ and all their main progenitor haloes in the previous snapshots, back to z = 0.904 (altogether 1224 haloes). Δc  μ(MΔ/Mvir)  σ(MΔ/Mvir)  μ(rΔ/rvir)  σ(rΔ/rvir)  200  0.848  0.097  0.807  0.082  5000  0.148  0.047  0.153  0.020  10 000  0.085  0.032  0.100  0.015  20 000  0.045  0.020  0.064  0.011  Δc  μ(MΔ/Mvir)  σ(MΔ/Mvir)  μ(rΔ/rvir)  σ(rΔ/rvir)  200  0.848  0.097  0.807  0.082  5000  0.148  0.047  0.153  0.020  10 000  0.085  0.032  0.100  0.015  20 000  0.045  0.020  0.064  0.011  View Large In the following sections we will compare our results with those of simulations of satellites in isolated host haloes (N17), in which the merger orbital parameters were measured at a radius rcen = 0.12rs, where rs is the halo scale radius. If we identify the truncation radius of N17 with rvir, we have rcen ≃ 0.024rvir and that the mass contained within rcen is Mcen ≃ 0.0075Mvir. Therefore, the region probed by N17 is somewhat smaller than the most central region here considered (Δc = 20 000) and roughly corresponds to an overdensity Δc = 50 000. It is useful to note that Kravtsov (2013) finds that, on average, the three-dimensional half-mass radius of the stellar distribution of observed CGs is r*, 1/2 ≈ 0.015r200c ≈ 0.012rvir (using the average ratio r200c/rvir ≃ 0.85 found for our sample of haloes; see Table 1). Therefore, in terms of r*, 1/2, we have r20000c ≈ 5r*, 1/2 and rcen ≈ 2r*, 1/2, which indicates that both r20000c and rcen probe the region of the halo occupied by the stellar distribution of the CG. 4.2 Mergers and fly-bys As mentioned in Section 2.2, we do not expect to have a rapid merger for all halo–halo encounters. Rapid mergers occur when the orbits are bound (E2b < 0), but also for unbound orbits (E2b ≥ 0), provided the orbital angular-momentum modulus L is sufficiently low (see section 7.4 of Binney & Tremaine 1987). For this reason, a convenient parameter that can be used to identify mergers is the orbit eccentricity   \begin{eqnarray} e = \sqrt{1+\frac{2{E_{\rm 2b}}L^2}{G^2{M}_{\rm 2b}^2}}, \end{eqnarray} (10)which, for E2b > 0, is an increasing function of both E2b and L. As in N17, we take as fiducial discriminating value of eccentricity ecrit = 1.5 and classify an encounter as a merger when e ≤ ecrit and as a fly-by when e > ecrit. The eccentricity distributions for our samples of halo–halo encounters with mass ratio 0.01 ≤ ξ < 0.1 are shown in Fig. 1 (upper panel) for Δc = Δvir, Δc = 5000, and Δc = 10 000 (corresponding to radii rvir, r5000c ≈ 0.15rvir and r10000c ≈ 0.1rvir, respectively; see Section 4.1). The distribution of the eccentricity for encounters with mass ratio ξ ≥ 0.1 is shown in the lower panel of Fig. 1 for Δc = Δvir and Δc = 20 000 (corresponding to radii rvir and r20000c ≈ 0.06rvir, respectively; see Section 4.1). In the same panel we plot also the distribution found by N17 for numerical models of satellites orbiting in isolated haloes with ξ ≃ 0.13 and ξ ≃ 0.67, measured at rcen ≈ 0.02rvir (see Section 4.1). From Fig. 1 it is clear that most of the encounters are indeed classified as mergers: the adopted cut in eccentricity allows us to effectively exclude the tail of high-eccentricity orbits, which are most likely fly-bys. The number of mergers and the total number of encounters for our sample are reported in Table 2 for different values of Δc and intervals of ξ. Figure 1. View largeDownload slide Upper panel: probability distribution p = dn/dx of the logarithm of the orbital eccentricity of halo–halo encounters computed in the two-body approximation (x = log10e), for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dotted histogram), and Δc = 10 000 (dashed histogram). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. The vertical dashed line (e = 1.5) discriminates mergers (e ≤ 1.5) and fly-bys (e > 1.5). Lower panel: same as the upper panel, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid histogram) and Δc = 20 000 (dot–dashed histogram). The dashed histogram represents the results obtained by N17 for encounters at rcen ≈ 0.02rvir, using simulations of satellites in isolated host haloes. Figure 1. View largeDownload slide Upper panel: probability distribution p = dn/dx of the logarithm of the orbital eccentricity of halo–halo encounters computed in the two-body approximation (x = log10e), for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dotted histogram), and Δc = 10 000 (dashed histogram). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. The vertical dashed line (e = 1.5) discriminates mergers (e ≤ 1.5) and fly-bys (e > 1.5). Lower panel: same as the upper panel, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid histogram) and Δc = 20 000 (dot–dashed histogram). The dashed histogram represents the results obtained by N17 for encounters at rcen ≈ 0.02rvir, using simulations of satellites in isolated host haloes. Table 2. Total number of encounters and number of encounters classified as mergers experienced by all the haloes in our sample (see Section 2.2). The data for rΔ = rcen refer to the results of N17. ξ  rΔ  Encounters  Mergers  0.005-0.05  r200c  1855  1733  0.01–0.1  rvir  1049  998  0.01–0.1  r5000c  456  330  0.01–0.1  r10000c  275  210  0.1–1  rvir  216  207  0.1–1  r20000c  98  69  0.1–1  rcen  82  44  ξ  rΔ  Encounters  Mergers  0.005-0.05  r200c  1855  1733  0.01–0.1  rvir  1049  998  0.01–0.1  r5000c  456  330  0.01–0.1  r10000c  275  210  0.1–1  rvir  216  207  0.1–1  r20000c  98  69  0.1–1  rcen  82  44  View Large As a quantitative test of our classification of mergers and fly-bys, we analysed the post-encounter evolution of the satellites in the Δc = 20 000 catalogue with mass ratio ξ ≥ 0.1. In practice, for each encounter occurring between the snapshots at redshifts zi − 1 and zi, we check whether the satellite and the main halo are distinct (i.e. the satellite has escaped) in the snapshot at redshift zi + 1 (clearly we exclude the case i = 13, because the snapshot at z13 = 0 is the last; see Section 2.2). We find that the satellite escapes in 80 per cent of the encounters classified as fly-bys and in 15 per cent of the encounters classified as mergers, which suggests that our classification is sufficiently accurate. We verified that the selection of mergers is not sensitive to the exact value of ecrit: the main results of the present work are essentially the same for values of ecrit in the range 1.25 ≲ ecrit ≲ 2. 4.3 Distribution of merger mass ratio There are good reasons to expect mergers on to CGs in clusters to be characterized by a distribution of mass ratios ξ different from that of cosmological halo–halo mergers measured at the virial radius. It is well known that dynamical friction, which is the main driver of galactic cannibalism, is more effective for more massive satellites, so we expect the typical mass ratio of mergers on to CGs to be higher than that of mergers at the virial radius of the host cluster. We can quantitatively explore this question by comparing the distributions of ξ in halo catalogues obtained for different values of Δc. The upper panel of Fig. 2 shows, as a function of the mass ratio ξ, the fraction Maccr( > ξ) of mass accreted in mergers with mass ratio larger than ξ, normalized to the total mass accreted in mergers with mass ratio ξ ≥ 0.01, for mergers measured at2rvir, r5000c, and r10000c. A clear trend emerges from this plot: in line with the expectations, mergers with higher mass ratios contribute more when more central halo regions are considered. The difference between the mergers measured at Δvir and those measured at higher overdensities becomes more and more important for ξ → 1 (see lower panel of Fig. 2). The median value ξmed, such that half of the mass is accreted in mergers with mass ratio larger than ξmed, is 0.26 at rvir, 0.37 at ≈0.15rvir and 0.53 at ≈0.1rvir. Another useful indicator of the characteristic mass ratio of mass accretion is the mass-weighted merger mass ratio 〈ξ〉M (see Nipoti et al. 2012), which can be written as   \begin{eqnarray} \left\langle \xi \right\rangle _{\rm M}=\frac{\left\langle \xi ^2\right\rangle _{\rm N}}{\left\langle \xi \right\rangle _{\rm N}}, \end{eqnarray} (11)where 〈⋅⋅⋅〉N is the number-weighted average. As shown in Fig. 2 (upper panel), 〈ξ〉M ≃ 0.38 at rvir, 〈ξ〉M ≃ 0.44 at ≈0.15rvir, and 〈ξ〉M ≃ 0.49 at ≈0.1rvir. For the innermost radius here probed (≈0.1rvir) the characteristic merger mass ratio is close to 1/2. Figure 2. View largeDownload slide Upper panel: fraction of mass accreted in mergers with mass ratio larger than ξ, relative to the total mass accreted in mergers with 0.01 ≤ ξ ≤ 1, for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dotted curve), and Δc = 10 000 (dashed curve). The measures are for mergers in the redshift interval 0 < z < 1 for our sample of dark-matter haloes with Mvir ≥ 1014M⊙ at z = 0. The horizontal lines indicate, for the distributions with the corresponding line styles, the fraction fmajor of mass accreted in major mergers, assuming major-merger mass-ratio threshold ξmajor = 1/3 (lower lines) or ξmajor = 1/4 (upper lines). The vertical lines indicate the mass-weighted average merger mass ratio (equation 11) for the distributions with the corresponding line styles. Lower panel: dotted curve: relative difference between the dotted and solid curves in the upper panel. Dashed curve: relative difference between the dashed and solid curve in the upper panel. Figure 2. View largeDownload slide Upper panel: fraction of mass accreted in mergers with mass ratio larger than ξ, relative to the total mass accreted in mergers with 0.01 ≤ ξ ≤ 1, for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dotted curve), and Δc = 10 000 (dashed curve). The measures are for mergers in the redshift interval 0 < z < 1 for our sample of dark-matter haloes with Mvir ≥ 1014M⊙ at z = 0. The horizontal lines indicate, for the distributions with the corresponding line styles, the fraction fmajor of mass accreted in major mergers, assuming major-merger mass-ratio threshold ξmajor = 1/3 (lower lines) or ξmajor = 1/4 (upper lines). The vertical lines indicate the mass-weighted average merger mass ratio (equation 11) for the distributions with the corresponding line styles. Lower panel: dotted curve: relative difference between the dotted and solid curves in the upper panel. Dashed curve: relative difference between the dashed and solid curve in the upper panel. Given a discriminant mass ratio ξmajor between major and minor mergers, we can define fmajor ≡ Maccr( ≥ ξmajor)/Maccr( ≥ ξmin) as the fraction of mass accreted in major mergers in the redshift range 0 < z < 1 (here the minimum mass ratio is ξmin = 0.01.). For, respectively, Δc = Δvir, 5000, and 10 000 we find fmajor = 0.48, 0.55, and 0.64 (assuming ξmajor = 1/3), and fmajor = 0.54, 0.62, and 0.69 (assuming ξmajor = 1/4). Taking the results for Δc = 10 000 as a proxy for accretion on to the CG, we can conclude that (at least in the explored mass ratio interval 0.01 ≤ ξ ≤ 1) more than 60 per cent of the mass accreted at z < 1 by CGs in clusters is due to major mergers. This conclusion is qualitatively consistent with previous observational (Lidman et al. 2013) and theoretical (Rodriguez-Gomez et al. 2016) results on the role of major mergers in the build-up of massive CGs. By definition, for given ξmajor, fmajor depends on the minimum mass ratio ξmin. Here, for the reasons explained in Section 2.2, we have fixed ξmin = 0.01, but of course also mergers with lower mass ratio contribute to the actual halo mass growth. The slopes at low ξ of the curves in the upper panel of Fig. 2 suggest that the relative contribution of mergers with ξ < 0.01 is more important at rvir than at r5000c and r10000c. To quantify this effect, we computed fmajor assuming ξmin = 0.005 (thus relaxing our requirement that the satellites have at least N ≈ 100 particles): in this case we get values of fmajor that are only slightly smaller than those obtained for ξmin = 0.01 (for instance, by ≲ 6 per cent for measures at rvir and by ≲ 3 per cent for measures at r10000c). Thus, in this respect, our conclusion about the predominance of major mergers in the z ≲ 1 build-up of cluster CGs appears robust. 4.4 Orbital parameters for mergers with mass ratio $$\boldsymbol{0.01\le \xi <0.1}$$ In this section we discuss the distribution of orbital parameters for mergers (i.e. encounters with eccentricity e ≤ 1.5) with mass ratio in the range 0.01 ≤ ξ < 0.1, comparing the results for Δc = Δvir, Δc = 5000, and Δc = 10 000. Fig. 3 shows, for these samples of mergers, the distributions of the two-body specific orbital energy E2b and of the modulus of the specific orbital angular momentum L. E2b is normalized to Ψ0 ≡ G(MΔ, host + MΔ, sat)/rΔ, host, which is the absolute value of the two-body gravitational potential of the encounter when the separation is rΔ, host. L is normalized to Lmax (equation 4), which is the modulus of the maximum angular momentum for given orbital energy E2b. We have fitted the distributions of E2b/Ψ0 with a Gaussian distribution   \begin{eqnarray} p(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp {\left[-\frac{(x-\mu )^2}{2\sigma ^2}\right]}, \end{eqnarray} (12)where μ is the mean and σ is the standard deviation, and the distributions of L/Lmax with a beta distribution   \begin{eqnarray} p(x)=\frac{x^{\alpha -1}(1-x)^{\beta -1}}{B(\alpha , \beta )}, \end{eqnarray} (13)where   \begin{eqnarray} B(\alpha , \beta )=\frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )} \end{eqnarray} (14)and Γ is the gamma function. In Fig. 4 we plot the distributions of the orbital parameters v/vcirc and |vr|/v, which, as pointed out in Section 3, are a pair of parameters, alternative to E2b and L, often used to characterize the orbits of galaxy and halo encounters. We emphasize that, in the present context, this pair of parameters does not carry exactly the same information as E2b and L: v/vcirc and |vr|/v are evaluated at a separation rΔ, host (equations 8 and 9), while E2b and L are evaluated at the snapshot before the merger. Moreover, while v is normalized to the main halo circular velocity vcirc, which is independent of the properties of the satellite, E2b is normalized to Ψ0, which depends also on the mass of the satellite (and therefore on the mass ratio ξ). The distributions of v/vcirc are fitted with a Gaussian (equation 12), while the distributions of |vr|/v are fitted with a beta distribution (equation 13). The best-fitting distributions of E2b/Ψ0, L/Lmax, v/vcirc, and |vr|/v are over-plotted in the corresponding panels of Figs 3 and 4, and their parameters are reported in Tables 3 and 4. Figure 3. View largeDownload slide Probability distribution p = dn/dx of the normalized orbital energy, computed in the two-body approximation (x = E2b/Ψ0; left-hand panel), and of the normalized angular-momentum modulus (x = L/Lmax; right-hand panel) for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dashed histogram), and Δc = 10 000 (dotted histogram). Here Ψ0 ≡ G(MΔ, host + MΔ, sat)/rΔ, host and Lmax is defined by equation (4). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Figure 3. View largeDownload slide Probability distribution p = dn/dx of the normalized orbital energy, computed in the two-body approximation (x = E2b/Ψ0; left-hand panel), and of the normalized angular-momentum modulus (x = L/Lmax; right-hand panel) for critical overdensities Δc = Δvir (solid histogram), Δc = 5000 (dashed histogram), and Δc = 10 000 (dotted histogram). Here Ψ0 ≡ G(MΔ, host + MΔ, sat)/rΔ, host and Lmax is defined by equation (4). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Figure 4. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and of the radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) when the satellite crosses the virial radius of the host rΔ, host for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dashed curve), and Δc = 10 000 (dotted curve). Here vcirc is the host circular velocity at rΔ, host. v/vcirc and |vr|/v are evaluated at rΔ, host as in equations (8) and (9). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Figure 4. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and of the radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) when the satellite crosses the virial radius of the host rΔ, host for critical overdensities Δc = Δvir (solid curve), Δc = 5000 (dashed curve), and Δc = 10 000 (dotted curve). Here vcirc is the host circular velocity at rΔ, host. v/vcirc and |vr|/v are evaluated at rΔ, host as in equations (8) and (9). The curves represent the best-fitting distributions of the histograms with the corresponding line styles (see Tables 3 and 4). Here we consider mergers with mass ratios 0.01 ≤ ξ < 0.1. Table 3. Mean (μ) and standard deviation (σ) of the best-fitting Gaussian distributions (equation 12) of the orbital parameters E2b/Ψ0 and v/vcirc, for different values of the overdensity radius rΔ and intervals of mass ratios ξ. The data for rΔ = rcen refer to the results of N17. Parameter  ξ  rΔ  μ  σ  E2b/Ψ0  0.01–0.1  rvir  −0.21  0.29  E2b/Ψ0  0.01–0.1  r5000c  −0.30  0.39  E2b/Ψ0  0.01–0.1  r10000c  −0.39  0.40  E2b/Ψ0  0.1–1  rvir  −0.24  0.23  E2b/Ψ0  0.1–1  r20000c  −0.31  0.37  E2b/Ψ0  0.1–1  rcen  −0.33  0.31  v/vcirc  0.01–0.1  rvir  1.17  0.25  v/vcirc  0.01–0.1  r5000c  0.86  0.31  v/vcirc  0.01–0.1  r10000c  0.76  0.32  v/vcirc  0.1–1  rvir  1.28  0.21  v/vcirc  0.1–1  r20000c  0.99  0.39  v/vcirc  0.1–1  rcen  1.39  0.35  Parameter  ξ  rΔ  μ  σ  E2b/Ψ0  0.01–0.1  rvir  −0.21  0.29  E2b/Ψ0  0.01–0.1  r5000c  −0.30  0.39  E2b/Ψ0  0.01–0.1  r10000c  −0.39  0.40  E2b/Ψ0  0.1–1  rvir  −0.24  0.23  E2b/Ψ0  0.1–1  r20000c  −0.31  0.37  E2b/Ψ0  0.1–1  rcen  −0.33  0.31  v/vcirc  0.01–0.1  rvir  1.17  0.25  v/vcirc  0.01–0.1  r5000c  0.86  0.31  v/vcirc  0.01–0.1  r10000c  0.76  0.32  v/vcirc  0.1–1  rvir  1.28  0.21  v/vcirc  0.1–1  r20000c  0.99  0.39  v/vcirc  0.1–1  rcen  1.39  0.35  View Large Table 4. Parameters α and β, mean (μ) and standard deviation (σ) of the best-fitting beta distributions (equation 13) of the orbital parameters L/Lmax and vr/v, for different values of the overdensity radius rΔ and intervals of mass ratios ξ. The data for rΔ = rcen refer to the results of N17. Parameter  ξ  rΔ  α  β  μ  σ  L/Lmax  0.01–0.1  rvir  1.58  1.38  0.53  0.25  L/Lmax  0.01–0.1  r5000c  1.01  0.71  0.59  0.30  L/Lmax  0.01–0.1  r10000c  0.96  0.69  0.58  0.30  L/Lmax  0.1–1  rvir  1.57  2.16  0.42  0.23  L/Lmax  0.1–1  r20000c  0.81  0.85  0.49  0.31  L/Lmax  0.1–1  rcen  1.99  0.68  0.75  0.23  vr/v  0.01–0.1  rvir  2.69  0.78  0.78  0.20  vr/v  0.01–0.1  r5000c  1.51  0.67  0.69  0.26  vr/v  0.01–0.1  r10000c  1.39  0.60  0.70  0.26  vr/v  0.1–1  rvir  3.70  0.67  0.85  0.16  vr/v  0.1–1  r20000c  1.24  0.43  0.75  0.27  vr/v  0.1–1  rcen  1.28  1.05  0.55  0.27  Parameter  ξ  rΔ  α  β  μ  σ  L/Lmax  0.01–0.1  rvir  1.58  1.38  0.53  0.25  L/Lmax  0.01–0.1  r5000c  1.01  0.71  0.59  0.30  L/Lmax  0.01–0.1  r10000c  0.96  0.69  0.58  0.30  L/Lmax  0.1–1  rvir  1.57  2.16  0.42  0.23  L/Lmax  0.1–1  r20000c  0.81  0.85  0.49  0.31  L/Lmax  0.1–1  rcen  1.99  0.68  0.75  0.23  vr/v  0.01–0.1  rvir  2.69  0.78  0.78  0.20  vr/v  0.01–0.1  r5000c  1.51  0.67  0.69  0.26  vr/v  0.01–0.1  r10000c  1.39  0.60  0.70  0.26  vr/v  0.1–1  rvir  3.70  0.67  0.85  0.16  vr/v  0.1–1  r20000c  1.24  0.43  0.75  0.27  vr/v  0.1–1  rcen  1.28  1.05  0.55  0.27  View Large The distributions of E2b/Ψ0 (left-hand panel in Fig. 3) suggest that for higher values of Δc (i.e. when more central regions of the haloes are considered) the orbits of mergers tend to be slightly more bound: the mean orbital energy for Δc = 10 000 is more negative than that for Δc = 5000, which in turn is more negative than that at Δc = Δvir (see Table 3). However, the distributions are relatively broad and there is substantial overlap. A qualitatively similar, but stronger trend can be seen in the distributions of v/vcirc (which is another measure of the binding energy of the orbit; left-hand panel in Fig. 4), which are characterized by larger offsets between the peak of the measures at rvir and those at r5000c or r10000c. The fact that the distributions are more offset in v/vcirc than in E2b/Ψ0 comes from the fact that the accretion history for higher Δc is characterized by higher merger mass ratios (see Section 4.3 and Fig. 2). As pointed out above, while vcirc ignores the properties of the satellite, the normalization potential Ψ0 accounts for the mass ratio. In this sense, E2b/Ψ0 should give a cleaner measure of the binding energy of the orbits, when samples with different mass-ratio distributions are compared. We also note that the distributions of both E2b/Ψ0 and v/vcirc have higher scatter (larger standard deviation; see Table 3) for increasing Δc. The distributions of L/Lmax (right-hand panel in Fig. 3) and |vr|/v (right-hand panel in Fig. 4) suggest that for higher values of Δc (i.e. when more central regions of the haloes are considered) the orbits of mergers tend to be significantly more tangential. Comparing the right-hand panels of Figs 3 and 4, it is apparent that, for Δc = 5000 and Δc = 10 000, the distributions of both L/Lmax = vtan/v and |vr|/v peak at ≈1. This might be counterintuitive, because |vr|/v → 0 when |vtan|/v → 1. However, the relation between |vr|/v and |vtan|/v, namely   \begin{eqnarray} \frac{|v_{\rm r}|}{v}=\sqrt{1-\left(\frac{v_{\rm tan}}{v}\right)^2}, \end{eqnarray} (15)is non-linear and such that, for instance, |vr|/v ≳ 0.9 corresponds to vtan/v ≲ 0.44. Thus, a peak at |vr|/v ≈ 1 does not necessarily imply a peak at L/Lmax ≈ 0. Moreover, we recall that while L/Lmax is evaluated at rsnap, |vr|/v is evaluated at rΔ, host, so the right-hand panels of Figs 3 and 4 do not contain exactly the same information. The distributions of L/Lmax and |vr|/v for Δc = 5000 and Δc = 10 000 are almost indistinguishable, but they are significantly different from those obtained for Δc = Δvir. For the higher overdensities, the mean values of L/Lmax and |vr|/v are, respectively, higher and lower than those obtained for Δc = Δvir (see Table 4). The difference between measures at rvir and those at ≲ 0.15rvir is best visualized in the right-hand panel of Fig. 3: the distribution of L/Lmax for measures at rvir drops for L/Lmax → 1, where instead the distributions for more central measures peak. The values of the standard deviation reported in Table 4 indicate that the distributions of L/Lmax and |vr|/v have larger scatter for Δc = 5000 and Δc = 10 000 than for Δc = Δvir. Overall, the results presented in this section lead us to conclude that, for mergers with 0.01 ≤ ξ < 0.1, the orbits of mergers at r10000c ≈ 0.1rvir are more bound and more tangential than those at rvir. 4.5 Orbital parameters for mergers with mass ratio $$\boldsymbol{\xi \ge 0.1}$$ We focus here on the distribution of orbital parameters for mergers with mass ratios in the range 0.1 ≤ ξ ≤ 1. As we limit ourselves to the mass ratios higher than ξ = 0.1, we can consider here the overdensity Δc = 20 000 (see Section 2.2), which probes the central region of the halo r20000c ≈ 0.06rvir (see Section 4.1). As done in Section 4.4, we compare the results obtained for this central halo region with those obtained at rvir. Moreover, in this case we can include in our analysis also the results of N17, who, using non-cosmological simulations, explored the distribution of orbital parameters at rcen ≈ 0.02rvir, roughly corresponding to Δc = 50 000, for merger mass ratios ξ ≃ 0.13 and ξ ≃ 0.67 (see Sections 4.1 and 4.2). Both r20000c ≈ 0.06rvir and rcen ≈ 0.02rvir can be considered proxies for the characteristic size of the CG in a cluster of galaxy (see Section 4.1). In Figs 5 and 6 we plot the distributions of E2b/Ψ0, L/Lmax, v/vcirc, and |vr|/v, together with their best fits. The parameters of the best fits (Gaussian distributions – equation 12 – for E2b/Ψ0 and v/vcirc, and beta distributions—equation 13—for L/Lmax and |vr|/v) are reported in Tables 3 and 4. The distribution of E2b/Ψ0 that we find at r20000c ≈ 0.06rvir has slightly more negative mean than the distribution measured at rvir (see the right-hand panel of Fig. 5), confirming the trend found in Section 4.4 for mergers with ξ < 0.1: the orbits of satellites accreting on to CGs tend to be slightly more bound than those of satellites accreting at the virial radius of the host cluster. The same result is visualized in Fig. 6 (left-hand panel), showing that the distribution of v/vcirc at r20000c peaks at lower values than the distribution of v/vcirc measured at rvir. Figure 5. View largeDownload slide Same as Fig. 3, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curve) and Δc = 20 000 (short-dashed curve). The long-dashed curve represents the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. Figure 5. View largeDownload slide Same as Fig. 3, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curve) and Δc = 20 000 (short-dashed curve). The long-dashed curve represents the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. Figure 6. View largeDownload slide Same as Fig. 4, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curves), and Δc = 20 000 (short-dashed curves). The long-dashed curves represent the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. Figure 6. View largeDownload slide Same as Fig. 4, but for merger mass ratios ξ ≥ 0.1, for critical overdensities Δc = Δvir (solid curves), and Δc = 20 000 (short-dashed curves). The long-dashed curves represent the results obtained by N17 for mergers at rcen ≈ 0.02rvir, using simulations of satellites orbiting in isolated host haloes. It is interesting to compare the results at r20000c ≈ 0.06rvir with those obtained at rcen ≈ 0.02rvir by N17. While the distributions of E2b/Ψ0 are very similar for rcen and r20000c, the distributions of v/vcirc are significantly offset: in the experiments of N17 the values of v/vcirc measured at rcen tend to be higher than those measured at rvir, in contrast with the results obtained here for r20000c. The different behaviour between the distributions of v/vcirc and E2b/Ψ0 can be explained as follows. Though both quantities measure the binding energy of the orbit, as pointed out in Section 4.4, they are normalized quite differently: while Ψ0 accounts for the merger mass ratio ξ, vcirc is independent of ξ. The experiments of N17 have average merger mass ratio 〈ξ〉N ≃ 0.62 higher than our samples of mergers at rvir and r20000c (〈ξ〉N ≈ 0.5 in the interval 0.1 ≤ ξ ≤ 1). Moreover, we recall that in this work E2b is measured at the snapshot before merger, while v is corrected to be evaluated at rΔ, host (see Section 3): this is another source of difference between the distributions of E2b/Ψ0 and v/vcirc. Based only on measures of v/vcirc, compared to those of J15 (measured at r200c), N17 concluded that the orbits for CG–satellite mergers tend to be less bound than those of cosmological halo–halo mergers. The present analysis reveals that the higher values of v/vcirc found in the simulations of N17 at least partly reflect a bias in the merger mass ratios, which tend to be higher than cosmologically motivated values. In any case, the simulations here considered should, in general, be more realistic than the idealized simulations of N17, thus we believe that the distribution of v/vcirc here obtained for r20000c should be more representative for real CGs than the distribution found for rcen in N17. Therefore, based on the results found for E2b/Ψ0 and v/vcirc, we can conclude that the orbits of satellites accreting on to CGs in clusters tend to be more bound than the orbits of satellites accreting on to the host cluster-sized haloes. The distributions of L/Lmax (right-hand panel in Fig. 5) and |vr|/v (right-hand panel in Fig. 6) indicate that, as it happens for mergers with ξ < 0.1, also for mass ratios ξ ≥ 0.1 the orbits measured at r20000c tend to be more tangential than those measured at rvir (see also Table 4). In particular, the probability density function of L/Lmax is flatter for Δc = 20 000 than for Δc = Δvir (the latter peaks at ≈0.3 and drops above ≈0.8); the probability density function of |vr|/v measured at r20000c has a strong peak at ≈1, which is absent for measures at rvir. As far as the eccentricity of the orbits is concerned, the results obtained by N17 are consistent with those obtained here: the distributions of L/Lmax and |vr|/v measured at rcen are biased towards, respectively, high and low values, even more than those found here for Δc = 20 000. Therefore the present results confirm and strengthen the finding of N17 that the orbits of satellites accreting on to CGs tend to be more tangential than those of cosmological halo–halo accretion at the virial radius. As far as the scatter in the distributions is concerned, the trend for ξ ≥ 0.1 is the same as that for ξ < 0.1: for all the considered parameters (E2b/Ψ0, v/vcirc, L/Lmax, |vr|/v) the scatter is larger for higher values of Δc (i.e. smaller radii; see Tables 3 and 4). The effect is strongest for v/vcirc, for which the best-fitting probability density function has standard deviation almost a factor of 2 higher for measures at ≈0.06rvir than for measures at rvir. This is qualitatively in agreement with the findings of N17: the standard deviations for measures at rcen are higher than those for measures at rvir, with the only exception of L/Lmax, for which the scatter is the same in the two cases (Tables 3 and 4). 5 CONCLUSIONS In this paper we have used the results of the dark-matter only cosmological N-body simulations Le SBARBINE (Despali et al. 2016) to study the statistical properties of mergers between central and satellite galaxies in galaxy clusters. In particular we selected a sample of 101 cluster-sized haloes at z = 0 from the simulations Ada and Bice and traced their merging history in the redshift interval 0 < z < 1. We constructed merger trees for different overdensities Δc. When we use the virial overdensity [Δc = Δvir, with 100 ≲ Δvir(z) ≲ 150] we probe the accretion of satellites at the cluster virial radius rvir (Giocoli et al. 2008; Giocoli 2010). When we use higher overdensities (Δc = 5000, 10 000, and 20 000) we probe the accretion of satellites in the central region of the cluster (at radii r5000c ≈ 0.15rvir, r10000c ≈ 0.1rvir, and r20000c ≈ 0.06rvir), which can be considered as a proxy for the accretion of satellite galaxies on to CGs. We measured the distributions of merger mass ratios and orbital parameters for these merger histories. The main results of this work are the following. Though minor mergers largely outnumber major mergers, the latter contribute to the mass accreted at z < 1 at least as much as minor mergers, for all values of Δc. The mass-weighted merger mass ratio 〈ξ〉M increases for increasing Δc, so major mergers are even more important for CGs than for the accretion at the cluster virial radius. In the mass-ratio interval 0.01 ≤ ξ ≤ 1, more than 60 per cent of the mass accreted by CGs at z < 1 is due to major mergers. For higher overdensities (i.e. more central regions), the orbits of the accreting satellites tend to be less bound and more tangential. Therefore, the orbits of satellites accreting on to CGs are characterized by higher specific orbital angular momentum and lower specific orbital energy than orbits of halo accretion at the virial radius. The scatter in the orbital parameters tends to be larger for accretion on to CGs than for accretion at the halo virial radius. In this respect, the strongest effect is found for the distribution of v/vcirc, which, for ξ ≥ 0.1, has standard deviation almost a factor of 2 higher at ≈0.06rvir than at rvir. We compared the results obtained at r20000c ≈ 0.06rvir in our cosmological simulations with those obtained by N17 at rcen ≈ 0.02rvir in idealized non-cosmological simulations. We found good agreement on the distribution of orbital angular momentum, but we revised N17's conclusions on the binding energy of the orbits, which were somewhat biased by the non-cosmological setting. We provided parameters of the analytic best-fitting distributions of the pairs of orbital parameters (E2b, L) and (v/vcirc, |vr|/v) for different values of Δc (Tables 3 and 4). The distributions obtained for Δc = 20 000 (i.e. measured at r20000c ≈ 0.06rvir) can be taken as reference for modelling accretion on to CGs in clusters. In particular, the provided analytic distributions could be included in models attempting to predict the evolution of the scaling relations of cluster CGs without resorting to hydrodynamic cosmological simulations (e.g. Bernardi et al. 2011; Volonteri & Ciotti 2013; Shankar et al. 2015). Acknowledgements We would like to thank Giuseppe Tormen (and the Physics and Astronomy Department of Padova) who provided the computational resources to run the simulations. CG acknowledges support from the Italian Ministry for Education, University and Research (MIUR) through the SIR individual grant SIMCODE, project number RBSI14P4IH. Footnotes 1 For instance, consider a halo with mass M at redshift zi that at redshift zi − 1 > zi splits in two haloes of mass M1 and M2 < M1, which, in turn, at redshift zi − 2 > zi − 1 split, respectively, in two haloes of mass M1, 1 and M1, 2 < M1, 1, and in two haloes of mass M2, 1 and M2, 2 < M2, 1. 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Specifically, we consider here the distributions of v/vcirc and |vr|/v found by J15 for dark-matter haloes of z = 0 mass M200c ≈ 1014M⊙ considering mergers with mass ratios in the range 0.005 ≤ ξ ≤ 0.05 in the redshift interval 0 < z < zHF, where zHF is the formation redshift of the halo (for M200c ≈ 1014M⊙ the distribution of zHF peaks between z = 0.5 and z = 1; see J15). In order to compare our results with those of J15, we built the Δc = 200 merger tree of our sample of 101 haloes with z = 0 mass Mvir ≥ 1014M⊙, taking all encounters with mass ratio 0.005 ≤ ξ ≤ 0.05. Altogether, in this way we select a sample of 1855 encounters (see Table 2). For these encounters we evaluated v/vcirc and vr/v at r200c using equations (8) and (9). The results are shown in Fig. A1: overall the agreement between our distributions and those of J15 is remarkable. The peaks and the widths of the two distributions of v/vcirc (left-hand panel of Fig. A1) almost coincide, while the tail at low values of v/vcirc is somewhat stronger in our distribution than in the distribution of J15. The two distributions of |vr|/v (right-hand panel of Fig. A1) are in excellent agreement over the entire range 0 ≤ |vr|/v ≤ 1. It must be noted that the time sampling of the DOVE simulation is significantly better than that of Le SBARBINE: for instance, in the redshift range 0 ≲ z ≲ 1 the number of available snapshots is 38 for DOVE and 13 for Le SBARBINE. Thus, the agreement in the distributions of v/vcirc and |vr|/v between J15's sample of encounters and ours suggests that the correction (Section 3) we applied to estimate the parameters at separation rΔ should be reliable. Figure A1. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) measured at r200c for halo–halo mergers in this work (solid curves) and in J15 (dashed curves). The solid curves are obtained for the cosmological simulations Ada and Bice considering mergers in the redshift range 0 < z < 1 for haloes with z = 0 mass Mvir ≥ 1014M⊙. The dashed curves are obtained for the cosmological simulation DOVE considering mergers from z = 0 up to the halo formation redshift (see text), for haloes with z = 0 mass M200c ≈ 1014M⊙. In both cases the satellite-to-host mass ratio is in the range 0.005 ≤ ξ ≤ 0.05. Figure A1. View largeDownload slide Probability distribution p = dn/dx of the relative speed (x = v/vcirc; left-hand panel) and radial-to-total relative velocity ratio (x = |vr|/v; right-hand panel) measured at r200c for halo–halo mergers in this work (solid curves) and in J15 (dashed curves). The solid curves are obtained for the cosmological simulations Ada and Bice considering mergers in the redshift range 0 < z < 1 for haloes with z = 0 mass Mvir ≥ 1014M⊙. The dashed curves are obtained for the cosmological simulation DOVE considering mergers from z = 0 up to the halo formation redshift (see text), for haloes with z = 0 mass M200c ≈ 1014M⊙. In both cases the satellite-to-host mass ratio is in the range 0.005 ≤ ξ ≤ 0.05. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

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Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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