Asteroseismology of KIC 7107778: a binary comprising almost identical subgiants

Asteroseismology of KIC 7107778: a binary comprising almost identical subgiants Abstract We analyse an asteroseismic binary system: KIC 7107778, a non-eclipsing, unresolved target, with solar-like oscillations in both components. We used Kepler short cadence time series spanning nearly 2 yr to obtain the power spectrum. Oscillation mode parameters were determined using Bayesian inference and a nested sampling Monte Carlo algorithm with the diamonds package. The power profiles of the two components fully overlap, indicating their close similarity. We modelled the two stars with mesa and calculated oscillation frequencies with gyre. Stellar fundamental parameters (mass, radius, and age) were estimated by grid modelling with atmospheric parameters and the oscillation frequencies of l = 0, 2 modes as constraints. Most l = 1 mixed modes were identified with models searched using a bisection method. Stellar parameters for the two sub-giant stars are MA = 1.42 ± 0.06 M⊙, MB = 1.39 ± 0.03 M⊙, RA = 2.93 ± 0.05 R⊙, RB = 2.76 ± 0.04 R⊙, tA = 3.32 ± 0.54 Gyr and tB = 3.51 ± 0.33 Gyr. The mass difference of the system is ∼1 per cent. The results confirm their simultaneous birth and evolution, as is expected from binary formation. KIC 7107778 comprises almost identical twins, and is the first asteroseismic sub-giant binary to be detected. binaries: general, stars: evolution, stars: oscillations 1 INTRODUCTION Binary star systems provide an ideal astronomical laboratory to study stellar structure and evolution. The fact that two components share same metal abundance and age provides powerful constraints on models. Eclipsing binaries are especially useful, since mass and radius can be directly measured from orbits and eclipses through spectroscopic observations and precise light-curve analysis (Andersen 1991). Asteroseismology ushers in a new way to study binaries (Huber 2015), even for unresolved ones (Miglio et al. 2014), by analysing the two components separately. Determining stellar mass and radius is feasible for single stars with solar-like oscillations (e.g. Kallinger et al. 2010; Chaplin et al. 2014). Sub-giant and red-giant oscillating stars displaying p (pressure dominated in the envelope) and g (gravity dominated in the core) mixed modes (Chaplin & Miglio 2013) are good indicators of evolutionary stages. Their great sensitivity to mass and age can produce precise estimates of stellar parameters (e.g. Metcalfe et al. 2010; Benomar et al. 2012). Binary systems with pulsators can be analysed from modulation of the pulsation frequencies (Shibahashi & Kurtz 2012; Murphy et al. 2014; Murphy & Shibahashi 2015), so that system parameters can be determined with radial velocity curves simply derived through photometry. White et al. (2017) pointed out that, until now, only five binary star systems have been detected with solar-like oscillations from both components. They are α Cen A and B (Bedding et al. 1998; Kjeldsen et al. 1999; Bouchy & Carrier 2001; Carrier & Bourban 2003), 16 Cyg A and B (KIC 12069424 and KIC 12069449; Metcalfe et al. 2012; Metcalfe, Creevey & Davies 2015), KIC 9139163 and KIC 9139151 (Appourchaux et al. 2012; Appourchaux et al. 2014), HD 177412 (KIC 7510397; Appourchaux et al. 2015), and HD 176465 (KIC 10124866, also known as Luke & Leia; White et al. 2017). The latter two systems are not resolved by Kepler, such that their light variations are mixed in a single time series. Both systems were analysed with one power spectrum, from which two sets of oscillation profiles were measured. Specifically, HD 177412 contains two main-sequence stars with mass difference ∼7.5 per cent, with separable oscillation frequency ranges, and HD 176465 contains two extremely similar main-sequence stars with mass difference ∼ 3 per cent, leading to two significantly overlapping oscillation ranges. In addition, KIC 10080943 is another unresolved binary system, comprising two δ Sct/γ Dor hybrid pulsators on the main sequence (Keen et al. 2015; Schmid et al. 2015). In this context, these binaries are not strictly ‘twins’ in that the masses are not equal to within 2 per cent (Lucy 2006; Simon & Obbie 2009). Twins are almost always found at small separations (P ≲ 25 d, Lucy & Ricco 1979), though not all close binaries are twins. Further, they are more common among lower mass systems, both in observational data (e.g. Tokovinin 2000; Simon & Obbie 2009) and in binary star formation simulations (e.g. Bate 2009). Their importance lies in their ability to discriminate between dominant physical processes operating in pre-main-sequence binaries, as those authors have discussed. Here we report on the widely separated twin binary system KIC 7107778, with mass difference ∼1 per cent based on our findings, an analogue to Luke & Leia (White et al. 2017). The system was not resolved by Kepler but was detected with oscillations from two stars in the mixed time series. It proves to be the first asteroseismic sub-giant binary system to be detected, with completely overlapping power spectra of two component stars. The paper will be structured as follows. Section 2 provides observations from previous literature and describes the processing of Kepler data. Section 3 illustrates the oscillation mode parameters. Section 4 details our asteroseismic analysis with stellar models of two stars, and is followed by discussions and conclusions in Section 5. 2 OBSERVATION AND DATA PROCESSING KIC 7107778 is a binary system observed as a single target, with Kepler magnitude Kp = 11.38. The first realization of its binary identity was from speckle interferometry (Horch et al. 2012). The angular separation of the two stars is ρ = 0.0288 arcsec, measured with filters whose central wavelengths are 692 and 880 nm. The physical separation can be determined by combining luminosities from models with the apparent magnitude; however, such estimation is only approximate because the apparent magnitude is a blended contribution from both stars. We took advantage of the parallax determined from Gaia (Gaia Collaboration 2016), 1.74 ± 0.43 mas, to infer the distance to Earth and physical projected separation between the two stars, which are $$d = 574^{+189}_{-113}$$ pc, and $$s = 16.52^{+5.44}_{-3.25}$$ au. Note that this estimation is very approximate because Gaia parallaxes are not yet fully reliable at the present mission stage (see Fabricius et al. 2016). Assuming the orbit is circular, and adopting Kepler's third law, it suggests the orbital period of this system is about 39 yr. Therefore, we would not expect to analyse its orbit using radial velocity curves. On the other hand, such a wide separation means fully independent evolution without tidal effects or mass transfer. In the following analysis, we treat them as two single stars that have not interacted. Several works have measured atmospheric parameters of this system. Effective temperatures measured by SDSS and IRFM are Teff = 5129 ± 82 K and Teff = 5045 ± 105 K, respectively. The binary system was also covered by the LAMOST-Kepler project, which used LAMOST low-resolution (R ≃ 1800 Å) optical spectra in the waveband of 3800–9000 Å and observed Kepler-field targets in 2014 September. The LAMOST Stellar Parameter Pipeline (Xiang et al. 2015) gives Teff = 5149 ± 150 K, and metallicity [Fe/H] = 0.11 ± 0.10 dex. Buchhave & Latham (2015) used the Tillinghast Reflector Echelle Spectrograph to obtain medium resolution (R ≃ 44 000Å) spectra in the waveband of 3800–9100 Å at Fred Lawrence Whipple Observatory. They measured the metallicity [Fe/H] = 0.05 ± 0.08 dex. However, all these observations treated two stars as one single target. We do not know to what extent the mixed value deviates from the real ones. Table 1 summarizes the observations in the literature. Table 1. Atmospheric parameters of KIC 7107778. Parameter  Value  Reference  Teff (K)  5129 ± 82  Drimmel, Cabrera-Lavers & López-Corredoira (2003)    5045 ± 105  Casagrande et al. (2010)    5149 ± 150  Xiang et al. (2015)  [Fe/H] (dex)  0.11 ± 0.10  Xiang et al. (2015)    0.05 ± 0.08  Buchhave & Latham (2015)  Parameter  Value  Reference  Teff (K)  5129 ± 82  Drimmel, Cabrera-Lavers & López-Corredoira (2003)    5045 ± 105  Casagrande et al. (2010)    5149 ± 150  Xiang et al. (2015)  [Fe/H] (dex)  0.11 ± 0.10  Xiang et al. (2015)    0.05 ± 0.08  Buchhave & Latham (2015)  View Large The Kepler mission observed the target in long-cadence mode (LC; 29.43 min sampling) over the whole mission and in short-cadence mode (SC; 58.84 s sampling) during Q2.1, Q5 and Q7–Q12 (Q represents three-month-long quarters). The pulsation frequency range is centred at 550 μHz, which is well above the Nyquist frequency of long cadence data (∼283 μHz). Therefore, we only considered the short-cadence time series. We concatenated the data and processed it following García et al. (2011), correcting outliers, jumps, and drifts. Then it passed through a high-pass filter which was based on a Gaussian smooth function with a width of 1 d. This largely minimized instrumental side effects and only affected frequencies lower than ∼12 μHz, far below the frequency range we intended to analyse. We obtained the power spectrum by applying a Lomb–Scargle Periodogram (Lomb 1976; Scargle 1982) to the time series with a frequency resolution ∼0.012 μHz. The power spectrum is shown in Figs 1 and 2 in both logarithmic and linear scales. Figure 1. View largeDownload slide Power spectrum on a logarithmic scale. The solid grey line, the solid black line, the dashed blue lines, the dot–dashed-dashed black line, and the solid green line, outline the original power spectrum, the 6 μHz smoothed power spectrum, the Harvey profile components, the white noise component, and the total fitted power spectrum, respectively. Figure 1. View largeDownload slide Power spectrum on a logarithmic scale. The solid grey line, the solid black line, the dashed blue lines, the dot–dashed-dashed black line, and the solid green line, outline the original power spectrum, the 6 μHz smoothed power spectrum, the Harvey profile components, the white noise component, and the total fitted power spectrum, respectively. Figure 2. View largeDownload slide Power spectrum around the oscillation range on a linear scale. The grey line denotes the original power spectrum superimposed with fitted l = 1 modes in black, l = 0 and l = 2 modes of star A in red, and those of star B in blue. Figure 2. View largeDownload slide Power spectrum around the oscillation range on a linear scale. The grey line denotes the original power spectrum superimposed with fitted l = 1 modes in black, l = 0 and l = 2 modes of star A in red, and those of star B in blue. The signature of solar-like oscillation is a Gaussian-like envelope located at νmax, the so-called frequency of maximum power, and is comprised of numerous oscillation modes. In main-sequence stars, p-mode oscillations are approximately equally spaced in frequency, as described by the asymptotic equation with radial orders n and spherical degrees l (Tassoul 1980):   \begin{equation} \nu _{nl} = \Delta \nu \left(n+\frac{l}{2}+\epsilon\right)-\delta \nu _{0l}, \end{equation} (1)where Δν is the large separation, which measures the spacing of adjacent modes with the same l, δν0l is the small separation, and ε is an offset. In more evolved stars, the core will have g-mode oscillations, which are approximately equally spaced in period. The analogous asymptotic equation is specified by order ng:   \begin{equation} \Pi _{nl} = \nu _{nl}^{-1} = \Delta \Pi _{l}(n_g+\epsilon _g), \end{equation} (2)where ΔΠl is the period spacing, and εg is an offset. As the star evolves off the main sequence, central hydrogen depletes, and p and g mixed modes of l ≥ 1 appear to have ‘avoided crossings’ (Aizenman, Smeyers & Weigert 1977), whereby oscillation frequencies are no longer equally spaced in either frequency or period. The l = 0 p modes are unaffected, which assisted us in determining the mean large separation 〈Δν〉 in an échelle diagram. The best value for 〈Δν〉 is the one that makes the l = 0 ridge vertical. Fig. 3 displays échelle diagrams for the two stars. To avoid ambiguity, the star with smaller 〈Δν〉 is named as KIC 7107778 A, and the larger one KIC 7107778 B. The best values are 〈Δν〉A = 31.83 ± 0.02 μHz and 〈Δν〉B = 34.55 ± 0.01 μHz. Figure 3. View largeDownload slide Échelle diagrams of star A (left) and star B (right). The circular and square open symbols represent identified l = 0 and l = 2 modes. Red and blue colours denote modes for star A and star B, respectively. The background grey-scale represents power density. Higher power density is shown darker. The unmarked peaks are l = 1 mixed modes. Figure 3. View largeDownload slide Échelle diagrams of star A (left) and star B (right). The circular and square open symbols represent identified l = 0 and l = 2 modes. Red and blue colours denote modes for star A and star B, respectively. The background grey-scale represents power density. Higher power density is shown darker. The unmarked peaks are l = 1 mixed modes. 3 MODE PARAMETERS To model the power spectrum, we used high-DImensional And multi-MOdal NesteD Sampling code (diamonds; Corsaro & De Ridder 2014). The diamonds code utilizes Bayes’ theorem:   \begin{equation} p(\theta \mid {D,M}) = \frac{{\mathcal {L}}(\theta \mid {D,M})\pi (\theta \mid {M})}{p(D\mid {M})}, \end{equation} (3)where θ = (θ1, θ2, ..., θk), $${\mathcal {L}}(\theta \mid {D,M})$$, π(θ∣M), p(θ∣D, M) are the parameter vector, likelihood function for a given model M and data set D, prior probability density function, and the posterior probability density function, respectively. diamonds uses a sampling algorithm, Nested Sampling Monte Carlo, to tackle the high-dimensional problem. We simultaneously fitted the two stars to separate their overlapping oscillations with the following steps. First, we modelled the power spectrum, P(ν), with   \begin{eqnarray} P(\nu ) &=& W+R\left(\nu \right)\left\lbrace \sum _{i = 1}^{k}\frac{2\sqrt{2}}{\pi }\frac{ a_i^2/b_i}{1+(\nu /b_i)^{4}}\right. \nonumber\\ &&\left. +\,H_0{\rm exp}\left[-\frac{(\nu -\nu _{\rm max})^2}{2\sigma ^2}\right]\right\rbrace , \end{eqnarray} (4)similar to the background models presented in Kallinger et al. (2014) and Corsaro, De Ridder & García (2015). The right-hand side of the equation comprises a flat white noise W, a sum of Harvey power profiles (Harvey 1985) with parameters (ai, bi), and a Gaussian envelope with (H0, νmax, σ). The Harvey profile models stellar background caused by granulation. In the case of KIC 7107778, the background was well fitted using three Harvey profiles, i.e. k = 3. The Gaussian envelope denotes the range of oscillation for two overlapping stars. All these components except the noise are modulated by the response function,   \begin{equation} R\left(\nu \right) = {\rm sinc}^2\left(\frac{\pi \nu }{2\nu _{\rm Nyq}}\right), \end{equation} (5)where νNyq = 8496.36 μHz denotes the Nyquist frequency. The total number of free variables is 1 + 2 × 3 + 3 = 10. The results are shown in Table 2. As argued by Kallinger et al. (2014), the granulation frequencies bi should scale with νmax. They provided empirical relations b2 = 0.317(νmax/μHz)0.970μHz and b3 = 0.948(νmax/μHz)0.992μHz, which result in b2 = 148.87 μHz and b3 = 511.87 μHz in our case. b2 is similar to our fit but not b3. This should be expected because the power spectrum is a superposition of the twins. b1 should be treated carefully because the light curve was processed by a high-pass filter, which affected the low-frequency spectrum. Table 2. Granulation background parameters. Parameter  Value  68.3 per cent credible  W (ppm2 μHz− 1)  11.927  0.100  a1 (ppm)  58.874  8.314  b1 (μHz)  5.203  1.460  a2 (ppm)  67.167  8.705  b2 (μHz)  149.144  32.207  a3 (ppm)  76.134  7.947  b3 (μHz)  400.646  64.318  H0 (ppm2 μHz− 1)  17.977  2.852  νmax (μHz)  568.051  7.935  σ (μHz)  55.811  8.690  Parameter  Value  68.3 per cent credible  W (ppm2 μHz− 1)  11.927  0.100  a1 (ppm)  58.874  8.314  b1 (μHz)  5.203  1.460  a2 (ppm)  67.167  8.705  b2 (μHz)  149.144  32.207  a3 (ppm)  76.134  7.947  b3 (μHz)  400.646  64.318  H0 (ppm2 μHz− 1)  17.977  2.852  νmax (μHz)  568.051  7.935  σ (μHz)  55.811  8.690  View Large Then, each mode was fitted with a Lorentzian profile with three free parameters: frequency centroid ν0, amplitude A, and linewidth Γ built on the background:   \begin{equation} L(\nu ) = R\left(\nu \right)\left[\frac{A^2/\pi \Gamma }{1+4\left(\nu -\nu _0\right)^2/\Gamma ^2}\right]. \end{equation} (6)The power spectrum was fitted with a sum of 32 Lorentzian profiles. The total number of free variables is 3 × 32 = 96. It is possible that the star rotates and lifts degeneracy of m-degree of the modes (Gizon & Solanki 2003). Thus we performed a hypothesis test with Bayesian evidence denoted by p(D∣M) in equation (3). The Bayesian evidence balances the goodness of fit and the need to fit, as it can be written as a product of the maximum of the likelihood and an Occam factor (Knuth et al. 2015). We adopted a splitting model of l = 1 modes   \begin{equation} L(\nu ) = R(\nu)\left[\sum _{m = -1}^{1}\frac{\xi _m A^2/\pi \Gamma }{1+4(\nu -\nu _0-m\delta \nu _{\rm split})^2/\Gamma ^2}\right], \end{equation} (7)where i is the inclination angle, ξ−1 = ξ1 = 0.5sin 2i, ξ0 = cos 2i, and δνsplit measures the extent of splitting. We fitted the splitting model to four l = 1 modes (mode frequency 477, 507, 539, and 620 μHz) individually. The four modes were selected because they do not have another mode in 3 μHz frequency range. We defined detection probability as p = p(D∣MB)/(p(D∣MA) + p(D∣MB)) (Jeffreys 1961; Corsaro & De Ridder 2014), where subscripts A and B refer to the non-splitting model and the splitting model. The detection probabilities are small: 10−1, 10−14, 10−9, and 10−2, respectively. We also fitted the four modes with a common inclination and the detection probability is 10−50. This indicates the non-splitting model gives a better depiction of the data. Hence, detecting the inclination angle i through rotation was not considered. Notice that only l = 0 and l = 2 modes can be identified straightforwardly and allocated to one of the two stars, since they are regularly spaced. The l = 1 mixed modes are strongly bumped and no clear patterns could be followed, so we used stellar models to help the identification, as discussed in Section 4. We also measured the frequency of maximum power, νmax, by fitting a Gaussian profile in power density to the l = 0 mode peaks. The results are νmax, A = 523 ± 16 μHz and νmax, B = 570 ± 18 μHz, where the uncertainties are half of 〈Δν〉. Fig. 3 shows the échelle diagram of both stars. The circular and square open symbols represent identified l = 0 and l = 2 modes. Red and blue colours denote star A and star B, respectively. The mode parameters are shown in Tables 6 and 7. The l = 1 modes in the tables are further discussed in Section 4.3. 4 ASTEROSEISMIC ANALYSIS As discussed in Section 2, KIC 7107778 has a long orbital period and no evidence of eclipses from which stellar fundamental parameters could be estimated. Therefore, estimating them through stellar models with asteroseismology is necessary. 4.1 Stellar models We constructed stellar models using Modules for Experiments in Stellar Astrophysics (mesa; Paxton et al. 2011; Paxton et al. 2013; Paxton et al. 2015). Paxton et al. (2011) have discussed the input physics of mesa. Here we list them for completeness. The equation of state was delivered by OPAL EOS tables (Rogers & Nayfonov 2002), SVCH tables (Saumon, Chabrier & van Horn 1995) at low temperatures and densities, and HELM (Timmes & Swesty 2000) and PC (Potekhin & Chabrier 2010) tables under other circumstances. Opacities were taken from Iglesias & Rogers (1996) at high temperature and Ferguson et al. (2005) at low temperature. Nuclear reaction rates were based on NACRE (Angulo et al. 1999) and CF88 (Caughlan & Fowler 1988) when NACRE was unavailable. The convection was implemented with mixing length theory (MLT) illustrated in Cox & Giuli (1968). The mixing length parameter αMLT = 1.917 was employed according to the mesa standard solar model (Paxton et al. 2011). Overshoot mixing was set according to Herwig (2000), with overshooting parameter fov = 0.016. The initial helium abundance was estimated through   \begin{equation} Y_{\rm ini} = Y_0+\frac{\Delta Y}{\Delta Z}\cdot {Z_{\rm ini}}, \end{equation} (8)where Y0 = 0.249 (Planck Collaboration XIII 2016) and ΔY/ΔZ = 1.33, calculated using the initial abundances of helium and heavy elements of the calibrated solar model (Paxton et al. 2011). The relation of metallicity and element abundance ratio we adopted here was   \begin{equation} \rm {[Fe/H]} = \log (\text{$Z$}/\text{$X$})-{\log }{(\text{$Z$}/\text{$X$})_{\odot }}, \end{equation} (9)where the solar value is (Z/X)⊙ = 0.022 93 (Grevesse & Sauval 1998). The varying input parameters for grid modelling are mass M and metallicity [Fe/H]. They were set as follows. First, the mass was set according to asteroseismic scaling relations. The mean large separation Δν and frequency of maximum power νmax are related to mean density ρ, surface gravity g, and effective temperature Teff: $$\Delta \nu \propto \sqrt{\rho }$$, $$\nu _{\rm {\rm max}}\propto {g/\sqrt{T_{\rm {eff}}}}$$ (Kjeldsen & Bedding 1995), i.e.   \begin{equation} \frac{\Delta \nu }{\Delta \nu _{\odot }}\approx \left(\frac{M}{\mathrm{M}_{\odot }}\right)^{1/2}\left(\frac{R}{\mathrm{R}_{\odot }}\right)^{-3/2}, \end{equation} (10)  \begin{equation} \frac{\nu _{\rm max}}{\nu _{\rm max,\odot }}\approx \left(\frac{M}{\mathrm{M}_{\odot }}\right)\left(\frac{R}{\mathrm{R}_{\odot }}\right)^{-2}\left(\frac{T_{\rm {eff}}}{T_{\rm {eff},\odot }}\right)^{-1/2}, \end{equation} (11)where Δν⊙ = 135.1 μHz, νmax, ⊙ = 3050 μHz (Chaplin & Miglio 2013) and Teff = 5777 K. Thus, the mass prescription can be deduced from equations (10) and (11):   \begin{equation} \frac{M}{\mathrm{M}_{\odot }}\approx \left(\frac{\Delta \nu }{\Delta \nu _{\odot }}\right)^{-4}\left(\frac{\nu _{\rm {max}}}{\nu _{\rm {max},\odot }}\right)^{3}\left(\frac{T_{\rm {eff}}}{T_{\rm {eff},\odot }}\right)^{3/2}. \end{equation} (12)Considering the uncertainties, the mass range of the grid should at least cover 1.34–1.60 M⊙. Secondly, we adopted metallicity [Fe/H] = 0.01, 0.11, and 0.21 according to spectral observations from LAMOST. Table 3 summarizes the input parameters of our grid modelling. Table 3. Input parameters of grid modelling.   Value  Step size  M (M⊙)  1.34–1.60  0.1  [Fe/H] (dex)  0.01–0.21  0.1  αMLT  1.917  Fixed  fov  0.016  Fixed    Value  Step size  M (M⊙)  1.34–1.60  0.1  [Fe/H] (dex)  0.01–0.21  0.1  αMLT  1.917  Fixed  fov  0.016  Fixed  View Large 4.2 Modelling the l = 0 and l = 2 modes We calculated oscillation frequencies for models that met the requirements of Δν and νmax with gyre (Townsend & Teitler 2013), which solves the adiabatic pulsation equations with stellar structure data. The calculated frequencies deviated from the observations due to the improper simulation of the stellar surface. Therefore, we followed the method described by Ball & Gizon (2014) to correct them. The correction formula we adopted here was   \begin{equation} \delta \nu = (a_{-1}\nu ^{-1}+a_3\nu ^3)/I, \end{equation} (13)where a−1 and a3 are coefficients determined through least-squares fit, and I is the mode inertia. We used χ2 = Σ(νobs − νmod)2/σ2 as the quality of the fit. Frequencies used in the χ2 calculation were all l = 0 and l = 2 modes. The smaller the χ2 of the model, the better the match. We took the best 10 per cent of the models, as measured by χ2, for further considerations. The choice of this criterion was a trade-off. Small values could be biased by fluctuations in limited samples. The problem was tackled by sorting samples according to the values of χ2, performing difference between two adjacent quantities. We found the trend around 10 per cent went smoothly, which ruled out the fluctuation effect. Including more models would make the selection less useful. This point was addressed by inspecting the value of χ2 around the 10 per cent cut-off and assuring that it was not too large. With the lowest 10 per cent χ2 models, we calculated the mean value of the stellar parameters as the centroid. Table 4 lists them with standard deviations. Additionally, Figs 4 and 5 display histograms of each stellar parameter, which reflect the distribution in the lowest 10 per cent χ2 models with [Fe/H] = 0.11. Red and blue indicate star A and star B, respectively. Figure 4. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star A grid models with [Fe/H] = 0.11 dex. Figure 4. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star A grid models with [Fe/H] = 0.11 dex. Figure 5. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star B grid models with [Fe/H] = 0.11 dex. Figure 5. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star B grid models with [Fe/H] = 0.11 dex. Table 4. Fundamental properties of models from grid modelling. Star  [Fe/H]  M  Age  Teff  L  R  log g  Δν  νmax    (dex)  (M⊙)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (μHz)  (μHz)  A  0.01  1.41 ± 0.06  3.178 ± 0.440  5216 ± 258  5.813 ± 1.508  2.925 ± 0.047  3.654 ± 0.005  32.01 ± 0.18  534 ± 8  B  0.01  1.38 ± 0.03  3.289 ± 0.263  5233 ± 132  5.140 ± 0.692  2.754 ± 0.025  3.699 ± 0.003  34.77 ± 0.13  592 ± 5  A  0.11  1.43 ± 0.07  3.266 ± 0.583  5184 ± 290  5.787 ± 1.663  2.945 ± 0.055  3.656 ± 0.007  31.98 ± 0.17  539 ± 9  B  0.11  1.38 ± 0.03  3.606 ± 0.313  5092 ± 105  4.594 ± 0.549  2.751 ± 0.025  3.700 ± 0.004  34.81 ± 0.12  601 ± 4  A  0.21  1.41 ± 0.07  3.643 ± 0.551  5037 ± 199  5.032 ± 1.160  2.926 ± 0.054  3.655 ± 0.006  32.06 ± 0.15  546 ± 6  B  0.21  1.40 ± 0.04  3.657 ± 0.317  5045 ± 106  4.473 ± 0.535  2.765 ± 0.028  3.701 ± 0.003  34.78 ± 0.13  606 ± 4  Star  [Fe/H]  M  Age  Teff  L  R  log g  Δν  νmax    (dex)  (M⊙)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (μHz)  (μHz)  A  0.01  1.41 ± 0.06  3.178 ± 0.440  5216 ± 258  5.813 ± 1.508  2.925 ± 0.047  3.654 ± 0.005  32.01 ± 0.18  534 ± 8  B  0.01  1.38 ± 0.03  3.289 ± 0.263  5233 ± 132  5.140 ± 0.692  2.754 ± 0.025  3.699 ± 0.003  34.77 ± 0.13  592 ± 5  A  0.11  1.43 ± 0.07  3.266 ± 0.583  5184 ± 290  5.787 ± 1.663  2.945 ± 0.055  3.656 ± 0.007  31.98 ± 0.17  539 ± 9  B  0.11  1.38 ± 0.03  3.606 ± 0.313  5092 ± 105  4.594 ± 0.549  2.751 ± 0.025  3.700 ± 0.004  34.81 ± 0.12  601 ± 4  A  0.21  1.41 ± 0.07  3.643 ± 0.551  5037 ± 199  5.032 ± 1.160  2.926 ± 0.054  3.655 ± 0.006  32.06 ± 0.15  546 ± 6  B  0.21  1.40 ± 0.04  3.657 ± 0.317  5045 ± 106  4.473 ± 0.535  2.765 ± 0.028  3.701 ± 0.003  34.78 ± 0.13  606 ± 4  Note . The models are selected with the lowest 10 per cent χ2. Column 2 is the input parameter for grid modelling. View Large Considering that [Fe/H] was estimated with uncertainties, we combined stellar parameters derived based on different [Fe/H] together, as the ultimate results. Here we list mass, radius, and age: MA = 1.42 ± 0.06 M⊙, MB = 1.39 ± 0.03 M⊙, RA = 2.93 ± 0.05 R⊙, RB = 2.76 ± 0.04 R⊙, tA = 3.32 ± 0.54 Gyr, and tB = 3.51 ± 0.33 Gyr. 4.3 Modelling the l = 1 mixed modes We next searched for the best models which could also fit the frequencies of l = 1 modes from both stars. KIC 7107778 contains two sub-giant stars. Tiny changes to the mass of the models influence oscillations greatly. Only extremely fine grids produce satisfying results, which makes the task demanding. Since our purpose was to find a pair of stellar models for both stars that fit observations best, we used the bisection method to search. The main idea of the bisection method is summarized as follows, similar to finding solutions for equation f(x) = 0. First, we started with two masses, M1 and M2, which lied on opposite sides of the best model. This choice was realized by visually inspecting oscillation frequencies on the échelle diagram. Secondly, we bisected this range, i.e. calculated frequencies of M3 = (M1 + M2)/2. Thirdly, we evaluated the result of M3 and determined M3 and whichever of M1 or M2 yield the best model. Fourthly, we kept bisecting the mass range until the difference became sufficiently small. We followed this search scheme under three different metallicities [Fe/H]: 0.01, 0.11, and 0.21. The maximum precision in calculation reaches 0.000 01 M⊙. The results revealed three combinations of best-fitting models for the two stars. Although they do still deviate from observed peaks, they give a reasonable fit to most l = 1 modes. Most model masses reach to 0.001 M⊙ precision. The reason of such small precision is that the mixed modes are very sensitive to the change of internal structure, and subgiants evolve very fast. Each combination has less than 0.2 Gyr difference in age, consistent with the idea that two components formed at the same time. Table 5 presents the fundamental parameters of the three model pairs, and Fig. 6 displays them on échelle diagrams. Open and filled symbols represent observational and theoretical frequencies. Red and blue indicate modes of star A and star B, respectively. Circles, squares, and triangles denote l = 0, l = 1, and l = 2 modes. Figure 6. View largeDownload slide Échelle diagrams of KIC 7107778. Number series (1)–(6) on the upper left corner of each panel corresponds to the series of models in Table 5, whose oscillation frequencies are displayed in the corresponding échelle. These models are without surface corrections. Open and filled symbols represent observational frequencies and theoretical ones. Red and blue represent modes of star A and star B, respectively. Circles, triangles, and squares denote l = 0, l = 1, and l = 2 modes. Figure 6. View largeDownload slide Échelle diagrams of KIC 7107778. Number series (1)–(6) on the upper left corner of each panel corresponds to the series of models in Table 5, whose oscillation frequencies are displayed in the corresponding échelle. These models are without surface corrections. Open and filled symbols represent observational frequencies and theoretical ones. Red and blue represent modes of star A and star B, respectively. Circles, triangles, and squares denote l = 0, l = 1, and l = 2 modes. Table 5. Fundamental properties of best models from bisection method. No.  Star  M  [Fe/H]  Age  Teff  L  R  log g  MH, core  Δν  νmax  #    (M⊙)  (dex)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (M⊙)  (μHz)  (μHz)  1  A  1.41  0.01  3.063  5130  5.342  2.929  3.655  0.165  32.04  540  2  B  1.39  0.01  3.229  5195  4.982  2.759  3.699  0.157  34.74  594  3  A  1.48  0.11  2.873  5120  5.464  2.975  3.661  0.170  32.00  548  4  B  1.47  0.11  2.929  5267  5.478  2.814  3.705  0.162  34.64  599  5  A  1.42  0.21  3.533  4987  4.763  2.927  3.656  0.165  32.10  549  6  B  1.41  0.21  3.548  5038  4.443  2.770  3.703  0.160  34.80  608  No.  Star  M  [Fe/H]  Age  Teff  L  R  log g  MH, core  Δν  νmax  #    (M⊙)  (dex)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (M⊙)  (μHz)  (μHz)  1  A  1.41  0.01  3.063  5130  5.342  2.929  3.655  0.165  32.04  540  2  B  1.39  0.01  3.229  5195  4.982  2.759  3.699  0.157  34.74  594  3  A  1.48  0.11  2.873  5120  5.464  2.975  3.661  0.170  32.00  548  4  B  1.47  0.11  2.929  5267  5.478  2.814  3.705  0.162  34.64  599  5  A  1.42  0.21  3.533  4987  4.763  2.927  3.656  0.165  32.10  549  6  B  1.41  0.21  3.548  5038  4.443  2.770  3.703  0.160  34.80  608  Note. Columns 3 and 4 are the input parameters for grid modelling. View Large Based on theoretical models, we found that most observed peaks could be matched with l = 1 modes. Tables 6 and 7 display the mode parameters in each Lorentzian profile we fitted to each peak, with associated l degrees. Modes which share the same peak on the power spectrum are labelled with an asterisk mark. Some modes from the two stars stand too close and cause ambiguity; they are labelled with a question mark. Here we remind readers that this solution is not unique, considering that the models still differ from the observation. In Fig. 7 , we show the mode linewidth as a function of the mode frequency. Mixed modes are expected to have smaller linewidths compared to radial modes because they have contributions from g modes trapped in the core, resulting a longer lifetime (Dupret et al. 2009; Benomar et al. 2013). This is strongest for the mode at 522.897 μHz. For other l = 1 modes, the linewidths are similar to l = 0 modes because they are less bumped and more p-like. Figure 7. View largeDownload slide Mode linewidth as a function of mode frequency. The circles and triangles represent the l = 0 and l = 1 modes respectively. The l = 1 modes shown here are only those which have relative certain identity (not associated with ‘?’ or ‘*’ in Tables 6 and 7). The l = 2 modes are instead not presented because each l = 2 mode region is fitted with a Lorentzian profile, the linewidth of which is not necessarily representing the real mode lifetime. Figure 7. View largeDownload slide Mode linewidth as a function of mode frequency. The circles and triangles represent the l = 0 and l = 1 modes respectively. The l = 1 modes shown here are only those which have relative certain identity (not associated with ‘?’ or ‘*’ in Tables 6 and 7). The l = 2 modes are instead not presented because each l = 2 mode region is fitted with a Lorentzian profile, the linewidth of which is not necessarily representing the real mode lifetime. Table 6. Mode parameters of KIC 7107778 A. l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  460.294  −0.022/ + 0.021  15.66  −0.80/ + 1.19  0.36  −0.09/ + 0.10      0  491.947  −0.013/ + 0.014  24.17  −1.29/ + 1.66  0.36  −0.10/ + 0.13      0  523.603  −0.012/ + 0.018  25.84  −1.85/ + 1.35  0.19  −0.07/ + 0.06      0  555.473  −0.133/ + 0.090  22.65  −5.44/ + 2.56  0.33  −0.16/ + 0.18      0  587.314  −0.420/ + 1.462  9.65  −2.24/ + 3.40  0.41  −0.17/ + 0.18      1  477.158  −0.031/ + 0.042  15.47  −0.18/ + 0.25  0.18  −0.02/ + 0.02      1  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    1  507.928  −0.014/ + 0.013  25.92  −0.83/ + 0.99  0.37  −0.06/ + 0.06      1  522.897  −0.002/ + 0.002  21.55  −0.75/ + 1.00  0.04  −0.01/ + 0.01      1  539.794  −0.011/ + 0.008  32.01  −1.36/ + 2.02  0.13  −0.04/ + 0.05      1  558.446  −0.172/ + 0.072  9.08  −3.06/ + 3.49  0.06  −0.02/ + 0.04    ?  1  572.522  −0.827/ + 0.844  9.03  −2.80/ + 4.84  0.24  −0.13/ + 0.09    ?  1  595.812  −0.040/ + 0.042  12.35  −1.44/ + 2.36  0.17  −0.11/ + 0.08    ?  2  457.106  −0.046/ + 0.041  11.10  −0.49/ + 0.40  0.41  −0.07/ + 0.07      2  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    2  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    2  553.122  −0.321/ + 0.354  16.27  −2.43/ + 3.94  0.43  −0.16/ + 0.14      2  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  460.294  −0.022/ + 0.021  15.66  −0.80/ + 1.19  0.36  −0.09/ + 0.10      0  491.947  −0.013/ + 0.014  24.17  −1.29/ + 1.66  0.36  −0.10/ + 0.13      0  523.603  −0.012/ + 0.018  25.84  −1.85/ + 1.35  0.19  −0.07/ + 0.06      0  555.473  −0.133/ + 0.090  22.65  −5.44/ + 2.56  0.33  −0.16/ + 0.18      0  587.314  −0.420/ + 1.462  9.65  −2.24/ + 3.40  0.41  −0.17/ + 0.18      1  477.158  −0.031/ + 0.042  15.47  −0.18/ + 0.25  0.18  −0.02/ + 0.02      1  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    1  507.928  −0.014/ + 0.013  25.92  −0.83/ + 0.99  0.37  −0.06/ + 0.06      1  522.897  −0.002/ + 0.002  21.55  −0.75/ + 1.00  0.04  −0.01/ + 0.01      1  539.794  −0.011/ + 0.008  32.01  −1.36/ + 2.02  0.13  −0.04/ + 0.05      1  558.446  −0.172/ + 0.072  9.08  −3.06/ + 3.49  0.06  −0.02/ + 0.04    ?  1  572.522  −0.827/ + 0.844  9.03  −2.80/ + 4.84  0.24  −0.13/ + 0.09    ?  1  595.812  −0.040/ + 0.042  12.35  −1.44/ + 2.36  0.17  −0.11/ + 0.08    ?  2  457.106  −0.046/ + 0.041  11.10  −0.49/ + 0.40  0.41  −0.07/ + 0.07      2  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    2  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    2  553.122  −0.321/ + 0.354  16.27  −2.43/ + 3.94  0.43  −0.16/ + 0.14      2  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    Note . Modes which share the same peak on the power spectrum are labelled with ‘*’ marks. Modes from two stars standing too close and causing ambiguity are denoted with ‘?’ marks. View Large Table 7. Mode parameters of KIC 7107778 B. l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  501.006  −0.019/ + 0.020  14.08  −0.55/ + 1.06  0.24  −0.07/ + 0.05      0  535.615  −0.016/ + 0.017  19.47  −0.66/ + 0.94  0.16  −0.03/ + 0.04      0  570.025  −0.146/ + 0.423  27.22  −6.39/ + 8.01  0.24  −0.11/ + 0.10      0  604.609  −0.026/ + 0.022  15.92  −2.44/ + 1.51  0.23  −0.08/ + 0.11      0  639.344  −0.076/ + 0.068  13.11  −0.74/ + 0.99  0.38  −0.11/ + 0.13      1  512.368  −0.008/ + 0.012  11.26  −0.93/ + 1.19  0.09  −0.06/ + 0.04      1  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    1  547.122  −0.735/ + 2.480  11.15  −2.42/ + 4.08  0.40  −0.16/ + 0.17      1  557.829  −0.177/ + 0.168  15.37  −5.30/ + 5.61  0.39  −0.19/ + 0.17    ?  1  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    1  596.595  −0.001/ + 0.001  18.12  −1.00/ + 1.37  0.02  −0.00/ + 0.01      1  620.979  −0.049/ + 0.040  16.97  −1.01/ + 1.17  0.33  −0.07/ + 0.10      2  497.849  −0.096/ + 0.091  10.21  −0.74/ + 0.89  0.38  −0.11/ + 0.12      2  532.691  −0.342/ + 0.315  11.42  −1.28/ + 2.08  0.41  −0.14/ + 0.14      2  566.534  −0.731/ + 0.830  10.72  −2.78/ + 7.32  0.44  −0.17/ + 0.30      2  601.946  −0.106/ + 0.109  14.16  −0.93/ + 1.22  0.36  −0.10/ + 0.10      2  636.541  −0.380/ + 0.493  9.04  −1.04/ + 1.20  0.44  −0.14/ + 0.13      l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  501.006  −0.019/ + 0.020  14.08  −0.55/ + 1.06  0.24  −0.07/ + 0.05      0  535.615  −0.016/ + 0.017  19.47  −0.66/ + 0.94  0.16  −0.03/ + 0.04      0  570.025  −0.146/ + 0.423  27.22  −6.39/ + 8.01  0.24  −0.11/ + 0.10      0  604.609  −0.026/ + 0.022  15.92  −2.44/ + 1.51  0.23  −0.08/ + 0.11      0  639.344  −0.076/ + 0.068  13.11  −0.74/ + 0.99  0.38  −0.11/ + 0.13      1  512.368  −0.008/ + 0.012  11.26  −0.93/ + 1.19  0.09  −0.06/ + 0.04      1  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    1  547.122  −0.735/ + 2.480  11.15  −2.42/ + 4.08  0.40  −0.16/ + 0.17      1  557.829  −0.177/ + 0.168  15.37  −5.30/ + 5.61  0.39  −0.19/ + 0.17    ?  1  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    1  596.595  −0.001/ + 0.001  18.12  −1.00/ + 1.37  0.02  −0.00/ + 0.01      1  620.979  −0.049/ + 0.040  16.97  −1.01/ + 1.17  0.33  −0.07/ + 0.10      2  497.849  −0.096/ + 0.091  10.21  −0.74/ + 0.89  0.38  −0.11/ + 0.12      2  532.691  −0.342/ + 0.315  11.42  −1.28/ + 2.08  0.41  −0.14/ + 0.14      2  566.534  −0.731/ + 0.830  10.72  −2.78/ + 7.32  0.44  −0.17/ + 0.30      2  601.946  −0.106/ + 0.109  14.16  −0.93/ + 1.22  0.36  −0.10/ + 0.10      2  636.541  −0.380/ + 0.493  9.04  −1.04/ + 1.20  0.44  −0.14/ + 0.13      Note . Modes which share the same peak on the power spectrum are labelled with ‘*’ marks. Modes from two stars standing too close and causing ambiguity are denoted with ‘?’ marks. View Large Fig. 8 is the Hertzsprung–Russel diagram where grid and additional models are displayed with dotted black lines. Specifically, the dashed lines indicate the best model tracks from the bisecting method with star A in red and star B in blue, respectively. The star symbols denote the best models. The boxes consisting of solid lines indicate the standard deviation of effective temperature Teff and luminosity L, shown in Table 4. The left-hand, middle, and right-hand panels represent models for metallicity [Fe/H] 0.01, 0.11, and 0.21. Figure 8. View largeDownload slide Hertzsprung–Russell diagram. Grid model tracks are displayed as black dotted lines. The best models, derived through bisection, are drawn with star symbols located on evolutionary tracks (dashed lines) in red for star A and in blue for star B. The solid lines indicate the standard deviation of effective temperature Teff and luminosity L, which are shown in Table 4. The left-hand, middle, and right-hand panel represent models for metallicity [Fe/H] 0.01, 0.11, and 0.21, respectively. Figure 8. View largeDownload slide Hertzsprung–Russell diagram. Grid model tracks are displayed as black dotted lines. The best models, derived through bisection, are drawn with star symbols located on evolutionary tracks (dashed lines) in red for star A and in blue for star B. The solid lines indicate the standard deviation of effective temperature Teff and luminosity L, which are shown in Table 4. The left-hand, middle, and right-hand panel represent models for metallicity [Fe/H] 0.01, 0.11, and 0.21, respectively. 5 CONCLUSION We applied asteroseismology to a binary target KIC 7107778, and confirmed that the two stars are in the sub-giant phase. We successfully identified their l = 0 and l = 2 oscillation modes and distinguished l = 1 modes to the greatest extent. We derived stellar fundamental parameters for the two stars: $$M_{\rm A} = 1.43^{+0.08}_{-0.08}$$ M⊙, $$M_{\rm B} = 1.40^{+0.05}_{-0.06}$$ M⊙, $$R_{\rm A} = 2.94^{+0.06}_{-0.06}$$ R⊙, $$R_{\rm B} = 2.77^{+0.04}_{-0.04}$$ R⊙, $$t_{\rm A} = 3.19^{+0.60}_{-0.64}$$ Gyr, and $$t_{\rm B} = 3.26^{+0.40}_{-0.40}$$ Gyr. All the evidence suggests that they formed at the same time and possess nearly equal masses. The results yield the similarity of masses for two stars, and the best models, derived through bisection modelling, determined the mass difference as 1.42 per cent, 0.68 per cent, and 0.70 per cent, from which we conclude it is ∼1 per cent. The H–R diagram verifies this, with extremely close tracks. Table 4 indicates the sensitivity of metallicity to age, and the two stars’ ages are equal within the error. We could conclude that they formed at the same time, as is expected by binary formation that they originate from the same protostellar cloud. The KIC 7107778 system contains two extremely similar components, with fully overlapping power spectra. This is one of the few identical twin systems to be found, proving the full potential of asteroseismology. Acknowledgements We acknowledge the data from the Kepler Discovery mission, whose funding is provided by NASA's Science Mission Directorate. This work has also made use of data from the European Space Agency (ESA) mission Gaia,1 processed by the Gaia Data Processing and Analysis Consortium (DPAC2). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. 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Asteroseismology of KIC 7107778: a binary comprising almost identical subgiants

Asteroseismology of KIC 7107778: a binary comprising almost identical subgiants

MNRAS 476, 470–481 (2018) doi:10.1093/mnras/sty222 Advance Access publication 2018 January 29 Asteroseismology of KIC 7107778: a binary comprising almost identical subgiants 1,2‹ 2,3‹ 2,3 1 Yaguang Li, Timothy R. Bedding, Tanda Li, Shaolan Bi, Simon 2,3 4 1 5 J. Murphy, Enrico Corsaro, Li Chen and Zhijia Tian Department of Astronomy, Beijing Normal University, Beijng 100875, China Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark INAF – Osservatorio Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy Department of Astronomy, Peking University, Beijing 100871, China Accepted 2018 January 23. Received 2018 January 23; in original form 2017 July 17 ABSTRACT We analyse an asteroseismic binary system: KIC 7107778, a non-eclipsing, unresolved target, with solar-like oscillations in both components. We used Kepler short cadence time series span- ning nearly 2 yr to obtain the power spectrum. Oscillation mode parameters were determined using Bayesian inference and a nested sampling Monte Carlo algorithm with the DIAMONDS package. The power profiles of the two components fully overlap, indicating their close simi- larity. We modelled the two stars with MESA and calculated oscillation frequencies with GYRE. Stellar fundamental parameters (mass, radius, and age) were estimated by grid modelling with atmospheric parameters and the oscillation frequencies of l = 0, 2 modes as constraints. Most l = 1 mixed modes were identified with models searched using a bisection method. Stellar parameters for the two sub-giant stars are M = 1.42 ± 0.06 M , M = 1.39 ± 0.03 M , A  B R = 2.93 ± 0.05 R , R = 2.76 ± 0.04 R , t = 3.32 ± 0.54 Gyr and t = 3.51 ± 0.33 Gyr. A  B  A B The mass difference of the system is ∼1 per cent....
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Abstract

Abstract We analyse an asteroseismic binary system: KIC 7107778, a non-eclipsing, unresolved target, with solar-like oscillations in both components. We used Kepler short cadence time series spanning nearly 2 yr to obtain the power spectrum. Oscillation mode parameters were determined using Bayesian inference and a nested sampling Monte Carlo algorithm with the diamonds package. The power profiles of the two components fully overlap, indicating their close similarity. We modelled the two stars with mesa and calculated oscillation frequencies with gyre. Stellar fundamental parameters (mass, radius, and age) were estimated by grid modelling with atmospheric parameters and the oscillation frequencies of l = 0, 2 modes as constraints. Most l = 1 mixed modes were identified with models searched using a bisection method. Stellar parameters for the two sub-giant stars are MA = 1.42 ± 0.06 M⊙, MB = 1.39 ± 0.03 M⊙, RA = 2.93 ± 0.05 R⊙, RB = 2.76 ± 0.04 R⊙, tA = 3.32 ± 0.54 Gyr and tB = 3.51 ± 0.33 Gyr. The mass difference of the system is ∼1 per cent. The results confirm their simultaneous birth and evolution, as is expected from binary formation. KIC 7107778 comprises almost identical twins, and is the first asteroseismic sub-giant binary to be detected. binaries: general, stars: evolution, stars: oscillations 1 INTRODUCTION Binary star systems provide an ideal astronomical laboratory to study stellar structure and evolution. The fact that two components share same metal abundance and age provides powerful constraints on models. Eclipsing binaries are especially useful, since mass and radius can be directly measured from orbits and eclipses through spectroscopic observations and precise light-curve analysis (Andersen 1991). Asteroseismology ushers in a new way to study binaries (Huber 2015), even for unresolved ones (Miglio et al. 2014), by analysing the two components separately. Determining stellar mass and radius is feasible for single stars with solar-like oscillations (e.g. Kallinger et al. 2010; Chaplin et al. 2014). Sub-giant and red-giant oscillating stars displaying p (pressure dominated in the envelope) and g (gravity dominated in the core) mixed modes (Chaplin & Miglio 2013) are good indicators of evolutionary stages. Their great sensitivity to mass and age can produce precise estimates of stellar parameters (e.g. Metcalfe et al. 2010; Benomar et al. 2012). Binary systems with pulsators can be analysed from modulation of the pulsation frequencies (Shibahashi & Kurtz 2012; Murphy et al. 2014; Murphy & Shibahashi 2015), so that system parameters can be determined with radial velocity curves simply derived through photometry. White et al. (2017) pointed out that, until now, only five binary star systems have been detected with solar-like oscillations from both components. They are α Cen A and B (Bedding et al. 1998; Kjeldsen et al. 1999; Bouchy & Carrier 2001; Carrier & Bourban 2003), 16 Cyg A and B (KIC 12069424 and KIC 12069449; Metcalfe et al. 2012; Metcalfe, Creevey & Davies 2015), KIC 9139163 and KIC 9139151 (Appourchaux et al. 2012; Appourchaux et al. 2014), HD 177412 (KIC 7510397; Appourchaux et al. 2015), and HD 176465 (KIC 10124866, also known as Luke & Leia; White et al. 2017). The latter two systems are not resolved by Kepler, such that their light variations are mixed in a single time series. Both systems were analysed with one power spectrum, from which two sets of oscillation profiles were measured. Specifically, HD 177412 contains two main-sequence stars with mass difference ∼7.5 per cent, with separable oscillation frequency ranges, and HD 176465 contains two extremely similar main-sequence stars with mass difference ∼ 3 per cent, leading to two significantly overlapping oscillation ranges. In addition, KIC 10080943 is another unresolved binary system, comprising two δ Sct/γ Dor hybrid pulsators on the main sequence (Keen et al. 2015; Schmid et al. 2015). In this context, these binaries are not strictly ‘twins’ in that the masses are not equal to within 2 per cent (Lucy 2006; Simon & Obbie 2009). Twins are almost always found at small separations (P ≲ 25 d, Lucy & Ricco 1979), though not all close binaries are twins. Further, they are more common among lower mass systems, both in observational data (e.g. Tokovinin 2000; Simon & Obbie 2009) and in binary star formation simulations (e.g. Bate 2009). Their importance lies in their ability to discriminate between dominant physical processes operating in pre-main-sequence binaries, as those authors have discussed. Here we report on the widely separated twin binary system KIC 7107778, with mass difference ∼1 per cent based on our findings, an analogue to Luke & Leia (White et al. 2017). The system was not resolved by Kepler but was detected with oscillations from two stars in the mixed time series. It proves to be the first asteroseismic sub-giant binary system to be detected, with completely overlapping power spectra of two component stars. The paper will be structured as follows. Section 2 provides observations from previous literature and describes the processing of Kepler data. Section 3 illustrates the oscillation mode parameters. Section 4 details our asteroseismic analysis with stellar models of two stars, and is followed by discussions and conclusions in Section 5. 2 OBSERVATION AND DATA PROCESSING KIC 7107778 is a binary system observed as a single target, with Kepler magnitude Kp = 11.38. The first realization of its binary identity was from speckle interferometry (Horch et al. 2012). The angular separation of the two stars is ρ = 0.0288 arcsec, measured with filters whose central wavelengths are 692 and 880 nm. The physical separation can be determined by combining luminosities from models with the apparent magnitude; however, such estimation is only approximate because the apparent magnitude is a blended contribution from both stars. We took advantage of the parallax determined from Gaia (Gaia Collaboration 2016), 1.74 ± 0.43 mas, to infer the distance to Earth and physical projected separation between the two stars, which are $$d = 574^{+189}_{-113}$$ pc, and $$s = 16.52^{+5.44}_{-3.25}$$ au. Note that this estimation is very approximate because Gaia parallaxes are not yet fully reliable at the present mission stage (see Fabricius et al. 2016). Assuming the orbit is circular, and adopting Kepler's third law, it suggests the orbital period of this system is about 39 yr. Therefore, we would not expect to analyse its orbit using radial velocity curves. On the other hand, such a wide separation means fully independent evolution without tidal effects or mass transfer. In the following analysis, we treat them as two single stars that have not interacted. Several works have measured atmospheric parameters of this system. Effective temperatures measured by SDSS and IRFM are Teff = 5129 ± 82 K and Teff = 5045 ± 105 K, respectively. The binary system was also covered by the LAMOST-Kepler project, which used LAMOST low-resolution (R ≃ 1800 Å) optical spectra in the waveband of 3800–9000 Å and observed Kepler-field targets in 2014 September. The LAMOST Stellar Parameter Pipeline (Xiang et al. 2015) gives Teff = 5149 ± 150 K, and metallicity [Fe/H] = 0.11 ± 0.10 dex. Buchhave & Latham (2015) used the Tillinghast Reflector Echelle Spectrograph to obtain medium resolution (R ≃ 44 000Å) spectra in the waveband of 3800–9100 Å at Fred Lawrence Whipple Observatory. They measured the metallicity [Fe/H] = 0.05 ± 0.08 dex. However, all these observations treated two stars as one single target. We do not know to what extent the mixed value deviates from the real ones. Table 1 summarizes the observations in the literature. Table 1. Atmospheric parameters of KIC 7107778. Parameter  Value  Reference  Teff (K)  5129 ± 82  Drimmel, Cabrera-Lavers & López-Corredoira (2003)    5045 ± 105  Casagrande et al. (2010)    5149 ± 150  Xiang et al. (2015)  [Fe/H] (dex)  0.11 ± 0.10  Xiang et al. (2015)    0.05 ± 0.08  Buchhave & Latham (2015)  Parameter  Value  Reference  Teff (K)  5129 ± 82  Drimmel, Cabrera-Lavers & López-Corredoira (2003)    5045 ± 105  Casagrande et al. (2010)    5149 ± 150  Xiang et al. (2015)  [Fe/H] (dex)  0.11 ± 0.10  Xiang et al. (2015)    0.05 ± 0.08  Buchhave & Latham (2015)  View Large The Kepler mission observed the target in long-cadence mode (LC; 29.43 min sampling) over the whole mission and in short-cadence mode (SC; 58.84 s sampling) during Q2.1, Q5 and Q7–Q12 (Q represents three-month-long quarters). The pulsation frequency range is centred at 550 μHz, which is well above the Nyquist frequency of long cadence data (∼283 μHz). Therefore, we only considered the short-cadence time series. We concatenated the data and processed it following García et al. (2011), correcting outliers, jumps, and drifts. Then it passed through a high-pass filter which was based on a Gaussian smooth function with a width of 1 d. This largely minimized instrumental side effects and only affected frequencies lower than ∼12 μHz, far below the frequency range we intended to analyse. We obtained the power spectrum by applying a Lomb–Scargle Periodogram (Lomb 1976; Scargle 1982) to the time series with a frequency resolution ∼0.012 μHz. The power spectrum is shown in Figs 1 and 2 in both logarithmic and linear scales. Figure 1. View largeDownload slide Power spectrum on a logarithmic scale. The solid grey line, the solid black line, the dashed blue lines, the dot–dashed-dashed black line, and the solid green line, outline the original power spectrum, the 6 μHz smoothed power spectrum, the Harvey profile components, the white noise component, and the total fitted power spectrum, respectively. Figure 1. View largeDownload slide Power spectrum on a logarithmic scale. The solid grey line, the solid black line, the dashed blue lines, the dot–dashed-dashed black line, and the solid green line, outline the original power spectrum, the 6 μHz smoothed power spectrum, the Harvey profile components, the white noise component, and the total fitted power spectrum, respectively. Figure 2. View largeDownload slide Power spectrum around the oscillation range on a linear scale. The grey line denotes the original power spectrum superimposed with fitted l = 1 modes in black, l = 0 and l = 2 modes of star A in red, and those of star B in blue. Figure 2. View largeDownload slide Power spectrum around the oscillation range on a linear scale. The grey line denotes the original power spectrum superimposed with fitted l = 1 modes in black, l = 0 and l = 2 modes of star A in red, and those of star B in blue. The signature of solar-like oscillation is a Gaussian-like envelope located at νmax, the so-called frequency of maximum power, and is comprised of numerous oscillation modes. In main-sequence stars, p-mode oscillations are approximately equally spaced in frequency, as described by the asymptotic equation with radial orders n and spherical degrees l (Tassoul 1980):   \begin{equation} \nu _{nl} = \Delta \nu \left(n+\frac{l}{2}+\epsilon\right)-\delta \nu _{0l}, \end{equation} (1)where Δν is the large separation, which measures the spacing of adjacent modes with the same l, δν0l is the small separation, and ε is an offset. In more evolved stars, the core will have g-mode oscillations, which are approximately equally spaced in period. The analogous asymptotic equation is specified by order ng:   \begin{equation} \Pi _{nl} = \nu _{nl}^{-1} = \Delta \Pi _{l}(n_g+\epsilon _g), \end{equation} (2)where ΔΠl is the period spacing, and εg is an offset. As the star evolves off the main sequence, central hydrogen depletes, and p and g mixed modes of l ≥ 1 appear to have ‘avoided crossings’ (Aizenman, Smeyers & Weigert 1977), whereby oscillation frequencies are no longer equally spaced in either frequency or period. The l = 0 p modes are unaffected, which assisted us in determining the mean large separation 〈Δν〉 in an échelle diagram. The best value for 〈Δν〉 is the one that makes the l = 0 ridge vertical. Fig. 3 displays échelle diagrams for the two stars. To avoid ambiguity, the star with smaller 〈Δν〉 is named as KIC 7107778 A, and the larger one KIC 7107778 B. The best values are 〈Δν〉A = 31.83 ± 0.02 μHz and 〈Δν〉B = 34.55 ± 0.01 μHz. Figure 3. View largeDownload slide Échelle diagrams of star A (left) and star B (right). The circular and square open symbols represent identified l = 0 and l = 2 modes. Red and blue colours denote modes for star A and star B, respectively. The background grey-scale represents power density. Higher power density is shown darker. The unmarked peaks are l = 1 mixed modes. Figure 3. View largeDownload slide Échelle diagrams of star A (left) and star B (right). The circular and square open symbols represent identified l = 0 and l = 2 modes. Red and blue colours denote modes for star A and star B, respectively. The background grey-scale represents power density. Higher power density is shown darker. The unmarked peaks are l = 1 mixed modes. 3 MODE PARAMETERS To model the power spectrum, we used high-DImensional And multi-MOdal NesteD Sampling code (diamonds; Corsaro & De Ridder 2014). The diamonds code utilizes Bayes’ theorem:   \begin{equation} p(\theta \mid {D,M}) = \frac{{\mathcal {L}}(\theta \mid {D,M})\pi (\theta \mid {M})}{p(D\mid {M})}, \end{equation} (3)where θ = (θ1, θ2, ..., θk), $${\mathcal {L}}(\theta \mid {D,M})$$, π(θ∣M), p(θ∣D, M) are the parameter vector, likelihood function for a given model M and data set D, prior probability density function, and the posterior probability density function, respectively. diamonds uses a sampling algorithm, Nested Sampling Monte Carlo, to tackle the high-dimensional problem. We simultaneously fitted the two stars to separate their overlapping oscillations with the following steps. First, we modelled the power spectrum, P(ν), with   \begin{eqnarray} P(\nu ) &=& W+R\left(\nu \right)\left\lbrace \sum _{i = 1}^{k}\frac{2\sqrt{2}}{\pi }\frac{ a_i^2/b_i}{1+(\nu /b_i)^{4}}\right. \nonumber\\ &&\left. +\,H_0{\rm exp}\left[-\frac{(\nu -\nu _{\rm max})^2}{2\sigma ^2}\right]\right\rbrace , \end{eqnarray} (4)similar to the background models presented in Kallinger et al. (2014) and Corsaro, De Ridder & García (2015). The right-hand side of the equation comprises a flat white noise W, a sum of Harvey power profiles (Harvey 1985) with parameters (ai, bi), and a Gaussian envelope with (H0, νmax, σ). The Harvey profile models stellar background caused by granulation. In the case of KIC 7107778, the background was well fitted using three Harvey profiles, i.e. k = 3. The Gaussian envelope denotes the range of oscillation for two overlapping stars. All these components except the noise are modulated by the response function,   \begin{equation} R\left(\nu \right) = {\rm sinc}^2\left(\frac{\pi \nu }{2\nu _{\rm Nyq}}\right), \end{equation} (5)where νNyq = 8496.36 μHz denotes the Nyquist frequency. The total number of free variables is 1 + 2 × 3 + 3 = 10. The results are shown in Table 2. As argued by Kallinger et al. (2014), the granulation frequencies bi should scale with νmax. They provided empirical relations b2 = 0.317(νmax/μHz)0.970μHz and b3 = 0.948(νmax/μHz)0.992μHz, which result in b2 = 148.87 μHz and b3 = 511.87 μHz in our case. b2 is similar to our fit but not b3. This should be expected because the power spectrum is a superposition of the twins. b1 should be treated carefully because the light curve was processed by a high-pass filter, which affected the low-frequency spectrum. Table 2. Granulation background parameters. Parameter  Value  68.3 per cent credible  W (ppm2 μHz− 1)  11.927  0.100  a1 (ppm)  58.874  8.314  b1 (μHz)  5.203  1.460  a2 (ppm)  67.167  8.705  b2 (μHz)  149.144  32.207  a3 (ppm)  76.134  7.947  b3 (μHz)  400.646  64.318  H0 (ppm2 μHz− 1)  17.977  2.852  νmax (μHz)  568.051  7.935  σ (μHz)  55.811  8.690  Parameter  Value  68.3 per cent credible  W (ppm2 μHz− 1)  11.927  0.100  a1 (ppm)  58.874  8.314  b1 (μHz)  5.203  1.460  a2 (ppm)  67.167  8.705  b2 (μHz)  149.144  32.207  a3 (ppm)  76.134  7.947  b3 (μHz)  400.646  64.318  H0 (ppm2 μHz− 1)  17.977  2.852  νmax (μHz)  568.051  7.935  σ (μHz)  55.811  8.690  View Large Then, each mode was fitted with a Lorentzian profile with three free parameters: frequency centroid ν0, amplitude A, and linewidth Γ built on the background:   \begin{equation} L(\nu ) = R\left(\nu \right)\left[\frac{A^2/\pi \Gamma }{1+4\left(\nu -\nu _0\right)^2/\Gamma ^2}\right]. \end{equation} (6)The power spectrum was fitted with a sum of 32 Lorentzian profiles. The total number of free variables is 3 × 32 = 96. It is possible that the star rotates and lifts degeneracy of m-degree of the modes (Gizon & Solanki 2003). Thus we performed a hypothesis test with Bayesian evidence denoted by p(D∣M) in equation (3). The Bayesian evidence balances the goodness of fit and the need to fit, as it can be written as a product of the maximum of the likelihood and an Occam factor (Knuth et al. 2015). We adopted a splitting model of l = 1 modes   \begin{equation} L(\nu ) = R(\nu)\left[\sum _{m = -1}^{1}\frac{\xi _m A^2/\pi \Gamma }{1+4(\nu -\nu _0-m\delta \nu _{\rm split})^2/\Gamma ^2}\right], \end{equation} (7)where i is the inclination angle, ξ−1 = ξ1 = 0.5sin 2i, ξ0 = cos 2i, and δνsplit measures the extent of splitting. We fitted the splitting model to four l = 1 modes (mode frequency 477, 507, 539, and 620 μHz) individually. The four modes were selected because they do not have another mode in 3 μHz frequency range. We defined detection probability as p = p(D∣MB)/(p(D∣MA) + p(D∣MB)) (Jeffreys 1961; Corsaro & De Ridder 2014), where subscripts A and B refer to the non-splitting model and the splitting model. The detection probabilities are small: 10−1, 10−14, 10−9, and 10−2, respectively. We also fitted the four modes with a common inclination and the detection probability is 10−50. This indicates the non-splitting model gives a better depiction of the data. Hence, detecting the inclination angle i through rotation was not considered. Notice that only l = 0 and l = 2 modes can be identified straightforwardly and allocated to one of the two stars, since they are regularly spaced. The l = 1 mixed modes are strongly bumped and no clear patterns could be followed, so we used stellar models to help the identification, as discussed in Section 4. We also measured the frequency of maximum power, νmax, by fitting a Gaussian profile in power density to the l = 0 mode peaks. The results are νmax, A = 523 ± 16 μHz and νmax, B = 570 ± 18 μHz, where the uncertainties are half of 〈Δν〉. Fig. 3 shows the échelle diagram of both stars. The circular and square open symbols represent identified l = 0 and l = 2 modes. Red and blue colours denote star A and star B, respectively. The mode parameters are shown in Tables 6 and 7. The l = 1 modes in the tables are further discussed in Section 4.3. 4 ASTEROSEISMIC ANALYSIS As discussed in Section 2, KIC 7107778 has a long orbital period and no evidence of eclipses from which stellar fundamental parameters could be estimated. Therefore, estimating them through stellar models with asteroseismology is necessary. 4.1 Stellar models We constructed stellar models using Modules for Experiments in Stellar Astrophysics (mesa; Paxton et al. 2011; Paxton et al. 2013; Paxton et al. 2015). Paxton et al. (2011) have discussed the input physics of mesa. Here we list them for completeness. The equation of state was delivered by OPAL EOS tables (Rogers & Nayfonov 2002), SVCH tables (Saumon, Chabrier & van Horn 1995) at low temperatures and densities, and HELM (Timmes & Swesty 2000) and PC (Potekhin & Chabrier 2010) tables under other circumstances. Opacities were taken from Iglesias & Rogers (1996) at high temperature and Ferguson et al. (2005) at low temperature. Nuclear reaction rates were based on NACRE (Angulo et al. 1999) and CF88 (Caughlan & Fowler 1988) when NACRE was unavailable. The convection was implemented with mixing length theory (MLT) illustrated in Cox & Giuli (1968). The mixing length parameter αMLT = 1.917 was employed according to the mesa standard solar model (Paxton et al. 2011). Overshoot mixing was set according to Herwig (2000), with overshooting parameter fov = 0.016. The initial helium abundance was estimated through   \begin{equation} Y_{\rm ini} = Y_0+\frac{\Delta Y}{\Delta Z}\cdot {Z_{\rm ini}}, \end{equation} (8)where Y0 = 0.249 (Planck Collaboration XIII 2016) and ΔY/ΔZ = 1.33, calculated using the initial abundances of helium and heavy elements of the calibrated solar model (Paxton et al. 2011). The relation of metallicity and element abundance ratio we adopted here was   \begin{equation} \rm {[Fe/H]} = \log (\text{$Z$}/\text{$X$})-{\log }{(\text{$Z$}/\text{$X$})_{\odot }}, \end{equation} (9)where the solar value is (Z/X)⊙ = 0.022 93 (Grevesse & Sauval 1998). The varying input parameters for grid modelling are mass M and metallicity [Fe/H]. They were set as follows. First, the mass was set according to asteroseismic scaling relations. The mean large separation Δν and frequency of maximum power νmax are related to mean density ρ, surface gravity g, and effective temperature Teff: $$\Delta \nu \propto \sqrt{\rho }$$, $$\nu _{\rm {\rm max}}\propto {g/\sqrt{T_{\rm {eff}}}}$$ (Kjeldsen & Bedding 1995), i.e.   \begin{equation} \frac{\Delta \nu }{\Delta \nu _{\odot }}\approx \left(\frac{M}{\mathrm{M}_{\odot }}\right)^{1/2}\left(\frac{R}{\mathrm{R}_{\odot }}\right)^{-3/2}, \end{equation} (10)  \begin{equation} \frac{\nu _{\rm max}}{\nu _{\rm max,\odot }}\approx \left(\frac{M}{\mathrm{M}_{\odot }}\right)\left(\frac{R}{\mathrm{R}_{\odot }}\right)^{-2}\left(\frac{T_{\rm {eff}}}{T_{\rm {eff},\odot }}\right)^{-1/2}, \end{equation} (11)where Δν⊙ = 135.1 μHz, νmax, ⊙ = 3050 μHz (Chaplin & Miglio 2013) and Teff = 5777 K. Thus, the mass prescription can be deduced from equations (10) and (11):   \begin{equation} \frac{M}{\mathrm{M}_{\odot }}\approx \left(\frac{\Delta \nu }{\Delta \nu _{\odot }}\right)^{-4}\left(\frac{\nu _{\rm {max}}}{\nu _{\rm {max},\odot }}\right)^{3}\left(\frac{T_{\rm {eff}}}{T_{\rm {eff},\odot }}\right)^{3/2}. \end{equation} (12)Considering the uncertainties, the mass range of the grid should at least cover 1.34–1.60 M⊙. Secondly, we adopted metallicity [Fe/H] = 0.01, 0.11, and 0.21 according to spectral observations from LAMOST. Table 3 summarizes the input parameters of our grid modelling. Table 3. Input parameters of grid modelling.   Value  Step size  M (M⊙)  1.34–1.60  0.1  [Fe/H] (dex)  0.01–0.21  0.1  αMLT  1.917  Fixed  fov  0.016  Fixed    Value  Step size  M (M⊙)  1.34–1.60  0.1  [Fe/H] (dex)  0.01–0.21  0.1  αMLT  1.917  Fixed  fov  0.016  Fixed  View Large 4.2 Modelling the l = 0 and l = 2 modes We calculated oscillation frequencies for models that met the requirements of Δν and νmax with gyre (Townsend & Teitler 2013), which solves the adiabatic pulsation equations with stellar structure data. The calculated frequencies deviated from the observations due to the improper simulation of the stellar surface. Therefore, we followed the method described by Ball & Gizon (2014) to correct them. The correction formula we adopted here was   \begin{equation} \delta \nu = (a_{-1}\nu ^{-1}+a_3\nu ^3)/I, \end{equation} (13)where a−1 and a3 are coefficients determined through least-squares fit, and I is the mode inertia. We used χ2 = Σ(νobs − νmod)2/σ2 as the quality of the fit. Frequencies used in the χ2 calculation were all l = 0 and l = 2 modes. The smaller the χ2 of the model, the better the match. We took the best 10 per cent of the models, as measured by χ2, for further considerations. The choice of this criterion was a trade-off. Small values could be biased by fluctuations in limited samples. The problem was tackled by sorting samples according to the values of χ2, performing difference between two adjacent quantities. We found the trend around 10 per cent went smoothly, which ruled out the fluctuation effect. Including more models would make the selection less useful. This point was addressed by inspecting the value of χ2 around the 10 per cent cut-off and assuring that it was not too large. With the lowest 10 per cent χ2 models, we calculated the mean value of the stellar parameters as the centroid. Table 4 lists them with standard deviations. Additionally, Figs 4 and 5 display histograms of each stellar parameter, which reflect the distribution in the lowest 10 per cent χ2 models with [Fe/H] = 0.11. Red and blue indicate star A and star B, respectively. Figure 4. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star A grid models with [Fe/H] = 0.11 dex. Figure 4. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star A grid models with [Fe/H] = 0.11 dex. Figure 5. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star B grid models with [Fe/H] = 0.11 dex. Figure 5. View largeDownload slide Parameter distributions for the lowest 10 per cent χ2 of star B grid models with [Fe/H] = 0.11 dex. Table 4. Fundamental properties of models from grid modelling. Star  [Fe/H]  M  Age  Teff  L  R  log g  Δν  νmax    (dex)  (M⊙)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (μHz)  (μHz)  A  0.01  1.41 ± 0.06  3.178 ± 0.440  5216 ± 258  5.813 ± 1.508  2.925 ± 0.047  3.654 ± 0.005  32.01 ± 0.18  534 ± 8  B  0.01  1.38 ± 0.03  3.289 ± 0.263  5233 ± 132  5.140 ± 0.692  2.754 ± 0.025  3.699 ± 0.003  34.77 ± 0.13  592 ± 5  A  0.11  1.43 ± 0.07  3.266 ± 0.583  5184 ± 290  5.787 ± 1.663  2.945 ± 0.055  3.656 ± 0.007  31.98 ± 0.17  539 ± 9  B  0.11  1.38 ± 0.03  3.606 ± 0.313  5092 ± 105  4.594 ± 0.549  2.751 ± 0.025  3.700 ± 0.004  34.81 ± 0.12  601 ± 4  A  0.21  1.41 ± 0.07  3.643 ± 0.551  5037 ± 199  5.032 ± 1.160  2.926 ± 0.054  3.655 ± 0.006  32.06 ± 0.15  546 ± 6  B  0.21  1.40 ± 0.04  3.657 ± 0.317  5045 ± 106  4.473 ± 0.535  2.765 ± 0.028  3.701 ± 0.003  34.78 ± 0.13  606 ± 4  Star  [Fe/H]  M  Age  Teff  L  R  log g  Δν  νmax    (dex)  (M⊙)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (μHz)  (μHz)  A  0.01  1.41 ± 0.06  3.178 ± 0.440  5216 ± 258  5.813 ± 1.508  2.925 ± 0.047  3.654 ± 0.005  32.01 ± 0.18  534 ± 8  B  0.01  1.38 ± 0.03  3.289 ± 0.263  5233 ± 132  5.140 ± 0.692  2.754 ± 0.025  3.699 ± 0.003  34.77 ± 0.13  592 ± 5  A  0.11  1.43 ± 0.07  3.266 ± 0.583  5184 ± 290  5.787 ± 1.663  2.945 ± 0.055  3.656 ± 0.007  31.98 ± 0.17  539 ± 9  B  0.11  1.38 ± 0.03  3.606 ± 0.313  5092 ± 105  4.594 ± 0.549  2.751 ± 0.025  3.700 ± 0.004  34.81 ± 0.12  601 ± 4  A  0.21  1.41 ± 0.07  3.643 ± 0.551  5037 ± 199  5.032 ± 1.160  2.926 ± 0.054  3.655 ± 0.006  32.06 ± 0.15  546 ± 6  B  0.21  1.40 ± 0.04  3.657 ± 0.317  5045 ± 106  4.473 ± 0.535  2.765 ± 0.028  3.701 ± 0.003  34.78 ± 0.13  606 ± 4  Note . The models are selected with the lowest 10 per cent χ2. Column 2 is the input parameter for grid modelling. View Large Considering that [Fe/H] was estimated with uncertainties, we combined stellar parameters derived based on different [Fe/H] together, as the ultimate results. Here we list mass, radius, and age: MA = 1.42 ± 0.06 M⊙, MB = 1.39 ± 0.03 M⊙, RA = 2.93 ± 0.05 R⊙, RB = 2.76 ± 0.04 R⊙, tA = 3.32 ± 0.54 Gyr, and tB = 3.51 ± 0.33 Gyr. 4.3 Modelling the l = 1 mixed modes We next searched for the best models which could also fit the frequencies of l = 1 modes from both stars. KIC 7107778 contains two sub-giant stars. Tiny changes to the mass of the models influence oscillations greatly. Only extremely fine grids produce satisfying results, which makes the task demanding. Since our purpose was to find a pair of stellar models for both stars that fit observations best, we used the bisection method to search. The main idea of the bisection method is summarized as follows, similar to finding solutions for equation f(x) = 0. First, we started with two masses, M1 and M2, which lied on opposite sides of the best model. This choice was realized by visually inspecting oscillation frequencies on the échelle diagram. Secondly, we bisected this range, i.e. calculated frequencies of M3 = (M1 + M2)/2. Thirdly, we evaluated the result of M3 and determined M3 and whichever of M1 or M2 yield the best model. Fourthly, we kept bisecting the mass range until the difference became sufficiently small. We followed this search scheme under three different metallicities [Fe/H]: 0.01, 0.11, and 0.21. The maximum precision in calculation reaches 0.000 01 M⊙. The results revealed three combinations of best-fitting models for the two stars. Although they do still deviate from observed peaks, they give a reasonable fit to most l = 1 modes. Most model masses reach to 0.001 M⊙ precision. The reason of such small precision is that the mixed modes are very sensitive to the change of internal structure, and subgiants evolve very fast. Each combination has less than 0.2 Gyr difference in age, consistent with the idea that two components formed at the same time. Table 5 presents the fundamental parameters of the three model pairs, and Fig. 6 displays them on échelle diagrams. Open and filled symbols represent observational and theoretical frequencies. Red and blue indicate modes of star A and star B, respectively. Circles, squares, and triangles denote l = 0, l = 1, and l = 2 modes. Figure 6. View largeDownload slide Échelle diagrams of KIC 7107778. Number series (1)–(6) on the upper left corner of each panel corresponds to the series of models in Table 5, whose oscillation frequencies are displayed in the corresponding échelle. These models are without surface corrections. Open and filled symbols represent observational frequencies and theoretical ones. Red and blue represent modes of star A and star B, respectively. Circles, triangles, and squares denote l = 0, l = 1, and l = 2 modes. Figure 6. View largeDownload slide Échelle diagrams of KIC 7107778. Number series (1)–(6) on the upper left corner of each panel corresponds to the series of models in Table 5, whose oscillation frequencies are displayed in the corresponding échelle. These models are without surface corrections. Open and filled symbols represent observational frequencies and theoretical ones. Red and blue represent modes of star A and star B, respectively. Circles, triangles, and squares denote l = 0, l = 1, and l = 2 modes. Table 5. Fundamental properties of best models from bisection method. No.  Star  M  [Fe/H]  Age  Teff  L  R  log g  MH, core  Δν  νmax  #    (M⊙)  (dex)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (M⊙)  (μHz)  (μHz)  1  A  1.41  0.01  3.063  5130  5.342  2.929  3.655  0.165  32.04  540  2  B  1.39  0.01  3.229  5195  4.982  2.759  3.699  0.157  34.74  594  3  A  1.48  0.11  2.873  5120  5.464  2.975  3.661  0.170  32.00  548  4  B  1.47  0.11  2.929  5267  5.478  2.814  3.705  0.162  34.64  599  5  A  1.42  0.21  3.533  4987  4.763  2.927  3.656  0.165  32.10  549  6  B  1.41  0.21  3.548  5038  4.443  2.770  3.703  0.160  34.80  608  No.  Star  M  [Fe/H]  Age  Teff  L  R  log g  MH, core  Δν  νmax  #    (M⊙)  (dex)  (Gyr)  (K)  (L⊙)  (R⊙)  (dex)  (M⊙)  (μHz)  (μHz)  1  A  1.41  0.01  3.063  5130  5.342  2.929  3.655  0.165  32.04  540  2  B  1.39  0.01  3.229  5195  4.982  2.759  3.699  0.157  34.74  594  3  A  1.48  0.11  2.873  5120  5.464  2.975  3.661  0.170  32.00  548  4  B  1.47  0.11  2.929  5267  5.478  2.814  3.705  0.162  34.64  599  5  A  1.42  0.21  3.533  4987  4.763  2.927  3.656  0.165  32.10  549  6  B  1.41  0.21  3.548  5038  4.443  2.770  3.703  0.160  34.80  608  Note. Columns 3 and 4 are the input parameters for grid modelling. View Large Based on theoretical models, we found that most observed peaks could be matched with l = 1 modes. Tables 6 and 7 display the mode parameters in each Lorentzian profile we fitted to each peak, with associated l degrees. Modes which share the same peak on the power spectrum are labelled with an asterisk mark. Some modes from the two stars stand too close and cause ambiguity; they are labelled with a question mark. Here we remind readers that this solution is not unique, considering that the models still differ from the observation. In Fig. 7 , we show the mode linewidth as a function of the mode frequency. Mixed modes are expected to have smaller linewidths compared to radial modes because they have contributions from g modes trapped in the core, resulting a longer lifetime (Dupret et al. 2009; Benomar et al. 2013). This is strongest for the mode at 522.897 μHz. For other l = 1 modes, the linewidths are similar to l = 0 modes because they are less bumped and more p-like. Figure 7. View largeDownload slide Mode linewidth as a function of mode frequency. The circles and triangles represent the l = 0 and l = 1 modes respectively. The l = 1 modes shown here are only those which have relative certain identity (not associated with ‘?’ or ‘*’ in Tables 6 and 7). The l = 2 modes are instead not presented because each l = 2 mode region is fitted with a Lorentzian profile, the linewidth of which is not necessarily representing the real mode lifetime. Figure 7. View largeDownload slide Mode linewidth as a function of mode frequency. The circles and triangles represent the l = 0 and l = 1 modes respectively. The l = 1 modes shown here are only those which have relative certain identity (not associated with ‘?’ or ‘*’ in Tables 6 and 7). The l = 2 modes are instead not presented because each l = 2 mode region is fitted with a Lorentzian profile, the linewidth of which is not necessarily representing the real mode lifetime. Table 6. Mode parameters of KIC 7107778 A. l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  460.294  −0.022/ + 0.021  15.66  −0.80/ + 1.19  0.36  −0.09/ + 0.10      0  491.947  −0.013/ + 0.014  24.17  −1.29/ + 1.66  0.36  −0.10/ + 0.13      0  523.603  −0.012/ + 0.018  25.84  −1.85/ + 1.35  0.19  −0.07/ + 0.06      0  555.473  −0.133/ + 0.090  22.65  −5.44/ + 2.56  0.33  −0.16/ + 0.18      0  587.314  −0.420/ + 1.462  9.65  −2.24/ + 3.40  0.41  −0.17/ + 0.18      1  477.158  −0.031/ + 0.042  15.47  −0.18/ + 0.25  0.18  −0.02/ + 0.02      1  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    1  507.928  −0.014/ + 0.013  25.92  −0.83/ + 0.99  0.37  −0.06/ + 0.06      1  522.897  −0.002/ + 0.002  21.55  −0.75/ + 1.00  0.04  −0.01/ + 0.01      1  539.794  −0.011/ + 0.008  32.01  −1.36/ + 2.02  0.13  −0.04/ + 0.05      1  558.446  −0.172/ + 0.072  9.08  −3.06/ + 3.49  0.06  −0.02/ + 0.04    ?  1  572.522  −0.827/ + 0.844  9.03  −2.80/ + 4.84  0.24  −0.13/ + 0.09    ?  1  595.812  −0.040/ + 0.042  12.35  −1.44/ + 2.36  0.17  −0.11/ + 0.08    ?  2  457.106  −0.046/ + 0.041  11.10  −0.49/ + 0.40  0.41  −0.07/ + 0.07      2  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    2  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    2  553.122  −0.321/ + 0.354  16.27  −2.43/ + 3.94  0.43  −0.16/ + 0.14      2  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  460.294  −0.022/ + 0.021  15.66  −0.80/ + 1.19  0.36  −0.09/ + 0.10      0  491.947  −0.013/ + 0.014  24.17  −1.29/ + 1.66  0.36  −0.10/ + 0.13      0  523.603  −0.012/ + 0.018  25.84  −1.85/ + 1.35  0.19  −0.07/ + 0.06      0  555.473  −0.133/ + 0.090  22.65  −5.44/ + 2.56  0.33  −0.16/ + 0.18      0  587.314  −0.420/ + 1.462  9.65  −2.24/ + 3.40  0.41  −0.17/ + 0.18      1  477.158  −0.031/ + 0.042  15.47  −0.18/ + 0.25  0.18  −0.02/ + 0.02      1  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    1  507.928  −0.014/ + 0.013  25.92  −0.83/ + 0.99  0.37  −0.06/ + 0.06      1  522.897  −0.002/ + 0.002  21.55  −0.75/ + 1.00  0.04  −0.01/ + 0.01      1  539.794  −0.011/ + 0.008  32.01  −1.36/ + 2.02  0.13  −0.04/ + 0.05      1  558.446  −0.172/ + 0.072  9.08  −3.06/ + 3.49  0.06  −0.02/ + 0.04    ?  1  572.522  −0.827/ + 0.844  9.03  −2.80/ + 4.84  0.24  −0.13/ + 0.09    ?  1  595.812  −0.040/ + 0.042  12.35  −1.44/ + 2.36  0.17  −0.11/ + 0.08    ?  2  457.106  −0.046/ + 0.041  11.10  −0.49/ + 0.40  0.41  −0.07/ + 0.07      2  488.837  −0.022/ + 0.019  10.70  −0.48/ + 0.64  0.26  −0.09/ + 0.07  *    2  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    2  553.122  −0.321/ + 0.354  16.27  −2.43/ + 3.94  0.43  −0.16/ + 0.14      2  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    Note . Modes which share the same peak on the power spectrum are labelled with ‘*’ marks. Modes from two stars standing too close and causing ambiguity are denoted with ‘?’ marks. View Large Table 7. Mode parameters of KIC 7107778 B. l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  501.006  −0.019/ + 0.020  14.08  −0.55/ + 1.06  0.24  −0.07/ + 0.05      0  535.615  −0.016/ + 0.017  19.47  −0.66/ + 0.94  0.16  −0.03/ + 0.04      0  570.025  −0.146/ + 0.423  27.22  −6.39/ + 8.01  0.24  −0.11/ + 0.10      0  604.609  −0.026/ + 0.022  15.92  −2.44/ + 1.51  0.23  −0.08/ + 0.11      0  639.344  −0.076/ + 0.068  13.11  −0.74/ + 0.99  0.38  −0.11/ + 0.13      1  512.368  −0.008/ + 0.012  11.26  −0.93/ + 1.19  0.09  −0.06/ + 0.04      1  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    1  547.122  −0.735/ + 2.480  11.15  −2.42/ + 4.08  0.40  −0.16/ + 0.17      1  557.829  −0.177/ + 0.168  15.37  −5.30/ + 5.61  0.39  −0.19/ + 0.17    ?  1  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    1  596.595  −0.001/ + 0.001  18.12  −1.00/ + 1.37  0.02  −0.00/ + 0.01      1  620.979  −0.049/ + 0.040  16.97  −1.01/ + 1.17  0.33  −0.07/ + 0.10      2  497.849  −0.096/ + 0.091  10.21  −0.74/ + 0.89  0.38  −0.11/ + 0.12      2  532.691  −0.342/ + 0.315  11.42  −1.28/ + 2.08  0.41  −0.14/ + 0.14      2  566.534  −0.731/ + 0.830  10.72  −2.78/ + 7.32  0.44  −0.17/ + 0.30      2  601.946  −0.106/ + 0.109  14.16  −0.93/ + 1.22  0.36  −0.10/ + 0.10      2  636.541  −0.380/ + 0.493  9.04  −1.04/ + 1.20  0.44  −0.14/ + 0.13      l  Frequency  68.3 per cent credible  Amplitude  68.3 per cent credible  Linewidth  68.3 per cent credible        (μHz)  (μHz)  (ppm)  (ppm)  (μHz)  (μHz)      0  501.006  −0.019/ + 0.020  14.08  −0.55/ + 1.06  0.24  −0.07/ + 0.05      0  535.615  −0.016/ + 0.017  19.47  −0.66/ + 0.94  0.16  −0.03/ + 0.04      0  570.025  −0.146/ + 0.423  27.22  −6.39/ + 8.01  0.24  −0.11/ + 0.10      0  604.609  −0.026/ + 0.022  15.92  −2.44/ + 1.51  0.23  −0.08/ + 0.11      0  639.344  −0.076/ + 0.068  13.11  −0.74/ + 0.99  0.38  −0.11/ + 0.13      1  512.368  −0.008/ + 0.012  11.26  −0.93/ + 1.19  0.09  −0.06/ + 0.04      1  520.851  −0.020/ + 0.024  19.66  −0.31/ + 0.64  0.19  −0.05/ + 0.04  *    1  547.122  −0.735/ + 2.480  11.15  −2.42/ + 4.08  0.40  −0.16/ + 0.17      1  557.829  −0.177/ + 0.168  15.37  −5.30/ + 5.61  0.39  −0.19/ + 0.17    ?  1  584.175  −0.393/ + 0.370  15.78  −2.33/ + 2.80  0.38  −0.13/ + 0.14  *    1  596.595  −0.001/ + 0.001  18.12  −1.00/ + 1.37  0.02  −0.00/ + 0.01      1  620.979  −0.049/ + 0.040  16.97  −1.01/ + 1.17  0.33  −0.07/ + 0.10      2  497.849  −0.096/ + 0.091  10.21  −0.74/ + 0.89  0.38  −0.11/ + 0.12      2  532.691  −0.342/ + 0.315  11.42  −1.28/ + 2.08  0.41  −0.14/ + 0.14      2  566.534  −0.731/ + 0.830  10.72  −2.78/ + 7.32  0.44  −0.17/ + 0.30      2  601.946  −0.106/ + 0.109  14.16  −0.93/ + 1.22  0.36  −0.10/ + 0.10      2  636.541  −0.380/ + 0.493  9.04  −1.04/ + 1.20  0.44  −0.14/ + 0.13      Note . Modes which share the same peak on the power spectrum are labelled with ‘*’ marks. Modes from two stars standing too close and causing ambiguity are denoted with ‘?’ marks. View Large Fig. 8 is the Hertzsprung–Russel diagram where grid and additional models are displayed with dotted black lines. Specifically, the dashed lines indicate the best model tracks from the bisecting method with star A in red and star B in blue, respectively. The star symbols denote the best models. The boxes consisting of solid lines indicate the standard deviation of effective temperature Teff and luminosity L, shown in Table 4. The left-hand, middle, and right-hand panels represent models for metallicity [Fe/H] 0.01, 0.11, and 0.21. Figure 8. View largeDownload slide Hertzsprung–Russell diagram. Grid model tracks are displayed as black dotted lines. The best models, derived through bisection, are drawn with star symbols located on evolutionary tracks (dashed lines) in red for star A and in blue for star B. The solid lines indicate the standard deviation of effective temperature Teff and luminosity L, which are shown in Table 4. The left-hand, middle, and right-hand panel represent models for metallicity [Fe/H] 0.01, 0.11, and 0.21, respectively. Figure 8. View largeDownload slide Hertzsprung–Russell diagram. Grid model tracks are displayed as black dotted lines. The best models, derived through bisection, are drawn with star symbols located on evolutionary tracks (dashed lines) in red for star A and in blue for star B. The solid lines indicate the standard deviation of effective temperature Teff and luminosity L, which are shown in Table 4. The left-hand, middle, and right-hand panel represent models for metallicity [Fe/H] 0.01, 0.11, and 0.21, respectively. 5 CONCLUSION We applied asteroseismology to a binary target KIC 7107778, and confirmed that the two stars are in the sub-giant phase. We successfully identified their l = 0 and l = 2 oscillation modes and distinguished l = 1 modes to the greatest extent. We derived stellar fundamental parameters for the two stars: $$M_{\rm A} = 1.43^{+0.08}_{-0.08}$$ M⊙, $$M_{\rm B} = 1.40^{+0.05}_{-0.06}$$ M⊙, $$R_{\rm A} = 2.94^{+0.06}_{-0.06}$$ R⊙, $$R_{\rm B} = 2.77^{+0.04}_{-0.04}$$ R⊙, $$t_{\rm A} = 3.19^{+0.60}_{-0.64}$$ Gyr, and $$t_{\rm B} = 3.26^{+0.40}_{-0.40}$$ Gyr. All the evidence suggests that they formed at the same time and possess nearly equal masses. The results yield the similarity of masses for two stars, and the best models, derived through bisection modelling, determined the mass difference as 1.42 per cent, 0.68 per cent, and 0.70 per cent, from which we conclude it is ∼1 per cent. The H–R diagram verifies this, with extremely close tracks. Table 4 indicates the sensitivity of metallicity to age, and the two stars’ ages are equal within the error. We could conclude that they formed at the same time, as is expected by binary formation that they originate from the same protostellar cloud. The KIC 7107778 system contains two extremely similar components, with fully overlapping power spectra. This is one of the few identical twin systems to be found, proving the full potential of asteroseismology. Acknowledgements We acknowledge the data from the Kepler Discovery mission, whose funding is provided by NASA's Science Mission Directorate. This work has also made use of data from the European Space Agency (ESA) mission Gaia,1 processed by the Gaia Data Processing and Analysis Consortium (DPAC2). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. 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Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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