Issue 
A&A
Volume 624, April 2019



Article Number  A140  
Number of page(s)  16  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201834095  
Published online  26 April 2019 
Two’s a crowd? Characterising the effect of photometric contamination on the extraction of the global asteroseismic parameter ν_{max} in redgiant binaries
^{1}
Instituut voor Sterrenkunde (IvS), KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
email: sanjay.sekaran@kuleuven.be
^{2}
Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain
^{3}
Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain
^{4}
Dept. of Astrophysics and Planetary Science, Villanova University, 800 Lancaster Ave, Villanova 19085, USA
Received:
15
August
2018
Accepted:
20
March
2019
Context. Theoretical scaling relations for solarlike oscillators and red giants are widely used to estimate fundamental stellar parameters. The accuracy and precision of these relations have often been questioned in the literature, with studies often utilising binarity for modelindependent validation. However, it has not been tested if the photometric effects of binarity introduce a systematic effect on the extraction of the seismic properties of the pulsating component(s).
Aims. In this paper, we present an estimation of the impact of a contaminating photometric signal with a distinct background profile on the global asteroseismic parameter ν_{max} through the analysis of synthetic redgiant binary light curves.
Methods. We generated the pulsational and granulation parameters for single red giants with different masses, radii and effective temperatures from theoretical scaling relations and use them to simulate single redgiant light curves with the characteristics of Kepler longcadence photometric data. These are subsequently blended together according to their light ratio to generate binary redgiant light curves of various configurations. We then performed a differential analysis to characterise the systematic effects of binarity on the extraction of ν_{max}.
Results. We quantify our methodological uncertainties through the analysis of single redgiant light curves, both in the presence and absence of granulation. This is used as a reference for our subsequent differential binary analysis, where we find that the ν_{max} extraction for redgiant power spectra featuring overlapping power excesses is unreliable if unconstrained priors are used. Outside of this scenario, we obtain results that are nearly identical to singlestar case.
Conclusions. We conclude that (i) the photometric effects of binarity on the extraction of ν_{max} are largely negligible as long as the power excesses of the individual components do not overlap, and that (ii) there is minimal advantage to using more than two superLorentzian components to model the granulation signal of a binary redgiant.
Key words: asteroseismology / binaries: general / stars: oscillations / stars: individual: Red Giants
© ESO 2019
1. Introduction
The revolution in redgiant asteroseismology began with the advent of the CoRoT space mission (Auvergne et al. 2009), which delivered highprecision and high dutycycle photometry. These data enabled the first detections of nonradial pulsations in redgiants (e.g. De Ridder et al. 2009; Kallinger et al. 2010) and gave rise to the birth of galactic archaeology (e.g. Miglio et al. 2013). The CoRoT data revolution had paved the way for the Kepler space mission (Borucki et al. 2010), producing data of an unprecedented precision with a fouryear nominal duty cycle. These new highprecision data have prompted numerous studies to test and improve asteroseismic scaling relations.
First introduced by Kjeldsen & Bedding (1995), these theoretical relations enable the determination of the masses and radii of stars exhibiting solarlike pulsations in a minimally modeldependent manner. Kjeldsen & Bedding (1995) had originally deduced these relations by calculating the global asteroseismic parameters known as the frequency of maximum oscillation power (ν_{max}) and the large frequency separation (Δν) from the fundamental parameters of a sample of stars with detected solarlike oscillations, including the Sun (used as the reference star).
Scaling relations take the form of a powerlaw with the effective temperature (T_{eff}), ν_{max} and Δν as inputs. It is most commonly formulated in the following manner, first introduced by Kallinger et al. (2010):
M and R are the stellar mass and radius, and the quantities with the ⊙ subscript refer to the solar reference values.
The large frequency separation (Δν) is the average difference in frequency between modes of the same spherical degree (ℓ) and consecutive radial orders (n). This quantity was first defined theoretically by Tassoul (1980) as a consequence of the asymptotic approximation. The asymptotic large frequency separation (Δν_{as}), which is directly proportional to the mean density () of the star, is related to the sound speed (c_{s}) and radius (R) of the star according to the following equation:
However, the observational large frequency separation (Δν_{obs}) is typically determined from the mean difference between frequencies of the modes (typically radial) of the same ℓ but different n (i.e. Δν_{obs} = < ν_{n, ℓ+1} − ν_{n, ℓ}>). This necessarily implies that the observational and asymptotic Δν are not equivalent, although the differences are small for red giants (see Mosser et al. 2013 for a detailed discussion on the observed vs asymptotic Δν).
The frequency of maximum power (ν_{max}) is an observational characterisation of the pulsational envelope of the star and was first used by Brown et al. (1991) to characterise the pulsations of Procyon, referring to it as an “envelope peak”. It it most commonly used to refer to the frequency centre of the Gaussian that is used to approximate the power excess displayed by all solarlike pulsators. Interestingly, there have been a few studies in the literature indicating that a Gaussian may not be the most optimal function to characterise the power excess for certain stars (e.g. Procyon, see Arentoft et al. 2008 for more details). However, the Gaussian still remains the most ubiquitouslyused function to characterise the oscillation power excess for solarlike oscillators.
Brown et al. (1991), and later Kjeldsen & Bedding (1995), theorised that ν_{max} was related to the acoustic wave cutoff frequency in an isothermal atmosphere, and therefore scales with the surface gravity (g) and the effective temperature (T_{eff}) according to the following approximation:
While studies such as Belkacem et al. (2011, 2013) provide a full theoretical formulation for ν_{max}, it has yet to be adopted ubiquitously in the current literature and is therefore mostly regarded as an observational quantity.
The accuracy of these scaling relations has often been tested in the literature, most commonly through ensemble studies (e.g. Chaplin et al. 2011, 2014; Huber et al. 2011; Kallinger et al. 2010, 2014). These studies are complemented by the TychoGaia astrometric solution (TGAS) parallaxes (Salgado et al. 2017), an offshoot of the data products provided by the recently launched Gaia space mission (Gaia Collaboration 2016). These parallaxes enable independent radii measurements that can be confronted to the scaling relation values, as demonstrated in De Ridder et al. (2016) and Huber et al. (2017).
Another method to test and verify scaling relations is through the confrontation of scaling relationderived masses and radii with those extracted from eclipsing binary dynamics (e.g. Frandsen et al. 2013; Gaulme et al. 2016; Brogaard et al. 2018; Themeßl et al. 2018), or otherwise constrained from spectroscopic binaries (e.g. Beck et al. 2018a). A notable example of the advantages offered by binarity is presented in Bellinger et al. (2017), where the application of binary constraints allowed for the successful execution of inversion techniques in the modelling of 16 Cyg A and B. The common occurrence of binaries with solarlike pulsating components has allowed for the critical evaluation of evolutionary models (Mathur et al. 2013; Appourchaux et al. 2015; Beck et al. 2014, 2018a; White et al. 2017; Li et al. 2018) and tidal theory (Beck et al. 2018b). The far lesscommon occurrence of eclipsing binaries with a solarlike pulsating red giant, however, requires more attention to detail to fully exploit the potential provided by dynamical mass and radius determinations.
The task of comparing asteroseismic and dynamical parameters is a complex one, with a number of requirements and drawbacks: For example, (i) long timeseries of both photometric, and highresolution, high signaltonoise spectroscopic observations (implying that said objects must be bright enough to be observed from the ground); (ii) small sample sizes (few objects display eclipses of sufficient depth for analysis); and (iii) complicated multistep analyses involving binary and asteroseismic modelling (see e.g. Rawls 2016). However, precision and accuracy (∼1% uncertainty; Torres et al. 2010) of the masses and radii derived from eclipse modelling provide highly stringent constraints that scan be used to confront redgiant asteroseismic results, as exemplified by studies such as Gaulme et al. (2013, 2014, 2016), Beck et al. (2014, 2018a), Huber (2015), Brogaard et al. (2016, 2018) and Themeßl et al. (2018).
Of particular concern are the claims of Gaulme et al. (2016), who had found an average overestimation of ∼15%/5% of redgiant mass/radius extracted from scaling relations when confronted with the mass/radius extracted from eclipsing binary dynamics. The departure of observationallyderived redgiant global asteroseismic parameters (ν_{max} and Δν) from theoretical ones is typically attributed to deficiencies in the scaling relations themselves. This phenomenon was most notably discussed in Mosser et al. (2013), who had proposed empirical corrections to account for this discrepancy. Indeed, many attempts to propose corrections to scaling relations have been published (e.g. White et al. 2011; Miglio et al. 2012; Guggenberger et al. 2016, 2017; Sharma et al. 2016; Rodrigues et al. 2017), typically involving new reference values for ν_{max, ⊙} and Δν_{⊙}, or empirical or modeldependent corrections to the derived ν_{max} and Δν, or both. In fact, the aforementioned studies of Brogaard et al. (2018) and Themeßl et al. (2018) also attempt to reconcile the discrepancy between eclipsing binaryanalysis derived and scalingrelation derived parameters through similar corrections. However, there is still no agreement on the exact form of the corrections to the scaling relations.
Most recently, nonlinear scaling relations that claim to solve the eclipsing binaryanalysis derived and scalingrelation derived parameter discrepancy have been proposed by Kallinger et al. (2018), based on the systems studied by Gaulme et al. (2016). However, this study, as well as the studies proposing scalingrelation corrections, still utilise the global asteroseismic parameter ν_{max} as an input parameter. So far, there have not been any studies that have attempted to quantify, in general for a wide range of stars, the impact (if any exists) that the light contribution of each component in a binary has on the extraction of these parameters.
This question can then be further generalised: what is the impact of a contaminating photometric signal (whether a background object or companion) with a distinct background profile on the extraction of global asteroseismic parameters? This knowledge will be of particular importance in the context of the recentlylaunched TESS space mission (Ricker et al. 2015), where the larger pixel sizes imply an increased likelihood of contamination from a background object. In this study, we attempt to answer the aforementioned question for the context of redgiant/redgiant binaries. This synergises with one of the suggestions in Beck et al. (2018a): reformulating scaling relations in terms of mass ratios and light factors in order to exploit the typical outputs of binary analysis. To that end, we simulated the light curves of redgiant/redgiant binaries, with the characteristics of Kepler longcadence photometric data, and extracted the values of the global asteroseismic parameter ν_{max} from our synthetic light curves, comparing them to the input values calculated from theoretical scaling relations.
Section 2 describes the methodology used in our study. Section 3 details the results of our single redgiant simulations, and Sect. 4 details the results of the binary redgiant simulations. We end by discussing our results and presenting our conclusions in Sect. 5.
2. Methodology
2.1. Redgiant signal components
The signals that constitute a typical red giant (RG) power spectral density (PSD) that is generated from Kepler longcadence photometric data can be separated into three distinct categories: (i) stellar pulsation, (ii) stellar background, and (iii) noise. The proper extraction of individual pulsational frequencies requires, and is strongly dependent on, the proper fitting and removal of the stellar background and noise.
The noise in the PSD is comprised of a frequencyindependent (white) photonshot noise component and a frequencydependent (coloured) instrumental noise component, where the instrumental noise is thought to be a result of aperture optimisation during the pointing of Kepler. This can be represented by the following equation:
P_{noise}(ν) is the PSD of the noise signal, W is the white noise, and is the coloured instrumental noise represented by a Lorentzian profile, with α and β representing the amplitude and turnover frequency of the instrumental noise. Kallinger et al. (2014) had shown that in practice, the coloured noise component is negligible, and therefore the noise can be assumed to be completely white.
The background signal, however, is not wellunderstood. The general consensus is that the background consists of frequencydependent (coloured) components that have been surmised to be the result of granulation occurring at different length and timescales. The first attempt to model this background signal was performed by Harvey (1985), who used a superLorentzian (or Harvey) profile that differed from a regular Lorentzian in that it had a larger exponent in the denominator. Since then, there have been several attempts to more accurately represent the background. Most notably, Kallinger et al. (2014) tested several background models using Kepler data and concluded that, at the data quality level provided by Kepler for a sufficiently bright target (i.e. brighter than K_{p} = 12, where K_{p} is the Kepler magnitude), a background model consisting of two superLorentzian profiles of the form
best reproduced the data. P_{bg}(ν) is the PSD of the background signal, and the sum of the two superLorentzian profiles represent granulation occurring at different timescales, each with their respective amplitudes (a_{i}) and turnover frequencies (b_{i}), and the normalisation factor^{1} . η^{2}(ν) is the sincsquared attenuation factor that accounts for the suppression of the amplitude of the oscillations and background occurring close to the Nyquist frequency due to the finite integration time of Kepler (García et al. 2011).
The pulsational signal present in RGs is understood to be a signature of stochasticallyexcited pulsations, with frequencydependent driving and damping rates (see Samadi et al. 2015 for a full description). This results in the formation of modes, which are typically described by spherical harmonics, around discrete frequencies dictated by the boundary conditions of stellar structure. The modes that result from this frequencydependent driving and damping can be modelled as damped harmonic oscillators, which manifest as simple Lorentzian functions in the Fourier domain. Therefore, we can model the PSD of this pulsational signal with a sum of m Lorentzian functions:
P_{puls}(ν) is the PSD of the pulsational signal, and A_{n}, Γ_{n}, and ν_{0 (n)} are the amplitude, width and central frequency of the nth Lorentzian profile respectively. The pulsational signal appears as a power excess in the PSD of a RG, and as mentioned is approximated by a Gaussian centred at ν_{max}, with an amplitude and width of A_{puls} and σ_{puls} respectively. The PSD of the Gaussian that represents the pulsational envelope (P_{env}) can be calculated using the following equation:
The standard practice in the literature is to fit the noise and background together with this Gaussian approximation of the pulsational power excess in order to constrain the morphology (or slope) of the background. The background fit is then removed from the PSD, and the individual peaks extracted by fitting Lorentzians to the residual signal.
2.2. Light curve synthesis
To synthesise the light curve of a RG, we need to simulate both the pulsational signal and the background signal. To simulate the pulsational signal, we first have to compile sets of template modes with known frequencies, amplitudes and mode lifetimes.
To that end, we created a grid of stellar models of RGs using the stellar evolution code MESA (revision 10348; Paxton et al. 2011, 2018). We adopted an exponential diffusive overshooting scheme (Herwig 2000), and a predictive mixing scheme as described in Paxton et al. (2018), with abundances based primarily on the work of Asplund et al. (2009), with morecurrent values for certain elements taken from Nieva & Przybilla (2012) and Przybilla et al. (2013). The full contents of the inlist used to generate our evolutionary models is specified in Appendix A. The models were made to evolve until the ignition of corehelium burning, resulting in a grid of RG models with masses of 1.0 M_{⊙}–2.0 M_{⊙} in steps of 0.2 M_{⊙}, and a ν_{max} of approximately 20 μHz–200 μHz in steps of 10 μHz for each mass value. These ν_{max} values were calculated from the M and R outputs of MESA by using the following equations (a result of Eqs. (3) and (4)) as follows:
The ν_{max} range was chosen in such a way as to cover a large portion of the red giant branch (RGB) and an upper limit of 200 μHz for ν_{max} was set to ensure that the pulsational power excesses were well within the Nyquist frequency threshold of 284 μHz for Kepler longcadence data. The ranges of the M, R, T_{eff} and ν_{max} values of our grid are listed in Table 1. This grid of RG models was then used as inputs for the stellar pulsation code GYRE (revision 5.1; Townsend & Teitler 2013), which we used to compute nonrotating pulsational models in the adiabatic framework. These pulsational models comprise the frequencies and normalised mode inertias of the ℓ = 0, 1, 2 and 3 modes for each MESA evolutionary model. The contents of the inlist used to generate our pulsational models are specified in Appendix B.
Ranges of M, R, T_{eff} and ν_{max} of the RG models used for the creation of our synthetic single and binaryRG light curves.
The corresponding mode amplitudes for each frequency were then generated using the following steps, similar to the methodology detailed in Sect. 4.3 of Kallinger et al. (2014) for the creation of synthetic RG data:
1. Initial amplitude ratios (A_{ratio}) between 0 and 1 were assigned to each frequency (ν_{i}) by using a Gaussian centred on ν_{max} with a standard deviation^{2} of 1.5Δν. This is represented by the following equation:
2. These amplitude ratios were then modulated by mode visibilities of 1.00, 1.35, 0.64 and 0.071 for the ℓ = 0, 1, 2 and 3 modes respectively, as reported by Mosser et al. (2012) for RGB stars.
3. The amplitude ratios of the ℓ = 1, 2 and 3 modes were then further modulated by the inverse of the local relative mode inertia^{3}.
4. To obtain realistic pulsational amplitudes, we refer to the characterisation of pulsational amplitudes reported in Corsaro et al. (2013), where different amplitude scalingrelation models were investigated in a Bayesian framework. We calculated the total bolometric pulsational amplitude A_{puls, bol} from the scaling relation given in Eq. (19) of Corsaro et al. (2013):
The values for the coefficients and exponents in the equation are as follows: d = e^{0.45}, p = 0.602, q = 1.31, and r = 5.87. These are taken from the longcadence expectation values of model ℳ_{4,d} of Corsaro et al. (2013), which they found to have the best Bayesian Information Criterion (BIC) value in their analysis.
5. Amplitudes (in ppm) for each frequency can then obtained by taking the product of the modulated amplitude ratios and A_{puls, bol}.
We generated mode lifetimes for each frequency based on the work of Vrard et al. (2018), who discussed both global (variation with stellar mass and temperature across all stars) and local (variation with frequency within a single star) modewidth variation in their study. Based on their results, we generated the modewidth at ν_{max} for each star by using the following equation:
Γ(ν_{max}) is the modewidth at ν_{max}, with the exponents of M and T_{eff} taken from the characterisation of RGB stars by Vrard et al. (2018). n is a multiplicative constant used to ensure that the modewidths occupy a range of 0.05–0.17 μHz, which is well within the distribution of modewidths displayed in Fig. 3 of Vrard et al. (2018).
Vrard et al. (2018) also investigated local modewidth variation for the radial modes of RGs as per the work of Appourchaux et al. (2014) for subgiants and Lund et al. (2017) for dwarfs. While they found a general increase in modewidth with frequency, they did not find the modewidth “dip” around ν_{max} (although they did find evidence of a reduction in the rate of increase around ν_{max}). Although their results suffer from large uncertainties, we note a general exponential morphology of the variation of modewidths with frequency. As such, we chose to adopt an exponential scaling of local modewidth with frequency as follows:
Γ(ν) is the modewidth at frequency ν, and z = 3.0 is a constant designed to keep the modewidth increase with frequency within the ranges reported by Vrard et al. (2018).
The frequencies, amplitudes and modewidths (see Fig. 1 for an example of a pulsation spectrum) are then fed to a firstorder autoregressive function to generate a synthetic timeseries, following the methodology laid out in De Ridder et al. (2006). This procedure transforms the “perfect” comb of our input pulsation spectrum into a morerealistic representation of a RG pulsation spectrum (cf. Fig. D.1). The time sampling of this function was setup in such a way as to produce a timeseries with the same timestamps as a full fouryear Kepler long cadence light curve with no gaps.
Fig. 1.
Example of a synthetic RG pulsation spectrum for a star with ν_{max} ∼ 100 μHz, used as an input to generates synthetic pulsational timeseries for the RG light curves. The ℓ = 0, 1, 2, and 3 modes are represented by Lorentzians and colourcoded as displayed in the legend. The presence of multiplets, particularly around the ℓ = 1 modes, correspond to mixed modes with the same pmode radial order but different gmode radial order. 
To simulate the granulation signal for our light curves, we first generated superLorentzian profiles as per Eq. (6). The granulation amplitudes (a_{i}) and turnover frequencies (b_{i}) were calculated from the scaling relations and coefficients listed in Table 2 of Kallinger et al. (2014), using the M values from MESA and our calculated ν_{max} values:
The coefficients k, s, and t for a_{1, 2} and b_{1, 2} are listed in Table 2. As the granulation signals of observational PSDs are often very noisy, we added a fixed amount of Gaussian noise to each granulation profile to make them more realistic.
Coefficients k, s, and t for the granulation amplitudes (a_{1, 2}) and turnover frequencies (b_{1, 2}).
To mimic the procedure that we had performed for the pulsation spectrum, we chose to generate synthetic timeseries for our granulation spectra. While it is possible to transform a simple Harvey profile into a timeseries using an autoregressive function, there is currently no analogue that can transform our granulation profiles into a timeseries. We therefore performed inverse Fourier transforms (iFTs) of our granulation PSDs in order to generate them. Similar to the pulsational timeseries, the iFTs are calculated in such a way that the resulting timeseries has the same timestamps as a full Kepler longcadence light curve.
We chose to include a luminositydependent white noise component as per the work of Pande et al. (2018), who characterised the variation of white noise with magnitude for approximately 2100 stars from shortcadence Kepler data. They reported an almostlinear variation of the logarithm of white noise with magnitude. We adapted this result and chose to scale the logarithm of white noise with the logarithm of luminosity as per the following equation:
L is the luminosity of star, which is one of the outputs of MESA, and c is a constant designed to ensure that the range of white noise amplitudes are within that of the stars in the 8–12 Kepler magnitude range as reported by Pande et al. (2018). This is within the range in which a two superLorentziancomponent model can accurately describe the granulation signal of a Kepler RG, as detailed in Kallinger et al. (2014). The implicit assumption being made here is that these stars are all at the same distance, which we adopt in order to investigate the effect of white noise on the extraction of ν_{max}. Similar to the granulation signal, iFTs were performed in order to produce white noise timeseries.
The three timeseries are then added to produce the final synthetic light curve for a single RG of mass M, radius R and effective temperature T_{eff}.
2.3. Synthetic PSD fitting
We fit the synthetic PSDs, generated by taking the Fourier transforms of our synthetic light curves, as per what is done in the literature (see Sect. 2.1). This fitting function, essentially a sum of the pulsational, granulation, and noise model functions, takes the following form:
The terms in the function echo those in Eqs. (5), (6) and (8): P_{model}(ν) is the PSD of the model; W is the white noise; represents the Gaussian power excess, with the amplitude, width and central frequency represented by A_{puls}, σ_{puls} and ν_{max} respectively; and are the two superLorentzian components representing the granulation signal, with the amplitude and characteristic timescale of the ith superLorentzian represented by A_{gran, i} and τ_{gran, i} respectively.
We perform our fit in a Bayesian framework using the Markov chain Monte Carlo (MCMC) routine emcee (ForemanMackey et al. 2013). emcee makes use of an ensemble sampler with numerous Markov chains to efficiently sample the posterior probability distribution as described by Bayes’ theorem:
P(H_{i}I) is known as the prior probability, which is the probability of the hypothesis H_{i} with knowledge of prior information I, but in the absence of the data D. P(H_{i}D, I) is known as the posterior probability, which is the probability of the hypothesis with knowledge of both I and D. The quantities P(DH_{i}, I) and P(DI) are the likelihood of H_{i} and the global likelihood of all hypotheses respectively.
For numerical simplicity, we evaluate Bayes’ theorem in logspace, where our loglikelihood function (ℒ) is
P_{input}(ν) is the PSD of the input light curve that has been smoothed by a Gaussian filter. We use a smoothed PSD to mitigate the impact of noise in the background profile on the fit. We adopt a convergence criterion based on the integrated autocorrelation time (τ_{autocorr}) as per Goodman & Weare (2010). We compute τ_{autocorr} for every 100 iterations and consider the MCMC routine to have achieved convergence if τ_{autocorr} changes by less than 1% between two successive computations.
After our MCMC routines have achieved convergence, we extract our fit parameters and determine ν_{max, fit} from the marginalised posterior distributions, comparing them with our input ν_{max, scaling} from scaling relations. An example fit of the PSD of one of our synthetic light curves is shown in Fig. 2.
Fig. 2.
Example of a singleRG PSD fit. The grey line is the unsmoothed PSD, the black line is the smoothed PSD, the red line represents the overall fit, the dashed green line represents the Gaussian used to fit the pulsational power excess, the blue dotted line represents the white noise, and the blue dashed lines represent the superLorentzians used to fit the granulation signal. 
2.4. Redgiant binary simulation setup
To create the light curve of a RG/RG binary, it was necessary to deal with the pulsations and granulation, and white noise separately. We first generated the white noise signal of the binary light curve according to Eq. (5) by using the total luminosity of the individual components as the input. We then scaled the pulsational and granulation signals of the individual components by their individual light contribution and added them together with the white noise according to the following equation:
F_{binary} is the flux of the binary light curve, and L_{A}, F_{puls, A}, F_{gran, A}, L_{B}, F_{puls, B}, and F_{gran, B} are the luminosities, and the pulsational and granulation fluxes of each binary component (A and B) respectively. Implicit in this methodology is that all orbital and tidal effects are ignored^{4}.
We created two sets of light curves with each set containing a specific pulsational configuration: (i) one pulsating component (e.g. Gaulme et al. 2014; Beck et al. 2014), and (ii) two pulsating components (e.g. Rawls et al. 2016; Beck et al. 2018a). For each set of binary light curves, we tested the impact of different granulation models by performing fits with (i) two superLorentzians (as in the singleRG case), and with (ii) four superLorentzians (two superLorentzians for each component), to model the granulation signal of each binary light curve.
3. Single redgiant simulations and results
To ensure the selfconsistency of our methodology and to establish a baseline for comparison with our subsequent binary results, we first simulated the light curves and fit the PSDs of the 114 single RGs that we created using our methodology. We investigated the effect of the granulation on the extraction of ν_{max} by performing one iteration of fits with only the pulsational signal (e.g. Fig. D.1), and another with both the pulsational and granulation signals (e.g. Fig. D.2) included in the light curves of the RGs. This is demonstrated in Fig. 3, showcasing the percentage difference between the fit (ν_{max, fit}) and the scaling relation (ν_{max, scaling}) input values. The mean percentage difference and the 1σ scatter of the ν_{max, fit} − ν_{max, scaling} values, and the mean precision of the extraction of ν_{max, fit} are detailed in Table 3.
Fig. 3.
Comparison of the ν_{max} values extracted from our synthetic singleRG PSDs. We tested synthetic light curves containing only a pulsational signal (triangles), and both a pulsational and a granulation signal (circles). The vertical axis corresponds to the scaling relation (ν_{max, scaling}) input values of the light curves, and the horizontal axis corresponds to the percentage difference between the fit (ν_{max, fit}) and the scaling relation (ν_{max, scaling}) values. The vertical dashed line represents the zeropoint of the difference between ν_{max, fit} and ν_{max, scaling}. The dashdotted lines represent the 1σ level about the mean scatter of the datapoints corresponding to the pulsational signal (red), and both the pulsational and granulation signals (green). The horizontal dotted lines connect the datapoints corresponding to the same RG model. The error bars correspond to the 68% Bayesian credible intervals of the marginalised posterior distributions of the fit parameters. The symbols are colourcoded according the logarithm of the white noise (log W) included in the light curves and PSDs. 
Mean percentage difference, 1σ scatter, and precision of the ν_{max} values extracted from our singleRG PSDs.
It can be seen that in general, for the iterations both with and without the inclusion of granulation in the PSDs, there is a systematic underestimation of the ν_{max, fit} compared to the ν_{max, scaling} values. This was the opposite of the expected result, which was a systematic overestimation due to an excess of power in the frequency domain above ν_{max, scaling} as the modewidths increase with increasing frequency.
However, there is an additional source of asymmetry in the input pulsational spectra: uneven distribution of modes and mode amplitudes. For example, in the input pulsation spectrum displayed in Fig. 1 (which correponds to a ν_{max, scaling} ∼ 100 μHz), there is a larger number of strong peaks in the frequency domain below ν_{max, scaling} than above it, particularly from the ℓ = 1 mixedmode multiplets. This distribution is a result of the mode inertia distribution (e.g. Fig. C.1) for the frequencies output by GYRE, which directly affects the amplitudes of the modes that were used in the input. This result indicates that the input pulsational spectra generally have more power in the frequency domain below ν_{max, scaling} than above it, and is the source of our systematic uncertainty. To confirm that this is indeed the reason for the obtained systematic effect, we performed a test by artificially symmetrising our pulsational inputs with respect to ν_{max, scaling} and repeating the same fitting procedure. We found that the systematic offset between the ν_{max, fit} and ν_{max, scaling} was no longer present, confirming our suspicion that its origins was indeed in the uneven distribution of modes and amplitudes.
Another result from these simulations is the difference in the ν_{max, fit} values extracted from PSDs with and without granulation. The ν_{max, fit} values for the iteration with granulation are slightly more underestimated on average compared to the ν_{max, scaling} values, seemingly exacerbating the asymmetry in the power excesses. In addition, the scatter of the ν_{max, fit} is greater, and the extraction of the ν_{max, fit} values is less precise for the iteration with granulation. These phenomena can be explained by the stochastic nature of the granulation signal and the degenerate nature of the background fit, where a smooth function (the two superLorentzians) are being fit to a rather noisy background. The effect of the inclusion and the fitting of the granulation seems to be rather unpredictable as the degree of underestimation of ν_{max, fit} is decreased in some cases when comparing the results with and without granulation.
It can also be seen in Fig. 3 that the precision of the extraction seems to decrease as the ν_{max, scaling} values decrease, and that this decrease seems to be correlated with a decrease in the amount of white noise. However, this is simply due to the fact that we represented our results in terms of percentages: the absolute errors for the different stars within a single iteration (i.e. whether with or without granulation) were very similar and similarly uncorrelated with the amount of white noise in the PSDs.
In addition, there seems to be a marked increase in the percentage difference and scatter of the ν_{max, fit} values, for the iterations with and without granulation, as the ν_{max, scaling} values decrease below ∼100 μHz. While asymmetrical mode distributions and our representation of the errors in percentages are partially to blame, there seems to be a trend in the scatter below ∼100 μHz that requires additional explanation. We posit that this result is a consequence of the large changes in the gradient of the granulation morphology (i.e. the “flattening” of the background slope) below ∼50 μHz (see e.g. Fig. D.2). Due to the fact that the pulsational power excesses lie partially on top of this “flattened” portion of the granulation morphology, the extraction of ν_{max, fit} between the two iterations shows greater deviance.
Overall, we observe systematic errors of the order of 1% in ν_{max}, 3% in M and 1% in R, with an intrinsic scatter of the order of 0.5% in ν_{max}, 1.5% in M and 0.5% in R. For all three parameters, the combined errors are smaller or of the order of the typical uncertainties of 4%/2% in mass/radius reported for the scaling relations in literature (e.g. Pérez Hernández et al. 2016; Kallinger et al. 2018).
The systematic errors and intrinsic scatter of the masses and radii follow from the scaling relations (Eqs. (1) and (2)) as expected, with ΔM = 3 ⋅ d(log ν_{max}) and ΔR = d(log ν_{max}). As such, in this and in the subsequent sections, we only present the mean percentage difference, 1σ scatter, and the mean precision of the ν_{max} values extracted from our PSDs, as those of the masses and radii can effectively be calculated from the ν_{max} results.
4. Redgiant binary simulations and results
We used a subset of the light curves generated using the methodology laid out in Sect. 2.2 for the creation of our binary grid, ensuring that the subset was relatively representative of the entire grid in terms of M, R, T_{eff} and ν_{max}. This was done in order to keep the binary grid, which comprises combinations of each star with every other star, relatively manageable in size. The subset comprises 30 stars, and their M, R, and T_{eff} distributions are displayed in Fig. 4.
Fig. 4.
M, R, and T_{eff} values of the individual components (highlighted in dark blue) used to create our binary grid. 
We therefore generated a total of two sets of 435 (i.e. ) binary light curves, with one set consisting of binaries with only one pulsating component, and the other consisting of binaries with both components pulsating. These were fit using different granulation model configurations (two and four superLorentzians), and we perform a differential analysis by comparing the ν_{max} values extracted from our binaryRG PSDs (ν_{max, binary}) with the maximum likelihood estimates of the ν_{max} values extracted from our singleRG PSDs (ν_{max, single}).
4.1. Results of RG binaries with two pulsating components
Figures 5 and 6 show the ν_{max} extraction results for a two superLorentzian granulation model and a four superLorentzian granulation model respectively, where both components (Star A and Star B) in the RG binary are pulsating. The mean percentage difference and the 1σ scatter of the ν_{max, binary} − ν_{max, single} values, and the mean precision of the ν_{max, binary} extraction are detailed in Table 4.
Fig. 5.
Comparison of the ν_{max} values of each component (Star A, left panel; and Star B, right panel) extracted from our synthetic RGbinary light curves where both components are pulsating, using a two superLorentzian granulation model. The vertical axis corresponds to the difference between the singleRG reference values for Star A (ν_{max, single (Star A)}) and Star B (ν_{max, single (Star B)}). The horizontal axis corresponds to the difference between the binary (ν_{max, binary}) and the singleRG (ν_{max, single}) reference values. The vertical dashed line represents the zeropoint of the difference between ν_{max, binary} and ν_{max, single}. The dashdotted green lines represent the interpolated 1 − σ level about the mean scatter of the datapoints grouped into 10 μHz bins along the vertical axis. The horizontal dashed line represents the zeropoint of ν_{max, single (Star A)} − ν_{max, single (Star B)}. The error bars correspond to the 68% Bayesian credible intervals of the marginalised posterior distributions of the binary fit parameters. The symbols are colourcoded according the percentage light contribution of each component to the binary light curve. 
Fig. 6.
Comparison of the ν_{max} values of each component (Star A, left panel; and Star B, right panel) extracted from our synthetic RGbinary light curves where both components are pulsating, using a four superLorentzian granulation model. The axes, symbols, error bars and colourcoding is the same as in Fig. 5. 
Mean percentage difference, 1σ scatter, and precision of the ν_{max, binary} values extracted from our RGbinary PSDs, using either two or four superLorentzians to fit the granulation signal.
The most obvious feature in Figs. 5 and 6 is the large scatter in the central region of the plot, which is where the ν_{max} values for Star A (ν_{max, single (Star A)}) and Star B (ν_{max, single (Star B)}) are very similar. This scatter is a result of fitting the binary star PSDs where the power excesses of each component overlap (e.g. Fig. D.3). This necessarily results in degeneracies during the fitting process that are difficult to alleviate without tight constraints on the prior distributions (we use relatively unconstrained priors). In addition, approximately 20% of the RG/RG binaries with overlapping power excesses displayed nonGaussian posterior distributions of ν_{max} values. This results in the asymmetric, and often large, error bars for the datapoints in the region around the zeropoint value of ν_{max, single (Star A)} − ν_{max, single (Star B)}.
We also observe a marked increase in the scatter of ν_{max, binary} − ν_{max, single} as the light contribution increases (see zoomed insets in Fig. 5). This is once again a consequence of representing our results in percentages, and is also observed in the singleRG case (see Sect. 3): The component with the higher light contribution tends to have the lower ν_{max} (as ν_{max} is inversely proportional to L), and therefore similar absolute ν_{max, binary} − ν_{max, single} values are larger in terms of percentage at lower ν_{max, single} values.
Our results shown in Table 4 indicate that there is no appreciable systematic offset in our ν_{max, binary} values when compared to our ν_{max, single} values. The small residual systematic offsets that remain are certainly well below the 1σ scatter reported in the singleRG case (cf. Table 3). However, the 1σ scatter of the ν_{max, binary} values is much higher, and the mean precision is much lower than in the singlestar case. It must be noted that these results represent the global properties of the entire grid, and are therefore largely biased by the datapoints corresponding to overlapping power excesses, where a large 1σ scatter and low precision are expected. For the cases where the power excesses are well separated (e.g. Fig. D.4), we clearly see a much smaller scatter that is quantitatively comparable to the single star case discussed in the Sect. 3.
Another interesting observation is that there is little difference in the overall results of the two and four superLorentzian component fits. This can be attributed once again to the degenerate nature of the background fit: adding additional superLorentzians does not seem to improve the quality of the fit. This is in agreement with the conclusions of Kallinger et al. (2014) and is demonstrated in Fig. 7, which shows the input superLorentzians for one of our synthetic PSDs compared with those derived from the maximumlikelihood values for the two and four superLorentzian component granulation models. The background fit is so degenerate that (i) two superLorentzians can still adequately model the background (verifying the approach of Beck et al. 2018a), and (ii) the input parameters are not recovered using the four superLorentzian model.
Fig. 7.
Comparison of the input granulation signal with those from the maximumlikelihood values of the two and four superLorentzian granulation model fits, for one of our binaryRG PSDs where both components are pulsating. The superLorentzians used as inputs are represented by the solid red curves, with the two and four superLorentzians fits represented by the dark blue and bluegreen dashdotted lines respectively. 
Overall, we observe an intrinsic scatter of the order of 2% in ν_{max}, corresponding to a scatter of 6% in M and 6% in R. We once again stress that the larger scatter in ν_{max} is a direct consequence of the additional fitting degeneracy under the condition of the overlapping power excesses of the two pulsating components: Our uncertainties are otherwise comparable to the singleRG case and do not exceed the typical uncertainties for the scaling relations reported in the literature (cf. Sect. 3).
4.2. Results of RG binaries with one pulsating component
Figure 8 shows the ν_{max} extraction results for the two and four superLorentzian component granulation models, where only one of the two components (Star A) in the RG binary is pulsating (e.g. Fig. D.5). The mean percentage difference and the 1σ scatter of the ν_{max, binary} − ν_{max, single} values, and the mean precision of the ν_{max, binary} extraction are also detailed in the last column of Table 4.
Fig. 8.
Comparison of the ν_{max} values for a two superLorentzian (left panel) and a four superLorentzian (right panel) granulation model, extracted from our synthetic RGbinary light curves where only one component (Star A) is pulsating. The axes, symbols, error bars and colourcoding is the same as in Fig. 5. 
Once again, our results show that there is no appreciable systematic offset in our ν_{max, binary} values when compared to our ν_{max, single} values. In fact, the results that we had obtained were very similar to that of the singleRG case: The 1σ scatter and precision of the ν_{max} extraction is almost identical to that of the singleRG case (detailed in Table 3), with perhaps a slight decrease in mean precision (0.3% vs. 0.2%). We do not observe the large 1σ scatter that was observed in the doublepulsator case, as was expected due to the absence of a second power excess that would add significant degeneracy in the overlapping case. We do, however, observe the increase in the scatter of ν_{max, binary} − ν_{max, single} as the light contribution increases (see zoomed insets in Fig. 8) as per the doublepulsator case, which is, as mentioned, due to the representation of our results in percentages.
Similar to our binaryRG results, we once again observe very small differences between the different granulation model configurations: a consequence of the degeneracy of the background fit. This is demonstrated in Fig. 9, which shows the input superLorentzians for one of our singlepulsator synthetic PSDs compared with those derived from the maximumlikelihood values for the two and four superLorentzian component granulation models. Once again, we observe that the background fit is so degenerate that (i) two superLorentzians can still adequately model the background, and (ii) the input parameters are not recovered using the four superLorentzian model. This is in line with the conclusions of Kallinger et al. (2014) that there is a limit to the number of granulation components that can be uniquely modelled for a sufficientlybright (K_{p} < 12) RG.
Fig. 9.
Comparison of the input granulation signal with those from the maximumlikelihood values of the two and four superLorentzian granulation model fits, for one of our binaryRG PSDs where only one component is pulsating. The superLorentzians used as inputs are represented by the solid red curves, with the two and four superLorentzian fits represented by the dark blue and bluegreen dashdotted lines respectively. 
Overall, we observe an intrinsic scatter of the order of 0.5% in ν_{max}, corresponding to a scatter of 1.5% in M and 0.5% in R, which is compatible with the singleRG case. It should be noted that the uncertainties typically reported for the scaling relations are larger than, or of the order of, the combined errors reported here.
4.3. Comparison between the different granulation model configurations
We performed a comparison of the two different granulation model configurations by taking the absolute difference of ν_{max, binary} values extracted from the fits using each type of granulation model, for light curves with two pulsating components and a single pulsating component. The left and middle panels of Fig. 10 show the difference in ν_{max, binary} for the RGbinary light curves where both components (Star A and Star B) are pulsating, and the right panel shows the difference in ν_{max, binary} for the RGbinary light curves where only one component (Star A) is pulsating. All of these results are detailed in the bottom row of Table 4, where we show the mean percentage difference and the 1σ scatter of the ν_{max, binary (2SL)} − ν_{max, binary (4SL)} values.
Fig. 10.
Difference in the ν_{max} values extracted from the fits using a two superLorentzian and a four superLorentzian granulation model. Left and middle panels: difference in ν_{max, fit} for the RGbinary light curves where both components (Star A and Star B) are pulsating. Right panel: difference in ν_{max, fit} for the RGbinary light curves where only one component (Star A) is pulsating. The axes, symbols, error bars and colourcoding is the same as in Fig. 5, except that the horizontal axis corresponds to the absolute percentage difference between the ν_{max, fit} values for the two superLorentzian (2SL) and four superLorentzian (4SL) cases. 
Once again, the greatest percentage difference between the ν_{max, fit} values extracted from fits using the two and four superLorentzian granulation models is in the region around the zeropoint value of ν_{max, single (Star A)} − ν_{max, single (Star B)}. As mentioned, the ν_{max} extraction from PSDs with overlapping power excesses is generally unreliable, and the different granulation models function as an additional confounding factor in these cases. It can also be seen that the differences between the different granulation model configurations increases as with increasing light contribution, which is once again a consequence of expressing our results in percentages.
In general, it can be seen that the difference between the ν_{max} values for the two and four superLorentzian granulation model configurations are small outside of the case of overlapping power excesses, with differences and scatter of the order of 0.5% in ν_{max}, corresponding to 1.5% in M and 0.5% in R.
5. Discussion and conclusions
This paper details the results of our study, where we attempted to estimate the impact of a contaminating photometric signal with a distinct background profile on the extraction of the global asteroseismic parameter ν_{max}. We developed a robust methodology for the simulation of the light curves of firstascent RGB single and binary stars, incorporating frequencydependent modewidths and luminositydependent white noise.
We simulated the light curves of single RGs, and RG/RG binaries with (i) one pulsating component and (ii) two pulsating components and extracted the ν_{max} values of the pulsating component. We also tested two different granulation model configurations for our RG binaries: (i) a two superLorentzian and (ii) a four superLorentzian granulation profile and computed the differences between the binaryRG and singleRG reference ν_{max} values using each type of granulation model configuration. A summary of these results is presented below:
Single RGs

Systematics: We find a systematic underestimation of 1% in ν_{max}, 2.5% in M and 1% in R. This is caused by asymmetric mode distribution in the power excesses (see Fig. 1), resulting in differences between the ν_{max} values derived from the fit and from scaling relations.

Granulation: We confirm the randomising effect of the granulation signal, slightly increasing the degree of systematic underestimation, the intrinsic scatter, and the precision of extraction. We also find that the ν_{max} values extracted from our singleRG PSDs with and without granulation often did not showcase agreement within the formal errors.

White Noise: We find that white noise has a negligible effect on the extraction of ν_{max} for stars with K_{p} < 12, as noted by Kallinger et al. (2014).
Binary RGs

Systematics: We find negligible systematic differences between the binaryRG and singleRG ν_{max} values, confirming that the systematic offset observed in the singleRG case is a methodological issue.

Two pulsating components: We find that the ν_{max} extraction for PSDs featuring overlapping power excesses was highly unreliable, as was expected due to the use of unconstrained Gaussians to fit the power excesses. Outside of this scenario, we find that our results are very similar to the singleRG case.

One pulsating component: We find that our results are very similar to the singleRG case, showing that the light contribution of the individual components has minimal effect on the ν_{max} extraction for cases where the light contribution of the component is above ∼10%.

Granulation model configurations: We find minimal difference in the results when using either two or four superLorentzian components to represent the granulation signal. This is due to the degeneracy of the granulation fits such that both model configurations are able to represent it equally well.
Our results indicate that binarity has a strong effect on the extraction of ν_{max} in configurations where both components are pulsating and their power excesses overlap in the frequency domain. These particular binary configurations need a special treatment when using binarity for testing and validating scaling relations (e.g. such as using methodology proposed by Beck et al. 2018a). Outside of the abovementioned scenario, irrespective of individual light contributions of the two stars, the extraction of ν_{max} is not affected by binarity and is only subject to systematic uncertainties due to the adopted methodology.
Our findings indicate that a systematic offset of observationallyderived parameters from scalingrelation derived parameters might be a result of the discrepancy between the observed ν_{max}, which we found to be highly sensitive to mode distribution in the power excess, and the theoretical ν_{max}. Our results also indicate that the ∼15%/5% overestimation of seismic versus dynamical mass/radius reported by Gaulme et al. (2016) is unrelated to the effect of photometric contamination due to binarity and hence requires alternative explanations. We also find that while granulation does function as a randomisation element, resulting in discrepancies in the extraction of ν_{max}, the differences in ν_{max} extraction between different methods (as reported in Hekker et al. 2011) are significantly larger than the granulation discrepancy. In addition, the minimal difference in our binary results when using either two or four superLorentzian components to represent the granulation indicates that the current physical interpretation of the granulation terms is suspect. Additional investigation of the physics of granulation and its observational signature would be required before the degeneracy in the granulation fit can be lifted.
Our results indicate that the presence of systematic errors in the extraction of ν_{max} are a consequence of the fitting methodology. We find that while these uncertainties are smaller than or of the order of typical uncertainties reported for the scaling relations in the literature, they still worth considering when interpreting results in the context of observed discrepancies between scaling relationsbased masses/radii and those inferred from binary dynamics. We can therefore conclude that photometric contamination, besides decreasing the signaltonoise ratios of the individual components in the binary PSD, would have a negligible effect on the extraction of the global asteroseismic parameter ν_{max} from TESS data.
See Kallinger et al. (2014) for a detailed explanation and mathematical background for the inclusion of a normalisation factor in the superLorentzian profiles.
We use a standard deviation of 1.5Δν, resulting in a fullwidth at halfmaximum (FWHM) of 3.5Δν, which is well within the FWHM ranges for RGs reported by Mosser et al. (2010).
The normalised mode inertias exhibit local minima in intervals of Δν, but display a global decreasing trend that can span more than one order of magnitude (e.g. Fig. C.1). This results in modes with high amplitudes clustered around Δν, which mirrors observational oscillation patterns. See Dupret et al. (2009) for a full description.
The observational signature and efficiency of dynamical and equilibrium tides in RGbinaries are discussed in Beck et al. (2018b).
Acknowledgments
The authors would like to thank the anonymous referee for the constructive criticism that had radically transformed the manuscript into the form seen here. The research leading to these results has received funding from the Fonds Wetenschappelijk Onderzoek – Vlaanderen (FWO) under the grant agreement G0H5416N (ERC Opvangproject), and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 670519: MAMSIE). PGB acknowledges the support of the MINECO under the program “Juan de la Cierva incorporacion” (IJCI201526034). The authors would also like to thank Prof. C. Aerts and the MAMSIE team for useful discussions.
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Appendix A: MESA inlist file
The running and output of the MESA stellar evolutionary code is configured by an inlist file. While the code does have a default configuration, any parameters that the user wishes to alter must be specified in the inlist file. The contents of the inlist file for creating our RG models are shown below:
&star_job
show_log_description_at_start = .false.
load_saved_model = .false.
create_pre_main_sequence_model = .true.
kappa_file_prefix = 'OP_a09_p13'
kappa_lowT_prefix = 'lowT_fa05_a09p'
kappa_CO_prefix = 'a09_p13_co'
initial_zfracs = 8
change_net = .true.
new_net_name =
'pp_cno_extras_o18_ne22_extraiso.net'
change_initial_net = .true.
pgstar_flag = .false.
pause_before_terminate = .false.
save_photo_when_terminate = .false.
save_model_when_terminate = .false.
write_profile_when_terminate = .false.
save_pulse_data_when_terminate = .false.
new_rotation_flag = .false.
change_rotation_flag = .false.
change_initial_rotation_flag = .false.
new_omega = 0
set_omega = .false.
set_initial_omega = .false.
/ !end of star_job namelist
&controls
star_history_name =
initial_mass =
mixing_length_alpha = 1.8
overshoot_f_above_burn_h_core = 0.02
overshoot_f_above_burn_he_core = 0.02
min_D_mix = 1.0
initial_y = 0.2485
initial_z = 0.018
varcontrol_target = 1d4
log_directory =
terminal_interval = 200
photo_interval = 1000000
photo_directory = './photos'
write_profiles_flag = .false.
history_interval = 1
write_pulse_data_with_profile = .true.
pulse_data_format = 'GYRE'
add_atmosphere_to_pulse_data = .true.
add_center_point_to_pulse_data = .true.
keep_surface_point_for_pulse_data = .true.
add_double_points_to_pulse_data = .true.
interpolate_rho_for_pulse_data = .true.
threshold_grad_mu_for_double_point = 5d0
alpha_bdy_core_overshooting = 5
he_core_boundary_h1_fraction = 1d2
hot_wind_scheme = 'Vink'
Vink_scaling_factor = 0.3d0
hot_wind_full_on_T = 1d4
xa_central_lower_limit_species(1) = 'he4'
xa_central_lower_limit(1) = 1d3
remove_small_D_limit = 1d6
use_Ledoux_criterion = .true.
num_cells_for_smooth_gradL_composition_
term = 0
alpha_semiconvection = 0d0
semiconvection_option =
'Langer_85 mixing; gradT = gradr'
thermohaline_coeff = 0d0
alt_scale_height_flag = .true.
MLT_option = 'Cox'
mlt_gradT_fraction = 1
okay_to_reduce_gradT_excess = .false.
set_min_D_mix = .true.
D_mix_ov_limit = 5d2
max_brunt_B_for_overshoot = 0
limit_overshoot_Hp_using_size_of_convection_
zone = .true.
overshoot_alpha = 1
predictive_mix(1) = .true.
predictive_zone_type(1) = 'burn_H'
predictive_zone_loc(1) = 'core'
predictive_bdy_loc(1) = 'any'
predictive_mix(2) = .true.
predictive_zone_type(2) = 'burn_He'
predictive_zone_loc(2) = 'core'
predictive_bdy_loc(2) = 'any'
predictive_mix(3) = .true.
predictive_zone_type(3) = 'nonburn'
predictive_zone_loc(3) = 'shell'
predictive_bdy_loc(3) = 'any'
predictive_mix(4) = .true.
predictive_zone_type(4) = 'burn_H'
predictive_zone_loc(4) = 'shell'
predictive_bdy_loc(4) = 'any'
conv_bdy_mix_softening_f0 = 0.002
conv_bdy_mix_softening_f = 0.001
conv_bdy_mix_softening_min_D_mix = 1d1
overshoot_f0_above_burn_h_core = 0.001
overshoot_f0_above_burn_h_shell = 0.001
overshoot_f_above_burn_h_shell = 0.005
overshoot_f0_below_burn_h_shell = 0.001
overshoot_f_below_burn_h_shell = 0.005
overshoot_f0_above_burn_he_core = 0.001
overshoot_f0_above_nonburn_shell = 0.001
overshoot_f_above_nonburn_shell = 0.001
overshoot_f0_below_nonburn_shell = 0.005
overshoot_f_below_nonburn_shell = 0.005
smooth_convective_bdy = .false.
do_element_diffusion = .false.
which_atm_option = 'simple_photosphere'
cubic_interpolation_in_X = .false.
cubic_interpolation_in_Z = .false.
num_cells_for_smooth_brunt_B = 0
interpolate_rho_for_pulsation_info = .true.
max_allowed_nz = 40000
mesh_delta_coeff = 0.4
mesh_adjust_use_quadratic = .true.
mesh_adjust_get_T_from_E = .true.
P_function_weight = 40
T_function1_weight = 110
T_function2_weight = 0
T_function2_param = 2d4
gradT_function_weight = 0
xtra_coef_os_above_burn_h = 0.1d0
xtra_dist_os_above_burn_h = 2d0
mesh_dlogX_dlogP_extra = 0.15
mesh_dlogX_dlogP_full_on = 1d6
mesh_dlogX_dlogP_full_off = 1d12
mesh_logX_species(1) = 'he4'
xtra_coef_czb_full_on = 1.0d0
xtra_coef_czb_full_off = 1.0d0
xtra_coef_a_l_hb_czb = 0.5d0
xtra_dist_a_l_hb_czb = 1d0
xtra_coef_b_l_hb_czb = 0.5d0
xtra_dist_b_l_hb_czb = 1d0
xtra_coef_a_l_hb_czb = 0.5d0
xtra_dist_a_l_hb_czb = 1d0
xtra_coef_b_l_hb_czb = 0.5d0
xtra_dist_b_l_hb_czb = 1d0
! nonburning zone
xtra_coef_a_l_nb_czb = 0.5d0
xtra_dist_a_l_nb_czb = 1d0
xtra_coef_b_l_nb_czb = 0.5d0
xtra_dist_b_l_nb_czb = 1d0
xtra_coef_a_l_nb_czb = 0.5d0
xtra_dist_a_l_nb_czb = 1d0
xtra_coef_b_l_nb_czb = 0.5d0
xtra_dist_b_l_nb_czb = 1d0
! He burning zone
xtra_coef_a_l_heb_czb = 0.5d0
xtra_dist_a_l_heb_czb = 1d0
xtra_coef_b_l_heb_czb = 0.5d0
xtra_dist_b_l_heb_czb = 1d0
xtra_coef_a_l_heb_czb = 0.5d0
xtra_dist_a_l_heb_czb = 1d0
xtra_coef_b_l_heb_czb = 0.5d0
xtra_dist_b_l_heb_czb = 1d0
xtra_coef_os_full_on = 1.0d0
xtra_coef_os_full_off = 1.0d0
xtra_coef_os_above_burn_h = 0.5d0
xtra_dist_os_above_burn_h = 0.5d0
xtra_coef_os_below_burn_h = 0.5d0
xtra_dist_os_below_burn_h = 0.5d0
xtra_coef_os_above_nonburn = 0.5d0
xtra_dist_os_above_nonburn = 0.5d0
xtra_coef_os_below_nonburn = 0.5d0
xtra_dist_os_below_nonburn = 0.5d0
xtra_coef_os_above_burn_he = 0.5d0
xtra_dist_os_above_burn_he = 0.5d0
xtra_coef_os_below_burn_he = 0.5d0
xtra_dist_os_below_burn_he = 0.5d0
/ ! end of controls namelist
Appendix B: GYRE inlist file
Similar to MESA stellar evolutionary code, the running and output of the GYRE stellar pulsation code is configured by an inlist file. The code will output the pulsational parameters of the modes specified in the input file for a specific input model. The contents of the inlist file for computing the pulsational frequencies and parameters for our input RG models are shown below:
&constants
/
&model
model_type = 'EVOL'
file =
file_format = 'MESA'
/
&mode
l = 0
m = 0
tag = 'l0m0' ! Tag for namelist matching
/
&mode
l = 1
m = 0
tag = 'l1m0' ! Tag for namelist matching
/
&mode
l = 2
m = 0
tag = 'l2m0' ! Tag for namelist matching
/
&mode
l = 3
m = 0
tag = 'l3m0'
/
&osc
inner_bound = 'REGULAR'
outer_bound = 'JCD'
variables_set = 'JCD'
inertia_norm = 'BOTH'
rotation_method = 'NULL'
/
&num
diff_scheme = 'MAGNUS_GL4'
n_iter_max = 50
/
&scan
grid_type = 'LINEAR'
grid_frame = 'INERTIAL'
freq_min =
freq_max =
freq_min_units = 'UHZ'
freq_max_units = 'UHZ'
freq_frame = 'INERTIAL'
n_freq = 400
/
&grid
alpha_osc = 10
alpha_exp = 5
n_inner = 5
alpha_thm = 0
alpha_str = 0
&ad_output
summary_file = 'Mini0100_profiles/
Mini0100_at_nu_max_0020.profile.freqs'
freq_units = 'UHZ'
summary_file_format = 'TXT'
summary_item_list = 'l,m,n_p,n_g,n_pg,freq,
E_norm'
/
&nad_output
Appendix C: The variation of the normalised mode inertia with frequency
Fig. C.1.
Variation of the normalised mode inertia values for each frequency output by GYRE for a star with ν_{max} ∼ 100 μHz. The ℓ = 1, 2 and 3 modes are colourcoded as displayed the legend. It should be noted that the local minima in mode inertia in intervals of Δν, and the general decreasing trend of mode inertia with frequency. 
Appendix D: Power spectrum fit examples
This section contains examples of the fits obtained for our synthetic single and binaryRG PSDs.
Fig. D.1.
Example of a singleRG PSD fit where light curve contains only the pulsational signal of the RG. The grey line is the unsmoothed PSD, the black line is the smoothed PSD, the red line represents the overall fit, the dashed green line represents the Gaussian used to fit the pulsational power excess, and the blue dotted line represents the white noise. 
Fig. D.2.
Example of a singleRG PSD fit where the light curve contains both the pulsational and granulation signals of the RG. The grey, black, red, green and blue lines represent the same signals as in Fig. D.1, with the addition of blue dashdotted lines to represent the superLorentzians. 
Fig. D.3.
Example of a RGbinary PSD fit where both components, with very similar ν_{max} values, are pulsating. Here we use four superLorentzians to fit the granulation signal in the PSD. The grey, black, red, green and blue lines (both dashed and dotted) represent the same signals as in Fig. D.2. 
Fig. D.4.
Example of a RGbinary PSD fit where both components, with very different ν_{max} values, are pulsating. Here we use four superLorentzians to fit the granulation signal in the PSD. The grey, black, red, green and blue lines (both dashed and dotted) represent the same signals as in Fig. D.2. 
Fig. D.5.
Example of a RGbinary PSD fit where only one component is pulsating. Here we use four superLorentzians to fit the granulation signal in the PSD. The grey, black, red, green and blue lines (both dashed and dotted) represent the same signals as in Fig. D.2. 
All Tables
Ranges of M, R, T_{eff} and ν_{max} of the RG models used for the creation of our synthetic single and binaryRG light curves.
Coefficients k, s, and t for the granulation amplitudes (a_{1, 2}) and turnover frequencies (b_{1, 2}).
Mean percentage difference, 1σ scatter, and precision of the ν_{max} values extracted from our singleRG PSDs.
Mean percentage difference, 1σ scatter, and precision of the ν_{max, binary} values extracted from our RGbinary PSDs, using either two or four superLorentzians to fit the granulation signal.
All Figures
Fig. 1.
Example of a synthetic RG pulsation spectrum for a star with ν_{max} ∼ 100 μHz, used as an input to generates synthetic pulsational timeseries for the RG light curves. The ℓ = 0, 1, 2, and 3 modes are represented by Lorentzians and colourcoded as displayed in the legend. The presence of multiplets, particularly around the ℓ = 1 modes, correspond to mixed modes with the same pmode radial order but different gmode radial order. 

In the text 
Fig. 2.
Example of a singleRG PSD fit. The grey line is the unsmoothed PSD, the black line is the smoothed PSD, the red line represents the overall fit, the dashed green line represents the Gaussian used to fit the pulsational power excess, the blue dotted line represents the white noise, and the blue dashed lines represent the superLorentzians used to fit the granulation signal. 

In the text 
Fig. 3.
Comparison of the ν_{max} values extracted from our synthetic singleRG PSDs. We tested synthetic light curves containing only a pulsational signal (triangles), and both a pulsational and a granulation signal (circles). The vertical axis corresponds to the scaling relation (ν_{max, scaling}) input values of the light curves, and the horizontal axis corresponds to the percentage difference between the fit (ν_{max, fit}) and the scaling relation (ν_{max, scaling}) values. The vertical dashed line represents the zeropoint of the difference between ν_{max, fit} and ν_{max, scaling}. The dashdotted lines represent the 1σ level about the mean scatter of the datapoints corresponding to the pulsational signal (red), and both the pulsational and granulation signals (green). The horizontal dotted lines connect the datapoints corresponding to the same RG model. The error bars correspond to the 68% Bayesian credible intervals of the marginalised posterior distributions of the fit parameters. The symbols are colourcoded according the logarithm of the white noise (log W) included in the light curves and PSDs. 

In the text 
Fig. 4.
M, R, and T_{eff} values of the individual components (highlighted in dark blue) used to create our binary grid. 

In the text 
Fig. 5.
Comparison of the ν_{max} values of each component (Star A, left panel; and Star B, right panel) extracted from our synthetic RGbinary light curves where both components are pulsating, using a two superLorentzian granulation model. The vertical axis corresponds to the difference between the singleRG reference values for Star A (ν_{max, single (Star A)}) and Star B (ν_{max, single (Star B)}). The horizontal axis corresponds to the difference between the binary (ν_{max, binary}) and the singleRG (ν_{max, single}) reference values. The vertical dashed line represents the zeropoint of the difference between ν_{max, binary} and ν_{max, single}. The dashdotted green lines represent the interpolated 1 − σ level about the mean scatter of the datapoints grouped into 10 μHz bins along the vertical axis. The horizontal dashed line represents the zeropoint of ν_{max, single (Star A)} − ν_{max, single (Star B)}. The error bars correspond to the 68% Bayesian credible intervals of the marginalised posterior distributions of the binary fit parameters. The symbols are colourcoded according the percentage light contribution of each component to the binary light curve. 

In the text 
Fig. 6.
Comparison of the ν_{max} values of each component (Star A, left panel; and Star B, right panel) extracted from our synthetic RGbinary light curves where both components are pulsating, using a four superLorentzian granulation model. The axes, symbols, error bars and colourcoding is the same as in Fig. 5. 

In the text 
Fig. 7.
Comparison of the input granulation signal with those from the maximumlikelihood values of the two and four superLorentzian granulation model fits, for one of our binaryRG PSDs where both components are pulsating. The superLorentzians used as inputs are represented by the solid red curves, with the two and four superLorentzians fits represented by the dark blue and bluegreen dashdotted lines respectively. 

In the text 
Fig. 8.
Comparison of the ν_{max} values for a two superLorentzian (left panel) and a four superLorentzian (right panel) granulation model, extracted from our synthetic RGbinary light curves where only one component (Star A) is pulsating. The axes, symbols, error bars and colourcoding is the same as in Fig. 5. 

In the text 
Fig. 9.
Comparison of the input granulation signal with those from the maximumlikelihood values of the two and four superLorentzian granulation model fits, for one of our binaryRG PSDs where only one component is pulsating. The superLorentzians used as inputs are represented by the solid red curves, with the two and four superLorentzian fits represented by the dark blue and bluegreen dashdotted lines respectively. 

In the text 
Fig. 10.
Difference in the ν_{max} values extracted from the fits using a two superLorentzian and a four superLorentzian granulation model. Left and middle panels: difference in ν_{max, fit} for the RGbinary light curves where both components (Star A and Star B) are pulsating. Right panel: difference in ν_{max, fit} for the RGbinary light curves where only one component (Star A) is pulsating. The axes, symbols, error bars and colourcoding is the same as in Fig. 5, except that the horizontal axis corresponds to the absolute percentage difference between the ν_{max, fit} values for the two superLorentzian (2SL) and four superLorentzian (4SL) cases. 

In the text 
Fig. C.1.
Variation of the normalised mode inertia values for each frequency output by GYRE for a star with ν_{max} ∼ 100 μHz. The ℓ = 1, 2 and 3 modes are colourcoded as displayed the legend. It should be noted that the local minima in mode inertia in intervals of Δν, and the general decreasing trend of mode inertia with frequency. 

In the text 
Fig. D.1.
Example of a singleRG PSD fit where light curve contains only the pulsational signal of the RG. The grey line is the unsmoothed PSD, the black line is the smoothed PSD, the red line represents the overall fit, the dashed green line represents the Gaussian used to fit the pulsational power excess, and the blue dotted line represents the white noise. 

In the text 
Fig. D.2.
Example of a singleRG PSD fit where the light curve contains both the pulsational and granulation signals of the RG. The grey, black, red, green and blue lines represent the same signals as in Fig. D.1, with the addition of blue dashdotted lines to represent the superLorentzians. 

In the text 
Fig. D.3.
Example of a RGbinary PSD fit where both components, with very similar ν_{max} values, are pulsating. Here we use four superLorentzians to fit the granulation signal in the PSD. The grey, black, red, green and blue lines (both dashed and dotted) represent the same signals as in Fig. D.2. 

In the text 
Fig. D.4.
Example of a RGbinary PSD fit where both components, with very different ν_{max} values, are pulsating. Here we use four superLorentzians to fit the granulation signal in the PSD. The grey, black, red, green and blue lines (both dashed and dotted) represent the same signals as in Fig. D.2. 

In the text 
Fig. D.5.
Example of a RGbinary PSD fit where only one component is pulsating. Here we use four superLorentzians to fit the granulation signal in the PSD. The grey, black, red, green and blue lines (both dashed and dotted) represent the same signals as in Fig. D.2. 

In the text 
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