11institutetext:  Instituut voor Sterrenkunde (IvS), KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium
11email<EMAIL_ADDRESS>22institutetext: Instituto de Astrofísica
de Canarias, E-38200 La Laguna, Tenerife, Spain 33institutetext: Departamento
de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
44institutetext: Villanova University, Dept. of Astrophysics and Planetary
Science, 800 Lancaster Ave, Villanova PA 19085

# Two’s a crowd? Characterising the effect of photometric contamination on the
extraction of the global asteroseismic parameter $\nu_{\text{max}}$ in red-
giant binaries

S. Sekaran Two’s a crowd? Characterising the effect of photometric
contamination on the extraction of the global asteroseismic parameter
$\nu_{\text{max}}$ in red-giant binariesTwo’s a crowd? Characterising the
effect of photometric contamination on the extraction of the global
asteroseismic parameter $\nu_{\text{max}}$ in red-giant binaries C. Johnston
Two’s a crowd? Characterising the effect of photometric contamination on the
extraction of the global asteroseismic parameter $\nu_{\text{max}}$ in red-
giant binariesTwo’s a crowd? Characterising the effect of photometric
contamination on the extraction of the global asteroseismic parameter
$\nu_{\text{max}}$ in red-giant binaries A. Tkachenko Two’s a crowd?
Characterising the effect of photometric contamination on the extraction of
the global asteroseismic parameter $\nu_{\text{max}}$ in red-giant
binariesTwo’s a crowd? Characterising the effect of photometric contamination
on the extraction of the global asteroseismic parameter $\nu_{\text{max}}$ in
red-giant binaries P. G. Beck Two’s a crowd? Characterising the effect of
photometric contamination on the extraction of the global asteroseismic
parameter $\nu_{\text{max}}$ in red-giant binariesTwo’s a crowd?
Characterising the effect of photometric contamination on the extraction of
the global asteroseismic parameter $\nu_{\text{max}}$ in red-giant
binariesTwo’s a crowd? Characterising the effect of photometric contamination
on the extraction of the global asteroseismic parameter $\nu_{\text{max}}$ in
red-giant binariesTwo’s a crowd? Characterising the effect of photometric
contamination on the extraction of the global asteroseismic parameter
$\nu_{\text{max}}$ in red-giant binaries A. Prša Two’s a crowd? Characterising
the effect of photometric contamination on the extraction of the global
asteroseismic parameter $\nu_{\text{max}}$ in red-giant binariesTwo’s a crowd?
Characterising the effect of photometric contamination on the extraction of
the global asteroseismic parameter $\nu_{\text{max}}$ in red-giant binaries K.
M. Hambleton Two’s a crowd? Characterising the effect of photometric
contamination on the extraction of the global asteroseismic parameter
$\nu_{\text{max}}$ in red-giant binariesTwo’s a crowd? Characterising the
effect of photometric contamination on the extraction of the global
asteroseismic parameter $\nu_{\text{max}}$ in red-giant binaries

(Received August 16, 2018; accepted March 20, 2019)

###### Abstract

Context. Theoretical scaling relations for solar-like 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 model-independent
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 $\nu_{\text{max}}$ through the analysis of synthetic
red-giant 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 red-giant light
curves with the characteristics of Kepler long-cadence photometric data. These
are subsequently blended together according to their light ratio to generate
binary red-giant light curves of various configurations. We then performed a
differential analysis to characterise the systematic effects of binarity on
the extraction of $\nu_{\text{max}}$.

Results. We quantify our methodological uncertainties through the analysis of
single red-giant 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 $\nu_{\text{max}}$ extraction for red-
giant 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 single-star case.

Conclusions. We conclude that i) the photometric effects of binarity on the
extraction of $\nu_{\text{max}}$ 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 super-Lorentzian components to model
the granulation signal of a binary red-giant.

###### Key Words.:

asteroseismology – binaries – stars: oscillations – stars: Red Giants

##  1 Introduction

The revolution in red-giant asteroseismology began with the advent of the
CoRoT space mission (Auvergne et al., 2009), which delivered high-precision
and high duty-cycle photometry. These data enabled the first detections of
non-radial pulsations in red-giants (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 four-year nominal duty cycle. These new high-precision 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 solar-
like pulsations in a minimally model-dependent manner. Kjeldsen & Bedding
(1995) had originally deduced these relations by calculating the global
asteroseismic parameters known as the frequency of maximum oscillation power
($\nu_{\text{max}}$) and the large frequency separation ($\Delta\nu$) from the
fundamental parameters of a sample of stars with detected solar-like
oscillations, including the Sun (used as the reference star). Scaling
relations take the form of a power-law with the effective temperature
($T_{\text{eff}}$), $\nu_{\text{max}}$ and $\Delta\nu$ as inputs. It is most
commonly formulated in the following manner, first introduced by Kallinger et
al. (2010):

$\hfill\frac{M}{M_{\odot}}=\left(\frac{\nu_{\text{max}}}{\nu_{\text{max,
}\odot}}\right)^{3}\left(\frac{\Delta\nu_{\odot}}{\Delta\nu}\right)^{4}\left(\frac{T_{\text{eff}}}{T_{\text{eff,
}\odot}}\right)^{\frac{3}{2}};\hfill$ (1)

$\hfill\frac{R}{R_{\odot}}=\left(\frac{\nu_{\text{max}}}{\nu_{\text{max,
}\odot}}\right)\left(\frac{\Delta\nu_{\odot}}{\Delta\nu}\right)^{2}\left(\frac{T_{\text{eff}}}{T_{\text{eff,
}\odot}}\right)^{\frac{1}{2}}.\hfill$ (2)

$M$ and $R$ are the stellar mass and radius, and the quantities with the
$\odot$ subscript refer to the solar reference values.

The large frequency separation ($\Delta\nu$) is the average difference in
frequency between modes of the same spherical degree ($\ell$) 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 ($\Delta\nu_{\text{as}}$), which is directly proportional
to the mean density ($\overline{\rho}$) of the star, is related to the sound
speed ($c_{\text{s}}$) and radius ($R$) of the star according to the following
equation:

$\hfill\Delta\nu_{\text{as}}=\left(2\int_{0}^{R}\frac{dr}{c_{\text{s}}}\right)^{-1}.\hfill$
(3)

However, the observational large frequency separation
($\Delta\nu_{\text{obs}}$) is typically determined from the mean difference
between frequencies of the modes (typically radial) of the same $\ell$ but
different $n$ (i.e. $\Delta\nu_{\text{obs}}=<\nu_{n\text{,
}\ell+1}-\nu_{n\text{, }\ell}>$). This necessarily implies that the
observational and asymptotic $\Delta\nu$ 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 $\Delta\nu$).

The frequency of maximum power ($\nu_{\text{max}}$) 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
center of the Gaussian that is used to approximate the power excess displayed
by all solar-like 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
ubiquitously-used function to characterise the oscillation power excess for
solar-like oscillators.

Brown et al. (1991), and later Kjeldsen & Bedding (1995), theorised that
$\nu_{\text{max}}$ was related to the acoustic wave cutoff frequency in an
isothermal atmosphere, and therefore scales with the surface gravity ($g$) and
the effective temperature ($T_{\text{eff}}$) according to the following
approximation:

$\hfill\nu_{\text{max}}\propto\frac{g}{\sqrt{T_{\text{eff}}}}.\hfill$ (4)

While studies such as Belkacem et al. (2011, 2013) provide a full theoretical
formulation for $\nu_{\text{max}}$, 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 Tycho-gaia astrometric solution (TGAS) parallaxes (Salgado
et al., 2017), an offshoot of the data products provided by the recently
launched gaia space mission (Gaia Collaboration et al., 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 relation-derived 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 solar-like pulsating components has allowed for the critical
evaluation of evolutionary models (Mathur et al., 2013; Appourchaux et al.,
2015; Beck et al., 2014; White et al., 2017; Beck et al., 2018a; Li et al.,
2018) and tidal theory (Beck et al., 2018b). The far less-common occurrence of
eclipsing binaries with a solar-like 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 time-series
of both photometric, and high-resolution, high signal-to-noise 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 multi-step analyses
involving binary and asteroseismic modelling (see e.g. Rawls 2016). However,
precision and accuracy ($\sim$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 red-giant 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), Brogaard et al. (2018)
and Themeßl et al. (2018).

Of particular concern are the claims of Gaulme et al. (2016), who had found an
average overestimation of $\sim$15%/5% of red-giant mass/radius extracted from
scaling relations when confronted with the mass/radius extracted from
eclipsing binary dynamics. The departure of observationally-derived red-giant
global asteroseismic parameters ($\nu_{\text{max}}$ and $\Delta\nu$) 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 $\nu_{\text{max, }\odot}$ and
$\Delta\nu_{\odot}$, or empirical or model-dependent corrections to the
derived $\nu_{\text{max}}$ and $\Delta\nu$, 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 binary-analysis
derived and scaling-relation derived parameters through similar corrections.
However, there is still no agreement on the exact form of the corrections to
the scaling relations.

Most recently, non-linear scaling relations that claim to solve the eclipsing
binary-analysis derived and scaling-relation 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
scaling-relation corrections, still utilise the global asteroseismic parameter
$\nu_{\text{max}}$ 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 recently-launched 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 red-giant/red-giant 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 red-
giant/red-giant binaries, with the characteristics of Kepler long-cadence
photometric data, and extracted the values of the global asteroseismic
parameter $\nu_{\text{max}}$ 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 red-giant simulations, and Section 4 details the results
of the binary red-giant simulations. We end by discussing our results and
presenting our conclusions in Section 5.

##  2 Methodology

###  2.1 Red-giant signal components

The signals that constitute a typical Red Giant (RG) power spectral density
(PSD) that is generated from Kepler long-cadence 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 frequency-independent (white) photon-
shot noise component and a frequency-dependent (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:

$\hfill
P_{\text{noise}}(\nu)=W+\frac{2\pi\alpha^{2}/\beta}{1+(\nu+\beta)^{2}}.\hfill$
(5)

$P_{\text{noise}}(\nu)$ is the PSD of the noise signal, $W$ is the white
noise, and $\frac{2\pi\alpha^{2}/\beta}{1+(\nu+\beta)^{2}}$ is the coloured
instrumental noise represented by a Lorentzian profile, with $\alpha$ and
$\beta$ 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 well-understood. The general consensus
is that the background consists of frequency-dependent (coloured) components
that have been surmised to be the result of granulation occurring at different
length- and time-scales. The first attempt to model this background signal was
performed by Harvey (1985), who used a super-Lorentzian (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 $K_{\text{p}}=12$, where $K_{\text{p}}$ is the Kepler
magnitude), a background model consisting of two super-Lorentzian profiles of
the form

$\hfill P_{\text{bg}}(\nu)=\eta^{2}(\nu)\left[\sum_{i}^{2}\frac{\xi
a_{\text{i}}^{2}/b_{\text{i}}}{1+(\nu/b_{i})^{4}}\right],\hfill$ (6)

best reproduced the data. $P_{\text{bg}}(\nu)$ is the PSD of the background
signal, and the sum of the two super-Lorentzian profiles
$\sum\limits_{i}^{2}\frac{\xi
a_{\text{i}}^{2}/b_{\text{i}}}{1+(\nu/b_{i})^{4}}$ represent granulation
occurring at different timescales, each with their respective amplitudes
($a_{\text{i}}$) and turnover frequencies ($b_{\text{i}}$), and the
normalisation factor111See Kallinger et al. (2014) for a detailed explanation
and mathematical background for the inclusion of a normalisation factor in the
super-Lorentzian profiles. $\xi=2\sqrt{2}/\pi$. $\eta^{2}(\nu)$ is the sinc-
squared 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
stochastically-excited pulsations, with frequency-dependent 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 frequency-dependent 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:

$\hfill
P_{\text{puls}}(\nu)=\sum_{n}^{m}\frac{A_{n}^{2}/(\pi\Gamma_{n})}{1+4(\nu-\nu_{0\
(n)}/\Gamma_{n})^{2}}.\hfill$ (7)

$P_{\text{puls}}(\nu)$ is the PSD of the pulsational signal, and $A_{n}$,
$\Gamma_{n}$, and $\nu_{0\ (n)}$ are the amplitude, width and central
frequency of the $n^{\text{th}}$ 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 $\nu_{\text{max}}$, with an
amplitude and width of $A_{\text{puls}}$ and $\sigma_{\text{puls}}$
respectively. The PSD of the Gaussian that represents the pulsational envelope
($P_{\text{env}}$) can be calculated using the following equation:

$\hfill
P_{\text{env}}(\nu)=A_{\text{puls}}\left[\text{exp}\left(\frac{-(\nu-\nu_{\text{max}})^{2}}{2\sigma_{\text{puls}}^{2}}\right)\right].\hfill$
(8)

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 more-current 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 core-helium burning, resulting in a grid of RG models with masses
of 1.0 $M_{\odot}$ to 2.0 $M_{\odot}$ in steps of 0.2 $M_{\odot}$, and a
$\nu_{\text{max}}$ of approximately 20 $\mu$Hz to 200 $\mu$Hz in steps of 10
$\mu$Hz for each mass value. These $\nu_{\text{max}}$ values were calculated
from the $M$ and $R$ outputs of mesa by using the following equations (a
result of Equations 3 and 4) as follows:

$\hfill\nu_{\text{max}}=\left(\frac{M}{M_{\odot}}\right)\left(\frac{R_{\odot}}{R}\right)^{2}\left(\frac{T_{\text{eff,
}\odot}}{T_{\text{eff}}}\right)^{\frac{1}{2}}\nu_{\text{max, }\odot};\hfill$
(9)

$\hfill\Delta\nu=\left(\frac{M}{M_{\odot}}\right)^{\frac{1}{2}}\left(\frac{R_{\odot}}{R}\right)^{\frac{3}{2}}\Delta\nu_{\odot}.\hfill$
(10)

The $\nu_{\text{max}}$ 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 $\mu$Hz for
$\nu_{\text{max}}$ was set to ensure that the pulsational power excesses were
well within the Nyquist frequency threshold of 284 $\mu$Hz for Kepler long-
cadence data. The ranges of the $M$, $R$, $T_{\text{eff}}$ and
$\nu_{\text{max}}$ 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 non-rotating
pulsational models in the adiabatic framework. These pulsational models
comprise the frequencies and normalised mode inertias of the $\ell=0$, 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.

Table 1: The ranges of $M$, $R$, $T_{\text{eff}}$ and $\nu_{\text{max}}$ of the RG models used for the creation of our synthetic single- and binary-RG light curves. |  $M$ [$M_{\odot}$] |  $R$ [$R_{\odot}$] |  $T_{\text{eff}}$ [K] |  $\nu_{\text{max}}$ [$\mu$Hz]
---|---|---|---|---
Min | 1.0 | 4.2 | 4192 | 20.2
Max | 2.0 | 19.0 | 4913 | 202.6

The corresponding mode amplitudes for each frequency were then generated using
the following steps, similar to the methodology detailed in Section 4.3 of
Kallinger et al. (2014) for the creation of synthetic RG data:

  1. 1.

Initial amplitude ratios ($A_{\text{ratio}}$) between 0 and 1 were assigned to
each frequency ($\nu_{\text{i}}$) by using a Gaussian centred on
$\nu_{\text{max}}$ with a standard deviation222We use a standard deviation of
$1.5\Delta\nu$, resulting in a full-width at half-maximum (FWHM) of
$3.5\Delta\nu$, which is well within the FWHM ranges for RGs reported by
Mosser et al. (2010). of $1.5\Delta\nu$. This is represented by the following
equation:

$\hfill
A_{\text{ratio}}=\text{exp}\left(\frac{-(\nu_{\text{i}}-\nu_{\text{max}})^{2}}{2(1.5\Delta\nu)^{2}}\right).\hfill$
(11)

  2. 2.

These amplitude ratios were then modulated by mode visibilities of 1.00, 1.35,
0.64 and 0.071 for the $\ell=0$, 1, 2 and 3 modes respectively, as reported by
Mosser et al. (2012) for RGB stars.

  3. 3.

The amplitude ratios of the $\ell=1$, 2 and 3 modes were then further
modulated by the inverse of the local relative mode inertia333The normalised
mode inertias exhibit local minima in intervals of $\Delta\nu$, but display a
global decreasing trend that can span more than one order of magnitude (e.g.
Figure 11). This results in modes with high amplitudes clustered around
$\Delta\nu$, which mirrors observational oscillation patterns. See Dupret et
al. (2009) for a full description..

  4. 4.

To obtain realistic pulsational amplitudes, we refer to the characterisation
of pulsational amplitudes reported in Corsaro et al. (2013), where different
amplitude scaling-relation models were investigated in a Bayesian framework.
We calculated the total bolometric pulsational amplitude $A_{\text{puls,
bol}}$ from the scaling relation given in Equation 19 of Corsaro et al.
(2013):

$\begin{split}\hfill A_{\text{puls,
bol}}=d\left(\frac{\nu_{\text{max}}}{\nu_{\text{max,
}\odot}}\right)^{2p-3q-r+0.2}&\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{4q-4p}\\\
\cdot&\left(\frac{T_{\text{eff}}}{T_{\text{eff,
}\odot}}\right)^{5p-1.5q-r+0.2}.\hfill\end{split}$ (12)

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
long-cadence expectation values of model $\mathcal{M}_{4\text{, }d}$ of
Corsaro et al. (2013), which they found to have the best Bayesian Information
Criterion (BIC) value in their analysis.

  5. 5.

Amplitudes (in ppm) for each frequency can then obtained by taking the product
of the modulated amplitude ratios and $A_{\text{puls, bol}}$.

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 $\nu_{\text{max}}$ for each star by using the
following equation:

$\hfill\Gamma(\nu_{\text{max}})=nM^{0.27}T_{\text{eff}}^{4.74}.\hfill$ (13)

$\Gamma(\nu_{\text{max}})$ is the modewidth at $\nu_{\text{max}}$, with the
exponents of $M$ and $T_{\text{eff}}$ 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 to 0.17 $\mu$Hz, which is well
within the distribution of modewidths displayed in Figure 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
$\nu_{\text{max}}$ (although they did find evidence of a reduction in the rate
of increase around $\nu_{\text{max}}$). 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:

$\hfill\Gamma(\nu)=\Gamma(\nu_{\text{max}})\left[\text{exp}\left(\frac{z(\nu-\nu_{\text{max}})}{\nu_{\text{max}}}\right)\right].\hfill$
(14)

$\Gamma(\nu)$ is the modewidth at frequency $\nu$, and $z=3.0$ is a constant
designed to keep the modewidth increase with frequency within the ranges
reported by Vrard et al. (2018).

Figure 1: An example of a synthetic RG pulsation spectrum for a star with
$\nu_{\text{max}}\sim 100$ $\mu$Hz, used as an input to generates synthetic
pulsational time-series for the RG light curves. The $\ell=0$, 1, 2, and 3
modes are represented by Lorentzians and colour-coded as displayed in the
legend. The presence of multiplets, particularly around the $\ell=1$ modes,
correspond to mixed modes with the same $p$-mode radial order but different
$g$-mode radial order.

The frequencies, amplitudes and modewidths (see Figure 1 for an example of a
pulsation spectrum) are then fed to a first-order auto-regressive function to
generate a synthetic time-series, following the methodology laid out in De
Ridder et al. (2006). This procedure transforms the ’perfect’ comb of our
input pulsation spectrum into a more-realistic representation of a RG
pulsation spectrum (c.f. Figure 13). The time sampling of this function was
setup in such a way as to produce a time-series with the same time-stamps as a
full four-year Kepler long cadence light curve with no gaps.

To simulate the granulation signal for our light curves, we first generated
super-Lorentzian profiles as per Equation (6). The granulation amplitudes
($a_{\text{i}}$) and turnover frequencies ($b_{\text{i}}$) 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
$\nu_{\text{max}}$ values:

$\hfill a_{\text{i}}=k\nu_{\text{max}}^{s}M^{t};\hfill$ (15)

$\hfill b_{\text{i}}=k\nu_{\text{max}}^{s}.\hfill$ (16)

The coefficients $k$, $s$, and $t$ for $a_{1\text{, }2}$ and $b_{1\text{, }2}$
are listed in Table 4. 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.

Table 2: The coefficients $k$, $s$, and $t$ for the granulation amplitudes
($a_{1\text{, }2}$) and turnover frequencies ($b_{1\text{, }2}$).

| $a_{1}$ | $a_{2}$ | $b_{1}$ | $b_{2}$
---|---|---|---|---
$k$ | 3710 | 2226 | 0.317 | 0.948
$s$ | -0.613 | -0.613 | 0.970 | 0.992
$t$ | -0.26 | -0.26 | – | –
444These coefficients were taken from Table 2 of Kallinger et al. (2014).

To mimic the procedure that we had performed for the pulsation spectrum, we
chose to generate synthetic time-series for our granulation spectra. While it
is possible to transform a simple Harvey profile into a time-series using an
auto-regressive function, there is currently no analogue that can transform
our granulation profiles into a time-series. We therefore performed inverse
Fourier Transforms (iFTs) of our granulation PSDs in order to generate them.
Similar to the pulsational time-series, the iFTs are calculated in such a way
that the resulting time-series has the same time-stamps as a full Kepler long-
cadence light curve.

We chose to include a luminosity-dependent 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 short-cadence Kepler data.
They reported an almost-linear 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:

$\hfill\text{log }W=-\text{log }L+c.\hfill$ (17)

$L$ 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 to 12 Kepler magnitude range as reported by
Pande et al. (2018). This is within the range in which a two super-Lorentzian-
component 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
$\nu_{\text{max}}$. Similar to the granulation signal, iFTs were performed in
order to produce white noise time-series.

The three time-series are then added to produce the final synthetic light
curve for a single RG of mass $M$, radius $R$ and effective temperature
$T_{\text{eff}}$.

###  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 Section
2.1). This fitting function, essentially a sum of the pulsational,
granulation, and noise model functions, takes the following form:

$\begin{split}P_{\text{model}}(\nu)=W&+A_{\text{puls}}\left[\text{exp}\left(\frac{-(\nu-\nu_{\text{max}})^{2}}{2\sigma_{\text{puls}}^{2}}\right)\right]\\\
&+\sum_{i}^{2}\frac{(2\sqrt{2}/\pi)\tau_{\text{gran, i}}A_{\text{gran,
i}}^{2}}{1+(\nu\tau_{\text{gran, i}})^{4}}.\end{split}$ (18)

The terms in the function echo those in Equations (5), (6) and (8):
$P_{\text{model}}(\nu)$ is the PSD of the model; $W$ is the white noise;
$A_{\text{puls}}\left[\text{exp}\left(\frac{-(\nu-\nu_{\text{max}})^{2}}{2\sigma_{\text{puls}}^{2}}\right)\right]$
represents the Gaussian power excess, with the amplitude, width and central
frequency represented by $A_{\text{puls}}$, $\sigma_{\text{puls}}$ and
$\nu_{\text{max}}$ respectively; and
$\sum\limits_{i}^{2}\frac{(2\sqrt{2}/\pi)\tau_{\text{gran, i}}A_{\text{gran,
i}}^{2}}{1+(\nu\tau_{\text{gran, i}})^{4}}$ are the two super-Lorentzian
components representing the granulation signal, with the amplitude and
characteristic timescale of the $i^{\text{th}}$ super-Lorentzian represented
by $A_{\text{gran, i}}$ and $\tau_{\text{gran, i}}$ respectively.

We perform our fit in a Bayesian framework using the Markov Chain Monte Carlo
(MCMC) routine emcee (Foreman-Mackey 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:

$\hfill
P(H_{\text{i}}|D,I)=\frac{P(H_{\text{i}}|I)P(D|H_{\text{i}},I)}{P(D|I)}.\hfill$
(19)

$P(H_{\text{i}}|I)$ is known as the prior probability, which is the
probability of the hypothesis $H_{\text{i}}$ with knowledge of prior
information $I$, but in the absence of the data $D$. $P(H_{\text{i}}|D,I)$ is
known as the posterior probability, which is the probability of the hypothesis
with knowledge of both $I$ and $D$. The quantities $P(D|H_{\text{i}},I)$ and
$P(D|I)$ are the likelihood of $H_{\text{i}}$ and the global likelihood of all
hypotheses respectively.

For numerical simplicity, we evaluate Bayes’ theorem in log-space, where our
log-likelihood function ($\mathcal{L}$) is

$\hfill\text{ln }\mathcal{L}=-\sum_{v}\left(\text{ln
}P_{\text{model}}(\nu)+\frac{P_{\text{input}}(\nu)}{P_{\text{model}}(\nu)}\right).\hfill$
(20)

$P_{\text{input}}(\nu)$ 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 ($\tau_{\text{autocorr}}$) as per
Goodman & Weare (2010). We compute $\tau_{\text{autocorr}}$ for every 100
iterations and consider the MCMC routine to have achieved convergence if
$\tau_{\text{autocorr}}$ changes by less than 1% between two successive
computations.

After our MCMC routines have achieved convergence, we extract our fit
parameters and determine $\nu_{\text{max, fit}}$ from the marginalised
posterior distributions, comparing them with our input $\nu_{\text{max,
scaling}}$ from scaling relations. An example fit of the PSD of one of our
synthetic light curves is shown in Figure 2.

Figure 2: An example of a single-RG 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 super-Lorentzians used to fit the granulation
signal.

###  2.4 Red-giant 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 Equation (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:

$\begin{split}F_{\text{binary}}=\frac{1}{L_{\text{A}}+L_{\text{B}}}\bigg{[}&L_{\text{A}}\left(F_{\text{puls,
A}}+F_{\text{gran, A}}\right)\\\ +&L_{\text{B}}\left(F_{\text{puls,
B}}+F_{\text{gran, B}}\right)\bigg{]}+W.\end{split}$ (21)

$F_{\text{binary}}$ is the flux of the binary light curve, and $L_{\text{A}}$,
$F_{\text{puls, A}}$, $F_{\text{gran, A}}$, $L_{\text{B}}$, $F_{\text{puls,
B}}$, and $F_{\text{gran, B}}$ 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
ignored555The observational signature and efficiency of dynamical and
equilibrium tides in RG-binaries are discussed in Beck et al. (2018b)..

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 super-
Lorentzians (as in the single-RG case), and with ii) four super-Lorentzians
(two super-Lorentzians for each component), to model the granulation signal of
each binary light curve.

##  3 Single red-giant simulations and results

To ensure the self-consistency 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 $\nu_{\text{max}}$ by performing one iteration of fits with only the
pulsational signal (e.g. Figure 13), and another with both the pulsational and
granulation signals (e.g. Figure 13) included in the light curves of the RGs.
This is demonstrated in Figure 3, showcasing the percentage difference between
the fit ($\nu_{\text{max, fit}}$) and the scaling relation ($\nu_{\text{max,
scaling}}$) input values. The mean percentage difference and the 1-$\sigma$
scatter of the $\nu_{\text{max, fit}}-\nu_{\text{max, scaling}}$ values, and
the mean precision of the extraction of $\nu_{\text{max, fit}}$ are detailed
in Table 6.

Figure 3: A comparison of the $\nu_{\text{max}}$ values extracted from our
synthetic single-RG 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
($\nu_{\text{max, scaling}}$) input values of the light curves, and the
horizontal axis corresponds to the percentage difference between the fit
($\nu_{\text{max, fit}}$) and the scaling relation ($\nu_{\text{max,
scaling}}$) values. The vertical dashed line represents the zero-point of the
difference between $\nu_{\text{max, fit}}$ and $\nu_{\text{max, scaling}}$.
The dash-dotted lines represent the 1-$\sigma$ 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 colour-coded
according the logarithm of the white noise ($\text{log }W$) included in the
light curves and 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 $\nu_{\text{max, fit}}$ compared to the $\nu_{\text{max, scaling}}$
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
$\nu_{\text{max, scaling}}$ 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 Figure 1 (which correponds to a
$\nu_{\text{max, scaling}}\sim 100$ $\mu$Hz), there is a larger number of
strong peaks in the frequency domain below $\nu_{\text{max, scaling}}$ than
above it, particularly from the $\ell=1$ mixed-mode multiplets. This
distribution is a result of the mode inertia distribution (e.g Figure 11) 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
$\nu_{\text{max, scaling}}$ 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 $\nu_{\text{max, scaling}}$ and repeating
the same fitting procedure. We found that the systematic offset between the
$\nu_{\text{max, fit}}$ and $\nu_{\text{max, scaling}}$ 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
$\nu_{\text{max, fit}}$ values extracted from PSDs with and without
granulation. The $\nu_{\text{max, fit}}$ values for the iteration with
granulation are slightly more underestimated on average compared to the
$\nu_{\text{max, scaling}}$ values, seemingly exacerbating the asymmetry in
the power excesses. In addition, the scatter of the $\nu_{\text{max, fit}}$ is
greater, and the extraction of the $\nu_{\text{max, fit}}$ 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 super-Lorentzians) 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 $\nu_{\text{max, fit}}$ is decreased in some cases when
comparing the results with and without granulation.

It can also be seen in Figure 3 that the precision of the extraction seems to
decrease as the $\nu_{\text{max, scaling}}$ 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 $\nu_{\text{max, fit}}$ values, for the iterations with and
without granulation, as the $\nu_{\text{max, scaling}}$ values decrease below
$\sim$100 $\mu$Hz. 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 $\sim$100 $\mu$Hz 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 $\sim$50 $\mu$Hz (see e.g. Figure
13). Due to the fact that the pulsational power excesses lie partially on top
of this ’flattened’ portion of the granulation morphology, the extraction of
$\nu_{\text{max, fit}}$ between the two iterations shows greater deviance.

Overall, we observe systematic errors of the order of 1% in
$\nu_{\text{max}}$, 3% in $M$ and 1% in $R$, with an intrinsic scatter of the
order of 0.5% in $\nu_{\text{max}}$, 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 (Equations 1 and 2) as expected, with $\Delta
M=3\cdot|\text{d(log }\nu_{\text{max}})|$ and $\Delta R=|\text{d(log
}\nu_{\text{max}})|$. As such, in this and in the subsequent sections, we only
present the mean percentage difference, 1-$\sigma$ scatter, and the mean
precision of the $\nu_{\text{max}}$ values extracted from our PSDs, as those
of the masses and radii can effectively be calculated from the
$\nu_{\text{max}}$ results.

Table 3: The mean percentage difference, 1-$\sigma$ scatter and precision of
the $\nu_{\text{max}}$ values extracted from our single-RG PSDs.

$\nu_{\text{max, fit}}-\nu_{\text{max, scaling}}$ [%]
---
| Pulsations Only | Pulsations+Granulation
Mean | -0.7 | -0.9
1-$\sigma$ | 0.4 | 0.6
Precision | 0.1 | 0.2
666The precision corresponds to the errors propagated from the 68% Bayesian
credible intervals of the marginalised posterior distributions of the fit
parameters.

##  4 Red-giant binary simulations and results

We used a subset of the light curves generated using the methodology laid out
in Section 2.2 for the creation of our binary grid, ensuring that the subset
was relatively representative of the entire grid in terms of $M$, $R$,
$T_{\text{eff}}$ and $\nu_{\text{max}}$. 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 $T_{\text{eff}}$ distributions are displayed in Figure 4.

We therefore generated a total of two sets of 435 (i.e. ${C^{30}_{2}}$) 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 super-Lorentzians), and we perform a differential analysis by
comparing the $\nu_{\text{max}}$ values extracted from our binary-RG PSDs
($\nu_{\text{max, binary}}$) with the maximum likelihood estimates of the
$\nu_{\text{max}}$ values extracted from our single-RG PSDs ($\nu_{\text{max,
single}}$).

###  4.1 Results of red-giant binaries with two pulsating components

Figure 6 and Figure 6 show the $\nu_{\text{max}}$ extraction results for a two
super-Lorentzian granulation model and a four super-Lorentzian granulation
model respectively, where both components (Star A and Star B) in the RG binary
are pulsating. The mean percentage difference and the 1-$\sigma$ scatter of
the $\nu_{\text{max, binary}}-\nu_{\text{max, single}}$ values, and the mean
precision of the $\nu_{\text{max, binary}}$ extraction are detailed in Table
7.

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

Figure 5: A comparison of the $\nu_{\text{max}}$ values of each component
(Star A, left panel; and Star B, right panel) extracted from our synthetic RG-
binary light curves where both components are pulsating, using a two super-
Lorentzian granulation model. The vertical axis corresponds to the difference
between the single-RG reference values for Star A ($\nu_{\text{max, single
(Star A)}}$) and Star B ($\nu_{\text{max, single (Star B)}}$). The horizontal
axis corresponds to the difference between the binary ($\nu_{\text{max,
binary}}$) and the single-RG ($\nu_{\text{max, single}}$) reference values.
The vertical dashed line represents the zero-point of the difference between
$\nu_{\text{max, binary}}$ and $\nu_{\text{max, single}}$. The dash-dotted
green lines represent the interpolated $1-\sigma$ level about the mean scatter
of the datapoints grouped into 10 $\mu$Hz bins along the vertical axis. The
horizontal dashed line represents the zero-point of $\nu_{\text{max, single
(Star A)}}-\nu_{\text{max, single (Star B)}}$. The error bars correspond to
the 68% Bayesian credible intervals of the marginalised posterior
distributions of the binary fit parameters. The symbols are colour-coded
according the percentage light contribution of each component to the binary
light curve.

Figure 6: A comparison of the $\nu_{\text{max}}$ values of each component
(Star A, left panel; and Star B, right panel) extracted from our synthetic RG-
binary light curves where both components are pulsating, using a four super-
Lorentzian granulation model. The axes, symbols, error bars and colour-coding
is the same as in Figure 6. Table 4: The mean percentage difference,
1-$\sigma$ scatter and precision of the $\nu_{\text{max, binary}}$ values
extracted from our RG-binary PSDs, using either two or four super-Lorentzians
to fit the granulation signal.

$\nu_{\text{max, binary}}-\nu_{\text{max, single}}$ [%]
---
|  Star A |  Star B |  Single Pulsator
2 Super-Lorentzians
Mean | -0.2 | -0.06 | -0.1
1-$\sigma$ | 2.1 | 1.7 | 0.5
Precision | 0.8 | 0.7 | 0.3
4 Super-Lorentzians
Mean | 0.008 | -0.15 | -0.05
1-$\sigma$ | 1.9 | 1.6 | 0.5
Precision | 0.8 | 0.7 | 0.3
$|\nu_{\mathrm{max,\ binary\ (2SL)}}-\nu_{\mathrm{max,\ binary\ (4SL)}}|$
Mean | 1.2 | 0.9 | 0.5
1-$\sigma$ | 2.3 | 1.8 | 0.4
777The precision corresponds to the errors propagated from the 68% Bayesian
credible intervals of the marginalised posterior distributions of the fit
parameters. The third row of the table showcases the absolute difference in
the $\nu_{\text{max}}$ values between the different granulation model
configurations (2SL and 4SL).

The most obvious feature in Figures 6 and 6 is the large scatter in the
central region of the plot, which is where the $\nu_{\text{max}}$ values for
Star A ($\nu_{\text{max, single (Star A)}}$) and Star B ($\nu_{\text{max,
single (Star B)}}$) are very similar. This scatter is a result of fitting the
binary star PSDs where the power excesses of each component overlap (e.g.
Figure 16). 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 non-Gaussian posterior distributions of $\nu_{\text{max}}$ values.
This results in the asymmetric, and often large, error bars for the datapoints
in the region around the zero-point value of $\nu_{\text{max, single (Star
A)}}-\nu_{\text{max, single (Star B)}}$.

We also observe a marked increase in the scatter of $\nu_{\text{max,
binary}}-\nu_{\text{max, single}}$ as the light contribution increases (see
zoomed insets in Figure 6). This is once again a consequence of representing
our results in percentages, and is also observed in the single-RG case (see
Section 3): The component with the higher light contribution tends to have the
lower $\nu_{\text{max}}$ (as $\nu_{\text{max}}$ is inversely proportional to
$L$), and therefore similar absolute $\nu_{\text{max, binary}}-\nu_{\text{max,
single}}$ values are larger in terms of percentage at lower $\nu_{\text{max,
single}}$ values.

Our results shown in Table 7 indicate that there is no appreciable systematic
offset in our $\nu_{\text{max, binary}}$ values when compared to our
$\nu_{\text{max, single}}$ values. The small residual systematic offsets that
remain are certainly well below the 1-$\sigma$ scatter reported in the single-
RG case (cf. Table 6). However, the 1-$\sigma$ scatter of the $\nu_{\text{max,
binary}}$ values is much higher, and the mean precision is much lower than in
the single-star 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
1-$\sigma$ scatter and low precision are expected. For the cases where the
power excesses are well separated (e.g. Figure 16), we clearly see a much
smaller scatter that is quantitatively comparable to the single star case
discussed in the Section 3.

Another interesting observation is that there is little difference in the
overall results of the two and four super-Lorentzian component fits. This can
be attributed once again to the degenerate nature of the background fit:
adding additional super-Lorentzians 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 Figure 7, which shows the input super-Lorentzians for
one of our synthetic PSDs compared with those derived from the maximum-
likelihood values for the two and four super-Lorentzian component granulation
models. The background fit is so degenerate that i) two super-Lorentzians 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 super-
Lorentzian model.

Figure 7: A comparison of the input granulation signal with those from the
maximum-likelihood values of the two and four super-Lorentzian granulation
model fits, for one of our binary-RG PSDs where both components are pulsating.
The super-Lorentzians used as inputs are represented by the solid red curves,
with the two and four super-Lorentzians fits represented by the dark blue and
blue-green dash-dotted lines respectively.

Overall, we observe an intrinsic scatter of the order of 2% in
$\nu_{\text{max}}$, corresponding to a scatter of 6% in $M$ and 6% in $R$. We
once again stress that the larger scatter in $\nu_{\text{max}}$ 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 single-RG case and do not exceed the typical
uncertainties for the scaling relations reported in the literature (cf.
Section 3).

Figure 8: A comparison of the $\nu_{\text{max}}$ values for a two super-
Lorentzian (left panel) and a four super-Lorentzian (right panel) granulation
model, extracted from our synthetic RG-binary light curves where only one
component (Star A) is pulsating. The axes, symbols, error bars and colour-
coding is the same as in Figure 6.

###  4.2 Results of red-giant binaries with one pulsating component

Figure 8 shows the $\nu_{\text{max}}$ extraction results for the two and four
super-Lorentzian component granulation models, where only one of the two
components (Star A) in the RG binary is pulsating (e.g. Figure 16). The mean
percentage difference and the 1-$\sigma$ scatter of the $\nu_{\text{max,
binary}}-\nu_{\text{max, single}}$ values, and the mean precision of the
$\nu_{\text{max, binary}}$ extraction are also detailed in the last column of
Table 7.

Once again, our results show that there is no appreciable systematic offset in
our $\nu_{\text{max, binary}}$ values when compared to our $\nu_{\text{max,
single}}$ values. In fact, the results that we had obtained were very similar
to that of the single-RG case: The 1-$\sigma$ scatter and precision of the
$\nu_{\text{max}}$ extraction is almost identical to that of the single-RG
case (detailed in Table 6), with perhaps a slight decrease in mean precision
(0.3% vs. 0.2%). We do not observe the large 1-$\sigma$ scatter that was
observed in the double-pulsator 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 $\nu_{\text{max,
binary}}-\nu_{\text{max, single}}$ as the light contribution increases (see
zoomed insets in Figure 8) as per the double-pulsator case, which is, as
mentioned, due to the representation of our results in percentages.

Similar to our binary-RG 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 Figure 9, which
shows the input super-Lorentzians for one of our single-pulsator synthetic
PSDs compared with those derived from the maximum-likelihood values for the
two and four super-Lorentzian component granulation models. Once again, we
observe that the background fit is so degenerate that i) two super-Lorentzians
can still adequately model the background, and ii) the input parameters are
not recovered using the four super-Lorentzian 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 sufficiently-bright
($K_{\text{p}}<12$) RG.

Overall, we observe an intrinsic scatter of the order of 0.5% in
$\nu_{\text{max}}$, corresponding to a scatter of 1.5% in $M$ and 0.5% in $R$,
which is compatible with the single-RG 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 $\nu_{\text{max, binary}}$
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 Figure 10 show the difference in $\nu_{\text{max,
binary}}$ for the the RG-binary light curves where both components (Star A and
Star B) are pulsating, and the right panel shows the difference in
$\nu_{\text{max, binary}}$ for the RG-binary light curves where only one
component (Star A) is pulsating. All of these results are detailed in the
bottom row of Table 7, where we show the mean percentage difference and the
1-$\sigma$ scatter of the $|\nu_{\text{max, binary (2SL)}}-\nu_{\text{max,
binary (4SL)}}|$ values.

Once again, the greatest percentage difference between the $\nu_{\text{max,
fit}}$ values extracted from fits using the two and four super-Lorentzian
granulation models is in the region around the zero-point value of
$\nu_{\text{max, single (Star A)}}-\nu_{\text{max, single (Star B)}}$. As
mentioned, the $\nu_{\text{max}}$ 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 $\nu_{\text{max}}$
values for the two and four super-Lorentzian granulation model configurations
are small outside of the case of overlapping power excesses, with differences
and scatter of the order of 0.5% in $\nu_{\text{max}}$, corresponding to 1.5%
in $M$ and 0.5% in $R$.

Figure 9: A comparison of the input granulation signal with those from the
maximum-likelihood values of the two and four super-Lorentzian granulation
model fits, for one of our binary-RG PSDs where only one component is
pulsating. The super-Lorentzians used as inputs are represented by the solid
red curves, with the two and four super-Lorentzian fits represented by the
dark blue and blue-green dash-dotted lines respectively. Figure 10: The
difference in the $\nu_{\text{max}}$ values extracted from the fits using a
two super-Lorentzian and a four super-Lorentzian granulation model. Left and
middle panels: The difference in $\nu_{\text{max, fit}}$ for the the RG-binary
light curves where both components (Star A and Star B) are pulsating. Right
panel: The difference in $\nu_{\text{max, fit}}$ for the RG-binary light
curves where only one component (Star A) is pulsating. The axes, symbols,
error bars and colour-coding is the same as in Figure 6, except that the
horizontal axis corresponds to the absolute percentage difference between the
$\nu_{\text{max, fit}}$ values for the two super-Lorentzian (2SL) and four
super-Lorentzian (4SL) cases.

##  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
$\nu_{\text{max}}$. We developed a robust methodology for the simulation of
the light curves of first-ascent RGB single- and binary stars, incorporating
frequency-dependent modewidths and luminosity-dependent 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
$\nu_{\text{max}}$ values of the pulsating component. We also tested two
different granulation model configurations for our RG binaries: i) a two
super-Lorentzian and ii) a four super-Lorentzian granulation profile and
computed the differences between the binary-RG and single-RG reference
$\nu_{\text{max}}$ 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 $\nu_{\text{max}}$,
2.5% in $M$ and 1% in $R$. This is caused by asymmetric mode distribution in
the power excesses (see Figure 1), resulting in differences between the
$\nu_{\text{max}}$ 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
$\nu_{\text{max}}$ values extracted from our single-RG 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 $\nu_{\text{max}}$ for stars with $K_{\text{p}}<12$, as noted by
Kallinger et al. (2014).

  * •

Binary RGs

    * –

Systematics: We find negligible systematic differences between the binary-RG
and single-RG $\nu_{\text{max}}$ values, confirming that the systematic offset
observed in the single-RG case is a methodological issue.

    * –

Two pulsating components: We find that the $\nu_{\text{max}}$ 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
single-RG case.

    * –

One pulsating component: We find that our results are very similar to the
single-RG case, showing that the light contribution of the individual
components has minimal effect on the $\nu_{\text{max}}$ extraction for cases
where the light contribution of the component is above $\sim$ 10%.

    * –

Granulation model configurations: We find minimal difference in the results
when using either two or four super-Lorentzian 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
$\nu_{\text{max}}$ 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
$\nu_{\text{max}}$ 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 observationally-derived
parameters from scaling-relation derived parameters might be a result of the
discrepancy between the observed $\nu_{\text{max}}$, which we found to be
highly sensitive to mode distribution in the power excess, and the theoretical
$\nu_{\text{max}}$. Our results also indicate that the $\sim$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 $\nu_{\text{max}}$, the differences in
$\nu_{\text{max}}$ 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 super-Lorentzian 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 $\nu_{\text{max}}$ 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 relations-based masses/radii and those inferred
from binary dynamics. We can therefore conclude that photometric
contamination, besides decreasing the signal-to-noise ratios of the individual
components in the binary PSD, would have a negligible effect on the extraction
of the global asteroseismic parameter $\nu_{\text{max}}$ from tess data.

###### Acknowledgements.

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’ (IJCI-2015-26034).
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 = 1d-4

        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 = 1d-2

        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) = 1d-3

        remove_small_D_limit = 1d-6
        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 = 5d-2
        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 = 1d-1

        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 = 1d-6
        mesh_dlogX_dlogP_full_off = 1d-12
        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

        ! non-burning 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 = 10ΨΨ
        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

Figure 11: The variation of the normalised mode inertia values for each
frequency output by gyre for a star with $\nu_{\text{max}}\sim 100$ $\mu$Hz.
The $\ell=1$, 2 and 3 modes are colour-coded as displayed the legend. It
should be noted that the local minima in mode inertia in intervals of
$\Delta\nu$, 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 binary-RG PSDs.

Figure 12: An example of a single-RG 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.

Figure 13: An example of a single-RG 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 Figure 13, with the
addition of blue dash-dotted lines to represent the super-Lorentzians.

Figure 14: An example of a RG-binary PSD fit where both components, with very
similar $\nu_{\text{max}}$ values, are pulsating. Here we use four super-
Lorentzians 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
Figure 13.

Figure 15: An example of a RG-binary PSD fit where both components, with very
different $\nu_{\text{max}}$ values, are pulsating. Here we use four super-
Lorentzians 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
Figure 13.

Figure 16: An example of a RG-binary PSD fit where only one component is
pulsating. Here we use four super-Lorentzians 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 Figure 13.