# Predicted asteroseismic detection yield for solar-like oscillating stars
with PLATO

M.J. Goupil  LESIA, Observatoire de Paris, PSL, CNRS, Sorbonne Université,
Université Paris Cité, 5 place Jules Janssen, 92195 Meudon, France  C. Catala
LESIA, Observatoire de Paris, PSL, CNRS, Sorbonne Université, Université Paris
Cité, 5 place Jules Janssen, 92195 Meudon, France  R. Samadi  LESIA,
Observatoire de Paris, PSL, CNRS, Sorbonne Université, Université Paris Cité,
5 place Jules Janssen, 92195 Meudon, France  K. Belkacem  LESIA, Observatoire
de Paris, PSL, CNRS, Sorbonne Université, Université Paris Cité, 5 place Jules
Janssen, 92195 Meudon, France  R.M. Ouazzani  LESIA, Observatoire de Paris,
PSL, CNRS, Sorbonne Université, Université Paris Cité, 5 place Jules Janssen,
92195 Meudon, France  D. R. Reese  LESIA, Observatoire de Paris, PSL, CNRS,
Sorbonne Université, Université Paris Cité, 5 place Jules Janssen, 92195
Meudon, France  T. Appourchaux  Institut d’Astrophysique Spatiale, UMR 8617,
Université Paris-Saclay, Bâtiment 121, 91045, Orsay  S. Mathur  Instituto de
Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Universidad
de La Laguna (ULL), Departamento de Astrofísica, E-38206 La Laguna, Tenerife,
Spain  J. Cabrera  Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR),
Institut für Planetenforschung, Rutherfordstr. 2, 12489 Berlin, Germany  A.
Börner  Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Institut für
Optische Sensorsysteme, Rutherfordstr. 2, 12489 Berlin, Germany  C. Paproth
Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Institut für Optische
Sensorsysteme, Rutherfordstr. 2, 12489 Berlin, Germany  N. Moedas  Instituto
de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das
Estrelas, 4150-762 Porto, Portugal  Departamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, s/n,
PT4169-007 Porto, Portugal  K. Verma  Department of Physics, Indian Institute
of Technology (BHU), Varanasi-221005, India  Y. Lebreton  LESIA, Observatoire
de Paris, PSL, CNRS, Sorbonne Université, Université Paris Cité, 5 place Jules
Janssen, 92195 Meudon, France  Université de Rennes, CNRS, IPR (Institut de
Physique de Rennes) – UMR 6251, 35000 Rennes, France  M. Deal  LUPM,
Université de Montpellier, CNRS, Place Eugène Bataillon, 34095 Montpellier,
France  J. Ballot  IRAP, Université de Toulouse, CNRS, CNES, UPS, 14 avenue
Edouard Belin, 31400 Toulouse, France  W. J. Chaplin  School of Physics &
Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK  J.
Christensen-Dalsgaard  Stellar Astrophysics Centre, Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  M.
Cunha  Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto,
CAUP, Rua das Estrelas, 4150-762 Porto, Portugal  A. F. Lanza  INAF-
Osservatorio Astrofisico di Catania, Via S. Sofia, 78 - 95123 Catania, Italy
A. Miglio  Dipartimento di Fisica e Astronomia, Università degli Studi di
Bologna, Via Gobetti 93/2, I-40129 Bologna, Italy  INAF - Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, I-40129
Bologna, Italy  T. Morel  Space sciences, Technologies and Astrophysics
Research (STAR) Institute, Université de Liège, Quartier Agora, Allée du 6
Août 19c, Bât. B5c, B4000 Liège, Belgium  A. Serenelli  Institute of Space
Sciences (ICE, CSIC), Carrer de Can Magrans S/N, 08193, Bellaterra, Spain
Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capita 4, 08034,
Barcelona, Spain  B. Mosser  LESIA, Observatoire de Paris, PSL, CNRS, Sorbonne
Université, Université Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
O. Creevey  Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4,
France  A. Moya  Departament d’Astronomia i Astrofísica, Universitat de
València, C. Dr. Moliner 50, 46100, Burjassot, Spain  R.A. Garcia  AIM, CEA,
CNRS, Univ. Paris-Saclay, Univ. Paris Diderot, Sorbonne Paris Cité, F-91191
M.B. Nielsen  School of Physics & Astronomy, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK  E. Hatt  School of Physics & Astronomy,
University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

In this work, we determine the expected yield of detections of solar-like
oscillations for the targets of the foreseen PLATO ESA mission. Our estimates
are based on a study of the detection probability, which takes into account
the properties of the target stars, using the information available in the PIC
1.1.0, including the current best estimate of the signal-to-noise ratio (S/N).
The stellar samples, as defined for this mission, include those with the
lowest noise level (P1 and P2 samples) and the P5 sample, which has a higher
noise level. For the P1 and P2 samples, the S/N is high enough (by
construction) that we can assume that the individual mode frequencies can be
measured. For these stars, we estimate the expected uncertainties in mass,
radius, and age due to statistical errors induced by uncertainties from the
observations only. We used a formulation from the literature to calculate the
detection probability. We validated this formulation and the underlying
assumptions with Kepler data. Once validated, we applied this approach to the
PLATO samples. Using again Kepler data as a calibration set, we also derived
relations to estimate the uncertainties of seismically inferred stellar mass,
radius, and age. We then applied those relations to the main sequence stars
with masses equal to or below 1.2 $\rm{M}_{\odot}$ belonging to the PLATO P1
and P2 samples and for which we predict a positive seismic detection. We found
that we can expect positive detections of solar-like oscillations for more
than 15 000 FGK stars in one single field after a two-year observation run.
Among them, 1131 main sequence stars with masses of $\leq 1.2\rm{M}_{\odot}$
satisfy the PLATO requirements for the uncertainties of the seismically
inferred stellar masses, radii, and ages. The baseline observation programme
of PLATO consists of observing two fields of similar size (one in the southern
hemisphere and one in the northern hemisphere) for two years apiece.
Accordingly, the expected seismic yields of the mission amount to over 30000
FGK dwarfs and subgiants, with positive detections of solar-like oscillations.
This sample of expected solar-like oscillating stars is large enough to enable
the PLATO mission’s stellar objectives to be amply satisfied. The PLATO
mission is expected to produce a catalog sample of extremely well seismically
characterized stars of a quality that is equivalent to the Kepler LEGACY
sample, but containing a number that is about 80 times greater, when observing
two PLATO fields for two years apiece. These stars are a gold mine that will
make it possible to make significant advances in stellar modelling.

##  1 Introduction

The PLAnetary Transits and Oscillations of stars (PLATO) mission is the ESA
Cosmic Vision M3 mission and its launch is scheduled for the end of 2026. Its
main objectives are 1) the detection and accurate and precise characterisation
of exoplanets down to Earth-size planets in the habitable zone of solar-like
stars, 2) the accurate and precise determinations of the basic parameters of
their host stars (mass, radius, age, etc), and 3) careful statistical analyses
of the above characteristics in order to better understand the formation and
evolution of planetary and stellar systems (hereafter, stellar systems).
Sufficiently precise determinations of the characteristics of these stellar
systems require very high-quality photometry carried out continuously over
long periods of time, hence, the need for a space mission. Furthermore, the
required accuracy calls for improvements of the stellar models used to
estimate the age of the star-planet system. Improving the processes of stellar
modelling is thus an intrinsic main objective of the PLATO mission (Rauer et
al., 2014, 2023). The science operation phase of PLATO is planned to last for
four years with a possible extension of 4.5 years. The baseline is two long-
pointings (LOPs) observing one field for two years apiece. PLATO will collect
high-precision photometric lightcurves of thousands of stars, which will be of
particular interest for asteroseismological studies. To reach its objectives,
the PLATO mission has defined a core programme with several types of stellar
samples. Here, we focus on the P1, P2, and P5 samples. The P1 and P2 samples
(hereafter, P1P2) consist in the brightest PLATO targets which will be
observed with a 25 s cadence. The P1 sample (resp., P2) includes at least 15
000 (resp., 1000) dwarf and subgiant stars (types F5 to K7), with $V\leq 11$
mag (resp. $V\leq 8.5$) observed over the mission and a noise level of
$<50~{}{\rm ppm~{}h^{1/2}}$. The noise level of those samples has been adapted
to enable precise seismological studies. The P5 sample contains at least 245
000 dwarf and subgiant stars (F5-K7), with $V\leq 13$ mag cumulative over two
target fields. Sampling of these light curves will be 600 s, but it is planned
to acquire light curves with a shorter cadence of 50 s for the brightest
targets in P5 or for targets of particular interest. For that reason, we also
consider the stars of the P5 sample. Input information about the stars in
those samples has been gathered in the Input PLATO Catalogue (PIC, Montalto et
al. (2021); Nascimbeni et al. (2022) ). Here, we use version PICv1.0.0. For
more details about the PLATO project, we refer to Rauer et al. (2014) and
Rauer et al. (2023).

Asteroseismology, namely, the detection, measurement, and analysis of stellar
oscillations, is one of the main tools that will be used in the framework of
the PLATO mission to achieve these objectives. The mission is indeed designed
to detect stellar oscillation modes for different classes of stars, including
solar-type ones which are its prime targets for exoplanet detection.
Asteroseismology can be used to infer the stellar properties, either by
measuring global seismic parameters such as $\nu_{\rm max}$, the frequency at
which the oscillation modes have their maximum amplitude, and/or $\Delta\nu$,
the large frequency separation, which characterizes the pattern of power
spectra of solar-like oscillations. In ideal conditions, we can precisely
measure the frequencies, amplitudes, and widths of individual oscillation
modes, which then provide tighter constraints on the stellar mass, radius, and
age.

Starting with the Sun, two decades of observations have demonstrated that
solar-like oscillations offer a powerful way to derive precise stellar masses,
radii, densities, and ages, provided that high-quality seismic parameters are
available (see for instance the reviews by Chaplin & Miglio, 2013;
Christensen-Dalsgaard, 2016, 2018; García & Ballot, 2019; Jackiewicz, 2021;
Serenelli et al., 2021). However, this requires short-cadence (less than 1 mn
for dwarfs and subgiants), ultra-high photometric precision (at the level of
parts-per-million or ppm), and nearly-uninterrupted long-duration (from months
to years) monitoring.

Here, we investigate the seismic performances of the core programme of the
PLATO mission; more precisely, our goal is twofold: 1) to obtain an estimate
of the number of stars of the PLATO catalogue for which solar-like
oscillations can be detected and 2) to obtain an estimate of the uncertainties
on the stellar mass, radius, and age inferences in cases where the data are of
high enough quality that individual modes can be detected and their
frequencies be measured. By uncertainties here we mean those statistical
errors which can only be decreased with higher quality observations (i.e.
higher S/N values and longer observation times). We then estimate the
(statistical) uncertainties on stellar mass, radius, and age, which result
from the propagation of observational errors on the seismic data. We stress
that systematic errors and/or biases must be added to the statistical errors
to obtain the final error budget. In the present case, systematic errors
and/or biases mostly depend on our ability to improve our stellar modelling.
This is only briefly discussed at the end of the paper.

The paper is organized as follows. In Sect. 2, we detail the theoretical
approach we used to derive the probability of detection of solar-like
oscillations. The calculation is based on (Chaplin et al., 2011b, hereafter
C11) ’s methodology which was developed for studying the asteroseismic
potential of the Kepler mission and later on used for the TESS (Campante et
al. (2016),Schofield et al. (2019)) and CHEOPS (Moya et al., 2018) missions.
In order to validate our own computations, we use several samples of stars
observed by the NASA Kepler mission (Borucki et al., 2010; Koch et al., 2010)
with and without detected solar-like oscillations. These test samples are
presented in Sect. 3, together with the results about the reliability of the
detection probability. In Sect. 4, we present our approach to estimate the
uncertainties in the seismic inferences of stellar masses, radii, and ages
(hereafter, MRA) when assuming that frequencies of individual modes can be
measured. In Sect. 5, we present the computations of the detection
probabilities for stars in the P1P2 and P5 samples. Those calculations predict
the number of stars in each sample for whichvwe expect to detect solar-like
oscillations in the core programme of PLATO. As assumed by the PLATO
consortium, we consider two observational conditions: 1) we take the adopted
noise level arising from observations from nominal 24 cameras at the beginning
of life (BOL), so there is no degradation of the instrument; and 2) we take
the noise level arising from observations by 22 cameras only at the end of of
life (EOL), allowing for some degradation of the instrument as usually taken
as reference by ESA. We use the noise level given in the PIC that takes into
account the fact that each target is observed by either 6, 12, 18, or 24
cameras (Montalto et al., 2021; Nascimbeni et al., 2022) . The calculations
are carried out for one LOP which will continuously observe the same field for
at least two years. In Sect. 6, for stars in the P1P2 samples with positive
seismic detection, we compute the expected uncertainties of the individual
frequencies and deduce the resulting MRA uncertainties using the approach
described in Sect. 4. A summary and some discussion are provided in Sect. 7.
Finally, we give our conclusions in Sect. 8.

##  2 Global solar-like oscillation detection level

###  2.1 Detection probability

In this section, we derive the formalism for computing the detection
probability of solar-like oscillations, $P_{\rm det}$, in the photometric
power density spectra. It is the probability that such oscillations are
detected globally in the power spectrum, not to be confused with the
probability to detect and measure the properties of individual oscillation
modes. $P_{\rm det}$ is calculated according to Eqs. 26 and 28 of C11. The
statistics of the power spectrum is a $\chi^{2}$ with $2N_{\rm b}$ degrees of
freedom, where $N_{\rm b}$ is the number of independent bins in the envelope
band, $\delta\nu_{\rm env}$, of the power spectrum. The probability of having
a peak above a given level $s$, due to noise only, in the binned power
spectrum is then given by Eq. 1 of Appourchaux (2004), namely,

$\displaystyle P(s^{\prime}>s,N_{b})=\int_{s}^{\infty}\frac{1}{\Gamma(N_{\rm
b})}~{}\frac{u^{N_{\rm b}-1}}{S^{N_{\rm b}}}~{}e^{-u/S}{\rm d}u\;,$ (1)

where $S$ is the mean of the power spectrum and $\Gamma(n)$ is the Gamma
function.

Here we seek the probability for any value $s^{\prime}$ to be larger than a
given level $s$ for a binned power spectrum normalised to the noise level,
$S/N$,

$\displaystyle P(s^{\prime}>s,N_{\rm
b})=\int_{s}^{\infty}\frac{1}{\Gamma(N_{\rm b})}~{}\frac{u^{N_{\rm
b}-1}}{(S/N)^{N_{\rm b}}}~{}e^{-u/(S/N)}{\rm d}u\;.$ (2)

With the change of variable $u^{\prime}=u/(S/N)$, the probability is given by

$\displaystyle P(x^{\prime}>x,N_{\rm
b})=\int_{x}^{\infty}\frac{1}{\Gamma(N_{\rm b})}~{}u^{\prime{N_{\rm
b}-1}}~{}e^{-u^{\prime}}{\rm d}u^{\prime}\;,$ (3)

which is the probability that the normalised power take any value
$x^{\prime}=(S/N)^{\prime}$ larger than a given level $x=S/N$. For
$x\rightarrow 0$, that-is $S\rightarrow 0$, corresponding to no signal in the
power spectrum, the above probability is $P=1$ as expected. We note that for
convenience, C11 considered a slightly different formulation and calculated
the probability that any value $x^{\prime}=[(S+N)/N]^{\prime}$ is larger than
a given level $x=(S+N)/N$, so that in absence of signal $S$, $x=1$, an
approach which we use in the following.

As in C11, using the above equation, we first calculate a signal threshold
$S_{\rm thres}$ such that the probability for any value $S\geq S_{\rm thres}$
to be due to pure noise is smaller than a predefined value $p_{\rm fa}$ (false
alarm probability). In practice, we remain very conservative and choose a very
small value for this false alarm probability, $p_{\rm fa}=0.1\%$.

In a second step, following C11, we now consider that there is some seismic
signal $S_{\rm mod}$ in the power spectrum. We obtain the probability of
detection of this signal within the false alarm probability $p_{\rm fa}$ as
the probability of being above the threshold defined previously (Eq. 3), but
with a new normalisation by $S_{\rm mod}+N$,

$\displaystyle P_{\rm det}\equiv P(s^{\prime}>S_{\rm thres}+N)$
$\displaystyle=\int_{S_{\rm thres}+N}^{\infty}\frac{1}{\Gamma(N_{\rm
b})}~{}\frac{u^{N_{\rm b}-1}}{(S_{\rm mod}+N)^{N_{\rm b}}}\,{\rm
e}^{-u/(S_{mod}+N)}\,{\rm d}u\,.$ (4)

Choosing as before $u^{\prime}=u/(S_{\rm mod}+N)$ as the new variable, we
obtain

$\displaystyle P_{\rm det}=\int_{u_{0}}^{\infty}\frac{1}{\Gamma(N_{\rm
b})}~{}\frac{[(S_{\rm mod}+N)u^{\prime}]^{N_{\rm b}-1}}{(S_{\rm
mod}+N)^{N_{\rm b}}}\,{\rm e}^{-u^{\prime}}(S_{\rm mod}+N)\,{\rm
d}u^{\prime}\,,$ (5)

which finally becomes

$\displaystyle P_{\rm det}=\int_{u_{0}}^{\infty}\frac{1}{\Gamma(N_{\rm
b})}~{}u^{\prime N_{\rm b}-1}\,{\rm e}^{-u^{\prime}}\,{\rm d}u^{\prime}\,,$
(6)

where

$\displaystyle u_{0}$ $\displaystyle=\frac{S_{\rm thres}+N}{S_{\rm
mod}+N}=\frac{1+(S/N)_{\rm thres}}{1+(S/N)_{\rm mod}},$ (7)

and $(S/N)_{\rm thres}$ is given by the solution of Eq. (3) for $P=p_{\rm
fa}$. We note that Eqs. (6)-(7) are equivalent to Eqs. (28) and (29) of C11.

In practice, the probability $P_{\rm det}$ can be computed for any particular
target of PLATO or any other mission, such as Kepler, as soon as we have
determined: (i) the signal-to-noise ratio (S/N) in the power spectrum
$(S/N)_{\rm mod}$ and (ii) the number $N_{\rm b}$ of bins to consider in the
envelope band of the oscillation modes. The derivation of these two quantities
is detailed below. The last technical difficulty is then the computation of
the $\Gamma$ function for the expected very large values of $N_{\rm b}$, which
we perform using the classical asymptotic approximation,

$\ln(\Gamma(n))\approx(n-1/2)\ln(n)-n+1/2\ln(2\pi),$

which is valid for large values of $n$.

We consider that there is detection of power excess due to solar-like
oscillations when the probability $P_{\rm det}>0.99$. Power spectra showing
peaks above $S_{\rm thres}$ as defined by Eq. (3) for $p_{\rm fa}=0.1\%$, but
with $P_{\rm det}\leq 0.99$ are considered as potentially indicating solar-
like oscillations, but with too little confidence to derive any global seismic
parameters, let alone properties of individual modes. This is a very
conservative position, so that the resulting detection probability can be
considered as a lower limit of what can be expected with PLATO.

##### 2.1.1 Global S/N in the power spectrum:

The global S/N entering Eq. (7) is calculated as

$\displaystyle(S/N)_{\rm mod}=\frac{P_{\rm tot}}{N_{\rm tot}}\,,$ (8)

where $P_{\rm tot}$ is the total power density in the oscillations and $N_{\rm
tot}$ the total power density in the noise, both quantities being estimated
near $\nu_{\rm max}$, the frequency where the oscillations reach their maximum
amplitude. In the following, we detail how these two quantities were
calculated.

##### 2.1.2 Oscillation power density $P_{\rm tot}$:

The calculation of the oscillation power density $P_{\rm tot}$ (in
ppm${}^{2}/\mu$Hz) can be performed using a formulation as established by C11
restricted to the calculation of the probability of a global detection of
power excess due to solar-like oscillations. The power density (in
ppm${}^{2}/\mu$Hz) is given by (Eq. 19 in C11):

$\displaystyle P_{\rm tot}\simeq\frac{1}{2}\,\frac{V_{\rm mod}^{2}{\rm A}_{\rm
max}^{2}}{\Delta\nu}\,,$ (9)

where $\Delta\nu$ is in $\mu$Hz , $V_{\rm mod}^{2}\simeq 3.1$ is the average
visibility calculated for a set of 4 modes $\ell=0-3$ (Ballot et al., 2011),
and $\rm{A}_{\rm max}$ (in ppm) is the maximum oscillation amplitude of modes,
reached at frequency $\nu\simeq\nu_{\rm max}$. The scaling laws for the
determination of $\Delta\nu$ are given in Appendix A. The estimate of
$\rm{A}_{\rm max}$ is discussed in Sect. 2.2 and Appendix A.

##### 2.1.3 Noise power density $N_{\rm tot}$:

We must also estimate the total noise power density$,N_{\rm tot}$. It is
composed on the one hand of the instrumental noise, $N_{\rm inst}$, which
includes photon noise and all other instrumental contributors to the noise,
and on the other hand of the stellar intrinsic noise, $N_{\rm gran}$, which we
assume is dominated by granulation noise at the relevant frequencies. The
total noise power density is then given by:

$\displaystyle N_{\rm tot}=N_{\rm inst}+N_{\rm gran}\,.$ (10)

Following (Samadi et al., 2019, hereafter S19), the granulation noise power
density, $N_{\rm gran}$, is calculated as a scaling power law of $\nu_{\rm
max}$ (i.e. Eq. 36 in S19) , as in Kallinger et al. (2014). The instrumental
noise, $N_{\rm inst}$, including photon noise as well as all other sources of
noise from the instrument or from the background (satellite jitter, readout
noise, digitisation noise, stellar background, zodiacal light, etc), will be a
major contributor to the total noise entering the calculation of the
$(S/N)_{\rm mod}$ term involved in Eq. (7). It depends on which sample of
stars is considered: Kepler stars or PLATO targets and will discussed in Sect.
2.2 below.

##### 2.1.4 Number of bins in the oscillation mode envelope:

Once $P_{\rm tot}$ and $N_{\rm tot}$, and thus $(S/N)_{\rm mod}$ are
determined, we only need to determine the number of frequency bins $N_{\rm b}$
in the oscillation envelope in order to apply Eqs. (3)-(6)-(7). This number is
given by

$N_{\rm b}=Int\left[T_{\rm obs}~{}\delta\nu_{\rm env}\times 10^{-6}\right]\,,$
(11)

where $T_{\rm obs}$ is the total time interval of the photometric monitoring,
in seconds, $\delta\nu_{\rm env}$ is the frequency range over which the
oscillations are present in the power spectrum, in $\mu$Hz, and $Int$ denotes
the integer part.

The parameter $\delta\nu_{\rm env}$ entering Eq. (11) is essential, because
with the observing time, it controls the number of degrees of freedom of the
$\chi^{2}$ statistics followed by the power spectrum.

###  2.2 Adopted inputs for the calculation of the detection probability
$P_{\rm det}$

In order to compute the global detection probability $P_{\rm det}$, we must
provide as input the S/N, namely, $(S/N)_{\rm mod}$, the width of the envelope
of the oscillations, $\delta\nu_{\rm env}$, and the observation time, $T_{\rm
obs}$, for each target. The S/N, here, $(S/N)_{\rm mod}$ in Eq. (8) involves
the power $P_{\rm tot}$. We estimate $P_{\rm tot}$ at its maximum namely, at
$\nu_{\rm max}$ leading $(S/N)_{\rm mod}$ at its maximum which we denote
hereafter $(S/N)_{\rm max}$ for clarity. We then use Eq. (9) assuming the
existence of a regular pattern in a power spectrum every $\Delta\nu$ and not
individual modes.

####  2.2.1 Amplitude at maximum power

The amplitude of maximum power, $\rm{A}_{\rm max}$, involved in $P_{\rm tot}$
is obtained in the literature under the form of an empirical relation
depending on some combination of the global parameters among stellar mass,
stellar radius, effective temperature and/or equivalently the seismic global
parameters $\nu_{\rm max}$, and $\Delta\nu$. We consider two relations,
respectively given by C11 and S19. Both relations are scaled to the solar
values. In Appendix A, we show that our adopted recalibrated theoretical
values for $\rm{A}_{\rm max}$, $\rm{A}_{\rm max,scal}$ as given in Table 1,
are in good agreement with the observed amplitudes at maximum power $A_{\rm
max,obs}$, as derived by Lund et al. (2017) (their Table 3) for the Kepler
LEGACY sample. In Table 1, we show how the stellar mass is derived from the
scaling relation Eq. (34) of Appendix A, relating the mass $M$ to the
effective temperature $T_{\rm eff}$ and $\nu_{\rm max}$, namely,

$\displaystyle\frac{\rm{M}}{\rm{M}_{\odot}}=\left(\frac{\nu_{\rm
max}}{\nu_{\rm max,\odot}}\right)^{-0.28}\left(\frac{\rm{T}_{\rm
eff}}{\rm{T}_{\rm eff,\odot}}\right)^{3/2}\,.$ (12)

For the Kepler samples, $\nu_{\rm max}$ values are taken from the measurements
by Lund et al. (2017). For the PLATO targets, in Sect. 5 and 6, we will take
the stellar radii and effective temperatures from PICv1.1.0 to derive
$\nu_{\rm max}$. For the solar values, we adopt $\rm{A}_{\rm
max,bol,\odot}=2.53~{}$ppm (rms value, see Michel et al., 2009), $\nu_{\rm
max,\odot}=3090\,\mu$Hz, $\Delta\nu_{\odot}=135.1~{}\mu$Hz, and $\rm{T}_{\rm
eff,\odot}=5777$ K throughout.

Table 1: Adopted amplitude formulation for $\rm{A}_{\rm max,scal}$ ($\rm{A}_{\rm max,C11}$ and $\rm{A}_{\rm max,S19}$ are obtained using the scaling relations as in C11 and S19, respectively in Appendix. A. As described in Sect. 2.2 and detailed in Appendix A, the proportionality factors are calibrated to the observations inferred by Lund et al. (2017) ($\nu_{\rm max}$ in $\mu$Hz, the mass, $M$, in solar units).  $\nu_{\rm max}$ | $\leq 2500$ | $>2500$
---|---|---
$M\leq 1.15$ | $\rm{A}_{\rm max,C11}*1.31$ | $\rm{A}_{\rm max,C11}*1.19$
$M>1.15$ |  $\rm{A}_{\rm max,S19}*0.95$ $\rm{A}_{\rm max,C11}*0.95$ |

####  2.2.2 Instrumental noise

As discussed in Sect. 2.1, we need to estimate the instrumental noise, $N_{\rm
inst}$, in the power spectrum. The Kepler instrumental noise used by C11 is
taken from Gilliland et al. (2010), namely,

$\displaystyle N_{\rm inst}=2~{}N_{\rm rand}^{2}~{}10^{-6}~{}dt\,,$ (13)

where $N_{\rm rand}$, the total random noise per time interval of the data
series (in ppm2) is given by

$\displaystyle N_{\rm
rand}^{2}=\alpha\left(1+0.1604\left(\frac{12}{K_{p}}\right)^{5}10^{-0.4~{}(12-Kp)}\right)\,,$
(14)

with

$\displaystyle\alpha=\frac{10^{5}}{1.28}10^{-0.4~{}(12-Kp)}\,.$ (15)

In the above equations, $K_{p}$ is the Kepler magnitude of the star, ${\rm
d}t$ is the cadence of the photometric series, namely, ${\rm d}t=58.85$
seconds for the Kepler short cadence mode. The remaining factor of 2 in Eq.
(13) accounts for the choice of a single-sided power spectrum in C11, a
convention that we adopt for the remainder of the paper.

For the PLATO calculations in Sects. 5 and 6, we use the noise level, $N_{\rm
inst}$, that is available for each target in the PLATO catalogue, taking into
account all instrumental sources of noise according to the most up-to-date
understanding of the instrument (Rauer et al. (2023) and Sect. 5).

####  2.2.3 Width of the envelope of the oscillations, $\delta\nu_{\rm env}$

The parameter $\delta\nu_{\rm env}$ can be measured by the width of the
assumed Gaussian-like shape envelope of the oscillations in the power
spectrum. Several formulations have been suggested for $\delta\nu_{\rm env}$
in the literature. They generally take the form of a scaling relation of the
type $\delta\nu_{\rm env}=a\nu_{\rm max}^{b}$. The coefficients $a$ and $b$
are obtained by fitting the Kepler data. Their values differ according to
whether one considers MS stars of spectral type G, K or hotter stars of
spectral type F or subgiants, or red giants (Kim & Chang (2021) and references
therein). For instance, Kim & Chang (2021) found slightly different values
depending on the formulation assumed for the granulation noise background. We
find that these different relationships remain within the upper and lower
limits $\delta\nu_{\rm env}=\nu_{\rm max}$ and $\delta\nu_{\rm env}=\nu_{\rm
max}/2$, respectively. Specificlly at low $\nu_{\rm max}$ (i.e. for more
evolved stars), the curves are close to $\nu_{\rm max}/2$, while the curves
approach $\nu_{\rm max}$ at higher $\nu_{\rm max}$ (i.e. for younger stars).
In the following, we therefore consider both cases $\delta\nu_{\rm
env}=\nu_{\rm max}$ and $\delta\nu_{\rm env}=\nu_{\rm max}/2$ , but keep the
conservative case $\delta\nu_{\rm env}=\nu_{\rm max}/2$ for our baseline and
estimate the changes in the detection predictions when using $\delta\nu_{\rm
env}=\nu_{\rm max}$.

####  2.2.4 Main sequence versus subgiant stars

For purpose of discussion presented later on in this work, we distinguish the
cases of main sequence stars (MS stars) and subgiants. From a stellar
evolutionary point of view, the subgiant phase starts when there is not enough
hydrogen left at the centre to produce nuclear energy and kinetic pressure to
sustain gravity. We use the central hydrogen mass fraction, $X_{c}$, to define
a threshold. The MS stars are then defined with $X_{c}\geq 10^{-6}$. According
to our stellar models for the range of mass of interest here and the adopted
solar chemical composition, the subgiants satisfy

$\log\left(\frac{L}{L_{\odot}}\right)>10\;\left(\log T_{\rm
eff}-3.7532\right)+0.25\,.$ (16)

The transition between MS tars and subgiants is located in a HR diagram in
Fig.11 in Appendix C.

Mass subsamples: In the present work, we consider that only MS stars with
$M\leq 1.6~{}\rm{M}_{\odot}$ can show solar-like oscillations whereas subgiant
being evolved and therefore cooler, can still oscillate with solar-like
oscillations while being more massive. We therefore study more carefully the
sample of MS stars with seismic masses below 1.6 $\rm{M}_{\odot}$ while no
mass restriction is made for the subgiants. The mass threshold corresponds
approximatively to the transition between stars with no convective core like
the Sun and stars with a convective core in the MS. We also draw a specific
attention to the subsample of main-sequence (MS) stars with seismic masses
$M\leq 1.2~{}\rm{M}_{\odot}$. The reason is that the stellar requirements of
the PLATO mission are established for a star like the Sun (in mass and age).

##  3 Validation of the calculation of the detection probability with Kepler
stars

The formalism described in Sect. 2 must be validated before being applied to
the stars of the PLATO Input catalogue. Here we use several samples of stars
observed by Kepler, in order to verify the performance of this formalism in
predicting detectability of solar-like oscillations. In other words, we
checked with which confidence level Eqs. (3)-(6)-(7), with the above
prescriptions for $P_{\rm tot}$, $N_{\rm tot}$ and $N_{\rm b}$, can predict
whether or not solar-like oscillations can be detected. For that purpose, we
use two types of Kepler stars, those for which such oscillations were or were
not detected. The confidence level will be measured in terms of false positive
and false negative predicted detections.

###  3.1 Kepler data sets for calibration and validation of theoretical
calculations

####  3.1.1 Sample 1: large sample of stars with solar-like oscillations
detected by Kepler

The first data set used to construct this sample is a compilation of Kepler
short cadence stars by (Serenelli et al., 2017, hereafter S17). It includes
415 stars with known detected oscillations, as already reported by C11. Since
2011, these stars were further observed over time intervals ranging from 40
days up to 1055 days. For most of these stars, only the global seismic
parameters $\nu_{\rm max}$ and $\Delta\nu$ are available.

A second Kepler data set used to construct our sample 1 is an updated
compilation by (Mathur et al., 2022, hereafter M22) of the Kepler short-
cadence stars with detected solar-like oscillations, derived on the basis of
samples from C11, Chaplin et al. (2014) and S17. It provides a homogeneous
catalog of global seismic parameters for 624 stars.

In order to build a final sample of stars with detected oscillations with all
the necessary parameters available, we considered the set of 413 stars common
to M22 (updated $\nu_{\rm max}$ and $\Delta\nu$) and S17 (observing intervals
and grid-based inferred stellar mass and radius). In the following we use the
updated $\nu_{\rm max}$ and $\Delta\nu$ from M22. In Appendix B, we look at
the impact of choosing the values of $\nu_{\rm max}$, $\Delta\nu$, and $T_{\rm
eff}$ from S17 instead of M22.

####  3.1.2 Sample 2: large sample of stars with no oscillations detected by
Kepler

As a second sample, we consider the list of 990 Kepler short-cadence main-
sequence solar-like stars for which analyses revealed no detected
oscillations, as published by (Mathur et al., 2019, hereafter M19). For each
star in sample 2, the value of $\nu_{\rm max}$ is computed according to

$\displaystyle\frac{\nu_{\rm max}}{\nu_{\rm max,\odot}}=\frac{\rm
g}{\rm{g}_{\odot}}\left(\frac{\rm{T}_{\rm eff}}{\rm{T}_{\rm
eff,\odot}}\right)^{-1/2},$ (17)

where the surface gravity $g=GM/R^{2}$ and $T_{\rm eff}$ are taken from M19.
111For the star KIC 4464952, we rather use the LAMOST value $T_{\rm
eff}=6161\pm 214$K.. Stellar masses are obtained from the seismic scaling law
Eq. (12). This provides $\delta\nu_{\rm env}$.The observation time, $T_{\rm
obs}$, is taken from KASOC (Kepler Asteroseismic Science Operations Center
provides asteroseismological data).

Here again, we focused on stars with masses $M<1.6~{}\rm{M}_{\odot}$. The
resulting set of 833 stars constitutes our final sample of Kepler non-
oscillating stars (sample 2).

####  3.1.3 Kepler LEGACY sample

Finally, we need to validate and calibrate the calculation of the oscillation
maximum amplitude $\rm{A}_{\rm max}$, as detailed in Appendix A, as well as
Libbrecht (1992)’s relation between individual mode frequencies, linewidths,
and S/Ns (see Appendix D), We then used the Kepler Legacy sample, which is
composed of 66 main-sequence stars with the highest quality of seismic data
(Lund et al., 2017) (in the following). For those stars, individual modes are
identified. Indeed solar-like oscillation modes can be described by spherical
harmonics with spherical degree $\ell$ and azimuthal order $m$ for their
surface geometry and by the radial order $n$ labelling the overtones of a
given $\ell,m$ mode. When rotation is not taken into account or cannot be
detected seismically, the modes are $m$-degenerate and the mode frequencies do
not depend on $m$. This is the case here, so for each individual mode
$\ell,n$, the frequency, amplitude and line width are measured with the
highest precision. For those stars, the observed values of $\nu_{\rm max}$ are
taken from L17.

###  3.2 Results of the validation: Performance of the detection probability
approach

Our approach for calculating the probability to globally detect solar-like
oscillations was tested against the above Kepler samples. Using the formalism
described in Sect. 2.1 and the various needed inputs as explained in Sect.
2.2, we assessed on the one hand the fraction of Kepler targets with detected
oscillations for which we predict no detection (false negatives) and on the
other hand the fraction of Kepler targets with no detected oscillations for
which we predict detection (false positives).

In prevision of the investigation for the PLATO case, we made as our baseline
the conservative choices of a positive detection when $P_{\rm det}>0.99$ and
$\delta\nu_{\rm env}=\nu_{\rm max}/2$. As summarized in Table 17 in Appendix
B, considering the total population of 1349 Kepler stars (MS stars with masses
$M<1.6\rm{M}_{\odot}$ and subgiants of all masses, hereafter $R_{1}$ sample)
with both predicted false seismic positive (186 stars) and negative (40 stars)
detections of oscillations in the baseline conditions leads to an
underestimate of the number of real detections (false negative) by $\sim 3\%$
for the PLATO samples. On the other hand, we can see in Table 17 that one
overestimates the number of real detection (false positive) by 14%. If one
considers only MS stars with masses $M\leq 1.2\rm{M}_{\odot}$, we overestimate
the number of real detection by 7%. These tests using Kepler results confirm
that our approach is valid within the quoted uncertainties and will be used
for the PLATO targets in Sect.5. The detailed results of the calculations of
the above results, as well as justifications of the choices made for defining
our baseline, are given in Appendix B.

We must stress here that the high percentage (14%) of false positive detection
for the $R_{1}$ sample is mostly due to subgiant stars with masses larger than
1.6 $M_{\odot}$. If we consider a subsample of stars including both MS stars
and subgiants with masses less than $1.6M_{\odot}$, the percentage of false
positive detection of 14% decreases to $\sim$ 9%.

False positive detections can be due to actual amplitudes being lower than
predicted. Several reasons have been put forward to explain lower-than-
expected amplitudes for Kepler stars, including significant magnetic activity
(Chaplin et al. (2011a); M19) and low metallicity (Samadi et al. (2010); M19).
This likely depends on the properties of the stars themselves (mass,
luminosity, temperature, rotation, magnetism, etc.). M19 provided the iron-to-
hydrogen mass fraction [Fe/H] and the photometric proxy for magnetic activity
$S_{ph}$ (Garcia et al., 2010; Mathur, 2014; Santos et al., 2023) which
measures the amplitude of the spot modulation in the light curves and must be
considered as a lower limit of the stellar activity (Salabert et al., 2016,
2017). M19 then found that the probability of non detection of solar-type
oscillations is $\sim 98.7\%$ when $S_{ph}>2000$ ppm (for reference, M19 gave
$S_{\rm ph,min}=67.4$ ppm and $S_{\rm ph,max}=314.5$ ppm at the minimum and
maximum of activity for the Sun and that the solar oscillation amplitudes
decrease by 12.5 % from minimum to maximum of activity). Concerning the impact
of metallicity, the magnitude of the amplitude decrease due to a low
metallicity remains uncertain and solar-like oscillations have been detected
for some metal poor stars. So there is no clear one-to-one correspondence
between metallicity and non detection of solar-like oscillation. Nevertheless
it is still of interest to look at the 146 stars with false positive detection
that have values for $Sph$ and [Fe/H] in the M19 sample. This subsample
includes 100 subgiants among which 72 with masses larger than 1.6 $M_{\odot}$.
We then note that none of those 72 subgiants have high stellar activity and
only 38 of them are metal poor compared to the Sun ([Fe/H]$<-0.1$). We also
considered the whole sample of 146 stars with false positive detection and
available values for $Sph$ and [Fe/H] and found 82 stars with a high activity
level ($S_{ph}>2000$) or [Fe/H] $<-0.1$. Removing those 82 stars from the
original sample of 186 stars with false positive detection leaves 104 stars.
Using 104 stars instead of the original 186 stars with false positive
detections, we find that the percentage of false positive for the $R_{1}$
sample drops to $\sim 7.7$%.

In any case, taking into account stellar activity and metallicity, and/or
additional specific properties of the stars to explain the whole sample of
false detections would deserve further investigation but is out of scope of
the present paper. Since such detailed information are not yet available for
the PIC 1.1.0 stars we will therefore keep a conservative value of 14% for the
false positive uncertainty. Accordingly we later give the values of $X$
together with their uncertainties under the form $X^{+3\%}_{-14\%}$ for the
PLATO subsample of MS stars with masses $M<1.6\rm{M}_{\odot}$ and subgiants of
all masses. However, as an optimist remark, let us stress that stars in the
M19 sample were observed over one month only. As the S/N increases over time,
we expect to reach smaller amplitudes, everything else equal, with the PLATO
mission and therefore a smaller percentage of false positive detection due to
too small oscillation amplitudes.

##  4 Uncertainties on stellar properties in cases of individual frequency
measurements

For estimating the seismic MRA inference performances, we go on to consider
the case when the mode frequencies can be measured individually. We derived
empirical relations giving the mass and radius relative uncertainties as a
function of the uncertainty $\delta\nu_{\ell=1,max}$ of the $\ell=1$ mode
closest to $\nu_{\rm max}$, the frequency at maximum power (those modes have
the smallest uncertainties). For that purpose, we used the stellar evolution
code CESTAM (Morel & Lebreton, 2008; Marques et al., 2013) to build a set of
stellar models of MS stars with masses of $M\leq 1.2\rm{M}_{\odot}$, for which
we numerically computed the individual frequencies using the ADIPLS code
(Christensen-Dalsgaard, 2008). The frequency uncertainties are obtained from a
rescaling of the frequency uncertainties derived for a ’degraded Sun’ (Lund et
al., 2017). We then used the above frequency set for each synthetic star in
the inference code AIMS (Rendle et al., 2019; Lund & Reese, 2018) updated for
the present purpose by one of the co-authors (D. Reese) in order to infer the
MRA and their statistical uncertainties. It is known that non seismic
constraints play only a minor role when the inference includes a large number
(e.g. a few dozen) of highly precise individual frequencies. We nevertheless
include uncertainties for non-seismic constraints: a generic 70 K as an
uncertainty for the effective temperature and 0.05 dex for the metallicity
expected from individual spectroscopic study (e.g see the PASTEL catalogue,
Soubiran et al. (2022)). For Sun-like stars, differential studies with respect
to the Sun are even more precise and accurate (Morel et al., 2021). On
purpose, we did not introduce any systematic errors, so that we can estimate
the seismic performances specifically due to the quality of the data. We then
established a correlation between the MRA uncertainties and the frequency
uncertainty $\delta\nu_{\ell=1,max}$ which was then fitted. We found the
following fitted relations (see Eq. (20) in Appendix C):

$\displaystyle\left\\{\begin{array}[]{ll}\delta
M/M&=2.083~{}\delta\nu_{\ell=1,max}+0.046\\\ \delta
R/R&=0.707~{}\delta\nu_{\ell=1,max}+0.149\,,\end{array}\right.$ (20)

where $\delta\nu_{\ell=1,max}$ is in $\mu$Hz. All details of these
calculations are presented in Appendix C. The MRA relative uncertainties (Eq.
20) can then be seen as a lower limit of what can be achieved given the
observational constraints (which depend on the observing conditions).
Realistic uncertainties require to ad d systematic errors to obtain the final
error budget. This is out of scope of the present work (but see the discussion
in Sect.7).

For the PLATO targets, we determine the (theoretical) uncertainties on the
individual frequencies using the Libbrecht (1992) formula (Eq. 49 in Appendix
D), $\sigma_{\rm Libb}$, which depends on the S/N for that particular mode and
on the duration of the observation and has proven to yield the right order of
magnitude. The S/N in such a case is given by the the power per resolved mode
-instead of the global power density in the oscillation envelope as before-
over the background noise. The power per resolved mode is related to the
height of the mode. Accordingly, the power density per resolved mode is
derived from (S19) and Lochard (2003) for a single-sided spectrum (see also
Appourchaux, 2004):

$\displaystyle P_{\rm mod}=2V_{1}\,\frac{A^{2}_{\rm max}}{\pi\Gamma_{\rm
max}}\,,$ (21)

where $\Gamma_{\rm max}$ is the mode linewidth at $\nu\approx\nu_{\rm max}$,
expressed in $\mu$Hz. For $\ell=1$ modes, the square visibility is $V_{1}=1.5$
(Ballot et al., 2011). The computation of $\rm{A}_{\rm max}$ is described in
Appendix A (see in particular Table 12). Estimates of the mode linewidths are
obtained by a fit of measurements in Lund et al. (2017) as a function of
effective temperature (see Appendix D, and in particular Table 19 for
details). The Libbrecht (1992) formula predicts the uncertainty for a single,
isolated mode peak such as the $\ell=0$ modes. For higher $\ell$ degree modes,
one should take into account the fact that they are comprised of multiple
components that might not be resolved; and moreover, the exact appearance of
the non-radial modes will depend on the angle of inclination presented by the
star, currently unknown for the PLATO targets. We therefore rather adopt an
empirical approach using real data from the Kepler mission: we establish in
Appendix D a relation between the theoretical Libbrecht uncertainty of a
$\ell=1$ mode at $\nu_{max}$, $\sigma_{\rm Libb,\ell=1,max}$, and the measured
frequency uncertainty $\delta\nu_{l=1,max}$ for the same mode for stars of the
Kepler Legacy sample. As found in Appendix D (Fig.16), the ratio
$\delta\nu_{\rm\ell=1,max}/\sigma_{\rm Libb,\ell=1,max}$ tends to decrease
with the effective temperature of the star and the decrease is significant
over the effective temperature interval found for the PLATO targets. The fit
of the ratio $\delta\nu_{\rm\ell=1,max}/\sigma_{\rm Libb,\ell=1,max}$ as a
function of effective temperature for the stars of Kepler LEGACY sample is
shown in Appendix D (see Eq. 51) and yields

$\displaystyle\delta\nu_{\rm\ell=1,max}=\sigma_{\rm
Libb,\ell=1,max}~{}\left(4.89-4.18~{}\frac{T_{\rm eff}}{6000}\right)\,,$ (22)

valid for $5000<T_{\rm eff}\leq 6200$ K. This roughly corresponds to the
effective temperature range of the PLATO targets for which we will compute
those uncertainties later on. For each PLATO target, we will compute
$\sigma_{\rm Libb,\ell=1,max}$ then derive $\delta\nu_{l=1,max}$ using Eq. 22,
before using Eq. (20) for the stellar mass and radius uncertainties.

##  5 Expected solar-like oscillations with PLATO

In this section, we estimate the number of stars for which solar-like
oscillations are expected to be detected with the PLATO mission. For this
purpose and as already mentioned, we took for each PLATO target star the
stellar radius and effective temperature from PICv1.1.0 (Montalto et al.,
2021; Nascimbeni et al., 2022). The luminosity is then derived as:

$\displaystyle\log\left(\frac{L}{L_{\odot}}\right)=2\log\left(\frac{R}{R_{\odot}}\right)+4~{}\log\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)\,.$ (23)

We consider, on one hand, the P1P2 sample and, on the other, the P5 sample,
both for one LOP. We removed the hot stars that appear in the instability
strip using the criterion from C11

$\displaystyle\log T_{\rm eff}>\log(8907)-0.093\log(L/L_{\odot}).$ (24)

This eliminated only a few stars because the criterion on the temperature of
the hot side adopted to construct the PIC is much more severe. We also removed
the early red giants from the PIC sample, based on their location in a
theoretical Hertzsprung-Russell (HR) diagram. These stars are included in a
specific scientific validation sub-catalogue of the PIC (Aerts et al., 2023).
PLATO performances for such evolved stars were assessed by Miglio et al.
(2017). These stars are located at the base of the red giant branch in a HR
diagram which we use as an empirical criterion to remove them. We define the
criterion in terms of luminosity and effective temperature by computing and
plotting a set of evolutionary tracks with different masses and locating the
onset of the red giant branch in the HR diagram. This leads us to remove stars
when they satisfy:

$\displaystyle\log T_{\rm eff}\leq 3.66+0.05\log\frac{L}{L_{\odot}}.$ (25)

After the removal of hot stars and evolved ones, we are left with a set of
7,009 stars in our P1P2 sample and 130 140 stars in our P5 sample for one LOP.
The stellar mass was derived from the seismic scaling relation Eq. (12) where
$\nu_{\rm max}$ is evaluated here according to Eq. (36):

$\displaystyle\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}=\left(\frac{R}{R_{\odot}}\right)^{-1.5625}\ \left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{0.78}.$ (26)

Appendix A offers more details. The mass is used only to consider various
subsamples of stars when analysing the results of the calculations. MS stars
with masses larger than 1.6 $\rm{M}_{\odot}$ are too hot and therefore
unlikely to show solar-like oscillations, except perhaps for the stars with
high metallicity. In absence of information about metallicity at the present
time, hereafter we exclude MS stars with predicted seismic masses
$M>1.6\rm{M}_{\odot}$.

The detection probability, $P_{\rm det}$, is obtained using Eq. (6) which
involves the (S/N)max (Eq. 8), the observing time $T_{\rm obs}$ and the width
of Gaussian-like envelope of the oscillation power spectrum $\delta\nu_{\rm
env}$. The amplitudes, $\rm{A}_{\rm max}$, used to compute $(S/N)_{\rm max}$
are taken according to Table 1. In the calculation of $N_{\rm tot}$ (Eq. 10),
we used for $N_{\rm inst}$ the PLATO (random and systematic residuals) noise
level included in the PIC1.1.0 , $N_{\rm PIC}$, for EOL conditions, which was
then converted in ppm${}^{2}/\mu$Hz. For the BOL conditions, we used the data
provided by one of the co-authors (Börner et al., 2023) . Because the
convention in our calculation is a single-sided spectrum as for the power
density, we take $N_{\rm inst}=2N_{\rm PIC}$. We then added the single-sided
stellar granulation background noise as used in S19 (see Sect. 2).

To remain conservative, we kept only those stars for which the probability of
the signal be due to noise is 0.1% or less and of those stars we kept only
stars for which the probability of the signal being due to solar-like
oscillation is larger than 99 %.

###  5.1 Expected solar-like oscillations within the PLATO P1P2 sample

The predicted numbers of P1P2 target stars with positive oscillation detection
obtained in different mass subsamples are collected in Tables 2 and 3. When we
apply the (1 LOP, BOL, $\delta\nu_{\rm env}=\nu_{\rm max}/2$) conditions after
two years of observation, we expect to detect solar-like oscillations for at
least 5839 stars (of which 2732 MS stars with $M\leq 1.6\rm{M}_{\odot}$ and
3107 subgiants.). Figure 1 show the distributions of those stars with expected
detected solar-like oscillations as a function of $\log T_{\rm eff}$, stellar
mass and radius. One expects to detect solar-like oscillations in a sub-sample
of 1245 stars with $M/\rm{M}_{\odot}\leq 1.2$ after two years of observation
in BOL conditions (Fig. 2). When restricted to MS stars with masses $\leq
1.2\rm{M}_{\odot}$, the subsample counts 1016 stars. Those stars are small and
therefore of the utmost interest for detecting small planets. On the stellar
side, more massive stars are likely prone to large systematic uncertainties
because they developed a convective core, the exact extent of which is
unknown. Stars with masses $M\sim 1.2\rm{M}_{\odot}$ might also develop a
convective core but it is small enough that convective overshoot does not
contribute significantly to the total error budget on the age.

Figure 1: Histograms of the number of stars from the P1P2 sample (subgiants of
all masses and MS stars with masses $M/\rm{M}_{\odot}<1.6$) with a probability
of $>$ 99 % of positive detection of solar-like oscillation in the case of (1
LOP, BOL,$\nu_{\rm env}=0.5\nu_{\rm max}$). Top: Distribution in $T_{\rm
eff}$, middle in stellar mass, bottom in stellar radius. The color code
represents the assumed observing duration.

Figure 2: Histograms of the number of stars from the P1P2 sample (with masses
$M/\rm{M}_{\odot}\leq 1.2$) with a probability $>$ 99% of positive detection
of solar-like oscillation in (1 LOP, BOL, $\delta\nu_{\rm env}=0.5\nu_{\rm
max}$) conditions. Top: Distribution in $T_{\rm eff}$; middle in stellar mass;
bottom: in stellar radius. The color code represents the assumed observing
duration.

  * •

Impact of BOL/EOL conditions: assuming EOL conditions instead of BOL ones, we
would lose a few tens of percent of stars, mostly at small mass and radius
(Table 2).

Table 2:  Numbers of stars in the P1P2 sample in 1 LOP with expected detection of solar-like oscillations after 730 days of observation and assuming $\delta\nu_{\rm env}=\nu_{\rm max}/2$. Stellar masses, $M$, and radii $R$ in solar units. cases | BOL | EOL
---|---|---
all | 5858 | 5553
MS stars | 2751 | 2449
$M<1.6$ | 4744 | 4439
$M<1.6$, MS stars | 2732 | 2430
$M\leq 1.2$ | 1245 | 1106
$M\leq 1.2$, MS stars | 1016 | 830
$R\leq 1.1$ | 269 | 203
  * •

Impact of $T_{\rm obs}$: as can be expected and be seen in Table 3, the
observing time plays an important role. The increase of the number of stars
with predicted positive seismic detection is a factor $\sim$ 5-7 greater when
increasing the observing time from 30 days to two years. After 30 days, one
expects 1877 stars (among which 1596 subgiants), number which increases up to
5858 stars (among which 3107 subgiants) after two years of observation. Not
surprisingly, stars for which we might not typically detect solar-like
oscillations for too short an observing time are low-mass, MS stars because
their oscillation amplitudes are too small. It is also interesting to consider
the distributions of stars with positive seismic detection of solar-like
oscillations with their magnitudes. Solar oscillations for stars with
magnitude $>10.5$ will be detected only after about one year of observations.

Table 3:  Number of stars in the P1P2 sample in 1 LOP with expected detection of solar-like oscillations ($P_{dec}>0.99$) assuming $\delta\nu_{\rm env}=\nu_{\rm max}/2$. The numbers without parenthesis correspond to stars with estimated seismic masses $\leq 1.2\rm{M}_{\odot}$, whereas the numbers in parenthesis correspond to stars with all masses.  | BOL |
---|---|---
$\delta\nu_{\rm env}$ | 730 days | 30 days
$\nu_{\rm max}/2$ | 1245 (5858) | 186 (1877)
$\nu_{\rm max}$ | 1541 (6387) | 329 (2811)
| EOL |
$\delta\nu_{\rm env}$ | 730 days | 30 days
$\nu_{\rm max}/2$ | 1106 (5553) | 151(1591)
$\nu_{\rm max}$ | 1389 (6131) | 267 (2399)
  * •

Impact of uncertainties in the probability calculations: the uncertainties on
the number of stars with positive seismic detection due to the uncertainty on
$\delta\nu_{\rm env}$ (see Sect.2.5) can be estimated from Table 3. Denoting
by $D$ the number of stars with positive seismic detection, the impact of
$\delta\nu_{\rm env}$ uncertainty can be estimated as $(D_{\nu_{\rm
max}/2}-D_{\nu_{\rm max}})/7009,$ where 7009 is the total number of stars in
the initial sample. This yields $\sim$ 4% and 7.5% when considering,
respectively the sample of stars with masses $\leq 1.2\rm{M}_{\odot}$ and the
sample of all mass stars with positive seismic detection over 730 days in BOL
conditions .

Table 4:  Uncertainties in the number of P1P2 stars with predicted positive seismic detection after 730 days of observations in (1 LOP, BOL) conditions. $\delta\nu_{\rm env}$ | $\nu_{\rm max}/2$ | $\nu_{\rm max}$
---|---|---
MS stars ($M<1.6$) and subgiants | $5839^{+175}_{-818}$ | $6997^{+70}_{-1679}$
$M\leq 1.2$ | $1245^{+37}_{-87}$ | $1541^{+14}_{-170}$

We obtain an order of magnitude of the uncertainties on the number of positive
detections, say $X$, by considering the underestimate due to false negative
and the overestimate due to false positive detections. We then used the false
negative and positive detection rates derived for our samples 1 and 2 of
Kepler stars in the option $\delta\nu_{\rm env}=\nu_{\rm max}/2$ (second
column of Table 17 in Appendix B). This yields $X^{+3\%}_{-14\%}$ for the
sample of MS stars with $M<1.6\rm{M}_{\odot}$ and subgiants and
$X^{+3\%}_{-7\%}$ for the sample of stars with $M\leq 1.2\rm{M}_{\odot}$ when
$\delta\nu_{\rm env}=\nu_{\rm max}/2$. Such uncertainties are provided in
Table 4. In percentages, Table 4 indicates a predicted seismic positive
detection rate for stars with $M\leq 1.2\rm{M}_{\odot}$ between $\sim$ 55-61%
(resp. 65-74%) of the whole sample of 2099 stars with masses $\leq 1.2{\rm
M}_{\odot}$ in BOL conditions after two years of observation taking into
account uncertainties due to $\nu_{\rm max}/2$ (resp. $\delta\nu_{\rm
env}=\nu_{\rm max}$). In the same conditions but for the sample of MS stars
with masses $M<1.6M_{\odot}$ and subgiants with all masses, the same
uncertainties yield predicted seismic positive detections at the level of
71-86% (resp. 76-100%) of the whole sample of 7009 stars in the initial sample
taking into account uncertainties due to $\nu_{\rm max}/2$ (resp.
$\delta\nu_{\rm env}=\nu_{\rm max}$)

We note that uncertainties in the probability calculation and the number of
stars with expected solar-like oscillation detection can also come from the
fact that we used PIC1.1.0 radius and effective temperature to compute the
global seismic parameters and derive the seismic mass.

###  5.2 Expected solar-like oscillations within PLATO P5 sample

We carried out the same probability calculation as for the P1P2 sample after
removing the same types of stars and assuming again a positive detection for
$P_{\rm det}>0.99$ (1 LOP, BOL, $\delta\nu_{\rm env}=\nu_{\rm max}/2)$
conditions. We found that the number of expected postive seismic detections
amounts to 9 486 for the sample of MS stars with $M/\rm{M}_{\odot}<1.6$ and
subgiants after two years of observation (Table 5).

Table 5:  Numbers of stars in the P5 sample with expected positive detection ($P_{dec}>0.99$) of solar-like oscillations in (1 LOP, BOL, $\delta\nu_{\rm env}=0.5\nu_{\rm max}$) conditions. Masses and radii are given in solar units. cases | 730d | 365d | 90d | 30d
---|---|---|---|---
all | 9491 | 5718 | 1599 | 380
MS stars ($M\leq 1.6\rm{M}_{\odot}$) | 9486 | 5716 | 1599 | 380
& subgiants of all masses |  |  |  |
Subgiants of all masses | 8877 | 5657 | 1599 | 380
$M\leq 1.2$ : | 878 | 392 | 81 | 21
MS stars | 250 | 43 | 0 | 0
subgiants | 628 | 349 | 81 | 21

Here again, we observe a drastic increase in positive seismic detections with
the observing time. This is illustrated in Fig. 3. This figure shows the
distributions in $T_{\rm eff}$, mass, and radius of P5 MS stars with masses
$M<1.6\rm{M}_{\odot}$ and subgiants with all masses with expected positive
detections for different observing durations. The number of stars
significantly increases when $T_{\rm obs}$ increases beyond 1 year, in
particular toward stars with smaller radii. The subgiants outnumber
significantly the MS stars for the whole sample of stars with
$M<1.6\rm{M}_{\odot}$ (Table 5). As expected, after only 90 days of
observation, only 81 stars with masses $M\leq 1.2\rm{M}_{\odot}$ have a
seismic positive detection, all subgiants because their amplitudes (roughly
$\propto L/M$) are the highest.

Figure 3: Histograms of the number of stars (MS stars with
$M<1.6\rm{M}_{\odot}$ and subgiants) from the P5 sample with an expected
detection of solar-like oscillations with at least 99% probability assuming
730, 365, 90 and 30 days of observations. Conditions are (1 LOP, BOL,
$\delta\nu_{\rm env}=0.5\nu_{\rm max}$). Figure 4: Distributions in $N_{PIC}$
(PLATO noise from the PIC1.1.0, in ppm $\rm{h}^{1/2}$) of MS stars with $M\leq
1.2\rm{M}_{\odot}$ from the P5 sample with an expected positive detection of
solar-like oscillations with a probability higher than 99% assuming 730 days
of observations. Conditions are (1 LOP, $\delta\nu_{\rm env}=\nu_{\rm
max}/2$), BOL (blue), EOL(red).

Figure 4 compares the distributions of the PLATO noise level, $N_{\rm PIC}$,
taken from the PIC1.1.0 between BOL and EOL conditions for stars with expected
positive detection of solar-like oscillations in the case of (1 LOP,
$\delta\nu_{\rm env}=\nu_{\rm max}/2$) for 730 days of observation. By
construction, following the PLATO ESA requirements, the stars belonging to the
P1P2 sample have $N_{\rm PIC}\leq 50$ ppm h1/2 whereas the stars with higher
noise levels constitute the P5 sample. This was based on the EOL conditions.
Assuming that the more optimistic BOL conditions hold, we find that 115 MS
stars with mass $M\leq 1.2\rm{M}_{\odot}$ in the P5 sample have $N_{\rm
PIC}\leq 50~{}$ppm.h1/2 and could be reclassified as P1 stars, increasing the
number of P1P2 stars from 1016 (Table 2) to 1131 -that-is an increase of the
positive detection rate from 15% to above 17%.

##  6 PLATO seismic performances for MRA inferences in the P1P2 sample

For the subset P1P2 stars with expected solar-like oscillations, the detection
and highly precise measurement of individual frequencies for a significant
number of modes is ensured by the selection of a high S/N by construction.
This will allow us to satisfy the requirements that must be achieved by the
PLATO mission (PLATO Science Requirements Document PTO-EST-SCI-RS-0150, ESA
document, June 2021) which are: a mass uncertainty better than 15%, a radius
uncertainty lower than 2% and an age uncertainty as low as 10% for a star like
the Sun or the PLATO reference star defined as $1\rm{M}_{\odot},1R_{\odot}$,
and $T_{\rm eff}=6000K$.

Figure 5: Distribution of frequency uncertainties $\delta\nu_{\ell=1,max}$ for
the sample of P1P2 stars with expected positive seismic detection. The color
code corresponds to different mass samples as indicated in the top right
panel. The top panels correspond to the frequency uncertainties computed with
Eq. (22) for observation lengths of 730d (top-left panel) and 90 days (top-
right panel). The bottom panels show the frequency uncertainties distribution
when one takes into account the scatter in the relation obtained with Eq. (22)
(left panel) or the uncertainties in the mode linewidths at low effective
temperature width, as per Appendix D (right panel).

###  6.1 PLATO seismic performances for oscillation frequencies for the P1P2
sample

For the PLATO targets, assuming individual frequencies are available, we can
only determine the (theoretical) uncertainty on the frequencies using the
Libbrecht (1992) formula (Eq. 49), $\sigma_{\rm Libb,\ell=1,max}$. We then use
Eq. (22) to relate $\delta\nu_{\rm\ell=1,max}$ to $\sigma_{\rm\ell=1,max}$ for
each target. We computed $\sigma_{\rm Libb,\ell=1,max}$ then
$\delta\nu_{\rm\ell=1,max}$ for each target of our P1P2 sample of stars with
expected positive seismic detection in BOL condition for 730 days and 90 days.
We also included the case where we take into account the scatter in the
relation Eq. (22) and the measurement uncertainties in the mode linewidths
derived from the Kepler stars which leads to multiplying by $\sqrt{3}$ the
Libbrecht’s uncertainties $\sigma_{\rm Libb,\ell=1,max}$ (Appendix D). The
corresponding distributions of frequency uncertainties
$\delta\nu_{\rm\ell=1,max}$ are shown in Fig. 5. In the case (BOL, 730 days,
$\delta\nu_{\rm env}=\nu_{\rm max}/2$) conditions, the bulk of uncertainties
are concentrated below 0.1 $\mu$Hz. Including the scatter in the relation Eq.
(22) shifts up the maxima of the distributions by about 0.07 $\mu$Hz. The
uncertainties remain below the PLATO requirement of frequency uncertainties
0.3-0.5 $\mu$Hz. In case of 3 months observations, the shift is higher, about
0.1-0.13 $\mu$Hz and the bulk of uncertainties reach 0.2-0.3 $\mu$Hz. When the
mode linewidth are increased by a factor $\sqrt{3}$, the uncertainties are
only slightly shifted with the bulk of uncertainties concentrating around 0.1
$\mu$Hz. We also computed the frequency uncertainties in EOL conditions for
730 days but the associated degradation of the signal has only a small impact
and is not shown.

###  6.2 PLATO seismic performances for stellar MRA inferences for the P1P2
sample

We now turn to the MRA uncertainties from seismic inferences resulting from
the error propagation due to $\delta\nu_{\rm\ell=1,max}$. We focused on stars
with masses $M\leq 1.2\rm{M}_{\odot}$. We used Eq. (20) to estimate the mass
and radius uncertainties as discussed in Sect. 4. We also used the constraint
$\delta\nu_{\rm\ell=1,max}$ as a proxy for the constraint on the age
uncertainty. The condition $\delta M/M\leq 3\%$ (often used to obtain an age
uncertainty at the level of 10%) can also be added as an additional
constraint.

We derived the numbers of P1P2 stars with a positive seismic detection while
adding successive constraints on the uncertainties giving rise to three cases
as follows:

  * •

case I : $\delta M/M\leq 15\%$ & $\delta R/R\leq 2\%$

  * •

case II : $\delta M/M\leq 15\%$ & $\delta R/R\leq 2\%$ &
$\delta\nu_{\rm\ell=1,max}\leq 0.2\mu$Hz

  * •

case III : $\delta M/M\leq 15\%$ & $\delta R/R\leq 2\%$ &
$\delta\nu_{\rm\ell=1,max}\leq 0.2\mu$Hz & $R/R_{\odot}\leq 1.1$

We consider the frequency uncertainties $\delta\nu_{\rm\ell=1,max}$ as given
by Eq. (22) or Eq. (51) without (case a) and with (case b) a $+0.5$ shift due
to the scatter in the fitted relation between $\delta\nu_{\ell=1,max}$ and
$\sigma_{Libb,\ell=1,max}$ (Appendix D). The results are listed in Table 6.

Table 6:  Number of MS dwarfs with $M\leq 1.2\rm{M}_{\odot}$ stars in the P1P2 sample with expected positive seismic detection and satisfying different cases of MRA uncertainties. Assumed conditions are (1 LOP, BOL, 730 days, $\delta\nu_{\rm env}=\nu_{\rm max}/2$). Labels a) and b) refer to frequency uncertainties assumed without and with a $+0.5$ shift due to the scatter in the fitted relation between $\delta\nu_{\rm\ell=1,max}$ and $\sigma_{\rm Libb,\ell=1,max}$ (Eq.D.3). case | BOL | EOL
---|---|---
Ia | 1016 | 880
IIa | 1016 | 880
IIIa | 260 | 195
Ib | 1016 | 880
IIb | 729 | 599
IIIb | 206 | 146

Figure 6 shows the distributions of MS stars with masses $M\leq
1.2\rm{M}_{\odot}$ corresponding to the cases listed in Table 6 as a function
of $T_{\rm eff}$, stellar mass and radius (1 LOP, BOL, $\delta\nu_{\rm
env}=0.5\nu_{\rm max}$ 730 days) conditions. We expect that the number of
stars decrease when adding new constraints. Cases I do not reduce the number
of stars compared with the initial sample of P1P2 MS stars with masses $M\leq
1.2\rm{M}_{\odot}$ with positive seismic detection. This means that the main
constraints are the S/Ns imposed by design and the detection probability. The
PLATO requirements for the mass, radius and age uncertainties (case II) of
P1P2 stars are automatically satisfied in the PIC, provided the oscillations
are detected.

Figure 7 shows the histogram of the evolution of the number of these stars
when the observation time increases from 90 days to 730 days, assuming (1 LOP,
BOL) conditions for MS stars with $M/\rm{M}_{\odot}\leq 1.2$, $\delta M/M\leq
15\%;\delta R/R\leq 2\%$, and $\delta\nu_{\rm\ell=1,max}\leq 0.2\mu$Hz. The
gain of stars satisfying the PLATO requirements is particularly significant
for stars like the reference star when going from 90 days to a year. We note
that no such star is found when observing over a short period of time of 30
days.

We also computed the numbers of stars satisfying other constraints such as the
case ($\delta M/M\leq 3\%$ ; $\delta R/R\leq 2\%$); the
case($\delta\nu_{\rm\ell=1,max}\leq 0.5\mu$Hz) or the case ($\delta M/M\leq
15\%$ ; $\delta R/R\leq 2\%$ & $\delta\nu_{\rm\ell=1,max}\leq 0.5\mu$Hz). The
number of stars remains the same as in the initial sample of P1P2 MS stars
with masses $M\leq 1.2\rm{M}_{\odot}$ with positive seismic detection.

Figure 6: Histograms in $T_{\rm eff}$ (top), and mass (bottom) for MS stars
with $M/\rm{M}_{\odot}\leq 1.2$ in the P1P2 sample assuming (1 LOP, BOL, 730
days, $\delta\nu_{\rm env}=\nu_{\rm max}/2$) conditions with expected
detection of solar-like oscillations and satisfying cases IIa and IIIa listed
in Table 6. Figure 7: Histogram in $T_{\rm eff}$ for the P1P2 subsample of MS
stars with $M/\rm{M}_{\odot}\leq 1.2$ assuming (1 LOP, BOL, $\delta\nu_{\rm
env}=0.5\nu_{\rm max}$) conditions and case IIa in Table 6.

##### Taking into account uncertainties in frequencies and mode linewidths
$\Gamma$ values:

The above numbers of stars remain unchanged when including $\sim$ 5-$\sigma$
uncertainties due to scatter in Eq. (20) (Fig. 14 in Appendix C): $\delta
M/M(1\pm 0.5\%)$; $\delta R/R(1\pm 0.1\%)$. In contrast, in the extreme case
where one takes into account a scatter of $+0.5$ in the
$\delta\nu_{\rm\ell=1,max}/\sigma_{\rm Libb,\ell=1,max}$ fitted relation for
all stars, the number of MS stars with $M\leq 1.2\rm{M}_{\odot}$ satisfying
the PLATO requirement decreases by about 25%.

The impact of degrading the frequency uncertainty $\delta\nu_{\ell=1,max}$ by
$\sqrt{3}$ for stars with $T_{\rm eff}\leq 5650$K (due to uncertainties in the
values the mode linewidths, as per Appendix D), instead of the original values
of $\delta\nu_{\rm\ell=1,max}$ \- leads to a decrease of the number of stars
by 38% in case IIb with (BOL, 730d, $\delta\nu_{\rm env}=\nu_{\rm max}/2$)
conditions.

We then conclude that even in the conservative case we considered for the
detection probability and in the above worst cases for the frequency
uncertainties, the PLATO mission should yield a set of MS stars with a seismic
characterisation between $\sim$ 80 and 100 times the Kepler Legacy sample
assuming two observing fields (i.e. doubling the number of stars obtained for
1 LOP) and depending on whether we assume $\delta\nu_{\rm env}=\nu_{\rm
max}/2$ or $\nu_{\rm max}$.

##  7 Summary and discussion

The present study is part of the scientific preparation for the ESA’s PLATO
mission, which will be launched towards the end of 2026. In this work, we
calculate the theoretical probability of detecting solar-like oscillations for
stars belonging to the PLATO Input Catalogue (PIC). More specifically, we
considered bright stars (magnitude of 11 and brighter) of the FGK spectral
type on the main sequence (masses lower than 1.6 $M_{\odot}$) and the subgiant
branch. The calculation takes into account the estimated noise level for each
individual star, provided by the PIC. Our results indicate that the proportion
of stars with positive detections of solar-like oscillations lies within a
range of 70% to 100% for a continuous observation of two years. The lower
(upper) value of this range comes from the assumed narrow (wide) bandwidth of
the oscillation spectrum in the Fourier domain for each star, which is the
main uncertainty in the calculation. It also depends on the beginning-of-life
or end-of-life conditions of the PLATO instrument.

The CoRoT and Kepler missions have taught us that individual oscillation modes
can be detected for stars with a noise level of 50 ppm in one hour or less.
For the stars in our sample that satisfy this criterion and with positive
seismic detection we have estimated the uncertainty in the individual
frequency measurements at the maximum of the power spectral density for each
star based on the results of the Kepler mission. This enabled us to assess the
propagation of this statistical error on the seismic inference of the mass,
radius, and age of each star. We found that $\sim$ 47- 61% of the sample of MS
stars with masses $M\leq 1.2\rm{M}_{\odot}$ with statistical uncertainties
below the PLATO requirements of 15%, 2% for the stellar masses and radii,
respectively and satisfying oscillation frequency uncertainty $\leq 0.2\mu$Hz
at maximum power density amplitude as a proxy for 10% uncertainty of the age
of a Sun-like star. Those uncertainties are small enough that they leave
margins for including systematic errors while still keeping the total error
budget satisfying the PLATO requirements. We note that the masses used to
define various mass samples, especially the sample represented by $M\leq
1.2\rm{M}_{\odot}$, are seismic masses derived from scaling relations. As
such, they are approximated as are the number of targets found in each mass
subsample but the order of magnitude remains correct.

We also stress that for a few stars, it may be expected that additional errors
can come from unexpected issues in the data acquisition or in the variable
behavior of the star (such as magnetic activity, Pérez Hernández et al., 2019;
Karoff et al., 2019; Thomas et al., 2021; Santos et al., 2023) that can alter
the measurements of the frequencies and therefore the MRA uncertainties. From
the Kepler experience, this could add an uncertainty up to 0.3 $\mu$Hz to the
statistical uncertainties but it is difficult at this stage to foresee for
which stars in the PIC this can happen and this was ignored here.

For the P5 sample, the noise level is higher- again by construction- than for
the P1P2 sample in the same (EOL or BOL) conditions. We find a percentage of
7.3-4.3 % P5 stars with expected positive seismic detection after 730 days of
observation in BOL and EOL conditions, respectively. Among those, a percentage
of 0.5-0.2% of P5 MS stars with masses $M\leq 1.2\rm{M}_{\odot}$ are expected
to show positive seismic detections. Among them, 115 P5 MS stars with masses
$M\leq 1.2\rm{M}_{\odot}$ and with a PIC noise level lower than 50 ppm . h1/2
in BOL conditions satisfy the above PLATO requirements, which means that those
stars could be re-classified as P1 stars.

Accordingly, and as a whole, the calculations yield a total of 1131 MS stars
with masses $M\leq 1.2\rm{M}_{\odot}$ for which one expects a positive seismic
detection and seismic analyses providing mass, radius, and age satisfying the
above PLATO requirements in BOL consitions after two years of observation for
one LOP. The stars of this sample are plotted in a HR diagram in Fig. 8. For
each target, the luminosity is derived with Eq. (23) and the stellar radius
and the effective temperature, and their uncertainties taken for the
PICv1.1.0. Uncertainties are plotted for three stars as representative of the
typical PIC uncertainties in the HR locations. Overplotted over the PLATO
targets locations, evolutionary tracks of stellar models cover the mass range
of $[0.8-1.2]\rm{M}_{\odot}$. The stellar models were built with the CESTAM
code (Morel & Lebreton, 2008; Marques et al., 2013) assuming the solar
relative chemical abundances AGSS09 (Asplund et al., 2009) with different
initial values for $X_{0},Y_{0},Z_{0}$ (namely, the mass fractions of hydrogen
$X_{0}$, helium $Y_{0}$, and metallicity representing all heavier chemical
elements collectively counted as $Z_{0}$). The convection is described with
the classical MLT formulation (Cox & Giuli, 1968) involving the mixing length
parameter, $\alpha_{MLT}$ (a free parameter representing the efficient of
convective transport in 1D stellar models). Otherwise, the input physical
assumptions are similar to those of the reference model A described in
Lebreton & Goupil (2014). The evolutionary tracks in Fig. 8 were computed
until an age of 14 Gyr (on purpose greater than the age of the Universe) for
the lowest mass stellar models or stopped at an arbitrary phase of the red
giant branch for the most massive ones. Hence assuming one single chemical
composition and $\alpha_{MLT}$ values- usually taken as for the Sun- would
clearly not reproduce the whole extended region in the HR occupied by the
PLATO targets with the lowest masses. This remains true even taking into
account the observational uncertainties in luminosity and effective
temperature and the fact that several stars might belong to binary systems.

Figure 8: P1,P2, P5 stars with $M\leq 1.2\rm{M}_{\odot}$, $N_{PIC}\leq 50$ppm
. h1/2 and satisfying constraint of case II in Tab.6. Crosses indicate the
(representative) PIC uncertainties for three target stars. Evolutionary tracks
of stellar models built with a given initial chemical composition and
$\alpha_{MLT}$ value are represented by the same color: cyan
($\alpha_{MLT}=1.845$, $Y_{0}=25$, $(Z/X)_{0}=0.0136$, no atomic diffusion
included, masses in the range [0.8 -1.10]$\rm{M}_{\odot}$ in steps of 0.05
$\rm{M}_{\odot}$) or red ($\alpha_{MLT}=1.642$, $Y_{0}=0.25$,
$(Z/X)_{0}=0.0246$, atomic diffusion included, masses in the range [0.8
-1.25]$\rm{M}_{\odot}$ in steps of 0.05 $\rm{M}_{\odot}$).

The anticipated PLATO sample of $\sim$ 2793 well characterized main sequence
stars with masses of $M<1.6M_{\odot}$ in one LOP will contribute to the PLATO
set of best seismically characterized stars and is roughly 42 times larger
than the Kepler Legacy sample and will complement the latter in providing
tight constraints on stellar modelling. All these results will be reviewed and
revised when confronted in 2.5 years from now with the real PLATO data after
launch.

In the present work, we have purposefully estimated only the statistical
uncertainties on the frequencies and resulting MRA seismic inferences in order
to appreciate the added value due to the expected high quality of the PLATO
photometric data. The total error budget, however, must include the
uncertainties due to the various systematic errors that can be identified but
not fully corrected. Of all stellar properties, the stellar age is by far the
most challenging to determine accurately. In most cases indeed, stellar ages
of single, field stars can only be determined through stellar modelling
(Soderblom, 2010; Christensen-Dalsgaard & Silva Aguirre, 2018); thus, their
accuracy strongly depends on the degree of reliability of the available
stellar models. Systematic errors are indeed expected to come mostly from
insufficiently realistic stellar modelling (e.g. Lebreton & Montalbán, 2010;
Christensen-Dalsgaard & Silva Aguirre, 2018; Lebreton et al., 2014b, a;
Salaris & Cassisi, 2017; Dupret, 2019; Buldgen, 2019).

Thanks to the high quality of data acquired by space missions such as CoRoT
(Baglin et al., 2009) and Kepler, and currently TESS (Ricker et al., 2015),
theoretical studies were initiated in order to identify and quantify the
impact of the main systematic errors that contribute the most to the error
budget. Studies have been carried out as pure theoretical investigations and
hare and hounds exercises (e.g. Appourchaux et al., 2006; Lebreton et al.,
2014a; Reese et al., 2016; Cunha et al., 2021) or by modelling specific or
sets of stars with the best seismic observations such as the small set of
CoRoT stars or the larger set of Kepler LEGACY stars (e.g. Appourchaux et al.,
2008; Benomar et al., 2010; Metcalfe et al., 2012; Lebreton & Goupil, 2014;
Silva Aguirre et al., 2015, 2017; Creevey et al., 2017; Bellinger et al.,
2017; Nsamba et al., 2018; Valle et al., 2020; Ong et al., 2021; Farnir et
al., 2020; Bétrisey et al., 2022, 2023, among others). At present, due to lack
of accuracy, the age uncertainties for solar-like filed stars can increase
roughly by $\sim$ 5 to 25% depending on poorly modelled physical processes and
unknown initial chemical composition and to $\sim$ 15%- 50% for a more massive
star with a convective core.

We consider below two illustrative cases : the degraded Sun and the two best
studied stars of the Legacy sample.

###  7.1  The ‘degraded’ Sun

A natural test of the accuracy of seismic modelling and characterisation is to
look at the Sun-as-a-star and compare the results of the seismic MRA
inferences to the independently known values of the mass, radius, and age of
the Sun. This is now done routinely when inferring the MRA for various sets of
Kepler stars. We therefore carried out several MRA inferences using the data
of the so-called ‘degraded Sun’ of Lund et al. (2017). The frequencies of the
’degraded’ Sun and their uncertainties were built to match the quality of the
Kepler Legacy sample ($\delta\nu_{\rm\ell=1,max}\sim 0.15\mu$Hz with
oscillation modes in the range
$\ell=0,n=16-27;\ell=1,n=15-27;\ell=2,n=16-24;\ell=3,n=20$ , (Lund et al.,
2017)). Here again we used a grid-based approach (GBM) with the AIMS code. The
observational constraints besides the oscillation frequencies and their
uncertainties were taken as: $T_{\rm eff}=5777\pm 77$K, $[Fe/H]=0\pm 0.1$,
$\nu_{\rm max}=3090\pm 3.9\mu$Hz. We compare the results obtained when using
two different grids of stellar models. The (Cunha et al., 2021, hereafter C21)
for which we recall that the stellar models were computed with MESA
evolutionary code (Paxton et al., 2018, and references therein) and the
frequencies were computed using with the oscillation code ADIPLS (Christensen-
Dalsgaard, 2008). The second grid of stellar models (hereafter Mo23) was
computed by one of the co-authors (namely, N. Moedas, for more details see
Moedas et al., in prep.) using also the MESA code and the frequencies were
computed using the GYRE oscillation code (Townsend & Teitler, 2013). The input
physics, values of free parameters and the reference solar relative chemical
abundances differ between both grids. This allows to assess the impact of the
main uncertainties in modelling solar-like stars. Intended to be applied to
real stars unlike C21, Mo23 used a more updated stellar physics (similar to
models D1 in Moedas et al. (2022), namely, it included atomic diffusion of
chemical elements that helioseismology taught us is crucial for the modelling
of solar-like stars. The reference solar abundances are AGSS09 (Asplund et
al., 2009) in Mo23 whereas it is GS98 (Grevesse & Sauval, 1998) in C21. For
sake of simplicity, Mo23 kept the mixing length value of $\alpha_{MLT}$ fixed
to the calibrated solar value whereas C21 let the convection parameter be
adjusted in the fitting process. Because here we dealt with real stars, we had
to include surface-effect corrections and adopted the Ball & Gizon (2014b)’s
correction in the AIMS inferences with both grids.

Following Reese et al. (2016) and C21, in the particular case of the Sun, we
can measure the biases (or departure from accuracy) with $d_{\rm
X,rel}\equiv(X_{\rm fit}-X_{\rm true})/X_{\rm true}$ for $X=M,R,A$. We wish
then to compare these values to the relative statistical uncertainties
$\delta_{X}\equiv|\delta X|/X_{\rm fit}$ (%) (where $|\delta X|$ corresponds
to one standard deviation) provided by the MRA inferences with the AIMS’ code.
Finally it is also informative to estimate how large are the departures from
accuracy compared to statistical uncertainties derived from the GBM approach
$d_{\rm X,norm}\equiv|(X_{\rm fit}-X_{\rm true})|/\delta X$ since we will have
only access to the last ones in most PLATO stars. The departure between the
seismically inferred values for the solar MRA and the known solar values
(taken here as $4.6\pm 0.4$ Gyr for the solar age (Houdek & Gough, 2007) as
measured by $d_{\rm X,rel}$ are given in Table 7 for the two grids. As
expected the accuracy is much higher in the case of the Mo23 grid than for the
C21 grid mainly because the C21 grid does not included atomic diffusion. These
figures are similar to the values derived in other works which all adopt
various different input assumptions and which fall in the range ($\sim
0.3-4$%) for the mass, ($\sim 0.1-2$%) for the radius, and for the age $\sim
2-9$% when atomic diffusion is included and $\sim 15-16$% when atomic
diffusion is not included (e.g. Silva Aguirre et al., 2017; Creevey et al.,
2017; Rendle et al., 2019; Jiang & Gizon, 2021; Aguirre Børsen-Koch et al.,
2022; Metcalfe et al., 2023).

On the other hand, the 1$\sigma$ uncertainties given by the inference
calculations as measured by $\delta_{X}$ are comparable between the two grids
(in the typical range 0.5-4% and 3-8% for the mass and age), relatively
independently of the accuracy. Accordingly the departures from accuracy as
measured in terms of $\delta X$ uncertainties ($d_{\rm X,norm}$, last column
of Table 7) significantly differ between the two calculations and between the
fitted parameters MRA. In addition in both cases, the inaccuracies of the
derived luminosities, which were not included as input constraints amount to
5% in both cases. As we want to decrease the inaccuracies at the level of or
below the statistical uncertainties, this shows that there is room for
improvements in the inference process or in the present solar modelling. This
also emphasized the importance of building a set of stars with determination
of mass and/or radius, and or age (benchmark stars) independently of stellar
modelling as is done for the GAIA mission (Heiter et al. (2015b) Heiter et al.
(2015a)) and in preparation for the PLATO mission (Maxted & Creevey, 2023).

Table 7:  MRA inference for the ’degraded Sun’ of one standard deviation or 68% credible intervals about the median values.  grid |  $\delta_{X}$ (%) |  $d_{X,rel}$ (%) | $d_{\rm X,norm}$
---|---|---|---
mass |  |  |
Mo23 | 0.20 | +0.74 | 3.7
C21 | 0.21 | -2.51 | 13
radius |  |  |
Mo23 | 0.07 | +0.02 | 0.32
C21 | 0.08 | -1.03 | 13
age |  |  |
Mo23 | 1.37 | \+ 2.1 | 0.97
C21 | 1.53 | \- 11 | 6.5
Validation |  |  |
luminosity |  |  |
Mo23 | 0.71 | 5.5 | 7.8
C21 | 0.88 | 5.3 | 6.35

In the case of the ‘degraded Sun’, the net error budget for the age remains
close to $\sim$ 10 % accuracy. One must nevertheless keep in mind that free
parameters entering the solar and stellar modelling (namely, the initial
helium abundance, convective efficiency parameter, $\alpha_{MLT}$) are
calibrated for the Sun so that the solar model reproduces the radius,
luminosity at the age of the Sun. On the seismic side, the surface corrections
of the frequency are designed for the theoretical frequencies of the stellar
models to match the observed ones. Inaccuracy in the solar modelling are then
either compensated or minimized by such procedures. This cannot be done for
other solar-like stars and one must either attribute arbitrarily the solar
values to the free parameters or adjust them during the fitting process for
other stars. This can lead to hidden inaccuracies. What can then be and is
done is rather studying the sensitivity of the fitted results to changes in
the physical description or the values of the free parameters. This is what
was carried out for the two brightest ($V\sim 6$) solar-like stars from the
Kepler LEGACY, which we discuss below.

###  7.2 Best studied stars of the Kepler Legacy

The two best studied stars of the Legacy sample, the stars 16 Cyg A
(KIC12069424) and B (KIC12069449) belong to a multiple system and show solar-
like oscillations (Metcalfe et al., 2012; Lund et al., 2017). They are bright
stars for which interferometric radii are available. We can also assume that
they were born with the same chemical composition and have the same age.
Unlike the Sun, we have no independent measurements of the masses and
independent determinations of their ages. On the other hand, the information
of a common age and interferometric radii can act as validation of the
inferred results and assessment of the accuracy of the MRA inferences. The
most recent studies dedicated to 16 Cyg A,B were those of Bazot (2020), Farnir
et al. (2020), Nsamba et al. (2022), Buldgen et al. (2022) who provided
references to former works. The uncertainties are found of the order of 4% and
15% for the masses and ages, respectively. The interferometric radii are well
reproduced with uncertainties of 2%. As an illustration, we carried out
seismic inferences for both stars with the two already mentioned grids C21 and
Mo23. We inferred the MRA for each Kepler star independently using again the
Ball & Gizon (2014a) surface-effect correction for the frequencies. The
observational constraints are listed in Table 8. The sample of frequencies are
those provided by Lund et al. (2017). As can be seen in Table 9, the
uncertainties derived from the MRA inference processes are small, namely,
$\sim$ 1% or below in all MRA cases and with both grids. They are smaller for
the C21 grid than the Mo23 grid. At that level of relative uncertainties, this
is likely due to a difference in the properties of the grids such as the
number of adjusted free parameters or to a difference in density of stellar
models in the parameter space around the studied stars. This would deserve
further investigation but is out of scope in the present study. As for the
accuracy, the relative departure of the inferred radius of each star from its
respective interferometric radius is slightly above $1\%$ for both grids
(Table 10), showing that the radius is well constrained by seismology, rather
independently of the physical description of the stellar models in the grids.
Table 10 also gives $\Delta A=(A_{X}-<A>)/\delta A$, the relative differences
between the age of each star and their average age, $(A_{X}-<A>)$ for
$X=16CygA$ and $16CygB$ and $<A>=(A_{16CygA}+A_{16Cyg_{B}})/2$ compared to the
relative uncertainties of the inferred ages, $\delta A$. This shows that the
departure from a common age is of the order of the inference uncertainties.

Table 8:  Observational constraints for the MRA inferences for 16 Cyg A and B. References: (a) Morel et al. (2021) (uncertainties increased arbitrarily); (b) Lund et al. (2017); (c) White et al. (2013) ; (d) Metcalfe et al. (2012).  | 16Cyg A | 16 Cyg B
---|---|---
$T_{\rm eff}^{(a]}$ (K) | 5800 (50) | 5750.0 (50)
$[Fe/H]^{(a]}$ (dex) | 0.11 (0.05) | 0.08 (0.05)
$\nu_{\rm max}^{(b)}$ ($\mu$Hz) | 2188.5 (4.6) | 2561.3 (5.6)
Validation |  |
$R/R_{\odot}^{(c)}$ | 1.22 (0.02) | 1.12 (0.02)
$L/L_{\odot}^{(d)}$ | 1.56 (0.05) | 1.27 (0.04)
Table 9:  Results of MRA inferences (%) for 16 Cyg A and B: 68% credible intervals about the median. 16 Cyg A $\delta\nu_{\rm\ell=1,max}=0.10\mu$Hz (53 modes); 16 Cyg B : $\delta\nu_{\rm\ell=1,max}=0.04\mu$Hz (52 modes) | $\delta M/M$ | $\delta R/R$ | $\delta A/A$
---|---|---|---
16Cyg A |  |  |
Mo23 | 0.33 | 0.12 | 1.01
C21 | 0.10 | 0.03 | 0.77
16Cyg B |  |  |
Mo23 | 0.43 | 0.14 | 0.85
C21 | 0.14 | 0.06 | 0.95
Table 10:  Relative differences (%) between the interferometric radius and the corresponding inferred radius $\Delta R=(R_{int}-R_{X})/R_{int}$ for each star. Relative differences between ages of both stars compared to the inference uncertainties $\Delta A\equiv(|A_{16CygX}-<A_{16Cyg}>|/\delta A$ where $<A_{16Cyg}>$ is the mean of ages of 16 Cyg A and 16Cyg B.  | Mo23 | C21
---|---|---
$\Delta R$ | 1.27-1.24 | 1.27-1.32
$\Delta A$ | 0.65-0.78 | 1.19-1.00

Along with the Sun, the 16Cyg system is often used to test the sensitivity of
the inferred results to the use of new/updated inference approaches or new and
updated physical processes implemented in stellar modelling (e.g. Bellinger et
al. (2016), Morel et al. (2021), Nsamba et al. (2021), Rendle et al. (2019),
Aguirre Børsen-Koch et al. (2022), Ong et al. (2021), Verma et al. (2022),
Farnir et al. (2023), Bétrisey et al. (2023), Metcalfe et al. (2023)). For
instance, Farnir et al. (2020) carried out a comprehensive study of the 16 Cyg
A,B binary system by estimating the sensitivity of several uncertainties of
the MRA inferences- each at a time- in the physical description of both stars.
Taking the extremum values about their centroid values of the full set of
calculations, the authors found relative differences of $\pm$ 3.7% and $\pm$
7% for the mass and age of 16 CygA and for CygB. The centroid values fall in
the same ranges as found by previous authors. However they were not able to
find stellar models of both stars with a common age and the same chemical
composition while assuming the same physical description. They had to give up
either the assumption of the same chemical composition or assume that the
stars undergo different efficiency of the atomic diffusion, probably
counteracted by additional transport processes yet to be identified. In our
illustrative case, we give in Table 9 the relative differences of the median
values for the mass, radius, and age of each star resulting from the
inferences using the two grids Mo23 and C21. As is well-known and already seen
above with the ‘degraded Sun’ discussion, the age is the most affected by
differences between the two grids. Here again the main reason is the inclusion
or not of atomic diffusion.

Table 11:  Sensitivity to different input physics, number and values of free parameters and reference solar chemical composition $DM\equiv(M_{Mo23}-M_{C21})/M_{Mo23};DR\equiv(R_{Mo23}-R_{C21})/R_{Mo23};DA\equiv(A_{Mo23}-A_{C21})/A_{Mo23}$ | 16 CygA | 16 CygB
---|---|---
DM (%) | 3.26 | 2.73
DR (%) | 1.14 | 0.97
DA (%) | -13.00 | -13.43

This illustrates the lessons we can learn from the study of seismically well
characterized stars.

##  8 Conclusion

We estimated the detection probability of solar-like oscillations for the
target stars of the ESA project PLATO as provided by the version 1.1.0 of the
PLATO input catalogue. The targets belong to different samples: stars with the
lowest expected noise level constitute the P1P2 sample (main sequence and
subgiant FGK stars with magnitude less or equal to 11) and the P5 sample
contains similar types of stars but with a higher noise level. A positive
detection was assumed whenever the probability that the signal is due to noise
is less or equal to 0.1% and the probability of the signal be due to solar
like oscillation is larger than 99%. We then found that we can expect positive
detections of solar-like oscillations for $5858^{+176}_{-879}$ stars in the
P1P2 sample in one single field after a two-year run of observation assuming
the instrument remains nominal over the two years. The given uncertainties are
due to false negative and false positive detections as calibrated with Kepler
data and likely mostly due the fact that we could not take into account the
stellar activity or a non solar chemical composition. For the P5 sample, we
find a positive detection of $9491^{+285}_{-1424}$ stars in the same observing
conditions and assuming the same relative uncertainty percentages. As a whole,
we can expect more than 15000 stars with solar-like oscillations to be
compared to the Kepler solar-like oscillating (main sequence and subgiant
stars) sample of 624 stars (M22).

The S/Ns of the targets in the P1P2 sample is (by construction of the sample)
high enough that individual mode frequencies can be measured with high
precision. For the P1P2 targets for which we predicted a positive seismic
detection, we computed the expected frequency uncertainties. We used the error
propagation due to those frequency uncertainties to estimate the relative
uncertainties that we must expect for the seismically inferred masses, radii,
and ages of those targets.

Focusing on main sequence stars with masses of $\leq 1.2M_{\odot}$, we found
that about 1131 stars satisfy the PLATO requirements for the uncertainties of
the seismically inferred stellar masses, radii, and ages in one single field
after a two-year run of observation. Those stars will constitute an enlarged
set of well characterized stars, compared to the Kepler LEGACY sample, which
contains 66 stars, out of which about 31 main sequence stars with mass $\leq
1.2M_{\odot}$.

We note that the PLATO mission is expected to operate for four years, with
possible extensions over 4.5 more years. This will make possible to more than
double the number of detection of solar-like oscillators or to increase
signicantly the precision of the measurements (of frequencies and then of MRA
inferences), depending on whether we observe several fields or remain longer
on one field.

We must stress that to the above uncertainties, we must add uncertainties due
to systematic errors that mostly arise from imperfect physical description of
our stellar models. Those can contribute up to 5 to 10% to the age
uncertainties depending on the type of stars. Ongoing theoretical works are
therefore currently addressing the main problems of inaccuracy. Tests and
validations of improvements in the physical description of stellar models must
use the best seismically characterized stars at our disposal. While the well-
characterized stars of the Kepler Legacy sample helped us to identify such
stellar modelling biases and offering a path to solving them, further advances
are currently limited by the small number of stars able to bring tight
constraints on the various modelling issues of solar-like oscillating stars.
It is therefore one of the key goals of the PLATO mission to collect a
sufficiently large number of stars with the highest quality data that can
serve as benchmark stars or calibration stars to improve stellar modelling.
The expected sample of PLATO solar-like oscillators will provide a much larger
diversity of well-characterized stars than that available today. This will
result in a larger and denser parameter space in terms of mass, age, chemical
composition, and rotation rate. This will then make it possible to reduce the
uncertainties in stellar modelling, particularly with regard to the internal
transport processes that mainly affect the determination of stellar ages.

##### acknowledgements

We would like to thank the anonymous referee for his pertinent comments, which
helped to improve the manuscript. This work presents results from the European
Space Agency (ESA) space mission PLATO. The PLATO payload, the PLATO Ground
Segment, and PLATO data processing are joint developments of ESA and the PLATO
Mission Consortium (PMC). Funding for the PMC is provided at national levels,
in particular by countries participating in the PLATO Multilateral Agreement
(Austria, Belgium, Czech Republic, Denmark, France, Germany, Italy,
Netherlands, Portugal, Spain, Sweden, Switzerland, Norway, and United Kingdom)
and institutions from Brazil. Members of the PLATO Consortium can be found at
https://platomission.com/. The ESA PLATO mission website is
https://www.cosmos.esa.int/plato. We thank the teams working for PLATO for all
their work. M.C. acknowledges the support of Fundação para a Ciência e
Tecnologia FCT/MCTES, Portugal, throughnational funds by these grants
UIDB/04434/2020, UIDP/04434/2020.FCT, 2022.03993.PTDC
(DOI:10.54499/2022.03993.PTDC) and CEECIND/02619/2017. T.M. acknowledges
financial support from Belspo for contract PLATO mission development. Funding
for the Stellar Astrophysics Centre was provided by The Danish National
Research Foundation (Grant DNRF106). M.J.G., C.C., R.S., K.B., R.M.O, D.R.,
Y.L., B.M., and J.B. acknowledge support from the Centre National d’Etudes
Spatiales (CNES). A.S. acknowledges grants PID2019-108709GB-I00 from Ministry
of Science and Innovation (MICINN, Spain), Spanish program Unidad de
Excelencia María de Maeztu CEX2020-001058-M, 2021-SGR-1526 (Generalitat de
Catalunya), and support from ChETEC-INFRA (EU project no. 101008324) and the
Plan de Recuperación, Transformación y Resiliencia (PRTR-C17.I1) A.M.
acknowledges support from the ERC Consolidator Grant funding scheme (project
ASTEROCHRONOMETRY, G.A. n. 772293.S.M. acknowledges support from the Spanish
Ministry of Science and Innovation (MICINN) with the Ramón y Cajal fellowship
no. RYC-2015-17697, the grant no. PID2019-107061GB-C66, and the grant no.
PID2019-107187GB-I00, and through AEI under the Severo Ochoa Centres of
Excellence Program 2020–2023 (CEX2019-000920-S).

## References

  * Aerts et al. (2023) Aerts, C., Brown, D., Cabrera, J., et al. 2023, Science Calibration Stars Input Catalogue Requirements Justification, PLATO-KUL-PMC-TN-0001
  * Aguirre Børsen-Koch et al. (2022) Aguirre Børsen-Koch, V., Rørsted, J. L., Justesen, A. B., et al. 2022, Mon. Not. Roy. Astron. Soc., 509, 4344
  * Appourchaux (2004) Appourchaux, T. 2004, Astron. Astrophys., 428, 1039
  * Appourchaux (2020) Appourchaux, T. 2020, PLATO noise background with the Normal Cameras: implication for age determination, PLATO-IAS-SCI-AN-001
  * Appourchaux et al. (2012) Appourchaux, T., Benomar, O., Gruberbauer, M., et al. 2012, Astron. Astrophys., 537, A134
  * Appourchaux et al. (2006) Appourchaux, T., Berthomieu, G., Michel, E., et al. 2006, in ESA Special Publication, Vol. 1306, The CoRoT Mission Pre-Launch Status - Stellar Seismology and Planet Finding, ed. M. Fridlund, A. Baglin, J. Lochard, & L. Conroy, 429
  * Appourchaux et al. (2008) Appourchaux, T., Michel, E., Auvergne, M., et al. 2008, Astron. Astrophys., 488, 705
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, Annual Review of Astronomy and Astrophysics, 47, 481
  * Baglin et al. (2009) Baglin, A., Auvergne, M., Barge, P., et al. 2009, in Transiting Planets, ed. F. Pont, D. Sasselov, & M. J. Holman, Vol. 253, 71–81
  * Ball & Gizon (2014a) Ball, W. H. & Gizon, L. 2014a, Astron. Astrophys., 568, A123
  * Ball & Gizon (2014b) Ball, W. H. & Gizon, L. 2014b, A new correction of stellar oscillation frequencies for near-surface effects (Corrigendum), Astronomy & Astrophysics, Volume 569, id.C2, 1 pp.
  * Ballot et al. (2011) Ballot, J., Barban, C., & van’t Veer-Menneret, C. 2011, Astron. Astrophys., 531, A124
  * Bazot (2020) Bazot, M. 2020, Astron. Astrophys., 635, A26
  * Bellinger et al. (2016) Bellinger, E. P., Angelou, G. C., Hekker, S., et al. 2016, Astrophys. J., 830, 31
  * Bellinger et al. (2017) Bellinger, E. P., Angelou, G. C., Hekker, S., et al. 2017, in European Physical Journal Web of Conferences, Vol. 160, European Physical Journal Web of Conferences, 05003
  * Benomar et al. (2010) Benomar, O., Baudin, F., Marques, J. P., et al. 2010, Astronomische Nachrichten, 331, 956
  * Bétrisey et al. (2023) Bétrisey, J., Buldgen, G., Reese, D. R., et al. 2023, Astron. Astrophys., 676, A10
  * Bétrisey et al. (2022) Bétrisey, J., Pezzotti, C., Buldgen, G., et al. 2022, Astron. Astrophys., 659, A56
  * Börner et al. (2023) Börner, A., Paproth, C., & Cabrera, P. G. e. a. 2023, Experimental Astronomy submitted
  * Borucki et al. (2010) Borucki, W. J., Koch, D., & Kepler Science Team. 2010, in AAS/Division for Planetary Sciences Meeting Abstracts, Vol. 42, AAS/Division for Planetary Sciences Meeting Abstracts #42, 47.03
  * Buldgen (2019) Buldgen, G. 2019, arXiv e-prints, arXiv:1902.10399
  * Buldgen et al. (2022) Buldgen, G., Farnir, M., Eggenberger, P., et al. 2022, Astron. Astrophys., 661, A143
  * Campante et al. (2016) Campante, T. L., Schofield, M., Kuszlewicz, J. S., et al. 2016, Astrophys. J., 830, 138
  * Chaplin et al. (2014) Chaplin, W. J., Basu, S., Huber, D., et al. 2014, The Astrophysical Journal Supplement Series, 210, 1
  * Chaplin et al. (2011a) Chaplin, W. J., Bedding, T. R., Bonanno, A., et al. 2011a, Astrophys. J. Lett., 732, L5
  * Chaplin et al. (2011b) Chaplin, W. J., Kjeldsen, H., Bedding, T. R., et al. 2011b, Astrophys. J., 732, 54
  * Chaplin & Miglio (2013) Chaplin, W. J. & Miglio, A. 2013, Annual Review of Astronomy and Astrophysics, 51, 353
  * Christensen-Dalsgaard (2008) Christensen-Dalsgaard, J. 2008, Astrophys. Space Sci., 316, 113
  * Christensen-Dalsgaard (2016) Christensen-Dalsgaard, J. 2016, arXiv e-prints, arXiv:1602.06838
  * Christensen-Dalsgaard (2018) Christensen-Dalsgaard, J. 2018, in Variability of the Sun and Sun-Like Stars: from Asteroseismology to Space Weather, ed. J. P. Rozelot & E. S. Babayev, 125
  * Christensen-Dalsgaard & Silva Aguirre (2018) Christensen-Dalsgaard, J. & Silva Aguirre, V. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte, 184
  * Corsaro et al. (2013) Corsaro, E., Fröhlich, H. E., Bonanno, A., et al. 2013, Mon. Not. Roy. Astron. Soc., 430, 2313
  * Cox & Giuli (1968) Cox, J. P. & Giuli, R. T. 1968, Principles of stellar structure
  * Creevey et al. (2017) Creevey, O. L., Metcalfe, T. S., Schultheis, M., et al. 2017, Astron. Astrophys., 601, A67
  * Cunha et al. (2021) Cunha, M. S., Roxburgh, I. W., Aguirre Børsen-Koch, V., et al. 2021, Mon. Not. Roy. Astron. Soc., 508, 5864
  * Dupret (2019) Dupret, M.-A. 2019, arXiv e-prints, arXiv:1901.08809
  * Farnir et al. (2020) Farnir, M., Dupret, M. A., Buldgen, G., et al. 2020, Astron. Astrophys., 644, A37
  * Farnir et al. (2023) Farnir, M., Valentino, A., Dupret, M. A., & Broomhall, A. M. 2023, Mon. Not. Roy. Astron. Soc., 521, 4131
  * García & Ballot (2019) García, R. A. & Ballot, J. 2019, Living Reviews in Solar Physics, 16, 4
  * Garcia et al. (2010) Garcia, R. A., Ballot, J., Mathur, S., Salabert, D., & Regulo, C. 2010, arXiv e-prints, arXiv:1012.0494
  * Gilliland et al. (2010) Gilliland, R. L., Jenkins, J. M., Borucki, W. J., et al. 2010, Astrophys. J. Lett., 713, L160
  * Grevesse & Sauval (1998) Grevesse, N. & Sauval, A. J. 1998, Space Sci. Rev., 85, 161
  * Heiter et al. (2015a) Heiter, U., Jofré, P., Gustafsson, B., et al. 2015a, Astron. Astrophys., 582, A49
  * Heiter et al. (2015b) Heiter, U., Lind, K., Asplund, M., et al. 2015b, Physica Scripta, 90, 054010
  * Hekker (2020) Hekker, S. 2020, Frontiers in Astronomy and Space Sciences, 7, 3
  * Houdek & Gough (2007) Houdek, G. & Gough, D. O. 2007, in American Institute of Physics Conference Series, Vol. 948, Unsolved Problems in Stellar Physics: A Conference in Honor of Douglas Gough, ed. R. J. Stancliffe, G. Houdek, R. G. Martin, & C. A. Tout, 219–224
  * Huber et al. (2011) Huber, D., Bedding, T. R., Stello, D., et al. 2011, Astrophys. J., 743, 143
  * Huber et al. (2017) Huber, D., Zinn, J., Bojsen-Hansen, M., et al. 2017, Astrophys. J., 844, 102
  * Jackiewicz (2021) Jackiewicz, J. 2021, Frontiers in Astronomy and Space Sciences, 7, 102
  * Jiang & Gizon (2021) Jiang, C. & Gizon, L. 2021, Research in Astronomy and Astrophysics, 21, 226
  * Kallinger et al. (2014) Kallinger, T., De Ridder, J., Hekker, S., et al. 2014, Astron. Astrophys., 570, A41
  * Karoff et al. (2019) Karoff, C., Metcalfe, T. S., Montet, B. T., et al. 2019, Mon. Not. Roy. Astron. Soc., 485, 5096
  * Kim & Chang (2021) Kim, K.-B. & Chang, H.-Y. 2021, New Astronomy, 84, 101522
  * Koch et al. (2010) Koch, D. G., Borucki, W. J., Basri, G., et al. 2010, Astrophys. J. Lett., 713, L79
  * Lebreton & Goupil (2014) Lebreton, Y. & Goupil, M. J. 2014, Astron. Astrophys., 569, A21
  * Lebreton et al. (2014a) Lebreton, Y., Goupil, M. J., & Montalbán, J. 2014a, in EAS Publications Series, Vol. 65, EAS Publications Series, ed. Y. Lebreton, D. Valls-Gabaud, & C. Charbonnel, 99–176
  * Lebreton et al. (2014b) Lebreton, Y., Goupil, M. J., & Montalbán, J. 2014b, in EAS Publications Series, Vol. 65, EAS Publications Series, ed. Y. Lebreton, D. Valls-Gabaud, & C. Charbonnel, 177–223
  * Lebreton & Montalbán (2010) Lebreton, Y. & Montalbán, J. 2010, Astrophys. Space Sci., 328, 29
  * Libbrecht (1992) Libbrecht, K. G. 1992, Astrophys. J., 387, 712
  * Lochard (2003) Lochard, J. 2003, in IAU Joint Discussion, Vol. 25, IAU Joint Discussion, E25
  * Lund & Reese (2018) Lund, M. N. & Reese, D. R. 2018, in Astrophysics and Space Science Proceedings, Vol. 49, Asteroseismology and Exoplanets: Listening to the Stars and Searching for New Worlds, ed. T. L. Campante, N. C. Santos, & M. J. P. F. G. Monteiro, 149
  * Lund et al. (2017) Lund, M. N., Silva Aguirre, V., Davies, G. R., et al. 2017, Astrophys. J., 835, 172
  * Marques et al. (2013) Marques, J. P., Goupil, M. J., Lebreton, Y., et al. 2013, Astron. Astrophys., 549, A74
  * Mathur (2014) Mathur, S. 2014, in SF2A-2014: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. J. Ballet, F. Martins, F. Bournaud, R. Monier, & C. Reylé, 257–262
  * Mathur et al. (2022) Mathur, S., García, R. A., Breton, S., et al. 2022, Astron. Astrophys., 657, A31
  * Mathur et al. (2019) Mathur, S., García, R. A., Bugnet, L., et al. 2019, Frontiers in Astronomy and Space Sciences, 6, 46
  * Maxted & Creevey (2023) Maxted, P. & Creevey, O. 2023, in sciencesconf.org:plato-stesci2023:469767
  * Metcalfe et al. (2012) Metcalfe, T. S., Chaplin, W. J., Appourchaux, T., et al. 2012, Astrophys. J. Lett., 748, L10
  * Metcalfe et al. (2023) Metcalfe, T. S., Townsend, R. H. D., & Ball, W. H. 2023, arXiv e-prints, arXiv:2307.16247
  * Michel et al. (2009) Michel, E., Samadi, R., Baudin, F., et al. 2009, Astron. Astrophys., 495, 979
  * Miglio et al. (2017) Miglio, A., Chiappini, C., Mosser, B., et al. 2017, Astronomische Nachrichten, 338, 644
  * Moedas et al. (2022) Moedas, N., Deal, M., Bossini, D., & Campilho, B. 2022, Astron. Astrophys., 666, A43
  * Montalto et al. (2021) Montalto, M., Piotto, G., Marrese, P. M., et al. 2021, Astron. Astrophys., 653, A98
  * Morel & Lebreton (2008) Morel, P. & Lebreton, Y. 2008, Astrophys. Space Sci., 316, 61
  * Morel et al. (2021) Morel, T., Creevey, O. L., Montalbán, J., Miglio, A., & Willett, E. 2021, Astron. Astrophys., 646, A78
  * Moya et al. (2018) Moya, A., Barceló Forteza, S., Bonfanti, A., et al. 2018, Astron. Astrophys., 620, A203
  * Nascimbeni et al. (2022) Nascimbeni, V., Piotto, G., Börner, A., et al. 2022, Astron. Astrophys., 658, A31
  * Nsamba et al. (2018) Nsamba, B., Campante, T. L., Monteiro, M. J. P. F. G., et al. 2018, Mon. Not. Roy. Astron. Soc., 477, 5052
  * Nsamba et al. (2022) Nsamba, B., Cunha, M. S., Rocha, C. I. S. A., et al. 2022, Mon. Not. Roy. Astron. Soc., 514, 893
  * Nsamba et al. (2021) Nsamba, B., Moedas, N., Campante, T. L., et al. 2021, Mon. Not. Roy. Astron. Soc., 500, 54
  * Ong et al. (2021) Ong, J. M. J., Basu, S., & McKeever, J. M. 2021, Astrophys. J., 906, 54
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, The Astrophysical Journal Supplement Series, 234, 34
  * Pérez Hernández et al. (2019) Pérez Hernández, F., García, R. A., Mathur, S., Santos, A. R. G., & Régulo, C. 2019, Frontiers in Astronomy and Space Sciences, 6, 41
  * Rauer et al. (2023) Rauer, H., Aerts, C., Cabrera, J., & et al. 2023, Astron. Astrophys. submitted
  * Rauer et al. (2014) Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249
  * Reese et al. (2016) Reese, D. R., Chaplin, W. J., Davies, G. R., et al. 2016, Astron. Astrophys., 592, A14
  * Rendle et al. (2019) Rendle, B. M., Buldgen, G., Miglio, A., et al. 2019, Mon. Not. Roy. Astron. Soc., 484, 771
  * Ricker et al. (2015) Ricker, R., Hendricks, S., Perovich, D. K., Helm, V., & Gerdes, R. 2015, Geophys. Res. Lett., 42, 4447
  * Salabert et al. (2016) Salabert, D., García, R. A., Beck, P. G., et al. 2016, Astron. Astrophys., 596, A31
  * Salabert et al. (2017) Salabert, D., García, R. A., Jiménez, A., et al. 2017, Astron. Astrophys., 608, A87
  * Salaris & Cassisi (2017) Salaris, M. & Cassisi, S. 2017, Royal Society Open Science, 4, 170192
  * Samadi et al. (2019) Samadi, R., Deru, A., Reese, D., et al. 2019, Astron. Astrophys., 624, A117
  * Samadi et al. (2010) Samadi, R., Ludwig, H. G., Belkacem, K., Goupil, M. J., & Dupret, M. A. 2010, Astron. Astrophys., 509, A15
  * Santos et al. (2023) Santos, A. R. G., Mathur, S., García, R. A., et al. 2023, Astron. Astrophys., 672, A56
  * Schofield et al. (2019) Schofield, M., Chaplin, W. J., Huber, D., et al. 2019, The Astrophysical Journal Supplement Series, 241, 12
  * Serenelli et al. (2017) Serenelli, A., Johnson, J., Huber, D., et al. 2017, The Astrophysical Journal Supplement Series, 233, 23
  * Serenelli et al. (2021) Serenelli, A., Weiss, A., Aerts, C., et al. 2021, Astron. Astrophys. Rev., 29, 4
  * Silva Aguirre et al. (2015) Silva Aguirre, V., Davies, G. R., Basu, S., et al. 2015, Mon. Not. Roy. Astron. Soc., 452, 2127
  * Silva Aguirre et al. (2017) Silva Aguirre, V., Lund, M. N., Antia, H. M., et al. 2017, Astrophys. J., 835, 173
  * Soderblom (2010) Soderblom, D. R. 2010, Annual Review of Astronomy and Astrophysics, 48, 581
  * Soubiran et al. (2022) Soubiran, C., Brouillet, N., & Casamiquela, L. 2022, VizieR Online Data Catalog, J/A+A/663/A4
  * Stello et al. (2009) Stello, D., Chaplin, W. J., Basu, S., Elsworth, Y., & Bedding, T. R. 2009, Mon. Not. Roy. Astron. Soc., 400, L80
  * Thomas et al. (2021) Thomas, A. E. L., Chaplin, W. J., Basu, S., et al. 2021, Mon. Not. Roy. Astron. Soc., 502, 5808
  * Townsend & Teitler (2013) Townsend, R. H. D. & Teitler, S. A. 2013, Mon. Not. Roy. Astron. Soc., 435, 3406
  * Valle et al. (2020) Valle, G., Dell’Omodarme, M., Prada Moroni, P. G., & Degl’Innocenti, S. 2020, Astron. Astrophys., 635, A77
  * Verma et al. (2022) Verma, K., Rørsted, J. L., Serenelli, A. M., et al. 2022, Mon. Not. Roy. Astron. Soc., 515, 1492
  * White et al. (2013) White, T. R., Huber, D., Maestro, V., et al. 2013, Mon. Not. Roy. Astron. Soc., 433, 1262

##  Appendix A Validation of adopted input for our theoretical calculations

We determine and validate our theoretical relations and calculations by
comparing them with the appropriate Kepler data sets described in Sect.3.1.
The goal is to choose the proper recipes for the input parameters that enter
the calculation of either the detection probability or the MRA uncertainties
for the PLATO target stars.

###  A.1 Global seismic parameters and validation of the scaling relation for
the seismic mass

The scaling properties of the global seismic parameters are discussed in
Chaplin & Miglio (2013); Hekker (2020) and references therein;
$\Delta\nu\propto\bar{\rho}^{1/2}\propto(M/R^{3})^{1/2}$ and $\nu_{\rm
max}\propto g/T_{\rm eff}^{1/2}$ where $\bar{\rho}$ and $g$ are the mean
density and the gravity, respectively. When scaled with the solar values, this
gives

$\displaystyle\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}=\left(\frac{M}{\rm{M}_{\odot}}\right)\left(\frac{R}{R_{\odot}}\right)^{-2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{-1/2}$ (27)
$\displaystyle\frac{\Delta\nu}{\Delta\nu_{\odot}}=\left(\frac{M}{\rm{M}_{\odot}}\right)^{1/2}\left(\frac{R}{R_{\odot}}\right)^{-3/2}\,.$
(28)

For the solar values, we adopt $\rm{A}_{\rm max,bol,\odot}=2.53~{}ppm$ (rms
value, see Michel et al. 2009), $\nu_{\rm max,\odot}=3090\mu$Hz,
$\Delta\nu_{\odot}=135.1\mu$Hz, and $T_{\rm eff,\odot}=5777$ K.

These relations can be inverted to provide the seismic mass and radius
provided that $\nu_{\rm max}$, $\Delta\nu$, and $\rm{T}_{\rm eff}$ are known.
The inverted relationships read as follows:

$\displaystyle\frac{M}{\rm{M}_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{3}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-4}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{3/2}$ (29)
$\displaystyle\frac{R}{R_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{-2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1/2}\,.$ (30)

Further, it is also well accepted that $\Delta\nu$ and $\nu_{\rm max}$ are
tightly correlated (Stello et al. 2009; Huber et al. 2011; Serenelli et al.
2017). Indeed a fit to the Kepler data yields

$\displaystyle\frac{\Delta\nu}{\Delta\nu_{\odot}}=a\left(\frac{\nu_{\rm
max}}{\nu_{\rm max,\odot}}\right)^{s}.$ (31)

We then use this scaling relation between $\nu_{\rm max}$ and $\Delta\nu$ into
Eq. (27) above and inverting the resulting equations yields the seismic mass
as

$\displaystyle\frac{M}{\rm{M}_{\odot}}=a^{-4}\left(\frac{\nu_{\rm
max}}{\nu_{\rm max,\odot}}\right)^{3-4s}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{3/2}.$ (32)

Similarly, the seismic radius is obtained as

$\displaystyle\frac{R}{R_{\odot}}=a^{-2}\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{1-2s}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{1/2}.$ (33)

For consistency, we rederived the relation Eq.(31) by fitting our Kepler data
set of solar-like oscillating stars. Figure 9 shows the variation of
$\Delta\nu$ as a function of $\nu_{\rm max}$ for the short cadence Kepler data
on one hand from the Legacy sample from Lund et al. (2017) (L17) and on the
other hand from the S17 catalogue (their Table 3). A fit of the S17’s data
gives $a=0.992$ and $s=0.805$ (assuming $\Delta\nu_{\odot}=135\mu$Hz and
$\nu_{\rm max,\odot}=3090\mu$Hz) while a fit to L17’s data gives $a=0.995$ and
$s=0.816$, both in satisfactory agreement with relations found in the
literature.

Figure 9: Scaled $\Delta\nu$ as a function of scaled $\nu_{\rm max}$ in
logarithmic scales with $\Delta\nu_{\odot}=135\mu$Hz and $\nu_{\rm
max}=3090\mu$Hz. Blue dots: Kepler short cadence data. Top: S17’s catalogue
(S17’s table 3). Bottom: Legacy data (L17). Dashed lines are linear fits.

We then adopt $s=0.82$ and $a=1$ in the rest of the paper. We then obtain,
from Eq. (32), the scaling for the seismic mass as:

$\displaystyle\frac{M}{\rm{M}_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{-0.28}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{3/2}.$ (34)

For the radius, we obtain, from Eq. (33):

$\displaystyle\frac{R}{R_{\odot}}=\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{-0.64}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{1/2}.$ (35)

It is known that the global seismic scaling relations Eqs. (27) and Eqs. (28)
are accurate at the level of a few percents (Huber et al. (2017), S17 and
references therein). This is enough for our purpose since masses and radii are
only used here to delimitate some sub-samples of stars of interest.

When $\nu_{\rm max}$ is not known, it is derived by inverting Eq. (35)

$\displaystyle\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}=\left(\frac{R}{R_{\odot}}\right)^{-1.5625}\ \left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{0.78}$ (36)
$\displaystyle\frac{\Delta\nu}{\Delta\nu_{\odot}}=\left(\frac{\nu_{\rm
max}}{\nu_{\rm max,\odot}}\right)^{0.82}$ (37)

The seismic parameter $\nu_{\rm max}$ can be obtained from a seismic analysis
(e.g. from Kepler data) and $T_{\rm eff}$ is obtained from a spectroscopic
study. Because for the PLATO targets, $\nu_{\rm max}$ are not yet known, we
use the stellar radius and $T_{\rm eff}$ from the PIC to derive $\nu_{\rm
max}$.

###  A.2 Oscillation power amplitude $\rm{A}_{\rm max}$

Both formulations, Eqs. (9) and (21), for the power, $P_{\rm tot}$, involve
the oscillation maximum amplitude $A_{\rm max}$. In order to adopt a
theoretical relation for the amplitudes, we compare the amplitudes given by
two semi-empirical relations derived in C11 and S19 with the measured
amplitudes, $\rm{A}_{\rm max,obs}$, of the Kepler Legacy sample by Lund et al.
(2017) (their Table 3). We recall that this sample is composed of 66 main-
sequence stars with the highest quality of seismic data, and for which
individual oscillation modes are identified and their frequencies, amplitudes,
and line widths are measured with the highest level of precision.

On the theoretical side, C11 derived the relation (Eq. 9 in C11):

$\displaystyle\frac{\rm{A}_{\rm max}}{\rm{A}_{\rm
max,bol,\odot}}=\beta_{is}~{}\left(\frac{R}{R_{\odot}}\right)^{2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1/2},$ (38)

where

$\displaystyle\beta_{is}=1-{\rm e}^{(T_{\rm eff}-T_{\rm red})/1550{\rm K}}$
(39)

and

$\displaystyle T_{\rm red}(K)=8907\left(\frac{L}{L_{\odot}}\right)^{-0.093},$
(40)
$\displaystyle\frac{L}{L_{\odot}}=\left(\frac{R}{R_{\odot}}\right)^{2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{4}\,.$ (41)

S19 used the relation between $\rm{A}_{\rm max},\nu_{\rm
max},\Delta\nu_{\odot}$, and $T_{\rm eff}$ established by Corsaro et al.
(2013) based on their model 4$\beta$ (Eq. 19 , Table 3 in Corsaro et al.
(2013))

$\displaystyle\frac{\rm{A}_{\rm max}}{\rm{A}_{\rm
max,bol,\odot}}=1.41~{}\left(\frac{\nu_{\rm max}}{\nu_{\rm
max,\odot}}\right)^{-2.314}\left(\frac{\Delta\nu}{\Delta\nu_{\odot}}\right)^{2.088}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{-2.235}\,,$ (42)

valid for $4000\mu$Hz $>\nu_{\rm max}>150\mu$Hz. We eliminate
$\Delta\nu/\Delta\nu_{\odot}$ with Eq. (31). With $s=0.82$, $a=1$, this yields
the second relation that we will consider (S19):

$\frac{\rm{A}_{\rm max}}{\rm{A}_{\rm max,bol,\odot}}=1.41\left(\frac{\nu_{\rm
max}}{\nu_{\rm max,\odot}}\right)^{-0.610}\left(\frac{T_{\rm eff}}{T_{\rm
eff,\odot}}\right)^{-2.235}\,.$ (43)

Both relations involve the effective temperature $T_{\rm eff}$, C11 also
involves the stellar radius and S19 involves $\nu_{\rm max}$. We also need the
stellar mass to discriminate between low and high mass cases. The stellar
radius and the stellar mass are given by the scaling relation Eq. (34).

Table 12: Adopted amplitude formulation for $\rm{A}_{\rm max,scal}$. Masses, $M,$ in solar units and $\nu_{\rm max}$ in $\mu$Hz.  $\nu_{\rm max}$ | $\leq 2500$ | $>2500$
---|---|---
$M\leq 1.15$ | $\rm{A}_{\rm max,C11}*1.31$ | $\rm{A}_{\rm max,C11}*1.19$
$M>1.15$ | $\rm{A}_{\rm max,S19}*0.95$ | $\rm{A}_{\rm max,C11}*0.95$
Figure 10: Comparison of the amplitudes $\rm{A}_{\rm max,scal}$ computed as
given in Table 12 with the observed amplitudes $\rm{A}_{\rm max,obs}$ for the
LEGACY sample from (Lund et al. 2017).

Figure 10 compares the amplitudes $\rm{A}_{\rm max,scal}$ as given in Table 12
to the observed amplitudes $\rm{A}_{\rm max,obs}$ for the Legacy sample. The
comparison leads us to adopt as best estimate the purely empirical amplitude
relations as given in Table 12. As can be seen in this figure, our empirical
calibration of $\rm{A}_{\rm max}$ is quite satisfactory. The rescaling factors
1.19 and 1.31 in Table 12 can be explained as arising from uncertainties in
the estimate of the Kepler noise. The threshold $M=1.15\rm{M}_{\odot}$ is
arbitrary but reflects the fact that the scatter in $\rm{A}_{\rm max}$ is
larger for stars in the high mass regime corresponding to higher effective
temperature for which measurements of frequencies, linewidths, and amplitudes
are more difficult (Appourchaux et al. 2012). On the theoretical side, the
$1.15\rm{M}_{\odot}$ threshold corresponds to the discrimination between main
sequence stars with and without a convective core.

##  Appendix B Validation for the probability of detection

In order to validate our calculations, we computed the detection probability
for the samples of Kepler stars with known detection (sample 1) on one hand
and the sample of stars for which no solar-like oscillations were detected
(sample 2) on the other hand. For sample 1, we estimate the percentages of
false negative detections (i.e. the number of stars which we predict no
seismic detection). For sample 2, we estimate the percentages of false
positive detections (i.e. the number of stars from which we predict a seismic
detection). For the results, we consider 3 types of mass regimes: R1 (MS stars
with $M<1.6M_{\odot}$ and subgiants all masses), R2 (stars with $M\leq 1.2$),
R3 (MS stars with $M\leq 1.2)$

###  B.1 Probability of detection for the sample 1 and sample 2

When computing the detection probability for sample 1 (413 stars), we assumed
as our baseline the following input assumptions: $\nu_{\rm max}$, $\Delta\nu$
and $T_{\rm eff}$ are from M22, masses are derived with Eq.29 and
$\delta\nu_{\rm env}=\nu_{\rm max}/2$. The yields are listed in Table 13. We
find, in the R1 mass regime when $P_{\rm det}\leq 0.99$, a percentage of about
$\sim 10\%$ of false negative detections when the detection probability is
$P_{\rm det}\leq 0.99$ and $\sim 5\%$ when $P_{\rm det}\leq 0.90$.

Table 13:  Numbers of false negative detections for sample 1 with known detected oscillations assuming the baseline conditions. Number of stars predicted with NO seismic detection. Masses in solar unit. mass regime | R1 | R2 | R3
---|---|---|---
nb of stars | 413 stars | 276 stars | 52 stars
$P_{\rm det}\leq 0.90$ | 20 | 9 | 7
|  $\sim$ 5% |  $\sim$ 3% |  $\sim$ 13%
$P_{\rm det}\leq 0.99$ | 40 | 16 | 11
|  $\sim$ 10% |  $\sim$ 6% |  $\sim$ 21%

The choice of $P_{\rm det}\leq 0.99$ obviously predicts a larger number of
false negative detections compared to the choice of $P_{\rm det}\leq 0.90$ and
is therefore more conservative for estimating the percentage of positive
detections. As a sanity check, we verified that all stars in the Legacy sample
are found with a detection probability equal to 1 as expected.

We next considered the 990 Kepler stars in sample 2. For that sample, we
assumed as our baseline the following input assumptions: $\nu_{\rm max}$
derived from $\log g$ and $T_{\rm eff}$ given by M19, $\Delta\nu$ from the
scaling relation Eq.31 with $a=1,s=0.82$, masses are derived with Eq.34 and
$\delta\nu_{\rm env}=\nu_{\rm max}/2$. Table 14 provides the yields of the
calculations of the detection probability for those stars. Assuming a seismic
positive detection when $P_{\rm det}>0.99$, we find that the percentage of
false positive detections is $\sim$ 20%, 11%, and 7% for the R1, R2, and R3
mass regimes, respectively .

Table 14: Numbers of false positive detections for the stars of sample 2 assuming the baseline case for that sample. Numbers of stars predicted with seismic positive detection. Masses are in solar units. mass regime | R1 | R2 | R3
---|---|---|---
nb of stars | 936 | 396 | 361
$P_{\rm det}>0.90$ | 279 | 64 | 44
|  $\sim$ 30% |  $\sim$ 16% |  $\sim$ 12%
$P_{\rm det}>0.99$ | 186 | 44 | 27
|  $\sim$ 20% |  $\sim$ 11% |  $\sim$ 7%

When smoothing the PSD in order to decrease the noise level in case of a low
S/N, it is found that the measured amplitudes are decreased by about 6%
compared to the true amplitudes (Lund et al. 2017). Assuming detection when
$P_{\rm det}>0.99$, the impact of decreasing our theoretical amplitudes by 6%
leads to a percentage of false positive detections $\sim$ 17%, 10%, 6% for the
R1, R2 and R3 mass regimes, respectively, which is slightly lower that for the
baseline case ($\sim$ 21%, 11% and 7% in Table 13). The impact is not
important and we will no longer consider the case of decreasing the
amplitudes. Here the choice of $P_{\rm det}>0.99$ rather than 0.90 provides a
lower number of false positive detections.

Hence, in both cases above, $P_{\rm det}>0.99$ is the conservative choice,
namely, lowering the number of positive detections and maximising the number
of false detections. We therefore adopted a detection threshold at 99% when
considering the PLATO targets.

There are a number of assumptions behind the above calculations which lead to
some uncertainties in the results. In the next sections, we estimate the
impact of the main sources of uncertainties in predicting the detection
probability (assuming the conservative case of a detection threshold at 99%).

###  B.2 Impact of uncertainties in the input quantities

We estimated the impact on the detection probability of the uncertainties in
the various input quantities. We carried out the calculations, changing one or
several input quantities at a time and the results must be compared to the
results obtained with the baseline cases.

We assessed first the impact on the number of false negative detections with
sample 1 when using the values of $\nu_{\rm max},\Delta\nu$, and $T_{\rm eff}$
from the S17 instead of using those of M22. The results are listed in Table
15. We note that the number of stars- in each mass regime- can change
depending on the adopted assumptions, specifically because the seismic mass is
computed using a scaling relation which involves $\nu_{\rm max},\Delta\nu$,
and $T_{\rm eff}$. The false negative detections in case of the S17 input
parameters are $\sim$ 11%, 14%, 16% for R1, R2, and R3, respectively (Table
15), namely, slightly greater than $\sim$ 10%, 6%, and 21%, respectively, when
using the set $\nu_{\rm max},\Delta\nu$ and $T_{\rm eff}$ from M22 in Table
14. The impact is higher when considering the smaller sample of MS stars with
$M\leq 1.2$ (R3) than the whole sample (R1).

Table 15:  False negative detections for sample 1. Each row corresponds to one change compared to the baseline case. Figures in parenthesis correspond to the total number of stars in each mass regimes and considered case. Masses, $M$, are in solar units.  mass regime | R1 | R2 | R3
---|---|---|---
$\nu_{\rm max,S17}$ | 44 (413) | 19 (276) | 17 (64)