11institutetext:  Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany<EMAIL_ADDRESS>22institutetext:
Institute of Planetary Research, German Aerospace Center, Rutherfordstrasse 2,
12489 Berlin, Germany<EMAIL_ADDRESS>33institutetext: LESIA,
Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université
Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon,
France<EMAIL_ADDRESS>
# Transit least-squares survey

IV. Earth-like transiting planets expected from the PLATO mission

René Heller 11 Jan-Vincent Harre 22 Réza Samadi 33

(Received 28 June 2021; Accepted 20 June 2022)

In its long-duration observation phase, the PLATO satellite (scheduled for
launch in 2026) will observe two independent, non-overlapping fields,
nominally one in the northern hemisphere and one in the southern hemisphere,
for a total of four years. The exact duration of each pointing will be
determined two years before launch. Previous estimates of PLATO’s yield of
Earth-sized planets in the habitable zones (HZs) around solar-type stars
ranged between 6 and 280. We use the PLATO Solar-like Light curve Simulator
(PSLS) to simulate light curves with transiting planets around bright
($m_{V}\leavevmode\nobreak\ {\leq}\leavevmode\nobreak\ 11$) Sun-like stars at
a cadence of 25 s, roughly representative of the $>15,000$ targets in PLATO’s
high-priority P1 sample (mostly F5-K7 dwarfs and subdwarfs). Our study
includes light curves generated from synchronous observations of 6, 12, 18,
and 24 of PLATO’s 12 cm aperture cameras over both 2 yr and 3 yr of continuous
observations. Automated detrending is done with the Wōtan software, and post-
detrending transit detection is performed with the transit least-squares (TLS)
algorithm. Light curves combined from 24 cameras yield true positive rates
(TPRs) near unity for planets ${\geq}1.2\,R_{\oplus}$ with two transits. If a
third transit is in the light curve, planets as small as $1\,R_{\oplus}$ are
recovered with TPR $\sim 100$ %. We scale the TPRs with the expected number of
stars in the P1 sample and with modern estimates of the exoplanet occurrence
rates and predict the detection of planets with $0.5\,R_{\oplus}\,\leq\,R_{\rm
p}\,\leq\,1.5\,R_{\oplus}$ in the HZs around F5-K7 dwarf stars. For the long-
duration observation phase (2 yr + 2 yr) strategy we predict 11–34 detections,
and for the (3 yr + 1 yr) strategy we predict 8–25 discoveries. These
estimates neglect exoplanets with monotransits, serendipitous detections in
stellar samples P2–P5, a dedicated removal of systematic effects, and a
possible bias of the P1 sample toward brighter stars and high camera coverage
due to noise requirements. As an opposite effect, Earth-sized planets might
typically exhibit transits around P1 sample stars shallower than we have
assumed since the P1 sample will be skewed toward spectral types earlier than
the Sun-like stars assumed in our simulations. Moreover, our study of the
effects of stellar variability on shallow transits of Earth-like planets
illustrates that our estimates of PLATO’s planet yield, which we derive using
a photometrically quiet star similar to the Sun, must be seen as upper limits.
In conclusion, PLATO’s detection of about a dozen Earth-sized planets in the
HZs around solar-type stars will mean a major contribution to this as yet
poorly sampled part of the exoplanet parameter space with Earth-like planets.

###### Key Words.:

methods: data analysis – occultations – planets and satellites: detection –
stars: solar-type – techniques: photometric

##  1 Introduction

Figure 1: Provisional pointing of PLATO’s northern (NPF) and southern (SPF)
long-duration phase fields in an all-sky Aitoff projection using Galactic
coordinates (an updated version is available in Nascimbeni et al., 2022). The
centers of the NPF and SPF are covered by 24 cameras (light blue).
Increasingly darker blue tones refer to a coverage by 18, 12, and 6 cameras,
respectively. Provisional step-and-stare field pointings (STEP01 – STEP10) use
dark blue tones for an overlap of 24 cameras and lighter blue tones for 18,
12, and 6 cameras, respectively. The fields of CoRoT (pink), Kepler (red), K2
(green), and the TESS continuous viewing zone (yellow) are also shown.

The science of extrasolar planets is driven to a large extent by transiting
planets, that is, planets passing in front of their host stars once every
orbit as seen from Earth (Charbonneau et al., 2000). In fact, most of the
planets known beyond the Solar System have been found via the transit method
in the long-term stellar light curves from space-based missions, starting with
37 planets and brown dwarfs from the CoRoT mission (2009-2013; Auvergne et
al., 2009; Deleuil et al., 2018), over 2700 exoplanets and about 2100
candidates yet to be validated111NASA Exoplanet Archive,
https://exoplanetarchive.ipac.caltech.edu, as of 20 June 2022 from the Kepler
primary mission (2009-2013; Borucki et al., 2010), more than 500 transiting
planets and almost 1000 candidates††footnotemark:  from the extended Kepler
mission, called K2 (2014-2018; Howell et al., 2014), and the addition of over
200 exoplanets††footnotemark:  discovered with the ongoing TESS mission (since
2018; Ricker et al., 2015).

Despite the rapid increase in the number of known planets by about three
orders of magnitude within three decades (Heller & Kiss, 2019), it remains
extremely challenging to find Earth-sized planets in Earth-like orbits around
Sun-like stars. As a result, it is hard to constrain their occurrence rates,
even with four years of continuous observations from the Kepler primary
mission (Mulders et al., 2019; Hsu et al., 2019; Kunimoto & Matthews, 2020;
Bryson et al., 2021). The key challenges for Kepler were the small number of
transits of these planets (Kepler required three), a systematic effect in the
Kepler data known as the rolling-band effect (Van Cleve & Caldwell, 2016;
Thompson et al., 2018; Bryson et al., 2021), and the fact that stars are more
variable than predicted via the scaling of the solar power spectrum (Gilliland
et al., 2011, 2015). Even detections of super-Earth-sized planets in one-year
orbits around Sun-like stars such as Kepler-452 b (Jenkins et al., 2015) and
KOI-456.04 (Heller et al., 2020) have been challenging due to these caveats
(Mullally et al., 2018).

The science objective of the PLATO mission (Rauer et al., 2014), with an
expected launch in December 2026, is to find terrestrial planets in the
habitable zones (HZs; Kasting et al., 1993) of solar-like stars and to
determine their bulk properties222PLATO Definition Study Report,
https://sci.esa.int/s/8rPyPew. To achieve this goal, PLATO’s 24 normal
cameras333The 24 “normal” cameras will observe stars with apparent visual
magnitudes $m_{V}\gtrsim 8$ in white light with a cadence of 25 s. Two
additional “fast” cameras will observe very bright stars with $4\lesssim
m_{V}\lesssim 8$ in two different color filters (one in a blue filter, the
other in a red filter) with a cadence of 2.5 s. The fast cameras will be used
for color refinements of astrophysical processes and fine guidance. will
observe over $15,000$ bright ($m_{V}\leq 11$) dwarfs and subgiants of spectral
types F5-K7 with a precision of 50 ppm in 1 hr of integration (Montalto et
al., 2021). This subset of the hundreds of thousands of stars that PLATO will
observe in total is referred to as the P1 stellar sample, and it will be given
the highest priority of all the samples. Over $5000$ stars with $m_{V}\leq 11$
will have their bulk densities measured using asteroseismology. The P1 sample
will be observed during the long-duration observation phase, which will take a
total of four years, and be compiled from the northern PLATO field (NPF) and
the southern PLATO field (SPF). Figure 1 shows the distribution of the PLATO
fields that include the preliminary NPF and SPF on the celestial plane. A more
up-to-date definition of these fields, which are referred to as Long-duration
Observation Phase North 1 (LOPN1) and Long-duration Observation Phase South 1
(LOPS1), is given in (Nascimbeni et al., 2022).

The Gaia mission has already provided highly accurate parallax and optical
photometry measurements of essentially all main-sequence stars with
$m_{V}\lesssim 20$ (Gaia Collaboration et al., 2016, 2021). In combination
with effective temperature measurements from PLATO’s ground-based high-
resolution spectroscopy follow-up, stellar radii will be determined with an
accuracy of 1–2 %, masses will be inferred with an accuracy of $10$ %, and
ages will be estimated using stellar evolution models with an accuracy of 10 %
for over 5 000 F5-K7 stars.

The PLATO Definition Study Report predicted PLATO’s yield of Earth-sized
transiting planets in the HZ around Sun-like stars to range between 6 and 280.
The details depend on PLATO’s observing strategy, that is, whether it observes
its two independent long-duration observation phase fields for 2 yr each (the
2 yr + 2 yr scenario) or one field for 3 yr and the other one for 1 yr (the 3
yr + 1 yr scenario). These yield predictions were derived with exoplanet
occurrence rates that were still very uncertain at the time (Fressin et al.,
2013). The PLATO Definition Study Report considered an occurrence rate of
planets in the HZ with radii smaller than two Earth radii ($R_{\rm
p}\leavevmode\nobreak\ <\leavevmode\nobreak\ 2\,R_{\oplus}$) of between 2 %
and 100 %, with a nominal value of 40 %. The most recent and comprehensive
statistical analysis of the completeness and reliability of the Kepler survey
by Bryson et al. (2021) suggests that the occurrence rate of planets with
$0.5\,R_{\oplus}\leavevmode\nobreak\ \leq\leavevmode\nobreak\ R_{\rm
p}\leavevmode\nobreak\ \leq\leavevmode\nobreak\ 1.5\,R_{\oplus}$ and orbital
periods between 340 d and 760 d (the conservative HZ; Kopparapu et al., 2014)
around G and K dwarf stars is closer to 60 %. Moreover, previous yield
predictions for PLATO were necessarily based on analytical estimates of the
expected signal-to-noise ratio (S/N) of the transits and taking the activity
of Sun-like stars observed with Kepler into account.

With the launch date of the PLATO mission approaching, great progress has been
made in the simulation of light curves that can be expected from PLATO. The
PLATOSim software (Marcos-Arenal et al., 2014) provides end-to-end simulations
from the charge-coupled device (CCD) level to the final light curve product.
The PLATO Solar-like Light curve Simulator (PSLS), following a much more
pragmatic and computationally efficient approach, allows simulations of a
large number of PLATO-like light curves with realistic treatment of the main
sources of astrophysical and systematic noise (Samadi et al., 2019). As for
transit detection, the Wōtan detrending software for stellar light curves has
been optimized with a particular focus on the preservation of exoplanet
transits (Hippke et al., 2019). This optimization was achieved based on
extended benchmarking tests invoking detrending and transit searches in light
curves from the Kepler primary mission, K2, and TESS. Finally, the release of
the transit least-squares (TLS) software provides enhanced detection
sensitivity for small planets over the traditional box least-squares (BLS)
algorithm (Kovács et al., 2002). As shown by Hippke & Heller (2019), for a
parameterization of TLS and BLS that produces a false positive rate (FPR) of 1
% in a search for transits of Earth-like planets around Sun-like stars, the
true positive rate (TPR) of TLS is about 93 %, while it is roughly 76 % for
BLS. This sensitivity gain of TLS over BLS will be vital for PLATO’s discovery
of Earth-sized planets in the HZs around Sun-like stars.

The key objectives of this paper are to update PLATO’s expected yield of
Earth-like planets in the HZs around Sun-like stars and to test if currently
available transit search software would be sufficient to find these planets in
PLATO data and enable PLATO to achieve its science objectives.

##  2 Methods

The principal approach of our study is as follows. We used PSLS to simulate a
sufficiently large number (${\sim}10,000$) of PLATO-like light curves for Sun-
like stars, including stellar variability and systematic effects, some with
transits of Earth-like planets, some without any transits. Then we used Wōtan
to automatically detrend the light curves from any stellar and systematic
variability while preserving any possible transit signals. To infer the
detectability of the injected transits as well as the corresponding false
alarm probability, we searched for the transits using TLS. A comparison of the
injected signals with our results from the recovery tests then yielded the
true and FPRs, which we scaled with literature values for the planet
occurrence rates to predict the number of Earth-like planets to be detected
with PLATO.

###  2.1 PLATO Solar-like Light Curve Simulator

Our analysis starts with the generation of synthetic PLATO-like light curves
with the publicly available444https://sites.lesia.obspm.fr/psls PSLS (v1.3)
Python software (Samadi et al., 2019). PSLS simulates Poisson noise
characteristic for a given stellar magnitude, instrumental effects, systematic
errors, and stellar variability.

In a first step, PSLS reads the input YAML file to simulate the stellar signal
in Fourier space. Then random noise is added (Anderson et al., 1990) to mimic
the stochastic behavior of the signal and finally the signal is transformed
back into the time domain to generate the light curve.

The stellar oscillation spectrum is computed as a sum of resolved and
unresolved differential modes, which are modeled with specific Lorentzian
profiles in the power spectral density space (Berthomieu et al., 2001; Samadi
et al., 2019). The mode frequencies, mode heights, and mode line widths for
main-sequence and subgiant stars are precomputed with the Aarhus adiabatic
pulsation package (ADIPLS) (Christensen-Dalsgaard, 2008). Stellar activity
phenomena such as magnetic activity (star spots), p-mode oscillations, and
granulation lead to time-dependent variability for the disk-integrated flux of
a solar-like star. The stellar activity component is simulated in PSLS with a
Lorentzian profile in frequency space, and it includes an amplitude
($\sigma_{A}$, subscript “A” referring to activity) and a characteristic
timescale ($\tau_{A}$), both of which can be adjusted in PSLS (Samadi et al.,
2019). Stellar granulation, caused by convection currents of plasma within the
star’s convective zone, occurs on a scale from granules the size of
${\sim}0.5\,$Mm (${\sim}0.08\,R_{\oplus}$) to supergranules with diameters of
${\sim}16\,$Mm (${\sim}2.5\,R_{\oplus}$), all of which appear stochastically
over time (Morris et al., 2020). Granulation is simulated in PSLS using the
two pseudo-Lorentzian functions with characteristic timescales $\tau_{1,2}$
(Samadi et al., 2019).

Systematic errors of the PLATO instrument are simulated in PSLS using the
Plato Image Simulator (PIS), developed at the Laboratoire d’études spatiales
et d’instrumentation en astrophysique (LESIA) at the Observatoire de Paris.
PIS models different sources of noise and other perturbations, like photon
noise, readout noise, smearing, long-term drift, satellite jitter, background
signal, intra-pixel response nonuniformity, pixel response nonuniformity,
digital saturation (Janesick, 2001; Marcos-Arenal et al., 2014), charge
diffusion (Lauer, 1999), and charge transfer inefficiency (Short et al.,
2013). In our simulations we used the beginning-of-life setup tables for
systematics, where charge transfer inefficiency is not included in simulating
systematics. We also turned off jitter in PIS as it demands substantial
amounts of CPU time. Furthermore, according to the PLATO Study Definition
Report, the pointing errors are expected to be sufficiently low that they will
be negligible to the overall noise (see also Marchiori et al., 2019). Light
curves for the P1 sample will not be extracted from aperture masks but from
the downloaded imagettes using the point-spread function fitting method. The
resulting systematic effects, including jumps in the light curves at the
quarterly repositioning of the stars on the PLATO CCDs after satellite
rotation, are properly taken into account in PSLS.

Finally, planetary transits can be automatically simulated in PSLS data using
the Mandel & Agol (2002) model. The actual implementation in PSLS is based on
the Python code of Ian Crossfield555https://www.mit.edu/ iancross/python. PSLS
users can specify the transit parameters, including the planet radius ($R_{\rm
p}$), the orbital period ($P$), the planet semimajor axis ($a$), and the
orbital inclination. Transits assume a quadratic limb darkening law as per
Mandel & Agol (2002), and the two quadratic limb darkening coefficients of the
star can be manually adjusted.

For all our simulations, we chose a solar-type star with solar radius and mass
and with Sun-like stellar activity to represent the expected 15,000 to 20,000
F5 - K7 stars in the P1 sample. As for the amplitude and characteristic
timescale of stellar photometric activity, we assumed activity parameters
close to those of 16 Cyg B, that is, $\sigma_{\rm A}=20$ ppm and $\tau_{\rm
A}=0.27$ d. This is similar to the values used in the default parameterization
of the PSLS YAML input file, as described in Samadi et al. (2019, Appendix A
therein). 16 Cyg B is a solar-type oscillating main-sequence star, for which
stellar activity, asteroseismology, and rotation periods have been well
constrained using Kepler data (Davies et al., 2015). In Appendix C we examine
the effects of different levels of stellar variability on transit detection
with PLATO.

The resulting PSLS light curves are derived as averages from 6, 12, 18, or 24
light curves (corresponding to the number of cameras) and have a cadence of 25
s, representative of the data that will be extracted for the P1 sample. For
reference, a PSLS light curve worth of 2 yr (3 yr) of simulated observations
contains about 2.5 (3.7) million data points.

###  2.2 Light curve detrending with Wōtan

Wōtan is a publicly available666https://github.com/hippke/wotan Python
software for the detrending of stellar light curves under optimized
preservation of exoplanet transit signatures (Hippke et al., 2019). The key
value of Wōtan is in its removal of instrumental and stellar variability from
light curves to prepare them for transit searches. Wōtan features many
different detrending methods. Among all these detrending filters, Hippke et
al. (2019) identified the biweight method as the optimal choice in most cases.

We thus use the biweight method in this work with a window size of three times
the expected maximum transit duration, as suggested by Hippke et al. (2019),
to preserve the transit signatures while removing any other type of
variability from the simulated light curves from PSLS. It is worth noting that
while we restrict our search to transits of planets in Earth-like orbits and,
hence, Earth-like transit durations, in an unrestricted search for transits of
different durations the window size of the detrending algorithm would need to
be adapted accordingly. Wōtan’s treatment of jumps in the PSLS light curves is
described in Sect. 4.3.

###  2.3 Transit Least-Squares

The transit search is executed with the publicly
available777https://github.com/hippke/tls TLS algorithm (Hippke & Heller,
2019). TLS is optimized for finding small planets. The principal sensitivity
gain over the BLS algorithm (Kovács et al., 2002) is in TLS’s accurate
modeling of the planetary ingress and egress and stellar limb-darkening. The
template of TLS’s transit search function is parameterized by two limb
darkening coefficients required to feed the analytical solution for the light
curve with a quadratic limb darkening law (Mandel & Agol, 2002).

TLS finds the minimum $\chi^{2}$ value for a range of trial orbital periods
($P$), transit times $t_{0}$, and transit durations $d$. The orbital
eccentricity is assumed to be zero, but the transit sequence of planets in
highly eccentric orbits are typically found with TLS as long as the transits
are periodic. TLS does not fit for transit timing variations (TTVs), but
transits are usually recovered as long as the TTV amplitude is smaller than
the transit duration.

The key search metric for TLS is the signal detection efficiency (SDE), which
is a measure for the significance of the $\chi^{2}$ minimum compared to the
surrounding $\chi^{2}$ landscape as a function of the period. Hippke & Heller
(2019), using simulated transits of Earth-like planets around Sun-like stars
in synthetic light curves with a white noise component of 110 ppm per 30 min
integration (corresponding to a photometrically quiet $m_{V}=12$ star observed
with Kepler), found that a detection threshold of ${\rm
SDE}\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 9$ results in an FPR
$<\leavevmode\nobreak\ 10^{-4}$ for TLS. At the same time, various studies
have demonstrated that this threshold is sufficiently low for TLS to detect
Earth-sized planets around low-mass stars in Kepler (Heller et al., 2019b, a)
and TESS (Trifonov et al., 2021; Rao et al., 2021; Feliz et al., 2021) data,
super-Earths around solar-type stars (Heller et al., 2020), and possibly even
Mars-sized planets in close orbits around subdwarfs from Kepler (Van Grootel
et al., 2021). Hence, we use ${\rm SDE}=9$ in this work as well. Moreover, we
require ${\rm S/N}>7$ for any candidate signal to count as a detection. This
threshold has been shown to yield one statistical false alarm in a sample of
100,000 photometrically quiet stars from the Kepler mission with normally
distributed noise (Jenkins et al., 2002, Sect. 2.3 therein). TLS calculates
the S/N as $(\delta/\sigma_{\rm o})\,n^{1/2}$, where $\delta$ is the mean
depth of the trial transit, $\sigma_{\rm o}$ is the standard deviation of the
out-of-transit data, and $n$ is the number of in-transit data points (Pont et
al., 2006). We do not use any binning of the light curves in TLS because our
focus is completeness rather than computational speed. We mask out any
residual jumps in the light curve after detrending using the transit_mask
parameter in TLS.

In Fig. 2 we summarize the sequence of steps executed with PSLS, Wōtan, and
TLS. In panel (a), trends are dominantly caused by spacecraft drift, inducing
changes of the camera pointings and a motion of the target star across CCD
pixels with different quantum sensitivities. Jumps result from satellite
rotation and the resulting repositioning of the star on the PLATO CCDs at the
beginning of each PLATO quarter. Panel (b) shows the detrended light curve,
which we obtain by dividing the simulated light curve from PSLS (black points
in (a)) by the trend computed with Wōtan (red line in (a)). Panel (c) presents
the phase-folded light curve of the transit detected with TLS, including the
phase-folded PSLS data (black dots), the TLS best fitting transit model (red
line), and the 21-bin walking median of the data (blue line). In this example,
TLS produces SDE = 7.3 and S/N = 24.5. We point out that the purpose of TLS is
not exoplanet characterization but exoplanet detection. A Markov chain Monte
Carlo fitting procedure that takes into account the stellar mass and radius as
well as the best-fit values for $t_{0}$, $P$, and $d$ from TLS as priors (as
in Heller et al., 2020) can improve the system parameter estimates
substantially.

Figure 2: Illustration of our transit injection and retrieval experiment.
Simulations include two transits of an Earth-sized planet with an orbital
period of 364 d in a 2 yr light curve of an $m_{V}=9$ Sun-like star. (a)
PLATO-like light curve generated with PSLS. The red line shows the trend
computed with Wōtan. (b) Light curve after detrending with Wōtan. The
simulated transits occur at about 91 d and 455 d (highlighted with green
points), that is, near quarterly reorientations of the spacecraft. This
example was deliberately chosen to illustrate that Wōtan and TLS can master
non-well-behaved systems. (c) Phase-folded light curve of the transit detected
with TLS. The best model fit is shown with a red line. The 21-bin walking
median (ten data points symmetrically around each data point) is shown with a
blue line.

###  2.4 Exoplanet occurrence rates and transit probability

Petigura et al. (2013) showed that the occurrence rate of exoplanets with
$2\,R_{\oplus}<R_{\rm p}<4\,R_{\oplus}$ and $200\,{\rm d}<P<400$ d around Sun-
like stars is $5\,(\pm\,2.1)\,\%$. The occurrence rate of smaller planets in
this period range was unconstrained. A thorough treatment of Kepler’s false
positive transit events by Bryson et al. (2021) resulted in an occurrence rate
of planets with radii $0.5\,R_{\oplus}\leavevmode\nobreak\
<\leavevmode\nobreak\ R_{\rm p}\leavevmode\nobreak\ <\leavevmode\nobreak\
1.5\,R_{\oplus}$ and inside the conservative HZs around GK stars close to
$\sim 60\,\%$, details depending on the parameterization of the HZ as a
function of the stellar properties and on the reliability of exoplanet
detections against systematic false positives (Bryson et al., 2020). For our
purpose of estimating PLATO’s Earth-like planet yield, we assume a 37 %
occurrence rate in the conservative scenario and 88 % in the optimistic
scenario, referring to stellar population Model 1 and hab2 stars in Table 3 of
Bryson et al. (2021).

PLATO’s P1 sample contains at least 15,000 and up to 20,000 dwarf and subgiant
bright ($m_{V}\leavevmode\nobreak\ {\leq}\leavevmode\nobreak\ 11$) stars with
spectral types F5-K7. Detailed histograms for the mass, radius, effective
temperature, distance, and Gaia magnitude distributions of the PLATO stellar
sample have recently been published by Montalto et al. (2021), but were not
available before the end of our simulations. We consider 15,000 stars for our
conservative scenario and 20,000 stars for our optimistic scenario. We assume
that the P1 stars will be equally distributed over the NPF and SPF during the
long-duration observation phase (Fig. 1), that is, we assume 7500 (or 10,000)
stars in each of the two fields. Hence, the (2 yr + 2 yr) strategy will
contribute 15,000 stars in the conservative and 20,000 stars in the optimistic
scenario. In contrast, the (3 yr + 1 yr) strategy will only contribute 7500
(or 10,000) stars to our experiment since we assume that the 1 yr field will
not yield any periodic transits (only mono transits) of Earth-like planets in
the HZs around Sun-like stars.

As for our neglect of finding mono transits in PLATO’s hypothetical 1 yr long-
duration observation field, TLS (like BLS and similar phase-folding
algorithms) is most sensitive to periodic signals. Although large-scale human
inspection of thousands of light curves from the Kepler (Wang et al., 2013,
2015) and K2 missions (LaCourse & Jacobs, 2018) have revealed about 200 mono
transit candidates in total, the corresponding planets are all substantially
larger than Earth. There have also been serendipitous mono transit discoveries
with dedicated follow-up observations to confirm rare long-period transiting
candidates as planets (Giles et al., 2018) and progress has been made on the
methodological side with the development of new automatized search tools for
mono transits (Foreman-Mackey et al., 2016; Osborn et al., 2016). But in none
of these cases mono transits from Earth-sized planets have been shown to be
detectable.

The geometric transit probability of an exoplanet can be approximated as
$P_{\rm geo}\leavevmode\nobreak\ {\sim}\leavevmode\nobreak\ R_{\rm s}/a$. In
our case of planets at 1 AU around solar-type stars we have $R_{\odot}/{\rm
1\,AU}\leavevmode\nobreak\ {\sim}\leavevmode\nobreak\ 0.5$ %. We thus expect
PLATO to observe (but not necessarily deliver detectable signal of)
$15,000\leavevmode\nobreak\ \times\leavevmode\nobreak\
37\,\%\leavevmode\nobreak\ \times\leavevmode\nobreak\ 0.5\,\%\,\sim\,28$
transiting planets with $0.5\,R_{\oplus}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ R_{\rm p}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 1.5\,R_{\oplus}$ and orbital periods between 340 d
and 760 d (within the conservative HZ) around F5-K7 dwarf stars in the
conservative scenario. In the optimistic scenario, we expect observations of
$20,000\leavevmode\nobreak\ \times\leavevmode\nobreak\
88\,\%\leavevmode\nobreak\ \times\leavevmode\nobreak\
0.5\,\%\leavevmode\nobreak\ \sim\leavevmode\nobreak\ 88$ such planets. The
question we want to answer with our transit injection and retrieval experiment
in simulated PLATO data is how many of these transits a state-of-the-art
transit detection software would be able to detect.

Figure 3: SDE vs. S/N distribution obtained with TLS for 100 simulated 2 yr
light curves from 24 PLATO cameras for bright ($m_{V}=8$) Sun-like stars. (a)
Light curves containing two transits of an $R_{\rm p}=0.8\,R_{\oplus}$ planet.
(b) Light curves containing two transits of an $R_{\rm p}=0.9\,R_{\oplus}$
planet. Black lines at $\mathrm{SDE}=9$ and $\mathrm{S/N}=7$ define our
nominal detection threshold. Black dots in quadrant I are true positives. Red
open circles in quadrant I are false positives that are not related to
quarterly jumps in the light curves. Red crossed circles in quadrant I are
related to false positives caused by quarter jumps (not present in this plot
but in similar plots in Appendix A). Gray dots outside of quadrant I are
recoveries of the injected transit sequence but with $\mathrm{SDE}<9$ or
$\mathrm{S/N}<7$ or both, that is, false negatives. Black open circles outside
of quadrant I are false negatives not related to the injected signals. Black
crossed circles outside of quadrant I are false negatives related to quarter
jumps. The “(i)” in the legend refers to the injected signal, and “(q)” refers
to quarter jumps. In the histograms, solid black bars count the number of
injected signals recovered in the respective SDE or S/N bins, whereas red line
bars count the number of false detections (both positives and negatives).

###  2.5 PLATO field of view and star count

For the purpose of computing PLATO’s expected planet yield, we weight the
planet detection rates for different camera counts (6, 12, 18, 24) with the
relative areas of the camera count coverage on the celestial plane. The
combined field of view of PLATO’s 24 normal cameras is 2132 deg2. The
provisional pointings of the long-duration observation fields (NPF and
SPF)888The provisional NPF has been designed to overlap with the field of the
Kepler primary mission. Though not a primary focus of PLATO, this overlap
might allow for the validation of a significant number of Kepler exoplanet
candidates and could help to secure exoplanet ephemeris. as well as the
provisional pointings during the step-and-stare phase are shown in Fig. 1.
Both the NPF and SPF are centered around a Galactic latitude of $30^{\circ}$.
Though not perfectly identical, the stellar densities in the NPF and SPF are
extremely similar (Fig. 8 in Nascimbeni et al., 2022). The central area of the
PLATO field, which is covered by all 24 normal cameras, is 294 deg2 ($13.8\%$
of the combined field of view). The area covered by 18 cameras has a size of
171 deg2 ($8.0\%$), by 12 cameras 796 deg2 ($37.3\%$), and by 6 cameras 871
deg2 ($40.9\%$) (Pertenais et al., 2021).

We also weight the planet detection rates with the relative star count as a
function of apparent magnitude in these areas. We examine the Gaia Early Data
Release 3 (EDR3; Gaia Collaboration et al., 2016, 2021) and find a total of
1,247,240 stars with a $G$-band magnitude $m_{G}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 11$ in the NPF and SPF. We then broke the Gaia EDR3
star counts into the following magnitude bins, assuming similar relative star
counts in the PLATO wavelength band:

$\displaystyle m_{V}$ $\displaystyle=8:\hskip 33.57404ptm_{G}<8.5$
$\displaystyle\Rightarrow\ \ \ \ 106,364\,\mathrm{stars}\ \ \ (8.5\,\%)$
$\displaystyle m_{V}$ $\displaystyle=9:\hskip 9.10509pt8.5\leq m_{G}<9.5$
$\displaystyle\Rightarrow\ \ \ \ 187,884\,\mathrm{stars}\ (15.1\,\%)$
$\displaystyle m_{V}$ $\displaystyle=10:\hskip 3.98337pt9.5\leq m_{G}<10.5$
$\displaystyle\Rightarrow\ \ \ \ 484,765\,\mathrm{stars}\ (38.9\,\%)$
$\displaystyle m_{V}$ $\displaystyle=11:10.5\leq m_{G}\leq 11$
$\displaystyle\Rightarrow\ \ \ \ 468,228\,\mathrm{stars}\ (37.5\,\%).$

The purpose of this binning was to obtain the weighting factors for the TPRs
from our injection-retrieval tests that include stars of different magnitude.
As for the spectral types, the distribution of stellar spectral types in the
NPF and SPF was unknown at the time of our writing, and the all-sky PLATO
input catalog has only been published very recently (Montalto et al., 2021).
It would not have been helpful to take spectral type counts from Gaia because
additional conditions are applied to the selection of the P1 sample stars in
addition to magnitude cuts. Hence, we chose a Sun-like star for reference.
This choice follows the procedure of using Sun-like benchmark targets in the
PLATO Definition Study Report (p. 111 therein). This approximation certainly
affects the expected planet yields significantly, the extent of which could be
examined in future studies.

###  2.6 Transit injection-and-retrieval experiment

We create two sets of light curves with PSLS. In the pollinated set, each
light curve contains a transit sequence of one planet. In the control set, no
light curve contains a transit. From the pollinated sample we determine the
TPR, FPR, and false negative rate (FNR). The control set is used to study the
FPR and the true negative rate (TNR).

For the pollinated set, we created 2 yr and 3 yr light curves with transits
for five planet radii $R_{\rm p}\leavevmode\nobreak\ \in\leavevmode\nobreak\
\\{0.8,0.9,1.0,1.2,1.4\\}\leavevmode\nobreak\ \times\leavevmode\nobreak\
R_{\oplus}$, four reference magnitudes $m_{V}\,{\in}\,\\{8,9,10,11\\}$, and
assuming 24 cameras, each combination of which is represented by 100 light
curves. This gives a total of $2\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 5\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 4\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 100\leavevmode\nobreak\ =\leavevmode\nobreak\
4000$ light curves.

For Earth-sized planets only, we also simulated 2 yr and 3 yr light curves for
all combinations of stellar apparent magnitudes
$m_{V}\,{\in}\,\\{8,9,10,11\\}$ and camera counts $n_{\rm
cams}\,{\in}\,\\{6,12,18,24\\}$, giving another $2\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 4\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 4\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 100\leavevmode\nobreak\ =\leavevmode\nobreak\
3200$ light curves. For the control set, which does not contain any transits,
we compute 2 yr and 3 yr light curves for $m_{V}\,{\in}\,\\{8,9,10,11\\}$ and
$n_{\rm cams}\,{\in}\,\\{6,12,18,24\\}$, leading again to
$2\leavevmode\nobreak\ {\times}\leavevmode\nobreak\ 4\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 4\leavevmode\nobreak\
{\times}\leavevmode\nobreak\ 100\leavevmode\nobreak\ =\leavevmode\nobreak\
3200$ light curves. All things combined, we generated $10,400$ light curves
with PSLS.

For all simulated transits, we set the transit impact parameter to zero, which
means that we assumed that all transits occur across the entire diameter. We
also used $t_{0}=91$ d and $P=364$ d throughout our simulations; the latter
choice was made to prevent a systematic effect at the Earth’s orbital period
of 365.25 d that was apparent in some PSLS light curves.

We execute our computations on a ($4{\times}12$)-core AMD Opteron(tm) 6174
processor (2200 MHz) on a GNU/Linux operating system with a 64-bit
architecture. Computation of a 2 yr (3 yr) light curve with PSLS takes about 5
min (9 min), depending on the configuration. The detrending of a 2 yr (3 yr)
light curve with Wōtan typically takes 15 min (25 min). The transit search in
each 2 yr (3 yr) light curve with TLS includes periods between 50.0 d and
364.75 d (547.25 d), the upper boundary chosen to embrace the most long-period
planets that exhibit at least two transits, and typically takes 4.5 h (10 h).
These numbers were measured during parallel processing of typically around ten
jobs, giving a total CPU time of about 270 d.

Figure 4: Similar to Fig. 3 but now for Earth-sized planets around bright
($m_{V}=8$) Sun-like stars and 24 PLATO cameras. Panel (a) is based on two
transits in 100 light curves of 2 yr length, and panel (b) shows the results
for three transits in 100 light curves that are 3 yr long. Figure 5: True
positive rate of our injection-and-retrieval experiment in simulated light
curves from 24 PLATO cameras as a function of injected transiting planet
radius. All stars are assumed to be Sun-like. A total of 100 light curves were
generated, detrended, and searched for transits for each histogram bar.
Detections are defined as any recovery of an injected signal within ${\pm}1$ d
of the injected $P$ or $T_{0}$ values and if the injected signal was detected
by TLS as the strongest signal with ${\rm SDE}\leavevmode\nobreak\
{\geq}\leavevmode\nobreak\ 9$ and ${\rm S/N}\leavevmode\nobreak\
{\geq}\leavevmode\nobreak\ 7$. Results for four apparent stellar magnitudes
are shown in different histogram shadings (see legend). In panel (a) simulated
light curves have a length of 2 yr. In panel (b) light curves have a length of
3 yr.

##  3 Results

###  3.1 Detection rates

In Fig. 3 we show the SDE versus S/N distribution obtained with TLS for 100
light curves that cover 2 yr from 24 PLATO cameras. Each light curve featured
one transiting planet in a 364 d orbit around an $m_{V}\leavevmode\nobreak\
=\leavevmode\nobreak\ 8$ Sun-like star. Panel (a) refers to planets with a
radius of $0.8\,R_{\oplus}$, panel (b) refers to $R_{\rm p}=0.9\,R_{\oplus}$.
Although TLS has the ability to search for successively weaker signals in a
given light curve, we stop our search after the first detection of the
strongest signal. Quadrant I, where $\mathrm{SDE}\leavevmode\nobreak\
{\geq}\leavevmode\nobreak\ 9$ and $\mathrm{S/N}\leavevmode\nobreak\
{\geq}\leavevmode\nobreak\ 7$, contains the true positives (black filled
dots), that is, the retrievals of the injected transits. We define two types
of false positives. Red open circles in quadrant I represent false positives
not related to quarter jumps and red crossed circles illustrate false
positives caused by quarter jumps. Moreover, we define three types of false
negatives. One such type is a detection of the injected signal with
$\mathrm{SDE}<9$ or $\mathrm{S/N}<7$ or both (gray filled dots). False
negatives can also be unrelated to the injected signal and instead be caused
by TLS’s detection of systematic or astrophysical variability below our
nominal SDE-S/N detection threshold. If such a false negative does not refer
to a detection of a quarter jump, we plot it with a black open circle. If a
false negative is caused by a quarter jump outside of quadrant I, we
illustrate it with a black crossed circle. The histograms along the ordinate
and abscissa illustrate the SDE and S/N distributions of the injected signal
recoveries (black filled bars) and false detections (red line bars),
respectively.

Figure 6: Summary of our transit injection-and-retrieval experiment with
PLATO-like light curves of Sun-like stars with an Earth-sized transiting
planet in a one-year orbit. As true positives, we count TLS detections that
fulfill the same criteria as those in Fig. 5. (a) Simulated light curves from
PSLS with a duration of 2 yr and containing two transits. (b) Simulated light
curves from PSLS with a duration of 3 yr and containing three transits.

In comparing Figs. 3(a) and (b), we recognize a substantial shift of the SDE
versus S/N distribution toward higher values, respectively. As a consequence,
the TPR increases from 3 % in panel (a) to 39 % in panel (b). The FPR (red
symbols in quadrant I) is 4 % in (a) and 6 % in (b). As a result, the
detection reliability, which we define as $R={\rm TPR/(TPR+FPR)}$, is $3/7\sim
43\,\%$ in panel (a) and $R=39/45\sim 87\,\%$ in panel (b).

In Fig. 4 we move on to Earth-sized planets ($R_{\rm p}=R_{\oplus}$) in one-
year orbits around bright ($m_{V}=8$) Sun-like stars.999The SDE vs. S/N
distributions obtained for injection and retrieval experiments with $R_{\rm
p}=R_{\oplus}$ for both 2 yr and 3 yr light curves of Sun-like stars with
$m_{V}\leavevmode\nobreak\ \in\leavevmode\nobreak\ \\{9,10,11\\}$, all of
which assumed 24 cameras, are available in Appendix A. Panel (a) represents
100 simulated light curves obtained with 24 PLATO cameras over 2 yr (similar
to Fig. 3), whereas panel (b) shows our results for 3 yr light curves. In our
analysis of the 2 yr light curves, each of which contains two transits, we
obtain TPR = 78 %, FNR = 17 %, FPR = 5 %, and $R=94\,\%$. For comparison, in
Fig. 4(b), where each light curve contained three transits, we find TPR = 100
% and $R=100\,\%$.

As a major result of this study, we find that all of the injected transits of
Earth-sized planets in Earth-like orbits around $m_{V}=8$ Sun-like stars are
recovered when three transits are available. In fact, the increase in both SDE
and S/N is significantly more pronounced when moving from two to three
transits compared to increasing the planetary radius from $0.8\,R_{\oplus}$ to
$1\,R_{\oplus}$. Moreover, our measurements of the detection reliability
suggests that $R\sim 100\,\%$ for Earth-sized planets and larger.

Figure 7: Summary of our transit injection-and-retrieval experiment with
PLATO-like light curves of Sun-like stars without any transiting planet. (a)
Simulated light curves from PSLS with a duration of 2 yr. (b) Simulated light
curves from PSLS with a duration of 3 yr.

In Fig. 5 and the following plots in this paper, we neglect any false
positives detected with periods near the PLATO quarters, that is, with
$P=90\leavevmode\nobreak\ ({\pm}1)$ d and full-integer multiples of that.
Wōtan does a decent job at removing quarterly jumps (see Fig. 2), but real
PLATO data will be cleaned from gaps and jumps prior to any transit search,
which is why this sort of false positives will be drastically reduced in the
analysis of real PLATO data. The histograms in Fig. 5 illustrate the TPRs as a
function of the injected planet radius ($R_{\rm p}\leavevmode\nobreak\
\in\leavevmode\nobreak\ \\{0.8,0.9,1.0,1.2,1.4\\}\leavevmode\nobreak\
\times\leavevmode\nobreak\ R_{\oplus}$). For each planetary radius, we show
four bars, which refer to $m_{V}\leavevmode\nobreak\ \in\leavevmode\nobreak\
\\{11,10,9,8\\}$ from left to right, respectively. All light curves were
simulated with 24 PLATO cameras. Panel (a) refers to 2 yr light curves with
two transits, and panel (b) refers to 3 yr light curves with three transits.

As a general trend, and as expected, we find that the TPRs increase with
increasing planetary radius in both panel (a) and (b). In panel (a), the TPRs
of the 2 yr light curves are on the percent level for the smallest planets
that we investigated ($R_{\rm p}\leavevmode\nobreak\ =\leavevmode\nobreak\
0.8\,R_{\oplus}$), irrespective of the stellar apparent magnitude. For $R_{\rm
p}=0.9\,R_{\oplus}$, the TPRs in (a) increase from $9\,$% for $m_{V}=11$ to
$20\,$% for $m_{V}=10$ and $50\,$% for $m_{V}=9$. Interestingly, the brightest
stars ($m_{V}=8$) do not yield the largest TPRs ($39\,$%), which we attribute
to the associated saturation of the CCD pixels. This loss of true positives
due to saturation is only present for sub-Earth-sized planets in panel (a).
For Earth-sized planets, the TPR is about 28 % for $m_{V}=11$ and near 80 %
for $m_{V}\leavevmode\nobreak\ \in\leavevmode\nobreak\ \\{10,9,8\\}$. For
planets as large as or larger than $1.2\,R_{\oplus}$ the TPRs are near 100 %.

For comparison, in panel (b) the TPRs of the 3 yr light curves are between
about 20 % and 45 % for $R_{\rm p}=0.8\,R_{\oplus}$, which represents a
substantial boost from the near zero TPRs derived for these small planets in
the 2 yr light curves in panel (a). Moreover, the TPRs in panel (b) reach 80 %
for $R_{\rm p}=0.9\,R_{\oplus}$ and between 91 % and 100 % for $R_{\rm
p}=\,R_{\oplus}$, details depending on the apparent stellar magnitude. For
planets as large or larger than $1.2\,R_{\oplus}$ the TPRs are 100 % for all
apparent stellar magnitudes.

In Fig. 6 we focus on Earth-sized planets in one-year orbits around bright
($m_{V}=8$) Sun-like stars, showing the TPRs, the FPRs, and the FNRs. Panel
(a) refers to 2 yr light curves with two transits, and panel (b) refers to 3
yr light curves with three transits. In each panel, four histograms illustrate
the respective rates for 6, 12, 18, and 24 PLATO cameras (see histogram
titles).

In general, the highest TPRs and lowest FNRs in Fig. 6 are obtained for the
largest camera counts and lowest apparent stellar magnitudes. As a general
result we find that TPRs are almost always $<75\,\%$ based on two transits in
the 2 yr light curves in panel (a), even for 24 cameras and
$m_{V}\leavevmode\nobreak\ \in\leavevmode\nobreak\ \\{8,9,10\\}$. For
comparison, for three transits in the 3 yr light curves in panel (b), ${\rm
TPR}\leavevmode\nobreak\ {\approx}\leavevmode\nobreak\ 100\,\%$ in almost all
the cases that we investigated, except for PSLS light curves simulated with 6
cameras of moderately dim stars with $m_{V}\leavevmode\nobreak\
\in\leavevmode\nobreak\ \\{10,11\\}$ and for light curves from 12 cameras and
$m_{V}=11$. For systems with three transits, TPRs are near 100 % for
$m_{V}\,\in\,\\{8,9\\}$ and about 75 % for $m_{V}\,=\,10$ even for as few as
six PLATO cameras. As a consequence, an increase in the camera count is only
beneficial to the TPRs of Earth-sized planets with three transits around stars
with $m_{V}\,=\,11$ (and likely for even dimmer stars).

Figure 6 also demonstrates that the TPRs increase substantially for targets
with low camera counts ($n_{\rm cams}\,{\in}\,\\{6,12\\}$) when adding a third
transit. This is particularly important given the fact that most of the PLATO
field of view is covered with 6 cameras ($40.9\%$) or 12 cameras ($37.3\%$)
(see Fig. 1 and Sect. 2.5). As a consequence, these areas have the greatest
impact on the expected planet yields. Adding observations of a third transit
is extremely beneficial to the planet yields in these regions of the PLATO
field of view that are covered by 6 or 12 cameras.

Figure 7 illustrates the TNRs and FPRs obtained with TLS from PSLS light
curves without any injected signal. Panel (a) refers to 2 yr light curves,
panel (b) refers to 3 yr light curves. As a general result, FPRs for the 2 yr
light curves in (a) are on the percent level at most and we do not observe any
significant correlation between the FPRs and the camera count or the FPRs and
the stellar apparent magnitude. For the 3 yr light curves in (b), our
interpretation is similar except for a dramatic increase in the FPRs to 100 %
for $n_{\rm cams}=6$ and $m_{V}=8$. In this particular configuration,
saturation is combined with a particularly weak transit signal and so TLS is
more sensitive to the quarter jumps than to the transit.

###  3.2 Earth-like planet yield

Next, we predict PLATO’s yield of Earth-like planets in the HZ around Sun-like
stars. To this purpose, we first interpolate the TPRs derived from light
curves with 24 PLATO cameras to cover the entire range of planetary radii we
are interested in. For the 2 yr light curves, we use TPR = 0 for $R_{\rm
p}\,\leq\,0.7\,R_{\oplus}$ and for the 3 yr light curves, we use TPR = 0 for
$R_{\rm p}\,\leq\,0.6\,R_{\oplus}$ for all magnitudes, respectively
(illustrated in Fig. 5). We then use our results for the TPRs of Earth-sized
planets (Fig. 6) to extrapolate the dependence of the TPRs on the camera
counts across all magnitudes and planetary radii. This results in a
distribution of the TPRs as a function of $R_{\rm p}$, $m_{V}$, and $n_{\rm
cams}$.

As an aside, we find that under the most unfavorable conditions of $n_{\rm
cams}=6$ and $m_{V}=11$, TPRs near 100 % are obtained for planets as small as
$1.5\,R_{\oplus}$ for two transits and as small as $1.3\,R_{\oplus}$ for three
transits.

We then multiply the resulting TPRs with our estimates of observable
transiting Earth-like planets from Sect. 2.5. We consider both the (2 yr + 2
yr) and the (3 yr + 1 yr) strategies for PLATO’s long-duration observation
phase (see Sect. 2.4). Although TLS can detect mono transits, its sensitivity
is strongly diminished compared to periodic signals. Hence, we neglect mono
transits in our analysis and assume that no transits will be detected during
the $1\,\mathrm{yr}$ observation phase (see Sect. 2.4).

Our predictions of PLATO’s Earth-like planet yield are shown in Tables 1 and
2. In Table 1, planet counts are itemized as per the apparent stellar
magnitude bins in which we predict them (see Sect. 2.5), whereas in Table 2
planet counts are shown as a function of the PLATO camera count. The
conservative and optimistic scenarios refer to different assumptions of the
star count and HZ exoplanet occurrence rate as detailed in Sect. 2.4. Although
we are fully aware that only a full-integer number of exoplanets can be found,
we chose to present our predictions including one decimal place given the low
number counts.

Table 1: Estimated yield of planets with $0.5\,R_{\oplus}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ R_{\rm p}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 1.5\,R_{\oplus}$ in the HZs around Sun-like stars
from PLATO’s P1 sample.

Observing | $m_{V}$ | Conservative | Optimistic
---|---|---|---
Strategy |  | scenario | scenario
2 yr + 2 yr | 8 | 1.2 | 3.4
| 9 | 2.2 | 6.9
| 10 | 4.4 | 14.0
| 11 | 3.0 | 9.6
| total | 10.7 | 33.8
3 yr + 1 yr | 8 | 0.8 | 2.4
| 9 | 1.4 | 4.3
| 10 | 3.4 | 10.8
| 11 | 2.5 | 7.9
| total | 8.0 | 25.4

101010The two different observing strategies assume that the two long-duration
observation phase fields are monitored for either (2 yr + 2 yr) or (3 yr + 1
yr), respectively. The conservative (optimistic) scenario assumes 15,000
(20,000) Sun-like stars in the P1 sample and an occurrence rate of 0.37 (0.88)
Earth-sized planets in the HZ per star. Planet yields are sorted in bins of
apparent stellar magnitudes (see Sect. 2.5). Numbers are rounded to one
decimal place, which leads to an offset of 0.1 between some of the column sums
and the respective total counts. Total counts are exact.

The key insight to be gained from Tables 1 and 2 is that the (2 yr + 2 yr)
observing strategy produces significantly higher planet yields than the (3 yr
+ 1 yr) observing strategy in any scenario. This interpretation is supported
by the total planet yield counts in both the conservative and optimistic
scenarios. The total count is 10.7 for the (2 yr + 2 yr) strategy compared to
8.0 for the (3 yr + 1 yr) strategy in the conservative scenario. In the
optimistic scenario, the (2 yr + 2 yr) strategy produces a predicted yield of
33.8, whereas the (3 yr + 1 yr) strategy yields 25.4 planets. Details of the
conservative versus optimistic scenarios aside, the yield of the (2 yr + 2 yr)
strategy is 133 % of the (3 yr + 1 yr) strategy.

In addition to these actual discoveries of small planets in the HZs around
Sun-like stars, our results suggest a detection reliability near 100 % for
Earth-sized and larger planets (see Sect. 3.1). Hence, we do not expect a
significant amount of statistical false detections, that is, false positives
caused by systematic, instrumental, or statistical effects for super-Earth-
sized planets. In fact, Hippke & Heller (2019) showed that an SDE threshold of
9 for TLS produces one statistical false positive in about 10,000 light curves
with normally distribute noise with an amplitude of 30 ppm. Consequently, the
P1 sample with its 15,000 - 20,000 stars will yield 1-2 statistical false
positives. That said, there will be about as many false detections of
transiting planets smaller than Earth as there will be genuine sub-Earth-sized
planets. And on top of that, there will also be false positives caused by
astrophysical effects such as blended eclipsing binaries.

##  4 Discussion

###  4.1 Effects of observing strategy, scenario, and detection thresholds

We find that the choice of the observing strategy is not as impactful as the
realization of the conservative versus the optimistic scenario. The
realization of the conservative or the optimistic scenario can only be
affected to a limited extent, for example through the choice of the PLATO
observing fields. Although neither the dilution of exoplanet transits due to
stellar blending nor the occurrence of astrophysical false positives (e.g.,
from blended eclipsing stellar binaries) have been taken into account in our
simulation, this issue has been taken care of by the PLATO consortium by
optimizing the trade-off between a high number of priority targets and a low
number of false-alarm detections due to crowding (Rauer et al., 2014;
Nascimbeni et al., 2022). The exoplanet occurrence rate, however, which also
goes into our definition of the conservative versus optimistic scenarios, is
an observational property and needs to be taken as is.

Table 2: Estimated planet yields as in Table 1 but yields are sorted in terms of the number of cameras used for the simulated light curves. Observing | cams | Conservative | Optimistic
---|---|---|---
Strategy |  | scenario | scenario
2 yr + 2 yr | 6 | 3.5 | 11.2
| 12 | 4.1 | 13.1
| 18 | 1.0 | 3.3
| 24 | 2.0 | 6.3
| total | 10.7 | 33.8
3 yr + 1 yr | 6 | 3.0 | 9.4
| 12 | 3.1 | 9.9
| 18 | 0.7 | 2.3
| 24 | 1.3 | 4.0
| total | 8.0 | 25.4

In our injection-and-retrieval experiment, we used SDE = 9 and S/N = 7 as our
nominal detection thresholds. The SDE versus S/N distributions in Figs. 4(b)
and Figs. 8(b)-10(b) as well as the summary plot in Fig. 5(b) show that these
thresholds are sufficient to detect Earth-sized (and larger) planets using
three transits around bright and moderately bright ($m_{V}\,{\leq}\,11$) Sun-
like stars with ${\rm TPR}\,>\,90\,\%$. Moreover, Figs. 11-14 illustrate that
most false signals achieve ${\rm SDE}\,<\,9$ in the first transit search with
TLS, although the S/N is often substantially above 10. The only type of false
alarm signal with ${\rm SDE}\,>\,9$ that we observed in our simulations is
quarter jumps, but these can be identified and dismissed. There will be other
sources of false positives for PLATO, but their quantification is beyond the
scope of this study. As a tribute to a rigid set of SDE and S/N thresholds,
sub-Earth-sized planets are hard to be discriminated from false alarms, as
becomes clear in a comparison of the SDE versus S/N distribution of the
injected signals in Fig. 3 with the SDE versus S/N distribution of the control
sample in Figs. 11-14. The same tribute is paid for Earth-sized planets with
only two transits (see Fig. 4(a) and Figs. 8(a)-10(a)). Machine learning
methods like self-organizing maps (Armstrong et al., 2017), random forest
(Armstrong et al., 2018), or convolutional neural networks (Pearson et al.,
2018; Osborn et al., 2020; Rao et al., 2021) might be helpful in the
separation of genuine exoplanet transit signals from false alarms, but for now
their advantage over smart-force search algorithms like TLS has not been
conclusively demonstrated.

###  4.2 Planet yields

Our focus on the strongest transit-like signal (true or false) and our
omission of an iterative transit search down to the detection threshold means
that we underestimate TLS’s capabilities to find shallow transits in PLATO
data. In fact, TLS can automatically search for successively weaker signals
(Hippke & Heller, 2019) and there are several ways to interpret an iterative
search in terms of true and FPRs. Though this is beyond the scope of this
study iterative transit searches will certainly be an important topic for
PLATO.

The TPRs for Earth-sized planets transiting $m_{V}\,=\,8$ Sun-like stars are
smaller than for more moderately bright stars with $m_{V}\,=\,9$ in Figs. 5
and 6(a). We attribute this effect to saturation, which results in higher-
than-realistic noise levels for the brightest stars. The resulting PSLS light
curves are thus not representative of real PLATO light curves. That said,
these stars are also the least abundant to be observed with PLATO (see Sect.
2.5). As a consequence, we expect the effect on our expected planets yields to
be minor and $\lesssim\,1$ in terms of number counts for all scenarios. For
details of the conversion between Johnson’s $V$-band magnitude ($m_{V}$) and
PLATO’s $P$ magnitude used in PSLS (see Marchiori et al. 2019).

A direct comparison of our predicted planet yields in the HZs around Sun-like
stars with those presented in the PLATO Definition Study Report is complex due
to several reasons. First, this report used analytical estimates of the
expected number of planets with ${\rm S/N}\,>\,10$ to predict PLATO’s planet
yield. For comparison, we used simulated PLATO light curves and a transit
injection-and-retrieval experiment. Second, we focused on the P1 stellar
sample and chose to represent its 15,000 - 20,000 F5-K7 stars with Sun-like
stars, including astrophysical variability. Instead, the PLATO Definition
Study Report included $m_{V}\leavevmode\nobreak\ {\leq}\leavevmode\nobreak\
11$ stars from both the P1 and P5 sample, the latter of which will comprise
${\geq}\,245,000$ F5-K7 dwarf and subgiant stars (assuming two long-duration
observation phase field pointings) with $m_{V}\leavevmode\nobreak\
{\leq}\leavevmode\nobreak\ 13$ and a cadence of 600 s in most cases. Third,
the estimate of 6 to 280 small planets in the HZs around
$m_{V}\leavevmode\nobreak\ \leq\leavevmode\nobreak\ 11$ stars given in the
PLATO Definition Study Report included all planets smaller than
$2\,R_{\oplus}$. By contrast, we derive exoplanet yields for
$0.5\,R_{\oplus}\,\leq\,R_{\rm p}\,\leq\,1.5\,R_{\oplus}$. Fourth, given the
large observational uncertainties at the time, the PLATO Definition Study
Report necessarily included a large range of the possible occurrence rates of
small planets in the HZ around Sun-like stars of between 2 % and 100 %. For
comparison, our planet yield predictions are based on updated occurrence rates
estimates (Bryson et al., 2021), which define our conservative scenario with
37 % and our optimistic scenario with 88 %.

Our yield estimates for planets with $0.5\,R_{\oplus}\,\leq\,R_{\rm
p}\,\leq\,1.5\,R_{\oplus}$ range between 11 in the conservative scenario of
the (2 yr + 2 yr) observing strategy (or 8 for the 3 yr + 1 yr observing
strategy) and 34 in the optimistic scenario of the (2 yr + 2 yr) observing
strategy (or 25 for the 3 yr + 1 yr observing strategy) (see Tables 1 and 2).
With all the caveats of a direct comparison in mind, our range of the
predicted yield of small planets in the HZ is much tighter than the previous
estimates from the PLATO Definition Study Report and tends to fall in the
lower regime of the previous planet yield estimates.

###  4.3 Methodological limitations and improvements

Our results demonstrate that the Wōtan detrending software efficiently removes
stellar and systematic variability while preserving transit signatures. That
said, in some cases we find that Wōtan does not effectively remove quarter
jumps from PSLS light curves. Wōtan has been designed to detrend jumps by
stitching the data prior and after gaps in the light curve, a functionality
that can be fine-tuned using the break_tolerance parameter. Real PLATO data
will indeed have gaps of at least several hours required for satellite
repositioning, which can be stitched with Wōtan. But PSLS does not include
temporal gaps at quarter jumps for now. This results in occasional false
positive detections with TLS at these quarter jumps, in particular for stars
with $m_{V}\,{\geq}\,10$ (see Figs. 10, 13, and 14).

In the final data products of the actual PLATO mission, quarter jumps will be
subjected to a dedicated light curve stitching. As a consequence, this type of
false positives will not, or very rarely, be an issue. As explained in Sect.
3.1, this is why we neglect quarterly false positives in our summary plots
(Figs. 5-7). Nevertheless, since we did not attempt to sort out false
positives detected with TLS at quarterly jumps and then rerun TLS, we can
expect that the TPRs derived with TLS in such an iterative manner could
actually be higher than shown in Figs. 5 and 6. As a consequence, the
application of TLS on a set of light curves that were properly corrected for
quarter jumps can be expected to produce slightly higher planet yields than in
Tables 1 and 2.

PSLS (v1.3) includes long-term drift correction. It corrects for the drift in
the CCD positions of the stars due to relativistic velocity aberration and
satellite motion. It does not currently take into account, however, any
detrending of the light curves from jitter, CCD positional changes from
thermal trends induced by spacecraft rotation every three months, regular
thermal perturbations caused by the daily data transfer, momentum wheel
desaturation, residual outliers not detected by the outlier detection
algorithms, or the stitching of parts of the light curves between mask updates
– the last of which is irrelevant for P1 sample stars since their photometry
will be extracted using a fitting of the point spread function. Although Wōtan
can be expected to remove most of these trends in a reasonable manner while
preserving transit signatures, the actual data products of the PLATO mission
will be subjected to a detailed detrending of systematic effects. In terms of
detrending of systematic effects (but not necessarily of astrophysical
variability), the real PLATO data will therefore have a somewhat better
quality for transit searches than the PSLS light curves that we used.

On the down side, our simulations assume near-continuous uptime and
uninterrupted functionality of the PLATO satellite. This might be overly
optimistic as demonstrated by the Kepler mission, which achieved an average of
${\sim}\leavevmode\nobreak\ 90\,\%$ time on target. Unplanned downtimes of
PLATO might outbalance the benefits of improved systematic detrending so that
our values in Tables 1 and 2 would remain almost unaffected.

We restricted our study to stars with solar-like activity, while the actual
stars to be observed in the P1 sample will naturally exhibit a range of
activity patterns. An analysis of the first six months of continuous
observations of moderately bright stars (Kepler magnitudes $10.5\leq{\it
Kp}\leq 11.5$) from the Kepler primary mission showed that solar-type stars
are intrinsically more active than previously thought (Gilliland et al.,
2011), a result later confirmed with the final four years of Kepler
observations (Gilliland et al., 2015). Our assumptions of solar-like activity
might thus be overly optimistic. This might have a negative effect on the
planet yields that we estimate since transit signatures of small planets are
harder to find around photometrically more active stars. Simulations of PLATO-
like light curves with more realistic intrinsic stellar variability is, in
principle, possible with PSLS. For now, PSLS requires a user-defined
parameterization of stellar activity but future versions are planned to
implement empirical descriptions of the magnetic activity to suit PLATO solar-
like oscillators (Samadi et al., 2019). Moreover, rotational modulation caused
by spots on the surface of the star are not yet implemented in PSLS.

We did not simulate a representative population of 15,000 to 20,000 F5-K7
stars with $m_{V}\,{\leq}\,11$ as will be observed in PLATO’s P1 stellar
sample. Instead, we assumed a solar-type star with solar radius and mass,
investigated four apparent stellar magnitudes $m_{V}\,\in\,\\{8,9,10,11\\}$
for reference simulations with PSLS, and weighted the abundance of stars in
one-magnitude bins around these reference magnitudes (Sect. 2.5). There are at
least three caveats with this assumption. First, the apparent magnitude
distribution of the P1 sample will likely differ from that of field stars,
with a drop between $m_{V}=10$ and $m_{V}=11$ since the outer, low-camera-
count regions of PLATO’s field of view are not able to meet the noise limit
requirement of 50 ppm per hour integration (V. Nascimbeni, priv. comm.).
Second, Sun-like stars will likely be underrepresented compared to more early-
type stars in the P1 sample. The median stellar radius in the P1 sample will
likely be closer to $1.3\,R_{\odot}$ (Montalto et al., 2021), roughly
corresponding to spectral type F0 on the main sequence. The HZ will be farther
away from the star than in our simulations and the orbital period will be
larger than 1 yr. And third, the P1 sample is not supposed to be evenly
distributed over PLATO’s NPF and SPF due to the noise limit requirement.
Instead, the P1 sample stars will be concentrated in the inner part of the NPF
and SPF, where they are covered by 18 or 24 telescopes. In summary, we expect
that (1) the apparent magnitude distribution of the P1 sample stars will be
skewed more toward brighter stars than based on our Gaia counts of fields
stars (Sect. 2.5), (2) transits of Earth-sized planets in the HZs around P1
sample stars will typically be shallower and have longer orbital periods than
the transits around nominal Sun-like stars in our simulations, and (3) the P1
sample stars will preferentially be observed with 18 or 24 cameras. Since
points (1) and (3) have an opposite effect on the planet yield to point (2) it
is impossible, based on the currently available data, to specify the resulting
effect on the actual planet yield presented in this paper.

In all our transit simulations, we assumed a transit impact parameter of zero,
that is, that the planet crosses the star along the apparent stellar diameter.
In reality, however, the average transit impact parameter for a large sample
of transiting planets is $\pi/4\sim 79\,\%$ of that value (Jenkins et al.,
2002). As a result, we overestimate the average transit duration (and
therefore the resulting S/N) around a large sample of Sun-like stars
systematically. That said, this effect is mitigated by the longer transit
durations expected for HZ planets in the P1 sample, as explained above.

##  5 Conclusions

We have developed a procedure to estimate the yield of Earth-like planets in
the HZs around Sun-like stars from the PLATO mission. In brief, we simulated
PLATO-like light curves, some of which included transits, with the PSLS
software, performed a detrending from systematic and astrophysical variability
with the Wōtan software, and then searched for the injected signals with the
TLS search algorithm. We combined our measurements of the TPRs with the
expected number of stars in PLATO’s P1 stellar sample of bright
($m_{V}\,{\leq}\,11$) stars and with modern estimates for the occurrence rate
of Earth-sized planets in the HZs around Sun-like stars to predict PLATO’s
exoplanet yield. We investigated the (2 yr + 2 yr) and the (3 yr + 1 yr)
observation strategies for PLATO’s long-duration observation phase fields.

We find that under the same simulation conditions the (2 yr + 2 yr) observing
strategy results in significantly enhanced planet yields compared to the (3 yr
+ 1 yr) strategy. Details of the exact numbers for both strategies depend on
the actual number of stars that will be observed in the P1 sample and on the
occurrence rate of small planets in the HZs around Sun-like stars.

Under the assumption of a Sun-like star with low stellar activity, we find
that PLATO can detect planets with radii ${\geq}\,1.2\,R_{\oplus}$ with TPR
${\sim}100$ % in the P1 sample ($m_{V}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 11$) if two transits can be observed synchronously by
24 PLATO cameras. When a third transit is added under otherwise unchanged
conditions, TPR ${\sim}100$ % is achieved for planet as small as Earth. True
positive rates of a few percent for planets as small as $0.8\,R_{\oplus}$ in
one-year orbits around bright Sun-like stars from the P1 sample suggest that
this is the minimum planet size that can be detected, in some rare and
photometrically optimal cases, if two transits are observed. If three transits
are available, planets as small as $0.7\,R_{\oplus}$ may be detectable in rare
cases.

Assuming the most unfavorable conditions in our setup with only six PLATO
cameras and transits in front of the dimmest Sun-like stars in PLATO’s P1
sample ($m_{V}=11$), TPRs near 100 % are nevertheless achieved for planets as
small as $1.5\,R_{\oplus}$ for two transits and as small as $1.3\,R_{\oplus}$
for three transits around solar-type stars. Again, these estimates all assume
low, Sun-like photometric variability.

Using the Sun as a proxy, we predict the detection of between 8 and 34 Earth-
like planets in the HZs around F5-K7 main-sequence stars with
$0.5\,R_{\oplus}\,\leq\,R_{\rm p}\,\leq\,1.5\,R_{\oplus}$. These estimates
should be considered an upper limit for several reasons. First, given that
PLATO’s P1 sample stars will typically be larger than the Sun-like benchmark
stars in our simulations (Montalto et al., 2021), the resulting transits of
Earth-like planets in the HZ will be shallower and less frequent than we
simulated. Second, astrophysical false positives, which we neglected, and as
yet unknown systematic effects of the PLATO mission might increase the FPR and
complicate the identification of genuine transits, in particular given the low
number of transits expected for Earth-like planets in the HZs around Sun-like
stars. Third, and maybe most importantly, all our estimates are based on
simulations of photometrically quiet Sun-like stars, whereas in reality most
F5-K7 main-sequence stars are photometrically more variable. On the other
hand, a more sophisticated correction of systematic effects and astrophysical
variability, more elaborate vetting than a mere SDE-S/N cut, a bias of the P1
stellar sample toward bright stars covered with 18 or 24 cameras, and
serendipitous discoveries in PLATO’s P2–P5 stellar samples could lead to
additional discoveries that are not considered in our estimates.

Our results suggest that PLATO can achieve its science objective of finding
Earth-like planets in the HZs around Sun-like stars. The prediction of the
discovery of 8–34 such worlds means a substantial step forward from the
previously available estimates that ranged between 6 and 280. Nevertheless,
our new estimates worryingly remind us of the initial predictions for the
number of Earth-like planets to be discovered with NASA’s Kepler mission,
which fluctuated around 50 over the years (Borucki et al., 1996; Borucki,
2016). These estimates for Kepler relied on the Sun as a proxy for stellar
variability, which turned out to be an overly optimistic approach. Hence, our
results require follow-up studies of PLATO’s expected planet yield with more
realistic stellar variability. If shallow transit detection can be achieved in
the presence of significant stellar variability, then our results suggest that
PLATO’s detections will mean a major contribution to this as yet poorly
sampled regime of the exoplanet parameter space with Earth-sized planets in
the HZs around solar-type stars.

###### Acknowledgements.

The authors thank Valerio Nascimbeni, Michael Hippke, Heike Rauer, Juan
Cabrera, and Carlos del Burgo for helpful comments on the manuscript. They are
also thankful to an anonymous referee for a thorough report. RH acknowledges
support from the German Aerospace Agency (Deutsches Zentrum für Luft- und
Raumfahrt) under PLATO Data Center grant 50OO1501. RS acknowledges financial
support by the Centre National D’Etudes Spatiales (CNES) for the development
of PSLS. 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. This work has made use of data
from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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##  Appendix A SDE versus S/N for Earth-sized planets and 24 PLATO cameras

As an extension of Fig. 4 we provide the SDE versus S/N distribution of
transiting Earth-like planets around more moderately bright Sun-like stars
from the P1 sample. All PSLS light curves analyzed for these plots assume
observations observed with 24 cameras. Figure 8 refers to $m_{V}=9$, Fig. 9
refers to $m_{V}=10$, and Fig. 10 refers to $m_{V}=11$ Sun-like stars.

Figure 8: SDE vs. S/N for 100 simulated light curves of an $m_{V}=9$ Sun-like
star with a $1\,R_{\oplus}$ transiting planet observed with 24 PLATO cameras.
Left: Each simulated 2 yr light curve contained two transits. Right: Each
simulated 3 yr light curve contained three transits.

Figure 9: Similar to Fig. 8 but for an $m_{V}=10$ Sun-like star.

Figure 10: Similar to Fig. 8 but for an $m_{V}=11$ Sun-like star.

##  Appendix B SDE versus S/N for 24 PLATO cameras without injections

In addition to the SDE versus SDE diagrams for our injection-and-retrieval
tests, we also generated similar plots for the control sample of light curves,
which did not contain any injected transits. These simulations are key to
determining the TNRs and FPRs shown in Fig. 7. The following plots are all
based on simulated PSLS observations with 24 PLATO cameras. Figure 11 refers
to $m_{V}=8$, Fig. 12 refers to $m_{V}=9$, Fig. 13 refers to $m_{V}=10$, and
Fig. 14 refers to $m_{V}=11$.

Figure 11: SDE vs. S/N for 100 simulated light curves of an $m_{V}=8$ Sun-like
star with no injected transiting planet observed with 24 PLATO cameras. Left:
2 yr light curves. Right: 3 yr light curves.

Figure 12: Similar to Fig. 11 but for an $m_{V}=9$ Sun-like star.

Figure 13: Similar to Fig. 11 but for an $m_{V}=10$ Sun-like star.

Figure 14: Similar to Fig. 11 but for an $m_{V}=11$ Sun-like star.

##  Appendix C Transits in the presence of stellar activity

Throughout our study we have assumed a Sun-like star with solar activity
levels. This assumption had also been made prior to the Kepler mission and
unfortunately the analysis of stellar activity levels of the Kepler stars has
shown that this assumption was too optimistic. As it turned out, the Sun is a
relatively quiet star, from the perspective of photometric variability
(Gilliland et al. 2011, 2015). The underestimation of stellar variability
prior to the Kepler mission is now seen as a main reason for why the mission
could not achieve the detection of an Earth-sized planet with three transits,
that is, in a one-year orbit around a solar-type star. PLATO will naturally
face the same challenges. Although a detailed investigation of the effects of
stellar variability on transit detectability is beyond the scope of this
manuscript, we have executed a limited qualitative study.

In discussing stellar activity in the context of exoplanet transits, various
metrics are in use throughout the community. For example, García et al. (2014)
measure ${\langle}S_{{\rm ph},k}{\rangle}$, the mean standard deviation in a
running window with a width corresponding to $k$ times the rotation period of
a given star. As such, ${\langle}S_{{\rm ph},k}{\rangle}$ correlates with the
amplitude of the stellar activity component used in the simulations with PSLS
($\sigma_{\rm A}$) (Douaglin 2018), which is given in Eq. (37) in Samadi et
al. (2019). Beyond stellar activity, ${\langle}S_{{\rm ph},k}{\rangle}$ takes
into account any kind of instrumental or systematic error in the light curves
and, hence, has the tendency to exceed the stellar activity component. In our
analysis we take a similar approach and measure the combined noise level as
the standard deviation in a sliding window with a width of 1 hr ($\sigma_{\rm
1\,hr}$), which is the reference timescale for the computation of the noise-
to-signal budgets in the PLATO mission (Rauer et al. 2014). As an aside,
another alternative metric was used by the Kepler mission, which applied the
“combined differential photometric precision” (Jenkins et al. 2010).

To examine transits in light curves with different stellar activity levels, we
simulated three stars of solar radius with increasing stellar activity levels
and on different characteristic timescales: (1) $\sigma_{\rm A}=40$ ppm and
$\tau_{\rm A}=0.8$ d; (2) $\sigma_{\rm A}=166$ ppm and $\tau_{\rm A}=0.5$ d;
and (3) $\sigma_{\rm A}=500$ ppm and $\tau_{\rm A}=0.3$ d. The choice of the
timescale is motivated by findings of Hulot et al. (2011), who measured the
characteristic timescales for the evolution of stellar activity. As an aside,
$\tau_{\rm A}$ used in PSLS corresponds to the timescales of Hulot et al.
(2011) divided by $2\pi$. As for the activity levels, we refer to García et
al. (2014), who determined for the Sun that ${\langle}S_{{\rm
ph},k}{\rangle}$, referred to as its photometric magnetic activity level,
ranges between 89 ppm and 259 ppm with a reference value of 166 ppm.

For each star, we generated 40 light curves with two transits of an Earth-like
planet and an equal amount of light curves without transits. In each light
curve we measured the sliding standard deviation in a 1 hr window as a proxy
for the combined activity on that timescale ($\sigma_{\rm 1\,hr}$). All
simulations assumed an apparent stellar magnitude of $m_{V}=9$ and coverage by
24 PLATO cameras. Our measurements of $\sigma_{\rm 1\,hr}$ for the three
benchmark stars are given in Table 3. In Figs. 15 \- 17 we show some examples
of the resulting light curves, each of which includes a transit of an Earth-
like planet. In our preliminary analysis, we found that the transit
detectability with TLS upon detrending with Wōtan depends sensitively on
$\sigma_{\rm A}$ but weakly on $\tau_{\rm A}$. As we quantified in a limited
injection-recovery test, shallow transits of Earth-sized planets are securely
recovered around the quiet benchmark star (Fig. 15) but a large fraction of
them gets lost in the stellar activity around even the moderately active
benchmark star (Fig. 16) and they become entirely undetectable around the more
active stars (Fig. 17). These findings illustrate that our estimates of PLATO
planet yield, which we derived using a photometrically quiet star, must be
seen as upper limits.

Table 3: Stellar activity measurements from simulated PSLS light curves.

$\sigma_{\rm A}$ | $\tau_{\rm A}$ | $\sigma_{\rm 1\,hr}$
---|---|---
40 ppm | 0.8 d | 173 ppm
166 ppm | 0.5 d | 178 ppm
500 ppm | 0.3 d | 198 ppm

111111The parameters $\sigma_{\rm A}$ and $\tau_{\rm A}$ are used in PSLS to
characterize the amplitude and timescale of stellar activity (Samadi et al.
2019). For comparison, $\sigma_{\rm 1\,hr}$ is the mean standard deviation
that we extracted in a running 1 hr window from the light curves. For each
combination of $\sigma_{\rm A}$ and $\tau_{\rm A}$ we simulated 40 light
curves.

Figure 15: Four example light curves generated with PSLS with a transit of an
Earth-like planet around a solar-sized star with weak photometric activity
($\sigma_{\rm A}=40$ ppm, $\tau_{\rm A}=0.8$ d). The light curves assumed
synchronous observations with 24 PLATO cameras and $m_{V}=9$. The duration of
the transit is indicated with two vertical black bars spanning a window of 12
hr around the mid-transit time.

Figure 16: Same as Fig. 15 but for a moderately active star ($\sigma_{\rm
A}=166$ ppm, $\tau_{\rm A}=0.5$ d).

Figure 17: Same as Fig. 15 but for an active star ($\sigma_{\rm A}=500$ ppm,
$\tau_{\rm A}=0.3$ d). Note the vastly different range along the ordinate
compared to Figs. 15 and 16.