# TIaRA TESS 1: Estimating exoplanet yields from Year 1 and Year 3 SPOC
lightcurves

Toby Rodel,1,2,3 , Daniel Bayliss1,2 , Samuel Gill1,2 , Faith Hawthorn1,2
1 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry,
CV4 7AL, UK
2 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, UK
3 Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, Belfast, BT7 1NN, UK
E-mail<EMAIL_ADDRESS>0009-0009-2175-7284 0000-0001-6023-1335
0000-0002-4259-0155 0000-0002-8675-182X

###### Abstract

We present a study of the detection efficiency for the TESS mission, focusing
on the yield of longer-period transiting exoplanets ($P>25$ days). We created
the Transit Investigation and Recoverability Application (TIaRA) pipeline to
use real TESS data with injected transits to create sensitivity maps which we
combine with occurrence rates derived from Kepler. This allows us to predict
longer-period exoplanet yields, which will help design follow-up photometric
and spectroscopic programs, such as the NGTS Monotransit Program. For the TESS
Year 1 and Year 3 SPOC FFI lightucurves, we find $2271^{+241}_{-138}$
exoplanets should be detectable around AFGKM dwarf host stars. We find
$215^{+37}_{-23}$ exoplanets should be detected from single-transit events or
"monotransits". An additional $113^{+22}_{-13}$ detections should result from
"biennial duotransit" events with one transit in Year 1 and a second in Year
3. We also find that K dwarf stars yield the most detections by TESS per star
observed. When comparing our results to the TOI catalogue we find our
predictions agree within $1\sigma$ of the number of discovered systems with
periods between 0.78 and 6.25 days and agree to $2\sigma$ for periods between
6.25 and 25 days. Beyond periods of 25 days we predict $403^{+64}_{-38}$
detections, which is 3 times as many detections as there are in the TOI
catalogue with $>3\sigma$ confidence. This indicates a significant number of
long-period planets yet to be discovered from TESS data as monotransits or
biennial duotransits.

###### keywords:

planetary systems – catalogs – surveys – planets and satellites: detection

††pubyear: 2023††pagerange: TIaRA TESS 1: Estimating exoplanet yields from
Year 1 and Year 3 SPOC lightcurves–B

##  1 Introduction

Transiting exoplanets are of exceptional scientific importance as they allow
for many parameters of a planetary system to be characterised. Transit
detections themselves allow for the orbital period and radius ratio of a
planet and its host to be constrained (Winn, 2014). If the host star is
sufficiently bright, spectroscopic radial velocity measurements can be used to
constrain the planetary mass. These can be combined with transit radius
measurements to constrain the planetary density and make inferences about
composition and internal structure. The brighest host stars also permit
atmospheric characterisation of planets via transmission spectroscopy (e.g.
Charbonneau et al., 2002; Madhusudhan et al., 2014a; Kreidberg, 2018; Kempton
et al., 2018; Madhusudhan, 2019).

Discoveries from transit surveys are biased towards shorter-period planets.
Shorter-period planets have a greater geometric probability of transit (Winn,
2014) and transit more frequently within any given monitoring campaign,
resulting in an increase in the signal-to-noise ratio (SNR) of the transit
signal. Despite this longer-period planet discoveries are valuable.

Longer-period planets yield insights into planet formation and migration
mechanisms. At the extremely small orbital separation of planets with $P<10$
days star-planet interactions may have a significant effect on dynamical
evolution (Valsecchi et al., 2015). This leads to very reduced eccentricities
(Albrecht et al., 2012; Winn, 2014), effectively removing any trace of past
dynamical interactions from the planet’s present day orbit. Planets with
longer-orbital periods have larger orbital separations and as such experience
weaker star-planet interactions, retaining more information on past dynamical
interactions. Giant planets on extremely short periods (i.e. "hot-Jupiters")
and those on longer periods ("warm-Jupiters") have been shown to exhibit
differences in their distributions of orbital eccentricity (Dong et al., 2021)
and the number of close orbital companions (Huang et al., 2016). This
indicates that giant planets with orbital periods longer than 10 days may
experience different migration pathways compared with short-period giant
planets (Wu & Lithwick, 2011; Petrovich, 2014; Mustill et al., 2017).
Furthermore, Rossiter-McLaughlin (RM; Rossiter, 1924; McLaughlin, 1924)
measurements of orbital obliquity allow further constraining of possible
migration mechanisms in the planet’s history (Triaud, 2018). Already, Rice et
al. (2022) have found a trend towards more aligned obliquities in some warm-
Jupiters compared to hot-Jupiters lthough the number of well-studied warm-
Jupiters is comparatively fewer.

Additionally, short-period planets experience a greater level of atmospheric
mass loss than those on longer-periods (Owen, 2019), meaning longer period
planets will retain more of their atmospheres. This makes long-period planets
promising targets for atmospheric studies to probe cooler atmospheres that
have undergone less photo-evaporation and to gain an accurate understanding of
their formation (Öberg et al., 2011; Madhusudhan et al., 2014b). However, in
spite of their promise, longer-period cooler planets are more difficult
targets for transmission spectroscopy than closer-in hotter planets due to
their generally smaller atmospheric scale height (Kempton et al., 2018).

Long-period planets also offer important targets in the search for exomoons
and exorings, which have not yet been detected but are theoretically
detectable around giant transiting planets (Barnes & Fortney, 2004; Kipping et
al., 2009; Simon et al., 2012; Aizawa et al., 2017). Exomoons and exorings are
predicted to be more stable and thus likelier to exist around longer-period
planets with greater orbital separations (Barnes & O’Brien, 2002; Barnes &
Fortney, 2004; Cassidy et al., 2009; Dobos et al., 2021; Makarov & Efroimsky,
2023).

Since 2018, the Transiting Exoplanet Survey Satellite (TESS; Ricker et al.,
2014) has been performing an all-sky survey searching for transiting
exoplanets around bright host stars. One of the primary science goals is to
find nearby planets amenable to atmospheric characterisation (Ricker et al.,
2014; Kempton et al., 2018). TESS observes each $24\degree\times\ 96\degree$
sector of sky for 27 days at a time, although overlap of some regions between
sectors mean that the observation baselines for some targets will be >300 days
near the Ecliptic poles in the Continuous Viewing Zone (CVZ; Ricker et al.,
2014). However, in the first year of TESS $\approx 75\%$ of target stars were
only observed in a single sector which means that longer period planets are
likely to only be observed as a single transit. Figure 1 shows the population
of TESS planet detections both confirmed and unconfirmed, showing the relative
lack of longer-period planets. Although such single-transiting candidates are
harder to detect, it is not impossible and the ability of TESS to do so has
been studied previously in Cooke et al. (2018, 2019); Villanueva et al.
(2019). Additionally, a number of long-period transiting planets have been
discovered via monotransits from the K2 (Howell et al., 2014) mission (e.g;
Vanderburg et al., 2015, 2016; Giles et al., 2018; Kruse et al., 2019).

Figure 1: The TESS planet sample. Full sample of TESS objects of interest
(TOI) are shown in grey and confirmed TOIs are coloured according to their
equilibrium temperature. The full 5910 TOIs consist of the TESS project
candidate list excluding those flagged as False Positives. The 273 confirmed
TESS planets include all published and confirmed exoplanets from TESS with
fully constrained, masses, radii and orbital periods. Both data sets were
downloaded from the NASA exoplanet archive (Akeson et al., 2013) on
2023-11-23.

Planets discovered from a single transit ("monotransits") do not have
constrained periods and require follow-up observations to rule out false
positive scenarios and constrain their periods. Such observations can be
performed using photometry or spectroscopic radial velocity (RV) measurements.

Photometric observations can be used to detect additional transit events,
which will either determine the period uniquely, or provide a discrete set of
aliases (e.g; Gill et al., 2020). This is one of the science goals of the Next
Generation Transit Survey (NGTS; Wheatley et al., 2018) monotransit working
group, which uses the 12-telescope NGTS facility in Paranal, Chile to monitor
TESS monotransit candidates that display transit events with depths 1000 ppm
or greater (Bayliss et al., 2020). This program has already successfully
confirmed "monotransits" from TESS data (e.g. Gill et al., 2020; Lendl et al.,
2020; Ulmer-Moll et al., 2022; Grieves et al., 2022). The TESS single transit
planet group has also been following up monotransits using a global network of
small telescopes (Dragomir et al., 2023).

For shallower TESS monotransits ($<$1000 ppm) usually associated with smaller
planets it is extremely difficult to detect additional transits from the
ground. In these cases space facilities such as CHEOPS (Benz et al., 2021) can
be used to detect further transits, e.g. (Tuson et al., 2023; Osborn et al.,
2023; Ulmer-Moll et al., 2023).

Spectroscopic monitoring of monotransits can also determine the true period of
a planet candidate and rule out some eclipsing binary stars which display very
large RV variations (e.g; Dragomir et al., 2019; Ulmer-Moll et al., 2022;
Eberhardt et al., 2022). However, this requires long-term spectroscopic
monitoring of candidates with high resolution instruments. Such spectrographs
have to be mounted on large telescopes for which observing time is limited.

Given the large-effort and hundreds of hours of telescope time involved with
such follow-up efforts, it is vital to inform these efforts with up-to-date
understanding of the sensitivity of TESS to monotransit events and the
expected yield of monotransits from the mission.

Previous studies of yields from the TESS mission have largely focused on
multi-event signals. Prior to the launch of TESS, yield estimates were
predicted in Sullivan et al. (2015); Bouma et al. (2017) based on the expected
performance of the mission given in Ricker et al. (2014).

Around the time that TESS was launched there were several studies that
estimated the potential yields from the TESS mission, including: Muirhead et
al. (2018); Huang et al. (2018); Barclay et al. (2018); Ballard (2019). These
studies were still largely based on the expected performance of the TESS
mission as the data from the mission was only just becoming available. More
recently, Kunimoto et al. (2022) have revised the yield estimates based on the
performance of TESS in its primary and first extended mission. All of these
studies were largely focused on planet discoveries from two or more transits.

The studies of Cooke et al. (2018) and Villanueva et al. (2019) were
specifically focused on the expected yield of monotransits from TESS. These
studies were mostly based on the expected performance of TESS from Ricker et
al. (2014) due to the relative lack of TESS data available at the time of
publication shortly after TESS had launched. Cooke et al. (2019) provided an
update to the yields from Cooke et al. (2018) using the performance of the
TESS primary mission and estimating the yield when TESS revisited the southern
ecliptic in year 3 during its first extended mission. All of these studies
found that a significant number of long-period planets will be detectable in
TESS lightcurves as monotransits.

In this study, we use the custom-made Transit Investigation and Recoverability
Application (TIaRA) pipeline to create sensitivity maps using real TESS
Science Processing Operations Centre (SPOC; Stassun et al., 2019) lightcurves
from the southern ecliptic hemisphere (years 1 and 3). We combine these with
occurrence rates to estimate a yield of planet discoveries.

We present the full details of the TIaRA pipeline in Section 2 of this paper.
In Section 3 we describe the application of the TIaRA pipeline to the SPOC FFI
lightcurves from Year 1 and Year 3 of the TESS mission. In Section 4 we
present sensitivity maps and estimates of exoplanet yields from our
simulation. In Section 5 we discuss interpretations of our results and some
potential caveats and compare them to both actual TESS discoveries and other
yield predictions. Finally in Section 6 we summarise our work and discuss the
potential for applying TIaRA to additional TESS data sets as well as other
transit surveys such as PLATO (Rauer et al., 2014).

##  2 The TIaRA pipeline

In order to estimate the discovery yield of planets from transit surveys such
as TESS, we need to simulate as many transiting exoplanet signals as possible
in a realistic manner. To do this, we base our simulation on the actual stars
that are monitored in the survey. We use the timestamps, measured noise
properties, and dilution factors that are recorded for each star. We also use
all available stellar properties such as radius (R⋆), temperature (Teff), and
mass (M⋆) - which are informed by Gaia data releases (Gaia Collaboration et
al., 2016, 2018, 2023).

In this Section we introduce the Transit Investigation and Recoverability
Application (TIaRA) pipeline, which has been developed to calculate the
sensitivity of a given transit survey to discovering transiting planets across
a range of orbital periods and planetary radii. When combined with occurrence
rates, this provides yield statistics for surveys that can be used to
understand and assess the completeness of discoveries from a survey and plan
future surveys.

###  2.1 Input of Lightcurves

TIaRA requires a catalogue of stars observed by a survey and their parameters.
Using these target lists we retrieve the relevant lightcurves for processing.
We use the timestamps in the lightcurve to calculate a window function for
each target star and to determine the number of in-transit datapoints when
calculating the signal-to-noise (SNR) for transit events. We also use
measurements of noise values and dilution ratios for each lightcurve to
calculate SNR for transit events. Using noise values calculated from real
lightcurves allows for us to realistically account for a range of
astrophysical and instrumental effects in the data. Similarly we use the
dilution ratio to account for blending of sources which is a very common
problem in wide field photometric transit surveys.

###  2.2 Simulating Transiting Planets

TIaRA simulates transiting exoplanets by generating planet parameters and then
calculates which timestamps are either occurring in the ingress, egress or
fully in-transit portion of the transit event.

####  2.2.1 Generation of planetary parameters

Thanks largely to the Kepler mission (Borucki et al., 2010), we have a good
understanding of the occurrence rates of exoplanets, particularly for those
with orbital periods less than $<$100 days (Fressin et al., 2013; Dressing &
Charbonneau, 2013, 2015; Hsu et al., 2019; Kunimoto & Matthews, 2020). For
example we know that super-Earth-sized exoplanets are far more common than gas
giant planets in short period orbits. We can take advantage of this
understanding to optimize TIaRA by simulating transiting planets in proportion
to the occurrence rate of those planets. The occurrence rates we used are
described in Section 3.7. Note that in including planets in the simulation
proportionally to occurrence rates is not an attempt to simulate the
underlying planetary population at this stage but instead is a measure to
optimize the simulation. We prioritise injection of more numerous types of
planets to avoid wasting simulation time on calculating a precise detection
efficiency for rare types of planets for which the yield calculation will
almost certainly be zero due to the small numbers in the underlying
population. This means that the sensitivity maps we produce are still
population agnostic, just with less precise values for rarer types of
exoplanets.

We simulate a large number of planets per star ($N_{\text{Pl}}$) to obtain a
more robust simulation. To choose the orbital period and radius of each of
these we first select a period-radius bin using the occurrence rates as a
weighted probability for a random draw. The exact period and radius of each
planet in the sample was taken from a uniform distribution between the upper
and lower limits of the chosen period and radius bins. We use the generated
period and the mass of the star to estimate the semi-major axis using Kepler’s
third law under the assumption that the mass of the planet is negligible
compared to that of the star.

In order to ensure our simulated planets have a realistic eccentricity
distribution, we randomly assign each planet an orbital eccentricity from a
beta distribution following the prescription set out in Kipping (2013). We
adopt the values of $\alpha=1.03$ and $\beta=13.6$ as proposed by Van Eylen &
Albrecht (2015).

We randomly assign a periastron angle ($\omega$) in radians, from a uniform
distribution over the full $\pi$ radian range.

For each planet generated, we calculated the geometric probability of transit
($p_{\text{tran}}$) following Winn (2014):

$p_{\text{tran}}=\left(\frac{\mbox{R${}_{\star}$}+\mbox{R\textsubscript{p}}}{a}\right)\left(\frac{1+e\sin\omega}{1-e^{2}}\right).$
(1)

For each simulated transiting planet we randomly generate $N_{b}$ different
impact parameters $(b)$ from a uniform distribution between the values of 0
and $1+\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}}$.

####  2.2.2 Simulation of transits

We calculate the total transit duration $(T_{14})$ including the ingress and
egress between the first and fourth points of intersection between the
planetary and stellar discs using the following equation from Winn (2014):

$T_{14}=\frac{P}{\pi}\arcsin\left[\frac{\mbox{R${}_{\star}$}}{a}\frac{\sqrt{\left(1+\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}}\right)^{2}-b^{2}}}{\sin
i}\right]\frac{\sqrt{1-e^{2}}}{1+e\sin\omega},$ (2)

where $i$ is the inclination of the orbital plane and $a$ is the semi-major
axis of the orbit.

In addition, we also calculate the duration of transit between the second and
third intersection $(T_{23})$ using the following equation from Winn (2014):

$T_{23}=\frac{P}{\pi}\arcsin\left[\frac{\mbox{R${}_{\star}$}}{a}\frac{\sqrt{\left(1-\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}}\right)^{2}-b^{2}}}{\sin
i}\right]\frac{\sqrt{1-e^{2}}}{1+e\sin\omega}.$ (3)

For grazing transits this does not give real solutions, and in such cases we
simply set the value of $T_{23}=0$.

To save computation time, we do not simulate signals with epochs that would
result with zero in-transit data points. To accomplish this we convert the
timestamps ingested by TIaRA into orbital phase using a chosen reference time
(TBJD=0, BJD=2457000 for TESS) and use these phase arrays to determine which
portions of the planets orbit are monitored by TESS. We then use these
observed stretches of phase to generate random epochs in the form of an offset
to the chosen reference time, expressed in orbital phase. These observed phase
arrays also allow us to calculate the probability that any transit would
randomly fall within an observed timespan of TESS ($p_{\text{obs}}$). We
generate $N_{\text{ph}}$ values of this phase offset for each inclination of
each planet we simulate resulting in $N_{b}\times N_{\text{ph}}$ total transit
signals per planet.

Using the values of $T_{14}$ and $T_{23}$ calculated from Equations 2 $\&$ 3
and the epoch of each signal, we count the number of points in the timestamps
of each lightcurve which are in ingress ($N_{\text{ingress}}$), egress
($N_{\text{egress}}$) or full transit ($N_{\text{full}}$). To obtain a more
realistic value of the signal to noise we use a trapezoidal approximation for
the shape of transit, and if the signal to noise calculation is likened to the
effective area of the transit curve, then the ingress and the egress are
triangular while the full transit is rectangular. Thus the effective count of
in-transit points for a trapezoidal transit is:

$N_{\text{trans,
eff}}=N_{\text{full}}+0.5(N_{\text{ingress}}+N_{\text{egress}}).$ (4)

It is worth noting that this trapezoidal model is a simplification of the
actual shape of a transit and does not account for effects such as limb
darkening(Espinoza & Jordán, 2016; Agol et al., 2020). We believe this
simplification is still robust enough to produce good yield estimates and
discuss these effects further in Section 5.6

###  2.3 Signal-to-Noise Calculation

A full injection and recovery test (e.g; Dressing & Charbonneau, 2015; Hippke
& Heller, 2019; Bryant et al., 2023) would require additional computation time
to run a detection algorithm (e.g. BLS; Kovács et al., 2002; Collier Cameron
et al., 2006) on the data and to initialise a full transit model such as those
produced by the batman package (Kreidberg, 2015). We instead calculate
detection based purely on the SNR for each signal based on the generated
planet properties and lightcurve properties.

The width of the transit is equal to the effective number of in-transit points
from Equation 4 multiplied by the time cadence of the lightcurve $(\Delta T)$
For non-grazing transits where $0\leq
b<1-\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}})$ the depth of
transit is simply:

$\delta=\left(\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}}\right)^{2}.$
(5)

For grazing transits
$(1-\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}}\leq b\leq
1+\frac{\mbox{R\textsubscript{p}}}{\mbox{R${}_{\star}$}})$ we calculate the
true anomaly at the time of mid transit in accordance with Maxted (2016). We
then use equation 5.63 from Hilditch (2001) to calculate the corresponding
projected orbital separation. We then calculate the depth of transit as the
overlap in area between the planetary and stellar disks at this projected
separation.

In either case, we then calculate SNR using the equation below:

$\text{SNR}=\frac{\delta}{1+C}\frac{\sqrt{N_{\text{trans, eff}}\Delta
T}}{\sigma},$ (6)

where the contamination ratio $(C)$ is the proportion of flux from background
objects rather than the target itself, $\Delta T$ is the time cadence of the
lightcurve and $\sigma$ is the lightcurve noise on the timescale of $\Delta
T$. This allows us to account for changes in cadence of observations such as
the change from 30 to 10 minute cadence between Year 1 and Year 3 of TESS full
frame images.

###  2.4 Detection Probability

To convert from SNR to a recovery rate, we need a function that encapsulates
the likelihood that a given signal with a certain SNR will be detected as a
transit candidate. Previous studies (Sullivan et al., 2015; Bouma et al.,
2017; Huang et al., 2018; Barclay et al., 2018; Cooke et al., 2018; Villanueva
et al., 2019) have used an SNR threshold of $\text{SNR}\geq 7.3$ to determine
whether a planet is detectable. This approach is essentially a step function,
where the probability of detecting a transit at $\text{SNR}\geq 7.3$ is unity
and the probability of detecting a transit at $\text{SNR}<7.3$ is zero.
Kunimoto et al. (2022) used an incomplete gamma function ($\gamma$) initially
fitted to the Kepler pipeline by Christiansen (2017); Hsu et al. (2019), to
characterise the probability of a planet being both detected and passing
vetting. The form of this function is:

$p_{\gamma}\left(\text{det}\right)=c_{N_{\text{tr}}}\times\gamma\left(\alpha_{N_{\text{tr}}},\frac{\text{S/N}}{\beta_{N_{\text{tr}}}}\right)$
(7)

where $c_{N_{\text{tr}}}$ is the maximum probability of detection a transit
signal with $N_{\text{tr}}$ observed transits can reach and
$\alpha_{N_{\text{tr}}},~{}\beta_{N_{\text{tr}}}$ are coefficients determined
from the value of $N_{\text{tr}}$ using Table 1.

For determining a detection probability, TIaRA uses the Kepler gamma functions
set out in Christiansen (2017) and as adopted by Hsu et al. (2019). For
testing and comparison purposes, TIaRA can also used a fixed detection
threshold such as $\text{SNR}\geq 7.3$.

In Hsu et al. (2019)there is no detection probability function defined for the
case of only one or two transit events in the lightcurve. However calculating
the yields for monotransits and duotransits is a key aspect of the TIaRA
pipeline. Therefore we perform a linear extrapolation of the coefficients
fitted by Hsu et al. (2019) to obtain values for $N_{\text{tr}}=1$ and
$N_{\text{tr}}=2$ resulting in Table 1. The incomplete gamma function is
represented graphically for each set of coefficients in Figure 2.

Applying a gamma function based on the detection efficiency of Kepler to other
surveys such as TESS comes with caveats as different detection pipelines will
have different performance for the same signal to noise value. The effects of
this on our yield predictions for TESS is discussed further in Section 5.7

Table 1: Gamma function coefficients. $N_{\text{tr}}$ | $\alpha_{N_{\text{tr}}}$ | $\beta_{N_{\text{tr}}}$ | $c_{N_{\text{tr}}}$ | Source
---|---|---|---|---
1 | 34.3932 | 0.254262 | 0.560547 | This work
2 | 33.8908 | 0.259367 | 0.629820 | This work
3 | 33.3884 | 0.264472 | 0.699093 | Hsu et al. (2019)
4 | 32.8860 | 0.269577 | 0.768366 | Hsu et al. (2019)
5 | 31.5196 | 0.282741 | 0.833673 | Hsu et al. (2019)
6 | 30.9919 | 0.286979 | 0.859865 | Hsu et al. (2019)
7-9 | 30.1906 | 0.294688 | 0.875042 | Hsu et al. (2019)
10-18 | 31.6342 | 0.279425 | 0.886144 | Hsu et al. (2019)
19-36 | 32.6448 | 0.268898 | 0.889724 | Hsu et al. (2019)
$\geq 37$ | 27.8185 | 0.32432 | 0.945075 | Hsu et al. (2019)
Figure 2: The set of incomplete gamma functions used as detection probability
functions in the TIaRA pipeline. Each function shows the probability a planet
will be detected given the signal-to-noise (SNR) of its transit signal in the
lightcurve. Each function is for a different number of individual transit
events in the lightcurve. The monotransits and duotransits are represented by
the blue N=1 and orange N=2 functions respectively.

###  2.5 Minimum detectable radius cutoff

Since we inject planets proportionally to their real occurrence rates, we will
simulate many more smaller radius planets than larger planets due to their
relative occurrence rates. However many of the smaller planets would produce
transits with extremely low SNR, which may be far below the level of
detectability in a given survey. Therefore to conserve computational resources
and increase the efficiency of the simulation we implement a radius cutoff to
the occurrence rates based on the precision of the given survey. To do this we
estimate the minimum detectable radius from a single transit.

To calculate the minimum detectable radius we assume a favourable set of
transit parameters which consist of $e=0$, $b=0$, and $T_{\text{dur}}$ set to
a "long" transit duration - defined to be the duration of a P=1000 days
transiting exoplanet of negligible radius around the given host star. Since
SNR increases with transit duration, this long duration - beyond what we
expect to realistically transit, was chosen so that small-radius planets which
could be detected from a long enough transit were not removed from the
simulation.

We choose a SNR threshold of 4 to determine the minimum detectable radius as
this correlates with a very low probability of detection: $<10^{-5}$ (see
Section 2.4).For planets with radii below the cutoff, we simply assign a
probability of detection equal to 0.

##  3 TESS simulations

###  3.1 TESS SPOC FFI lightcurves

In this study we apply TIaRA to the Year 1 and Year 3 Full Frame Image (FFI;
Ricker et al., 2014) lightcurves from the TESS Science Processing Operations
Center (SPOC; Jenkins et al., 2016). The SPOC FFI lightcurve sample is a high
quality, homogeneous and readily availiable dataset representing TESS targets
which are most amenable to exoplanet detection and thus is appropriate for our
purposes (Caldwell et al., 2020). Additionally the NGTS monotransit working
group uses the SPOC FFI lightcurves in their search for TESS monotransits
(e.g; Gill et al., 2020) which allows for easy comparison of our results. We
compiled the SPOC FFI target lists per TESS sector from MAST availiable at
https://archive.stsci.edu/hlsp/tess-spoc into a single list containing the
TESS Input Catalogue (TICv8; Stassun et al., 2019) numbers of all the SPOC FFI
target stars and the TESS sectors they were observed in. We also removed all
stars with radii $<0.1$R⊙ and $>4$R⊙ from the sample. This results in a sample
of 1323228 stars across Year 1 and Year 3. For every star in this list, we
obtained the relevant lightcurve FITS files via the MAST portal at
https://archive.stsci.edu/.

###  3.2 Stellar parameters

We extracted the stellar radius (R⋆), TESS magnitude $(T_{\text{mag}})$, and
effective temperature (Teff) directly from the FITS file headers: These
stellar parameters are themselves sourced from the TICv8 (see Stassun et al.,
2019).

To calculate a semi-major axis for our injected planets, we require a stellar
mass (M⋆). Due to a large number of targets lacking a value of the stellar
mass in the lightcurve FITS file headers, we estimate the stellar mass using
the following power law, which is approximated from the mass and radius of
binary stars (Harmanec, 1988):

$\mbox{M${}_{\star}$}=\mbox{R${}_{\star}$}^{1.25},$ (8)

where M⋆ and R⋆ are both in solar units. The dependence of the semi-major axis
on stellar mass is relatively weak and thus has a minor effect on planet
detectability, therefore this power law is adequate for the main-sequence
dwarf stars in the SPOC FFI sample. We assign a spectral type to each star
based on Teff using definitions from Pecaut & Mamajek (2013). Stars with Teff
hotter than the maximum cutoff for A type stars (10050 K) were marked as
spectral type OB.

###  3.3 Window functions

In order to use the real window function for each star in the SPOC FFI list,
we determine blocks of continuous TESS observations for each target using the
timestamps from the FITS file TIME header for each TESS sector. We only
consider good quality data where the data quality flag (QUALITY) = 0. We use
the first and last timestamps of each Sector to define the block of continuous
data for each Sector. To identify gaps within Sectors, we search for instances
where the difference between two consecutive timestamps is greater than 0.5
days. Most of these gaps are the mid-Sector gaps, which occur during the
perigee of the 13.7 day TESS orbit when data is downloaded to ground-stations
(Ricker et al., 2014). Other data gaps are caused by technical issues with the
specific TESS camera, the entire TESS spacecraft, or the variety of reasons
that give rise to non-zero data quality flags (e.g. stray light,cosmic rays,
spacecraft momentum dumps and pointing issues).

To illustrate these data gaps, we plot the lightcurve of a typical SPOC FFI
target (TIC-261236954) in Figure 3. TIC-261236954 is in the TESS CVZ, so can
be used to illustrate gaps over the duration of Year 1 of the TESS mission.
For this star the blocks of continuous photometry account for 79.3% of the
total Year 1 duration, while the gaps make up 20.78% of that year.

In order to demonstrate the effect of these gaps in the TESS window functions,
we calculate the fraction of orbital phase covered by TESS observations as a
function of the orbital period of a simulated planet for two scenarios: (1) an
idealised 27 day window function, and (2) the window function of TIC-261236954
(the lightcurve for which is set out in Fig 3. This allows us to compare the
effect of gaps for up to 13 consecutive Sectors. The results are set out in
Figure 4, and show the significant difference between the idealised 27 day
Sectors and the real TIC-261236954 lightcurve window function. This highlights
the need to account for the gaps in the TESS data for each star as we do in
the TIaRA pipeline. We note that different pipelines may use different data
quality flags, which may slightly alter the position and/or duration of some
of the data gaps.

Figure 3: An example SPOC FFI lightcurve from a star (TIC-261236954) in the
CVZ of TESS in Year 1 of the mission, illustrating gaps in the data. The blue
shaded regions show data gaps longer than 0.5 days. In this example we find
that over the course of one year 79.3% of time is photometrically monitored,
while the gaps account for 20.7% of the year.

Figure 4: Completeness of phase coverage as a function of planetary orbital
period. Left: Idealised 27 day coverage (no gaps) for durations ranging from 1
to 13 consecutive sectors. Right: The real phase coverage function for an
example SPOC FFI target (TIC-261236954) in the TESS CVZ.

###  3.4 Signal to Noise

To calculate the SNR (Equation 6) for each simulated transiting planet, we
take the noise of the lightcurve ($\sigma$) to be the Combined Differential
Photometric Precision (CDPP; Christiansen et al., 2012) 2hr noise as produced
by the SPOC FFI pipelilne and recorded in the FITS headers as CDPP2_0. This
provides us with a readily available pre-calculated noise metric, which saves
computation time. Similarly we take the source-to-background brightness ratio
$(C+1)$ from the SPOC FFI pipeline recorded in the FITS headers as CROWDSAP.
The time cadence $(\Delta T)$ has changed over the course of the TESS mission.
For Year 1 the time cadence was 30 mins, while for Year 3 the time cadence was
10 mins.

###  3.5 Probability of detection

We assume that the detection and vetting of transiting exoplanet signals in
TESS is similar to that of the Kepler mission, and we therefore use the the
modified version of the Kepler gamma function (Section 2.4) to define our
probability of detection for each simulated transiting planet. For comparative
purposes, we also record detections via two threshold SNR criteria: at
SNR$\geq$7.3 (to match some previous yield studies) and SNR$\geq$20 (to match
monotransit searches that typically require such a high SNR for robust
detections).

###  3.6 Creating sensitivity maps

For each transit signal that we generate, we convert the SNR into detection
probability as described in Section 3.5. Each of these probabilities is then
multiplied by the probability of observation for each signal calculated from
its window function (see Section 3.3) to give a probability of observation and
detection. These values are again multiplied by the geometric probability of
transit for each signal to give a probability of transit, observation and
detection by TESS for each signal. We take the binned averages for all signals
using the same period and radius bins as Kunimoto & Matthews (2020).

We treat each of our detection probabilities as a Poisson statistic such that
the error in each is equal to its value divided by the square root of the
total count of signals in that bin.

###  3.7 Occurrence Rates

In order to convert our sensitivity maps to yield estimates for the Year 1 and
Year 3 TESS SPOC FFI sample, we need to know the occurrence rates for the
underlying population of planets orbiting each type of star in the sample. For
F,G and K dwarf stars we use the occurrence rates calculated in Kunimoto &
Matthews (2020). For A type stars we do not have robust occurrence rates, so
we use the same grid as for F type stars.

Kunimoto & Matthews (2020) did not calculate occurrence rates for M dwarfs, so
for consistency and ease of comparison between our results we rebin the M
dwarf occurrence rate grid from Dressing & Charbonneau (2015) to use the same
bins as the grids used in Kunimoto & Matthews (2020). To do this we use a
bivariate spline approximation over a rectangular mesh using the
RectBivariateSpline interpolator from the python module; scipy.interpolate
(Virtanen et al., 2020). We take the midpoints of each period and radius bin
from the Dressing & Charbonneau (2015) grid and use these for the coordinates
of each grid value. We then used the median value, lower bound, and upper
bound of each grid cell with these coordinates to create three separate
interpolators for each value. We then feed the midpoints of all the Kunimoto &
Matthews (2020) bins with Rp$\leq 4$R⊕ and an orbital period $\leq$200 days
into these to obtain the values for the new grid. For the 200-400 day period
bin we assume the values were identical to the 100-200 day bin. For all bins
above 4R⊕ we use the values for K dwarf stars but reduced by a factor of 0.5
to account for the generally lower occurrence rates of giant planets around M
dwarfs (Sabotta et al., 2021; Bryant et al., 2023; Gan et al., 2023).

To account for the fact that some bins in the grid from Dressing & Charbonneau
(2015) are unconstrained and contain only an upper bound on the occurrence
rate we manually set the equivalent grid cells in our new grid to be
unconstrained as well. For all such grid cells, we set the median and lower
bound values to zero and the upper bound value to the interpolated median
value added to the interpolated upper bound value. We consider the new grid
cells between 0.78-1.56 days and 2-4R⊕, 6.25-25 days and 1-1.41 R⊕, 50-200
days and 0.5-1R⊕ and 2.83-4R⊕, to be equivalent to the old grid cells between
0.5-1.7 days and 2-4R⊕, 5.5-18.2 days and 1.0-1.5 R⊕, 60.3-200 days and
0.5-1R⊕and 3.5-4R⊕respectively for the purposes of considering them
unconstrained. Furthermore we also consider all the grid cells between 200-400
days and <4R⊕ to be unconstrained as these are extrapolated and not
interpolated from the measured Kepler rates by Dressing & Charbonneau (2015)
and such planets are shown to be very rarely detected by TESS if at all (NASA
Exoplanet Archive; Akeson et al., 2013).

Our resulting M-dwarf occurrence rate grid is set out in Figure 5.

The Dressing & Charbonneau (2015) occurrence rates were calculated prior to
Gaia, and therefore the stellar parameters of candidates are not as robust as
post-Gaia studies such as Hsu et al. (2019); Kunimoto & Matthews (2020).
Recently; Bergsten et al. (2023) have updated the Dressing & Charbonneau
(2015) occurrence rates for small radius planets (Rp$<$2 R⊕). These small
radii planets are unlikely to be detectable in the TESS data, so this update
will not significantly impact the results we present in this paper. However,
we plan to include these updated occurrence rates in future TIaRA simulations.

Figure 5: Percentage occurrence rates of planets around M dwarfs. Values for
R$<$4R⊕ taken from Dressing & Charbonneau (2015), rebinned to our standardised
grid. Values for R$>$4R⊕ from the a scaled K dwarf occurrence rate from
Kunimoto & Matthews (2020).

####  3.7.1 Use as input priority metrics

As set out in Section 2.2.1 we also use occurrence rate grids as a weighted
distribution to select periods and radii for planets we inject into the
simulation. This requires us to set a median value for unconstrained grid
cells that possess only an upper bound. For the A, F, G and K dwarf grids we
perform a linear extrapolation of each grid row and column and then take the
mean of these as the input grid. For the M dwarf grid we simply take the upper
bound of unconstrained grid cells as the median value.

###  3.8 Yield Estimates

To calculate our final yield estimates we multiply the occurrence rates by the
detection efficiencies for each spectral type, and then multiply those rates
and their uncertainties by the number of each type of star in the Year 1 and
Year 3 SPOC FFI sample. This gives us a grid of expected yields for each
period-radius bin with uncertainties combined from those of the detection
sensitivities and occurrence rates. The error in occurrence rate is
considerably larger and so dominates the overall uncertainty in predicted
yield.

##  4 Results

###  4.1 TESS Sensitivity

The TIaRA pipeline creates binned sensitivity maps on the same grid as the
occurrence rates from Kunimoto & Matthews (2020). These show the probability
that a transiting exoplanet of a given radius and orbital period is observed
and detected by TESS in the Year 1 and Year 3 SPOC FFI lightcurves. We compute
separate sensitivity maps for each of A,F,G,K, and M dwarfs, and these are set
out in Appendix A Figure 15. We also plot the average sensitivity for AFGKM
dwarfs in the SPOC FFI sample in Figure 6. The uncertainties in sensitivity
are very small, this is largely on account of the large numbers of signals we
inject in the simulation, which allows for such precise calculation.

Figure 6: Sensitivity map showing the probability of a transiting planet in
each period/radius bin being observed and detected around an AFGKM dwarf star
in the TESS Year 1 and Year 3 SPOC FFI lightcurves. The percentage sensitivity
values are shown in each grid cell. Where no uncertainties are given the
uncertainty is 0% to two decimal places.

Naturally, there is a trend of higher sensitivity towards larger radii planets
as these planets produce deeper transits and thus a higher SNR signal. We find
that TESS should detect over 80% of very short period (P$<$6.25 day),
transiting giant planets (R$>$8R⊕). This probability drops below 50% for small
transiting planets (R$<$4R⊕), and is below 1% for Earth-radii transiting
planets.

Additionally, we are more sensitive to shorter period planets due to the
greater number of observed transits in any given monitoring duration. This
both produces a higher SNR value when these transits are summed and allows for
greater confidence in the detection as accounted for by the gamma function we
use to calculate probability of detection (see Section 3.5). For giant planets
(R$>$8R⊕), we move from approximately 80% completeness for short periods
(P$<$6.25 day) down to less than 10% completeness for periods between 200 and
400 days.

As the TESS mission continues, these sensitivity maps will evolve as a
function of the window function for each star, and the sensitivity will
improve over the entire grid. TIaRA will be able to re-calculate these
sensitivity maps using the timestamps from TESS lightcurves as the extended
TESS mission continues to gather data. For the southern ecliptic hemisphere,
this will be the update provided by the Year 5 lightcurves (Sectors 61-69). It
is also possible to simulate timestamps for future sectors based on our
knowledge of previous sectors with some assumptions needed for distribution of
the future data gaps and FFI cadence.

###  4.2 Predicted Yield

Figure 7: Predicted exoplanet yield from Year 1 and 3 SPOC FFI lightcurves of
AFGKM type stars. Note that cells where the median value is 1 have been
displayed as $\leq 1$ to account for the inherent uncertainty in a predicted
yield of 1.

(a) Distribution of orbital periods of predicted yields

(b) Distribution of planetary radius of predicted yields

(c) Expected yield by spectral type

(d) Expected yield per star by spectral type

Figure 8: Predicted TIaRA transiting exoplanet yields from the TESS Year 1 and
Year 3 SPOC FFI lightcurves (solid blue bars). Also displayed are actual TESS
discoveries (transparent black outlined bars) calculated using TOI catalogue
downloaded from the NASA exoplanet archive (Akeson et al., 2013) on 2023-06-15
(excluding flagged false positives) and matched to TIC IDs of southern
ecliptic SPOC FFI sample.

We present the TIaRA predicted yield of transiting exoplanet discoveries from
TESS Year 1 and Year 3 SPOC FFI lightcurves in Figure 7. This grid is the
summation of all spectral types, however we also present the yields broken
down by spectral type in Appendix B Figure 16.

Overall we predict a yield of $2271^{+241}_{-138}$ exoplanets detected around
AFGKM dwarf host stars. We set out the yield distributions of orbital period,
planet radius, and host spectral type in Figure 8. We find the discoveries
should peak at orbital periods between 3.13 and 6.25 days. However there are a
significant number of longer period planets in our predicted yield, with
$403^{+64}_{-38}$ planets with orbital periods greater than 25 days and
$113^{+23}_{-17}$ will have orbital periods greater than 100 days.
Interestingly the distribution of planet radii is quite flat, with the
exception of giant planets (R$>$11.31R⊕), which are twice as numerous as other
radius bins. Most discoveries are predicted to be around G dwarf stars,
although K dwarfs provide the most number of discoveries per star monitored.

To compare our predicted yield to the actual discoveries, we crossmatch the
TOI catalogue as of 2023-06-15 with the SPOC FFI target lists for Year 1 and 3
to find transiting exoplanet discoveries from our sample. The distributions of
these TOIs and our predicted yields are shown in Figure 8.

###  4.3 Monotransits

(a) Period distribution of predicted monotransit detections

(b) Radius distribution of predicted monotransit detections

Figure 9: Predicted yields of monotransits, shown as binned distributions in
orbital period (a) and radius (b).

We predict that $215^{+37}_{-23}$ of the transiting planet discoveries will
only have one transit event in the TESS Year 1 or Year 3 data - i.e. are
"monotransits". This makes up a relatively small fraction (9%) of the total
predicted yield, but is a significiant fraction of the long-period detections.
$202^{+36}_{-22}$ (50%) of the planets with $P>25$ days are monotransits,
while $86^{+20}_{-14}$ (76%) of the planets with $P>100$ days are
monotransits. Figure 9 shows the the distributions of orbital period and
planetary radius for the predicted monotransit yield. As expected there are
very few monotransits with orbital periods less than the sector length of TESS
(27 days). However beyond that the distribution of orbital period is
remarkably flat out to our final bin of 400 days. The distribution of
planetary radii for the monotransits are much more heavily skewed towards
larger radii planets than typical multi-transit detections. 73% of all
detections are for planets with $Rp>4$R⊕, while this fraction is 94% of just
the monotransit detections.

Figure 10: Comparison of the distribution in period of the yields of
monotransits from Year 1 alone (green) compared to Year 1 and Year 3 combined
(orange).

In addition to the monotransit yield for Years 1 and 3 together, we also
estimated the yield for Year 1 alone in order to investigate the effect of
reobserving the field on the monotransit yield. The results are shown in
Figure 10. We find that the number of monotransits in Year 1 alone is
$227^{+36}_{-20}$, compared to the Year 1 and 3 combined yield of to
$215^{+37}_{-23}$. Assuming that all of the Year 1 monotransits that have an
additional Year 3 transit observed become duotransits (see Section 4.4), then
this leaves $114^{+42}_{-24}$ detected monotransits observed in Year 1 and not
Year 3 and $101^{+55}_{-33}$ observed in Year 3 and not Year 1. The
distribution of orbital periods changes to favour longer periods with 40% of
Year 1 and Year 3 monotransits having P$>$100 days compared to 26% of Year 1
only monotransits. The distribution of monotransit detections in period also
becomes significantly flatter at longer-periods when Year 3 is added to Year
1.

###  4.4 Duotransits

(a) Period distribution of predicted biennial duotransit yield

(b) Radius distribution of predicted biennial duotransit yield

Figure 11: Predicted yields of biennial duotransit planets detected from one
transit in Year 1 and another in Year 3. Shown in distributions of period (a)
and radius (b). Plot (b) also shows a comparison with the radii of biennial
duotransits from Hawthorn et al. (2023) excluding candidates over 16R⊕ which
we do not simulate.

As TESS reobserved the southern ecliptic hemisphere in Year 3 of the mission,
many Year 1 monotransit signals will have an additional transit observed in
Year 3, and become "biennial duotransits". We define these duotransits as
"biennial" to distinguish them from other duotransits in which the two transit
events happen in the same year of TESS data. . We predict a total of
$170^{+29}_{-18}$ duotransits in total with $113^{+22}_{-13}$ of these being
biennial duotransits.

The distribution of biennial duotransits is shown in Figure 11. Compared to
monotransit detections (see Section 4.3) we find that the sample of planets
detected from biennial duotransits show a flatter distribution in radius, with
21% of biennial duotransit detections coming from planets below 4R⊕ compared
to 6% of monotransits, although radii $>$11.31R⊕ are still the largest bin of
detections. We find the period of expected biennial duotransits peaks between
25 and 50 days and falls off more rapidly than monotransits for longer
periods, with 6% of biennial duotransits coming from periods $>$100 days
compared to 40% of monotransits.

###  4.5 Breakdown by Spectral Type

As shown in Figure 8(c), we find G dwarfs should be host to the most
detections $(1005^{+143}_{-103})$, followed by K $(551^{+123}_{-54})$ and F
$(500^{+130}_{-67})$ dwarfs in quick succession with a reasonable number from
M dwarfs $(151^{+66}_{-20})$ and a very small number from A dwarfs
$(64^{+21}_{-11})$. This distribution is largely due to the numbers of each
spectral type represented in the SPOC FFI sample with G types being the most
numerous and thus host to the most detected planets.

Figure 8(d) shows the yield from each spectral type divided by the number of
stars of that spectral type in the SPOC sample. From this it becomes apparent
that K dwarfs are the most numerous hosts of detections per star with a rate
of $(3.4^{+0.8}_{-0.3})\times 10^{-3}$ transiting planets per star observed. G
dwarfs are the next most efficient sources of detections with
$(2.4^{+0.3}_{-0.2})\times 10^{-3}$ predicted per star, followed by M with
$(1.2^{+0.5}_{-0.2})\times 10^{-3}$, F with $({0.99}^{+0.26}_{-0.13})\times
10^{-3}$ and A with $(0.72^{+0.24}_{0.12})\times 10^{-3}$.

K dwarfs sit at the best crossover between occurrence rates and TESS’
detection efficiency, and thus have the highest rate of detections per star.
Despite M dwarfs having higher planetary occurrence rates than AFGK stars
(Dressing & Charbonneau, 2015) and their small size causing deeper transits
due to a higher planet-star radius-ratio, they display a relatively low rate
of detections per star. M dwarfs host many small radius planets compared with
AFGK dwarfs, but fewer giant planets. These small planets produce low SNR
signals that result in low detection probabilities for M dwarf host stars.
This results in fewer detections per star for M dwarfs.

Figure 12: Comparison of the proportions of predicted long-period planet yield
by spectral type. Planets with periods less than 25 days are shown in blue,
periods between 25 and 100 days are shown in orange and periods longer than
100 days are shown in green. The predicted yield numbers for each spectral
type and period range are printed on the plot, along with the associated
uncertainties.

Figure 12 shows the proportion of different period ranges in the predicted
yield by spectral type. We find a reasonably flat trend in the distribution of
long-period discoveries for each spectral type with a possible minor trend
towards shorter-period discoveries being favoured at later spectral types,
although increasingly small numbers create uncertainty in such a trend.

###  4.6 Comparison on Detection Probability Functions

(a) Period distribution of predicted yields for three detection criteria

(b) Radius distribution of predicted yields for three detection criteria

Figure 13: Comparison of the yields predicted using three different detection
criteria, a gamma cumulative distribution (blue), and signal to noise
thresholds of 7.3 (green) and 20 (red). Distributions are shown in period (a)
and radius (b).

As well as the gamma function described in Section 2.4 we also calculate
yields using simple signal to noise threshholds of 7.3 and 20. Figure 13 shows
a comparison of the results from these three approaches.

The predicted total yield using a $\text{SNR}\geq 20$ threshold of
$1555^{+216}_{-126}$ shows a strong agreement with the total number of TOIs
from the Year 1 and 3 SPOC FFI discoveries (1666). However Figure 13
demonstrates that the predicted distribution in periods and radii from this
criterion does not match the proportions of the TOI sample as well as other
methods. We see a significant under-prediction of planet yields for radii
below 5.66R⊕ which increases in disagreement for decreasing radii from this
method as shown in Figure 13(b). We also find that the yields for short period
planets ($P<25$ days) are under-predicted by this method, with the largest
deficit being between 3.13 and 6.25 days

Using a smaller SNR threshold of 7.3 as many yield studies have done
previously (e.g; Sullivan et al., 2015; Bouma et al., 2017; Barclay et al.,
2018; Cooke et al., 2018, 2019; Villanueva et al., 2019), we find a
significantly greater yield prediction than either the threshold of SNR$\geq
20$ or the gamma function. The predicted yield of planets with SNR$\geq 7.3$
($3334^{+309}_{-177}$) is more than double the size of the TOI discoveries and
the yield predicted by SNR$\geq 20$ and around 1.5 times greater than that
predicted by the gamma function. A large fraction of this additional predicted
yield comes from small planets and longer period planets as shown in Figure
13.

##  5 Discussion

###  5.1 Comparison to actual TESS detections

Comparison to the population of TESS transiting planet discoveries via the TOI
catalogue is a relatively straightforward and robust check for the TIaRA yield
results. We cross-match the TIC IDs of the TOI catalogue with stars we
simulated from the SPOC FFI sample to compare against our predicted yields as
shown in Figure 8.

While we use the SPOC FFI lightcurves as a homogeneous sample it must be noted
that the TOI sample is not generated homogeneously. While we limit the TOI
catalogue to just those which have SPOC FFI lightcurves which we include in
our simulation, many of these will still have been discovered in lightcurves
produced by other pipelines such as the SPOC 2 minute postage stamp or Quick
look lightcurves (QLP; Huang et al., 2020). These lightcurves may have
different duty cycles, noise levels and systematics to the SPOC FFI sample we
use. Different quality flags in these pipelines may also affect the location
and duration of data gaps between data products. Additionally the automated
TOI search carried out is different between pipelines, which means TOIs from
different pipelines are affected by different biases. It is for this reason
that we stress that our comparison is not specifically an attempt to assess
the completeness of the SPOC FFI pipeline TOI search.

Our total predicted yield of $2271^{+241}_{-138}$ is $\sim 4.4\sigma$ greater
than the total number of TOIs (1666) as of 2023-06-15. However, for orbital
periods less than 6.25 days the TIaRA predictions agree with he discoveries to
within uncertainties. For periods between 6.25 and 25 days the TIaRA
predictions are not more than $2\sigma$ above the discoveries. However for
planets with periods greater than 25 days the deficit between TIaRA
predictions and TOI discoveries becomes increasingly significant as shown in
Figure 14.

Figure 14: Percentage deficit between the discovered TOIs and our predicted
yield for the TESS SPOC FFI lightcurves from Year 1 and Year 3.

The SPOC Transiting Planet Search (TPS; Jenkins et al., 2016; Guerrero et al.,
2021) relies on phase folding data to search for periodic signals (Guerrero et
al., 2021). This becomes increasingly difficult for longer-periods as such
planets may have transits in different TESS sectors or only one transit which
is impossible to phase fold. This suggests that the difference between our
predicted yield and the TOI sample may be due to TESS discoveries being
currently incomplete for longer-period planets. Since the majority of
candidates are given TOI status by the TPS (Guerrero et al., 2021), many
monotransit and biennial duotransit events in the SPOC FFI lightcurves may
have yet to be detected.

It is difficult to uniformly simulate the planet detection process that makes
up the TOI catalogue, as the TOI catalogue is not built up from a single
uniform search of the SPOC FFI lightcurves. Instead the SPOC Transiting Planet
Search only runs on selected postage stamp lightcurves (around 20,000 per
sector) while the rest of the SPOC FFI targets are searched by Quick Look
Pipeline (QLP; Huang et al., 2020). However, the QLP operates at a hard
magnitude cutoff of T=13.5, and some of the SPOC FFI targets are fainter than
this cut-off. Therefore some of the lightcurves in the SPOC FFI sample have
not been searched in a systematic manner. Additionally, third party groups
will have their own methods of searching lightcurves for candidates which can
lead to additional TOIs being created via the Community TOI process (CTOI;
Guerrero et al., 2021). These subtleties could contribute to a small
discrepancy between our predictions and the actual TOI catalogue, although
significant differences are likely still due to incompleteness as described
previously.

We can gain an indication of this by comparing our predicted yield with and
without monotransits to the TOI sample. When we remove monotransits from our
predicted yield we find $2056^{+245}_{-140}$ detected exoplanets and excluding
biennial duotransits we further reduce this to $1943^{+246}_{-141}$. This
brings our yield into closer alignment with the TOI detections (Taken from
NASA exoplanet archive; Akeson et al., 2013) from $4.4\sigma$ to $2.8\sigma$
and $2.0\sigma$ respectively.

This could also explain the seeming overabundance of giant planets as seen in
Figure 8(b) as long-period monotransit and duotransit detections are
particularly biased towards larger radii (see Figures 9(b) and 11(b)), and
hence the undiscovered long-period planets make up the majority of the
undiscovered giant planets as well. Additionally it could explain the
overestimation of planets detected around G and K dwarfs shown in Figures 8(c)
and 8(d). As these stars make up the largest numbers of stars in the SPOC FFI
sample and have the highest detection rate per star, it it likely that most of
the long-period and other planets not yet found would be around these.

###  5.2 Comparison to Biennial Duotransit search

We are able to compare our predicted yield of biennial duotransit detections
with a real search of the Year 1 and Year 3 SPOC FFI lightcurves for biennial
duotransits set out in Hawthorn et al. (2023). Figure 11(b) shows the
distribution of the TIaRA predicted yields and the sample from Hawthorn et al.
(2023) in radius. The periods of the majority of candidates in Hawthorn et al.
(2023) are currently unknown and so cannot be compared directly to our
prediction. Our predicted yield of biennial duotransits is $113^{+22}_{-13}$,
which is $3.5\sigma$ greater than the 68 candidates in the real sample. The
numbers are in close agreement for radii between 8.0 and 11.31R⊕ and near
agreement for 11.31 to 16R⊕. However for smaller radii planets (R$<4$R⊕),
TIaRA predicts many more biennial duotransits than than are detected in
Hawthorn et al. (2023), especially between 2.83 and 4 R⊕. This is to be
expected as Hawthorn et al. (2023) only consider high SNR biennial
duotransits, which will be produced by larger planets. We note that Hawthorn
et al. (2023) also have a significant number of candidates with radii larger
than 16R⊕, however we do not consider such large radius objects in our TIaRA
simulation.

We also note the strong agreement between the predicted number of biennial
duotransits with SNR$\geq 20$ of $86^{+25}_{-13}$ and the discovery of 68
biennial duotransits from Hawthorn et al. (2023). The effect of using
different probability of detection functions is discussed further in Section
5.7.

###  5.3 Comparison to previously predicted yield

In addition to comparing our results to the TOI catalogue (see Section 5.1) we
can also check the validity of our results by comparing them to previous
attempts at predicting yield. This is especially useful as our methods include
monotransits, which are not included in studies such as Kunimoto et al.
(2022).

Kunimoto et al. (2022) use a considerably larger population of stars. We have
simulated the yield from the $\sim 1.3$ million stars in the Year 1 and Year 3
SPOC FFI sample. Kunimoto et al. (2022) use 4 million stars from the TESS
Candidate Target List (CTL; Stassun et al., 2019). Therefore we normalise each
set of results by the number of stars in each sample and compare in terms of
number of planets predicted per star (similarly to in Figure 8(d)) for ease of
comparison. However, it should be noted that the selection criteria for each
stellar sample, especially for SPOC stars (Caldwell et al., 2020), introduces
biases in the average detectability for each stars, so this is somewhat of a
crude comparison.

Kunimoto et al. (2022) predict a total of $2532\pm 189$ exoplanets found
around 4066063 stars in Year 1 and $1748\pm 103$ around 4021948 stars in Year
3. This gives a total of $(1.06\pm 0.05)\times 10^{-3}$ detections per star.
This is 6.5 $\sigma$ below our estimate of $(1.55^{+0.18}_{-0.11})\times
10^{-3}$ per star excluding monotransits. It is to be expected that our yield
prediction will be greater than Kunimoto et al. (2022) since we use the SPOC
FFI sample, which has been explicitly created to target stars that are most
likely to have detectable transits with TESS Caldwell et al. (2020).

###  5.4 Eccentricity distribution

As in Cooke et al. (2018, 2019) we use a beta distribution to assign
eccentricity to planets we inject into our simulation and use values for
$\alpha$ and $\beta$ taken from Van Eylen & Albrecht (2015). However, this
study was focused on super-Earths and Mini-Neptunes from Keplerand not longer-
period giant planets which are the focus of this paper. Warm-Jupiters have
been found to often show much higher orbital eccentricities than smaller
and/or closer-in planets (Dong et al., 2021) and therefore we are likely
underestimating the eccentricity of the Warm-Jupiter population in our
simulation. However eccentric long-period planets are more likely to transit
at periastron when closer to their host stars and hence this means we are
likely underestimating the detection sensitivity of TESS towards such planets.

###  5.5 Occurrence Rates

The TIaRA yield predictions rely on the occurrence rates from Kunimoto &
Matthews (2020) for AFGK stars and a modified occurrence rate from Dressing &
Charbonneau (2015) for M dwarfs.

The occurrence rates we use from Kunimoto & Matthews (2020) do not include any
planets larger than 16R⊕ although larger planets are known to exist and are
likely to be detectable by TESS even if the occurrence rates from Kepler are
known to be low (Fressin et al., 2013; Hsu et al., 2019; Kunimoto & Matthews,
2020). Future yield estimate studies could benefit from studies into the
occurrence rate of planets larger than 16R⊕which could be carried out on
samples of planets from TESS.

Due to the lack of occurrence rate studies on A and earlier spectral-type
stars, we assume the occurrence rates of planets around A and earlier type
stars are equal to that of F type stars (see Section 3.7). In reality these
may be different as the higher mass and temperature of A type stars are
expected to have significant effects on planet formation and thus the
occurrence rate (Johnson et al., 2010). Future studies could benefit from TESS
derived occurrence rates of A type stars (e.g; Zhou et al., 2019; Johnson,
2023).

For M dwarfs we rebin the values found by Dressing & Charbonneau (2015) onto
the same grid used by Kunimoto & Matthews (2020). While this does, of course,
change the exact values for M dwarf occurrence rates, due to the large errors
on the values from Dressing & Charbonneau (2015) we believe our values for
occurrence rates within the original bounds of this grid
(0.5R⊕$\leq$Rp$\leq$4R⊕, 0.5 days$\leq P\leq$200 days) are robust. As
discussed in Section 3.7 we expect the yields of planets with periods between
200-400 days from TESS to be low and thus treat these bins as unconstrained
with only an upper bound value. Furthermore we estimate values for planets
between 4 and 16 R⊕ around M-dwarfs using the values found for K dwarf stars
by Kunimoto & Matthews (2020) but reduced by a factor of one half to account
for the expected lower occurrence rate of giant planets around M dwarfs
(Sabotta et al., 2021; Bryant et al., 2023; Gan et al., 2023). To improve the
accuracy of future yield estimations, it would be useful to constrain giant
planet occurrence rates around M dwarfs as discussed in Bryant et al. (2023);
Gan et al. (2023) and Sabotta et al. (2021). It is also worth noting that by
rebinning the rates for M dwarfs we increase the shortest possible period of
planet we simulate from 0.5 days to 0.78 days, this means we are likely to
underestimate yields in this region of period-space as TESS has already found
candidates (NASA Exoplanet Archive; Akeson et al., 2013) and 9 confirmed
planets (Vanderspek et al., 2019; Giacalone et al., 2022; Luque & Pallé, 2022;
Essack et al., 2023; Goffo et al., 2023) in this region.

###  5.6 Transit model

We use a Trapezoidal model to approximate the shape of transits in our
simulation as described in Section 2.2.2, this assumes a flat-bottomed shape
to the transit and ignores the effects of limb-darkening (Espinoza & Jordán,
2015, 2016; Agol et al., 2020) which may slightly increase the measured depth
of a transit and thus increase signal to noise. However we expect this effect
to be small in magnitude and therefore did not use further computational
resources to include it in our simulation. This may lead to our sensitivity
values for TESS towards all exoplanets to be slightly conservative.

Additionally, this model assumes the contributions to SNR from Ingress and
Egress are effectively half that of the full transit. This approximation works
well enough for our purposes but use of the more accurate trapezoid model put
forward in Kipping (2023) may improve our results. However we expect the
magnitude of this effect on our estimated SNR to be negligible.

While both of these effects are small, their inclusion in future iterations of
TIaRA could improve results, and may be worth including with more
computational resources.

###  5.7 Probability of detection

As set out in Section 2.4, we use an incomplete gamma function to estimate the
probability of detection in the TIaRA simulation following Christiansen
(2017); Hsu et al. (2019) and Kunimoto et al. (2022). As discussed in Kunimoto
et al. (2022), there are some caveats associated with the use of this
probability of detection function:

  1. 1.

Kepler monitored a different population of stars and had different light curve
properties to TESS. While our use of real TESS lightcurves allows us to
account for many of the unique properties of TESS data we still are applying a
probabilty of detection function developed for Kepler to a very different
mission. It would be useful for work similar to that of Christiansen (2017)
and Hsu et al. (2019) on Kepler DR25 to be performed on TESS to gain a more
accurate understanding of the false positive rate and detection efficiency of
TESS.

  2. 2.

We perform a linear extrapolation for the detection efficiency of duotransit
and monotransit events in TESS. While we believe the result to be reasonable
as it produces the lower detection efficiency we expect for such events it is
not based on the same rigorous testing done by Christiansen (2017) and Hsu et
al. (2019) for 3 transit and greater events. Furthermore monotransits are
often not found by the same algorithms that search for multitransit events and
thus the probability of detection may not scale down linearly with number of
transits down to the level of a monotransit. For this reason it is also worth
considering a simple SNR$\geq 20$ threshold for monotransits as discussed in
Section 4.6.

Compared with Kepler, Vetting and confirmation of TESS candidates is easier at
the same SNR due to the fact that TESS observes brighter stars and benefits
from the Gaia data releases that better characterise host stars. Therefore it
is likely that, with the caveats listed above, the use of the gamma function
for Kepler on TESS data is likely to lead to more conservative (i.e. lower)
estimates of sensitivity and therefore yield. However, quantifying the exact
difference is difficult and therefore we continue to use the Kepler function.

Additionally, as shown in Section 4.6 the use of a gamma function, and SNR
thresholds of 7.3 and 20 all recover a deficit in the number of long-period
planets compared with the sample of TOI discoveries. This provides additional
evidence for the potential of undiscovered long-period planets in TESS data.

###  5.8 Use of Year 1 and Year 3 SPOC FFI Lightcurves

In our simulation we use Year 1 and Year 3 SPOC FFI lightcurves which gives us
a stellar population of $\sim 1.3$ million stars in the southern ecliptic
hemisphere. This sample is selected in part due to computational limits of
SPOC (Caldwell et al., 2020), and TESS observes many more stars at sufficient
precision to detect transiting exoplanets. Thus the total planetary yield from
the TESS mission will be higher than those from just the SPOC FFI targets as
we simulate here. However, the SPOC FFI sample prioritises bright, main
sequence targets with low levels of dilution from other sources to fill the
maximum of 160000 targets per sector (Caldwell et al., 2020). The Candidate
Target List (CTL; Stassun et al., 2019), as used by Kunimoto et al. (2022),
uses a similar prioritisation metric but with a higher number of selected
targets. This difference results in a smaller sample for SPOC, but with a
greater proportion of targets most amenable to small planet detection. In
future, it would be interesting to run TIaRA on a larger sample of
lightcurves, such as those generated by the Quick Look Pipeline (QLP; Huang et
al., 2020).

##  6 Summary and Conclusions

We develop the TIaRA pipeline to simulate the sensitivity of transit surveys
to detecting transiting exoplanets. We focus in this paper on the Year 1 and
Year 3 SPOC FFI sample from the TESS mission. In particular we are interested
in the sensitivity and yield for longer-period planets that present as
monotransits or biennial duotransits in the data. Our simulations is based on
the actual stars monitored by TESS and the real TESS window functions
(accounting for discontinuities in observations). We also use the actual noise
properties of the lightcurves for each star.

We find a total of $2271^{+241}_{-138}$ exoplanets should be detected around
AFGKM dwarf host stars in the Year 1 and Year 3 SPOC FFI lightcurves. Of these
$403^{+64}_{-38}$ will have orbital periods greater than 25 days and
$113^{+23}_{-17}$ will have orbital periods greater than 100 days. We find
4.4$\sigma$ more predicted detections than the current TOI sample size of
1666. We find an increasing disparity between our predictions and actual TESS
discoveries from the TOI catalogue at longer-periods, suggesting that the TESS
discovery sample is incomplete at longer-periods and more long-period planets
remain to be discovered in TESS data.

These additional planets will require concentrated follow-up efforts to
confirm as the majority of them will be initially detected as some of the
$215^{+37}_{-23}$ predicted TESS monotransits. 50% of planet detections with
periods above 25 days and 76% of planet detections above 100 days will be
monotransits. Aside from monotransits, a large portion of the remaining long-
period planets will be found as biennial duotransit events with one transit in
Year 1 and an additional in Year 3, with $113^{+22}_{-13}$ of these
discoveries detected.

The TIaRA pipeline developed for this project can be applied to additional
TESS data sets such as the northern ecliptic SPOC FFI lightcurves from Years 2
and 4 and the recently released Year 5 SPOC FFI lightcurves. We intend to
perform follow-up simulations using these additional SPOC data in the future.
The QLP lightcurves (Huang et al., 2020) could also be analysed using TIaRA.
Reasonable assumptions of the observing strategy and current performance of
TESS could also be used to create simulated lightcurves and make predictions
of future TESS extended missions. With simulated data, TIaRA can easily be
used to predict the yields from future missions, in particular the upcoming
PLAnetary Transits and Oscillations of stars (PLATO; Rauer et al., 2014)
mission.

## Acknowledgements

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program. Resources supporting this
work were provided by the NASA High-End Computing (HEC) Program through the
NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the
production of the SPOC data products. The TESS team shall assure that the
masses of fifty (50) planets with radii less than 4R⊕ are determined.

We acknowledge the use of public TESS Alert data from pipelines at the TESS
Science Office and at the TESS Science Processing Operations Center.

This paper includes data collected by the TESS mission that are publicly
available from the Mikulski Archive for Space Telescopes (MAST).

TR and FR are supported by STFC studentships. The contributions at the
University of Warwick by SG and DB have been supported by STFC through
consolidated grants ST/P000495/1, ST/T000406/1 and ST/X001121/1.

We thank Dr Chelsea X. Huang and Dr Thomas G. Wilson for their helpful
comments when reviewing this work in the form of an MSc thesis. We also would
like to thank the assistant editor, Dr Sean Hodges and the anonymous reviewer
for their helpful comments.

## Data Availability

The TESS data is accessible via the MAST (Mikulski Archive for Space
Telescopes) portal at
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html. The binned
sensitivity and yield maps used to produce Figures 6,7,15 and 16 are available
in a machine readable format from an open GitHub repository at
https://github.com/TobyRodel/TIaRA-TESS-Yields.

## References

  * Agol et al. (2020) Agol E., Luger R., Foreman-Mackey D., 2020, AJ, 159, 123
  * Aizawa et al. (2017) Aizawa M., Uehara S., Masuda K., Kawahara H., Suto Y., 2017, AJ, 153, 193
  * Akeson et al. (2013) Akeson R. L., et al., 2013, PASP, 125, 989
  * Albrecht et al. (2012) Albrecht S., et al., 2012, ApJ, 757, 18
  * Ballard (2019) Ballard S., 2019, AJ, 157, 113
  * Barclay et al. (2018) Barclay T., Pepper J., Quintana E. V., 2018, ApJS, 239, 2
  * Barnes & Fortney (2004) Barnes J. W., Fortney J. J., 2004, ApJ, 616, 1193
  * Barnes & O’Brien (2002) Barnes J. W., O’Brien D. P., 2002, ApJ, 575, 1087
  * Bayliss et al. (2020) Bayliss D., et al., 2020, The Messenger, 181, 28
  * Benz et al. (2021) Benz W., et al., 2021, Experimental Astronomy, 51, 109
  * Bergsten et al. (2023) Bergsten G. J., Pascucci I., Hardegree-Ullman K. K., Fernandes R. B., Christiansen J. L., Mulders G. D., 2023, AJ, 166, 234
  * Borucki et al. (2010) Borucki W. J., et al., 2010, Science, 327, 977
  * Bouma et al. (2017) Bouma L. G., Winn J. N., Kosiarek J., McCullough P. R., 2017, arXiv e-prints, p. arXiv:1705.08891
  * Bryant et al. (2023) Bryant E. M., Bayliss D., Van Eylen V., 2023, MNRAS,
  * Caldwell et al. (2020) Caldwell D. A., et al., 2020, Research Notes of the American Astronomical Society, 4, 201
  * Cassidy et al. (2009) Cassidy T. A., Mendez R., Arras P., Johnson R. E., Skrutskie M. F., 2009, ApJ, 704, 1341
  * Charbonneau et al. (2002) Charbonneau D., Brown T. M., Noyes R. W., Gilliland R. L., 2002, ApJ, 568, 377
  * Christiansen (2017) Christiansen J. L., 2017, Planet Detection Metrics: Pixel-Level Transit Injection Tests of Pipeline Detection Efficiency for Data Release 25, Kepler Science Document KSCI-19110-001, id. 18. Edited by Michael R. Haas and Natalie M. Batalha
  * Christiansen et al. (2012) Christiansen J. L., et al., 2012, PASP, 124, 1279
  * Collier Cameron et al. (2006) Collier Cameron A., et al., 2006, MNRAS, 373, 799
  * Cooke et al. (2018) Cooke B. F., Pollacco D., West R., McCormac J., Wheatley P. J., 2018, A&A, 619, A175
  * Cooke et al. (2019) Cooke B. F., Pollacco D., Bayliss D., 2019, A&A, 631, A83
  * Dobos et al. (2021) Dobos V., Charnoz S., Pál A., Roque-Bernard A., Szabó G. M., 2021, PASP, 133, 094401
  * Dong et al. (2021) Dong J., et al., 2021, ApJS, 255, 6
  * Dragomir et al. (2019) Dragomir D., Huang C. X., Kane S., Dalba P. A., Günther M. N., 2019, in AAS/Division for Extreme Solar Systems Abstracts. p. 103.04
  * Dragomir et al. (2023) Dragomir D., et al., 2023, in American Astronomical Society Meeting Abstracts. p. 318.03
  * Dressing & Charbonneau (2013) Dressing C. D., Charbonneau D., 2013, ApJ, 767, 95
  * Dressing & Charbonneau (2015) Dressing C. D., Charbonneau D., 2015, ApJ, 807, 45
  * Eberhardt et al. (2022) Eberhardt J., et al., 2022, in Bulletin of the American Astronomical Society. p. 102.385
  * Espinoza & Jordán (2015) Espinoza N., Jordán A., 2015, MNRAS, 450, 1879
  * Espinoza & Jordán (2016) Espinoza N., Jordán A., 2016, MNRAS, 457, 3573
  * Essack et al. (2023) Essack Z., et al., 2023, AJ, 165, 47
  * Fressin et al. (2013) Fressin F., et al., 2013, ApJ, 766, 81
  * Gaia Collaboration et al. (2016) Gaia Collaboration et al., 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2018) Gaia Collaboration et al., 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2023) Gaia Collaboration et al., 2023, A&A, 674, A1
  * Gan et al. (2023) Gan T., et al., 2023, AJ, 165, 17
  * Giacalone et al. (2022) Giacalone S., et al., 2022, AJ, 163, 99
  * Giles et al. (2018) Giles H. A. C., et al., 2018, A&A, 615, L13
  * Gill et al. (2020) Gill S., et al., 2020, ApJ, 898, L11
  * Goffo et al. (2023) Goffo E., et al., 2023, ApJ, 955, L3
  * Grieves et al. (2022) Grieves N., et al., 2022, arXiv e-prints, p. arXiv:2209.14830
  * Guerrero et al. (2021) Guerrero N. M., et al., 2021, ApJS, 254, 39
  * Harmanec (1988) Harmanec P., 1988, Bulletin of the Astronomical Institutes of Czechoslovakia, 39, 329
  * Hawthorn et al. (2023) Hawthorn F., et al., 2023, TESS Duotransit Candidates from the Southern Ecliptic Hemisphere (arXiv:2310.17268)
  * Hilditch (2001) Hilditch R. W., 2001, An Introduction to Close Binary Stars. Cambridge University Press, doi:10.1017/CBO9781139163576
  * Hippke & Heller (2019) Hippke M., Heller R., 2019, A&A, 623, A39
  * Howell et al. (2014) Howell S. B., et al., 2014, PASP, 126, 398
  * Hsu et al. (2019) Hsu D. C., Ford E. B., Ragozzine D., Ashby K., 2019, AJ, 158, 109
  * Huang et al. (2016) Huang C., Wu Y., Triaud A. H. M. J., 2016, ApJ, 825, 98
  * Huang et al. (2018) Huang C. X., et al., 2018, arXiv e-prints, p. arXiv:1807.11129
  * Huang et al. (2020) Huang C. X., et al., 2020, Research Notes of the American Astronomical Society, 4, 204
  * Jenkins et al. (2016) Jenkins J. M., et al., 2016, in Chiozzi G., Guzman J. C., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 9913, Software and Cyberinfrastructure for Astronomy IV. p. 99133E, doi:10.1117/12.2233418
  * Johnson (2023) Johnson M., 2023, in American Astronomical Society Meeting Abstracts. p. 401.02
  * Johnson et al. (2010) Johnson J. A., Aller K. M., Howard A. W., Crepp J. R., 2010, PASP, 122, 905
  * Kempton et al. (2018) Kempton E. M. R., et al., 2018, PASP, 130, 114401
  * Kipping (2013) Kipping D. M., 2013, MNRAS, 434, L51
  * Kipping (2023) Kipping D., 2023, MNRAS, 523, 1182
  * Kipping et al. (2009) Kipping D. M., Fossey S. J., Campanella G., 2009, MNRAS, 400, 398
  * Kovács et al. (2002) Kovács G., Zucker S., Mazeh T., 2002, A&A, 391, 369
  * Kreidberg (2015) Kreidberg L., 2015, PASP, 127, 1161
  * Kreidberg (2018) Kreidberg L., 2018, in Deeg H. J., Belmonte J. A., eds, , Handbook of Exoplanets. Springer International Publishing, Cham, pp 2083–2105, doi:10.1007/978-3-319-55333-7_100, https://doi.org/10.1007/978-3-319-55333-7_100
  * Kruse et al. (2019) Kruse E., Agol E., Luger R., Foreman-Mackey D., 2019, ApJS, 244, 11
  * Kunimoto & Matthews (2020) Kunimoto M., Matthews J. M., 2020, AJ, 159, 248
  * Kunimoto et al. (2022) Kunimoto M., Winn J., Ricker G. R., Vanderspek R. K., 2022, AJ, 163, 290
  * Lendl et al. (2020) Lendl M., et al., 2020, MNRAS, 492, 1761
  * Luque & Pallé (2022) Luque R., Pallé E., 2022, Science, 377, 1211
  * Madhusudhan (2019) Madhusudhan N., 2019, ARA&A, 57, 617
  * Madhusudhan et al. (2014a) Madhusudhan N., Knutson H., Fortney J. J., Barman T., 2014a, in Beuther H., Klessen R. S., Dullemond C. P., Henning T., eds, Protostars and Planets VI. pp 739–762 (arXiv:1402.1169), doi:10.2458/azu_uapress_9780816531240-ch032
  * Madhusudhan et al. (2014b) Madhusudhan N., Amin M. A., Kennedy G. M., 2014b, ApJ, 794, L12
  * Makarov & Efroimsky (2023) Makarov V. V., Efroimsky M., 2023, A&A, 672, A78
  * Maxted (2016) Maxted P. F. L., 2016, A&A, 591, A111
  * McLaughlin (1924) McLaughlin D., 1924, ApJ, 60
  * Muirhead et al. (2018) Muirhead P. S., Dressing C. D., Mann A. W., Rojas-Ayala B., Lépine S., Paegert M., De Lee N., Oelkers R., 2018, AJ, 155, 180
  * Mustill et al. (2017) Mustill A. J., Davies M. B., Johansen A., 2017, MNRAS, 468, 3000
  * Öberg et al. (2011) Öberg K. I., Murray-Clay R., Bergin E. A., 2011, ApJ, 743, L16
  * Osborn et al. (2023) Osborn H. P., et al., 2023, MNRAS, 523, 3069
  * Owen (2019) Owen J. E., 2019, Annual Review of Earth and Planetary Sciences, 47, 67
  * Pecaut & Mamajek (2013) Pecaut M. J., Mamajek E. E., 2013, ApJS, 208, 9
  * Petrovich (2014) Petrovich C., 2014, in AAS/Division for Planetary Sciences Meeting Abstracts #46. p. 301.03
  * Rauer et al. (2014) Rauer H., et al., 2014, Experimental Astronomy, 38, 249
  * Rice et al. (2022) Rice M., et al., 2022, AJ, 164, 104
  * Ricker et al. (2014) Ricker G. R., et al., 2014, in Oschmann Jacobus M. J., Clampin M., Fazio G. G., MacEwen H. A., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 9143, Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave. p. 914320 (arXiv:1406.0151), doi:10.1117/12.2063489
  * Rossiter (1924) Rossiter R., 1924, ApJ, 60
  * Sabotta et al. (2021) Sabotta S., et al., 2021, A&A, 653, A114
  * Simon et al. (2012) Simon A. E., Szabó G. M., Kiss L. L., Szatmáry K., 2012, MNRAS, 419, 164
  * Stassun et al. (2019) Stassun K. G., et al., 2019, AJ, 158, 138
  * Sullivan et al. (2015) Sullivan P. W., et al., 2015, ApJ, 809, 77
  * Triaud (2018) Triaud A. H. M. J., 2018, in Deeg H. J., Belmonte J. A., eds, , Handbook of Exoplanets. Springer International Publishing, Cham, pp 1375–1401, doi:10.1007/978-3-319-55333-7_2, https://doi.org/10.1007/978-3-319-55333-7_2
  * Tuson et al. (2023) Tuson A., et al., 2023, MNRAS, 523, 3090
  * Ulmer-Moll et al. (2022) Ulmer-Moll S., et al., 2022, A&A, 666, A46
  * Ulmer-Moll et al. (2023) Ulmer-Moll S., et al., 2023, A&A, 674, A43
  * Valsecchi et al. (2015) Valsecchi F., Rappaport S., Rasio F. A., Marchant P., Rogers L. A., 2015, ApJ, 813, 101
  * Van Eylen & Albrecht (2015) Van Eylen V., Albrecht S., 2015, ApJ, 808, 126
  * Vanderburg et al. (2015) Vanderburg A., et al., 2015, ApJ, 800, 59
  * Vanderburg et al. (2016) Vanderburg A., et al., 2016, ApJ, 827, L10
  * Vanderspek et al. (2019) Vanderspek R., et al., 2019, ApJ, 871, L24
  * Villanueva et al. (2019) Villanueva Steven J., Dragomir D., Gaudi B. S., 2019, AJ, 157, 84
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Wheatley et al. (2018) Wheatley P. J., et al., 2018, MNRAS, 475, 4476
  * Winn (2014) Winn J. N., 2014, arXiv e-prints, p. arXiv:1001.2010v5
  * Wu & Lithwick (2011) Wu Y., Lithwick Y., 2011, ApJ, 735, 109
  * Zhou et al. (2019) Zhou G., et al., 2019, AJ, 158, 141

##  Appendix A Sensitivity by spectral types

(a) A type dwarf stars

(b) F type dwarf stars

(c) G type dwarf stars

(d) K type dwarf stars

(e) M type dwarf stars

Figure 15: Sensitivity maps arranged by spectral type showing the probability
of a transiting exoplanet being observed by TESS and then detected in the SPOC
FFI lightcurves for Years 1 and 3. Numbers in each grid cell are the
percentage sensitivity values. Where no uncertainties are given the
uncertainty is 0% to two decimal places.

##  Appendix B Yield by spectral type

(a) A type dwarf stars

(b) F type dwarf stars

(c) G type dwarf stars

(d) K type dwarf stars

(e) M type dwarf stars

Figure 16: Predicted yields of exoplanets from the Years 1 and 3 TESS SPOC FFI
lightcurves divided by spectral type.