# PLATO on the shoulders of TESS: analyzing mono-transit planet candidates in
TESS data as a prior knowledge for PLATO observations

Christian Magliano 1,2,3, Giovanni Covone1,2,3, Valerio Nascimbeni4,5, Laura
Inno3,6, Jose I. Vines 7, Veselin Kostov,8,9, Stefano Fiscale6,3, Valentina
Granata5,4, Marco Montalto5,4, Isabella Pagano10, Giampaolo Piotto5,4, Vito
Saggese 1,2.
1Dipartimento di Fisica "Ettore Pancini", Università di Napoli Federico II,
Napoli, Italy
2INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via
Cintia Edificio 6, 80126 Napoli, Italy
3INAF - Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131
Napoli, Italy
4INAF – Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122
Padova, Italy
5Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi
di Padova, vicolo dell’Osservatorio 3, 35122 Padova, Italy
6Science and Technology Department, Parthenope University of Naples, Naples
80143 Italy
7Instituto de Astronomía, Universidad Católica del Norte, Angamos 0610,
1270709, Antofagasta, Chile
8NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771,
USA
9GSFC Sellers Exoplanet Environments Collaboration, USA
10INAF - Osservatorio Astrofisico di Catania, Catania 95123 ,Italy
E-mail<EMAIL_ADDRESS>0000-0001-6343-4744 0000-0002-2553-096X
0000-0001-9770-1214 0000-0002-0271-2664 0000-0002-1896-2377
0000-0001-9786-1031 0000-0001-8371-8525 0000-0002-1425-4541
0000-0002-7618-8308 0000-0001-9573-4928 0000-0002-9937-6387

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Transiting Exoplanet Survey Satellite (TESS) and the upcoming PLATO
mission (PLAnetary Transits and Oscillations of stars) represent two space-
based missions with complementary objectives in the field of exoplanet
science. While TESS aims at detecting and characterizing exoplanets around
bright and nearby stars on a relative short-period orbit, PLATO will discover
a wide range of exoplanets including rocky planets within the habitable zones
of their stars. We analyze mono-transit events in TESS data around stars that
will or could be monitored by the PLATO mission, offering a unique opportunity
to bridge the knowledge gap between the two missions and gain deeper insights
into exoplanet demographics and system architectures. We found $48$ TESS mono-
transit events around stars contained in the all-sky PLATO Input Catalog; of
these, at least four will be imaged on the first long-pointing PLATO field,
LOPS2. We uniformly vetted this sample to rule out possible false positive
detections thus removing $10$ signals from the original sample. We developed
an analytic method which allows us to estimate both the orbital period and
inclination of a mono-transit planet candidate using only the shape of the
transit. We derived the orbital period and inclination estimates for $30$ TESS
mono-transit planet candidates. Finally, we investigated whether these
candidates are amenable targets for a CHEOPS observing campaign.

###### keywords:

planets and satellites: general - planets and satellites: detection -
techniques: photometric

††pubyear: 2023††pagerange: PLATO on the shoulders of TESS: analyzing mono-
transit planet candidates in TESS data as a prior knowledge for PLATO
observations–5

##  1 Introduction

The continuous advancements in exoplanet science have significantly expanded
our understanding of the vast exoplanetary population, prompting the need for
comprehensive surveys to unveil the intricate details of exoplanet
demographics. The TESS space-based mission (Ricker et al., 2015), launched by
NASA in 2018, has been designed to conduct an all-sky survey, observing more
than 200,000 bright stars across the sky during its two-year primary mission.
TESS completed its primary mission in July 2020 and its primary extended
mission in September 2022. Now it has just entered into the second extended
mission which will last almost three years. Equipped with four wide-field
cameras, it monitors the sky into 26 sectors and observes each sector for
approximately 27 days. Using this observing strategy, TESS has enabled the
detection and characterization of more than three hundred exoplanets,
fostering our understanding of the diverse exoplanetary population. The
drawback of TESS’s nearly complete coverage of the sky is that it is more
sensitive towards short period planets, which limits the opportunities to
discover exoplanets orbiting the so-called habitable zone of Sun-like stars.
When the orbital period $P$ is longer than the total observational baseline,
the probability a planet transits only one time its star decreases as
$P^{-5/3}$ (Yee & Gaudi, 2008). Therefore, as most of the sky is continuously
observed by TESS for less than 27 days the expected yield of mono-transit
events detected by TESS is larger if compared with previous missions like
Kepler and K2 (Cooke et al., 2018; Villanueva et al., 2019). According to the
simulations by Cooke et al. (2018), about $10\%$ of predicted TESS discoveries
will be mono-transit candidates. Using a semianalytic technique, Villanueva et
al. (2019) estimated TESS will find over one thousand mono-transit events,
with about $90\%$ of the sample accessible to photometric and spectroscopic
follow-up investigations from the ground (Yao et al., 2019; Bayliss et al.,
2020; Gill et al., 2020). Taking into account the primary TESS mission
extension, Cooke et al. (2019) updated the findings of their previous work.
Combining TESS Year 1 and Year 4 observations, they predicted a total of $140$
mono-transit events in the Southern Ecliptic Hemisphere. In addition, they
claimed to recover $189$ duo-transit events during the time span between the
TESS Year 1 and 4 campaigns. For these systems, it is possible to constrain
their orbital period through the period aliasing technique (Cooke et al.,
2021).

Mono-transits events are more challenging to confirm as bonafide planets than
planets transiting two or more times within the observing time window. Since
we are not able to fold multiple transits, their physical properties are
poorly constrained and there is a higher probability of false positive
detections from sporadic events. Nevertheless, these events are worthy of
deeper scientific investigation because, if confirmed as true planet, they: i)
would further enrich our understanding of exoplanetary systems (Alibert et
al., 2013); ii) may be long period worlds orbiting the habitable zone of their
host stars. Both of these crucial points would provide valuable insights into
whether configurations as the Solar System are rare occurrences in the Galaxy
or not (e.g., Kipping et al. 2016). This is important in particular
considering the fact that planets with optimal conditions for life might orbit
K or G-type stars (e.g., Covone et al., 2021). Different facilities spread all
over the world could play a crucial role in recovering the TESS mono-transit
events (e.g., Gill et al. 2019) thus enabling the confirmation of these events
as genuine planets while also providing improved characterization of their
orbital parameters.

In this context, the ESA space-based mission PLATO (Rauer et al., 2014),
slated for launch in 2026, will build upon the foundations laid by TESS and
Kepler/K2 missions. PLATO aims at conducting a comprehensive survey of
exoplanets, particularly targeting Earth-like planets around Sun-like stars.
In fact, by leveraging on its precise photometry and long-duration
observations, it will delve into the statistical properties of exoplanets,
providing invaluable insights into their occurrence rates, orbital
architectures and habitability potential. Therefore, PLATO will provide a
unique opportunity to fill the void left by TESS. The synergistic
collaboration between TESS and PLATO would yield mutual benefits, as it allows
for the optimal utilization of TESS data and enhances the observational
capabilities of PLATO by providing prior knowledge of the mono-transit systems
under scrutiny. In summary, mono-transit events in TESS data could represent a
boost for the scientific return of the PLATO mission.

In this paper, we focus our attention on mono-transit planet candidates
detected by TESS around the stars that PLATO will monitor in the future. We
propose an analytic approach to retrieve an estimate of the orbital period and
inclination of these candidates, based on the shape of the transit. The
outline of the paper is the following. In Sect. 2 we describe the analytic
technique used to retrieve an approximation of a mono-transit event’s orbital
period and inclination. Then, in Sect. 3 we construct the sample of mono-
transit TESS candidates orbiting those stars that could be also monitored by
PLATO. We discuss the results obtained by applying this technique to our
sample in Sect. 4. Finally, we summarize our conclusions in Sect. 5.

##  2 Methods

The most challenging aspect when investigating mono-transit events is to
retrieve the orbital period of the planet. The period is not only a key
parameter to investigate the formation pathway of an exoplanet, but it is also
crucial to schedule follow-up observations. A mono-transit event is by
definition a single dip in the light curve of a given star, thus making the
use of Lomb-Scargle periodogram (Lomb 1976; Scargle 1982) or periodic transit
fitting algorithms (e.g. Kovács et al. 2002; Hippke & Heller 2019) completely
useless. Indeed, the primary purpose of the aforementioned techniques is to
identify periodic signals in time-series photometric data. Once a mono-transit
event is detected, Bayesian methods usually offer a valuable tool to analyze
them. By incorporating prior knowledge, such as stellar properties and
observational constraints, Bayesian analysis can provide quite accurate and
informative estimates of the planet’s parameters (e.g., Osborn et al. 2016;
Incha et al. 2023).

However, Seager & Mallén-Ornelas (2003) (S03, hereafter) showed that it is
possible to relate the orbital parameters of a transiting planet with the
stellar properties and the transit features. Specifically, when assuming a
circular orbit and the absence of contamination in the measured star flux from
other sources, there exists a relationship between the total transit duration
$t_{T}$ of a planet and its orbital period $P$ given by:

$t_{T}=\dfrac{P}{\pi}\arcsin{\left(\dfrac{R_{*}}{a}\left[\dfrac{[1+\sqrt{\delta}]^{2}-[a\cos
i/R_{*}]^{2}}{1-\cos^{2}(i)}\right]^{1/2}\right)}\,,$ (1)

where $a$ is the semi-major axis of the planet, $\delta$ is the transit depth
when neglecting limb-darkening effects, $i$ the orbital inclination and
$R_{*}$ the stellar radius. Using Kepler’s third law and assuming that the
mass of the planet is small (i.e., $M_{p}\ll M_{*}$, where $M_{*}$ is the
stellar mass), equation (1) can be written as

$t_{T}-\dfrac{P}{\pi}\arcsin{\left[\left(\dfrac{\Lambda^{-2/3}_{*}P^{-4/3}(1+\sqrt{\delta})^{2}-\cos^{2}(i)}{1-\cos^{2}(i)}\right)^{1/2}\right]}=0\,,$
(2)

where we defined $\Lambda_{*}\equiv G\rho_{*}/3\pi$ with $\rho_{*}=3M_{*}/4\pi
R_{*}^{3}$ the mean stellar density111Note we used a slightly different
definition of $\rho_{*}$ with respect to S03. and $G$ the gravitational
constant. Under the approximation $\pi t_{T}/P\ll 1$ and $\cos i\ll 1$ (which
are reasonable for likely long-period transiting planets) equation (2) can be
cast in the following form:

$\cos^{2}(i)\,P^{2}-\Lambda^{-2/3}_{*}(1+\sqrt{\delta})^{2}\,P^{2/3}+\pi^{2}\,t_{T}^{2}=0\,.$
(3)

Equations (2), or its approximated form (3), can be used to estimate the
orbital period of mono-transit events (e.g., Yee & Gaudi 2008; Wang et al.
2015; Uehara et al. 2016). For a fixed value of the orbital inclination $i$,
if the mean stellar density of the host star $\rho_{*}$ is calculated from an
independent observation, one can solve equations (2) and (3) in terms of the
orbital period $P$. Equation (2) is a transcendental equation for which only
numeric solving approaches are possible. Equation (3), instead, is a quadratic
and fractional power equation, so that an analytic resolution can be derived
(see Sect. 2.1). The uncertainty in the solutions to equations (2) and (3)
will be the result of the combination of the uncertainty in the stellar
density and the orbital parameters (see equations A18-A24 in Appendix A for
details on the error propagation). In general, for $i\approx 90^{\circ}$ the
orbital period of a mono-transit event in a star’s light curve can be
approximated either through the numerical solution of equation (2) or through
the analytical solution of equation (3). For $i$ exactly $90^{\circ}$ , i.e.
in the edge-on (EO, $i=90^{\circ}$) condition, the quadratic term in the
equation (3) vanishes and the orbital period $P_{\text{EO}}$ is given by:

$P_{\text{EO}}=\left(\dfrac{\pi\,t_{T}}{1+\sqrt{\delta}}\right)^{3}\Lambda_{*}\,.$
(4)

Equation (4) shows that the accuracy to which we can determine the orbital
period depends on the accuracy on which we measure the mean stellar density
(Wang et al., 2015). By combining Gaia parallaxes with isochrone fitting,
Sandford et al. (2019) estimated the stellar densities for a selection of K2
stars hosting planets. Using the third Kepler’s law to infer the orbital
period, their analysis resulted in a fractional uncertainty of $15_{-6}^{+30}$
per cent for the orbital period, assuming a circular orbit.

However, the PLATO’s asteroseismology programme aims at reducing the
uncertainty on stellar mean density determinations to about $10\%$ for G0V
stars with a visual magnitude of 10 (Goupil, 2017). In light of this, the
estimate of the orbital period could be updated using cutting-edge estimates
of the stellar parameters. Finally, as we will show in Sect. 2.1, even a
slight change on the orbital inclination can lead to very different values for
the period. Consequently, the edge-on assumption can be considered a first-
order approximation of the orbital period of a mono-transit planet candidate.

###  2.1 Polynomial equation

For now, let us assume that the inclination of the orbit is known. In Sect.
2.2 we will discuss the realistic scenario where the orbital inclination is an
additional unknown in the equations (2) and (3). Let $\alpha$, $\beta$ and
$\gamma$ be defined as

$\alpha\equiv\cos^{2}(i),\\\
\beta\equiv\Lambda^{-2/3}_{*}(1+\sqrt{\delta})^{2},\\\
\gamma\equiv\pi^{2}t_{T}^{2}\,.$ (5)

Then equation (3) can be written as

$f(P)\equiv\,\alpha\,P^{2}-\beta\,P^{2/3}+\gamma=0\,.$ (6)

For a transiting planet, equation (6) has two solutions $P_{1}$ and $P_{2}$
(degeneracy issue, see Sect. 2.3), given by:

$P_{1}=2^{3/2}\left(\dfrac{\beta}{3\alpha}\right)^{3/4}\cos^{3/2}\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]},$
(7)
$P_{2}=2^{3/2}\left(\dfrac{\beta}{3\alpha}\right)^{3/4}\cos^{3/2}\Bigg{[}\dfrac{\pi}{3}-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\,.$
(8)

The comprehensive calculations are available in the Appendix A. Moreover,
equations (7) and (8) have physical meaning only if the argument of the square
root is non negative, that is

$\dfrac{2\sqrt{3}}{3\pi}\dfrac{(1+\sqrt{\delta})^{3}}{G\,t_{T}^{2}\,\rho_{*}\,\cos
i}\geq 1\,.$ (9)

As demonstrated in Appendix A, when $(4\beta^{3}/27\alpha\gamma^{2})<1$,
equation (6) does not yield any physically meaningful solutions, while in the
case of $(4\beta^{3}/27\alpha\gamma^{2})=1$, only one solution is obtained.

In order to understand how changing the three coefficients affects the
solutions to equation (6), in Fig. 1 we show the relative variation of the two
solutions $P_{1}$ and $P_{2}$ against the variation of one of the fundamental
parameters. We examined four specific scenarios, in each one we made a small
adjustment to one of the parameters $i$, $\rho_{*}$, $\delta$, or $t_{T}$,
while keeping the other three constant. To make our simulations as realistic
as possible, for each parameter, we have considered the maximum variation to
be the median relative percentage error observed in the TESS photometric data
publicly available on the ExoFOP
archive222https://exofop.ipac.caltech.edu/tess/. In particular, we defined the
TESS relative percentage error on each of the aforementioned parameters as the
median of the distribution in order to damp the effects of the outliers.

Figure 1: Variation of the solutions to equation (6). On top left we kept
$\beta$ and $\gamma$ constant while varying $\alpha$ through a slight
adjustment in the orbital inclination $i$ by a few percentage points. In the
top right and bottom left plots, we fixed $\alpha$ and $\gamma$ steady while
modifying $\beta$ by respectively altering the mean stellar density $\rho_{*}$
(top right) and adjusting the transit depth $\delta$ (bottom left). On bottom
right we fixed $\alpha$ and $\beta$ while varying $\gamma$ by changing the
total duration of the transit $t_{T}$.

As we can see in Fig. 1, the orbital inclination (or $\alpha\equiv\cos^{2}i$)
is the primary physical parameter to which the function $f(P)$ exhibits the
highest sensitivity. In fact, even if the inclination undergoes a small
variation ($\Delta i<1\%$), $P_{1}$ and $P_{2}$ can span a broad spectrum of
values. In this specific case, while the range of $P_{1}$ extends over a few
dozen of percentages due to a $\Delta i$ up to $0.5\%$, $P_{2}$ covers five
orders of magnitudes with the same variation. The second parameter to which
$f(P)$ is most sensitive is the duration of the transit $t_{T}$ (or $\gamma$)
which causes a rigid translation upwards (if $t_{T}$ increases) or downwards
(if $t_{T}$ decreases) of $f(P)$. TESS’s typical $10\%$ variation in $t_{T}$
leads to an approximate $30\%$ change in $P_{1}$, while $P_{2}$ experiences a
fluctuation of a few percentage points. Furthermore, we observed that a $30\%$
change in stellar density $\rho_{*}$ corresponds to a $35\%$ variation in
$P_{1}$ and a $15\%$ variation in $P_{2}$. It is noteworthy that the $10\%$
accuracy on $\rho_{*}$ that will be achieved by PLATO’s asteroseismology
analysis would lead to a reduction of $10\%$ and $5\%$ in the variation of
$P_{1}$ and $P_{2}$, respectively. Finally, we remark that the solution
$P_{1}$ and $P_{2}$ exhibit the lowest sensitivity to the variations in
transit depth $\delta$. A typical $1\%$ variation in $\delta$ results in a
minor change of a few tenths of a percentage in the period. Fig. 1 clearly
shows that $P_{2}$ is more robust than $P_{1}$ against variations of
$\rho_{*}$, $\delta$ and $t_{T}$, however it is highly affected by tiny
changes in the orbital inclination which dominates the uncertainty on $P_{2}$.

This section focuses on solving the polynomial equation (3), but the findings
are also qualitatively valid for equation (2)’s solutions since the condition
$\cos i\ll 1$ is basically satisfied for all transiting planets. In Fig. 2 we
show the distribution of TESS confirmed exoplanets according to their orbital
period. As we can see from the histogram, about $70\%$ of the total sample has
inclinations within $87^{\circ}\leq i\leq 90.5^{\circ}$.

Figure 2: Distribution of TESS confirmed exoplanets according to their orbital
inclination $i$. The dotted and dashed vertical lines represent respectively
the mean ($87^{\circ}$) and the median ($88.2^{\circ}$) of the distribution.

Throughout our discussion we ignored that the limb-darkening affects the shape
of the transit: i) it affects the transit depth $\delta$ as a function of the
orbital inclination $i$, ii) it reduces the elapsed time from the second to
the third contact ($t_{F}$, hereafter) by rounding the flat bottom part of the
transit and iii) it darkens the distinction between the ingress/egress phases
and the flat bottom region. Finally, we are assuming that the light curve is
not contaminated by other sources. However, since the big TESS pixel scale,
this assumption may not be always met (more on this in Sect.3.2). Thus the
findings of this method also rely on the angular resolution of the telescope
used for the observation.

###  2.2 The orbital inclination issue

Despite the majority of transiting exoplanets having almost $i=90^{\circ}$, a
very small discrepancy ($\Delta i\approx 0.5\%$) in the orbital inclination
can result in a drastic change in the orbital period solutions to equation (6)
(as shown in Sect. 2.1). In general, when the orbital inclination is close to
$90^{\circ}$ the exoplanet’s transit across the stellar disk is longer since
it covers a larger distance. This leads to a longer transit duration. On the
contrary, when the inclination deviates from $90^{\circ}$, the exoplanet’s
transit is shorter as it covers a smaller distance, resulting in a shorter
transit duration. Moreover, as mentioned above, the orbital inclination would
also affect the transit duration when accounting for the limb-darkening
effect. Therefore, we assume here that the errors on the orbital parameters
$\delta$ and $t_{T}$ depend only on TESS photometric precision and also the
lack of information on the orbital inclination of the system under examination
assuming the limb-darkening effect is negligible. The orbital inclination of a
mono-transit planet candidate is not known because a mono-transit event only
provides information about the total duration of the transit $t_{T}$ and the
transit depth $\delta$. Therefore, in order to consider the orbital
inclination as an additional unknown in equation (3), we will solve it for a
grid of possible inclination values such that the equation (9) is satisfied,
that is

$i\leq\arccos\left(\dfrac{2\sqrt{3}}{3\pi}\dfrac{(1+\sqrt{\delta})^{3}}{G\,t_{T}^{2}\,\rho_{*}}\right).$
(10)

Thus we obtain two solutions, $P_{1}(i)$ and $P_{2}(i)$, as a function of the
orbital inclination that describes all the possible combinations $(P,i)$ of
the mono-transit event within the error bars.

###  2.3 Degeneracy of the solutions

As observed in Sect. 2.1, equations (2) and (3) may admit two solutions
leading to a degeneracy. A possible way to break this degeneracy comes from
the relationship that exists between the total transit duration $t_{T}$ and
the elapsed time from the second to the third contact $t_{F}$, in the non-
limb-darkening assumption (equations (9) and (7) of S03) and considering
$M_{p}\ll M_{*}$, that is

$\dfrac{M_{*}}{R_{*}^{3}}=\left(\dfrac{4\pi^{2}}{P^{2}G}\right)\Biggl{\\{}\dfrac{(1+\sqrt{\delta})^{2}-b^{2}(1-\sin^{2}(t_{T}\pi/P)}{\sin^{2}(t_{T}\pi/P)}\Biggr{\\}}^{3/2}\,,$
(11)

where

$b=\sqrt{\dfrac{(1-\sqrt{\delta})^{2}-\left[\sin^{2}(t_{F}\pi/P)/\sin^{2}(t_{T}\pi/P)\right](1+\sqrt{\delta})^{2}}{1-\left[\sin^{2}(t_{F}\pi/P)/\sin^{2}(t_{T}\pi/P)\right]}}\,.$
(12)

If we can measure $t_{F}$ from the transit shape, then we are able to
numerically resolve equation (11) in terms of the period $P$ for each value of
the orbital inclination in the given grid. Thus from a theoretical point of
view, equations (3) (or eq.(2)) and (11) form a system of equations that
allows us to break the degeneracy of the solutions.

S03 also showed that equation (11) can take a simpler form under the
assumption $R_{*}\ll a$, or equivalently $\pi\,t_{T}/P\ll 1$, that is always
satisfied for long-period planets. In this approximation the orbital period
$P$ is given by

$P=\dfrac{G\pi^{2}}{24}\rho_{*}\dfrac{(t_{T}^{2}-t_{F}^{2})^{3/2}}{\delta^{3/4}}.$
(13)

Hereafter, we will refer to $P_{t_{F}}$ when the orbital period of the mono-
transit planet candidate is calculated by means of the equation (13). While
$P_{1}(i)$ and $P_{2}(i)$ solutions depend on the orbital inclination,
$P_{t_{F}}$ represents an independent estimate which theoretically breaks the
$P_{1}-P_{2}$ degeneracy. However, if the uncertainty $\Delta P_{t_{F}}$ over
$P_{t_{F}}$ is large enough, its estimate could be compatible with both
$P_{1}(i)$ and $P_{2}(i)$ branches. In Sect. 2.5, we will demonstrate how
equation (13) helps in constraining the orbital period within a range of
values and also allows to establish limits on the physically possible values
of the orbital inclination.

We note that, even if we did not measure $t_{F}$, by putting $t_{F}=0$,
equation (13) gives the upper limit for the orbital period $P_{\text{max}}$,
that is

$P_{\text{max}}=\dfrac{G\pi^{2}}{24}\rho_{*}\dfrac{t_{T}^{3}}{\delta^{3/4}}\,.$
(14)

Despite $t_{F}=0$ being a more theoretical limit than a realistic scenario,
grazing transiting planets are such that $t_{F}/t_{T}\ll 1$, so
$P_{\text{max}}$ represents a conservative estimate of the true maximum
orbital period of the planet. Using equation (14), Cacciapuoti et al. (2022)
estimated the upper period of the candidate planet TOI-411 d to be $46\pm 4$
days. Based on the combined TESS and CHEOPS observations, Garai et al. (2023)
determined the true period to be $47.42489\pm 0.00011$ days. Thus, if we found
degenerate solutions to equation (2) and (3) we could check whether one of
these is larger than $P_{\text{max}}$ and rule it out, even before resolving
equation (11). We note that equations (13) and (14) differ from equations
(27)-(28) of S03 because of our different definition of the mean stellar
density.

All the equations presented in this work are summarised in Table 5 in Appendix
B, together with the assumptions under which they can be applied.

###  2.4 Beyond the circular orbit approximation

Our framework highly relies on the circular orbit approximation but exoplanets
may not have a null eccentricity. The orbital eccentricity affects the shape
of a planetary transit (e.g., Winn et al. 2007). Tingley & Sackett (2005) were
the firsts to examine the influence of the planetary eccentricity on the
transit light curve. Furthermore, Barnes (2007) demonstrated that if the star
radius is constrained by an independent measurement, transit photometry can
provide a lower limit on planetary eccentricity.

Planets on eccentric orbits have a higher transit probability than planets in
circular orbit with the same semi-major axis (Barnes, 2007). Moreover, transit
events for exoplanets on eccentric orbits may exhibit asymmetric shapes or
varying durations compared to those on circular orbits (Yee & Gaudi, 2008). As
a result, the duration of the transit depends on where the planet is in its
orbit. Thus, for eccentric planets, equation (1) involves the planetary
eccentricity $e$ and the argument of periastron $\omega$ and becomes

$t_{T}=\dfrac{P}{\pi}\arcsin{\left(\dfrac{R_{*}}{a}\left[\dfrac{[1+\sqrt{\delta}]^{2}-[a\cos
i/R_{*}]^{2}}{1-\cos^{2}i}\right]^{1/2}\right)}\dfrac{\sqrt{1-e^{2}}}{1+e\sin\omega}.$
(15)

If we define

$\tau\equiv t_{T}\left(\dfrac{1+e\sin\omega}{\sqrt{1-e^{2}}}\right),$ (16)

then equation (15) can be written as

$\tau=\dfrac{P}{\pi}\arcsin{\left(\dfrac{R_{*}}{a}\left[\dfrac{[1+\sqrt{\delta}]^{2}-[a\cos
i/R_{*}]^{2}}{1-\cos^{2}i}\right]^{1/2}\right)},$ (17)

which is structurally identical to equation (1). Thus equation (17) admits the
solutions $P_{1}$ and $P_{2}$ given by equations (7) and (8) where
$t_{T}\rightarrow\tau$.

In Fig. 3 we show how the $P_{1}$ and $P_{2}$ solutions depend on the
planetary eccentricity $e$ for different values of the argument of periastron
$\omega$ and transit duration $t_{T}$, once the orbital inclination $i$, the
stellar density $\rho_{*}$ and the transit depth $\delta$ are fixed. Firstly,
we observed that the $P_{2}$ solution in the circular orbit approximation is
systematically underestimated with respect to that for non-zero planetary
eccentricity, and viceversa for the $P_{1}$ solution. Furthermore, we found
that the second solution, $P_{2}$, is more stable than $P_{1}$ under the
circular orbit approximation. While the former can vary by up to $\approx
20\%$ from its value in the circular orbit approximation, the latter can
change by more than $100\%$ from its prior value. Thus, if the evaluation of
the orbital period using equation (13) alongside its uncertainty rules out the
$P_{1}$ solution, then our prediction is based on $P_{2}$. In this case we
expect a discrepancy of up to $20\%$ in case of non-zero eccentricity.
Conversely, the most problematic scenario is when the retrieved orbital period
comes from the $P_{1}$ solution or, if the uncertainty $\Delta P_{t_{F}}$ over
$P_{t_{F}}$ is large enough, a combination of $P_{1}$ and $P_{2}$.

When the planet orbits on a mild eccentric orbit (i.e. $e<0.1$) both solutions
vary by less than $30\%$ from the circular orbit case. To date $\approx
50\%$333https://exoplanetarchive.ipac.caltech.edu/ of exoplanets with known
eccentricity have $e<0.1$. Furthermore, once the argument of periastron
$\omega$ is fixed, increasing the transit duration $t_{T}$ leads to a narrower
range of potential eccentricities and a more pronounced deviation of the
$P_{2}$ solution from the circular orbit case.

Lastly, we also analysed the behaviour of $P_{1}$ and $P_{2}$ as function of
the argument of periastron $\omega$. In particular, since
$\tau\propto\sin\omega$, it is sufficient to restrict our analysis in the
ranges $0^{\circ}\leq\omega\leq 90^{\circ}$ and $180^{\circ}\leq\omega\leq
270^{\circ}$. We found that when $\omega=0^{\circ}$ increases to
$\omega=90^{\circ}$ the range of possible eccentricities diminishes while the
deviation from the circular case increases for both $P_{1}$ and $P_{2}$.
Conversely we did not find appreciable differences when $\omega=180^{\circ}$
increases to $\omega=270^{\circ}$.

Figure 3: The solutions $P_{1}$ (solid lines) and $P_{2}$ (dashed lines) as
function of the planetary eccentricity $e$ and argument of periastron $\omega$
for different values of the transit duration $t_{T}$ (black $2$ hours, red $3$
hours and blue $4$ hours). The quantity $\Delta
P_{e}\equiv(P_{e}-P_{0})/P_{0}$, where $P_{e}$ is the orbital period for non-
zero eccentricity and $P_{0}$ corresponds to the orbital period if the planet
is on a circular orbit. Each panel refers to a different value of the argument
of periastron $\omega$. In each panel the vertical blue line indicates
$e=0.1$.

A non-zero eccentricity not only has a significant impact on the transit
parameters, but it may also result in false positive scenarios that are not
geometrically feasible under the circular orbit approximation as we will
discuss in Section 3.2.

###  2.5 Comparative analysis

To evaluate the effectiveness of the procedure outlined in Sections 2.1, 2.2,
and 2.3, we here show its application to TOI 216.02, a giant warm Jupiter
($R_{P}=8.0^{+3.0}_{-2.0}\,R_{\oplus}$) orbiting its host star in
$34.556_{-0.010}^{+0.014}$ days on a $89.89_{-0.12}^{+0.11}$ degree inclined
and mild eccentric ($e=0.160^{+0.003}_{-0.002}$) orbit (Kipping et al., 2019).
It has been detected by TESS for the first time in Sector 1 at $\approx
1331.26$ BTJD. We then used this single transit measurement to perform a best-
fitting trapezoid model as shown in Fig. 4. The best-trapezoid transit fit
allows us to estimate $t_{T},t_{F}$ and $\delta$. We retrieved the mean
stellar density $\rho_{*}$ from the Gaia DR3 archive (Gaia Collaboration et
al., 2016, 2022). Next, we calculated $P_{1}$ and $P_{2}$ (from equations
(7)-(8)) for each inclination value given by the inequality (10). If either
$P_{1}$ or $P_{2}$ exceeded the threshold $P_{\text{max}}$, we discarded them
as viable solutions. Ultimately, our procedure yielded the functions
$P_{1}(i)$ and $P_{2}(i)$. Fig. 4 depicts the outcomes of our method when
applied to TOI 216.02. We observe that both $P_{1}(i)$ and $P_{2}(i)$ branches
cover a broad range of periods, extending from approximately $32$ days to
around $120$ days, corresponding to orbital inclinations in the range of
$89.5^{\circ}$ to $90.0^{\circ}$. The issue of degeneracy becomes apparent,
resulting in two potential orbital periods for a given inclination. To break
the degeneracy and put constraints on both $P$ and $i$ at the same time, we
computed a preliminary estimate of the orbital period using equation (13)
obtaining $P_{t_{F}}=44\pm 11$ days. This strongly constrains the possible
orbital periods and inclinations. As shown in Fig. 4, the $P_{t_{F}}$ estimate
and its related uncertainty intersect the $P_{1}$ branch breaking the
$P_{1}-P_{2}$ degeneracy. The uncertainty $\Delta P_{t_{F}}$ defines a range
of possible values for the orbital period and inclination. We take the median
of their respective ranges as the best estimate value for $P$ and $i$, while
their uncertainties correspond to the semi-difference of this range. Thus, the
orbital period and inclination ranges are narrower as $\Delta P_{t_{F}}$
decreases. Furthermore, from the best-trapezoid fit, we also obtain an
estimate of the planetary radius which is $R_{p}=11.22\pm 0.81\,R_{\oplus}$.
In summary, our approach yields an orbital period of $P=42.1\pm 8.6$ days and
an orbital inclination of $i=89.6\pm 0.1^{\circ}$. We note that our
overestimation by $\approx 20\%$ from the actual orbital period is expected
since the solution $P_{1}$ is more sensitive to non-zero eccentric orbit as
discussed in Section 2.4. We here stress that the precision of our approach
strongly depends on the quality of the transit fit. Specifically, the
precision of parameters $t_{T}$, $t_{F}$, and $\delta$ directly impacts the
uncertainty associated with $P_{t_{F}}$ as discussed in Appendix A. A more
precise determination of these parameters will result in a smaller uncertainty
on $P_{t_{F}}$ which in turn leads to a narrower range of orbital periods and
inclinations.

Figure 4: Results of our method applied to the giant warm Jupiter TOI 216.02.
Left: the TOI 216.02 transit in Sector 1 alongside the best-fitting trapezoid
model. This model allows us to estimate $T_{1},T_{2},T_{3}$ and $T_{4}$ which
respectively correspond to the first, second, third and four contact in the
no-limb darkening assumption. The total transit duration is
$t_{T}=T_{4}-T_{1}$ while the elapsed time from the second to the third
contact $t_{F}=T_{3}-T_{2}$. Right: the solid and dashed black lines represent
the solutions $P_{2}$ and $P_{1}$ respectively as function of the orbital
inclination $i$. There is a gap between the two functions between around $57$
and $65$ days. The blue solid line represents the value of $P_{t_{F}}$ and the
blue shaded area is its correspondent uncertainty. The range of possible
periods, along with a spectrum of inclinations, is defined by the intersection
between the blue shaded area and $P(i)$.

Subsequently, we applied our methodology to several mono-transit events
identified in the TESS data that were later confirmed as genuine planets
through subsequent observations. The outcomes of our comparative analysis are
presented in Table 1 and shown in Fig. 5. There is a good agreement between
our method and the results from the literature when the assumption of a
circular orbit is met, as seen in the cases of TOI 199 b, TOI 1847 b, TOI 1899
b, TOI 4406 b and TOI 5153 b. However, the method fails to retrieve the
orbital parameters in cases where the assumption of a circular orbit is not
satisfied. This was expected since the circular orbit assumption is the first
and the most relevant hypothesis at the core of our framework. We remark the
only exception in our sample test is TOI 2338 b for which we found a good
agreement despite its high-eccentric orbit. Furthermore, we observed that a
simple trapezoid fit was inadequate for cases with asymmetric transits caused
by high-eccentricity orbits, as exemplified by TOI 5152 b.

TOI | Period (days) | $i(^{\circ})$ | $e$ | Source
---|---|---|---|---
199 b | $104.87236\pm 0.00005$ | $89.81\pm 0.02$ | $0.09^{+0.01}_{-0.02}$ | Hobson et al. 2023
| $107\pm 11$ | $89.63\pm 0.01$ | - | This work
1847 b | $35.45533\pm 0.00019$ | $89.16^{+0.20}_{-0.29}$ | $0.13^{+0.10}_{-0.09}$ | Gill et al. 2020
| $40\pm 18$ | $89.14\pm 0.16$ | - | This work
1899 b | $29.090312^{+0.000036}_{-0.000036}$ | $89.64^{+0.07}_{-0.08}$ | $0.044^{+0.029}_{-0.027}$ | Lin et al. 2023
| $34\pm 9$ | $89.7\pm 0.3$ | - | This work
2180 b | $260.79^{+0.59}_{-0.58}$ | $89.955^{+0.032}_{-0.044}$ | $0.3683\pm 0.0073$ | Dalba et al. 2022
| $1044\pm 158$ | $89.86\pm 0.04$ | - | This work
2338 b | $22.65398\pm 0.00002$ | $89.52\pm 0.02$ | $0.676\pm 0.002$ | Brahm et al. 2023
| $15\pm 6$ | $88.3\pm 0.2$ | - | This work
2589 b | $61.6277\pm 0.00002$ | $89.17\pm 0.01$ | $0.522\pm 0.006$ | Brahm et al. 2023
| $108\pm 22$ | $89.4\pm 0.1$ | - | This work
4127 b | $56.39879\pm 0.00010$ | $89.30^{+0.46}_{-0.60}$ | $0.7471^{+0.0078}_{-0.0086}$ | Gupta et al. 2023
| $17\pm 6$ | $87.92\pm 0.28$ | - | This work
4406 b | $30.08364\pm{+0.00005}$ | $88.44\pm 0.04$ | $0.1^{+0.04}_{-0.05}$ | Brahm et al. 2023
| $25\pm 8$ | $88.6\pm 0.3$ | - | This work
5152 b | $54.18915\pm 0.00015$ | $88.4\pm 0.6$ | $0.432\pm 0.023$ | Ulmer-Moll et al. 2022
| - | - | - | This work
5153 b | $20.33003\pm 0.00007$ | $88.27\pm 0.14$ | $0.0910^{+0.0240}_{-0.0260}$ | Ulmer-Moll et al. 2022
| $22\pm 10$ | $88.27\pm 0.25$ | - | This work
Table 1: Comparison between the results of our method and values from
literature on mono-transit TESS events later confirmed as true planets by
follow-up observations. Figure 5: Comparative analysis of our method applied
on mono-transit TESS signals confirmed as true planets by follow-up
observations. For each planet of the sample test, in the left panel we show
our predicted period against its true value, while in the right panel we
depict our predicted orbital inclination against its true value. Each planet
is colored according to its eccentricity. In both plots, the black dashed line
represents the agreement between our prediction and the literature. We remark
that our method mainly fails when the circular orbit assumption is not
satisfied.

##  3 The sample

Cross-matching mono-transit TESS candidates with the all-sky PLATO Input
Catalog (asPIC; Montalto et al. 2021) can aid in the planning and
prioritization of follow-up observations. If a mono-transit candidate is
associated with an asPIC entry, it indicates the potential future availability
of additional high-precision photometry and asteroseismic data, should a LOP
(long-observing phase; at least two years of continuous coverage) or a SOP
field (short-observing phase; at least two months) be overlapped with it. This
additional information can guide the selection of suitable candidates for
further ground-based observations, such as radial velocity measurements or
atmospheric characterization, increasing the chances of confirming and
studying the exoplanet.

As of May 2023, TESS detected $83$ mono-transit events whose key features are
publicly available in the ExoFOP archive. Out of these $83$ events, $45$ have
been detected by the Science Processing Operations Center pipeline (SPOC,
Jenkins et al. 2016), $24$ by the Quick-look Pipeline (QLP, Huang et al.
2020a, b) and the last $14$ are Community TOIs (Guerrero et al., 2021). TOI
289.01 is the oldest mono-transit event of the sample to be detected (3rd
October 2018), while the latest candidates have been detected on the 2nd
February 2023. However, $23$ of these signals have been ruled out as false
positives or false alarms by the TESS Follow-up Observing Program Working
Group (TFOPWG). We decided to remove them from our analysis to avoid
duplicated work. We also filtered out all the transit events with $S/N<7.3$,
which is a standard threshold value used to refer to a planet as detectable
(see e.g. Sullivan et al., 2015; Bouma et al., 2017; Barclay et al., 2018;
Villanueva et al., 2019). Thus, we selected 57 TESS mono-transit planet
candidates with $S/N\geq 7.3$. Then, we cross-matched this sample with the
asPIC (Montalto et al., 2021) to focus on those mono-transit TESS candidates
orbiting a star that could also be monitored by PLATO in the future. After
performing the cross-matching using topcat we are left with a catalog of $48$
mono-transit events detected by TESS that are included in the asPIC. They are
plotted as labelled red circles in the sky map of Fig. 6 and listed in Table
2.

###  3.1 PLATO LOP fields

It has been already formally announced that PLATO will start its nominal
mission at the end of 2026 by pointing a LOP field in the Southern hemisphere
for at least two continuous years. This field, named LOPS2, is a slightly
modified version of the LOPS1 field identified by Nascimbeni et al. (2022) and
will be described in detail in a forthcoming paper (Nascimbeni et al., in
prep.). Four mono-transit candidates lie within the LOPS2 boundaries (Fig. 7,
right plot): TOI-429 and TOI-2529 in the 12-telescope regions, TOI-2447 and
TOI-2490 in the six-telescope regions. A fifth candidate, TOI-2423, is located
within a few arcminutes from the LOPS2 outer rim, and could be possibly fall
on silicon depending on the actual pointing and co-alignment perfomances of
PLATO.

A second, northern LOP field (LOPN1) has been identified as well by Nascimbeni
et al. (2022). While it is not guaranteed that it will be observed, we did the
same excercise as above, by identifying eight mono-transit targets within its
boundaries (Fig. 7, left plot). One of those, TOI-4585, lies in the
24-telescope region, i.e., where the field of view of all the PLATO normal
cameras (NCAMs) is overlapped.

Figure 6: All-sky Aitoff projection in Galactic coordinates, showing the
footprints of the two LOP fields (LOPS2 and LOPN1; color-coded in blue shades
according to the number of co-pointing cameras), the mono-transit targets
identified in this work (red points with the TOI IDs as labels), the Ecliptic
(yellow line) and the celestial equator (green line). A zoom-in version
centered on the two LOP fields is given in Fig. 7.

Figure 7: As in Fig. 6, but zoomed in on the PLATO long-duration fields LOPN1
(left panel) and LOPS2 (right panel). The latter is the LOP field formally
selected as the first one to be pointed by PLATO in 2027. Projection is
orthographic in Galactic coordinates.

###  3.2 Vetting

The next step in our analysis consists in uniformly vetting the $48$ mono-
transit events in order to produce a clean catalog. Vetting is a crucial step
in the process of confirming the nature of candidate exoplanets. It involves a
rigorous analysis of various observational and statistical factors to
distinguish genuine exoplanets from false positives, which are astrophysical
signals (eclipsing binary stars, stellar spots etc.) or instrumental artefacts
(e.g., jitter noise and momentum dumps) that mimic the signals of transiting
exoplanets (Ciardi et al., 2018). Planet candidates are generally vetted
firstly by an-eye inspection at both the light curve and the pixel level (see
for example Magliano et al. 2023a, b); then if the candidate passes this first
investigation, vetters try to statistically validate it calculating the false
positive probability by means of tools like VESPA (Morton, 2012, 2015) and
TRICERATOPS (Giacalone et al., 2021). Following this procedure for a mono-
transit planet candidate poses inherent challenges due to limited available
data and the associated uncertainties. Insufficient observations and data gaps
hinder a comprehensive understanding of these events, making it difficult to
confidently ascertain the true nature of mono-transit signals. Hence, it is
only more likely for a single event in a photometric survey to be a false
alarm compared to a periodic signal. Moreover, a single measurement may not
provide robust statistical evidence to classify that event as a false
positive. For this reason, in our vetting analysis, we opted for an extremely
cautious approach considering signals as potential planet candidates even if
they exhibited minor flags (e.g., low signal-to-noise ratio detections that
can be challenging to interpret, particularly based on a single measurement
alone). Nevertheless, it is still possible to rule out a candidate if a clear
systematic in the field-of-view (FoV) occurs at the time of the transit. The
presence of a sudden flux change in the background, occurring at or close to
the transit event, can introduce false signals into the light curve,
effectively imitating or distorting the transit signal. Moreover, we also
looked for any centroid offset (CO) which hints the transit could have
originated from a nearby source that falls within the aperture mask used to
extract the target’s light curve. Therefore, we thoroughly examined both the
light curve and the surrounding background flux for each candidate with a
custom pipeline alongside the LATTE web interface 444http://latte-
online.flatironinstitute.org/app(Eisner et al., 2020). Any anomalies, such as
atypical characteristics or discontinuities in the background flux, prompt us
to flag the transit with a potential issue. Lastly, as anticipated in Section
2.4, under certain geometrical configurations ($e>0,180^{\circ}\leq\omega\leq
360^{\circ}$ and $i\neq 90^{\circ}$) an eclipsing binary could present only
the secondary eclipse (Santerne et al., 2013) while the primary falls outside
the line of sight. If the secondary eclipse is shallow enough, it can mimic a
planetary transit leading to a false positive detection (e.g., KOI-419 and
KOI-698 binary system discussed in Santerne et al. 2012). Santerne et al.
(2013) showed that secondary-only eclipsing binary systems produce a V-shaped
transit profile which however could be also caused by a grazing transiting
giant planet. This devious false positive scenario can be ruled out if a
radial velocity follow-up measurement finds a strong variation in anti-phase
with the photometric ephemeris. Since in this work we are just limiting our
analysis to the light curve of the star, we cannot inspect this possibility
but only flag whether the transit exhibits a V-shaped feature.

At the end of the vetting process we passed 38 signals as Planet Candidate
(PC) while ruling out 10 events as False Positive (FP). For each signal, we
also checked whether any other sources fall within the same pixel of the
target by consulting stellar catalogs (Wenger et al., 2000; Gaia Collaboration
et al., 2022). In this case, we flagged the candidate with the acronym FSCP
(Field Star in Central Pixel). This is a common issue when dealing with TESS
due to its low angular resolution ($1\text{px}\approx 21^{\prime\prime}$)
which imposes an inherent limitation on the vetting process. Blended sources
falling within the same pixel can not only contaminate the starlight,
resulting in a shallower transit but can also cause the observed transit.
However, this flag alone is not sufficient to rule out a transiting-like event
as a FP. Nevertheless, it is important to keep this in mind because, as
mentioned in Sect. 2, our method relies on the assumption of no contamination
by other sources. Thus, a FSCP flag indicates the possibility the orbital
period estimates could be revised using a less contaminated light curve (Han &
Brandt, 2023). Table 2 presents a comprehensive list of the $48$ TESS mono-
transit candidates analyzed in this study, including their disposition and
corresponding comments following our thorough vetting analysis. After removing
the false positive signals, our final dataset contains 38 good mono-transit
TESS candidates.

TOI | Disposition | Comments | TOI | Disposition | Comments | TOI | Disposition | Comments
---|---|---|---|---|---|---|---|---
225.01 | FP | Background event | 2134.02 | PC | High SNR | 4321.01 | PC | LowSNR
289.01 | PC | High SNR | 2263.01 | PC | FSCP, lowSNR | 4326.01 | PC | LowSNR
429.01 | PC |  | 2270.01 | FP | NT in SAP light curve | 4465.01 | PC |
772.02 | PC | Multiple system | 2298.01 | PC | FSCP, lowSNR | 4585.01 | PC | FSCP, low SNR
1301.02 | FP | NT† in SAP light curve | 2300.03 | PC | High SNR, Long duration | 4862.01 | PC | FSCP
1386.01 | PC | FSCP | 2423.01 | PC |  | 5523.01 | PC | Long duration
1563.01 | PC | FSCP | 2433.01 | PC | FSCP, low SNR | 5563.01 | PC |
1581.01 | FP | V-shaped transit | 2435.01 | FP | NT in SAP light curve | 5564.01 | PC | FSCP
1772.02 | PC | High SNR | 2436.01 | PC | Low SNR | 5626.01 | PC |
1812.01 | PC | High SNR | 2447.01 | PC | High SNR | 5719.01 | PC | LowSNR
1835.02 | PC | Low SNR | 2456.01 | FP | Background event | 5745.01 | PC |
1892.01 | PC | Low SNR,FSCP | 2472.01 | PC | Low SNR | 5870.01 | PC |
2007.01 | FP | CO | 2490.01 | PC | High SNR | 5893.01 | PC |
2009.01 | FP | FSCP, V-shaped transit | 2529.01 | PC | High SNR | 5967.01 | PC | LowSNR
2085.01 | PC | FSCP | 2534.01 | PC | FSCP | 5968.01 | PC | LowSNR
2087.01 | FP | NT in SAP light curve | 3724.01 | PC |  | 6092.01 | FP | FSCP, CO

  * †

No Transit.

Table 2: Summary vetting report of the 48 monotransit TESS planet candidates
orbiting stars that will be monitored by PLATO. We ruled out 10 of them as
false positives (FP) while keeping 38 as planet candidate (PC). The acronyms
CO and FSCP stand for "Centroid Offset" and "Field Star in Central Pixel"
respectively.

###  3.3 Stellar parameters

We obtained the stellar properties of the 38 host stars by utilizing ARIADNE
(Vines & Jenkins, 2022) which models the stellar Spectral Energy Distribution
(SED). ARIADNE integrates six atmospheric model grids which have been
previously convolved with various filters’ across multiple bandpasses. The
utilized models include Phoenix v2 (Husser et al., 2013), BT-Settl, BT-Cond
(Allard et al., 2011), BT-NextGen (Hauschildt et al., 1999; Allard et al.,
2011), Kurucz (1993) and Castelli & Kurucz (2003). ARIADNE uses the (Dynamic)
Nested Sampling algorithm (Skilling, 2004, 2006; Higson et al., 2019) via
dynesty (Speagle, 2020) for each of the aforementioned models. It conducts
Bayesian Model Averaging over all fitted models to derive the final set of
parameters for a given star. Initially, ARIADNE seeks broadband photometry
through astroquery (Astropy Collaboration et al., 2013, 2018, 2022), accessing
MAST and VizieR archives to query catalogs such as Tycho-2 (Høg et al., 2000),
ASCC (Kharchenko, 2001), 2MASS (Skrutskie et al., 2006), GLIMPSE (Churchwell
et al., 2009), ALL-WISE (Wright et al., 2010), GALEX (Bianchi et al., 2011),
APASS DR9 (Henden & Munari, 2014), SDSS DR12 (Alam et al., 2015), Strömgren
Photometric Catalog (Hauck & Mermilliod, 1998; Paunzen, 2015), Pan-STARRS1
(Chambers et al., 2016) and Gaia DR2 and DR3 (Gaia Collaboration et al., 2018,
2023). Model selection is based on an initial effective temperature estimation
from Gaia DR2, as Kurucz (1993) and Castelli & Kurucz (2003) exhibit poor
performance for stars with $T_{\text{eff}}<4000K$. Priors for $T_{\text{eff}}$
and [Fe/H] are derived from the RAVE catalog (Shank et al., 2022), log $g$
uses a uniform prior ranging from 3.5 to 6. Radius and distance priors follow
ARIADNE’s defaults, while interstellar extinction $A_{v}$ relies on a prior
drawn from the Bayestar 3D dustmaps (Green et al., 2019) or a uniform prior
from $0$ to the maximum line-of-sight extinction from the SFD dustmaps
(Schlegel et al., 1998; Schlafly & Finkbeiner, 2011) when the former is
unavailable.

In Table 3 we report the stellar parameters used in this work. In Fig. 8 we
show the distribution of the 38 stars within our sample according to their
mean stellar density and their temperature as retrieved by ARIADNE. The peak
of the two distributions is around $\rho_{\odot}$ and $T_{\odot}$ respectively
as illustrated in Fig. 8.

TOI | $V_{\text{mag}}{\textsuperscript{\textdagger}}$ | $T_{*}(K)$ | $R_{*}(R_{\odot})$ | $\log g$ | $M_{*}(M_{\odot})$ | $\rho_{*}(\rho_{\odot})$
---|---|---|---|---|---|---
289 | $12.05\pm 0.32$ | $5676^{+34}_{-53}$ | $1.76^{+0.015}_{-0.034}$ | $3.95^{+0.10}_{-0.08}$ | $1.01^{+0.04}_{-0.06}$ | $0.18^{+0.01}_{-0.02}$
429 | $10.91\pm 0.12$ | $5245^{+25}_{-67}$ | $0.946^{+0.020}_{-0.017}$ | $4.48^{+0.07}_{-0.1}$ | $0.99^{+0.06}_{-0.06}$ | $1.17^{+0.14}_{-0.13}$
772 | $11.68\pm 0.19$ | $4760^{+20}_{-25}$ | $0.937^{+0.012}_{-0.014}$ | $4.57^{+0.03}_{-0.05}$ | $1.19^{+0.04}_{-0.05}$ | $1.45^{+0.10}_{-0.13}$
1386 | $10.67\pm 0.16$ | $5940^{+115}_{-70}$ | $1.023^{+0.009}_{-0.013}$ | $4.44^{+0.07}_{-0.06}$ | $1.05^{+0.04}_{-0.04}$ | $0.98^{+0.06}_{-0.07}$
1563 | $10.46\pm 0.15$ | $4662^{+30}_{-17}$ | $0.712^{+0.010}_{-0.007}$ | $4.60^{+0.03}_{-0.03}$ | $0.74^{+0.02}_{-0.02}$ | $2.0^{+0.1}_{-0.1}$
1772 | $9.96\pm 0.14$ | $5683^{+18}_{-18}$ | $0.958^{+0.007}_{-0.006}$ | $4.49^{+0.08}_{-0.08}$ | $1.04^{+0.03}_{-0.03}$ | $1.18^{+0.06}_{-0.06}$
1812 | $12.46\pm 0.12$ | $4918^{+31}_{-21}$ | $0.758^{+0.007}_{-0.009}$ | $4.58^{+0.07}_{-0.03}$ | $0.8^{+0.03}_{-0.02}$ | $1.84^{+0.12}_{-0.11}$
1835 | $8.38\pm 0.12$ | $5310^{+25}_{-26}$ | $0.783^{+0.009}_{-0.008}$ | $4.58^{+0.04}_{-0.04}$ | $0.85^{+0.03}_{-0.02}$ | $1.77^{+0.12}_{-0.10}$
1892 | $10.61\pm 0.12$ | $5427^{+21}_{-19}$ | $0.797^{+0.007}_{-0.007}$ | $4.58^{+0.07}_{-0.06}$ | $0.88^{+0.03}_{-0.03}$ | $1.74^{+0.11}_{-0.11}$
2085 | $10.62\pm 0.14$ | $5232^{+67}_{-48}$ | $0.792^{+0.01}_{-0.013}$ | $4.57^{+0.06}_{-0.06}$ | $0.85^{+0.03}_{-0.04}$ | $1.71^{+0.13}_{-0.16}$
2134 | $8.93\pm 0.15$ | $4483^{+27}_{-31}$ | $0.744^{+0.012}_{-0.012}$ | $4.61^{+0.03}_{-0.04}$ | $0.82^{+0.03}_{-0.03}$ | $1.99^{+0.17}_{-0.17}$
2263 | $10.34\pm 0.15$ | $5718^{+69}_{-49}$ | $0.853^{+0.007}_{-0.009}$ | $4.53^{+0.04}_{-0.04}$ | $0.9^{+0.02}_{-0.03}$ | $1.45^{+0.07}_{-0.09}$
2298 | $12.44\pm 0.17$ | $4351^{+18}_{-45}$ | $0.629^{+0.01}_{-0.008}$ | $4.67^{+0.03}_{-0.03}$ | $0.68^{+0.03}_{-0.02}$ | $2.73^{+0.25}_{-0.18}$
2300 | $12.58\pm 0.25$ | $5577^{+118}_{-94}$ | $0.793^{+0.009}_{-0.011}$ | $4.58^{+0.04}_{-0.03}$ | $0.87^{+0.03}_{-0.03}$ | $1.74^{+0.12}_{-0.13}$
2423 | $10.04\pm 0.12$ | $6003^{+30}_{-25}$ | $1.194^{+0.012}_{-0.012}$ | $4.37^{+0.08}_{-0.08}$ | $1.22^{+0.05}_{-0.05}$ | $0.72^{+0.05}_{-0.05}$
2433 | $14.56\pm 0.32$ | $3238^{+51}_{-31}$ | $0.324^{+0.007}_{-0.010}$ | $4.99^{+0.13}_{-0.07}$ | $0.37^{+0.03}_{-0.03}$ | $11.0^{+1.7}_{-1.7}$
2436 | $9.02\pm 0.15$ | $6230^{+37}_{-29}$ | $1.238^{+0.014}_{-0.012}$ | $4.3^{+0.16}_{-0.16}$ | $1.12^{+0.07}_{-0.06}$ | $0.59^{+0.06}_{-0.05}$
2447 | $10.59\pm 0.12$ | $5821^{+41}_{-32}$ | $1.037^{+0.013}_{-0.013}$ | $4.44^{+0.10}_{-0.07}$ | $1.08^{+0.05}_{-0.04}$ | $0.97^{+0.08}_{-0.07}$
2472 | $9.52\pm 0.14$ | $5829^{+35}_{-30}$ | $1.686^{+0.014}_{-0.017}$ | $4.06^{+0.08}_{-0.09}$ | $1.19^{+0.04}_{-0.05}$ | $0.25^{+0.01}_{-0.02}$
2490 | $11.95\pm 0.15$ | $5372^{+30}_{-64}$ | $1.183^{+0.048}_{-0.010}$ | $4.37^{+0.16}_{-0.05}$ | $1.20^{+0.08}_{-0.08}$ | $0.7^{+0.1}_{-0.1}$
2529 | $11.4\pm 0.22$ | $5684^{+147}_{-110}$ | $1.759^{+0.023}_{-0.03}$ | $4.06^{+0.16}_{-0.14}$ | $1.3^{+0.09}_{-0.09}$ | $0.24^{+0.03}_{-0.03}$
2534 | $11.23\pm 0.09$ | $6465^{+33}_{-35}$ | $1.616^{+0.019}_{-0.017}$ | $4.11^{+0.08}_{-0.07}$ | $1.23^{+0.05}_{-0.05}$ | $0.29^{+0.02}_{-0.02}$
3724 | $12.31\pm 0.60$ | $5927^{+139}_{-94}$ | $1.455^{+0.019}_{-0.018}$ | $4.13^{+0.06}_{-0.07}$ | $1.04^{+0.04}_{-0.04}$ | $0.34^{+0.03}_{-0.03}$
4321 | $7.78\pm 0.13$ | $6175^{+34}_{-28}$ | $2.418^{+0.026}_{-0.024}$ | $3.87^{+0.08}_{-0.09}$ | $1.58^{+0.07}_{-0.07}$ | $0.11^{+0.01}_{-0.01}$
4326 | $8.25\pm 0.14$ | $5940^{+29}_{-24}$ | $1.123^{+0.012}_{-0.012}$ | $4.3^{+0.07}_{-0.06}$ | $0.92^{+0.03}_{-0.03}$ | $0.65^{+0.04}_{-0.04}$
4465 | $10.5\pm 0.17$ | $5783^{+111}_{-77}$ | $0.986^{+0.009}_{-0.012}$ | $4.42^{+0.05}_{-0.05}$ | $0.93^{+0.03}_{-0.03}$ | $0.97^{+0.06}_{-0.07}$
4585 | $10.22\pm 0.15$ | $5871^{+30}_{-30}$ | $1.29^{+0.009}_{-0.011}$ | $4.25^{+0.06}_{-0.05}$ | $1.08^{+0.03}_{-0.03}$ | $0.5^{+0.02}_{-0.03}$
4862 | $12.38\pm 0.16$ | $5368^{+47}_{-38}$ | $0.915^{+0.015}_{-0.011}$ | $4.48^{+0.09}_{-0.08}$ | $0.92^{+0.05}_{-0.04}$ | $1.2^{+0.12}_{-0.10}$
5523 | $15.64\pm 0.33$ | $3679^{+1}_{-33}$ | $0.613^{+0.001}_{-0.008}$ | $4.69^{+0.01}_{-0.01}$ | $0.67^{+0.02}_{-0.02}$ | $2.91^{+0.2}_{-0.2}$
5563 | $11.41\pm 0.15$ | $5130^{+14}_{-68}$ | $1.243^{+0.065}_{-0.002}$ | $4.44^{+0.11}_{-0.01}$ | $1.56^{+0.20}_{-0.01}$ | $0.81^{+0.23}_{-0.01}$
5564 | $12.36\pm 0.36$ | $5787^{+156}_{-101}$ | $0.882^{+0.014}_{-0.016}$ | $4.48^{+0.09}_{-0.07}$ | $0.86^{+0.04}_{-0.04}$ | $1.25^{+0.12}_{-0.13}$
5626 | $10.42\pm 0.15$ | $5684^{+32}_{-23}$ | $0.977^{+0.009}_{-0.011}$ | $4.46^{+0.07}_{-0.05}$ | $1.01^{+0.03}_{-0.03}$ | $1.08^{+0.06}_{-0.07}$
5719 | $10.13\pm 0.14$ | $6296^{+32}_{-31}$ | $1.107^{+0.012}_{-0.011}$ | $4.39^{+0.09}_{-0.10}$ | $1.1^{+0.05}_{-0.05}$ | $0.81^{+0.06}_{-0.06}$
5745 | $10.62\pm 0.13$ | $4948^{+25}_{-15}$ | $0.301^{+0.005}_{-0.007}$ | $4.51^{+0.02}_{-0.02}$ | $0.11^{+0.01}_{-0.01}$ | $4.03^{+0.20}_{-0.65}$
5870 | $11.29\pm 0.24$ | $5827^{+40}_{-7}$ | $1.678^{+0.011}_{-0.022}$ | $4.16^{+0.03}_{-0.01}$ | $1.49^{+0.04}_{-0.04}$ | $0.31^{+0.02}_{-0.02}$
5893 | $12.08\pm 0.22$ | $5853^{+51}_{-79}$ | $1.161^{+0.015}_{-0.022}$ | $4.42^{+0.07}_{-0.09}$ | $1.3^{+0.05}_{-0.08}$ | $0.83^{+0.06}_{-0.10}$
5967 | $10.73\pm 0.15$ | $4883^{+55}_{-47}$ | $1.103^{+0.012}_{-0.015}$ | $4.38^{+0.02}_{-0.05}$ | $1.07^{+0.03}_{-0.04}$ | $0.8^{+0.05}_{-0.06}$
5968 | $10.06\pm 0.13$ | $5997^{+58}_{-56}$ | $1.042^{+0.011}_{-0.009}$ | $4.39^{+0.06}_{-0.06}$ | $0.97^{+0.03}_{-0.03}$ | $0.86^{+0.05}_{-0.05}$

  * †

Visual apparent magnitude in the Johnson-Cousin system.

Table 3: Stellar parameters of the 38 stars hosting mono-transit planet
candidates used in this work. Figure 8: Upper panel: distribution of the mean
stellar density $\rho_{*}$ of the 38 stars in our sample; lower panel:
distribution of the effective temperature of the 38 stars in our sample. The
vertical dashed black line represents the effective temperature of the Sun,
i.e. $5772\,K$.

###  3.4 Notes on false positives

The mono-transit planet candidate TOI 225.01 detected by the SPOC turned out
to be a false alarm due to the passage of a solar-system object (maybe an
asteroid) in the field-of-view of TESS. TOI 225 is an early K star observed by
TESS in sectors 2 and 29 in 2018 and 2020, respectively. The SPOC pipeline
detected a mono-transit event at $\approx 1355.119$ BTJD with $S/N\approx
22.75$. When inspecting whether any background anomalies could have led to a
false-alarm detection we found that is the case, as shown in Figs. 9 and 10.
TOI 225.01 is not an isolated occurrence within our dataset; strikingly, we
have observed a similar event in the case of TOI 2456.01. Thus we removed
these mono-transit events as they originated from non-planetary scenarios.

Figure 9: The background flux analysis of planet candidate TOI 225.01 orbiting
TOI 225 (or TIC 47525799) observed in sector 2 by TESS. The upper panel
presents an enlarged view of the simple aperture photometry flux (depicted in
red) during the transit time-frame. The lower panel illustrates the simple
aperture photometry background flux (shown in blue) inside the aperture used
to calculate the SAP flux, spanning the same time window (Thompson et al.,
2016; Lightkurve Collaboration et al., 2018). The vertical black line
demarcates the timing of the transit events. We clearly notice a sudden spike
in the background flux at the exact time of the transit. Hence, we conclude
this is a false alarm signal. Figure 10: This plot shows the original CCD
pixel TESS observations of TOI 225 from which the light curves of Figure 9
have been extracted. The red contour represents the aperture mask used by the
SPOC pipeline. The left panel shows the FoV $\approx 2.3$ hours before the
transit; the middle panel shows the FoV at the time of the transit; the right
panel shows the FoV $\approx 2.3$ hours after the transit. The orange circle
in each panel tracks the presence of a bright object passing in front of the
camera. In the transit time-frame, the brightness of this object contaminates
the pixels within the aperture mask used to extract the light curve thus
mimicking a transit during the pre-processing of the signal.

In addition, we excluded the planet candidate TOI 6092.01 from our sample due
to a distinct centroid offset observed during the transit event. As
illustrated in Fig. 11, the Pixel Level light curve exhibits a transit-like
feature not on the pixel corresponding to our target, but rather on a
neighboring pixel. Consequently, we ruled out TOI 6092.01 as a false positive
due to the presence of a nearby transiting system within the aperture mask.
Furthermore, the transit’s V-shaped profile and significant depth suggest the
possibility of it being a nearby eclipsing binary system. For the same reason,
we also removed TOI 2007.01 from the list.

Figure 11: The Pixel Level light curve plot for planet candidate TOI 6092.01.
It shows the light curve for each individual pixel of the corresponding target
pixel file in a 3-day time-window centered at the time of transit $T0$. Clear
eclipses are seen in the bottom right pixels near the lower edge of the
aperture mask (red contour). Hence, the source of the transit is not TOI
6092.01.

Despite V-shaped transits might be caused by grazing transiting exoplanets,
they most likely are eclipsing binary systems. In addition, the best-fitting
trapezoid model is inadequate for capturing the characteristics of a V-shaped
transit. Due to their pronounced V-shaped transit profiles, we opted to
exclude TOI 1581.01 and 2009.01.

Finally, we also removed from our sample TOI 1301.02, TOI 2087.01, TOI
2270.01, and TOI 2435.01 for a common reason. In each case, we observed a
transit-like feature in the Pre-search Data Conditioning Simple Aperture
Photometry (PDCSAP) light curve that disappeared upon verifying the Simple
Aperture Photometry (SAP) light curve (Kinemuchi et al., 2012). Furthermore,
while the SAP light curves do not exhibit any significant transit, the PDCSAP
light curves display a noticeable correlation with background flux modulation
as shown in Fig. 12. Consequently, we eliminated these suspicious cases from
our sample to ensure we were working with the cleanest dataset possible.

Figure 12: TOI 2435.01 is a mono-transit event detected by the SPOC pipeline
in the TESS sector 16 around TOI 2435 (or TIC 298164705). In top panel the
PDCSAP light curve shows a transit at $\approx 1743.3$ BTJD, which completely
vanishes in the SAP light curve in the bottom panel. In the middle panel, the
background flux exhibits peculiar ringing modulations. We not only noticed the
transit in the PDCSAP light curve occurred in correspondence with a dim in the
background flux but also it has a fictitious spike at $\approx 1744.5$. We
suspect that TOI 2435.01 could be a result of an aggressive pre-processing
procedure that created an artificial transit-like feature in the PDCSAP light
curve.

##  4 Results

Among the $48$ mono-transit TESS candidates, 38 have successfully passed our
rigorous vetting procedure, thus having a promising likelihood of confirmation
by ground- or space-based missions. We applied the technique discussed in the
Sect. 2 to our sample of 38 mono-transit candidates in order to retrieve an
estimate of their orbital period. While effective for high SNR transits, the
simple trapezoid model is not intended for low SNR transit signals which are
dominated by systematic noise sources (e.g., stellar activity, instrumental
effects, flux contamination) and usually affect smaller planets orbiting
larger stars. Gaussian Process (GP) modeling offers a viable alternative to
the best-fitting trapezoid model due to its capability in capturing the effect
of stellar variability during transit observations (Rasmussen & Williams,
2005). However, performing a comprehensive fitting of all the transit events
using GP processes is computationally demanding and time-consuming. Hence, as
a pragmatic approach, in this work we have chosen to exclude low SNR signals
from our sample that cannot be accurately fitted by a simple trapezoid model.
Table 4 presents the outcomes of our approach applied to the subset of $30$
mono-transit events from our sample that could be effectively fitted using a
trapezoid model.

TOI | $R_{p}(R_{\oplus})$ | Period (days) | $i(^{\circ})$
---|---|---|---
289.01 | $16.42\pm 0.73$ | $28\pm 10$ | $87.7\pm 0.4$
429.01 | $9.03\pm 0.22$ | $415\pm 94$ | $89.83\pm 0.01$
772.02 | $7.16\pm 0.21$ | $55\pm 28$ | $89.40\pm 0.07$
1386.01 | $6.83\pm 0.20$ | $180\pm 74$ | $89.61\pm 0.08$
1563.01 | $5.25\pm 0.11$ | $242\pm 57$ | $89.78\pm 0.02$
1772.02 | $6.15\pm 0.12$ | $120\pm 44$ | $89.55\pm 0.07$
1812.01 | $9.18\pm 0.18$ | $208\pm 48$ | $89.76\pm 0.01$
2085.01 | $3.72\pm 0.15$ | $551\pm 454$ | $89.9\pm 0.1$
2134.02 | $8.02\pm 0.17$ | $101\pm 19$ | $89.63\pm 0.02$
2300.03 | $9.06\pm 0.19$ | $2736\pm 424$ | $89.96\pm 0.02$
2423.01 | $14.03\pm 0.24$ | $28\pm 4$ | $88.58\pm 0.08$
2433.01 | $2.21\pm 0.27$ | $583\pm 456$ | $89.95\pm 0.05$
2436.01 | $2.66\pm 0.15$ | $1302\pm 1234$ | $89.8\pm 0.2$
2447.01 | $9.63\pm 0.19$ | $84\pm 19$ | $89.8\pm 0.2$
2490.01 | $11.7\pm 0.4$ | $108\pm 42$ | $89.53\pm 0.03$
2529.01 | $11.55\pm 0.36$ | $43\pm 20$ | $89.4\pm 0.6$
2534.01 | $12.40\pm 0.25$ | $39\pm 15$ | $89.5\pm 0.5$
3724.01 | $14.14\pm 0.52$ | $45\pm 22$ | $88.7\pm 0.2$
4326.01 | $2.71\pm 0.13$ | $142\pm 134$ | $89.2\pm 0.7$
4465.01 | $11.83\pm 0.19$ | $253\pm 50$ | $89.88\pm 0.05$
4862.01 | $10.20\pm 0.27$ | $113\pm 31$ | $89.54\pm 0.05$
5523.01 | $11.11\pm 0.23$ | $742\pm 73$ | $89.98\pm 0.02$
5563.01 | $8.76\pm 0.79$ | $78\pm 29$ | $89.2\pm 0.2$
5564.01 | $11.77\pm 0.42$ | $262\pm 107$ | $89.76\pm 0.03$
5626.01 | $9.86\pm 0.20$ | $9\pm 2$ | $87.4\pm 0.3$
5719.01 | $2.59\pm 0.21$ | $1667\pm 1586$ | $89.8\pm 0.2$
5745.01 | $1.07\pm 0.08$ | $853\pm 773$ | $89.91\pm 0.09$
5893.01 | $10.44\pm 0.36$ | $104\pm 49$ | $89.4\pm 0.1$
5967.01 | $5.57\pm 0.28$ | $368\pm 176$ | $89.71\pm 0.08$
5968.01 | $3.23\pm 0.17$ | $573\pm 533$ | $89.75\pm 0.24$
Table 4: The results of our methodology detailed in Sect. 2 on 30 TOIs that we
were able to fit with a simple trapezoid model.

We show the distribution of these 30 mono-transit planet candidates within the
$(P,R)$ diagram in Fig. 13. What stands out in this figure is that our mono-
transit planet candidates fill a region of the $(P,R)$ space-parameter that is
mostly devoid of TESS confirmed exoplanets. The majority of candidates in our
sample consist of warm and cold gas giants. As already discussed in Sect. 1,
this was expected due to the TESS observing strategy. We also observe that the
uncertainty over the orbital period is higher for the sub-Neptunes candidates
in our sample, highlighting that our method loses its power for low SNR mono-
transit events.

Figure 13: Distribution of the 30 mono-transit candidates within the $(P,R)$
diagram alongside their uncertainties over the orbital period and planetary
radius. The cyan dots represent all the confirmed planets discovered by TESS.

For each of the 30 mono-transit candidates, we checked whether they reside
within the habitable zone of their host stars. Specifically, we computed the
circumstellar temperate zone boundaries based on the runaway greenhouse effect
and the maximum greenhouse effect, as defined by Kopparapu et al. (2013) for
different values of the planetary albedo. We found that TOI 2134.02, TOI
1812.01, TOI 5564.01 and TOI 4465.01 potentially orbit within the
circumstellar temperate zone when a high albedo value is assumed, as
illustrated in Fig. 14. However, when considering lower albedo values similar
to those found in the Solar System (e.g., 0.3), among the previous candidates
only TOI 1812.01 is still found orbiting the circumstellar temperate zone. In
addition, we also found that TOI 429.01, TOI 1563.01, TOI 5967.01 and TOI
5968.01 potentially lie within the circumstellar temperate zone. We highlight
that out of the candidates potentially in the circumstellar temperate zone of
their host star, TOI 429.01 is the only one that is additionally in the field
of view of LOPS2.

Figure 14: Distribution of the $30$ mono-transit TESS candidates (yellow dots)
alongside the conservative habitable zone (green shaded area) of their host
star for different values of the bond albedo $A$ ($A=0.3$ in the upper panel,
$A=0.7$ in the bottom panel). The red and blue dashed lines represent
respectively the runaway greenhouse effect and the maximum greenhouse effect
edges (Kopparapu et al., 2013).

###  4.1 Interesting targets

With a planetary radius of $R_{p}=16.42\pm 0.73\,R_{\oplus}$, TOI 289.01 is
the largest candidate in our sample. It revolves around its star in $28\pm 10$
days. Even though there is a higher probability for very large transiting
objects to be brown dwarfs rather than planets (i.e., the FP probability
increases towards a bigger radius, see for example Magliano et al. 2023b), TOI
289.01 did not show any clue of an eclipsing binary scenario during our
photometric analysis. However, since TOI 289 is contained in the Gaia Non-
Single stars catalogue, Tarrants & Mendes (2023) recently confirmed it as
being a spectroscopic binary with a $P=224.706\pm 0.906$ days. Using the
orbital parameters obtained by Holl et al. (2023) from Gaia data, Tarrants &
Mendes (2023) found that the single transit detected by TESS closely matches
with the secondary eclipse of the system. Thus, the companion of TOI 289 is a
$1.4R_{J}$ object on an eccentric orbit $e=0.35848\pm 0.08394$.

TOI 2300.03 is found to be the furthest candidate orbiting its star in
$2736\pm 424$ days at the equilibrium temperature of $\approx 125\,{\rm K}$,
assuming a null planetary albedo. What makes this candidate particularly
interesting is that it could belong to a multi-planetary system composed of a
candidate super-Earth (TOI 2300.01) and a candidate Neptune (TOI 2300.02),
both on close-in orbits, and a much further away Jupiter-like companion (TOI
2300.03). This system is a valuable planetary system to test models of
planetary evolution. Finally, we remark that if TOI 2300.03 is on a high-
eccentric orbit, its orbital period could be drastically reduced as we have
seen for TOI 2180 b in Sect. 2.5.

A similar scenario holds for TOI 5523.01, a Jupiter-like candidate that
completes a full orbit around its star (K2-43) in $742\pm 73$ days. Notably,
K2-43 has been monitored during the first K2 campaign in 2014 (Crossfield et
al., 2016; Dressing et al., 2017; Hedges et al., 2019; Kruse et al., 2019). We
inspected the K2 light curve to check whether a previous transit event was
registered in the past but the time spanning by the K2 observations did not
overlap with the predicted transit of TOI 5523.01. It is important to mention
that K2-43 is already recognized as the host of two confirmed exoplanets,
K2-43 b and K2-43 c, on short-period orbits. Therefore, if TOI 5523.01 is
confirmed as a bonafide planet through subsequent observations as K2-43 d, it
would become the third planet in this planetary system.

The sub-Jovians TOI 772.02 and TOI 1772.02 2 also could form two-planet
systems with either another sub-Jovian companion candidate (TOI 772.01) or a
mini-Neptune candidate (TOI 1772.01).

Rescigno et al. (2023) have recently confirmed that the candidate TOI 2134.02
is a sub-Saturn planet orbiting its K-dwarf star in $95.50^{+0.36}_{-0.25}$
days on a highly eccentric orbit ($e=0.64^{+0.05}_{-0.06}$). We here stress
that our $101\pm 19$ days prediction, which comes from the $P_{2}$-branch,
deviates from the actual orbital planet by $6\%$ due to its eccentric orbit.
The obtained overestimate is consistent with our findings discussed in Sect.
2.4.

TOI 1812.01 is a $9.18\,R_{\oplus}$ candidate on a $(208\pm 48)$ days orbit
that could form a three-planet system with its candidate companions, the mini-
Neptune TOI 1812.02 and the Neptune-sized TOI 1812.03 which orbit their host
star in $\approx 11$ and $\approx 48$ days, respectively.

The giant TOI 5564.01 ($R_{p}=11.77\pm 0.42$) orbiting its star in $262\pm
107$ days could be the outer companion of the planet candidate TOI 5564.02, a
sub-Jupiter planet on a $45$ days orbit.

Similarly, TOI 5968.01 ($R_{p}=3.23\pm 0.17$) completing an orbital in $573$
days ($\approx 93\%$ uncertainty) could form a two-planet system with the
mini-Neptune planet candidate TOI 5968.02 which completes its orbit in
$\approx 9$ days.

TOI 5626.01 has been detected by the SPOC pipeline at $\approx 2660.04$ BTJD
in Sector 49 which spans a time-window of $[2637.47,2664.32]$ BTJD with a gap
in $[2651.42,2655.83]$ BTJD. This is a puzzling candidate because we found it
orbits its variable host star in $9\pm 2$ days, thus it should have been
transited at least once in the time-frame $[2637.47,2651.42]$ BTJD. We
actually found a possible additional transiting feature at $2643.96$ BTJD
which unfortunately occurred just before a TESS momentum dump making the
measurement unreliable. However, it is worth noting that TOI 5626 was also
observed by TESS in Sector 23, during which no evident transiting-like feature
was observed. These circumstances raise doubts regarding the true nature of
TOI 5626.01. Additionally, although TOI 5626.01 successfully passed our
vetting procedure, it is important to consider the possibility that it may
possess a non-negligible eccentricity. Such eccentricity can significantly
impact the outcomes and interpretations derived from our approach.

A similar case is TOI 2423.01, detected by the SPOC pipeline in Sector 29 at
$\approx 2097.06$ BTJD. Our analysis determined an orbital period of $28\pm 4$
days. However, despite further observations of the star in Sectors 2, 4, 30,
and 31, no additional analogous transits were observed. Hence, we are quite
suspicious about the true nature of TOI 2423.01 but it could also be revolving
around its star on an eccentric orbit.

In a recent work, Hawthorn et al. (2023) found one additional transit for both
TOI 2490.01 and TOI 2529.01, by performing a systematic duotransit search.
Following the same methodology discussed in Osborn et al. (2022) they found
the most probable orbital period for TOI 2490.01 to be
$P=38.01^{+14.0}_{-15.0}$ days, while $P=65.0^{+24.0}_{-5.0}$ days for TOI
2529.01.

###  4.2 Follow-up efforts

We have also investigated the feasibility of follow-up observing campaigns on
these targets using the CHaracterising ExOPlanets Satellite (CHEOPS, Broeg et
al. 2013; Fortier et al. 2014) telescope. CHEOPS is an ESA space-based mission
launched on December 18, 2019, whose main goal is to perform ultrahigh
precision photometry on bright stars already known to host planets.

Thus we checked whether the $30$ mono-transit planet candidates in our sample
would be amenable targets of a CHEOPS’ observer programme following the steps
of Cooke et al. (2020). Due to its specific orbit around the Earth,
observations from CHEOPS experience interruptions along its trajectory leading
to different sky coverage depending on the requested time per CHEOPS orbit
(observing efficiency). The optimal requested time per orbit is given by the
best trade-off between maximizing both the sky and temporal coverage. In this
work, we opted for a minimum observing time of $59$ minutes per orbit
(equivalent to approximately $60\%$ of the observing efficiency) as it matches
quite well with the typical ingress/egress duration of our candidates. We plot
the distribution of the $30$ mono-transit planet candidates over the CHEOPS
sky coverage using an observation efficiency of $60\%$ in Fig. 15. As we can
see from this chart, almost $36\%$ of our final sample falls within the
uncovered sky region. Moreover, the most promising stars that CHEOPS would
monitor for $50\div 60$ days in $1$ year are TOI $772$, TOI $4465$, TOI
$5893$, TOI $523$ and TOI $4862$. As found by Cooke et al. (2020), we also
highlight that none of our TESS mono-transit candidates lie in the region of
the sky CHEOPS monitors for the longest.

Despite the feasibility to schedule photometric follow-up campaigns, we
acknowledge that RV follow-up observations could significantly accelerate the
determination of orbital periods for mono-transit candidates. In comparison to
photometric observations, RV measurements may give the advantage of acquiring
a data point for a single candidate in a single observing night. Thus, the
addition of RV follow-up programmes would significantly improve the efficiency
of our mono-transit candidate validation procedure.

Figure 15: The $30$ TESS mono-transit planet candidates in our sample
(represented by red star symbols) on the CHEOPS sky coverage map using an
observing efficiency of $\approx 60\%$. Each star that falls in the contours
would be monitored by CHEOPS for a minimum of $59$ minutes per orbit. The
color bar represents the total observing time in days using CHEOPS over a
period of 1 year.

##  5 Conclusions

In this study, we have investigated the mono-transit events observed by the
TESS mission around stars that will be further monitored by the PLATO mission.
We developed an analytic approach that enables us to estimate the orbital
period and inclination of a mono-transit signal by exploiting only the shape
of the single transit. The effectiveness of the method proposed in this work
depends on both the photometric accuracy of the measurements and the
uncertainty associated with the stellar parameters under the assumption of a
circular orbit. We applied the method on a sample of $38$ uniformly vetted
TESS mono-transit planet candidates orbiting stars that could be potentially
PLATO targets, out of the initial 48 detected signals. We constrained the
orbital period and inclination for $30$ objects in our sample. These planet
candidates encompass a region of the space of parameters still unexplored by
the TESS mission. Among these targets, at least four will be observed on the
first LOP field, LOPS2, starting from the end of 2026. Since our vetting
procedure is based only on the single observed transit, these candidates are
still away from a robust validation, thus follow-up observations are needed to
validate and properly characterize them. We also checked whether the
candidates in our sample could be amenable targets for a CHEOPS observing
campaign.

Through the cross-matching of TESS data with the PLATO Input Catalog, we have
demonstrated a potential synergy between the two missions. As TESS continues
to operate and collect data, its discoveries set the stage for further
investigations and pave the way for future missions, such as PLATO. The
availability of PLATO observations will play a crucial role in validating the
planetary nature of mono-transit candidates and discerning them from false
positives, such as blended stars or background eclipsing binaries.

As the PLATO mission continues to observe the stars monitored by TESS, we
anticipate significant advancements in the confirmation, characterization, and
statistical analysis of mono-transit events.

## Acknowledgements

We would like to thank the anonymous referee for their careful reading and
their insightful comments which contributed to improve the quality of this
manuscript.

This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.

We acknowledge the use of the Exoplanet Follow-up Observation Program website,
which is operated by the California Institute of Technology, under contract
with the National Aeronautics and Space Administration under the Exoplanet
Exploration Program. The TFOP is led by the Smithsonian Astrophysical
Observatory (SAO), in coordination with MIT, as part of the TESS Science
Office.

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. JSJ greatfully acknowledges
support by FONDECYT grant 1201371 and from the ANID BASAL projects ACE210002
and FB210003. JIV acknowledges support of CONICYT-PFCHA/Doctorado
Nacional-21191829.

Software: topcat (Taylor, 2005), lightkurve (Lightkurve Collaboration et al.,
2018), ARIADNE (Vines & Jenkins, 2022).

## Data Availability

The data underlying this article are taken from the literature.

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##  Appendix A Formal solutions to polynomial equation

In this Appendix we show the key steps to find the analytic solutions to
equation (6) that here we write again

$\alpha P^{2}-\beta P^{2/3}+\gamma=0.$ (18)

Initially, we introduce the notation $P^{2/3}\equiv y$ to conveniently reframe
equation (18) in the subsequent manner,

$\alpha y^{3}-\beta y+\gamma=0\quad\Leftrightarrow\quad y^{3}+py+q=0,$ (19)

where we set $p\equiv-\beta/\alpha$ and $q\equiv\gamma/\alpha.$ The equation
(19) is a cubic equation whose solutions can be found using the so called
Cardano’s formula. We first define the discriminant $\Delta$ as

$\Delta\equiv\dfrac{q^{2}}{4}+\dfrac{p^{3}}{27}=\dfrac{1}{\alpha^{2}}\left(\dfrac{\gamma^{2}}{4}-\dfrac{\beta^{3}}{27\alpha}\right).$
(20)

Depending on the value of the discriminant $\Delta$, we have the following
cases.

#### Case 1 - $\Delta>0$.

In this case we have three different solutions to equation (19):

$\displaystyle y_{1}$ $\displaystyle=u+v,$ (21) $\displaystyle y_{2}$
$\displaystyle=u\left(-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right)+v\left(-\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}\right),$
(22) $\displaystyle y_{3}$
$\displaystyle=u\left(-\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}\right)+v\left(-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right),$
(23)

where

$u=\left(-\dfrac{q}{2}+\sqrt{\Delta}\right)^{1/3}\quad\text{and}\quad
v=\left(-\dfrac{q}{2}-\sqrt{\Delta}\right)^{1/3}\,.$ (24)

We note that among the solutions $y_{1}$, $y_{2}$, and $y_{3}$, only $y_{1}$
is real and has a physical meaning. Nevertheless, as $p\equiv-\beta/\alpha<0$,
it becomes evident that both $u$ and $v$ assume negative values. Consequently,
$y_{1}<0$, but since the physical variable of interest is the orbital period
denoted by $P\equiv y^{3/2}$, we can conclude that equation (18) does not
yield a viable physical solution in this particular scenario.

#### Case 2 - $\Delta<0$.

In this scenario, it is necessary to compute the true anomaly $\theta$ of the
complex number $\left(-q/2+i\sqrt{-\Delta}\right)$, which can be determined
using the following expression:

$\theta=\pi-\arctan\left(2\sqrt{-\Delta}/q\right).$ (25)

Then there are three solutions to equation (19),

$\displaystyle y_{1}$
$\displaystyle=2\sqrt{\dfrac{-p}{3}}\cos\left(\dfrac{\theta}{3}\right),$ (26)
$\displaystyle y_{2}$
$\displaystyle=2\sqrt{\dfrac{-p}{3}}\cos\left(\dfrac{\theta+2\pi}{3}\right),$
(27) $\displaystyle y_{3}$
$\displaystyle=2\sqrt{\dfrac{-p}{3}}\cos\left(\dfrac{\theta+4\pi}{3}\right).$
(28)

As $\arctan(2\sqrt{-\Delta}/q)\in\left[0,\pi/2\right]$, it follows that the
true anomaly $\theta\in\left[\pi/2,\pi\right]$. Consequently, we can readily
observe that only solutions $y_{1}$ and $y_{3}$ yield positive values, while
$y_{2}<0$. By substituting the expression of $\theta$ into $y_{1}$ and
$y_{3}$, we can derive the following results:

$\displaystyle y_{1}$
$\displaystyle=2\sqrt{\dfrac{-p}{3}}\cos\left(\dfrac{\pi-\arctan\left(2\sqrt{-\Delta}/q\right)}{3}\right),$
(29) $\displaystyle y_{3}$
$\displaystyle=2\sqrt{\dfrac{-p}{3}}\cos\left(\dfrac{5\pi-\arctan\left(2\sqrt{-\Delta}/q\right)}{3}\right).$
(30)

Then one can retrieve the formula for the orbital periods $P_{1}=y_{3}^{3/2}$
and $P_{2}=y_{1}^{3/2}$ shown in the Sect.2.1, thus

$\displaystyle P_{1}$
$\displaystyle=2^{3/2}\left(\dfrac{\beta}{3\alpha}\right)^{3/4}\cos^{3/2}\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]},$
(31) $\displaystyle P_{2}$
$\displaystyle=2^{3/2}\left(\dfrac{\beta}{3\alpha}\right)^{3/4}\cos^{3/2}\Bigg{[}\dfrac{\pi}{3}-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\,.$
(32)

The uncertainties $\Delta P_{1}$ and $\Delta P_{2}$ associated respectively to
$P_{1}$ and $P_{2}$ are given by

$\displaystyle\Delta P_{1}$
$\displaystyle=P_{1}(\sigma_{\alpha,1}+\sigma_{\beta,1}+\sigma_{\gamma,1}),$
(33) $\displaystyle\Delta P_{2}$
$\displaystyle=P_{2}(\sigma_{\alpha,2}+\sigma_{\beta,2}+\sigma_{\gamma,2}),$
(34)

where we posed

$\displaystyle\sigma_{\alpha,1}$
$\displaystyle\equiv\dfrac{\Delta\alpha}{4\alpha}\Biggl{\\{}3+\dfrac{\sqrt{\Delta}}{\Delta+1}\Bigg{|}\tan\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\Bigg{|}\Biggr{\\}},$
(35) $\displaystyle\sigma_{\alpha,2}$
$\displaystyle\equiv\dfrac{\Delta\alpha}{4\alpha}\Biggl{\\{}3+\dfrac{\sqrt{\Delta}}{\Delta+1}\Bigg{|}\tan\Bigg{[}\dfrac{\pi}{3}-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\Bigg{|}\Biggr{\\}},$
(36) $\displaystyle\sigma_{\beta,1}$
$\displaystyle\equiv\dfrac{3\Delta\beta}{4\beta}\Biggl{\\{}1+\dfrac{\sqrt{\Delta}}{\Delta+1}\Bigg{|}\tan\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\Bigg{|}\Biggr{\\}},$
(37) $\displaystyle\sigma_{\beta,2}$
$\displaystyle\equiv\dfrac{3\Delta\beta}{4\beta}\Biggl{\\{}1+\dfrac{\sqrt{\Delta}}{\Delta+1}\Bigg{|}\tan\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\Bigg{|}\Biggr{\\}},$
(38) $\displaystyle\sigma_{\gamma,1}$
$\displaystyle\equiv\dfrac{\Delta\gamma}{2\gamma}\dfrac{\sqrt{\Delta}}{\Delta+1}\Bigg{|}\tan\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\Bigg{|},$
(39) $\displaystyle\sigma_{\gamma,2}$
$\displaystyle\equiv\dfrac{\Delta\gamma}{2\gamma}\dfrac{\sqrt{\Delta}}{\Delta+1}\Bigg{|}\tan\Bigg{[}\dfrac{\pi}{3}-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\Bigg{|},$
(40)

with $\Delta\alpha,\Delta\beta$ and $\Delta\gamma$ the uncertainties on the
parameters $\alpha,\beta$ and $\gamma$ respectively.

#### Case 3 - $\Delta=0$.

We address this particular scenario to ensure comprehensive coverage; however,
it is important to note that in practical situations, we would not encounter
an exact value of $\Delta=0$. Nonetheless, even in this hypothetical case,
equation (19) still yields three solutions.

$\displaystyle y_{1}$ $\displaystyle=-2\left(\dfrac{q}{2}\right)^{1/3},$ (42)
$\displaystyle y_{2}$ $\displaystyle=y_{3}=\left(\dfrac{q}{2}\right)^{1/3}.$
(43)

We rule out the negative solution $y_{1}$ and so equation (18) admits a single
solution

$P=y_{2}^{3/2}=\sqrt{\dfrac{q}{2}}=\sqrt{\dfrac{\gamma}{2\alpha}}\,.$ (44)

##  Appendix B Equations and approximations

In this Appendix we make a summary of the all equations and approximations we
used in this work alongside, where available, their analytic solutions.

Approximation | Simplified Equation | Analytic solutions
---|---|---
$\begin{aligned} &\text{Circular orbit}\\\ &\text{No flux contamination}\\\ &\text{No limb-darkening}\end{aligned}$ | $t_{T}=\dfrac{P}{\pi}\arcsin{\left(\dfrac{R_{*}}{a}\left[\dfrac{[1+\sqrt{\delta}]^{2}-[a\cos i/R_{*}]^{2}}{1-\cos^{2}i}\right]^{1/2}\right)}$ | -
$M_{p}\ll M_{*}$ | $t_{T}=\dfrac{P}{\pi}\arcsin{\left[\left(\dfrac{\Lambda^{-2/3}_{*}P^{-4/3}(1+\sqrt{\delta})^{2}-\cos^{2}(i)}{1-\cos^{2}(i)}\right)^{1/2}\right]}$ | -
$\begin{aligned} &\pi t_{T}/P\ll 1\\\ &\cos(i)\ll 1\end{aligned}$ | $\cos^{2}(i)\,P^{2}-\Lambda^{-2/3}_{*}(1+\sqrt{\delta})^{2}\,P^{2/3}+\pi^{2}\,t_{T}^{2}=0$ | $\begin{aligned} P_{1}=&2^{3/2}\left(\dfrac{\beta}{3\alpha}\right)^{3/4}\cos^{3/2}\Bigg{[}\dfrac{5}{3}\pi-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\\\ P_{2}=&2^{3/2}\left(\dfrac{\beta}{3\alpha}\right)^{3/4}\cos^{3/2}\Bigg{[}\dfrac{\pi}{3}-\dfrac{1}{3}\arctan\left(\sqrt{\dfrac{4\beta^{3}}{27\alpha\gamma^{2}}-1}\right)\Bigg{]}\end{aligned}$
$i=90^{\circ}$ | $\Lambda^{-2/3}_{*}(1+\sqrt{\delta})^{2}\,P^{2/3}-\pi^{2}\,t_{T}^{2}=0$ | $P_{\text{EO}}=\left(\dfrac{\pi\,t_{T}}{1+\sqrt{\delta}}\right)^{3}\Lambda_{*}$
Approximation | Simplified Equation | Analytic solutions
$\begin{aligned} &\text{Circular orbit}\\\ &\text{No flux contamination}\\\ &\text{No limb-darkening}\\\ &M_{p}\ll M_{*}\\\ \end{aligned}$ | $\dfrac{M_{*}}{R_{*}^{3}}=\left(\dfrac{4\pi^{2}}{P^{2}G}\right)\Biggl{\\{}\dfrac{(1+\sqrt{\delta})^{2}-b^{2}(1-\sin^{2}(t_{T}\pi/P)}{\sin^{2}(t_{T}\pi/P)}\Biggr{\\}}^{3/2}$ | -
$\pi t_{T}/P\ll 1$ | $\dfrac{M_{*}}{R_{*}^{3}}=\dfrac{32\delta^{3/4}P}{G\pi(t_{T}^{2}-t_{F}^{2})^{3/2}}$ | $P_{t_{F}}=\dfrac{G\pi^{2}}{24}\rho_{*}\dfrac{(t_{T}^{2}-t_{F}^{2})^{3/2}}{\delta^{3/4}}$
Table 5: Summary of equations used in this work and their analytic solutions.