# Understanding the Planetary Formation and Evolution in Star
Clusters(UPiC)-I: Evidence of Hot Giant Exoplanets Formation Timescales

Yuan-Zhe Dai  School of Astronomy and Space Science, Nanjing University, 163
Xianlin Avenue, Nanjing, 210023, People’s Republic of China  Key Laboratory of
Modern Astronomy and Astrophysics, Ministry of Education, Nanjing, 210023,
People’s Republic of China  Hui-Gen Liu  School of Astronomy and Space
Science, Nanjing University, 163 Xianlin Avenue, Nanjing, 210023, People’s
Republic of China  Key Laboratory of Modern Astronomy and Astrophysics,
Ministry of Education, Nanjing, 210023, People’s Republic of China  Jia-Yi
Yang  School of Astronomy and Space Science, Nanjing University, 163 Xianlin
Avenue, Nanjing, 210023, People’s Republic of China  Key Laboratory of Modern
Astronomy and Astrophysics, Ministry of Education, Nanjing, 210023, People’s
Republic of China  Ji-Lin Zhou  School of Astronomy and Space Science, Nanjing
University, 163 Xianlin Avenue, Nanjing, 210023, People’s Republic of China
Key Laboratory of Modern Astronomy and Astrophysics, Ministry of Education,
Nanjing, 210023, People’s Republic of China

###### Abstract

Planets in young star clusters could shed light on planet formation and
evolution since star clusters can provide accurate age estimation. However,
the number of transiting planets detected in clusters was only $\sim 30$, too
small for statistical analysis. Thanks to the unprecedented high-precision
astrometric data provided by Gaia DR2 and Gaia DR3, many new Open
Clusters(OCs) and comoving groups have been identified. The UPiC project aims
to find observational evidence and interpret how planet form and evolve in
cluster environments. In this work, we cross-match the stellar catalogs of new
OCs and comoving groups with confirmed planets and candidates. We carefully
remove false positives and obtain the biggest catalog of planets in star
clusters up to now, which consists of 73 confirmed planets and 84 planet
candidates. After age validation, we obtain the radius–age diagram of these
planets/candidates. We find an increment of the fraction of Hot Jupiters(HJs)
around 100 Myr and attribute the increment to the flyby-induced high-e
migration in star clusters. An additional small bump of the fraction of HJs
after 1 Gyr is detected, which indicates the formation timescale of HJ around
field stars is much larger than that in star clusters. Thus, stellar
environments play important roles in the formation of HJs. The hot-Neptune
desert occurs around 100 Myr in our sample. A combination of photoevaporation
and high-e migration may sculpt the hot-Neptune desert in clusters.

exoplanet,

††journal: AJ††software: astropy (Astropy Collaboration et al., 2013),
matplotlib (Hunter, 2007), pandas Wes McKinney (2010).

##  1 Introduction

Open Clusters(OCs) in the Milky Way are the collection of stars formed from
the same molecular cloud and gravitationally bound together, Thus, sharing
similar specific characteristics, e.g. age, distance, reddening, metal
abundance, etc. OCs provide an ideal laboratory for studying star formation
and evolution. Recent studies based on Kepler data show that nearly 50% of
stars host planets (Zhu & Dong, 2021). Since most stars form in clusters (Lada
& Lada, 2003), many exoplanets are formed in cluster environments. Then the
majority of stars will eventually become field stars as clusters are
dissociated. Detecting exoplanets in OCs can provide an ideal sample for
studying planet formation and evolution.

The first planet in OCs, $\epsilon$ tau b, was detected by Sato et al. (2007)
via Radial Velocity. Kepler-66b and Kepler-67b are the first cluster planets
discovered by transit (Meibom et al., 2013). Thanks to Kepler/K2 and TESS,
tens of planets in clusters have been discovered, and the number is growing.
There are several programs focusing on planets in star clusters, especially
young exoplanets. Zodiacal Exoplanets In Time (ZEIT) collaboration uses K2
data to monitor young open clusters and associations in the ecliptic plane and
found planets in the Hyades, Praesepe, Upper Sco, and Taurus (Mann et al.
2016a, b, 2017, 2018; Rizzuto et al. 2017; Gaidos et al. 2017; Vanderburg et
al. 2018; Rizzuto et al. 2018). With the help of enable extensive follow-up
observations, The TESS Hunt for Young and Maturing Exoplanets (THYME)
collaboration has reported on planets in Upper Sco (Rizzuto et al., 2020), the
Tuc-Hor association (Newton et al., 2019), the Ursa Major moving group (Mann
et al., 2020), and the Pisces Eridanus stream (Newton et al., 2021). Bouma et
al. (2019, 2020) begun a Cluster Difference Imaging Photometric Survey(CDIPS)
to discover giant transiting planets with known ages, and to provide light
curves suitable for studies in stellar astrophysics. Nardiello et al. 2019,
2020; Nardiello 2020; Nardiello et al. 2021 use a PSF-based Approach to TESS
High-quality data of star clusters (PATHOS) and find 90 planet candidates.

Hitherto, there are many surveys and projects focusing on the young planets in
clusters, but the number of reported planets in clusters is limited and is not
enough to support statistical work, $\sim 30$ according to Nardiello et al.
(2021). To enlarge the number of planets in open clusters, both expanding the
number of cluster members and identifying new clusters are available in Gaia
Era.

The Gaia DR2/EDR3 catalog (Gaia Collaboration et al., 2018; Lindegren et al.,
2021) presents more than 1.3 billion stars with unprecedented high-precision
astrometric and photometric data, greatly improving the reliability of stellar
membership determination and characterization of a large sample of stellar
groups including star clusters, association, and other comoving groups. The
recent analysis of Gaia Data has greatly expanded our knowledge of stellar
groups (e.g., Cantat-Gaudin et al., 2018; Kounkel & Covey, 2019; Kerr et al.,
2021). In previous knowledge, a star cluster is a set of stars that are
gravitationally bound to one another (Portegies Zwart et al., 2010). However,
the recent discovery of stars in diffuse regions reminds us that we need to
extend the original definition of star clusters. Because these stars in
diffuse regions, i.e. not gravitational bound, are proved to have the same age
as core cluster members through analyses of color-absolute magnitude diagrams
(CAMDs; Kounkel & Covey, 2019; Meingast & Alves, 2019; Bouma et al., 2021). On
top of that, these stars in diffuse regions also share a similar distribution
with core cluster members in stellar rotation periods (Bouma et al., 2021) and
chemical abundances (Arancibia-Silva et al., 2020; Hawkins et al., 2020).
Therefore, these stars in diffuse regions are probably coeval. In this series
of papers, we extend the definition of open clusters to those stars in diffuse
regions, i.e. comoving groups, diffuse streams, tidal tails, etc., not only
the core cluster members. Thus, the number of stars in open clusters can be
extremely enlarged.

There are many previous works that identified new OCs in the Milky Way using
different algorithms, e.g. Cantat-Gaudin et al. (2018) applied the UPMASK
algorithm to select cluster members and provided an updated catalog of 1229
OCs; Cantat-Gaudin & Anders (2020) found 582 new cluster candidates located in
the low galactic latitude area using an algorithm named Density-Based Spatial
Clustering of Applications with Noise(DBSCAN), and etc. Using Hierarchical
Density-Based Spatial Clustering of Applications with Noise (HDBSCAN, McInnes
et al., 2017), Kounkel & Covey (2019) systematically clustered Gaia DR2 data
within 1 kpc and identified 1640 populations containing a total of 288,370
stars. In their recent work, Kounkel et al. (2020) (hereafter K2020), they
extended the distance from 1 to 3 kpc and identified 8292 comoving groups
consisting of 987,376 stars.

Utilizing the enlarged stellar population in clusters, the UPiC (Understanding
Planetary Formation and Evolution in Star Clusters) project focuses on the
planets in open clusters, including association and other comoving groups. We
are trying to find evidence that how planets form and evolve in cluster
environments. Lots of works have shown that both dynamical (Spurzem et al.,
2009; Liu et al., 2013; Cai et al., 2017; Hamers & Tremaine, 2017; Rodet et
al., 2021; Li et al., 2023) and radiation (Johnstone et al., 1998; Matsuyama
et al., 2003; Dai et al., 2018; Winter et al., 2018) environments in clusters
can influence the planet’s formation and evolution. As the initial work of
UPiC, this paper collects the largest transiting planet sample in open
clusters and aims to analyze the correlation between the planetary radius and
cluster ages, which is crucial for planet formation timescales. For example,
(Szabó & Kiss, 2011; Beaugé & Nesvorný, 2013; Mazeh et al., 2016) discovered
the hot Neptune desert in planetary mass-period and radius-period
distribution. The boundaries of the desert can be explained by
photoevaporation and high-e migration Owen & Lai (2018). These two mechanisms
have different timescales. Thus, the age of the planets can distinguish
different mechanisms and help us understand the time-evolution of planet
radius.

This paper is arranged as follows. In section 2, we describe methods,
including data collection, age validation, and sample cut. In section 3, we
use these refined data to study planet radius – age diagram and estimate the
evolution of three different planet populations. In section 4, we discuss how
the statistical results constrain the planet formation mechanisms, including
high-e migration and photoevaporation. Last, we summarize and discuss our
major conclusions in section 5.

##  2 Catalog of Transiting Planet in OCs

###  2.1 Data collection

There are many works that use Gaia data to identify new OCs. If we combine all
the catalogs of star clusters, we can definitely get more OCs, so as the stars
in OCs. However, data selection criteria and clustering algorithms vary in
different works, which will increase the inhomogeneity of the combined
catalog. Therefore, to maximize the number of stars in OCs and make the sample
as homogeneous as possible, we adopt the catalog from K2020, the largest
catalog of stars with age estimations in comoving groups.

K2020 identified 8292 comoving groups within 3 kpc and galactic latitude
$|b|<30^{\circ}$ by applying the unsupervised machine learning algorithm
HDBSCAN on Gaia DR2’s 5D data. We use the stellar catalog of K2020 to cross-
match with the host stars of confirmed transiting planets and planet
candidates. In this section, we use planets/candidates from Kepler, K2, and
TESS. Since we are concerned about the radius of planets, we do not consider
planets detected via the Radial Velocity method. The following subsections
will briefly introduce how we select the planets and cut the planet sample to
exclude some observation biases.

Table 1: Planets in clusters

Plname | $R_{\rm p}$ | Period | OName | Group | Gaia DR2 | Age | Validation | $T_{\rm eff}$ | logg | Stmass | Flag
---|---|---|---|---|---|---|---|---|---|---|---
| ($\rm R_{\oplus}$) | (days) |  |  |  | (Myr) |  | (K) |  | (M⊙) |
TOI 520.01 | 1.49${}^{+0.66}_{-0.66}$ | 0.524 |  $\cdots$ | Group 95 | 5576476552334683520 | 30.2${}^{+5.3}_{-4.5}$ | NO | 7450 | 4.34 | 1.66 | TESS
TOI 626.01 | 19.74${}^{+0.66}_{-0.66}$ | 4.40 |  $\cdots$ | Group 449 | 5617241426979996800 | 195${}^{+80}_{-57}$ | NO | 8489 | 4.03 | 2.11 | TESS
TOI 2453.01 | 3.02${}^{+0.20}_{-0.20}$ | 4.44 | Hyades | Group 1004 | 3295485490907597696 | 646${}^{+113}_{-96}$ | NO | 3609 | 4.73 | 0.50 | TESS
TOI 2519.01 | 2.29${}^{+0.20}_{-0.20}$ | 6.96 | Columba | Group 208 | 2924619634745251712 | 263${}^{+109}_{-77}$ | NO | 4742 | 4.57 | 0.76 | TESS
TOI 2640.01 | 7.39${}^{+0.50}_{-0.50}$ | 0.911 | IC 2602 | Group 92 | 5404579488593432576 | 45${}^{+14}_{-11}$ | NO | 2999 | 4.95 | 0.25 | TESS
TOI 2646.01 | 8.10 | 0.313 | NGC 2516 | Group 613 | 5288535107223500928 | 145${}^{+37}_{-30}$ | NO | 5202 | 4.53 | 0.88 | TESS
TOI 2822.01 | 11.53${}^{+0.72}_{-0.72}$ | 2.88 |  $\cdots$ | Group 5076 | 5597777288033556480 | 537${}^{+94}_{-80}$ | NO | 6086 | 4.04 | 1.14 | TESS
TOI 3077.01 | 13.67${}^{+0.72}_{-0.72}$ | 6.36 |  $\cdots$ | Group 3176 | 5307513536932390272 | 186${}^{+130}_{-77}$ | NO | 7689 | 4.06 | 1.80 | TESS
TOI 3335.01 | 11.60${}^{+0.70}_{-0.70}$ | 3.61 |  $\cdots$ | Group 550 | 5903623451060661504 | 214${}^{+75}_{-55}$ | NO | 6071 | 4.04 | 1.13 | TESS
TOI 1097.02 | 12.7${}^{+1.0}_{-1.0}$ | 2.27 |  $\cdots$ | Group 1502 | 6637496339607744768 | 3020${}^{+447}_{-390}$ | NO | 5568 | 3.89 | 0.98 | TESS
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$

Note. — Due to the space limitation, we only show the first 10 rows of the
table. Here, “OName” shows the name of the parental cluster, “Group” shows the
corresponding group number in K2020, “Flag” shows the name of the mission that
detects the planets, and “Validation” shows whether this planet/candidate has
the convincing age estimation (i.e. “YES”, “NO”, and “Excluded”). The machine-
readable table can be obtained in the following link:
https://github.com/astrodyz/UPiC

####  2.1.1 K2020 cross-matching with confirmed exoplanets

The number of confirmed planets from the NASA exoplanets
archive111https://exoplanetarchive.ipac.caltech.edu is 5347 up to now
(05/2023). After cross-matching with K2020, we find 76 confirmed planets in
clusters. To study the planet size-age distribution, we select 51 transiting
planets with known planet radii.

The properties of planets and their host stars, e.g. planet radius, orbital
periods, effective temperature, surface gravity, and stellar mass are adopted
from the table of NASA exoplanets archive. We adopt the age of the host stars
from K2020 temporarily, which is completed. The adopted ages from K2020 will
be validated via comparison in subsection 2.2.

####  2.1.2 K2020 cross-matching with KOIs

The number of KOIs from Kepler DR25 (Thompson et al., 2018) is 8445, including
confirmed planets and candidates. After cross-matching with K2020, we find 98
KOIs in clusters. Since some of these KOIs may be False Positives, such as
eclipsing binaries in the background of the targets, or physically bound to
them which can mimic the photometric signal of a transiting planet. Then, we
exclude the sources flagged as False Positive. There are 26 confirmed planets
and 17 planet candidates in clusters for Kepler sources. Here, we use the
Gaia-Kepler Stellar Properties Catalog (Berger et al., 2020) to update and
obtain the accurate and precise properties of these 43 selected KOIs and their
host stars. The age information is also adopted from K2020.

####  2.1.3 K2020 cross-matching with K2

For K2 sources, we use the catalog of K2 including both confirmed planets and
candidates from the NASA exoplanet archive to cross-match with K2020. We get
only 25 matching sources. After excluding 9 candidates flagged as “FALSE
POSITIVE”, we obtain 9 confirmed planets and 7 planet candidates with planet
radius measurements. The properties of planets and planets’ host stars are
taken from the NASA exoplanet archive. The age information is taken from
K2020.

####  2.1.4 K2020 cross-matching with TOIs

Up to 05/2023, there are 6586 TOIs detected by TESS. After cross-matching with
K2020, there are 116 TOIs left. There are many False Positives in TOIs (Gan et
al., 2023). Thus, we select the TOIs carefully. Firstly, we exclude some
sources flagged as “FA”, “APC”, and “FP”, which means false alarm, ambiguous
planet candidate, and false positive respectively. After exclusion, 68 TOIs
are left. Besides, based on publicly available observational notes on
ExoFOP222https://exofop.ipac.caltech.edu/tess/, we also remove some TOIs with
comments like “centroid offset333The light from nearby eclipsing binaries
within 1’ may pollute the aperture and cause transit-like signals on the
target light curve, especially in a crowded field like open clusters”,
“V-shaped”, “Likely eclipsing binary(EB)”, and “odd-even”. For example, the
comment of TOI 1376.01 is “centroid offset on TIC 190743999 in spoc-s56”. So
after removing these TOIs, we finally select 42 TOIs, i.e. 12 confirmed
planets and 30 candidates. The properties of these 42 selected TOIs and their
host stars are taken from the table of TOIs. The age information is taken from
K2020.

####  2.1.5 other sources

K2020 focuses on the stars with galactic latitude $|b|<30^{\circ}$. Actually,
there are many other clusters out of such range, and so do the planets in
clusters. We add the planets and planet candidates in the PATHOS project to
include more planets at higher galactic latitudes. Table 6 of PATHOS - IV
(Nardiello et al., 2021), provides 33 confirmed planets in clusters. Although
11 confirmed planets are repeated selected planets in section 2.1.1, 14 of
them are with $|b|>30^{\circ}$, and 8 of them are uncross-matched with K2020.
Additionally, the PATHOS project has found 90 planet candidates in clusters,
which are not included in TOI completely. After cross-matching with K2020, we
get the age information of 40 planet candidates with $|b|<30^{\circ}$. Here,
we do not include the candidates of the PATHOS project with $|b|>30^{\circ}$,
because they do not have the age measurements. Nardiello et al. (2021) provide
false positive probabilities for the PATHOS candidates. We remove 8 sources
with high false positive probabilities. Most of them are likely eclipsing
binaries. Thus, we add 32 PATHOS candidates. The stellar properties are taken
from the TESS Input Catalog v8.0 (TIC-8, Stassun et al., 2019), e.g. stellar
mass, stellar effective temperature, and surface gravity.

####  2.1.6 The catalog of transiting planets in star clusters

After the cross-matching, we check the catalog of transiting planets in
clusters and exclude the repeated planets. Finally, there are 73 confirmed
planets and 84 planet candidates in 86 clusters. Table 1 shows all these
planets in 133 planetary systems. Planets detected by different missions are
flagged as “Kepler”, “K2”, “TESS”, and “other”. Here, “other” means sources
detected by other facilities, e.g. CoRoT (Léger et al., 2009) and ground-based
telescopes.

This is the largest catalog of the planets and planet candidates in star
clusters. Due to space constraints, we only list 10 sources in Table 1. The
whole table can be downloaded from the web: https://github.com/astrodyz/UPiC

Figure 1: The comparison of age estimation from different catalogs. Panel (a)
uses data from the cross-matching catalog of K2020 and He2022. Panel (b) uses
data from K2020 and the NASA exoplanet archive. Gray shadows are 1-sigma and
3-sigma areas considering the average uncertainty of age estimation of
comoving groups within 1kpc in K2020. A black solid diagonal line is plotted
for reference.

###  2.2 Age validation

Since we focus on the age-size distribution of planets in clusters, the
accuracy, and precision of age measurements are essential. In this section, we
will compare the age measurement of stars from He et al. (2022) (hereafter
He2020), K2020, and the table of the NASA exoplanet archive to illustrate
whether the age estimation in K2020 is robust. Kounkel et al. (2020) adopted a
neural network called Auriga to robustly estimate the ages of the individual
groups they identified. The uncertainty of log(Age) of comoving groups within
1 kpc in K2020 is similar to Kounkel & Covey (2019), i.e. $\sim$ 0.15 dex. In
He2022, they use the isochrone fitting to derive the ages of 886 nearby
clusters and candidates within 1.2 kpc.

Although Kounkel et al. (2020) discuss the contamination and demonstrate that
the vast majority of the age are well consistent with the results of isochrone
fitting, the age estimation in K2020 may still have some systematic biases
compared to He2022. On top of that, the difference in identifying clusters may
also influence the final age estimation. K2020 used unsupervised machine
learning HDBSCAN to identify clusters, while He2022 used DBSCAN. HDBSCAN has a
better performance on the data with different density structures than DBSCAN,
i.e. prefer to reveal more fine structures. Therefore, both the age estimation
and the cluster membership identification will lead to systematic biases.

Then, we cross-matched the catalog of K2020 and He2022 to present whether the
bias of age estimation in K2020 is non-negligible compared to other age
sources. In panel (a) of Figure 1, red hollow dots are the 36
planets/candidates in 22 clusters that both have the age estimation in K2020
and He2022. Only one planet candidate has a large difference of age
estimation, out of the 3 sigma limit, i.e. the PATHOS 64 in King-6. Therefore,
we assume that the majority of the age estimation in K2020 is relatively
robust. To validate the ages of this cluster, we refer to the ages from other
previous works. In K2020, the estimation of the age of King-6 is
204${}^{+91}_{-63}$ Myr which is consistent with the previous result of Ann et
al. (2002)(250$\pm$50 Myr), while in He2022, the age of King-6 is 44 Myr
without uncertainty. Thus, we adopt 204${}^{+91}_{-63}$ Myr as the final age
of King-6.

Additionally, we also compare the age of the K2020 and the NASA exoplanet
archives. In panel (b) of Figure 1, there are 29 planet host stars(44 planets)
in our catalog with both ages from K2020 and NASA exoplanet archives. Eleven
host stars are out of the 3 sigma limit in the age measurement, i.e. CoRoT-22,
HATS-47, HD 110113, KELT-20, Kepler-1062, Kepler-1118, Kepler-1502,
Kepler-411, Kepler-968, TOI-1937 A, and TOI-4145 A. All the ages of these
planets’ host stars from NASA exoplanet archives are much larger than the ages
from K2020. E.g. the age of CoRoT-22 is 3.3$\pm$2.0 Gyr in Moutou et al.
(2014), while in K2020 is 339$\pm$ 100 Myr; the age of HATS-47 in Hardegree-
Ullman et al. (2020) is 8.10${}^{+2.90}_{-4.30}$ Gyr, while in K2020 is
589${}^{+243}_{-171}$ Myr; the age HD 110113 Osborn et al. (2021) is 4.0$\pm$
0.5 Gyr, while in K2020 is 645${}^{+245}_{-178}$ Myr.

Because the individual age estimation of these stars depends on the models and
methods, which are inhomogeneous in the NASA exoplanet archive. Strictly, we
use different ways to validate the ages of the eleven host stars.

Firstly, some stars may have several($\geq$ 3) age measurements, which we can
evaluate through majority voting. For instance, the age estimation of KELT-20
is $\leq$0.6 Gyr according to Lund et al. (2017). However, according to Talens
et al. (2018), KELT-20 is 200 ${}^{+100}_{-50}$ Myr, which is consistent with
K2020’s result, i.e. 166${}^{+58}_{-43}$ Myr. The age estimation of
Kepler-1118 from Morton et al. (2016)(4.07 Gyr) is nearly ten times of that in
K2020(490${}^{+155}_{-118}$ Myr). We strengthen that Morton et al. set the age
prior to 1-15 Gyr, which means the stellar age in their catalog is
artificially larger than 1 Gyr. As the host cluster of Kepler-1118, NGC 6866
has the age of 705$\pm$ 140 Myr from Janes et al. (2014)). Therefore, we adopt
the age in K2020 for KELT-20 and Kepler-1118.

Secondly, if stars do not have independent and consistent age measurements, we
estimate the age via the gyrochronological relation. Kepler-411 is a special
case that three age measurements are significantly different. Sun et al.
(2019) use gyrochronological relation (Barnes (2007)) and estimate an age of
212 $\pm$ 31 Myr, K2020 estimate an age of 794${}^{+302}_{-219}$ Myr, and
Morton et al. (2016) estimate an age of 2.69${}^{+2.67}_{-1.10}$ Gyr.) To
validate the age of Kepler-411, we adopt the new gyrochronological relation,
which includes the empirical mass dependence of the rotational coupling
timescale, developed by Spada & Lanzafame (2020). The stellar rotation period
of Kepler 411, 10.4 days, is taken from McQuillan et al. (2013). Then, we
estimate an age of 770 Myr for Kepler 411 system which is consistent with the
age estimation in K2020. Thus, we adopt 794${}^{+302}_{-219}$ Myr as the age
of Kepler-411.

The same with Kepler-411, we use the rotating period and stellar mass, and
obtain the ages of HATS-47, HD 110113, Kepler-1062, and Kepler 968, through
the new gyrochronological relation, i.e. $<0.1$ Gyr, $\sim$ 3 Gyr, $\sim$ 1.3
Gyr, and $\sim$ 0.7 Gyr, respectively. These age estimations are significantly
different from K2020 (i.e. 589${}^{+243}_{-171}$ Myr, 645${}^{+245}_{-178}$
Myr, 21${}^{+4}_{-4}$ Myr, and 181${}^{+52}_{-41}$Myr, respectively). We
speculate HATS-47HD 110113, Kepler-1062, and Kepler 968 may be the
contaminating stars in star cluster identification. They may be field stars
having similar kinematic properties compare to the comoving stellar groups in
their proximity coincidently. Therefore, we exclude these three potential
contaminating sources in cluster identification. We also remove HATS-47
because it does not have a convincing age measurement.

Thirdly, for CoRoT-22, Kepler-1502, TOI-1937 A b, and TOI-4145 A b which have
inconsistent age measurements and lack of stellar rotation measurements, we
can hardly validate their age. Additionally, Yee et al. (2023) suspects that
TOI-1937 A and TOI-4145 A may be the field star because of the poorly
constrained cluster membership identification. Therefore, we directly exclude
these four systems.

To sum up, we validate the age measurement of 70 planets/candidates in star
clusters, obtain more convinced ages of three host stars via either literature
or the new gyrochronological relation, and exclude eight planetary systems
without convincing age estimations. If we assume those eight host stars are
field stars, the contamination rate of our catalog is about 6%, which is
consistent with that in Kounkel & Covey (2019), i.e., 5%-10%.

###  2.3 Sample cut

In section 2.2, we obtain 63 planets and 84 planet candidates in star clusters
with relatively robust age estimation. We aim to obtain the planet radius
evolution, i.e. planet radius–age distribution. The accuracy of planet radius
and age measurement will significantly influence our results. Therefore, we
need to do the sample cut to minimize the influence of observational biases.

Here, we list the steps of sample cut in Table 2. Without the mass
measurement, we can hardly determine whether the planet candidates are planets
or brown dwarfs. Planet candidates with large radii are unlikely planets.
Thus, we exclude 23 planets/candidates, with $R_{\rm p}>2.5\,\rm R_{\rm J}$
(the same criteria described in Nardiello et al. (2021)). Brown dwarfs with
larger masses can induce the motion of the photon center. Berger et al. (2020)
suggest that stars with high re-normalized unit-weight error (RUWE$>$1.4), are
likely to be binaries. Thus, we adopt the criteria, RUWE$<$1.4, to exclude 9
planet candidates in potential binary systems.

Since planets with poor radius measurements may contaminate the results, we
exclude 7 samples with relative radius errors larger than 50%. Due to the
precision of TESS and the stellar noise of young stars, small planets detected
by TESS and planets around young stars are less complete. Thus, we need to
constrain the lower limits of the planet radius to exclude the bias of
completeness. As shown in Appendix A, planets with radius $R_{\rm p}>2\rm
R_{\oplus}$ and period $P<20$ days are considered to be detectable (SNR$>$7.1)
via both Kepler and TESS. Thus, we cut the sample via $R_{\rm p}>2\rm
R_{\oplus}$ and $P<20$ days.

After the sample cut, there are 66 planets/candidates left. In the next
section, we mainly use the sample to do the analysis.

Table 2: Sample Cut of Planets/Candidates in clusters  Criterion | Planets | Planet candidates
---|---|---
The whole number | 73 | 84
Age validation | 63 | 84
$R_{\rm p}<2.5\,\rm R_{\rm J}$ | 62 | 62
RUWE $<1.4$ | 62 | 53
$\sigma_{\rm R_{\rm p}}/R_{\rm p}<0.5$ | 58 | 50
$P<20$ days | 37 | 44
$R_{\rm p}>2\rm R_{\oplus}$ | 30 | 36

##  3 Planet Radius–Age Distribution

Figure 2: The planetary radius – age distribution of 37 planets and 44 planet
candidates in star clusters(Table 2). Different colors show the planets in
clusters detected by different missions. Red, yellow, blue, and purple symbols
are planets/candidates detected by different missions, i.e. Kepler, K2, TESS,
and other ground-based telescopes, respectively. The gray dots are confirmed
transiting planets whose host stars have age measurements.

###  3.1 Planet Radius–Age Diagram

Figure 2 shows the planetary size – age distribution of 37 planets and 44
planet candidates in star clusters(15 planets/planet candidates with $R_{\rm
p}<2\,\rm R_{\oplus}$ and 66 planets/planet candidates with $2\,\rm
R_{\oplus}<R_{\rm p}<2.5\,R_{\rm J}$).

Here, we classify planets into three groups by size for the sake of
simplicity:

  * •

Sub-Neptunes, i.e. planets of $2\,\rm R_{\oplus}<R_{\rm p}<4\,\rm R_{\oplus}$,

  * •

Sub-Jupiters, i.e. planets of $4\,\rm R_{\oplus}<R_{\rm p}<8\,\rm R_{\oplus}$,

  * •

Jovian planets, i.e. planets of $8\,\rm R_{\oplus}<R_{\rm p}<2.5\,\rm R_{\rm
J}$.

There are only 5 Jovian planets younger than 100 Myr, while dozens beyond 100
Myr. So, it seems that there is a gap in the planet radius–age diagram for the
Jovian planets younger than 100 Myr. Additionally, there may be another gap
for Sub-Jupiters with ages between 50–200 Myr. Before 50 Myrs, there are
several Sub-Jupiters, while between 50–200 Myr, the number of Sub-Jupiters
declines to nearly none. On top of that, there are nearly no Sub-Neptunes
between 50–100 Myr.

Therefore, it seems that all of the planets disappear between 50–100 Myr.
However, due to the small number of planets/candidates(i.e. the large
statistical error), whether the gap is real can not be easily demonstrated. In
order to avoid observational bias, in the next subsection, we will take into
the age error and the radius error of the planets to obtain the time-dependent
relation for the proportion (instead of the number) of different-sized planets
in star clusters.

Figure 3: The time-dependent relation of the proportions of planets(in star
clusters) of different sizes. Different colors show planets of different
sizes. Panel (a) and (b) are different in numbers of age bins, i.e. 4 age bins
between 10 to 1000 Myr under log scale in panel (a), and 8 age bins in panel
(b). Figure 4: The time-dependent relation of the proportions of planets(both
in star clusters and probably around field stars) of different sizes.
Different colors show planets of different sizes. Panel (a) and (b) are
different in the number of age bins, i.e. 6 age bins between 10 to 10000 Myr
under log scale in panel (a), and 12 age bins in panel (b). We add a cut of
effective temperature in panel (c) and an additional period cut in panel (d),
i.e. $P<10$ days. For panel (e), we show the result for planets with $P<200$
days as a comparison. Two shadow regions around 100 Myr and 2 Gyr are over-
plotted to emphasize two typical timescales.

###  3.2 The Evolution of Planet Radius

To investigate the time-dependent relation of different-sized planets in star
clusters. We defined the proportions of planets with different sizes and ages.
To determine the proportions and the uncertainties, we randomize the ages and
radii of all the planets 100,000 times, assuming the Gaussian distribution.
Then, we can obtain the proportions of each time, denoted as $f_{\rm i}$, via
the formula:

$f_{\rm i}=\frac{N_{\rm i}}{N_{\rm SubN}+N_{\rm SubJ}+N_{\rm J}},$ (1)

where $N_{i}$ is the number of planets in star clusters with different sizes,
i.e. $N_{\rm SubN},N_{\rm SubJ}$ or $N_{\rm J}$ corresponding to Sub-Neptunes,
Sub-Jupiters, and Jovian planets in section 3.1, respectively. After
calculating 100,000 times, we obtain the distribution of $f_{\rm i}$ and adopt
the lower limits, median values, and the upper limit, according to the 16, 50,
and 84 percentiles of $f_{\rm i}$, respectively. The results are as shown in
Figure 3. Note the age is cut by $\leq$ 1 Gyr because most of the selected
planets/candidates in star clusters are younger than 1 Gyr(Figure 2). Panel
(a) and (b) are different in numbers of age bins, i.e. 4 and 8 age bins
between 10 to 1000 Myr under the log scale, respectively. Both two panels show
that the proportion of Jovian planets (red diamonds, $f_{\rm J}$) increases
before 100 Myr, and then declines after 200 Myrs, i.e. a peak occurs between
100 and 200 Myr. The proportion of Sub-Jupiters (blue squares, $f_{\rm SubJ}$)
declines around 100 Myr. The proportion of Sub-Neptunes (gray circles, fSubN)
shows a clear increase after 100 Myr. Here, we also consider the Poisson error
because of the small number of planets in each bin, which has similar features
with Figure 3(See Figure B1 in Appendix B)

To extend the evolution of planets/candidates older than 1 Gyr, we calculate
the proportion of planets both in star clusters and around field stars, with
different sizes. Here, we add $\sim$ 871 confirmed planets with age
measurements from the NASA Exoplanet Archive. These confirmed planets share
the same cut with planets/candidates in star clusters, i.e. 2$\rm
R_{\oplus}<R_{\rm p}<2.5\rm R_{\rm J}$, and $P<20$ days. Using the same
estimating procedures as those in Figure 3, we obtained the proportion varying
with age, as shown in Figure 4.

In the panel (a) and (b) of Figure 4, the proportion of Jovian planets (red
diamonds, $f_{\rm J}$), Sub-Jupiters (blue squares, $f_{\rm SubJ}$), and Sub-
Neptunes (gray circles, $f_{\rm SubN}$) show the similar time-dependent
relation within 1 Gyr to that in Figure 3. I.e. $f_{\rm J}$ reaches maximum
between 100 and 200 Myr, $f_{\rm SubJ}$ rapidly declines around 100 Myr, and
$f_{\rm SubN}$ increases after 100 Myr. Because Figure 4 has a longer time
span than Figure 3, there are more substructures in Figure 4. For example, all
of panels in Figure 4 show a tiny bump of $f_{\rm J}$ around 2 Gyr, which is
anti-correlated with $f_{\rm SubN}$, i.e. a small dip of $f_{\rm SubN}$ around
2 Gyr. These two timescales, i.e. 100 Myr and 2 Gyr (gray shadow regions), may
correspond to different planet formation environments (see discussion in 4.2).
Because the majority of planets younger than 300 Myr are in star clusters,
while most of the planets older than 1 Gyr are around field stars.

To illustrate that our results are robust, we should exclude the influence of
some other stellar parameters. For example, some of the planet host stars are
very hot, i.e. $T_{\rm eff}>7500$ K, especially for the candidates detected by
TESS. For main-sequence stars, hotter stars usually have larger stellar radii
than cooler ones. Therefore, the transit method tends to find larger planet
candidates around hotter stars. In panel (c), we add another criterion for
planets/candidates in star clusters and around field stars, i.e. $T_{\rm
eff}<7500$ K. Although the proportion of Jovian planets $f_{\rm J}$ around 200
Myr is smaller than that in panel (b) because of the additional sample cut,
$f_{\rm J}$ still continuously increases between 100 and 400 Myr, i.e., the
peak moves backward to around 400 Myr. The proportion of $f_{\rm SubJ}$
rapidly decreases around 100 Myr, then remain at $\sim$ 0.1 after 100 Myr,
which is similar to that in panel (b). $f_{\rm SubN}$ does not show an obvious
increasing/decreasing after 100 Myr.

The widely used definition of Hot Jupiters(HJs) is Jupiter-sized planets
within 10 days Dawson & Johnson (2018). Here, in panel (d), as a comparison,
we also show the results of the conventional hot planets within 10 days. For
Jovian planets and sub-Jupiters, panel (d) shows similar results to panel (b).
Therefore, in the following, we call the Jovian planets within 20 days as HJs
for simplicity (if without additional annotation). For sub-Neptunes, the
increasing tendency after 100 Myr is ambiguous.

In panel (e), we show the results of the time-dependent relation of planet
radius for planets within 200 days. Because including some warm planets, the
increment of $f_{\rm J}$ around 100 Myr in panel (e) becomes less than that in
panel (b). As the majority of the warm planets are sub-Neptunes, $f_{\rm
SubN}$ in panel (e) is systematically higher than that in panel (b) after 100
Myr. In turn, the $f_{\rm J}$ in panel (e) is systematically lower than that
in panel (b) after 100 Myr. The time-dependent relation of $f_{\rm SubJ}$ in
panel (e) is similar to other panels.

In Figure 4, we can not find an obvious increase trend of the proportion of
sub-Neptunes within 1 Gyr as shown in Figure 3. Because it seems sensitive to
the parameter cut.

To summarize this section, we obtain the time-dependent relation of planet
radius for planets/candidates in star clusters and around both cluster members
and field stars, i.e.

  * •

The proportion of Jovian planets $f_{\rm J}$ increases around 100 Myr and
reaches maximum between 100 Myr and 200 Myr, which is mainly attributed to the
HJs in star clusters. The tiny bump of $f_{\rm J}$ around 2 Gyr is attributed
to the HJs around field stars.

  * •

The proportion of Sub-Jupiters $f_{\rm SubJ}$ declines rapidly around 100 Myr,
then remains at a low value. The declination of $f_{\rm SubJ}$ is mainly
attributed to the hot Sub-Jupiters in star clusters.

##  4 Constraints of Hot giant planets formation timescale

Based on the statistical results above, we try to explain or constrain the
timescales of hot giant planets’ formation mechanisms in star clusters. Here,
the hot giant planets mean HJs and hot Sub-Jupiters (or hot Neptunes).

There are several formation scenarios of HJ, i.e. in-situ formation when disk
mass is large, or ex-situ formation then undergoing disk migration or high-e
migration. The time scales of the first two HJ formation scenarios are mainly
limited by the lifetime of the gas disk, which is typically $\sim$10 Myr.
Therefore, if the in-situ formation and disk migration are the dominant
channels of HJ formation, the number of HJs will not significantly change
after 10 Myrs. However, Figure 3 and 4 show that the proportion of HJs($f_{\rm
J}$) has an obvious increment around $\sim$ 100 Myrs, which is probably
attributed to the high-e migration.

Note, we do not exclude the possibility of HJs forming through the in-situ
formation and disk migration. However, with a lack of clusters younger than 10
Myr, we can hardly constrain the in-situ formation mechanism and the fraction
of such planets.

In the following discussion, we mainly focus on the increment of $f_{\rm J}$
and the rapid declination of $f_{\rm SubJ}$ around 100 Myrs in star clusters.
More specifically, in section 4.1, we estimate the timescales of flyby-induced
high-e migrations in star clusters, using typical parameters. In section 4.2,
we try to explain the tiny bump of HJs, as well as the small dip of sub-
Neptune. The Hot-Neptune desert is also discussed in section 4.3.

###  4.1 Flybys induced high-e migration in open clusters within 200 Myr

Recently, several observation works have shown environments in clusters can
influence planet formation and evolution (Winter et al. 2020; Dai et al.
2021). In star clusters, especially dense clusters, close stellar flybys may
occur frequently. A series of theoretical works have shown that the HJ
formation can be triggered by stellar flybys in star clusters (Wang et al.
2020, 2022; Li et al. 2020; Rodet et al. 2021; Li et al. 2023). Similar to
previous works, we consider hierarchical planet systems with both Jovian
planets and an out companion (e.g. a cold giant planet, sub-stellar, or
stellar companion). The high-e migrations of the Jovian planet induced by
flybys can be described as follows. During a close flyby event, a flyby star
exchanges the angular momentum with the outer companion and excites its
eccentricity and inclinations. Consequently, the eccentricities of the Jovian
planets will be highly excited through the von Zeipel–Lidov–Kozai mechanism
(ZKL, von Zeipel, 1910; Kozai, 1962; Lidov, 1962). Finally, tidal
circularization leads to the inward migration of the Jovian planet.

There are three factors determining the formation timescale of HJs under
flybys-induced high-e migration in star clusters, i.e. the timescale of the
close flyby ($\tau_{\rm flyby}$), the ZLK mechanism ($\tau_{\rm ZKL}$), and
the tidal circularization ($\tau_{\rm tidal}$). Roquette et al. (2021)
demonstrates that an effective stellar flyby means flyby with small periastron
$q$, which can trigger the ZKL oscillation successfully, and subsequently
excite the high-eccentricity of the innerJovian planet. Here, we combine the
equation (17),(18), and (19) in Roquette et al. (2021), to get an estimation
of $\tau_{\rm flyby}$, the timescale of an effective flyby, i.e.,

$\tau_{\rm flyby}=\left(\frac{10^{3}\,\rm
pc^{-3}}{n_{*}}\right)\left(\frac{2\,\rm M_{\odot}}{M_{\rm
tot}}\right)\left(\frac{50\,\rm AU}{a_{\rm
out}}\right)\left(\frac{\sigma_{*}}{1\,\rm km/s}\right)\rm Gyr,$ (2)

where $n_{*}$ is stellar density in clusters, $M_{\rm tot}$ is the total mass
of the hierarchical three-body system, $a_{\rm out}$ is the semi-major axis of
the outer companion, and $\sigma_{*}$ is the velocity dispersion of star
clusters. Here, we assume a solar-Jupiter system plus a solar-like companion,
i.e. $M_{\rm tot}\sim 2\rm\,M_{\odot}$. Other parameter setting are as
follows, $n_{*}=10^{4}$ stars pc-3, $a_{\rm out}$ = 50 AU, and $\sigma_{*}$ =
1 km/s. The setting of $\sigma_{*}$=1 km/s and $a_{\rm out}$ = 50 AU are the
same as Rodet et al. (2021). For stellar density $n_{*}$, some earlier works
(e.g. Adams 2010; Malmberg et al. 2011; Craig & Krumholz 2013) assume that
high stellar density can only exist in the high-mass clusters. However, the
recent work Pfalzner & Govind (2021) revealed that low-mass clusters share a
similar stellar density around 104 pc-3 as high-mass clusters (e.g. the
central area of the Orion nebula cluster) at least in the early stage of
cluster evolution. Therefore, we set the typical density of star clusters to
104 stars pc-3. Then, the timescale of an effective flyby $\tau_{\rm flyby}$
is about 100 Myr,

The ZLK timescale($\tau_{\rm ZKL}$) can be estimated as Wang et al. (2022):

$\tau_{\rm ZKL}=P_{\rm in}\left(\frac{M_{\rm tot,in}}{M_{\rm
out}}\right)\left(\frac{a_{\rm out}}{a_{\rm in}}\right)^{3}(1-e^{2}_{\rm
out})^{3/2}$ (3)

where $P_{\rm in}$ is the orbital period of the inner Jovian planet, $M_{\rm
tot,in}$ is the total mass of the Sun-Jupiter system, $M_{\rm out}$ is the
mass of outer companion, and the $e_{\rm out}$ is the eccentricity of the
outer companion. If we assume that the inner Jupiter form outside the water
ice line (around 2.7 AU), i.e. $a_{\rm in}\geq$ 2.7 AU and $P_{\rm in}\geq$
4.4 yr, the typical ZLK timescale $\tau_{\rm ZKL}$ is $\lesssim$ 0.3 Myr,
which is much shorter than $\tau_{\rm flyby}$.

According to Figure 2 in Rodet et al. (2021), an effective flyby can
successfully lead to the high-eccentric orbit of the inner planet, typically
larger than 0.99. For HJs in star clusters, the median value of the orbital
period is around 3 days, which corresponds to 0.04 AU around a Solar-like
star. If we set the final semi-major axis ($a_{f}=2q_{\rm in}$, Owen & Lai,
2018) after tidal circularization of a Jovian planet as 0.04 AU, the
periastron of the inner planet $q_{\rm in}$ is about 0.02 au. According to
Figure 1 or Equation 5 in Wang et al. (2022), the typical tidal dissipation
timescale of such a system is $\lesssim$ 100 Myr. Therefore, the typical
formation timescale of HJs ($\tau_{\rm HJC}$) through flyby-induced high-e
migration is $\lesssim$ 200 Myr under our parameter settings. Due to the
uncertainty of the mass of outer companions, $\tau_{\rm HJC}$ may move
backward to several hundreds of Myr.

Dawson & Johnson (2018) provides that the occurrence rate of HJs is 0.5-1%,
which is ten times of estimation by Roquette et al. (2021). Interestingly, if
we adopt the new stellar density value in Pfalzner & Govind (2021) (i.e.
$n_{*}=10^{4}\,\,\rm pc^{-3}$), the observed HJs can be successfully explained
by flyby-induced high eccentricity migration in star clusters. This may
indicate that the flyby-induced high-e migration is the dominant formation
scenario of HJ in star clusters. As described of the high-e migration,
Jupiter-sized planets (i.e. warm-Jupiters and cold-Jupiters) may migrate
inward and become HJs. Therefore it also results in the decline of warm-
Jupiters and cold-Jupiters after $\sim$100 Myr. In Figure 5, we show the
planetary radius–age distribution of 107 planets/candidates within 200 days in
star clusters. There are 7 young Warm-Jupiters in clusters(WJ, $20<P<200$
days, age $<$ 100 Myr, green circles). However, there are no WJs in clusters
with ages between 100 Myr and 1000 Myr. This may be another hint of the flyby-
induced high-e migration in star clusters.

Note there are dozens of WJs with age measurements from the NASA exoplanet
archive ($20<P<200$ days, 8$\rm R_{\oplus}<R_{\rm p}<2.5\rm{R_{\rm J}}$, blue
diamonds in Figure 5). The majority of these WJs around field stars are older
than 1 Gyr. The absence of WJs between 100–1000 Myr may indicate that WJs are
hardly sustained in cluster environments, i.e. such relatively dense
environments prefer the formation of HJs. However, due to the small number of
warm plants in star clusters, we need more observation data.

Figure 5: The planetary radius – age distribution of 107 planets/candidates
(within 200 days, red squares and green circles) in clusters and 1991
confirmed planets (within 200 days, gray dots) around field stars. Here, we
also show the 39 warm Jupiters around field stars with age measurements from
the NASA exoplanet archive ($20<P<200$ days, blue diamonds).

###  4.2 High-e migration of HJs around field stars beyond 1 Gyr

Figure 6: The eccentricity-age relation of HJs. We select 336 known HJs
($P<20$ days, $\sigma_{R_{p}}/R_{\rm p}<0.5$) with age and eccentricity
measurements. Panel (a) shows the entire sample in the eccentricity-age
diagram. Panel (b) represents the ratio of high-eccentricity HJs changing with
age. The blue and purple dots are the ratio of HJs with $e>0.1$ and $e>0.2$
changing with age, respectively. The shadow region is over-plotted to
emphasize the peak of the ratio.

In Figure 4, we find a tiny bump of proportion of Jovian planets $f_{\rm J}$
around 2 Gyr, which is anti-correlated with $f_{\rm SubN}$, i.e. a small dip
of $f_{\rm SubN}$ around 2 Gyr.

High-e migration can explain the anti-correlation. During the inward migration
of the Jovian planet, the inner planets can be ejected from the system due to
the planet-planet interaction (Mustill et al., 2015). I.e. HJs after high-e
migration are usually lonely (Steffen et al., 2012; Hord et al., 2021). Thus,
the proportion of smaller planets, e.g. Sub-Neptunes and Super-Earths, have a
declination.

Due to tidal dissipation during the high-e migration, the eccentricity of the
HJs will decrease with time. Therefore, if the tiny bump in the proportion of
Jovian planets $f_{\rm J}$ around 2 Gyr is attributed to the HJs that form
through high-e migration, we may also see a small bump in the eccentricity-age
diagram around 2 Gyr.

To test this conjecture, in the following, we select 336 HJs both having
eccentricity and age measurements from the NASA exoplanet archive. We
constrain the period of the HJs to be shorter than 20 days and the relative
uncertainty of planet radii is no more than 50%. Figure 6 shows the
eccentricity-age distribution of the 289 HJs. In Panel (a) red dots shows the
median eccentricity of each age bin changing with age. The error bar is
calculated according to the 16 and 84 percentiles of the eccentricities of
each age bin. Because the majority of the HJs have low eccentricity, the
median eccentricity of each age bin is nearly a zero constant, i.e. unchanging
with age.

However, red dots have larger error bars in the older region. It seems that
the number of HJs with high eccentricity increases beyond 1 Gyr. Therefore, we
calculate the relative ratio of High-eccentricity HJs in each age bin, as
shown in panel (b) of Figure 6. Similar to the previous analysis, this ratio
is calculated with the assumption of the Gaussian distribution of age and
eccentricity of each HJ. We find that both the ratio of HJs with $e>0.1$ and
$e>0.2$ rapidly increase beyond 1 Gyrs, i.e. the difference between the two
data points before and after 1 Gyr is 2 $\sigma$ and 12 $\sigma$,
respectively. After reaching the maximum of around 2 Gyr, the ratio of HJs
with high-eccentricity declines with age due to the tidal dissipation as
expected. Therefore, the eccentricity evolution also supports the high-e
migration for these HJs older than 1 Gyr.

Because HJs around field stars are the dominant population beyond 1 Gyr, as
shown in Figure 2, we can conclude that the bump of $f_{\rm J}$ around 2 Gyr
is likely due to the formation of HJs around field stars through high-e
migration. Since the timescale of 2 Gyr is ten times longer than the formation
timescale of HJs through flyby-induced high-e migration in star clusters
($\tau_{\rm HJC}\lesssim$ 200 Myr). We explain the large differences via the
different environments of stellar density. The HJs form around field stars
after 1 Gyr, which escape the cluster environments at the early stage ($<100$
Myr). Because the less dense dynamical environments lead to longer flyby
timescale, the trigger time of high-e migration is probably much longer than
that in cluster environments.

###  4.3 Formation of Hot Neptune desert around 100 Myr

Several previous works find the hot Neptune desert in planetary mass-period
distribution and radius-period distribution, i.e. Szabó & Kiss (2011); Beaugé
& Nesvorný (2013); Mazeh et al. (2016). In Figure 4, we find that the
proportion of the Sub-Jupiters within 20 days, $f_{\rm SubJ}$, rapidly
declines around 100 Myr, then remains at the low value. The declination is
correlated to the hot Neptune desert and may indicate the formation timescale
of such desert.

However, the Sub-Jupiters classified via radius and period independently, are
not in the hot-Neptune desert exactly. According to Mazeh et al. (2016), the
borders of the desert are period-dependent. I.e. at large radii, the planet’s
radius decreases with increasing period, while at small radii, the radius
increases with increasing period (the dashed lines in Figure 7). Using the
same region with Mazeh et al. (2016), we compare the time-dependent ratio of
the number of planets inside and outside the hot Neptune desert, to constrain
the formation timescale of the hot Neptune desert.

Figure 7: The scatter plot of planets in planets radius-period plane. Red dots
are hot and young planets in star clusters($P<20$ days, age $<$ 100 Myr).
While black dots are hot and old planets in star clusters ($P<20$ days, age
$>$ 100 Myr). Orange squares are planets in clusters with periods beyond 20
days. Two black dashed lines are boundaries of the Hot Neptune desert
described in Mazeh et al. (2016). The gray line of P=20 days and the gray line
of $R_{\rm p}$=2 R⊕ are over-plotted as reference.

In Figure 7, we show 107 planets/candidates in star clusters and 1991 other
confirmed planets around field stars in the radius–period plane. We divide the
planets/candidates in star clusters within 20 days into two groups. One is
planets younger than 100 Myr (red circles). The other is planets older than
100 Myr (green diamonds).

We calculate the ratio of the number of planets inside and outside the hot
Neptune desert, i.e. $N_{\rm in}/N_{\rm out}$, for both the younger and older
groups. Then, we use the Monte Carlo simulation to get a distribution of
$N_{\rm in}/N_{\rm out}$ under the assumption of the Gaussian distribution of
planet radius, period, and age. The error of $N_{\rm in}/N_{\rm out}$ is
adopted according to the 16 and 84 percentiles of the ratio distribution. We
find that the younger group have much higher $N_{\rm in}/N_{\rm out}$
(0.80${}^{+0.20}_{-0.19}$) than older groups (0.19${}^{+0.03}_{-0.03}$) in
nearly 3.0 $\sigma$ confidence level. If we add a planet radius cut for
detection completeness, i.e. $R_{\rm p}>2\rm R_{\oplus}$ (horizontal shadow
line), the result is similar. I.e. the younger group has higher $N_{\rm
in}/N_{\rm out}$ (0.67${}^{+0.17}_{-0.13}$) than older group
(0.18${}^{+0.03}_{-0.01}$) in 3.67 $\sigma$ confidence level. Alternatively,
if we use 300 Myr to distinguish the younger and older group, the difference
of $N_{\rm in}/N_{\rm out}$ between the young and old group will decrease,
which is smaller than the 100-Myr case. I.e. the difference between the median
value of the young group’s $N_{\rm in}/N_{\rm out}$
(0.48${}^{+0.07}_{-0.05}$), and old group’s $N_{\rm in}/N_{\rm out}$
(0.31${}^{+0.04}_{-0.04}$). Therefore, we can conclude that the rapid
declination of $f_{SubJ}$ around 100 Myr corresponds to the formation of the
hot-Neptune desert around 100 Myr.

Owen & Lai (2018) explains two boundaries of the hot-Neptune desert in a
combination of photoevaporation and high-e migration. For the lower boundary,
photoevaporation functions effectively in the first several hundred of million
years and will trigger the atmospheric mass loss of Neptune-sized planets,
which is consistent with the formation timescale we obtained. If the high-e
migration sculpts the lower boundary, the formation timescale will be much
larger than 100 Myr, because the hot-Neptunes usually experience longer tidal
circularization ($\gtrsim$ 1 Gyr). We prefer that photoevaporation sculpts the
lower boundary of the hot-Neptune desert around 100 Myr.

For the upper boundary, photoevaporation seems not a suitable explanation.
Because several works show that the massive planets(M${}_{p}>$ 0.5 MJ) can
resist photoevaporation even at extremely short periods (e.g. Owen & Jackson,
2012; Tripathi et al., 2015; Owen & Alvarez, 2016). Therefore,
photoevaporation will predict a lower upper boundary(Figure 4 in Owen & Lai
(2018)). However, in some specific cases, the rapid declination of the radius
of giant planets may explain the upper boundary of the Hot-Neptune desert.
E.g. Thorngren et al. (2023) developed a model including radius inflation,
photoevaporative mass loss, and Roche lobe overflow, which can trigger a
runaway mass loss of a puff hot Saturn around 400 Myr. Because the desert
already existed around 100 Myr, we do not consider such a mechanism as the
dominant.

High-e migration can deliver Jovian planets from the outer region to the inner
orbit. Then, the subsequent decay due to stellar tides will further sculpt the
upper boundary. More specifically, both the tidal circularization timescale
(planetary tides) and tidal decay timescale (stellar tides) decrease with
increasing planetary radius or mass. As described in section 4.1, the typical
formation timescale of HJs in clusters is $\lesssim$ 200 Myr, which is
consistent with the formation timescale of the Hot-Neptune desert. Therefore,
flyby-induced high-e migration could sculpt the upper boundary of the hot-
Neptune desert in clusters.

The formation timescale of the Hot-Neptune desert in star clusters is around
100 Myr. When it comes to field stars, the formation timescale of the Hot-
Neptune desert may move backward to several Gyr because of the relatively slow
high-e migration for HJs around field stars. However, due to the limited
sample of young planets around field stars, we do not find any evidence.

Because the scenario of flyby-induced high-e migration can not only explain
the increment of $f_{\rm J}$ around 100 Myr, but may also sculpt the upper
boundary of the Hot-Neptune desert around 100 Myr. Therefore, we prefer to the
scenario of flyby-induced high-e migration, i.e. a combination of
photoevaporation and flyby-induced high-e migration could sculpt the hot-
Neptune desert around 100 Myr.

##  5 Conclusion & Discussion

Planets in young star clusters could help us understand the planet’s formation
and evolution because of the accurate age estimation. In section 2, we collect
the largest catalog of 73 planets and 84 candidates in star clusters by cross-
matching with K2020 and planets/candidates from the NASA exoplanet archive. We
validate the age estimation of 70 planets/candidates in star clusters, obtain
more convinced ages of three host stars via either literature or the new
gyrochronological relation, and exclude eight planetary systems with no robust
age estimations.

In section 3, we use this catalog to study the planet radius – age relation.
The main statistical results are as follows:

  * •

The proportion of Jovian planets $f_{\rm J}$ increases around 100 Myr and
reaches maximum between 100 Myr and 200 Myr, which is mainly attributed to the
HJs in star clusters. The bump of $f_{\rm J}$ around 2 Gyr is attributed to
the HJs around field stars.

  * •

The proportion of Sub-Jupiters $f_{\rm SubJ}$ declines rapidly around 100 Myr,
then remains at a low value. The declination of $f_{\rm SubJ}$ is mainly
attributed to the hot Sub-Jupiters in star clusters.

After discussing several possible scenarios to explain the results, we give
two constraints on the hot giant exoplanet formation timescales in section 4:

  * •

HJs likely form through flyby-induced high-e migration in star clusters within
200 Myr.

  * •

A combination of photoevaporation and flyby-induced high-e migration in star
clusters can sculpt the hot-Neptune desert around 100 Myr.

We find that flyby-induced high-e migration may be the dominant formation
channel of HJs in star clusters. As described in section 4.1, those HJs in
star clusters will accompany an outer companion, which is an effective angular
momentum transmitter during a close flyby event. Therefore, we hope to
discover some outer companions beyond these HJs with the Radial Velocity
observations from ground-based telescopes and the astrometric data from the
future data release of Gaia.

Different from HJs in star clusters, HJs around field stars may have much
longer formation timescale ($\sim$ 2 Gyr), which can be attributed to the
different dynamical environments (section 4.2).

Note in this paper, we mainly focus on the ZLK mechanism to excite the high
eccentricity of the inner Jovian planet. Actually, other mechanisms such as
planet-planet scattering can trigger the high-eccentric orbit too. Wang et al.
(2020) demonstrates that a very small fraction of HJs can form from the fly-
induced planet-planet scattering channel, i.e. ZLK mechanism may be the
dominant scenario of eccentricity excitation. However, we could not exclude
the possibility of planet-planet scattering. One way to distinguish these two
mechanisms definitely is the stellar obliquity, the angle between a planet’s
orbital axis and its host star’s spin axis. ZLK mechanism predicts a bimodal
stellar obliquity distribution, concentrated at 40∘ and 140∘ (e.g. Fabrycky &
Tremaine, 2007). While the planet-planet scattering after a convergent disk
migration predicts a concentration of stellar obliquity around 90∘ (Nagasawa &
Ida, 2011).

Additionally, a hint from the absence of WJs around field stars between 100
Myr and 1000 Myr also supports the scenario of flyby-induced high-e migration
(section 4.1). However, this absence may be due to the observation bias. For
instance, TESS prefers to discover HJs instead of WJs, because of the
relatively short time span.

In the future, with the extended mission of TESS, the Earth 2.0 mission
(ET2.0, Ge et al., 2022), the Chinese Space Station Telescope (CSST, Zhan,
2011; Gong et al., 2019), and PLATO (Rauer et al., 2014), we hope to detect
more young planets both in star clusters and around field stars. The
subsequent astrometry data from Gaia and the follow-up Radial-Velocity
observation (including the Rossiter-McLaughlin effect) from ground-based
telescopes can also provide more information about warm planets and even outer
companions. A larger sample of planets in clusters will benefit us to test
different formation scenarios of HJs, as well as hot-Neptune deserts.

We thank Prof. Dr. Bo Ma for helpful recommendations to improve the paper.
This work is supported by the National Natural Science Foundation of China
(grant Nos. 11973028, 11933001, 1803012, 12150009) and the National Key R&D
Program of China (2019YFA0706601). We also acknowledge the science research
grants from the China Manned Space Project with No. CMS-CSST-2021-B12 and CMS-
CSST-2021-B09, as well as Civil Aerospace Technology Research Project
(D050105).

##  Appendix A SNR-age relation

In young clusters, stars are more active and may hide the transiting events,
especially for small planets. Thus, the detection of small planets in young
clusters is uncompleted. To study the selection effects, we try to derive the
empirical relations between SNR and stellar age for planets of different
sizes. Then, we can select suitable criteria of planet radius to cut our
samples.

The calculation of SNR of a transiting planet is calculated as follows:

$\rm{SNR}=\frac{\delta n^{0.5}}{\sigma_{*}\left(t_{\rm dur}\right)}$ (A1)

where $\delta=(R_{p}/R_{*})$ is the transit depth of the star and $n$ is the
transit number. $\sigma_{*}$ is the stellar photometric noise. The transit
duration time $t_{\rm dur}$ is given by:

$t_{\rm dur}=\frac{PR_{*}\sqrt{1-e^{2}}}{\pi a}$ (A2)

where $P$ is the planet’s orbital period and $a$ is the semi-major axis. In
the calculation of SNR, we assume that host stars are solar-like (i.e.
$M_{*}=1\rm M_{\odot}$ and $R_{*}=1\rm R_{\odot}$ ) and set $P$=20 days
(because most of our select planets in star clusters are within 20 days). The
same as the assumption of the Mulders et al. (2015) to assume that the stellar
noise change with time as a simple power law distribution:

$\sigma_{*}\left(t\right)=\sigma_{\rm LC}\left(\frac{t}{t_{\rm
LC}}\right)^{ind_{\rm CDPP}},$ (A3)

where we normalize the noise ($\sigma_{\rm LC}$) at the period in the long
cadence, 1765.5 s ($t_{\rm LC}$). $ind_{\rm CDPP}$ is the power law index. We
use Combined Differential Photometric Precision (CDPP, Christiansen et al.,
2012) from Kepler DR25, which characterizes the noise level in Kepler
lightcurves. Then, using the stellar kinematic age from Chen et al. (2021), we
can obtain $ind_{\rm CDPP}$–age relation and SNR–age relation.

For a rough calculation, we assume that the stars observed by Kepler and TESS
are similar, i.e. the stellar noise evolution is similar. The major difference
in SNR of planets of different sizes lies in the observation time $t_{obs}$
which determines the transit numbers. Here, the observation time of the Kepler
stars is $\sim 1450$ days and TESS is roughly two observation sectors, i.e.
$\sim 54$ days.

Figure A1 shows the calculated SNR of planets changing with age. Red, orange,
and black hollow dots and dashed lines present the results of planets of
different sizes (i.e. 1 R⊕, 2 R⊕, and 2.5 R⊕) for TESS. The purple hollow dots
and dashed line show the result of planets of 1 R⊕ for Kepler. Since the data
from Chen et al. (2021) does not provide the CDPP of stars younger than 300
Myr, we simply extend the relationship to 10 Myr through a log-linear
exploration. The blue horizontal line is the SNR of 7.1 above which we
consider TESS or Kepler can detect planets. To sum up, planets with a radius
larger than 2 R⊕ can be detected both by TESS and Kepler. Thus, in section
2.2, we focus on the planets with a radius larger than 2 R⊕ to exclude the
uncompleted detection due to the stellar noise of young stars.

Figure A1: The SNR-age relation of planets of different sizes. Red, orange,
and black hollow dots and dashed lines are SNR-age relations of planets of 1,
2, and 2.5 R⊕ for TESS. The purple presents the SNR-age relation of planets of
1 R⊕ for Kepler. The blue horizontal line (SNR=7.1) is over-plotted as a
reference.

##  Appendix B Poisson Error

The ’standard’ confidence interval for a Poisson parameter Figure B1 shows the
time-dependent relation of the proportions of planets of different sizes,
adopted Poisson error. The error bar, i.e. standard confidence interval
related to a Poisson parameter, is calculated through the chi-square
distribution. Panel (a) includes planets/candidates in star clusters,
consistent with the results of Figure 3, although with more significant
uncertainties. Panel (b) includes planets/candidates in star clusters and
around field stars, consistent with the results of Figure 4. I.e. the $f_{J}$
increases before 100 Myr and then decreases around 1-2 Gyr.

Figure B1: The time-dependent relation of the proportions of planets of
different sizes. Here, the error is adopted via Poisson distribution. Panel
(a) corresponds to Figure 3 and panel (b) corresponds to Figure 4.

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