# The Period Distribution of Hot Jupiters is Not Dependent on Host Star
Metallicity

Samuel W. Yee Joshua N. Winn Department of Astrophysical Sciences, Princeton
University, 4 Ivy Lane, Princeton, NJ 08544, USA

###### Abstract

The probability that a Sun-like star has a close-orbiting giant planet (period
$\lesssim$ 1 year) increases with stellar metallicity. Previous work provided
evidence that the period distribution of close-orbiting giant planets is also
linked to metallicity, hinting that there two formation/evolution pathways for
such objects, one of which is more probable in high-metallicity environments.
Here, we check for differences in the period distribution of hot Jupiters
($P<10$ days) as a function of host star metallicity, drawing on a sample of
232 transiting hot Jupiters and homogeneously-derived metallicities from Gaia
Data Release 3. We found no evidence for any metallicity dependence; the
period distributions of hot Jupiters around metal-poor and metal-rich stars
are indistinguishable. As a byproduct of this study, we provide
transformations between metallicities from the Gaia Radial Velocity
Spectrograph and from traditional high-resolution optical spectroscopy of
main-sequence FGK stars.

##  1 Introduction

One of the goals of exoplanet demographics – measuring planet occurrence rates
and the statistical distributions of their properties – is to improve our
understanding of planet formation. One of the earliest demographic discoveries
was the strong and positive association between stellar metallicity and giant-
planet occurrence (Gonzalez, 1997; Santos et al., 2004; Fischer & Valenti,
2005). This “metallicity effect” is often interpreted as evidence in favor of
the core-accretion theory of planet formation, and is observed to be stronger
for hot Jupiters than for smaller or longer-period planets (Petigura et al.,
2018).

Another important topic in giant-planet demographics is the measurement of
their orbital period distribution. Early results from the radial-velocity
surveys (e.g., Udry et al., 2003; Butler et al., 2006; Cumming et al., 2008)
found that the period distribution has a peak at 3–5 days, falling at
intermediate periods followed by a gradual rise as the period increases
further. Early transit surveys found that the very shortest-period giant
planets ($P<3$ days) are rarer than those with periods between 3 and 10 days
(e.g., Gaudi et al., 2005), after correcting for detection biases. These
results gave rise to the notion of a “three-day pile-up” in the period
distribution of giant planets, Later, Santerne et al. (2016) used the Kepler
data to detect a 2-$\sigma$ peak centered on $P\approx$ 3–5 days in the giant-
planet occurrence rate, supporting the general notion of a “pile-up”, although
we caution that this term has not been used consistently in the literature.
Some authors have in mind a narrow peak at 3-days wholly contained within the
$P<10$ day range of hot Jupiters, while others are referring to a general
excess of hot Jupiters relative to warm Jupiters.

Dawson & Murray-Clay (2013) had the insight to search for a connection between
the metallicity effect and the giant-planet period distribution. They examined
all of the giant-planet candidates with $P\lesssim 500$ days that had been
detected by NASA’s Kepler mission, and examined the period distributions
separately for stars with super-solar and sub-solar metallicities. They found
that the distributions differed significantly, reporting that “only metal-rich
stars host a pile-up of hot Jupiters.” These findings, along with evidence
that high eccentricities of warm Jupiters ($0.1<a<1$ AU) are associated with
high metallicities, led Dawson & Murray-Clay (2013) to propose that both disk-
driven migration and high-eccentricity migration produce hot Jupiters. In this
picture, disk-driven migration does not produce a period pile-up and depends
weakly if at all on metallicity; high-eccentricity migration produces a pile-
up of giant planets at short orbital periods and occurs more often in metal-
rich disks, where multiple giant planets are likelier to form and trigger a
dynamical instability.

Although the statistical tests performed by Dawson & Murray-Clay (2013) were
based on giant planets with periods ranging out to 500 days, if their
interpretation is correct, one might expect differences between the metal-rich
and metal-poor samples even when restricting the period range to $P<10$ days.
With this in mind, Hellier et al. (2017) examined period distribution of all
the transiting giant planets with $P<22$ days that had been confirmed with
mass measurements and with measured host star metallicities. Their sample of
271 planets consisted mainly of hot Jupiters discovered by the ground-based
transit surveys, as opposed to Kepler, because most of the Kepler hot Jupiters
had not been confirmed (and remain unconfirmed today). They found no evidence
for any statistical difference between the period distributions of giant
planets orbiting metal-rich and metal-poor stars within this period range.
However, the sample was drawn from a heterogeneous collection of surveys,
involved a wide range of stellar types, and adopted metallicities derived by
many different authors.

We and others have been assembling a magnitude-limited sample of several
hundred transiting hot Jupiters ($P<10$ days), by combining the past two
decades of discoveries with new discoveries from the NASA Transiting Exoplanet
Survey Satellite (TESS; Ricker et al., 2015). Our contribution goes by the
name of the Grand Unified Hot Jupiter Survey (Yee et al., 2022, 2023). Objects
in this sample orbit brighter stars and have been vetted more thoroughly than
the Kepler sample, while being more complete and homogeneous than the
collection studied by Hellier et al. (2017). Spectroscopic metallicities are
available in almost all cases from ground-based high-resolution optical
spectroscopy and/or from the Radial Velocity Spectrometer (RVS) aboard the ESA
Gaia spacecraft (Gaia Collaboration et al., 2016; Cropper et al., 2018; Gaia
Collaboration et al., 2022). Given the increased size, reliability and
completeness (§3.2) of the currently available sample, we decided to revisit
the relationship between the metallicity effect and the period distribution of
hot Jupiters.

Figure 1: Left: Period distribution of confirmed hot Jupiters in the NASA
Exoplanet Archive orbiting FGK stars brighter than $G=12.0$ mag, split by
metallicity relative to the median. The metallicities were drawn from the
literature (§2). Right: Period distribution of an enlarged sample of hot
Jupiters with $G<12$, split by metallicity relative to the median. In this
case, the metallicities are based on calibrated Gaia spectroscopic (GSP-Spec)
metallicities (§3). In both cases, the observed distribution has a peak at 3–5
days, although we caution that this does not necessarily correspond to a peak
in the intrinsic occurrence rate; the distributions were not corrected for
detection efficiency. K-S tests do not reject the hypothesis that the period
distributions of the metal-poor and metal-rich samples are drawn from the same
distribution.

##  2 A Large Heterogeneous Sample

For a first glimpse at the results, we used data from the NASA Exoplanet
Archive (2022) (NEA).111https://exoplanetarchive.ipac.caltech.edu/ We queried
the database on November 2, 2022 for transiting hot Jupiters, defined as a
planet with a radius between 8 and 24 $R_{\oplus}$ and an orbital period
$P<10$ days. We also required the star to have an effective temperature
between 4400 and 6600 K, a radius smaller than 2.2$\,R_{\odot}$, and an
apparent Gaia magnitude brighter than 12, the limiting magnitude for which
completeness is expected to be high (Yee et al., 2021). This resulted in a
sample of 202 confirmed transiting hot Jupiters around main-sequence FGK
stars.

Using the “default” NEA metallicities, which are drawn from the various
sources in the literature, we split the sample into low- and high-metallicity
subsamples relative to the median metallicity of [Fe/H]$=+0.09$ dex. The left
panel of Figure 1 shows the orbital period distribution of each subsample.
Visually, the two distributions do not seem significantly different. Both
distributions exhibit a peak at 3–5 days, although it is premature to conclude
that the intrinsic period distribution has any such peak, because the transit
detection efficiency declines steeply with increasing period (see, e.g. Gaudi
et al., 2005). A two-sample Kolmogorov-Smirnov (K-S) test yielded a $p$-value
of 0.31, and therefore cannot reject the null hypothesis that the orbital
periods are drawn from the same parent distribution. We repeated the same test
after splitting the sample at $\mathrm{[Fe/H]}=0.0$ dex, instead of the median
metallicity, to match the breakpoint chosen by Dawson & Murray-Clay (2013) and
Hellier et al. (2017). Again, we found that the period distributions are
indistinguishable via the K-S test, with $p=0.90$.

##  3 A More Homogeneous Sample

Next, we assembled a more homogeneous dataset of host star metallicities
(§3.1), and considered the effects of sample incompleteness (§3.2).

###  3.1 Gaia Metallicities

Figure 2: The Gaia GSP-Spec metallicities are biased as a function of the
inferred metallicity (top left panel). After fitting a linear function to the
residuals between the SPOCS and Gaia $\mathrm{[M/H]}$ values, the remaining
residuals still exhibit a trend as a function of Gaia
$G_{\mathrm{BP}}-G_{\mathrm{RP}}$ color (bottom left panel). We removed this
trend by fitting a piecewise linear function with a breakpoint at 0.95. Using
the calibration coefficients from a joint fit to these variables, the rms
residual between the SPOCS and calibrated Gaia $\mathrm{[M/H]}$ values is
0.064 dex (right panels).

The NEA metallicities were derived using many different instruments and
analysis methods. Given the varying resolution, spectral coverage, and signal-
to-noise ratios of the spectroscopic observations, and the variety of codes
and spectral libraries that were employed, we expect systematic offsets on the
order of 0.1–0.2 dex that prevent more precise comparisons between
metallicities in the sample (e.g., Petigura et al., 2017). Some metallicities
were also derived from the less accurate method of fitting stellar-
evolutionary models to broad-band photometric measurements.

Gaia Data Release 3 (DR3; Gaia Collaboration et al., 2022) provides a new and
more homogeneous source of spectroscopic metallicities. The RVS instrument is
a spectrograph with $R\approx 11{,}500$ covering the wavelength range 846–870
nm (Cropper et al., 2018). Stellar atmospheric parameters are available for
about 5.6 million stars, including almost all of the stars brighter than
$G=12$ mag (Recio-Blanco et al., 2022). The Gaia DR3 AtmosphericParameters
table contains $T_{\mathrm{eff}}$, $\log{g}$, $\mathrm{[M/H]}$,
$\mathrm{[}\alpha\mathrm{/Fe]}$, and up to 15 individual elemental abundances
derived from the GSP-Spec MatisseGaugin workflow, which used an iterative
procedure to match the observed spectra to a grid of synthetic templates.
Confidence intervals for these parameters were determined through Monte-Carlo
resampling of the flux uncertainties on each wavelength pixel.

The large size and uniformity of this catalog made it attractive for our work,
but first we checked for biases and offsets relative to other spectroscopic
catalogs. Recio-Blanco et al. (2022) found that the RVS $\log{g}$ and
$\mathrm{[M/H]}$ determinations have a gravity-dependent bias and proposed
polynomial corrections based on comparisons with the APOGEE, GALAH, and RAVE
spectroscopic surveys. We decided to perform our own calibration focused on
the main-sequence FGK stars.

We took as our ground truth the Spectroscopic Properties of Cool Stars (SPOCS)
sample from Brewer et al. (2016), who analyzed high-resolution, high signal-
to-noise Keck/HIRES spectra of 1,617 FGK stars. They used the same line list
in all cases, and fitted each spectrum with the same code (SME, Valenti &
Piskunov, 1996) to derive spectroscopic properties and abundances for these
targets. They achieved precisions of 25 K in $T_{\mathrm{eff}}$, 0.01 dex in
$\mathrm{[M/H]}$, and 0.028 dex in $\log{g}$, as determined from separate
observations of the same star. For this analysis, we restricted the SPOCS
sample to the 1,275 stars with $\log{g}\geq 3.8$.

We downloaded the GSP-Spec atmospheric parameters for these stars from the
Gaia archive.222https://gea.esac.esa.int/archive/ Gaia parameters were
available for 1,150, or 90%, of the SPOCS sample. In some cases, the
parameters had quality flags indicating possible problems. We imposed the
strictest quality cuts, requiring all quality flags to be 0 except for the
fluxNoise flag which was allowed to be 0 or 1 (see Recio-Blanco et al. (2022)
for details on each flag). This left 677 remaining stars.

After some experimentation, we found that the best calibration was achieved by
fitting the SPOCS $\mathrm{[M/H]}$ values with a combination of a linear
function of the GSP-Spec $\mathrm{[M/H]}$ and a piecewise linear function of
the $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ color (Figure 2):

$\displaystyle\mathrm{[M/H]}_{\mathrm{cal}}=\,$ $\displaystyle
a+b\mathrm{[M/H]}_{\mathrm{GSP\mathchar 45\relax
Spec}}+c(\mathrm{BP\text{--}RP})$ (1)
$\displaystyle+d\left(\mathrm{BP\text{--}RP}-0.95\right)\cdot
H(\mathrm{BP\text{--}RP}-0.95),$

where $H(x)$ is the Heaviside function. Thus, the breakpoint of the piecewise
linear function is at $G_{\mathrm{BP}}-G_{\mathrm{RP}}=0.95$. The need for a
broken linear correction in color is likely due to temperature-dependent
systematic differences in the model atmospheres used, with the breakpoint
corresponding roughly to the color stars of solar temperature. The best-fit
calibration coefficients are $a=0.384$, $b=0.761$, $c=-0.392$, and $d=0.708$.
Applying this calibration reduced the root-mean-squared (rms) difference
between SPOCS and GSP-Spec metallicities from 0.10 dex to 0.064 dex. We saw no
correlations in the residuals with the SPOCS $T_{\mathrm{eff}}$ or $\log{g}$
parameters. This calibration was used to correct the Gaia GSP-Spec parameters
for our sample of hot Jupiter hosts, providing a more homogeneous catalog of
spectroscopic metallicities.

###  3.2 Sample Completeness

Most of the hot Jupiters in the NEA were discovered in wide-field ground-based
transit surveys that were subject to complex selection effects, such as a
strong bias favoring the detection of short-period planets (Gaudi et al.,
2005). In addition, the parent population of stars that were thoroughly
searched for hot Jupiters is not well characterized. In Yee et al. (2021), we
simulated the expected characteristics of a magnitude-limited sample, based on
Kepler statistics, and found that the collection of hot Jupiters known at the
time was probably only $\approx$45% complete down to $G=12$. A sample with low
completeness is still acceptable for comparing the period distributions of
metal-poor and metal-rich stars are still valid and interpretable, but only if
the selection effects do not depend on metallicity. For example, although the
geometric transit probability distorts the observed period distribution away
from the intrinsic distribution, this applies equally to all planets, and will
not affect our comparison apart from determining the overall number of planets
in the sample.

However, it is important to check on the possibility that our ability to
detect a transiting giant planet at fixed orbital period and apparent
magnitude depends on the host star’s metallicity. For this purpose, we used
data from NASA’s Transiting Exoplanet Survey Satellite (TESS; Ricker et al.
2015). We downloaded all the light-curves generated by the TESS Quick Look
Pipeline (QLP; Huang et al. 2020a, b; Fausnaugh et al. 2020) for FGK stars
with $G<12$ mag that were observed in Sector 5 (chosen arbitrarily). For each
star, we used the lightkurve package (Lightkurve Collaboration et al., 2018)
to compute the Combined Differential Photometric Precision (CDPP; Christiansen
et al. 2012; van Cleve et al. 2016) on a 2-hour timescale. The transit
detection probability is chiefly a function of CDPP.

Figure 3 shows the CDPP as a function of Gaia apparent magnitude. Also shown
are CDPP quantiles for the metal-poor and metal-rich stars separately, based
on the calibrated Gaia GSP-Spec metallicities. We found no significant
difference in the photometric noise properties of the two subsamples. This
eliminates the possibility that metal-rich stars have systematically noisier
light curves than metal-poor stars, or vice versa.

Figure 3: Estimated 2-hr CDPP values for FGK stars brighter than $G=12$ mag
observed by TESS in Sector 5 (blue dots), as a function of apparent magnitude.
The solid and dashed lines show the 90th and 50th percentiles in magnitude
bins of width 0.05. The black and red lines represent high and low metallicity
subsamples, respectively, divided at the median metallicity of the hot Jupiter
host sample.

Despite this reassurance, there may be other sources of metallicity-dependent
period biases. Using a more complete sample of planets would reduce the
possibility that a large number of undetected planets exist that have a
different metallicity/period distribution. Constructing a magnitude-limited
sample of hot Jupiters is the goal of the Grand Unified Hot Jupiter Survey
(Yee et al., 2022, 2023). Although this goal has not yet been achieved,
completeness has been substantially increased over the last few years. There
are also many hot Jupiters for which confirmation has been completed but has
only been described in preprints or works in preparation, and are therefore
not included in the NEA.

We decided to augment the NEA-based sample with 16 hot Jupiters described only
in preprints (Anderson et al., 2014, 2018; Bakos et al., 2016; Brown et al.,
2019; Rodriguez et al., 2023; Sha et al., 2022; Yee et al., 2023) at the time
of archive query, and 22 that will be described in forthcoming papers by
Schulte et al., Yee et al., and Quinn et al. that meet the same criteria on
planet and stellar properties described in §2. This brought the total sample
size to 240. Based on the simulations in Yee et al. (2021), we expect that
this enlarged sample is $\approx$80% complete to $G=12$, and therefore that
any biases are limited to the 10–20% level.

Gaia GSP-Spec metallicities are available for 232 (97%) of the host stars in
the sample. After performing the correction described in Section 3.1, the
median metallicity is $+0.171$ dex, consistent with previous homogeneous
studies of giant-planet hosts (Buchhave et al., 2018; Petigura et al., 2018).
Thus, the high completeness, high reliability, and homogeneous metallicity
scale for this sample improves over the previous works by Dawson & Murray-Clay
(2013) and Hellier et al. (2017).

The right panel of Figure 1 displays the period distributions of the high-
metallicity and low-metallicity halves of the sample, which appear to be
similar. A K-S test gives a $p$-value of 0.46. When the sample is split at
$\mathrm{[M/H]}=0$ instead of the median value of $+0.171$, the $p$-value is
0.64. In neither case can we reject the hypothesis that the periods in each
subsample are drawn from the same distribution.

##  4 Kepler Sample

Figure 4:  Period distributions of hot-Jupiter candidates from the first eight
quarters of the Kepler mission (the same catalog used by Dawson & Murray-Clay
(2013)). Within the $P<10$ day range, the period distribution for planets
orbiting metal-rich stars (top) appears distinct from that for metal-poor
stars (bottom), but the number of objects is too small to draw any statistical
conclusions (K-S test $p$-value = 0.24). Furthermore, when excluding known
false positives (FPs, shaded region), leaving only planet candidates (PCs),
the differences between the two distributions are reduced.

With our larger and purer sample of hot Jupiters, we did not find any evidence
that their orbital period distribution depends on the metallicity of their
host stars. We sought to understand if this finding is discrepant with the
period distribution of Kepler hot Jupiters.

Dawson & Murray-Clay (2013) found a significant difference when comparing the
period distributions of all the Kepler giant-planet candidates, split into
metal-rich and metal-poor samples. This is in contrast to our comparisons,
which were restricted to $P<10$ days. Nonetheless, it is worth investigating
if our result is corroborated in the Kepler data, which has the advantage of
being from a single homogeneous survey that would have detected essentially
all the transiting short-period giant planets orbiting FGK stars for which
data were collected.

We downloaded the list of planet candidates from the first eight quarters of
the Kepler mission (Burke et al., 2014),the same catalog examined in Dawson &
Murray-Clay (2013). We made the same cuts on planet and stellar host
properties as in §2, restricted the sample to $P<10$ days, and split the
sample according to the metallicities provided in the Kepler Input Catalog
(KIC; Brown et al. 2011) , using a boundary of $\mathrm{[Fe/H]}=0.0$. Figure 4
shows the period distribution of these two subsamples. The period distribution
for planets orbiting metal-rich stars appears to have a peak at $\sim 3$ days,
while the number of planets orbiting metal-poor stars increases steadily out
to $P=10$ days.

However, even though Kepler searched $\sim 10^{5}$ stars, the intrinsic
occurrence rate of hot Jupiters is low, resulting in a small sample. The full
Q1-Q8 Kepler sample contains only 56 hot Jupiter candidates. A K-S test
comparing only the Kepler hot Jupiters gives a $p$-value of 0.24, too large to
conclude that the period distributions differ between metal-poor and metal-
rich hosts.

Furthermore, most of the stars that were searched by the Kepler mission are
too faint for detailed follow-up and characterization of planet candidates.
Even today, most of the planets remain only as “validated” (deemed likely to
be planets based on statistical considerations) as opposed to “confirmed” by
Doppler spectroscopy. Of the 56 hot Jupiter candidates, seven were
subsequently determined to be false positives, and five others were still
unvalidated candidates as of the end of the mission (Twicken et al., 2016). If
we remove the known false positives from the orbital period distribution
(shaded histogram in Figure 4), the differences between the two subsamples
diminish further. Thus, our results do not contradict the results of the
Kepler survey.

##  5 Discussion

Does the lack of any detectable difference between the orbital period
distributions of hot Jupiters orbiting metal-rich stars and metal-poor stars
allow us to place any constraints on the relative fraction of hot Jupiters
that formed in different ways? Since we are not aware of any theory that
provides specific quantitative predictions for the period distribution, we
answered this question using a relatively simple parametric model inspired by
the work of Ford & Rasio (2006) and Nelson et al. (2017).

If hot Jupiters migrate to their current orbital positions through high-
eccentricity migration, the minimum periastron distance they can reach without
becoming tidally disrupted is the Roche distance, $a_{\mathrm{R}}$. Planetary
orbits that initially have large semimajor axes (large enough to extend beyond
the “ice line” where giant planet formation is expected) and a periastron
distance of $a_{\mathrm{R}}$ will end up on circular orbits of radius
$2a_{\mathrm{R}}$, after tidal circularization and assuming no loss of angular
momentum. Thus, in this scenario, we expect the distribution of orbital
distances to cut off below $2a_{\mathrm{R}}$, corresponding to a minimum
orbital period as a function of the planet’s mean density (Rappaport et al.,
2013). In contrast, if hot Jupiters arise from disk-driven migration, they may
be able to migrate all the way to a circular orbit of radius $a_{\mathrm{R}}$.

Based on this reasoning, Nelson et al. (2017) modeled the giant planet
population using a two-component truncated power law model, $dN/dx\propto
x^{\gamma-1}$, where $x\equiv a/a_{\mathrm{R}}$ and the lower truncation limit
is $x=1$ for disk-driven migration and $x=2$ for high-eccentricity migration.
They found that the data could be explained if the disk-driven migration
component (Pop. 1) follows a shallow power law, and the high-eccentricity
migration component (Pop. 2) follows a steeper power law. They found the
fraction of planets belonging to Pop. 1 to be 0.15–0.35, depending on the
sample used. They also found that such a model was preferred over a single-
component power law.

In this picture, one might expect the occurrence of Pop. 1 planets to have a
relatively weak dependence on metallicity, while the occurrence of Pop. 2
planets is higher for metal-rich stars that can form multiple giant planets
that eventually undergo scattering events. We simulated both populations of
planets, sampling $x$ from power law distributions with parameters fixed at
the best-fit values from Nelson et al. (2017)
($x_{l,1}=1,\gamma_{1}=-0.04,x_{l,2}=2,\gamma_{2}==1.38$), and planet masses
and radii drawn from those of the known hot Jupiters. We then drew random
samples of 240 planets each from the two populations, varying the relative
fraction $f_{1}$ of planets in Pop. 1 between 0.1 and 0.5, and accounting for
transit probability (Figure 5, top panel). We assigned planets from Pop. 1 to
stars brighter than $G<12$ from the Gaia DR3 catalog (Gaia Collaboration et
al., 2022) independently of metallicity. Meanwhile, planets from Pop. 2 were
assigned to stars with probability $\mathrm{Pr.}\propto
10^{\beta\mathrm{\mathrm{[Fe/H]}}}$, the exponential relation seen for the
overall hot Jupiter population (e.g., Valenti & Fischer, 2005). We chose
$\beta=3.4$ based on the study of Kepler planet occurrence by Petigura et al.
(2018).

Using these simulated planet samples, we performed the same K-S tests that we
performed on the actual sample of planets, and repeated this process 1,000
times for each choice of the relative fraction $f_{1}$. As $f_{1}$ is
increased, the peak of the orbital period distribution in the metal-poor
subsample shifts to smaller orbital periods. The lower panel of Figure 5 shows
the fraction of simulations in which the low-metallicity and high-metallicity
subsamples had differences in orbital period distribution larger than that
seen in the observed planet sample as measured by the K-S statistic. We also
counted the fraction of trials in which the K-S test could reject with 95%
confidence the null hypothesis that the metal-rich and metal-poor subsamples
are drawn from the same distribution. When the sample comprises a roughly
equal admixture of both populations ($0.35\lesssim f_{1}\lesssim 0.75$), the
null hypothesis was rejected in the majority of simulations. When the relative
fraction of either population is smaller ($f_{1}\lesssim 0.3$ or $f_{1}\gtrsim
0.8$), there is no longer a significant difference in the period distributions
of the two subsamples.

Thus, in the context of our adopted model, the absence of a detectable
difference between the period distributions of hot Jupiters orbiting metal-
rich and metal-poor stars rules out $0.35\lesssim f_{1}\lesssim 0.75$, and
allows for smaller relative fractions of either population. However, we stress
that this analysis is based on a simple parametric model in which most of the
constraining power arises from the shortest period planets; more detailed
theoretical predictions of the form of the orbital period distribution arising
from these two migration pathways would likely be more informative, even when
applied to the same sample.

Figure 5: Top: Example Monte Carlo sample of planets. The blue histogram is
for the disk-driven migration component, obtained by drawing $x\equiv
a/a_{\mathrm{R}}$ from a shallow power-law distribution ($\gamma_{1}=-0.04$)
truncated at $x=1$., blue histogram) The orange histogram is for the high-
eccentricity migration component, obtained by drawing $x$ from a steeper
power-law distribution($\gamma_{2}=-1.38$) truncated at $x=2$. Both
distributions were further attenuated by the transit probability,
$R_{\star}/a$. Bottom: Blue line: The fraction of simulations where the K-S
statistic is larger than that for the observed hot Jupiters. Orange line: The
fraction of simulations in which the null hypothesis that the period
distribution is the same for high- and low-metallicity stars could be ruled
out by the K-S test with $>$95% confidence.

##  6 Conclusion

Studies of the metallicity distribution of planet-hosting stars is a promising
method for distinguishing planet populations that formed in different ways
(see, e.g., Winn et al., 2017; Schlaufman, 2018; Buchhave et al., 2018). We
revisited the possibility that the orbital period distribution of hot Jupiters
depends on their host stars’ metallicities, but found no evidence for such an
effect. This corroborates the results of Hellier et al. (2017) with a more
homogeneous and complete dataset.

Dawson & Murray-Clay (2013) used the Kepler data to argue that the short-
period pile-up of giant planets is a feature only of metal-rich host stars.
The lack of difference in period distribution for planets within the $P<10$
day range weakens that line of argument. Nonetheless, Dawson & Murray-Clay
(2013) presented two other lines of evidence for this formation pathway: that
gas giants orbiting metal-rich stars span a range of eccentricities in
contrast to those orbiting metal-poor stars, and that most eccentric proto-hot
Jupiters appear to be orbiting metal-rich stars. These were based on studies
of planets found by the radial-velocity and ground-based transit surveys, and
are unaffected by our work.

The lack of a detectable metallicity dependence of the period distribution may
itself be a clue about the formation of hot Jupiters. Within the context of
the two-pathway model of Nelson et al. (2017) — in which a metallicity-
dependent formation pathway brings hot Jupiters close to the Roche limit,
while a metallicity-indepedent pathway delivers hot Jupiters to distances no
closer than twice the Roche limit — the data imply that one of these pathways
must produce $\gtrsim 70\%$ of the population. More quantitative theoretical
work to predict the period distribution from different formation pathways
(e.g., Heller, 2019) would be required to place more reliable constraints.
Alternatively, jointly modeling the orbital period and eccentricity
distributions may help us identify the planets arising from each pathway.

Thanks to the success of all-sky survey missions such as TESS and Gaia, hot
Jupiter demographics are coming into sharper focus. Previous work was based on
heterogeneous samples from ground-based transit surveys, or the smaller number
of planets drawn from Kepler or radial-velocity searches. The combination of
TESS and Gaia is allowing a much larger statistical sample to be assembled.
Combining a large sample of true planets with an assessment of the detection
efficiency of TESS will also enable a measurement of occurrence rates (Zhou et
al., 2019; Beleznay & Kunimoto, 2022) and their intrinsic period distribution.

We thank the anonymous referee for helpful suggestions which helped improve
and clarify this manuscriipt. This research 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. This research also made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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