# Statistical Analysis of the Dearth of Super-eccentric Jupiters in the Kepler
Sample

Jonathan M. Jackson Astronomy Department and Van Vleck Observatory, Wesleyan
University, 96 Foss Hill Drive, Middletown, CT 06459, USA  Department of
Astronomy & Astrophysics, Center for Exoplanets and Habitable Worlds, The
Pennsylvania State University, University Park, PA 16802, USA  Rebekah I.
Dawson Department of Astronomy & Astrophysics, Center for Exoplanets and
Habitable Worlds, The Pennsylvania State University, University Park, PA
16802, USA  Billy Quarles Department of Physics, Astronomy, Geosciences and
Engineering Technology, Valdosta State University, Valdosta, GA 31698, USA
Center for Relativistic Astrophysics, School of Physics, Georgia Institute of
Technology, Atlanta, GA 30332, USA  Jiayin Dong Flatiron Research Fellow
Department of Astronomy & Astrophysics, Center for Exoplanets and Habitable
Worlds, The Pennsylvania State University, University Park, PA 16802, USA
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA Jonathan M. Jackson<EMAIL_ADDRESS>
###### Abstract

Hot Jupiters may have formed in situ, or been delivered to their observed
short periods through one of two categories of migration mechanisms: disk
migration or high-eccentricity migration. If hot Jupiters were delivered by
high-eccentricity migration, we would expect to observe some “super-eccentric”
Jupiters in the process of migrating. We update a prediction for the number of
super-eccentric Jupiters we would expect to observe in the Kepler sample if
all hot Jupiters migrated through high-eccentricity migration and estimate the
true number observed by Kepler. We find that the observations fail to match
the prediction from high-eccentricity migration with 94.3% confidence and show
that high-eccentricity migration can account for at most $\sim 62\%$ of the
hot Jupiters discovered by Kepler.

##  1 Introduction

Hot Jupiters are challenging to form on the orbits we observe them on today
(e.g., Bodenheimer et al. 2000; Rafikov 2006). There are two main families of
theories used to explain the hypothesized migration of these planets: (1)
migration via interactions with the protoplanetary disk (Goldreich & Tremaine,
1980; Ward, 1997; Alibert et al., 2005; Ida & Lin, 2008; Bromley & Kenyon,
2011; Baruteau et al., 2014), and (2) high-eccentricity migration (HEM) (Wu &
Murray, 2003). In the latter case, the migrating planet is excited to large
eccentricity by von Zeipel-Lidov-Kozai oscillations (von Zeipel, 1910; Kozai,
1962; Lidov, 1962) via interactions with a binary star (Wu & Murray, 2003;
Fabrycky & Tremaine, 2007; Naoz et al., 2012) or planetary (Naoz et al., 2011;
Lithwick & Naoz, 2011) companion, planet-planet scattering (Rasio & Ford,
1996; Ford & Rasio, 2006, 2008; Chatterjee et al., 2008; Jurić & Tremaine,
2008; Matsumura et al., 2010; Nagasawa & Ida, 2011; Beaugé & Nesvorný, 2012;
Boley et al., 2012), or secular chaos (Wu & Lithwick, 2011), and then tidal
dissipation in the planet shrinks and circularizes its orbit.

If hot Jupiters are being delivered by HEM – migrating from extremely
eccentric long period orbits to shorter periods and intermediate
eccentricities to, finally, very short periods and circular orbits (i.e., hot
Jupiters) – we might catch them in the act at high and moderate
eccentricities. Planets along HEM ‘tracks’, defined by their final orbital
period at zero eccentricity, have approximately constant angular momentum and
transit probability. Socrates et al. (2012b) showed that the number of super-
eccentric Jupiters with $e>0.9$ along a given migration track can be predicted
from the number of moderately-eccentric Jupiters ($0.2<e<0.6$) on the same
track and that if HEM is the dominant channel for hot Jupiter delivery, the
Kepler mission should have detected a handful of these super-eccentric
Jupiters. Dawson et al. (2015) updated that prediction, accounting for Poisson
counting uncertainties and incompleteness, and then compared it to the
observed Kepler sample using the “photoeccentric effect” (e.g., Dawson &
Johnson, 2012) to estimate the eccentricities of the transiting Jupiters.
Dawson et al. (2015) found a paucity of super-eccentric Jupiters indicating
that their results were inconsistent with the prediction from HEM with 96.9%
confidence. Moreover, they calculated a two-sigma statistical upper limit on
the fraction of hot Jupiters that could have migrated through the super-
eccentric warm Jupiter parameter space of 44%. Thus, most hot Jupiters must
have bypassed this region of parameter space, perhaps by migrating smoothly
through disk migration or engaging in HEM with chaotic tides (e.g., Mardling
1995a, b; see Section 5.3).

Here we update the photoeccentric effect calculation from Dawson et al. (2015)
using new Kepler light curve fits (Quarles, in prep.), updated stellar
parameters from Gaia Data Release 2 (Berger et al., 2020), and hot Jupiters
discovered (or re-classified as hot Jupiters) after that study, resulting in a
45% larger sample. We then compare the predicted number of super-eccentric
Jupiters to those observed to produce a posterior for the fraction of hot
Jupiters that can be explained by HEM. In Section 2, we discuss the steady-
state HEM prediction from Socrates et al. (2012b), which was subsequently
updated by Dawson et al. (2015). We discuss the construction of our updated
sample of Kepler Jupiters and calculate eccentricities in Section 3. In
Section 4, we compare the expected number of super-eccentric Jupiters to those
observed in our updated data set. In Section 5, we discuss these results in
the context of other possible migration mechanisms and, in Section 6, we
summarize our work and discuss our conclusions.

##  2 Prediction for the expected number of super-eccentric Jupiters in the
Kepler sample

Here we update the prediction (Socrates et al., 2012b; Dawson et al., 2015)
for the expected number of super-eccentric Jupiters that should be found in
the Kepler sample if all the moderately eccentric Jupiters were delivered by a
steady-state flow of high-eccentricity migration (HEM). To predict the number
of super-eccentric Jupiters ($e>0.9$, $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{sup}}$) expected in
the Kepler sample, we must assume a population of migrating proto-hot Jupiters
being tidally circularized along a track of constant angular momentum defined
by $a_{\rm{final}}=a(1-e^{2})$ or, equivalently,
$P_{\rm{final}}=P(1-e^{2})^{3/2}$, where $a$, $e$, and $P$ represent the
current semi-major axis, eccentricity, and orbital period of the migrating
planet respectively. The number of super-eccentric Jupiters is then related to
the number of moderately-eccentric Jupiters ($0.2<e<0.6$, $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod}}$) along the
same track by a constant ratio ($r$) of time spent at super eccentricities to
moderate eccentricities. The ratio $r$ (defined below) depends on the assumed
functional form (but not absolute normalization) for the circularization
timescale.

Since most Kepler Jupiters do not have measured eccentricities, $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod}}$ is unknown
and we must approximate it using a calibration sample of non-Kepler Jupiters,
plotted in Figure 1. We update the calibration sample compiled by Socrates et
al. (2012b) and Dawson et al. (2015) to account for newly discovered planets
and updates to their parameters (See Appendix A). We include $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod},0}=23$
moderately eccentric calibration Jupiters and $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{P=P_{\rm{final}},0}=176$ calibration Jupiters with $2.8<P<10$ days,
roughly doubling the size of the Dawson et al. (2015) calibration sample.

Figure 1: Eccentricity vs. semi-major axis distribution of Jupiter-mass
($>0.25M_{\rm{Jup}}$) confirmed planets orbiting main sequence stars (see
Section 3 for exact cuts). Here $1-e^{2}$ is plotted on the y-axis to
emphasize elliptical orbits. Hot Jupiters are plotted as red circles,
moderately eccentric ($0.2<e<0.6$) Jupiters on tidal migration tracks are
plotted as blue squares in the orange-shaded region, and more distant Jupiters
($P>10$ days) are plotted as gray circles. Approximate periods shown on the
upper axis are calculated assuming $M_{*}=1M_{\sun}$. The range of potential
tidal migration tracks is outlined with dashed lines and two intervals of
migration tracks, defined by $2.8<P_{\rm{final}}<5$ and $5<P_{\rm{final}}<10$
days, are delineated by the dotted line. The super-eccentric region ($e>0.9$)
within these tracks is shaded blue. The upper limit on the completeness of the
Kepler sample (2 years) is shown by the vertical purple line. Error bars have
been removed for clarity, but see Dawson & Johnson (2018) Figure 4 for a
similar plot with uncertainties included. Data taken from the NASA Exoplanet
Archive on 2022 May 6 (NASA Exoplanet Science Institute, 2020).

Assuming the fraction of moderately eccentric Jupiters to hot Jupiters is the
same between the calibration and Kepler samples, we can then combine this
fraction with the number of Kepler Jupiters with $P=P_{\rm{final}}$ ($\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{P=P_{\rm{final}}}$) to
calculate the expected number of super-eccentric Jupiters,

$\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{\rm{sup}}=\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{\rm{mod}}r(e_{\rm{max}})=\frac{\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod},0}}{\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{P=P_{\rm{final}},0}}\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{P=P_{\rm{final}}}r(e_{\rm{max}}),$ (1)

where $e_{\rm{max}}=[1-(P_{\rm{final}}/P_{\rm{max}})^{2/3}]^{1/2}$ is set by
the maximum observable period, $P_{\rm{max}}$. The bar over each $N$ indicates
that they are mean numbers. We treat the observationally counted numbers as
samples from a Poisson distribution with an unknown mean. For each sample, we
assume a Jeffrey’s prior for the mean and calculate a posterior distribution
of Poisson means, following the procedure outlined in Dawson et al. (2015)
Appendix B. We summarize the means and 95% confidence intervals of these
distributions in Table 1.

Accounting for incompleteness, under the tidal time lag approximation
(Eggleton et al., 1998; Hansen, 2010; Socrates & Katz, 2012; Socrates et al.,
2012a, b; Dawson et al., 2015), the ratio of time spent at super
eccentricities to moderate eccentricities for a given $P_{\rm{final}}$ is
defined as:

$r(e_{\rm{max}})=\frac{\int_{0.9}^{e_{\rm{max}}}C_{\rm{comp,sampled}}(e)|\dot{e}|^{-1}de}{\int_{0.2}^{0.6}|\dot{e}|^{-1}de},$
(2)

where $C_{\rm{comp,sampled}}(e)$ is the completeness accounting for missing
data, or the fraction of phases for which we would observe three or more
transits during Q1-Q17 of Kepler. See Dawson et al. (2015) for further
discussion of the completeness of the sample.

Dawson et al. (2015) calculate $r$ to be 0.913, 0.668, 0.422 for
$P_{\rm{final}}=$ 2.8, 5, 10 days, respectively. We can plug these values into
Equation 1 along with our posteriors for $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod},0}$, $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{P=P_{\rm{final}},0}$,
and $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{P=P_{\rm{final}}}$ to produce an approximate $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{sup}}$. Figure 2
shows the posteriors for $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{sup}}$ in the
$P_{\rm{final}}$ ranges 2.8-5, 5-10, and 2.8-10 days. The predicted number
observed (bottom panel) is sampled from the distribution of Poisson means (top
panel) calculated by Equation 1. With a significantly larger calibration
sample than Dawson et al. (2015), we predict a similar number super-eccentric
Jupiters expected in the Kepler sample with expectation value $\mkern
1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{\rm{sup}}=4.5^{+1.5}_{-1.1}$. In Section 4, we will use this prediction
for $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{sup}}$
in the Kepler sample to set limits on the HEM process.

Figure 2: Top: Histogram depicting the relative probability distribution of the predicted mean number of super-eccentric Jupiters in $P_{\rm{final}}$ interval 1 (red, dotted), interval 2 (blue, dashed), and combined (black). Bottom: Histogram depicting the relative probability distribution of the expected number of observed super-eccentric Jupiters, created by sampling from the above distribution of Poisson means.  Table 1: Counted Planets $e$ | Interval |  | Counted |  |  | MeanaaMedian, with 68.3% confidence interval, of the posterior of Poisson means, each defining a Poission distribution from which the counted number may be sampled. |  | SamplebbKep: Kepler sample; Cal: calibration non-Kepler sample
---|---|---|---|---|---|---|---|---
| (days) |  |  |  |  |  |  |
$0.2<e<0.6$ | 1: 2.8-5 | $N_{\rm{mod},0}$ | = | $9$ | $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod},0}$ | = | $9^{+4}_{-3}$ | Cal
| 2: 5-10 | $N_{\rm{mod},0}$ | = | $16$ | $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{\rm{mod},0}$ | = | $16^{+5}_{-4}$ | Cal
unspecified | 1: 2.8-5 | $N_{P=P_{\rm{final}},0}$ | = | $134$ | $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{P=P_{\rm{final}},0}$ | = | $134^{+12}_{-11}$ | Cal
| 1: 2.8-5 | $N_{P=P_{\rm{final}}}$ | = | $24$ | $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{P=P_{\rm{final}}}$ | = | $24^{+5}_{-4}$ | Kep
| 2: 5-10 | $N_{P=P_{\rm{final}},0}$ | = | $42$ | $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{P=P_{\rm{final}},0}$ | = | $42^{+7}_{-6}$ | Cal
| 2: 5-10 | $N_{P=P_{\rm{final}}}$ | = | $15$ | $\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu_{P=P_{\rm{final}}}$ | = | $15^{+4}_{-3}$ | Kep

##  3 Observations

Here we update and expand the sample of Kepler warm Jupiter systems from
Dawson et al. (2015) using new stellar parameters from Gaia (Berger et al.,
2020) and improved transit light curve fitting (Quarles, in prep.). These
updates improve the precision of the measured stellar densities and light
curve fit parameters, which we use to compute probability distributions for
each planet candidate’s eccentricity.

Following Dawson et al. (2015), we identify planet candidates in the Kepler
sample with radii ($R_{p}$) between $8R_{\earth}$ and $22R_{\earth}$
(including candidates whose 95% radius confidence intervals only slightly
overlap with this range), periods ($P$) between 34 days and 2 years, stellar
temperatures ($T_{\rm{eff}}$) between $4500K$ and $6500K$, and stellar surface
gravities ($\rm{log}g$) $>4$ (or consistent with 4 within the uncertainty). We
also require the candidates to transit 3 times in Q1-Q17, have signal-to-noise
ratios above 10, and pass basic false-positive vetting according to the Kepler
data validation report. Several candidates whose planetary radius posteriors
only slightly overlap with our desired range are included, but will be
weighted proportionally in our statistical analyses in Section 4. These cuts
produce 45 candidates in our sample, an increase of 45% from the Dawson et al.
(2015) sample. Many of these initially failed the planet radius cut, but have
seen their estimated radii increase with updated stellar parameters from Gaia.
Table 2 lists all of our candidates and their properties. We note that,
although the sample has grown substantially, many of the newly included planet
candidates have nearby companions, making them unsuitable for high-
eccentricity migration (e.g., Mustill et al. 2015), and are excluded from our
calculations in Section 4, as well as Table 2. We also exclude Kepler-767 b
because it has an interior debris disk that would overlap with the planet’s
orbit if the planet were super eccentric (Lawler & Gladman, 2012). Kepler-419
b, Kepler-539 b, and Kepler 1657 b are also excluded because $e>0.9$ has been
ruled out by radial velocity followup and modeling (Dawson et al., 2014;
Mancini et al., 2016; Almenara et al., 2018; Jackson et al., 2019; Hébrard et
al., 2019).

We discuss the updates to this sample from the Dawson et al. (2015) sample in
Appendix A. We also update the sample of Kepler hot Jupiters, which we
describe in Appendix A. The net change in the counted number of hot Jupiters,
listed in Table 1, is 1.

The eccentricity of a planet can affect the observed light curve in many ways
and, most notably, can have a detectable effect on the transit duration
(Barnes, 2007; Ford et al., 2008; Kipping, 2008). We look for highly
elliptical planet candidates by combining directly observed parameters from
the transit light curve and the star itself. By fitting the light curve under
the assumption of a circular orbit, we can produce an estimate of the stellar
density under this assumption, $\rho_{\rm{circ}}$, which is related to the
true stellar density, $\rho_{*}$, by:

$\rho_{*}(e,\omega)g^{3}(e,\omega)=\rho_{\rm{circ}},$ (3)

for which,

$g(e,\omega)=\frac{1+e\sin{\omega}}{\sqrt{1-e^{2}}}$ (4)

is the approximate ratio of the observed transit speed to the transit speed
the planet would have on a circular orbit with the same orbital period for a
given argument of pericenter, $\omega$ (Kipping, 2010; Dawson & Johnson,
2012).

We combine the posteriors of $\rho_{\rm{circ}}$ from Kepler light curves
(Quarles, in prep.) and $\rho_{*}$ from stellar parameters including Gaia
measurements (Berger et al., 2020) into a single $\rho_{\rm{circ}}/\rho_{*}$
distribution for each of the planets in our sample, marginalized over all
other parameters. We then plot the resulting values in Figure 3 over the
2-dimensional probability distribution of the predicted super-eccentric warm
Jupiters. Red bars represent candidates without known companions and blue bars
represent planets with known transiting planet companions. We omit candidates
from the plot whose radius posteriors contain $<10\%$ overlap with our warm
Jupiter radius constraints. All of the multiplanet systems have interior
companions or would result in unstable orbit crossings if the warm Jupiter had
an eccentricity $>0.9$.

Based on the prediction in Section 2, we would expect to see $\sim 4$-5
candidates in the high-density region of Figure 3 if all the moderately
eccentric Jupiters were delivered through a steady-state HEM process. While no
planets appear to lie directly in the high-probability density region, several
planets have some small probability of having $e>0.9$ and
$2.8<P_{\rm{final}}<10$. In fact, 6 candidates have mean
$\rho_{\rm{circ}}/\rho_{*}>10$, indicating their eccentricities are unlikely
to be zero. We calculate the probability that each candidate is super-
eccentric and compare the observations to our predictions in Section 4.

Table 2: Warm Jupiter Planet Candidate Sample KIC | KOI | Kepler | Period |  $\rho_{\rm{circ}}/\rho_{*}$bbPosterior median and 68% confidence interval of the ratio of mean $\rho_{\rm{circ}}$ to mean $\rho_{*}$ | Radius | WeightccFraction of planetary radius posterior that falls within 8-22$R_{\earth}$ | Estimated $e$ddSee Section 4 | ProbabilityeePosterior probability of $e>0.9$ and $2.8<P_{\rm{final}}<10$ days. Values marked with ∗ indicate candidates those that appear in the eclipsing binary catalog (Kirk et al. 2016; see Section 4.1).
---|---|---|---|---|---|---|---|---
numberaaKIC number of host star | number | number | (days) |  | ($R_{\earth}$) |  |  |
3534076 | 1209.01 | - | 272.09 | $5.54_{-2.22}^{+4.20}$ | $5.95^{+2.67}_{-0.53}$ | 0.427 | $0.36_{-0.17}^{+0.22}$ | 1.3%
3942446 | 1193.01 | - | 119.06 | $22.46_{-5.30}^{+7.20}$ | $10.40^{+2.82}_{-1.89}$ | 0.392 | $0.84_{-0.06}^{+0.09}$ | 7.1%
4155328 | 1335.01 | 820b | 127.83 | $2.91_{-0.74}^{+0.47}$ | $9.46^{+1.56}_{-2.14}$ | 0.899 | $0.48_{-0.16}^{+0.30}$ | 4.2%
4544670 | 815.01 | - | 35.74 | $5.47_{-1.77}^{+3.52}$ | $7.61^{+1.72}_{-0.91}$ | 0.553 | $0.64_{-0.15}^{+0.20}$ | 0.1%
4566848 | 5071.01 | - | 180.41 | $14.43_{-3.33}^{+5.37}$ | $8.86^{+2.60}_{-0.86}$ | 0.099 | $0.78_{-0.08}^{+0.14}$ | 1.0%
4918309 | 1582.01 | - | 186.44 | $1.74_{-0.54}^{+1.62}$ | $11.32^{+1.80}_{-3.35}$ | 0.049 | $0.38_{-0.26}^{+0.34}$ | 0.1%
5184584 | 1564.01 | 891b | 53.45 | $9.96_{-2.25}^{+2.67}$ | $5.82^{+1.49}_{-0.50}$ | 1.000 | $0.73_{-0.09}^{+0.14}$ | 4.7%
6504954 | 1790.01 | 952b | 130.35 | $5.69_{-0.95}^{+0.76}$ | $7.70^{+2.53}_{-0.84}$ | 0.351 | $0.62_{-0.09}^{+0.21}$ | 1.8%∗
6766634 | 1375.01 | 1746b | 321.21 | $18.15_{-3.78}^{+6.99}$ | $7.25^{+1.16}_{-1.17}$ | 0.990 | $0.81_{-0.07}^{+0.11}$ | 5.4%
7211141 | 1355.01 | 827b | 51.93 | $5.85_{-2.09}^{+1.65}$ | $8.49^{+1.40}_{-1.86}$ | 0.445 | $0.62_{-0.13}^{+0.19}$ | 1.7%
7619236 | 682.01 | - | 562.70 | $1.67_{-0.21}^{+0.27}$ | $10.90^{+1.96}_{-1.72}$ | 0.992 | $0.31_{-0.14}^{+0.37}$ | 1.0%
7811397 | 1477.01 | - | 169.50 | $1.16_{-0.67}^{+0.53}$ | $9.06^{+0.90}_{-0.28}$ | 0.907 | $0.27_{-0.21}^{+0.35}$ | 1.6%
7951018 | 1553.01 | 890b | 52.76 | $3.14_{-0.41}^{+0.48}$ | $8.76^{+2.91}_{-0.97}$ | 1.000 | $0.51_{-0.12}^{+0.29}$ | 2.7%
7984047 | 1552.01 | - | 77.63 | $4.56_{-0.67}^{+0.57}$ | $9.40^{+0.67}_{-0.92}$ | 1.000 | $0.58_{-0.11}^{+0.26}$ | 3.8%
8672910 | 918.01 | 725b | 39.64 | $0.75_{-0.05}^{+0.06}$ | $10.98^{+0.72}_{-0.99}$ | 1.000 | $0.22_{-0.12}^{+0.41}$ | 0.6%
8813698 | 1268.01 | - | 268.94 | $6.14_{-1.01}^{+1.65}$ | $13.57^{+2.30}_{-2.31}$ | 0.998 | $0.66_{-0.11}^{+0.22}$ | 4.5%
8827930 | 3801.01 | - | 288.31 | $14.75_{-3.33}^{+3.45}$ | $13.21^{+2.07}_{-2.30}$ | 0.998 | $0.79_{-0.08}^{+0.12}$ | 5.9%
9147029 | 5626.01 | - | 397.50 | $0.41_{-0.09}^{+0.45}$ | $6.57^{+3.9}_{-0.91}$ | 0.004 | $0.42_{-0.07}^{+0.0}$ | 0.1%
9425139 | 1411.01 | - | 305.08 | $4.17\pm 0.82$ | $7.82^{+1.25}_{-0.84}$ | 0.023 | $0.58_{-0.16}^{+0.21}$ | 0.1%
9480535 | 3901.01 | 1527b | 160.13 | $0.04_{-0.01}^{+0.03}$ | $6.67^{+1.19}_{-1.47}$ | 0.004 | $0.74_{-0.02}^{+0.20}$ | 0.1%∗
9512981 | 1466.01 | - | 281.56 | $1.85_{-0.16}^{+0.18}$ | $10.83^{+0.37}_{-0.70}$ | 0.995 | $0.36_{-0.14}^{+0.34}$ | 2.0%
9958387 | 2677.01 | - | 237.79 | $0.44_{-0.08}^{+0.09}$ | $9.91^{+1.84}_{-4.28}$ | 0.971 | $0.44_{-0.17}^{+0.31}$ | 2.0%
10265602 | 2689.01 | - | 165.34 | $78.73_{-61.44}^{+87.19}$ | $6.98^{+1.53}_{-0.82}$ | 0.901 | $0.85^{+0.11}_{-0.10}$ | 17.3%∗
10656508 | 211.01 | - | 124.04 | $21.87_{-3.09}^{+3.42}$ | $11.13\pm 1.80$ | 1.000 | $0.84_{-0.05}^{+0.09}$ | 14.4%∗
10795103 | 3683.01 | 1515b | 214.31 | $1.05_{-0.08}^{+0.10}$ | $9.17^{+1.64}_{-0.93}$ | 1.000 | $0.10_{-0.08}^{+0.41}$ | 1.1%
11027624 | 1439.01 | 849b | 394.62 | $0.90_{-0.13}^{+0.14}$ | $7.79^{+1.27}_{-1.90}$ | 0.768 | $0.15_{-0.12}^{+0.41}$ | 0.7%
11075279 | 1431.01 | - | 345.16 | $1.17_{-0.12}^{+0.15}$ | $7.79^{+1.13}_{-0.36}$ | 0.355 | $0.17_{-0.12}^{+0.35}$ | 0.2%
11449844 | 125.01 | 468b | 38.48 | $1.73_{-0.09}^{+0.12}$ | $13.26^{+1.78}_{-0.59}$ | 1.000 | $0.34_{-0.14}^{+0.39}$ | 0.5%∗
Figure 3: Expected 2D posterior for orbital period $P$ vs.
$\rho_{circ}/\rho_{*}$ of super-eccentric Jupiters. The 45 observed candidates
in our sample are overplotted with blue/red bars for planet candidates
with/without companions. If the observations matched the prediction, we would
expect to see approximately four candidates in the high-probability density
region.

##  4 Comparison between predictions and observations

While we can visually see in Figure 3 that our prediction of $\sim 4$-5 super-
eccentric Jupiters does not appear to be matched by the observations, we must
now calculate the probability that each observed planet candidate lies in the
super-eccentric range of parameter space and total these probabilities to
determine if the observations fail to match the prediction with statistical
significance. We perform this calculation in Section 4.1 and use these results
to estimate an upper limit on the fraction of hot Jupiters that may have
arrived via HEM in Section 4.2.

###  4.1 Consistency of the Observations with the Prediction

While none of our candidates appear to lie in the high-probability density
region of Figure 3, each has some small chance of truly being super-eccentric
($e>0.9$ and $2.8<P_{\rm{final}}<10$ days). A super-eccentric orbit is a
possibility because $\rho_{\rm{circ}}/\rho_{*}$ is a function of both $e$ and
$\omega$ (see Equation 3). For very particular ranges of $\omega$, these
candidates could have extreme eccentricities while maintaining
$\rho_{\rm{circ}}/\rho_{*}$ close to 1. To produce an eccentricity posterior
distribution for each planet candidate,we run a Markov Chain Monte Carlo
(MCMC) exploration of $\rho_{\rm{circ}}$, $\rho_{*}$, $e$, and $\omega$,
following the procedure outlined in Dawson & Johnson (2012) Section 3.4.

To estimate $N_{\rm{sup}}$, we conduct a separate Monte Carlo process in which
we draw from the eccentricity posteriors of each candidate $10^{6}$ times. In
each trial, we compute $P_{\rm{final}}$ for each candidate and count
$N_{\rm{sup}}$, the number of planets with $e>0.9$ and $P_{\rm{final}}$
falling within each of our two intervals (2.8-5 or 5-10 days), weighted by the
candidate’s overlap with the warm Jupiter radius range. We then draw from the
predicted $N_{\rm{sup}}$ posterior from Section 2 for each interval. If the
observed $N_{\rm{sup}}$ in each interval is greater than or equal to the
predicted $N_{\rm{sup}}$ in a given trial, we count that trial as a success.

We measure an $N_{\rm{sup}}$ posterior with a median of
$N_{\rm{sup}}=1^{+1}_{-1}$ and find that we detected too few super-eccentric
Jupiters to match our prediction with 94.3% confidence. Figure 4 shows the
distribution of counted super-eccentric planets in our trials (bottom panel)
compared to the predicted number (top panel) from Figure 2. We list the
posterior probability that each planet in our sample is super-eccentric in
Table 2. The highest probability of super-eccentricity is KOI-2689.01 with
$17.3\%$ and only five candidates have probabilities $\geq 5\%$. We plot the
measured $68\%$ eccentricity confidence interval of each of these five
candidates in Figure 5. If the true eccentricity of each of these planets lies
in the blue shaded region, our prediction would be confirmed by the
observations and HEM could not be ruled out as a delivery mechanism for all
hot Jupiters.

We note that five candidates in our sample, including KOI-2689.01 and
KOI-211.01, are listed in the Kepler eclipsing binary catalog (Kirk et al.,
2016). These candidates may still be planets, but have been identified as
presenting features indicative of eclipsing binaries in their transit light
curves. Because we cannot confirm these candidates as false positives, we
include them in our analysis (as did Dawson et al. 2015) and mark them as
potential eclipsing binaries in Table 2. Rerunning our calculations without
these systems, we find that we detected too few super-eccentric Jupiters to
match our prediction with 95.9% confidence.

Figure 4: Top: Histogram depicting the relative probability distribution of
the expected number of super-eccentric Jupiters from our prediction in Section
2 (identical to bottom panel of Figure 2). Bottom: histogram depicting the
relative probability distribution of super-eccentric Jupiters observed in the
Kepler sample. The clear difference between these distributions demonstrates
that the observations deviate from the prediction.

Our result is consistent with that of Dawson et al. (2015), but with slightly
lower confidence (94.3% vs. 96.9%). Despite the higher $\rho_{*}$ precision of
our updated sample, The addition of new candidates with non-zero probability
of being super-eccentric into the sample means that the overall probability we
are observing too few super-eccentric planets has decreased.

Figure 5: Eccentricity vs. semi-major axis of the 5 planet candidates with the
greatest probability of being super-eccentric Jupiters (blue shaded region).
From left to right, these are KOI-1193.01, KOI-211.01, KOI-2689.01,
KOI3801.01, and Kepler-1746 b. The error bars represent the $68\%$ confidence
interval for each candidate’s eccentricity. Three of these confidence
intervals cross into the super-eccentric region of parameter space. Shaded
regions and deliminator lines are the same as in Figure 1.

###  4.2 Upper limit on hot Jupiter delivery from HEM

We now set an upper limit on the delivery of hot Jupiters via the HEM process
based on our observations of the Kepler sample. To do this, we rewrite
Equation 1 to include the fraction of hot Jupiters that arrived by HEM,
$f_{*}$.

$\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{\rm{sup}}=f_{*}\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern
1.5mu_{\rm{mod}}r(e_{\rm{max}})$ (5)

Previously, this equation assumed that all hot Jupiters came from HEM so that
we could test the prediction derived from this assumption against the
observations. Now we can rerun the Monte Carlo calculation from Section 4.1
with our new factor included. We implement a modified Jeffrey’s prior on
$f_{*}$ following Dawson et al. (2015) such that it is uniform between 0 and
5% and scales with $1/f_{*}$ above $f_{*}=5\%$. We then draw from the
eccentricity posteriors of each candidate $10^{6}$ times and count the number
of super-eccentric Jupiters ($N_{\rm{sup}}$). We consider the trials in which
the observed $N_{\rm{sup}}$ meets that drawn from the modified prediction
(Equation 5) to be successes and derive a final posterior for $f_{*}$ from
these trials.

We find a median of 10% and a two-sigma upper limit of 62% on $f_{*}$, or the
fraction of hot Jupiters that could have been delivered by HEM. This limit is
larger than that found by Dawson et al. (2015), but still implies at least
$\sim 1/3$ of hot Jupiters must have been delivered by other mechanisms. We
discuss caveats to this conclusion in Section 5. If the candidates on the
eclipsing binary catalog are removed, the two-sigma upper limit on $f_{*}$
drops to 52%.

##  5 Discussion

In Section 4, we showed that HEM, the process by which a highly eccentric
giant planet is tidally circularized during close stellar passages resulting
in a hot Jupiter, cannot account for all of the observed hot Jupiters. Here we
discuss alternatives to HEM (Section 5.1), an important caveat to our primary
result (Section 5.2), and some ways that migrating planets may be able to move
through the super-eccentric phase more quickly than expected (Section 5.3).

###  5.1 Alternatives to HEM

If hot Jupiters were not delivered by HEM, they must have either (a) formed in
situ, (b) migrated via disk-driven migration, or (c) been implanted directly
through planet-planet scattering. The final of these is unlikely to account
for the majority of hot Jupiters due to the large change in orbital energy
required. Both disk migration and in situ formation remain under investigation
as plausible origins mechanisms. In situ formation of hot Jupiters is
challenging due to the large quantity of solid material needed to form a
planetary core (Rafikov, 2006). In situ formation is also difficult to
reconcile with the lack of nearby planets in the majority of hot Jupiter
systems (Huang et al., 2016; Poon et al., 2021). Disk migration, in which
competing torques in a gas disk may result in a net inward migration of a
growing gas giant (Goldreich & Tremaine, 1980; Ward, 1997; Alibert et al.,
2005; Ida & Lin, 2008; Bromley & Kenyon, 2011), has been a common explanation
for hot Jupiters. However, some studies have found that disk conditions may
not be conducive to inward migration of gas giants (e.g., Fung & Lee 2018).
Another challenge with the in situ formation and disk scenarios is that
planet-planet scattering (Petrovich et al., 2014; Anderson et al., 2020) and
planet-disk interactions (Duffell & Chiang, 2015) are not able to excite the
observed moderate eccentricities so close to the star.

###  5.2 The Effect of Planet-planet von Zeipel-Lidov-Kozai Migration

Warm Jupiters undergoing planet-planet von Zeipel-Lidov-Kozai oscillations
leading to HEM can remain unquenched until they reach very short periods if
the perturber is nearby (Dong et al., 2014; Jackson et al., 2021), meaning we
will tend to observe them at eccentricities below their tidal migration
tracks. This does not affect the total number of super-eccentric Jupiters
predicted by Equation 1 because that equation uses the ratio of time spent at
super-eccentricities to moderate eccentricities while on the track,
irrespective of how long a planet spends off of it. However, as Dawson et al.
(2015) argued, it is possible that some of the moderately eccentric planets
included in the calibration sample are undergoing von Zeipel-Lidov-Kozai
oscillations and tidally migrating on a track with $P_{\rm{final}}<2.8$ days,
and thus do not truly belong in the $2.8<P_{\rm{final}}<10$ days interval. For
some moderately eccentric Jupiters, existing radial-velocity observations can
rule out such companions (Dong et al., 2014) – for example, a Jupiter mass
companion would need to be located within 0.6 au to remain coupled to a hot
Jupiter located at 0.05 au – but others need a longer baseline of
observations.

However, we note that the timescale for tidal circulation is very strongly
dependent on $P_{\rm{final}}$. If migrating Jupiters with $P_{\rm{final}}<2.8$
days that have not yet circularized are common, it is unlikely that hot
Jupiters with periods between 2.8 and 10 days could have circularized over the
age of the universe. Thus, we still need some explanation other than HEM to
account for these hot Jupiters and any observed moderately eccentric Jupiters
not currently undergoing planet-planet von Zeipel-Lidov-Kozai oscillations.

###  5.3 Theoretical Methods of Bypassing the Super-eccentric Phase

An alternative explanation for the dearth of super-eccentric Jupiters in the
Kepler sample is that hot Jupiters are migrating via HEM, but are bypassing
the super-eccentric phase more quickly than that assumed from equilibrium
tides. Chaotic or diffusive tides (e.g., Mardling 1995a, b) may also
contribute to the dissipation inside the planet. Recent efforts to model
chaotic tides in migrating Jupiters show that, if the large amount of energy
produced can be efficiently dissipated in the planet, the HEM process can be
sped up in its early stages at high eccentricities (Ivanov & Papaloizou, 2004;
Wu, 2018; Vick & Lai, 2018; Vick et al., 2019; Yu et al., 2021). However, this
mechanism is most effective for planets undergoing periapse passages very
close to the star and thus may not have operated for many of the moderately
eccentric Jupiters in our sample, particularly those with $P_{\rm{final}}>5$
days.

Another possibility is that migrating super-eccentric Jupiters are thermally
inflated due to residual heat from their core-accretion phase that has not yet
dissipated. Rozner et al. (2021) show that Jupiters undergoing “inflated
eccentric migration” would move through the early stages of HEM more quickly,
and predict warm Jupiters with $e>0.9$ to be very rare.

A related explanation is that moderately eccentric hot Jupiters may begin HEM
much closer to the star (e.g., within $\sim 0.5\rm{au}$), following in situ
formation or disk migration, never traveling through much of the blue region
in Figure 1.

##  6 Conclusions

Dawson et al. (2015) showed that there is a paucity of observed super-
eccentric Jupiters in the Kepler sample if all or most hot Jupiters are
delivered via HEM (Socrates et al., 2012b). We have updated this calculation
using improved light curve fits (Quarles, in prep.) and stellar parameters
from Gaia (Berger et al., 2020). We confirm the paucity of super-eccentric
Jupiters in the Kepler sample with 94.3% confidence and set a 95% upper limit
of 62% on the fraction of hot Jupiters that could have arrived through HEM,
assuming steady tidal circularization with equilibrium tides. Thus, some other
mechanism must account for at least $\sim 1/3$ of the known hot Jupiters.

In Section 5, we discussed some possible alternatives to the standard HEM
picture we assumed for our calculations. Hot Jupiters may have formed in situ
or been delivered by disk migration. In this case, planet-planet scattering or
some other mechanism must be invoked to explain the observed moderately
eccentric Jupiters on tidal migration tracks. Another alternative is that most
hot Jupiters did arrive through HEM, but bypassed the super-eccentric phase
through chaotic tides, inflated eccentric migration, or eccentricity
excitation after forming as in situ warm Jupiters or following disk migration.

While the TESS mission will not provide a population of Jupiters complete
enough to large periods on which to repeat a similar calculation, it may
provide some new eccentric planets to improve our constraints on the minimum
contribution of HEM. For example, Dong et al. (2021) recently discovered
TOI-3362b, a massive planet candidate on a 18.1 day orbital period with
$e=0.815^{+0.023}_{-0.032}$, which would be very difficult to explain with
planet-planet scattering. Radial velocity follow up of TESS hot and warm
Jupiters will help us better understand the eccentricity and multiplicity
distributions of these planets, which may improve our constraints on HEM as a
hot Jupiter delivery mechanism.

We thank the anonymous referee for their helpful comments on this paper. We
gratefully acknowledge support from grant NNX16AB50G awarded by the NASA
Exoplanets Research Program and the Alfred P. Sloan Foundation’s Sloan
Research Fellowship. Computations for this research were performed on the
Pennsylvania State University’s Institute for Computational and Data Sciences’
Roar supercomputer. This research was supported in part through research
cyberinfrastructure resources and services provided by the Partnership for an
Advanced Computing Environment (PACE) at the Georgia Institute of Technology.
The Center for Exoplanets and Habitable Worlds is supported by the
Pennsylvania State University, the Eberly College of Science, and the
Pennsylvania Space Grant Consortium. This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program.

##  Appendix A Updates to the sample of giant planet candidates

Here we describe our updates to the various samples of giant planet candidates
from Dawson et al. (2015) that we use in our calculations, including the
sample of long-period Kepler Jupiters, the sample of short-period Kepler
Jupiters in each $p_{\rm{final}}$ interval, the sample of short-period non-
Kepler Jupiters in $p_{\rm{final}}$ interval, and the sample of moderately
eccentric Jupiters in each $p_{\rm{final}}$ interval. For each sample, we
began with every planet candidate that fell within our planetary radius and
stellar parameter cuts and checked for false positive signatures including
secondary eclipses inconsistent with planethood, large centroid-transit
correlations, different odd/even eclipse depths, strong centroid-flux
correlations, and large position offsets from the KIC catalog. Below are the
additions to and removals from the Dawson et al. (2015) sample using our
updated knowledge of the candidates. All totals excluding the long-period
Jupiters are listed in Table 1.

  1. 1.

Long-period giant planet candidates: We exclude KOI-1483.01 and KOI-5241.01,
which appeared in the Dawson et al. (2015) sample, because they have since
been identified as false positives (Morton et al., 2016). We also exclude
KOI-458.01, which orbits an evolved star, and KOI-1587.01, whose radius has
been revised and is now too large (Berger et al., 2020). We exclude KIC
9413313 because it does not appear in the Berger et al. (2020) catalog or on
the NASA Exoplanet Archive. All of our additions to the (Dawson et al., 2015)
sample had estimated radii below $8R_{\earth}$ but were revised to fall within
our radius cut by Berger et al. (2020).

  2. 2.

Moderately eccentric Jupiters ($N_{\rm{mod},0}$) with $2.8<P_{\rm{final}}<5$
days: We add WASP-150 b, WASP-186 b, and NGC 2682 YBP 1514 b to the sample
from Dawson et al. (2015), which have all been discovered since its
publication, for a net increase of 3 candidates.

  3. 3.

Moderately eccentric Jupiters ($N_{\rm{mod},0}$) with $5<P_{\rm{final}}<10$
days: We add CoRoT-20 b, TIC 237913194 b, TOI-172 b, TOI-677 b, WASP-117 b,
WASP-148 b, WASP-162 b, WASP 185 b, and NGC 2682 YBP 1194 b to the sample from
Dawson et al. (2015), which have all been discovered since its publication or
have had their eccentricities constrained, for a net increase of 9 candidates.

  4. 4.

Non-Kepler Jupiters with $2.8<P<5$ days ($N_{P=P_{\rm{final}},0}$): We remove
HAT-P-35 b from the Dawson et al. (2015) sample because its radius estimate
has changed with new Gaia data and we remove OGLE-TR-211 b because its most
recent eccentricity estimate is ambiguous. We add CoRoT-27 b, CoRoT-29 b,
HAT-P-3 b, HAT-P-42 b, HAT-P-44 b, HAT-P-45 b, HAT-P-50 b, HAT-P-51 b,
HAT-P-55 b, HAT-P-58 b, HAT-P-59 b, HAT-P-63 b, HAT-P-64 b, HAT-P-9 b, HATS-12
b, HATS-13 b, HATS-22 b, HATS-25 b, HATS-29 b, HATS-30 b, HATS-36 b, HATS-47
b, HATS-48 A b, HATS-5 b, HATS-55 b, HATS-60 b, HATS-63 b, HATS-65 b, HATS-68
b, KELT-8 b, NGTS-13 b, TOI-1296 b, TOI-1298 b, TOI-628 b, TOI-905 b, TrES-1
b, TrES-4 b, WASP-101 b, WASP-120 b, WASP-124 b, WASP-126 b, WASP-13 b,
WASP-153 b, WASP-20 b, WASP-28 b, WASP-39 b, WASP-53 b, WASP-56 b, WASP-57 b,
WASP-60 b, WASP-69 b, WASP-70 A b, WASP-83 b, WASP-94 A b, WASP-96 b, WASP-98
b, WTS-1 b, XO-1 b, XO-4 b, XO-5 b, XO-7 b, K2-29 b, K2-30 b, K2-34 b, HD
103720 b, and HD 202772 A b, which have all been discovered since 2015 or have
had their eccentricities constrained, for a net increase of 65 candidates.

  5. 5.

Non-Kepler Jupiters with $5<P<10$ days ($N_{P=P_{\rm{final}},0}$): We add
CoRoT-20 b, CoRoT-30 b, HATS-61 b, HATS-72 b, KELT-6 b, TOI-1268 b, TOI-172 b,
TOI-559 b, WASP-106 b, WASP-129 b, WASP-148 b, WASP-150 b, WASP-162 b,
WASP-185 b, WASP-186 b, WASP-58 b, WASP-68 b, WASP-84 b, WASP-99 b, HD 103774
b, HD 168746 b, HD 285507 b, HIP 91258 b, NGC 2682 YBP 1194 b, and NGC 2682
YBP 1514 b to the sample from Dawson et al. (2015), which have all been
discovered since its publication or have had their eccentricities constrained,
for a net increase of 24 candidates.

  6. 6.

Kepler Jupiters with $2.8<P<5$ days ($N_{P=P_{\rm{final}}}$): We add
Kepler-1935 b to the sample from Dawson et al. (2015), whose radius posterior
now overlaps with our target range (Berger et al., 2020). We also remove
KOI-838.01, which has been identified as a false positive, and are left with a
total $N_{P=P_{\rm{final}}}$ of 24 for a net change of 0.

  7. 7.

Kepler Jupiters with $5<P<10$ days ($N_{P=P_{\rm{final}}}$): We add Kepler-677
b to the sample from Dawson et al. (2015), whose radius posterior now overlaps
with our target range (Berger et al., 2020). We also remove KOI-774.01 and
KOI-1391.01, which have been identified as false positives, and are left with
a total $N_{P=P_{\rm{final}}}$ of 15 for a net change of 1 fewer candidate.

## References

  * Alibert et al. (2005) Alibert, Y., Mordasini, C., Benz, W., & Winisdoerffer, C. 2005, A&A, 434, 343, doi: 10.1051/0004-6361:20042032
  * Almenara et al. (2018) Almenara, J. M., Díaz, R. F., Hébrard, G., et al. 2018, A&A, 615, A90, doi: 10.1051/0004-6361/201732500
  * Anderson et al. (2020) Anderson, K. R., Lai, D., & Pu, B. 2020, MNRAS, 491, 1369, doi: 10.1093/mnras/stz3119
  * Barnes (2007) Barnes, J. W. 2007, PASP, 119, 986, doi: 10.1086/522039
  * Baruteau et al. (2014) Baruteau, C., Crida, A., Paardekooper, S.-J., et al. 2014, Protostars and Planets VI, 667, doi: 10.2458/azu_uapress_9780816531240-ch029
  * Beaugé & Nesvorný (2012) Beaugé, C., & Nesvorný, D. 2012, ApJ, 751, 119, doi: 10.1088/0004-637X/751/2/119
  * Berger et al. (2020) Berger, T. A., Huber, D., van Saders, J. L., et al. 2020, AJ, 159, 280, doi: 10.3847/1538-3881/159/6/280
  * Bodenheimer et al. (2000) Bodenheimer, P., Hubickyj, O., & Lissauer, J. J. 2000, Icarus, 143, 2, doi: 10.1006/icar.1999.6246
  * Boley et al. (2012) Boley, A. C., Payne, M. J., & Ford, E. B. 2012, ApJ, 754, 57, doi: 10.1088/0004-637X/754/1/57
  * Bromley & Kenyon (2011) Bromley, B. C., & Kenyon, S. J. 2011, ApJ, 735, 29, doi: 10.1088/0004-637X/735/1/29
  * Chatterjee et al. (2008) Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008, ApJ, 686, 580, doi: 10.1086/590227
  * Dawson & Johnson (2012) Dawson, R. I., & Johnson, J. A. 2012, ApJ, 756, 122, doi: 10.1088/0004-637X/756/2/122
  * Dawson & Johnson (2018) —. 2018, ARA&A, 56, 175, doi: 10.1146/annurev-astro-081817-051853
  * Dawson et al. (2015) Dawson, R. I., Murray-Clay, R. A., & Johnson, J. A. 2015, ApJ, 798, 66, doi: 10.1088/0004-637X/798/2/66
  * Dawson et al. (2014) Dawson, R. I., Johnson, J. A., Fabrycky, D. C., et al. 2014, ApJ, 791, 89, doi: 10.1088/0004-637X/791/2/89
  * Dong et al. (2021) Dong, J., Huang, C. X., Zhou, G., et al. 2021, ApJ, 920, L16, doi: 10.3847/2041-8213/ac2600
  * Dong et al. (2014) Dong, S., Katz, B., & Socrates, A. 2014, ApJ, 781, L5, doi: 10.1088/2041-8205/781/1/L5
  * Duffell & Chiang (2015) Duffell, P. C., & Chiang, E. 2015, ApJ, 812, 94, doi: 10.1088/0004-637X/812/2/94
  * Eggleton et al. (1998) Eggleton, P. P., Kiseleva, L. G., & Hut, P. 1998, ApJ, 499, 853, doi: 10.1086/305670
  * Fabrycky & Tremaine (2007) Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298, doi: 10.1086/521702
  * Ford et al. (2008) Ford, E. B., Quinn, S. N., & Veras, D. 2008, ApJ, 678, 1407, doi: 10.1086/587046
  * Ford & Rasio (2006) Ford, E. B., & Rasio, F. A. 2006, ApJ, 638, L45, doi: 10.1086/500734
  * Ford & Rasio (2008) —. 2008, ApJ, 686, 621, doi: 10.1086/590926
  * Fung & Lee (2018) Fung, J., & Lee, E. J. 2018, ApJ, 859, 126, doi: 10.3847/1538-4357/aabaf7
  * Goldreich & Tremaine (1980) Goldreich, P., & Tremaine, S. 1980, ApJ, 241, 425, doi: 10.1086/158356
  * Hansen (2010) Hansen, B. M. S. 2010, ApJ, 723, 285, doi: 10.1088/0004-637X/723/1/285
  * Hébrard et al. (2019) Hébrard, G., Bonomo, A. S., Díaz, R. F., et al. 2019, A&A, 623, A104, doi: 10.1051/0004-6361/201834333
  * Huang et al. (2016) Huang, C., Wu, Y., & Triaud, A. H. M. J. 2016, ApJ, 825, 98, doi: 10.3847/0004-637X/825/2/98
  * Ida & Lin (2008) Ida, S., & Lin, D. N. C. 2008, ApJ, 673, 487, doi: 10.1086/523754
  * Ivanov & Papaloizou (2004) Ivanov, P. B., & Papaloizou, J. C. B. 2004, MNRAS, 347, 437, doi: 10.1111/j.1365-2966.2004.07238.x
  * Jackson et al. (2021) Jackson, J. M., Dawson, R. I., Shannon, A., & Petrovich, C. 2021, AJ, 161, 200, doi: 10.3847/1538-3881/abe61f
  * Jackson et al. (2019) Jackson, J. M., Dawson, R. I., & Zalesky, J. 2019, AJ, 157, 166, doi: 10.3847/1538-3881/ab09eb
  * Jurić & Tremaine (2008) Jurić, M., & Tremaine, S. 2008, ApJ, 686, 603, doi: 10.1086/590047
  * Kipping (2008) Kipping, D. M. 2008, MNRAS, 389, 1383, doi: 10.1111/j.1365-2966.2008.13658.x
  * Kipping (2010) —. 2010, MNRAS, 407, 301, doi: 10.1111/j.1365-2966.2010.16894.x
  * Kirk et al. (2016) Kirk, B., Conroy, K., Prša, A., et al. 2016, AJ, 151, 68, doi: 10.3847/0004-6256/151/3/68
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591, doi: 10.1086/108790
  * Lawler & Gladman (2012) Lawler, S. M., & Gladman, B. 2012, ApJ, 752, 53, doi: 10.1088/0004-637X/752/1/53
  * Lidov (1962) Lidov, M. L. 1962, Planet. Space Sci., 9, 719, doi: 10.1016/0032-0633(62)90129-0
  * Lithwick & Naoz (2011) Lithwick, Y., & Naoz, S. 2011, ApJ, 742, 94, doi: 10.1088/0004-637X/742/2/94
  * Mancini et al. (2016) Mancini, L., Lillo-Box, J., Southworth, J., et al. 2016, A&A, 590, A112, doi: 10.1051/0004-6361/201526357
  * Mardling (1995a) Mardling, R. A. 1995a, ApJ, 450, 722, doi: 10.1086/176178
  * Mardling (1995b) —. 1995b, ApJ, 450, 732, doi: 10.1086/176179
  * Matsumura et al. (2010) Matsumura, S., Peale, S. J., & Rasio, F. A. 2010, ApJ, 725, 1995, doi: 10.1088/0004-637X/725/2/1995
  * Morton et al. (2016) Morton, T. D., Bryson, S. T., Coughlin, J. L., et al. 2016, ApJ, 822, 86, doi: 10.3847/0004-637X/822/2/86
  * Mustill et al. (2015) Mustill, A. J., Davies, M. B., & Johansen, A. 2015, ApJ, 808, 14, doi: 10.1088/0004-637X/808/1/14
  * Nagasawa & Ida (2011) Nagasawa, M., & Ida, S. 2011, ApJ, 742, 72, doi: 10.1088/0004-637X/742/2/72
  * Naoz et al. (2011) Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. 2011, Nature, 473, 187, doi: 10.1038/nature10076
  * Naoz et al. (2012) Naoz, S., Farr, W. M., & Rasio, F. A. 2012, ApJ, 754, L36, doi: 10.1088/2041-8205/754/2/L36
  * NASA Exoplanet Science Institute (2020) NASA Exoplanet Science Institute. 2020, Planetary Systems Table, IPAC, doi: 10.26133/NEA12.  https://catcopy.ipac.caltech.edu/dois/doi.php?id=10.26133/NEA12
  * Petrovich et al. (2014) Petrovich, C., Tremaine, S., & Rafikov, R. 2014, ApJ, 786, 101, doi: 10.1088/0004-637X/786/2/101
  * Poon et al. (2021) Poon, S. T. S., Nelson, R. P., & Coleman, G. A. L. 2021, MNRAS, 505, 2500, doi: 10.1093/mnras/stab1466
  * Rafikov (2006) Rafikov, R. R. 2006, ApJ, 648, 666, doi: 10.1086/505695
  * Rasio & Ford (1996) Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954, doi: 10.1126/science.274.5289.954
  * Rozner et al. (2021) Rozner, M., Glanz, H., Perets, H. B., & Grishin, E. 2021, arXiv e-prints, arXiv:2111.12718.  https://arxiv.org/abs/2111.12718
  * Socrates & Katz (2012) Socrates, A., & Katz, B. 2012, arXiv e-prints, arXiv:1209.5723.  https://arxiv.org/abs/1209.5723
  * Socrates et al. (2012a) Socrates, A., Katz, B., & Dong, S. 2012a, ArXiv e-prints.  https://arxiv.org/abs/1209.5724
  * Socrates et al. (2012b) Socrates, A., Katz, B., Dong, S., & Tremaine, S. 2012b, ApJ, 750, 106, doi: 10.1088/0004-637X/750/2/106
  * Vick & Lai (2018) Vick, M., & Lai, D. 2018, MNRAS, 476, 482, doi: 10.1093/mnras/sty225
  * Vick et al. (2019) Vick, M., Lai, D., & Anderson, K. R. 2019, MNRAS, 484, 5645, doi: 10.1093/mnras/stz354
  * von Zeipel (1910) von Zeipel, H. 1910, Astronomische Nachrichten, 183, 345, doi: 10.1002/asna.19091832202
  * Ward (1997) Ward, W. R. 1997, Icarus, 126, 261, doi: 10.1006/icar.1996.5647
  * Wu (2018) Wu, Y. 2018, AJ, 155, 118, doi: 10.3847/1538-3881/aaa970
  * Wu & Lithwick (2011) Wu, Y., & Lithwick, Y. 2011, ApJ, 735, 109, doi: 10.1088/0004-637X/735/2/109
  * Wu & Murray (2003) Wu, Y., & Murray, N. 2003, ApJ, 589, 605, doi: 10.1086/374598
  * Yu et al. (2021) Yu, H., Weinberg, N. N., & Arras, P. 2021, arXiv e-prints, arXiv:2111.04649.  https://arxiv.org/abs/2111.04649