# The Aligned Orbit of the Eccentric Warm Jupiter K2-232 b

Songhu Wang Department of Astronomy, Indiana University, Bloomington, IN 47405
<EMAIL_ADDRESS>Joshua N. Winn Department of Astrophysical Sciences, Princeton
University, 4 Ivy Lane, Princeton, NJ 08544  Brett C. Addison University of
Southern Queensland, Centre for Astrophysics, West Street, Toowoomba, QLD 4350
Australia  Fei Dai Division of Geological and Planetary Sciences 1200 E
California Blvd, Pasadena, CA 91125  Malena Rice Department of Astronomy, Yale
University, New Haven, CT 06511  Bradford Holden UCO/Lick Observatory,
Department of Astronomy and Astrophysics, University of California at Santa
Cruz,Santa Cruz, CA 95064  Jennifer A. Burt Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove drive, Pasadena CA 91109
Xian-Yu Wang National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100012, China  University of Chinese Academy of Sciences, Beijing,
100049, China  R. Paul Butler Earth and Planets Laboratory, Carnegie
Institution for Science, 5241 Broad Branch Road, NW, Washington, DC 20015
Steven S. Vogt UCO/Lick Observatory, Department of Astronomy and Astrophysics,
University of California at Santa Cruz,Santa Cruz, CA 95064  Gregory Laughlin
Department of Astronomy, Yale University, New Haven, CT 06511

(Received ?; Revised ?; Accepted ?)

###### Abstract

Measuring the obliquity distribution of stars hosting warm Jupiters may help
us to understand the formation of close-orbiting gas giants. Few such
measurements have been performed due to practical difficulties in scheduling
observations of the relatively infrequent and long-duration transits of warm
Jupiters. Here, we report a measurement of the Rossiter-McLaughlin effect for
K2-232 b, a warm Jupiter on an $11.17$-day orbit with an eccentricity of
$0.26$. The data were obtained with the Automated Planet Finder during two
separate transits. The planet’s orbit appears to be well-aligned with the spin
axis of the host star, with a projected spin-orbit angle of $\lambda=-11.1\pm
6.6^{\circ}$. Combined with the other available data, we find that high
obliquities are almost exclusively associated with planets that either have an
orbital separation greater than 10 stellar radii or orbit stars with effective
temperatures hotter than 6,000 K. This pattern suggests that the obliquities
of the closest-orbiting giant planets around cooler stars have been damped by
tidal effects.

planetary alignment (1243), exoplanet dynamics (490), star-planet interactions
(2177), exoplanets (498), planetary theory (1258), exoplanet systems (484)

††journal: AAS

##  1 Introduction

In the Solar System, the orbits of the major planets are all aligned with the
net angular momentum vector of the Solar System planets to within a few
degrees, and the Sun’s equator is tilted by only $6^{\circ}$ (Souami &
Souchay, 2012) relative to the invariable plane. The coplanarity of the Solar
System was part of the original evidence leading to the proposal that the
planets formed within a flat disk surrounding the Sun (Kant, 1755; de Laplace,
1796).

Figure 1: An up-to-date mass-period diagram of currently known exoplanets.
Planets with Rossiter-McLaughlin (including Doppler tomography) measurements
drawn from the TEPCat catalog (Southworth, 2011) are shown as points color-
coded by their observed spin-orbit angles, while planets without stellar
obliquity measurements are depicted as gray dots. The majority of planets with
existing Rossiter-McLaughlin measurements are hot Jupiters, which span a wide
range of stellar obliquities. In this work, we have expanded the list of spin-
orbit measurements to include a new warm Jupiter system: K2-232.

In contrast to the picture presented by our Solar System, observations of the
Rossiter–McLaughlin effect (Rossiter, 1924; McLaughlin, 1924) occurring during
exoplanetary transits have revealed that a considerable fraction of hot
Jupiters are on orbits that are misaligned with the equators of their host
stars (Winn et al., 2010; Albrecht et al., 2012). The origins of large spin-
orbit misalignments for hot Jupiters are still unclear. The current
theoretical explanations fall into two categories:

  1. 1.

High-eccentricity migration, in which dynamical interactions tilt the orbit of
the planet away from its initial plane. These theories invoke phenomena such
as planet-planet scattering (Rasio & Ford, 1996; Ford & Rasio, 2008), Lidov-
Kozai cycles with tidal friction (Wu & Murray, 2003; Fabrycky & Tremaine,
2007; Naoz, 2016), and secular interactions (Wu & Lithwick, 2011; Petrovich,
2015).

  2. 2.

Obliquity excitation via mechanisms that are unrelated to planet migration.
These include chaotic star formation (Bate et al., 2010), stochastic internal
gravity waves (Rogers et al., 2012), magnetic torques between a young star and
its protoplanetary disk (Lai et al., 2011), and gravitational torques from
distant companions (Batygin et al., 2011; Storch et al., 2014).

Observations of “warm Jupiters” — giant planets with orbital periods longer
than about 10 days — may be helpful in evaluating these formation scenarios.
Unlike hot Jupiters, longer-period warm Jupiters experience relatively weak
tidal interactions and can only form through high-eccentricity migration under
certain conditions (Dong et al., 2013). Therefore, if high obliquities are
indeed associated with high-eccentricity migration, then large spin-orbit
misalignments should be confined to hot Jupiters, while warm Jupiters should
have orbits that are roughly aligned with the stellar equators. If the true
explanation is found in the second category of theories, spin-orbit
misalignments should occur not only in hot Jupiter systems, but also in a
broader class of planetary systems, including warm Jupiters.

Furthermore, the distribution of spin-orbit angles of warm Jupiter hosts might
be easier to interpret than that of hot Jupiter hosts because warm Jupiters
are unlikely to have influenced the stellar rotation through tides or other
proximity effects.

An effective method for measuring or placing bounds on a star’s obliquity is
to observe the Rossiter-McLaughlin effect, which requires spectroscopy
throughout a transit. This requirement helps to explain why there are
relatively few observations of the Rossiter-McLaughlin effect for warm
Jupiters. Compared to hot Jupiters, the warm Jupiters have lower transit
probabilities. Even for the warm Jupiters that do transit, the transits are
less frequent, and the transit durations are longer. This makes it challenging
to schedule transit observations from a single observatory, and it renders the
observations, which require several consecutive hours of reliable data, more
vulnerable to disruptions by bad weather. Therefore, while stellar obliquities
of warm Jupiters may provide important clues to understand the diverse
architectures of exoplanet systems, the existing measurements are sparse.

Here, we present observations of the Rossiter-McLaughlin effect for K2-232 b,
a warm Jupiter of mass $0.39~{}M_{\rm Jup}$ and radius $1.06~{}R_{\rm Jup}$ on
an 11.17-day orbit with an eccentricity of 0.26 (Brahm et al., 2018; Yu et
al., 2018). In what follows, we describe our observations (§2), the
parameterized model we used to determine the spin-orbit angle (§3), and the
possible implications (§4).

##  2 Observations

Figure 2: Spectroscopic radial velocities of K2-232 measured with the APF, as
a function of orbital phase (minutes from mid-transit) along with the best-
fitting Rossiter-McLaughlin model (red-dashed line). The radial velocity
offsets in each of the datasets have been removed prior to fitting the
combined and phased data. The green and maroon points are the radial
velocities from the 29 October 2019 and 4 January 2020 transit observations,
respectively. The lower panel shows the residuals between the observed data
and the best-fitting model. The small structures remaining in the residuals
are likely caused by stellar noise and variations in atmospheric extinction
due to the presence of clouds at the end of night on 4 January 2020

.

Figure 3: Posterior distributions of the projected spin-orbit angle
($\lambda$) and projected stellar rotational velocity ($v\sin i_{\star}$) for
K2-232 from the Markov Chain Monte Carlo Rossiter-McLaughlin simulation. The
posterior points inside the blue, yellow, and red contours lie within the
$1\sigma$, $2\sigma$, and $3\sigma$ confidence regions, respectively. We have
marginalized over $\lambda$ and $v\sin i_{\star}$ and fit each with a
Gaussian. The red square denotes the preferred model solution for $\lambda$
and $v\sin i_{\star}$ as given in Table 2.

K2-232 was observed with the Automated Planet Finder (APF) telescope (Vogt et
al., 2014) on UT 29 October 2019 and UT 04 January 2020, covering two transit
events of the warm Jupiter K2-232 b. Each transit was observed as a series of
20-minute exposures using the APF’s $1\times 3\arcsec$ slit, which produces a
typical spectral resolution of 90,000. The APF uses the iodine-cell technique
for radial-velocity determination: a container of gaseous iodine is placed in
the converging beam of the telescope, imprinting the 5000–6200 Å region of the
incoming stellar spectra with a dense forest of absorption lines that acts as
a wavelength calibrator and provides a means of determining the spectrometer’s
point-spread function (PSF). The October exposures have an average of $2630$
counts/pixel in the iodine region of the spectrum, while the January exposures
have a lower average of $1494$ counts/pixel due to a combination of clouds and
worse atmospheric seeing ($1.52^{\prime\prime}$ and $1.98^{\prime\prime}$ for
October and January data, respectively).

Once the iodine region of the spectrum has been extracted, it is divided into
2 Å chunks. Each chunk is analyzed using the spectral synthesis technique
originally described by Butler et al. (1996), which disentangles the stellar
spectrum from the iodine absorption lines and produces an independent measure
of the wavelength, instrument PSF, and Doppler shift. The Doppler velocity is
calculated as the weighted mean of the velocities that are determined from
each of the $\approx$700 chunks. The final internal uncertainty of each
velocity is the standard deviation of the mean of all 700 chunk velocities.
The October and January exposure series have mean internal uncertainties of
4.44 and 8.20 m s-1, respectively. The radial velocities with uncertainties
can be found in Table 1.

##  3 Spin-Orbit Angle Determination

Figure 4: Key variables that may affect tidal dissipation rates, provided for
all the transiting giant planets ($M_{\rm p}>0.3\,M_{\rm Jup}$) for which the
Rossiter-McLaughlin effect has been reported. The horizontal axis shows
effective temperature of the host star, and the vertical axis shows the
dimensionless orbital separation (semimajor axis divided by stellar radius).
Red points indicate the sky-projected obliquity exceeds 0.1 rad (5.7 deg), and
blue encircled points indicate the orbital eccentricity exceeds 0.1, in both
cases with at least 3-$\sigma$ confidence. The scarcity of misaligned or
eccentric systems with $a/R_{\star}\lesssim 10$ and $T_{\rm eff}\lesssim
6000\,{\rm K}$ suggests that tidal effects have damped the obliquity and
eccentricity of at least some of those systems.

We determined the sky-projected spin-orbit angle ($\lambda$) for K2-232 b
using the Allesfitter code111https://github.com/MNGuenther/allesfitter
(Günther & Daylan, 2020). We simultaneously modeled the K2 photometry of the
target; the in-transit APF radial-velocity time series (which exhibit the
Rossiter-McLaughlin effect); and the out-of-transit radial velocities
available from the discovery papers (Yu et al., 2018; Brahm et al., 2018),
taken with the APF, FEROS, CORALIE, and HARPS spectrographs.

The model parameters included the orbital period ($P$), time of mid-transit at
a reference epoch ($T_{0}$), cosine of the orbital inclination ($\cos{i}$),
planet-to-star radius ratio ($R_{P}/R_{\star}$), sum of radii divided by the
orbital semi-major axis ($(R_{\star}+R_{P})/a$), radial-velocity semi-
amplitude ($K$), parameterized eccentricity and argument of periastron
($\sqrt{e}\,\cos{\,\omega}$, $\sqrt{e}\,\sin{\,\omega}$), quadratic limb
darkening coefficients ($q_{1}$, $q_{2}$), sky-projected spin-orbit angle
($\lambda$), and sky-projected stellar rotational velocity ($v\sin
i_{\star}$). Uniform priors were adopted for all of these parameters. Initial
guesses for $P$, $T_{0}$, $\cos{i}$, $R_{P}/R_{\star}$, $(R_{\star}+R_{P})/a$,
$K$, $\sqrt{e}\,\cos{\,\omega}$, and $\sqrt{e}\,\sin{\,\omega}$ were set to
the values reported by Yu et al. (2018). The initial guess for each limb
darkening coefficient was $0.5$. We included standard instrument offsets
between radial velocities obtained from each spectrograph in the fitting. We
also placed uniform priors on the additive radial velocity offsets between the
time series obtained on each transit night and the radial velocities gathered
outside of transit on other nights. The additive offsets account for any
additional astrophysical or instrumental noise on timescales longer than $\sim
6\,$hours but shorter than the total time span of all observations
($2\,$years). These priors were bounded by reasonable intervals of $\pm 1000$
m s-1. Table 2 summarizes the model parameters, priors, and results.

We conducted three joint fits. First, we fit two separate models to
independently obtain a Rossiter-McLaughlin measurement from each of the two
transit events. Then, we modeled the combined dataset. Together with the in-
transit data, each fit incorporated the $K2$ photometry and out-of-transit
radial velocity data. For each fit, we sampled the posterior distributions of
the model parameters using the Metropolis-Hastings Markov Chain Monte Carlo
(MCMC) algorithm with 100 independent walkers each with 300,000 total accepted
steps. The results, as listed in Table 2, provided solutions for transit and
radial velocity parameters, as well as the sky-projected spin-orbit angle
$\lambda$, the sky-projected stellar rotational velocity $v\sin i_{\star}$,
and the associated $1\sigma$ uncertainties. The radial-velocity and transit
parameters obtained from our analysis are in good agreement with the values
from Brahm et al. (2018) and Yu et al. (2018).

The observations and the resulting best-fit model for the phased transit are
shown in Figure 2, and Figure 3 shows the posterior distributions of $\lambda$
and $v\sin i_{\star}$. Figure 3 reveals a nearly ideal 2D Gaussian
distribution, suggesting that $\lambda$ and $v\sin i_{\star}$ are not strongly
correlated with each other. This is generally the case for transiting systems
with moderate impact parameters ($a\cos i/R_{\star}\approx 0.5$) as is the
case for K2-232 b (Gaudi & Winn, 2007). The small fractional uncertainty in
the $v\sin i_{\star}$ prior ($\sim 12\%$) further helps in determining
$\lambda$.

We find the best-fit projected spin-orbit angle and projected stellar
rotational velocity from the phased transit to be $\lambda=-11.1\pm
6.6^{\circ}$ and $v\sin i_{*}=5.15\pm 0.62$ km s-1, which agree with the
corresponding values from independent fits to the two transit events. Our
results suggest that the orbit of K2-232 b is aligned with the spin axis of
its host star.

##  4 Discussion

Previous studies have indicated that obliquity excitation is not specific to
hot Jupiters: giant planets with moderately wider orbits — wide enough to
expect tidal dissipation to be insignificant — have also been observed with
large orbital inclinations relative to the stellar equator. However, the
number of warm Jupiters with measured spin-orbit angles is small, meaning that
it is not yet clear how prevalent these large misalignments are across the
longer-period giant planet population. The low obliquity of K2-232 b provides
an example of a relatively well-aligned system among the longer-period giant
planets. Additional observations are necessary to draw robust conclusions
regarding the distribution of spin-orbit angles for warm Jupiters.

While K2-232 b is well-aligned with its host star, the planet’s orbit is
moderately eccentric. One might expect eccentricity and inclination to be
coupled, in the sense that many interactions that would excite eccentricity
would also excite inclination, and vice versa. Some examples of such
interactions are planet-planet perturbations and Lidov-Kozai oscillations.
Among the observations, we sometimes see orbits with high eccentricities and
high inclinations, such as HD 80606 b (Winn et al., 2009b) and WASP-8 b
(Queloz et al., 2010). There are also cases of low eccentricities and high
inclinations, such as HAT-P-7 b (Winn et al., 2009a), CoRoT-1 b (Pont et al.,
2010), and KELT-9 (Gaudi et al., 2017). And, there are cases of high
eccentricities and low inclinations (at least as projected on the sky) similar
to K2-232 b, such as HAT-P-2 b (Winn et al., 2007), HD 17156 b (Narita et al.,
2009), and HAT-P-34 b (Albrecht et al., 2012).

The absence of a firm relationship between inclination and eccentricity is a
clue that there are multiple physical processes at play, some of which are
specific to eccentricity or to inclination. For examples, disk–planet
interactions almost always damp inclinations, but they can result in a growth
of eccentricity under the action of Lindblad resonances (Goldreich & Sari,
2003). As for the reverse, some secular effects, such as nodal precession due
to an external inclined perturber, can modify inclination without necessarily
modifying eccentricity (Innanen et al., 1997). More measurements may be needed
to delineate the relative contributions of each effect.

One pattern that is already clear is that the closest-orbiting planets
($a/R_{\star}\lesssim 10$) around cool stars ($T_{\rm{eff}}\lesssim$ 6,000 K)
tend to have both circular and well-aligned orbits. Figure 4 shows
$a/R_{\star}$ versus stellar $T_{\rm eff}$ for all of the transiting giant
planets for which $\lambda$ has been reported based on observations of the
Rossiter-McLaughlin effect. The data were drawn from the TEPCAT
database222https://www.astro.keele.ac.uk/jkt/tepcat/ and restricted to cases
for which the planet is more massive than Saturn (0.3 $M_{\rm Jup}$). The blue
encircled points are those systems for which Doppler observations have shown
that the eccentricity exceeds 0.1 and differs from zero with at least
3-$\sigma$ confidence. The red filled points are those for which Rossiter-
McLaughlin observations have shown that $\lambda$ exceeds 0.1 radians (5.7∘)
and differs from zero with at least 3-$\sigma$ confidence.

Table 1: Radial velocities for the K2-232 system collected with the APF in this work  Time (BJD) | RV ( m s-1) |  $\sigma_{\rm RV}$ ( m s-1)
---|---|---
Oct. 29, 2019
2458785.76135 | 54.43 | 6.22
2458785.77494 | 58.95 | 4.97
2458785.78924 | 46.95 | 4.93
2458785.80668 | 51.24 | 5.0
2458785.82189 | 40.96 | 4.69
2458785.83645 | 36.59 | 4.67
2458785.85075 | 17.47 | 4.15
2458785.86519 | 19.14 | 4.21
2458785.8791 | 0.0 | 4.39
2458785.89446 | 17.18 | 4.55
2458785.90838 | 9.12 | 4.32
2458785.92253 | 10.52 | 4.06
2458785.93707 | 22.14 | 4.04
2458785.95148 | 25.33 | 3.95
2458785.96582 | 19.29 | 4.56
2458785.98007 | 24.96 | 3.62
2458785.99483 | 20.63 | 4.26
2458786.00914 | 31.08 | 4.38
2458786.02352 | 23.42 | 4.19
2458786.03779 | 3.06 | 4.16
2458786.05224 | 30.56 | 4.07
2458786.06682 | 13.6 | 4.18
Jan. 4, 2020
2458852.62955 | 9.26 | 7.32
2458852.64381 | 5.75 | 10.65
2458852.65918 | 12.45 | 6.25
2458852.67522 | 10.99 | 5.22
2458852.68760 | -1.73 | 5.16
2458852.70191 | 4.28 | 4.99
2458852.71790 | 3.62 | 4.65
2458852.73021 | -3.69 | 4.68
2458852.74530 | 5.81 | 4.29
2458852.76125 | 20.99 | 4.89
2458852.77514 | 18.67 | 5.77
2458852.78896 | 10.38 | 5.71
2458852.80286 | 42.77 | 7.80
2458852.81951 | 21.92 | 5.60
2458852.83234 | 29.54 | 4.96
2458852.84758 | 12.73 | 6.02
2458852.86249 | -19.92 | 8.31
2458852.87701 | -34.37 | 9.8
Table 2: System Parameters, Priors, and Results for K2-232 Parameter | PriorsaaThe uniform priors are presented in the form of three numbers: the first number is the lower bound, the middle number is the initial guess, and the last number is the upper bound. | Results 29 Oct. 2019 | Results 4 Jan. 2020 | Preferred Solution
---|---|---|---|---
|  |  |  | Results combined transits
Fitted Parameters: |  |  |  |
Orbital period, $P$ (days) | $10;11.168454;12$ | $11.168452\pm 0.000025$ | $11.168466\pm 0.000024$ | $11.168455\pm 0.000023$
Mid-transit epoch (2450000-BJD), $T_{0}$ | $7825.3\tablenotemark{b}$ | $8305.59490\pm 0.00099$ | $8339.1008\pm 0.0010$ | $8339.10040\pm 0.0010$
Cosine of the orbital inclination, $\cos{i}$ | $0;0.015;1$ | $0.0345_{-0.0024}^{+0.0028}$ | $0.0344_{-0.0024}^{+0.0029}$ | $0.0344_{-0.0023}^{+0.0028}$
Planet-to-star radius ratio, $R_{P}/R_{\star}$ | $0;0.08868;1$ | $0.09142_{-0.00049}^{+0.00052}$ | $0.09138_{-0.00048}^{+0.00052}$ | $0.09140_{-0.00048}^{+0.00051}$
Sum of radii divided by the orbital semimajor axis, $(R_{\star}+R_{P})/a$ | $0;0.05654;1$ | $0.0676_{-0.0023}^{+0.0026}$ | $0.0676_{-0.0023}^{+0.0027}$ | $0.0675_{-0.0022}^{+0.0026}$
RV semi-amplitude, $K$ ($\rm m\ s^{-1}$) | $0;33;1000$ | $32.4\pm 2.5$ | $32.3\pm 2.6$ | $32.4\pm 2.5$
Eccentricity parameter 1, $\sqrt{e}$ cos$\omega$ | $-1.0;-0.453;1.0$ | $-0.441_{-0.030}^{+0.035}$ | $-0.440_{-0.031}^{+0.036}$ | $-0.440_{-0.030}^{+0.035}$
Eccentricity parameter 2, $\sqrt{e}$ sin$\omega$ | $-1.0;-0.231;1.0$ | $-0.217_{-0.061}^{+0.076}$ | $-0.217_{-0.062}^{+0.081}$ | $-0.221_{-0.060}^{+0.074}$
Limb-darkening coefficient 1, $q_{1}$ | $0;0.5;1.0$ | $0.538\pm 0.068$ | $0.544\pm 0.067$ | $0.540\pm 0.068$
Limb-darkening coefficient 2, $q_{2}$ | $0;0.5;1.0$ | $0.187_{-0.043}^{+0.054}$ | $0.183_{-0.041}^{+0.054}$ | $0.185_{-0.043}^{+0.054}$
Stellar rotation velocity, $v\sin i_{\star}$ ($\rm km\ s^{-1}$) | $0;5;10$ | $5.23\pm 0.72$ | $5.5_{-1.6}^{+1.8}$ | $5.15\pm 0.62$
Projected spin-orbit angle, $\lambda$ (deg) | $-180;0;180$ | $-14.5\pm 7.7$ | $-2_{-16}^{+12}$ | $-11.1\pm 6.6$
Relative RV Offset for in-transit APF, Oct. 29 2019 ( m s-1) | $-1000;0;1000$ | $32.0\pm 3.1$ | … | $32.8\pm 2.8$
Relative RV Offset for in-transit APF, Jan. 4 2020 ( m s-1) | $-1000;0;1000$ | … | $8.6\pm 4.3$ | $7.1_{-3.5}^{+3.2}$
Relative RV Offset for out-of-transit APF ( m s-1) | $-1000;0;1000$ | $-98.7\pm 1.9$ | $98.7_{-1.8}^{+1.9}$ | $-98.7_{-1.7}^{+1.9}$
Relative RV Offset for Coralie ( m s-1) | $-1000;0;1000$ | $-57_{-22}^{+19}$ | $-57_{-22}^{+19}$ | $-58_{-22}^{+19}$
Relative RV Offset for FEROS ( m s-1) | $-1000;0;1000$ | $3.5\pm 2.1$ | $3.4\pm 2.1$ | $3.5\pm 2.1$
Relative RV Offset for HARPS ( m s-1) | $-1000;0;1000$ | $-0.7\pm 2.7$ | $-0.6_{-2.5}^{+2.7}$ | $-0.8_{-2.5}^{+2.7}$
Derived Parameters: |  |  |  |
Planetary radius $R_{\mathrm{b}}$ ($\mathrm{R_{jup}}$) | … | $0.380_{-0.044}^{+0.049}$ | $0.378_{-0.045}^{+0.049}$ | $0.378_{-0.045}^{+0.047}$
Planetary Mass $M_{\mathrm{b}}$ ($\mathrm{M_{jup}}$) | … | $1.097\pm 0.025$ | $1.096\pm 0.025$ | $1.097\pm 0.024$
Impact parameter $b$ | … | $0.587\pm 0.014$ | $0.586\pm 0.013$ | $0.587\pm 0.013$
Transit duration $T_{\mathrm{14}}$ (h) | … | $5.2893\pm 0.010$ | $5.2893\pm 0.010$ | $5.2895\pm 0.0099$
Transit depth $\delta$ | … | $0.009331\pm 0.000019$ | $0.009330\pm 0.000018$ | $0.009330_{-0.000021}^{+0.000019}$
Inclination $i$ (∘) | … | $88.02_{-0.16}^{+0.14}$ | $88.03_{-0.17}^{+0.14}$ | $88.03_{-0.16}^{+0.13}$
Eccentricity $e$ | … | $0.245_{-0.022}^{+0.020}$ | $0.246_{-0.022}^{+0.020}$ | $0.247_{-0.021}^{+0.020}$
Argument of periastron $\omega$ (deg) | … | $206.3_{-9.4}^{+8.0}$ | $206.3_{-9.8}^{+7.8}$ | $206.7_{-9.0}^{+7.8}$
Limb darkening; $u_{\mathrm{1}}$ | … | $0.276_{-0.053}^{+0.061}$ | $0.272_{-0.052}^{+0.058}$ | $0.273_{-0.053}^{+0.059}$
Limb darkening; $u_{\mathrm{2}}$ | … | $0.461_{-0.10}^{+0.098}$ | $0.468_{-0.10}^{+0.096}$ | $0.464_{-0.10}^{+0.097}$

Figure 4 shows a mixture of red points, blue points, red and blue points, and
uncolored points, illustrating the point that eccentricity and inclination are
not always observed to go together. The subject of this paper, K2-232, is
represented by a blue point with $a/R_{\star}\approx 19$ and $T_{\rm
eff}\approx 6154$ K.

Note, too, that the red and blue points are almost all found outside of the
lower left corner. The systems with $a/R_{\star}\lesssim 10$ and $T_{\rm
eff}\lesssim 6000$ K are not demonstrably inclined or eccentric (although in
some cases the uncertainties are large). Apparently, the hottest Jupiters
around relatively cool stars have lower eccentricities and inclinations than
the other members of the sample. This may be evidence for tidal dissipation,
which acts to dynamically cool the system. Dissipation rates are strong
functions of orbital separation and may also depend sensitively on the
internal structure of the star (although for eccentricity damping, tidal
dissipation within the planet is expected to be at least as important the
dissipation within the star).

Thus, the current dataset, limited though it might be, provides evidence that
tidal dissipation has reoriented and circularized the orbits of hot Jupiters.

The authors thank Maximilian N. G${\rm\ddot{u}}$enther for helpful discussions
on the Allesfitter and are also grateful to the anonymous referee for their
constructive comments and suggestions. J.N.W. thanks the Heising-Simons
Foundation for support. M.R. is supported by the National Science Foundation
Graduate Research Fellowship Program under Grant Number DGE-1752134. We are
grateful to Metrics for providing helpful suggestions on the figure design.
Part of this research was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration (80NM0018D0004). This work is supported
by China National Astronomical Data Center (NADC), Chinese Virtual Observatory
(China-VO), Astronomical Big Data Joint Research Center, co-founded by
National Astronomical Observatories, Chinese Academy of Sciences and Alibaba
Cloud.

## References

  * Albrecht et al. (2012) Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2012, ApJ, 757, 18
  * Bate et al. (2010) Bate, M. R., Lodato, G., & Pringle, J. E. 2010, MNRAS, 401, 1505
  * Batygin et al. (2011) Batygin, K., Morbidelli, A., & Tsiganis, K. 2011, A&A, 533, A7
  * Brahm et al. (2018) Brahm, R., Espinoza, N., Jordán, A., et al. 2018, MNRAS, 477, 2572
  * Butler et al. (1996) Butler, R. P., Marcy, G. W., Williams, E., et al. 1996, PASP, 108, 500
  * de Laplace (1796) de Laplace, P. S. 1796, Exposition du système du monde, doi:10.3931/e-rara-497
  * Dong et al. (2013) Dong, S., Katz, B., & Socrates, A. 2013, The Astrophysical Journal Letters, 781, L5
  * Fabrycky & Tremaine (2007) Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298
  * Ford & Rasio (2008) Ford, E. B., & Rasio, F. A. 2008, ApJ, 686, 621
  * Gaudi & Winn (2007) Gaudi, B. S., & Winn, J. N. 2007, ApJ, 655, 550
  * Gaudi et al. (2017) Gaudi, B. S., Stassun, K. G., Collins, K. A., et al. 2017, Nature, 546, 514
  * Goldreich & Sari (2003) Goldreich, P., & Sari, R. 2003, ApJ, 585, 1024
  * Günther & Daylan (2020) Günther, M. N., & Daylan, T. 2020, arXiv e-prints, arXiv:2003.14371
  * Innanen et al. (1997) Innanen, K. A., Zheng, J. Q., Mikkola, S., & Valtonen, M. J. 1997, AJ, 113, 1915
  * Kant (1755) Kant, I. 1755, Allgemeine Naturgeschichte und Theorie des Himmels
  * Lai et al. (2011) Lai, D., Foucart, F., & Lin, D. N. C. 2011, MNRAS, 412, 2790
  * McLaughlin (1924) McLaughlin, D. 1924, The Astrophysical Journal, 60
  * Naoz (2016) Naoz, S. 2016, ARA&A, 54, 441
  * Narita et al. (2009) Narita, N., Hirano, T., Sato, B., et al. 2009, PASJ, 61, 991
  * Petrovich (2015) Petrovich, C. 2015, ApJ, 799, 27
  * Pont et al. (2010) Pont, F., Endl, M., Cochran, W. D., et al. 2010, MNRAS, 402, L1
  * Queloz et al. (2010) Queloz, D., Anderson, D. R., Collier Cameron, A., et al. 2010, A&A, 517, L1
  * Rasio & Ford (1996) Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954
  * Rogers et al. (2012) Rogers, T. M., Lin, D. N. C., & Lau, H. H. B. 2012, ApJ, 758, L6
  * Rossiter (1924) Rossiter, R. 1924, The Astrophysical Journal, 60
  * Souami & Souchay (2012) Souami, D., & Souchay, J. 2012, A&A, 543, A133
  * Southworth (2011) Southworth, J. 2011, MNRAS, 417, 2166
  * Storch et al. (2014) Storch, N. I., Anderson, K. R., & Lai, D. 2014, Science, 345, 1317
  * Vogt et al. (2014) Vogt, S. S., Radovan, M., Kibrick, R., et al. 2014, PASP, 126, 359
  * Winn et al. (2010) Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010, ApJ, 718, L145
  * Winn et al. (2009a) Winn, J. N., Johnson, J. A., Albrecht, S., et al. 2009a, ApJ, 703, L99
  * Winn et al. (2007) Winn, J. N., Johnson, J. A., Peek, K. M. G., et al. 2007, ApJ, 665, L167
  * Winn et al. (2009b) Winn, J. N., Howard, A. W., Johnson, J. A., et al. 2009b, ApJ, 703, 2091
  * Wu & Lithwick (2011) Wu, Y., & Lithwick, Y. 2011, ApJ, 735, 109
  * Wu & Murray (2003) Wu, Y., & Murray, N. 2003, ApJ, 589, 605
  * Yu et al. (2018) Yu, L., Rodriguez, J. E., Eastman, J. D., et al. 2018, AJ, 156, 127