# A Tendency Toward Alignment in Single-Star Warm Jupiter Systems

Malena Rice 51 Pegasi b Fellow  Department of Physics and Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Department of Astronomy, Yale University, New Haven,
CT 06511, USA  Songhu Wang Department of Astronomy, Indiana University,
Bloomington, IN 47405, USA  Xian-Yu Wang National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 10010, China  University of the Chinese
Academy of Sciences, Beijing, 100049, China  Guðmundur Stefánsson Henry Norris
Russell Fellow  Princeton University, Department of Astrophysical Sciences, 4
Ivy Lane, Princeton, NJ 08540, USA  Howard Isaacson Department of Astronomy,
University of California Berkeley, Berkeley CA 94720, USA  Centre for
Astrophysics, University of Southern Queensland, Toowoomba, QLD, Australia
Andrew W. Howard Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Sarah E. Logsdon NSF’s National Optical-Infrared
Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA
Heidi Schweiker NSF’s National Optical-Infrared Astronomy Research Laboratory,
950 N. Cherry Ave., Tucson, AZ 85719, USA  Fei Dai Division of Geological and
Planetary Sciences 1200 E California Blvd, Pasadena, CA 91125  Casey Brinkman
NSF Graduate Research Fellow  Institute for Astronomy, University of Hawai’i
at Manoa, Honolulu, HI 96822, USA  Steven Giacalone Department of Astronomy,
University of California Berkeley, Berkeley, CA 94720-3411, USA  Rae Holcomb
Department of Physics and Astronomy, University of California Irvine, Irvine,
CA 92697, USA Malena Rice<EMAIL_ADDRESS>
###### Abstract

The distribution of spin-orbit angles for systems with wide-separation,
tidally detached exoplanets offers a unique constraint on the prevalence of
dynamically violent planetary evolution histories. Tidally detached planets
provide a relatively unbiased view of the primordial stellar obliquity
distribution, since they cannot tidally realign within the system lifetime. We
present the third result from our Stellar Obliquities in Long-period Exoplanet
Systems (SOLES) survey: a measurement of the Rossiter-McLaughlin effect across
two transits of the tidally detached warm Jupiter TOI-1478 b with the
WIYN/NEID and Keck/HIRES spectrographs, revealing a sky-projected spin-orbit
angle $\lambda=6.2^{+5.9}_{-5.5}\,{}^{\circ}$. Combining this new measurement
with the full set of archival obliquity measurements, including two previous
constraints from the SOLES survey, we demonstrate that, in single-star
systems, tidally detached warm Jupiters are preferentially more aligned than
closer-orbiting hot Jupiters. This finding has two key implications: (1)
planets in single-star systems tend to form within aligned protoplanetary
disks, and (2) warm Jupiters form more quiescently than hot Jupiters, which,
in single-star systems, are likely perturbed into a misaligned state through
planet-planet interactions in the post-disk-dispersal phase. We also find that
lower-mass Saturns span a wide range of spin-orbit angles, suggesting a
prevalence of planet-planet scattering and/or secular mechanisms in these
systems.

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

††software: numpy (Oliphant, 2006; Walt et al., 2011; Harris et al., 2020),
matplotlib (Hunter, 2007), pandas (McKinney et al., 2010), scipy (Virtanen et
al., 2020), allesfitter (Günther & Daylan, 2020), emcee (Foreman-Mackey et
al., 2013), MESA (Paxton et al., 2010, 2013, 2015)

##  1 Introduction

In contrast with the quiescent picture presented by the solar system,
exoplanets have been discovered orbiting forwards, sideways, and potentially
even backwards around their host stars. The degree of spin-orbit alignment of
a system is quantified by its “stellar obliquity,” which is the angle between
the system’s net orbital angular momentum vector and the spin axis of the host
star. The maximum stellar obliquity attained by a system during its lifetime
provides a unique constraint on its evolutionary history, offering a window
into that system’s dynamical past.

Stellar obliquities are commonly constrained using the Rossiter-McLaughlin
effect (Rossiter, 1924; McLaughlin, 1924), which offers an avenue to measure
the 2D, sky-projected spin-orbit angle of an occulting body. The Rossiter-
McLaughlin effect describes the phenomenon in which an occulter (in exoplanet
systems, a transiting planet) blocks out different red- or blue-shifted
components of its host star’s light as it occults the stellar disk, producing
radial velocity (RV) shifts across the transit.

Because the Rossiter-McLaughlin effect is observed across individual transit
events, the vast majority of stellar obliquity measurements to date have been
made in systems with massive, close-orbiting planets due to the relatively
short, frequent, and deep transits observed for these systems (see Albrecht et
al. (2022) for a review of existing constraints). However, planets orbiting at
small orbital separation are heavily impacted by tidal interactions with the
host star, which can erase the signatures of misalignments over time.

By contrast, “tidally detached” planets at wider orbital separation remain
relatively unaffected by interactions with the host star over the age of the
Universe. As a result, tidally detached planets provide a clearer view of the
maximum stellar obliquity obtained in each system. Tidally detached planets
can also help to distinguish whether misalignments typically occur at the
protoplanetary disk stage – that is, whether all planetary systems span a
similar range of misalignments due to formation in misaligned disks – or
whether misalignments typically occur in the post-disk-dispersal phase.

The Stellar Obliquities in Long-period Exoplanet Systems (SOLES) survey (Rice
et al., 2021; Wang et al., 2022) is, for the first time, systematically
accruing a set of stellar obliquity constraints for the tidally detached
population of exoplanets. These constraints will help to delineate the true
rate of dynamically violent phases during planetary systems’ evolution. By
distinguishing the maximum misalignments typically attained by planetary
systems as a function of orbital properties and planet mass, the SOLES survey
will help to delineate the range of formation channels producing different
types of planetary systems.

An immediate outcome of this effort is a new set of constraints on the origins
of hot and warm Jupiters. In this work, we present the third result from the
SOLES survey: a sky-projected stellar obliquity measurement for the tidally
detached warm Jupiter TOI-1478 b. We present two observations of the Rossiter-
McLaughlin effect across two transits of TOI-1478 b, taken with the NEID
spectrograph (Schwab et al., 2016) on the WIYN 3.5-meter telescope and the
High Resolution Echelle Spectrometer (HIRES; Vogt et al., 1994) on the
10-meter Keck I telescope.

TOI-1478 b is a confirmed, tidally detached ($a/R_{*}=18.54_{-0.6}^{+0.7}$)
giant planet ($M_{p}=0.851^{+0.052}_{-0.047}M_{J}$) initially identified
through an analysis of full-frame images obtained by the Transiting Exoplanet
Survey Satellite (TESS; Ricker et al., 2015). TOI-1478 b lies on a circular or
near-circular orbit ($e=0.024^{+0.032}_{-0.017}$), with orbital period
$P=10.180249\pm 0.000015$ days (Rodriguez et al., 2021). The planet is hosted
by a solar-mass ($M_{*}=0.946^{+0.059}_{-0.041}M_{\odot}$) star, TOI-1478 (TIC
409794137), with $V=10.81$, $T_{\mathrm{eff}}=5595\pm 83$ K, and age
$\tau=9.2^{+3.1}_{-3.9}$ Gyr (Rodriguez et al., 2021).

Using our new observations, we find that TOI-1478 b is consistent with
alignment, with $\lambda=6.2^{+5.9}_{-5.5}\,{}^{\circ}$. Combining this
finding with the previous SOLES results and archival measurements, we then
demonstrate for the first time that warm Jupiters in single-star systems are
preferentially more aligned than their hot Jupiter analogues. We conclude with
implications of this finding, which suggests that (1) planetary systems around
single stars are typically formed in aligned disks and (2) hot Jupiters are
likely misaligned after disk dispersal, while warm Jupiters form quiescently.

##  2 Observations

###  2.1 WIYN/NEID Observations

We obtained $20$ radial velocity measurements of TOI-1478 with the high-
resolution mode of the NEID spectrograph ($R\sim 110,000$) from 03:18-09:49 UT
on Feb. 5, 2022. Exposure times were fixed at 1000 seconds. Observations were
interrupted during the post-transit baseline from 08:39-08:57 UT to obtain a
set of intermediate calibration frames, including a Fabry-Pérot etalon frame
and three laser frequency comb frames. Excluding the intermediate calibration
window, our observations spanned the full transit with an additional $\sim 18$
and $\sim 70$ minutes of pre- and post-transit observations, respectively, to
constrain the RV baseline. Seeing was variable due to windy conditions,
ranging from $1.0\arcsec-2.5\arcsec$. We provide NEID RV measurements and
uncertainties in Table 1.

To extract precise stellar parameters, we also obtained one 52-minute NEID
exposure of TOI-1478 on March 6, 2022 (seeing ranged from
$1.3\arcsec-1.9\arcsec$) and one 1-hour NEID exposure on March 12, 2022
(seeing stable at $\sim$1.0$\arcsec$). All data were reduced using the NEID
Data Reduction Pipeline111More information on the NEID data reduction pipeline
can be found here: https://neid.ipac.caltech.edu/docs/NEID-DRP/ and retrieved
from the NExScI NEID Archive222https://neid.ipac.caltech.edu/.

Table 1: NEID radial velocities for the TOI-1478 system. Time (BJD) | RV (m/s) |  $\sigma_{\rm RV}$ (m/s)
---|---|---
2459615.655110 | 20799.5 | 5.0
2459615.665986 | 20804.0 | 4.4
2459615.678704 | 20803.8 | 4.6
2459615.708742 | 20805.4 | 4.1
2459615.734892 | 20792.6 | 4.2
2459615.746750 | 20797.1 | 5.0
2459615.759267 | 20795.1 | 5.2
2459615.771037 | 20799.7 | 5.3
2459615.782804 | 20788.8 | 4.9
2459615.794706 | 20791.3 | 4.8
2459615.806261 | 20793.2 | 5.1
2459615.819525 | 20785.7 | 4.6
2459615.830167 | 20787.1 | 4.1
2459615.841928 | 20792.4 | 3.5
2459615.853901 | 20795.9 | 4.0
2459615.866358 | 20789.7 | 3.4
2459615.890445 | 20788.7 | 3.0
2459615.901364 | 20790.7 | 3.9
2459615.915296 | 20790.2 | 4.2
2459615.927363 | 20783.5 | 3.5

###  2.2 Keck/HIRES Observations

We obtained 39 radial velocity measurements of TOI-1478 with the HIRES
spectrograph from 06:00-13:00 UT on Feb. 15, 2022. The median exposure time
was $592$ seconds, with 100k exposure meter counts per spectrum. Our
observations spanned the full 4-hour transit of TOI-1478 b, with an additional
$\sim$100 and $\sim$45 minutes of pre- and post-transit observations,
respectively, to constrain the RV baseline. Seeing was stable throughout the
observing period, ranging from $1.2\arcsec-1.4\arcsec$.

The moon phase was two days from full during our observation, with a
separation of $36.6\arcdeg-36.7\arcdeg$ from TOI-1478 throughout the night. We
used the C2 decker ($14\arcsec\times 0.861\arcsec,R=60,000$) for all
Keck/HIRES RV observations, enabling direct sky subtraction to better account
for scattered light from the moon. Our dataset was reduced using the
California Planet Search pipeline (Howard et al., 2010). We provide HIRES RV
measurements, S-index values, and associated uncertainties in Table 2.

To calibrate our observations and extract precise stellar parameters, we also
obtained a 21-minute iodine-free HIRES template exposure of TOI-1478 using the
B3 decker ($14.0\arcsec\times 0.574\arcsec,R=72,000$) on UT Feb. 21, 2022. We
collected 250k exposure meter counts for the template spectrum, which was
observed in good conditions with $1.2\arcsec$ seeing.

Table 2: HIRES radial velocities for the TOI-1478 system. Time (BJD) | RV (m/s) |  $\sigma_{\rm RV}$ (m/s) | S-index | $\sigma_{S}$
---|---|---|---|---
2459625.763126 | 7.29 | 1.37 | 0.153 | 0.001
2459625.770626 | 9.27 | 1.38 | 0.155 | 0.001
2459625.777964 | 4.25 | 1.32 | 0.155 | 0.001
2459625.785418 | 8.66 | 1.44 | 0.155 | 0.001
2459625.792536 | 1.63 | 1.36 | 0.155 | 0.001
2459625.799538 | 6.65 | 1.34 | 0.153 | 0.001
2459625.806147 | 8.54 | 1.31 | 0.152 | 0.001
2459625.812859 | 6.17 | 1.34 | 0.153 | 0.001
2459625.819630 | 7.46 | 1.28 | 0.153 | 0.001
2459625.826262 | 6.78 | 1.32 | 0.154 | 0.001
2459625.832894 | 3.34 | 1.31 | 0.155 | 0.001
2459625.839537 | 3.57 | 1.31 | 0.153 | 0.001
2459625.846632 | -0.22 | 1.47 | 0.153 | 0.001
2459625.854098 | 3.47 | 1.39 | 0.152 | 0.001
2459625.861366 | 4.48 | 1.26 | 0.151 | 0.001
2459625.868333 | 8.35 | 1.38 | 0.151 | 0.001
2459625.875300 | 6.66 | 1.30 | 0.151 | 0.001
2459625.881851 | 10.19 | 1.39 | 0.152 | 0.001
2459625.888981 | 1.70 | 1.33 | 0.150 | 0.001
2459625.896273 | 2.31 | 1.32 | 0.152 | 0.001
2459625.903645 | 4.20 | 1.36 | 0.151 | 0.001
2459625.911388 | 2.25 | 1.42 | 0.151 | 0.001
2459625.918667 | 2.11 | 1.37 | 0.153 | 0.001
2459625.926156 | 1.24 | 1.44 | 0.151 | 0.001
2459625.933066 | -2.40 | 1.50 | 0.152 | 0.001
2459625.940647 | -2.17 | 1.45 | 0.150 | 0.001
2459625.948228 | -4.86 | 1.29 | 0.152 | 0.001
2459625.955716 | -10.76 | 1.26 | 0.150 | 0.001
2459625.970819 | -6.96 | 1.32 | 0.151 | 0.001
2459625.963181 | -8.71 | 1.22 | 0.151 | 0.001
2459625.978389 | -15.52 | 1.26 | 0.149 | 0.001
2459625.985738 | -12.07 | 1.33 | 0.148 | 0.001
2459625.993365 | -13.67 | 1.34 | 0.149 | 0.001
2459626.001456 | -10.52 | 1.42 | 0.149 | 0.001
2459626.010009 | -3.19 | 1.45 | 0.149 | 0.001
2459626.018040 | -7.20 | 1.46 | 0.149 | 0.001
2459626.026131 | -3.47 | 1.51 | 0.149 | 0.001
2459626.035286 | -9.51 | 1.43 | 0.149 | 0.001
2459626.044105 | -7.98 | 1.44 | 0.148 | 0.001

##  3 Stellar Obliquity Modeling

We applied the allesfitter Python package (Günther & Daylan, 2020) to jointly
model our in-transit Rossiter-McLaughlin data together with the photometric
and radial velocity datasets presented in the TOI-1478 b discovery paper
(Rodriguez et al., 2021). These additional datasets include photometry from
TESS, PEST, and FLWO/KeplerCam, as well as radial velocities from the TRES
(Fűrész, 2008), CHIRON (Tokovinin et al., 2013), FEROS (Kaufer et al., 1999),
and CORALIE (Queloz et al., 2001) spectrographs.

Our model parameters, each initialized with uniform priors provided in Table
3, include the orbital period ($P$), transit mid-times ($T_{0}$), 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
($(R_{\star}+R_{P})/a$), RV semi-amplitude ($K$), parameterized eccentricity
and argument of periastron ($\sqrt{e}\,\cos{\,\omega}$,
$\sqrt{e}\,\sin{\,\omega}$), sky-projected spin-orbit angle ($\lambda$), and
sky-projected stellar rotational velocity ($v\sin i_{\star}$). The two limb
darkening coefficients were initialized as $q_{1}=q_{2}=0.5$.

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 drawn from Rodriguez et al. (2021). We applied
a quadratic model to fit for additive RV offsets between each spectrograph,
and we allowed $\lambda$ to vary between $-180\arcdeg$ and $+180\arcdeg$.

To thoroughly sample the posterior distribution for each model parameter, we
ran an affine-invariant Markov Chain Monte Carlo (MCMC) analysis with 100
walkers. We ensured that each chain was fully converged by running all Markov
chains to over 30$\times$ their autocorrelation length, which corresponded to
at minimum 200,000 accepted steps per walker.

The best-fit model parameters are provided in Table 3 together with their
associated 1$\sigma$ uncertainties. The resulting best-fit Rossiter-McLaughlin
model from each of our three runs is shown in Figure 1, with the residuals
provided below each model fit. We ultimately find that TOI-1478 is an aligned
system, with sky-projected stellar obliquity
$\lambda=6.2_{-5.5}^{+5.9}\,{}^{\circ}$. An analysis of photometry from the
TESS, WASP (Pollacco et al., 2006), and KELT (Pepper et al., 2007, 2012, 2018)
surveys has revealed no periodicities that are robustly associated with the
stellar spin rate (Rodriguez et al., 2021), preventing a further constraint on
the 3D stellar obliquity $\psi$.

Figure 1: Best-fitting models for the Rossiter-McLaughlin observations
obtained in this work. Left: Best fit to the WIYN/NEID observations taken from
the TOI-1478 b transit on UT Feb. 5, 2022. Center: Best fit to the Keck/HIRES
observations taken from the TOI-1478 b transit on UT Feb. 15, 2022. Right:
Joint fit incorporating both datasets. Table 3: System Parameters, Priors, and
Results for TOI-1478 b

| HIRES Spectrum | NEID Spectrum | Rodriguez+ 2021 | Global RM fit: NEID+HIRES |
---|---|---|---|---|---
| The Cannon | iSpec and PARAM 1.3 | EXOFASTv2 | (Preferred Solution) |
Stellar Parameters: |
$M_{*}$ ($\,{\rm M_{\sun}}$) | - | $0.907\pm 0.022$ | $0.947^{+0.059}_{-0.041}$ | - |
$R_{*}$ ($\,{\rm R_{\sun}}$) | - | $1.034\pm 0.023$ | $1.048^{+0.030}_{-0.029}$ | - |
$\log{g}$ (cgs) | $4.231\pm 0.164$ | $4.14\pm 0.14$ | $4.374^{+0.039}_{-0.032}$ | - |
$[\rm{Fe/H}]$ (dex) | $-0.03\pm 0.06$ | $-0.17\pm 0.07$ | $0.078^{+0.072}_{-0.065}$ | - |
$T_{\rm eff}$ (K) | $5500\pm 132$ | $5496\pm 110$ | $5597^{+83}_{-82}$ | - |
$v\sin i_{\star}$ (km/s) | $3.00\pm 0.98$ | $2.22\pm 0.82$ | $4.3\pm 0.5^{a}$ | $1.24\pm 0.16$ |
| Priors for global fit | Global fit 1: NEID | Global fit 2: HIRES | Global fit 3: NEID+HIRES |
|  |  |  | (Preferred Solution) |
Fitted parameters: |
$R_{P}/R_{\star}$ | $\mathcal{U}(0.1004;0.0;1.0)$ | $0.1021_{-0.0012}^{+0.0010}$ | $0.1020_{-0.0013}^{+0.0010}$ | $0.1020_{-0.0012}^{+0.0011}$ |
$(R_{\star}+R_{p})/a$ | $\mathcal{U}(0.04937;0.0;1.0)$ | $0.0691_{-0.0038}^{+0.0042}$ | $0.0686_{-0.0040}^{+0.0044}$ | $0.0683\pm 0.0032$ |
$\cos{i}$ | $\mathcal{U}(0.026;0.0;1.0)$ | $0.0425_{-0.0049}^{+0.0058}$ | $0.0418_{-0.0053}^{+0.0060}$ | $0.0414\pm 0.0044$ |
$T_{0}\ (\rm BJD_{\rm TDB}-2450000)$ | $\mathcal{U}(9066.014585;9065.0;9067.0)$ | $9066.01871\pm 0.00031$ | $9066.01870\pm 0.00032$ | $9066.01870\pm 0.00031$ |
$P$ (days) | $\mathcal{U}(10.180249;9.0;11.0)$ | $10.1802104\pm 0.0000072$ | $10.1802116\pm 0.0000071$ | $10.1802117\pm 0.0000071$ |
$K$ ($\rm m\ s^{-1}$) | $\mathcal{U}(82.5;0.0;1000.0)$ | $81.7\pm 3.2$ | $81.8\pm 3.2$ | $81.5\pm 2.8$ |
$\sqrt{e}\cos{\omega}$ | $\mathcal{U}(-0.053;-1.0;1.0)$ | $-0.003_{-0.078}^{+0.073}$ | $-0.003\pm 0.077$ | $-0.011\pm 0.078$ |
$\sqrt{e}\sin{\omega}$ | $\mathcal{U}(-0.145;-1.0;1.0)$ | $0.236_{-0.16}^{+0.092}$ | $0.239_{-0.16}^{+0.093}$ | $0.222_{-0.11}^{+0.065}$ |
$q_{\rm 1:HIRES}$ | $\mathcal{U}(0.5;0;1)$ | $-$ | $0.54\pm 0.31$ | $0.49_{-0.30}^{+0.32}$ |
$q_{\rm 2:HIRES}$ | $\mathcal{U}(0.5;0;1)$ | $-$ | $0.42_{-0.29}^{+0.36}$ | $0.39_{-0.27}^{+0.36}$ |
$q_{\rm 1:NEID}$ | $\mathcal{U}(0.5;0;1)$ | $0.41_{-0.29}^{+0.37}$ | $-$ | $0.45_{-0.31}^{+0.35}$ |
$q_{\rm 2:NEID}$ | $\mathcal{U}(0.5;0;1)$ | $0.48_{-0.33}^{+0.35}$ | $-$ | $0.52_{-0.36}^{+0.33}$ |
$\lambda$ (deg) | $\mathcal{U}(0;-180;+180)$ | $23\pm 41$ | $6.9_{-9.9}^{+10.}$ | $6.2_{-5.5}^{+5.9}$ |
$v\sin i_{\star}$ ($\rm km\ s^{-1}$) | $\mathcal{U}(4.3;0.0;10)$ | $0.78_{-0.49}^{+0.53}$ | $1.31_{-0.19}^{+0.20}$ | $1.24\pm 0.16$ |
Derived parameters: |
$R_{P}\ (\,{\rm R_{\rm J}})$ | - | $1.07_{-0.10}^{+0.16}$ | $1.07_{-0.10}^{+0.16}$ | $1.07_{-0.10}^{+0.16}$ |
$M_{P}\ (\,{\rm M_{\rm J}})$ | - | $0.90\pm 0.13$ | $0.89\pm 0.13$ | $0.88_{-0.12}^{+0.11}$ |
$b$ | - | $0.641_{-0.034}^{+0.027}$ | $0.636_{-0.038}^{+0.029}$ | $0.636_{-0.035}^{+0.029}$ |
$T_{\rm 14}$ | - | $4.140\pm 0.042$ | $4.129\pm 0.042$ | $4.131\pm 0.041$ |
$\delta_{\rm FLWO}$ | - | $11.52\pm 0.18$ | $11.51_{-0.16}^{+0.19}$ | $11.52\pm 0.18$ |
$\delta_{\rm PESTR}$ | - | $11.02_{-0.23}^{+0.28}$ | $11.02_{-0.24}^{+0.30}$ | $11.00_{-0.26}^{+0.28}$ |
$\delta_{\rm TESS}$ | - | $10.95\pm 0.10$ | $10.942\pm 0.095$ | $10.936_{-0.087}^{+0.10}$ |
$a$ | - | $0.0808_{-0.0087}^{+0.013}$ | $0.0814_{-0.0090}^{+0.013}$ | $0.0818_{-0.0085}^{+0.013}$ |
$i$ (deg) | - | $87.56_{-0.33}^{+0.28}$ | $87.60_{-0.35}^{+0.30}$ | $87.63\pm 0.25$ |
$e$ | - | $0.061_{-0.043}^{+0.050}$ | $0.062_{-0.044}^{+0.051}$ | $0.055\pm 0.032$ |
$\omega$ | - | $92_{-18}^{+35}$ | $92_{-18}^{+34}$ | $94_{-20.}^{+26}$ |
$u_{\rm 1:HIRES}$ | - | $-$ | $0.54_{-0.38}^{+0.53}$ | $0.47_{-0.34}^{+0.50}$ |
$u_{\rm 2:HIRES}$ | - | $-$ | $0.10\pm 0.44$ | $0.13\pm 0.42$ |
$u_{\rm 1:NEID}$ | - | $0.52_{-0.38}^{+0.58}$ | $-$ | $0.59_{-0.42}^{+0.57}$ |
$u_{\rm 2:NEID}$ | - | $0.02\pm 0.41$ | $-$ | $-0.02_{-0.41}^{+0.43}$ |

aafootnotetext: Rodriguez et al. (2021) measured $v\sin i_{\star}$ for
TOI-1478 from TRES data using the methodology in Zhou et al. (2018).

##  4 Stellar Parameters

Next, we analyzed individual iodine-free spectra of TOI-1478 to extract the
system’s stellar parameters directly from the HIRES and NEID observations
described in this work. The observed RM semi-amplitude ($A_{\mathrm{RM}}$) was
significantly smaller than anticipated based on previous constraints from
Rodriguez et al. (2021): rather than detecting the anticipated signal with RM
semi-amplitude $28.2\pm 4.5$ m/s (obtained by simulating the system with the
allesfitter Python package), we observed an RM semi-amplitude of only $6.6\pm
0.9$ m/s.

The theoretical semi-amplitude of a Rossiter-McLaughlin signal is given by

$A_{\mathrm{RM}}\simeq\frac{2}{3}\Big{(}\frac{R_{p}}{R_{*}}\Big{)}^{2}v\sin
i_{*}\sqrt{1-b^{2}},$ (1)

where $R_{p}$ and $R_{*}$ are the planet and stellar radius, respectively,
$v\sin i_{*}$ is the projected stellar rotational velocity, and $b$ is the
impact parameter of the planetary companion’s transit (Triaud, 2017). The
transit depth $\frac{R_{p}}{R_{*}}$ and the impact parameter $b$ are each
well-constrained by the observed transit, suggesting that an overestimated
value of $v\sin i_{*}$ may have caused the observed discrepancy. As described
in the remainder of Section 4, we obtain lower $v\sin i_{*}$ estimates from
the HIRES and NEID data, which helps to account for the lower observed RM
semi-amplitude.

###  4.1 Stellar Parameters from Keck/HIRES

First, we modeled the stellar parameters of TOI-1478 by analyzing the iodine-
free Keck/HIRES template spectrum obtained in this work. We applied the
methods described in Rice & Brewer (2020), which transfers stellar labels
using the generative machine learning program The Cannon (Ness et al., 2015)
after training on Keck/HIRES spectra of 1,202 FGK stars from the Spectral
Properties of Cool Stars (SPOCS) catalogue (Valenti & Fischer, 2005; Brewer et
al., 2016). The SPOCS catalogue includes 18 precisely determined stellar
labels: 3 global stellar parameters ($T_{\mathrm{eff}}$, $\log g$, $v\sin
i_{\star}$), and 15 elemental abundances: C, N, O, Na, Mg, Al, Si, Ca, Ti, V,
Cr, Mn, Fe, Ni, and Y.

We fit and divided out the continuum baseline of our Keck/HIRES template
spectrum using the Alpha-shape Fitting to Spectrum (AFS) algorithm described
in Xu et al. (2019). Then, we calibrated the wavelength solution by cross-
correlating the normalized spectrum with the solar atlas provided by Wallace
et al. (2011). These two steps placed our spectrum on a normalized scale with
a continuum baseline of unity for direct comparison with the training set.

To estimate the uncertainties in our results, we trained and applied our model
separately using each of the 16 echelle orders in our Keck/HIRES spectrum. We
then extracted the mean and standard deviation determined for each parameter.
Our derived parameters and associated uncertainties are provided in Table 3.
For direct comparison with previous results, we report only $\log g$, [Fe/H],
$T_{\mathrm{eff}}$, and $v\sin i_{*}$ in this work. From our spectroscopic
modeling of our Keck/HIRES spectrum, we obtained $v\sin i_{*}=3.00\pm 0.98$
km/s.

###  4.2 Stellar Parameters from WIYN/NEID

Next, we co-added our two NEID template spectra from Mar. 6 and 12, 2022, and
we analyzed the combined spectrum to independently extract stellar parameters
for TOI-1478. For this spectrum, we obtained a signal-to-noise ratio (SNR) of
58 for the wavelength range $480-680$ nm. We adopted the synthetic spectral
fitting technique implemented in the iSpec Python package (Blanco-Cuaresma et
al., 2014; Blanco-Cuaresma, 2019) to determine the stellar parameters.

Within iSpec, we selected the SPECTRUM radiative transfer code (Gray &
Corbally, 1994) and the MARCS333https://marcs.astro.uu.se/ model atmospheres
and solar abundances from Grevesse et al. (2007). We adopted the sixth version
of the GES atomic line list (Heiter et al., 2015). iSpec incorporates these
constraints to iteratively generate and compare model spectra with the input
spectrum, minimizing the difference using the nonlinear least-squares
Levenberg-Marquardt fitting algorithm (Markwardt, 2009).

To reduce the computation time for this algorithm, we optimized specific
spectral regions geared towards our parameters of interest. These include the
wings of the H$\alpha$, H$\beta$, and Mg I triplet lines, which are sensitive
to $T_{\mathrm{eff}}$ and $\log g$, as well as the Fe I and Fe II lines that
can precisely constrain [Fe/H] and $v\sin i_{*}$. We used the derived
$T_{\mathrm{eff}}$ and [Fe/H] values, as well as the $V$ magnitude ($V=10.81$)
and corrected Gaia parallax (Stassun & Torres, 2018) for TOI-1478, as inputs
to the Bayesian stellar parameter estimation code PARAM
1.3444http://stev.oapd.inaf.it/cgi-bin/param_1.3 (da Silva et al., 2006) to
extract the stellar mass ($M_{*}$) and radius ($R_{*}$). All of the parameters
derived from this study are provided in Table 3. From this analysis, we
obtained $v\sin i_{*}=2.22\pm 0.82$ km/s.

###  4.3 Stellar Parameters from Global Fitting

We also obtained an estimated $v\sin i_{*}$ value from our global fitting,
which incorporated our full photometric and radial velocity datasets (see
Section 3). Based on this global model, we determined that $v\sin
i_{*}=1.24\pm 0.16$ km/s. This lower $v\sin i_{*}$ value, which we adopt as
our preferred solution, can fully account for the observed discrepancy between
the expected and observed RM semi-amplitude.

###  4.4 Comparison of Stellar Parameters

All of our derived stellar parameters agree with previous results from
Rodriguez et al. (2021) at the 1$\sigma$ level, with the exception of [Fe/H]
and $v\sin i_{\star}$. By comparison with the results from Rodriguez et al.
(2021), we obtained a lower value of both [Fe/H] and $v\sin i_{\star}$ from
the NEID spectrum. All results from the NEID spectrum are consistent with our
results from the HIRES spectrum within 1$\sigma$.

Our derived values for $v\sin i_{*}$ each suggest a lower projected spin rate
than the value reported in Rodriguez et al. (2021). This discrepancy in $v\sin
i_{*}$ measurements likely results from the comparatively lower resolving
power of the TRES spectrograph ($R=44,000$), by contrast with the NEID
($R=110,000$) and HIRES ($R=72,000$) spectrographs.

Measurements of $v\sin i_{*}$ are limited by the resolving power of the
instrument: for example, previous work has demonstrated that the IGRINS
spectrograph ($R=45,000$; Park et al., 2014), with a comparable resolving
power to that of TRES, has a lower measurement limit of $v\sin i_{*}\sim 3-4$
km/s (Kesseli et al., 2018; Nofi et al., 2021). As a result, the higher
measured $v\sin i_{*}$ value in Rodriguez et al. (2021) likely reflects the
limited resolving power of TRES.

##  5 Population Analysis

Results from the SOLES survey, together with archival measurements, are
beginning to accrue into a statistical sample of stellar obliquities in
tidally detached exoplanet systems. This enables, for the first time,
comparative studies between distinct populations as the set of available
constraints expands.

Most stellar obliquity measurements to date provide constraints on only a 2D,
sky-projected version of the true stellar obliquity. These measurements,
including that of TOI-1478 b, are not fully representative of the true stellar
obliquity for each individual system: systems that are aligned in their sky-
projection may still be misaligned along the line of sight. However,
population studies can provide useful insights into the typical trends across
a group of planets, even while individual systems require further scrutiny to
understand the typical degree of misalignment.

In this section, we examine the sky-projected stellar obliquity distribution
in relatively isolated, single-star systems. We demonstrate that, in single-
star systems, tidally detached Jovian planets are preferentially more aligned
than their closer-orbiting counterparts. This result has two key implications
that we discuss in this section: (1) protoplanetary disks are typically
aligned in single-star systems, and (2) warm Jupiters typically form
quiescently, while this is not necessarily true for hot Jupiters.

We emphasize that this trend toward alignment does not necessarily extend to
tidally detached Jovian planets in binary and multi-star systems. Our choice
to focus on only single-star systems excludes some highly misaligned warm
Jupiters in systems with more than one star (e.g. K2-290 c (Hjorth et al.,
2021), Kepler-420 b (Santerne et al., 2014), and WASP-8 b (Queloz et al.,
2010; Bourrier et al., 2017)). However, other warm Jupiters in binary or
multi-star systems have instead been observed in or near alignment (e.g. V1298
Tau b (Gaidos et al., 2022; Johnson et al., 2022) and WASP-11 b (Mancini et
al., 2015)). The set of obliquity constraints in binary star systems so far
appears to be more heterogeneous than in the sample of single-star systems,
and further measurements are required to delineate the dominant trends in this
population.

###  5.1 Sample Selection

####  5.1.1 Data Sources

Our sample consists of the set of systems included in both the NASA Exoplanet
Archive “Planetary Systems” list (NASA Exoplanet Archive, 2022) and the TEPCat
catalogue (Southworth, 2011), which compiles published stellar obliquity
measurements into a single database. Both datasets were downloaded on Feb. 19,
2022. We drew $\lambda$ and $T_{\mathrm{eff}}$ from the TEPCat catalogue, and,
where possible, all other system parameters were drawn from the default
parameter set of the NASA Exoplanet Archive. When parameters were unavailable
in the NASA Exoplanet Archive, values were supplemented from the Extrasolar
Values Encyclopaedia555http://exoplanet.eu where possible.

We used the most recently obtained stellar obliquity measurement for most
stars, with the exception of systems in which previous measurements were much
more precise. To maximize the uniformity of our sample, we also favored
measurements of the Rossiter-McLaughlin effect or Doppler tomography over
constraints from starspot crossings. Where only upper planet mass and/or
eccentricity limits were provided, we adopted the upper limit as the fiducial
value. The full set of parameters used in this work is provided in Table 4.

####  5.1.2 Stellar Multiplicity

To develop a clean sample of single-star systems, we then evaluated the
stellar multiplicity of each system with a measured $\lambda$ value. We first
cross-matched our sample with with the Catalogue of Exoplanets in Binary Star
Systems (Schwarz et al., 2016), which includes binary and multiple-star
systems with confirmed exoplanets. We removed all systems with companions
listed either in this catalogue or elsewhere from a literature search of each
system.

Next, we searched for any comoving companions in the SIMBAD database (Wenger
et al., 2000), which includes parallaxes and proper motions drawn from Gaia
eDR3 (Brown et al., 2021). We followed the methods of El-Badry et al. (2021)
to identify neighboring sources, with details summarized here for convenience.

We first queried SIMBAD for all stars with a projected separation $\theta$
within 5 pc of the host star, or

$\theta\leq 17.19\,\mathrm{arcmin}\times(\varpi\,\mathrm{mas}^{-1})$ (2)

for parallax $\varpi$. For each candidate companion star, we checked the
following conditions to verify whether it is a comoving companion:

Condition 1: The candidate companion must have a proper motion $\mu$ within 5
km/s of the planet-hosting star, or

$\sqrt{\Delta\mu_{\mathrm{RA}}^{2}+\Delta\mu_{\mathrm{Dec}}^{2}}\leq
1.05\,\mathrm{mas/yr}\times(\varpi_{1}-2\sigma_{\varpi_{1}})\,\mathrm{mas}^{-1}$
(3)

$\Delta\mu_{\mathrm{RA}}=\mu_{\mathrm{RA},1}-\mu_{\mathrm{RA},2}$ (4)

$\Delta\mu_{\mathrm{Dec}}=\mu_{\mathrm{Dec},1}-\mu_{\mathrm{Dec},2}.$ (5)

Condition 2: The candidate companion must have a parallax consistent with that
of the planet-hosting star within 5$\sigma$, or

$|\varpi_{1}-\varpi_{2}|\leq
5\sqrt{\sigma_{\varpi_{1}}^{2}+\sigma_{\varpi_{2}}^{2}}.$ (6)

We note that El-Badry et al. (2021) required a 2$\sigma$ agreement for
Condition 2, whereas we set a more conservative 5$\sigma$ requirement to
ensure a clean sample of single stars. We removed all systems from our sample
in which one or more neighboring stars was identified as a comoving companion
(that is, all systems in which at least one companion met both of the
conditions above).

Afterwards, we cross-validated our results with the the visual binary
catalogue published by Fontanive & Bardalez Gagliuffi (2021), which identified
exoplanetary systems in binary or multiple-star configurations based on
results from Gaia DR2 (Gaia Collaboration et al., 2016, 2018). All of our
results match with the classifications determined by Fontanive & Bardalez
Gagliuffi (2021). Table 4 includes the full set of single-star systems
retained within our population study.

###  5.2 Stellar Obliquities Comparison

To compare the rate of misalignments across populations, we assigned a label
of either “aligned” or “misaligned” to each system in our sample. We consider
“aligned” planets to be those with $|\lambda|\leq 20\arcdeg$, while planets
with $|\lambda|>20\arcdeg$ are considered “misaligned.” We use this limit as a
comparative diagnostic, but we acknowledge that the typical maximum
misalignment in single-star warm Jupiter systems is not yet well-determined
enough to distinguish between an upper limit of e.g. $|\lambda|=20\arcdeg$ vs.
$|\lambda|=25\arcdeg$. Our results do not substantially change based on the
exact limit that is selected.

####  5.2.1 Initial Trends

Figure 2 places the first three SOLES systems into context as some of the
widest-separation tidally detached exoplanets with measured $|\lambda|$
values. All three of these systems are consistent with alignment. Furthermore,
all of the widest-orbiting ($a/R_{*}>11$), high-mass ($M\geq 0.4M_{J}$)
planets in single-star systems are, so far, consistent with alignment. We
refer to this set of tidally-detached, high-mass planets as “warm Jupiters”
throughout this work.

Figure 2: Distribution of measured $|\lambda|$ values for hot and warm
Jupiters as a function of the orbital separation $a/R_{*}$. Measurements in
systems below the Kraft break are shown as circular markers, while those above
the Kraft break are displayed as triangular markers. The new measurement of
TOI-1478 b from this work is shown in yellow, and the two previous SOLES
measurements are shown in peach. Saturn-mass planets are shown in gray for
reference; we note that WASP-148 b falls into this category, while K2-140 b
and TOI-1478 b are more massive warm Jupiters.

By contrast, many hot Jupiter systems ($a/R_{*}<11$, $M\geq 0.4M_{J}$) are
inconsistent with alignment. In particular, hot Jupiters orbiting hot stars
(above the Kraft break at $T_{\mathrm{eff}}\approx 6100$ K; (Kraft, 1967))
have been commonly discovered in misaligned configurations (e.g. Albrecht et
al., 2012). Furthermore, recent work has demonstrated that, when their 3D
stellar obliquities have been constrained, misaligned hot Jupiters are often
found on polar orbits (Albrecht et al., 2021).

This discrepancy points toward a distinct formation channel for hot Jupiters,
which are likely dominated by a dynamically violent formation mechanism, as
compared with warm Jupiters, which likely form quiescently (Wu, Rice, & Wang,
in review). The apparent distinction between these two populations is
demonstrated in Figure 3: hot Jupiters orbiting hot stars span a wide range of
$|\lambda|$ values, while warm Jupiters have so far all been observed near
alignment.

Figure 3: Distribution of sky-projected spin-orbit angles $|\lambda|$ observed
for hot Jupiters (HJs), warm Jupiters (WJs), and Saturns. Unlike the hot
Jupiters and Saturns, the warm Jupiters are confined to low values
$|\lambda|\leq 20\,^{\circ}$ irrespective of their host star effective
temperature.

In parallel, we also find that lower-mass Saturns, with $0.2M_{J}\leq M_{\rm
pl}<0.4M_{J}$, span a wider range of sky-projected spin-orbit angles than
their higher-mass counterparts. While we turn our focus to the higher-mass
systems for the remainder of our population study, we revisit the implications
of this Saturn-mass population in Section 6.2.

####  5.2.2 Statistical Significance: Divided Comparison

Noting the visible distinction between hot and warm Jupiters, we tested the
significance with which tidally detached Jupiters are more aligned than
closer-in hot Jupiters. To accomplish this, we compared the number of
misaligned warm Jupiters with the number of misaligned hot Jupiters obtained
from random draws.

In total, our sample of warm Jupiters includes 12 systems, none of which are
misaligned. For a direct comparison with this population, we iteratively drew
12 random $|\lambda|$ values without replacement from the full sample of hot
Jupiters with measured $|\lambda|$ values.

The sample $|\lambda|$ values were each allowed to vary within a Gaussian
distribution characterized by the reported mean and uncertainty in $|\lambda|$
for that system. Asymmetric uncertainties were accounted for by first flipping
a coin to determine whether we would sample from the upper or lower half of
the distribution. We then sampled from a Gaussian distribution characterized
by the corresponding uncertainty, and we calculated the distance of the
sampled value from the mean. The final value was set to match that distance
from the mean on the side of the distribution selected in the first step.

We completed our comparative test with 30,000 iterations, then fit a Gaussian
function to the resulting distribution of misaligned hot Jupiter draws. The
result is shown in the left panel of Figure 4.

Figure 4: Significance with which warm Jupiters in single-star systems are
more aligned than analogous hot Jupiters, considering three segmentations.
Each panel shows the result of 30,000 iterations in which we sample $N_{\rm
WJ}$ random $|\lambda|$ values without replacement from the set of hot Jupiter
systems. The resulting distribution of misaligned hot Jupiter draws is shown
in blue, in comparison with the number of misaligned warm Jupiter systems that
have been observed (purple). The number of samples, $N_{\rm WJ}$, is
determined by the number of observed warm Jupiters in each segmentation. Left:
The full population of warm Jupiters ($N_{\rm WJ}=12$) compared with the full
population of hot Jupiters. Center: The full population of warm Jupiters
($N_{\rm WJ}=12$) compared with the population of hot Jupiters orbiting hot
stars. Right: The population of warm and hot Jupiters orbiting cool stars
($N_{\rm WJ}=10$).

This initial test demonstrates that, assuming no systematic differences
between the two populations, warm Jupiters in single-star systems are more
aligned than hot Jupiters in single-star systems to a significance of
$\Delta=3.3\sigma$ (99.96% of HJ trials included at least one misaligned
planet). Therefore, warm Jupiters appear to be more aligned in these systems
based on the current distribution of $|\lambda|$ measurements.

A shortcoming of this global population analysis, however, is that it does not
consider the differential tidal effects in systems with different host star
types. Short-period exoplanets orbiting stars with convective exteriors – that
is, cool stars below the Kraft break – may be susceptible to tidal realignment
through star-planet interactions. This effect should not extend to tidally
detached warm Jupiter systems, since they orbit at a wide enough star-planet
separation to evade strong tidal interactions.

In the tidal realignment framework, it would be more appropriate to compare
the population of warm Jupiters with the population of hot Jupiters orbiting
hot stars. Hot Jupiters orbiting stars above the Kraft break ($T_{\rm eff}\geq
6100$ K) are less susceptible to tidal realignment and therefore may be more
representative of the maximum stellar obliquity reached by each system.
Repeating the same experiment but drawing hot Jupiters only from the hot star
population ($T_{\mathrm{eff}}\geq 6100$ K), we found that warm Jupiters in
single-star systems are more aligned than hot Jupiters around single hot stars
to $\Delta=4.0\sigma$ (100% of HJ trials included at least one misaligned
planet; see the central panel of Figure 4).

A final possibility is that the misalignment production mechanism may be
unrelated to the dynamical evolution of planets within the system. Alternative
mechanisms may act as a function of only the stellar properties: for example,
if misalignments are caused by internal gravity waves within the host star
(Rogers et al., 2012), then the prevalence of misalignments should depend only
on the host star’s internal structure. In this case, misalignments should
occur as a function of stellar type and should only be affected by the
neighboring planet masses and orbital separations insofar as different stellar
types produce different types of planets.

We examined this possibility by directly comparing the populations of hot and
warm Jupiters around cool stars ($T_{\mathrm{eff}}\leq 6100$ K). To remove
potential biases from tidal realignment within the hot Jupiter population, for
this test we excluded systems with the largest ratio between their measured
age and tidal realignment timescale. The tidal realignment timescale
$\tau_{R}$ for cool stars, drawn from Albrecht et al. (2012), can be roughly
estimated as

$\tau_{R}=10\Big{(}\frac{a/R_{*}}{40}\Big{)}^{6}\Big{(}\frac{M_{pl}}{M_{*}}\Big{)}^{-2}\,\,\mathrm{Gyr},$
(7)

where this expression has been scaled to match synchronized binary star
systems with an assumed 10 Gyr main-sequence lifetime.

We note that the tidal dissipation scaling for planets may substantially
differ from that of stars. Furthermore, the ages within our sample have been
heterogeneously measured, with large uncertainties in many cases. As a result,
we conservatively excluded all systems from our sample with an estimated age
greater than 10% the tidal realignment timescale, $\tau_{R}$, of that system.
This cut to the sample leaves 15 of the original 23 hot Jupiters around cool
stars.

The results of this final test are shown in the right panel of Figure 4. Warm
Jupiter systems with cool, single host stars are more aligned than analogous
hot Jupiter systems with a significance of $\Delta=3.1\sigma$ (99.88% of HJ
trials included at least one misaligned planet). The difference between these
two populations appears to be significant irrespective of the host star’s
internal structure: that is, warm Jupiter systems are more aligned than hot
Jupiter systems, even when considering only planets orbiting cool stars. This
suggests that mechanisms that act primarily to misalign hot star systems, such
as internal gravity waves (Rogers et al., 2012), are likely not the sole
driver of the observed misalignments.

However, further measurements are needed to verify the robustness of this
final result. For example, Appendix C of Albrecht et al. (2022) describes
potential caveats to the $|\lambda|$ measurements of the CoRoT-1 and CoRoT-19
systems. Excluding these, the significance of each result in Figure 4 is
adjusted to $3.2\sigma$, $4.0\sigma$, and $2.4\sigma$ in the left, center, and
right panel, respectively. While the first two results remain strong, the
significance of the third result is weakened when these two systems are
excluded.

“Misaligned” hot Jupiter systems around single, cool stars have also typically
been interpreted as being misaligned due to the absence of a large signal,
rather than due to the presence of a large signal clearly indicating
misalignment. This may signify a true, polar misalignment– or, alternatively,
it may instead indicate an underestimated $v\sin i_{*}$ value for the system,
which can result from a measurement made using a spectrograph with
insufficient resolving power (see Section 4.4). We discuss this point and
further uncertainties within our analysis in greater detail in Section 5.4.

Because there are only two warm Jupiter systems with $|\lambda|$ measurements
around hot stars, we are unable to repeat this experiment for hot star
systems. Additional measurements are necessary to conclusively demonstrate
whether the warm Jupiter population around hot stars substantially differs
from its hot Jupiter counterpart.

####  5.2.3 Statistical Significance: Summed Comparison

To further examine the robustness of our result, we also compared the summed
$|\lambda|$ values between the single-star warm Jupiter population and the
single-star hot Jupiter population. The benefit of this test is that it
requires no specific assumptions regarding the cutoff between “aligned” and
“misaligned” planets. Instead, we directly compare the summed $|\lambda|$
measurements to demonstrate population-wide differences.

We used the same three populations for comparison that were described in
Section 5.2.2. However, in the summed tests, we added together the 12 randomly
drawn $|\lambda|$ values from each population rather than assigning a label of
“aligned” or “misaligned.” To account for uncertainties, we iteratively drew
each $|\lambda|$ value from a Gaussian distribution characterized by the
reported central value and its uncertainty (see Section 5.2.2 for our
treatment of asymmetric uncertainties). The resulting distributions are shown
in Figure 5.

Figure 5: Comparison of the summed sky-projected misalignments,
$\sum|\lambda|$, in warm Jupiter (purple) vs. hot Jupiter (blue) systems. The
three segmentations shown are the same as those in Figure 4; however, here we
compare the distribution of summed sky-projected spin-orbit angles
$\sum|\lambda|$, without assigning a label of “aligned” or “misaligned” to
each system. Each distribution includes 30,000 draws, where each draw consists
of $N_{\rm WJ}$ summed $|\lambda|$ measurements ($N_{\rm WJ}$=12 in the left
and middle panels, and $N_{\rm WJ}$=10 in the right panel). Each sampled
$|\lambda|$ value was allowed to vary about a Gaussian distribution
characterized by its measured error bars.

We fit a Gaussian function to each distribution and applied the two-sided
Kolmogorov-Smirnov (K-S) test to demonstrate the likelihood that, in each
panel of Figure 5, the warm and hot Jupiter systems were drawn from the same
underlying distribution (the null hypothesis). The two-sided Kolmogorov-
Smirnov statistic $D_{n,m}$ is given by

$D_{n,m}=\mathrm{sup_{x}}|F_{1,n}(x)-F_{2,m}(x)|,$ (8)

where $F_{1,n}(x)$ and $F_{2,m}(x)$ are the two cumulative distributions with
size $n$ and $m$, respectively, and $D_{n,m}$ is the supremum of the distances
between these two distributions. The null hypothesis can be rejected with a
significance characterized by $\alpha$ in the case that

$D_{n,m}>\sqrt{-\frac{1}{2}\Big{(}\ln{\Big{(}\frac{\alpha}{2}\Big{)}}\Big{)}\Big{(}\frac{n+m}{nm}\Big{)}}.$
(9)

A smaller $\alpha$ value corresponds to a lower likelihood that the null
hypothesis stands. That is, $\alpha$ quantifies the probability that the two
sets of data would differ as much as they do if they were both randomly
sampled from the same underlying distribution.

The cumulative distribution function (CDF) of a Gaussian is given by

$F(x)=\frac{1}{2}\Big{[}1+\mathrm{erf}\Big{(}\frac{x-\mu}{\sigma\sqrt{2}}\Big{)}\Big{]},$
(10)

where $\mu$ and $\sigma$ are the mean and standard deviation, respectively, of
the Gaussian. The CDF of each sample is shown in an inset on each panel, with
the associated $D_{n,m}$ value provided to the right.

We rule out the null hypothesis with a confidence level of $\alpha<10^{-16}$
in all cases, demonstrating that the two $|\lambda|$ distributions are
distinct. In other words, warm Jupiter systems are preferentially more
aligned, with a lower $\sum|\lambda|$, than hot Jupiter systems.

###  5.3 Tidal Evolution of Stellar Obliquities

Next, we directly examined the role of tidal realignments that may have
altered the stellar obliquity distribution of single-star systems over time.
One way to study this effect is by considering the $|\lambda|$ distribution as
a function of the system age. If systems typically begin misaligned and are
realigned by star-planet tidal interactions over time, as in the tidal
realignment scenario, then older systems should, all else equal, be
systematically more aligned than younger systems.

The ages in our sample were, like other parameters described in Section 5.1,
drawn from the NASA Exoplanet Archive and supplemented with values either
directly from the literature or from the Extrasolar Values Encyclopaedia where
possible. As shown in Figure 6, hot Jupiters in single-star systems are
typically aligned at ages $\tau>5$ Gyr, while they are sometimes misaligned at
$\tau\leq 5$ Gyr. By contrast, all warm Jupiters in our sample are aligned
irrespective of their age, including several systems with $\tau\leq 5$ Gyr.

The age distribution of hot Jupiter misalignments, paired with the absence of
this trend in tidally detached warm Jupiter systems, is consistent with the
framework of tidal realignment. However, degeneracies between stellar $T_{\rm
eff}$ and age complicate the interpretation of Figure 6. For example, cool
stars are observed at a wider range of ages than hot stars due to their long
main-sequence lifetimes – reflected by the near-absence of hot stars with
$\tau>5$ Gyr in our sample. Hot stars also cool over time after they finish
burning hydrogen in their cores, such that initially hot stars may be observed
with $T_{\rm eff}<6100$ K at later ages (Triaud, 2011). Therefore, the
apparent trend with age observed among the hot Jupiter population may instead
be more telling of intrinsic differences between hot and cool star systems
(Safsten et al., 2020).

Figure 6: Sky-projected spin-orbit angles $|\lambda|$ as a function of the
system age. Warm Jupiter systems (purple) are aligned irrespective of their
age, consistent with a quiescent formation mechanism. By contrast, hot Jupiter
systems (blue) are more commonly misaligned at young system ages. We emphasize
that most observed misalignments are found in hot star systems, such that the
apparent trend with age may instead indicate a trend with $T_{\rm eff}$.

To account for the heterogeneous stellar types within our sample, we next
considered metrics for the tidal realignment timescale, characterized by the
dimensionless tidal parameter $\xi=(M_{pl}/M_{*})^{-2}(a/R_{*})^{6}$. As shown
in Figure 7, hot Jupiters – particularly those orbiting cool stars – can have
relatively short tidal realignment timescales, whereas warm Jupiters and
Saturn-mass planets have comparatively longer tidal realignment timescales.

Figure 7: Measured $|\lambda|$ values as a function of the dimensionless tidal
parameter $\xi=(M_{pl}/M_{*})^{-2}(a/R_{*})^{6}$, which is used as a proxy for
the tidal realignment timescale. For clarity, we divide the full sample into
hot star (triangles; $T_{\mathrm{eff}}\geq 6100$ K) and cool star (circles;
$T_{\mathrm{eff}}<6100$ K) systems; the same color scheme is applied to both
panels. Warm Jupiters have comparatively long tidal realignment timescales,
suggesting a primordial origin of their aligned orbits.

In spite of their long tidal realignment timescales, warm Jupiters in single-
star systems have not been found with large misalignments
($|\lambda|>20\,^{\circ}$). Importantly, the scale factor for the planetary
realignment timescale is not well-constrained in the equilibrium tides
framework considered here. However, the young ages of many warm Jupiter
systems (six with estimated ages $<2$ Gyr; see Figure 6), paired with the
prevalence of aligned warm Jupiters with relatively large $\xi$ – despite the
observation of some misaligned hot Jupiters with smaller $\xi$ values –
suggests that these systems were likely not all realigned over their
lifetimes. Several systems containing Saturn-mass planets have been found with
substantial misalignments, suggesting that these planets, which have some of
the longest tidal realignment timescales in our sample, may be more
susceptible to misalignment via dynamical interactions with neighboring
planets.

###  5.4 Caveats and Uncertainties

We emphasize that our sample is heterogeneous: the measurements within our
population study are drawn from many individual studies. Certain measurements
would be particularly helpful to verify with additional observations: for
example, our results are largely driven by the small sample of misaligned
systems that have been observed, but the Rossiter-McLaughlin measurements of
two misaligned hot Jupiter measurements in cool star systems include little to
no egress data (e.g. CoRoT-19 b (Guenther et al., 2012) and HATS-14 b (Zhou et
al., 2015)), providing weaker constraints on the obliquity of each system.
Furthermore, underestimated $v\sin i_{*}$ values can mimic a low-amplitude
Rossiter-McLaughlin signal, which may be incorrectly interpreted as a polar
orbit. Additional tests to verify and more precisely constrain the obliquities
of misaligned systems will be crucial to confirm or to challenge the results
presented in this work.

Our cutoff between “hot” and “warm” Jupiters at $a/R_{*}>11$ was selected
empirically and is not necessarily the “true” cutoff between the two
populations. A shifted cutoff of $a/R_{*}>10$, which expands the sample of
warm Jupiters by four systems (two with $|\lambda|>20\,^{\circ}$), still
reveals a tentative trend toward warm Jupiters being more aligned than hot
Jupiters, with an adjusted significance of $2.7\sigma$, $3.7\sigma$, and
$3.7\sigma$ for the left, center, and right panel of Figure 4, respectively.
Different sets of stellar parameters may also shift individual systems
slightly in $a/R_{*}$ space; for example, KELT-6 b is listed with
$a/R_{*}=11.17$ in Albrecht et al. (2012).

A transition range may exist between the hot and warm Jupiter populations,
with a rate of misalignments that is reduced relative to that of the close-in
hot Jupiters but that is higher than that of the tidally detached warm
Jupiters. Additional $|\lambda|$ measurements for planets near $a/R_{*}\sim
11$ will help to better sculpt the landscape of this parameter space and to
potentially constrain the tidal realignment timescale for planets.

##  6 Discussion

###  6.1 Implications for Hot and Warm Jupiter Evolution

We have demonstrated that, in single-star systems, the tidally detached warm
Jupiter population is substantially more aligned than the hot Jupiter
population, despite hot Jupiters’ bias toward potentially realigned systems.
This result has two key implications: (1) planets in single-star systems
typically form within aligned protoplanetary disks, and (2) warm Jupiters form
quiescently, whereas hot Jupiters are likely misaligned after the
protoplanetary disk has dispersed.

Quiescently formed warm Jupiters should remain roughly within the plane of
their natal protoplanetary disk. As a result, if misalignments typically
originate at the protoplanetary disk stage, then quiescently formed warm
Jupiters should be observed with a similarly high rate of misalignments to
that of the hot Jupiter population. In this scenario, observational biases
should favor a higher rate of misaligned warm Jupiters, since these tidally
detached systems are unable to realign over timescales within the age of the
Universe.

By contrast, the population of tidally detached warm Jupiters is significantly
more aligned than the population of close-orbiting hot Jupiters. We conclude
that warm Jupiters in single-star systems typically form within disks that are
aligned with the host star’s equator, whereas misalignments in hot Jupiter
systems typically arise after the disk dispersal phase. This framework is
consistent with the high rate of observed planetary companions in quiescent
warm Jupiter systems (Huang et al., 2016), by contrast with a much lower
companion rate in hot Jupiter systems (Steffen et al., 2012) that may have
lost their companions during the misalignment process.

This framework is also supported by previous observations from the Atacama
Large Millimeter Array (ALMA) that have suggested that most, if not all,
protoplanetary disks are aligned with the spin axis of their host stars
(Davies, 2019). Furthermore, no misaligned planets have been found in very
young systems with $\tau<100$ Myr (Albrecht et al., 2022). While the sample of
planets in this parameter space is small (only 6 to date), the absence of
misaligned planets in young systems is consistent with our framework in which
misalignments originate later in the systems’ evolution.

Previous results have demonstrated that hot Jupiters’ observed misalignments
are fully consistent with a dominant formation mechanism that is dynamically
violent (Rice et al., 2022). Many hot Jupiters that are currently observed in
aligned configurations may have been misaligned earlier in their evolution and
subsequently realigned over time through star-planet tidal interactions. The
closest-orbiting hot Jupiters around the coolest stars are typically all
aligned, suggesting that tidal interactions have played an important role in
the orbital evolution of these systems (Wang et al., 2021).

Some promising hot Jupiter misalignment mechanisms, such as the Kozai-Lidov
mechanism in stellar binaries (Naoz et al., 2012), require the presence of a
stellar companion. However, the high rate of hot Jupiter misalignments, even
in single-star systems, indicates that hot Jupiters cannot be misaligned
exclusively through interactions with a stellar companion.

Instead, planet-planet interactions likely play an important role in the
production of hot Jupiters’ misalignments. Alternative mechanisms invoking one
or more planetary companions include secular chaos in multiplanet systems with
overlapping resonances (Wu & Lithwick, 2011; Teyssandier et al., 2019) and
planet-planet scattering (Rasio & Ford, 1996), particularly when combined with
the Kozai-Lidov mechanism (Naoz et al., 2011; Nagasawa & Ida, 2011).

###  6.2 The Evolution of Extrasolar Saturns

While examining the hot and warm Jupiter populations, we also found that
Saturn-mass planets, including those at a wide orbital separation
$a/R_{*}>11$, span a wide range of spin-orbit angles. Three of the four
misaligned Saturns in our sample (TOI-1268 b,666Dong et al. (2022) includes 3
separate $|\lambda|$ measurements for TOI-1268 b. While we adopt the
$|\lambda|=40^{+7.2}_{-9.9}\,{}^{\circ}$ value included in the TEPcat
catalogue, we note that the other two provided measurements each indicate a
lower sky-projected obliquity ($|\lambda|=14^{+14}_{-10}\,{}^{\circ}$ and
$|\lambda|=25\pm 13\,^{\circ}$, from two separate reductions of the same NEID
dataset). As a result, this system may not be strongly misaligned. Kepler-63
b, and WASP-117 b) have significant projected eccentricities ($e>0.1$),
suggesting that their eccentricities and stellar obliquities may have been
jointly elevated. Three processes that can produce eccentric, misaligned
Saturns include the adiabatic disk-dispersal resonance proposed by Petrovich
et al. (2020), Kozai-Lidov interactions (Naoz et al., 2011), and secular chaos
(Wu & Lithwick, 2011).

####  6.2.1 Disk-Dispersal Resonance

The Petrovich et al. (2020) disk-dispersal resonance was originally proposed
as a mechanism to naturally produce polar Neptunes—a growing population of
observed systems (Stefansson et al., 2021; Albrecht et al., 2022). This
resonance occurs when the nodal precession rate of an outer planetary
companion, induced by interactions with the protoplanetary disk, reaches
commensurability with the nodal precession rate of the inner planet, induced
by general relativistic effects and the stellar quadrupole field.

The same mechanism can also extend to higher-mass planets, with the caveat
that a larger initial mutual inclination is required between the inner planet
and its outer companion to ensure conservation of the angular momentum deficit
$\Theta\prime$:

$\Theta\prime=L_{\rm in}(1-\cos i_{\rm in})+L_{\rm out}(1-\cos i_{\rm out}).$
(11)

Here, $L_{\rm in}$ and $L_{\rm out}$ are the orbital angular momenta of the
inner and outer planet, while $i_{\rm in}$ and $i_{\rm out}$ are the planets’
inclinations.

The mutual inclination $i_{\rm mut}$ required to conserve $\Theta\prime$ is
given by

$i_{\rm mut}\geq\Big{(}\frac{m_{\rm in}}{m_{\rm
out}}\Big{)}^{1/2}\Big{(}\frac{a_{\rm in}}{a_{\rm out}}\Big{)}^{1/4}.$ (12)

Here, $m_{\rm in}$ and $m_{\rm out}$ are the masses of the inner and outer
planet, respectively, while $a_{\rm in}$ and $a_{\rm out}$ are the
corresponding semimajor axes. For a Neptune-mass planet at $a_{\rm in}=0.1$ au
with a $4M_{J}$ companion at $a_{\rm out}=5$ au, the required mutual
inclination is only $i_{\rm mut}\geq 2.5\,^{\circ}$. By contrast, $i_{\rm
mut}\geq 5.9\,^{\circ}$ if the inner planet is set to be Saturn-mass.

Therefore, the population of misaligned Saturns may represent the upper mass
limit of inner planets to which this mechanism applies. This would also imply
a limit to the typical $i_{\rm mut}$ values attained during disk dispersal,
with a larger $i_{\rm mut}$ limit required for lower-mass outer companions.

####  6.2.2 Kozai-Lidov

The Saturns in our sample may instead be misaligned by the Kozai-Lidov
mechanism. In this framework, a third perturber with a significant initial
mutual inclination ($i_{\rm mut}>39.2\,^{\circ}$) is required to initialize a
secular exchange of eccentricity and inclination between hierarchical orbits
(Naoz, 2016). In single-star systems, this additional perturber could be a
companion planet within the system. However, this raises the question of why
Saturn-mass planets would be misaligned, while warm Jupiters would not.

If Saturns are misaligned through Kozai-Lidov oscillations, this would imply
that Saturns more commonly appear in systems that meet the mechanism’s
required initial conditions. Planet-planet scattering simulations have offered
evidence that, in systems with equal-mass planets, lower-mass planets
preferentially reach higher mutual inclinations from planet-planet scattering.
For example, Raymond et al. (2010) found that 10% of unstable planets in their
disk-free simulations reached inclinations larger than 39$\,{}^{\circ}$ in
J-J-J systems (3 Jupiters), 42$\,{}^{\circ}$ in S-S-S systems (3 Saturns), and
49$\,{}^{\circ}$ in N-N-N systems (3 Neptunes). This increase in mutual
inclinations with decreasing planet mass reflects the ease with which higher-
mass planets can eject each other from the system, reducing the total number
of close encounters over the systems’ evolution. Therefore, the misaligned
Saturns within our sample may represent a mass cutoff below which scattering
processes can produce mutual inclinations $i_{\rm mut}>39.2\,^{\circ}$.

So far, no planetary companions have been found in any of the single-star
Saturn systems that show evidence for misalignment. However, long-period
Saturn-mass companions also cannot be ruled out based on current radial
velocity surveys (e.g. Figure 1 in Fulton et al. (2021)). If the relatively
high occurrence rate of misaligned Saturn-mass planets emerges due to planet-
planet scattering combined with the Kozai-Lidov mechanism, we anticipate that
outer planetary companions should be prevalent in misaligned Saturn systems.

####  6.2.3 Secular Chaos

Secular chaos may also play a role in misaligning the Saturns observed in our
sample. Secular chaos requires the presence of at least $2-3$ planets in an
individual system, and it is produced by overlapping, nonlinear secular
resonances in which the precession rates of different planets reach
commensurability (Lithwick & Wu, 2011). Previous work has demonstrated that
secular chaos can produce substantial spin-orbit misalignments in multiplanet
systems (Wu & Lithwick, 2011).

If Saturns form with neighboring companion planets, they may be misaligned
through secular chaos in certain cases. However, spin-orbit angles larger than
90$\,{}^{\circ}$ are difficult to generate from secular chaos (Wu & Lithwick,
2011; Teyssandier et al., 2019), such that highly misaligned systems (for
example, Kepler-63, with $\lambda=-110^{+22}_{-14}\,{}^{\circ}$ (Sanchis-Ojeda
et al., 2013)) likely could not have been misaligned by this mechanism.
Furthermore, the secular chaos framework does not explain the heightened rate
of spin-orbit misalignments for Saturn-mass planets as compared with warm
Jupiters.

Because secular chaos takes place over timescales similar to or longer than
the secular precession timescale, this framework would imply that misaligned
Saturns should typically be observed at older stellar ages. Misaligned hot
Jupiters have recently been found to be more common at older stellar ages
(Hamer & Schlaufman, 2022), providing evidence that secular chaos may be
prevalent in at least some misaligned giant planet systems.

##  7 Conclusions

In this work, we have measured the Rossiter-McLaughlin effect across the
transit of tidally detached warm Jupiter TOI-1478 b using the Keck/HIRES and
WIYN/NEID instruments. This is the third measurement made as part of the SOLES
survey, which is designed to constrain planetary evolution models by studying
stellar obliquities in long-period exoplanet systems.

We combined this measurement with archival stellar obliquity constraints in
single-star systems to conduct a population study demonstrating the
statistical significance with which warm Jupiters are more aligned than
analogous hot Jupiters. Our findings are summarized as follows:

  * •

The TOI-1478 system is consistent with alignment, with sky-projected spin-
orbit angle $\lambda=6.2_{-5.5}^{+5.9}\,{}^{\circ}$.

  * •

The previous $v\sin i_{*}$ value for TOI-1478 may have been overestimated, as
indicated by the low Rossiter-McLaughlin amplitude of the system.

  * •

The current set of stellar obliquity constraints indicates that warm Jupiters
in single-star systems are preferentially more aligned than analogous hot
Jupiters (at $3.3\sigma$ when considering the full population, and at
$4.0\sigma$ when comparing to hot stars that presumably have not experienced
tidal realignment).

  * •

Even when considering only systems around cool stars, warm Jupiters still
appear to be more misaligned than analogous hot Jupiters (at $3.1\sigma$).
However, further measurements are needed to better constrain the prevalence of
misaligned hot Jupiters in cool, single-star systems and to verify the
robustness of this result.

  * •

Saturn-mass planets span a wide range of sky-projected spin-orbit angles, even
at wide orbital separation.

Based on these findings, we conclude that (1) planets in single-star systems
tend to form within aligned protoplanetary disks and (2) warm Jupiters tend to
form quiescently, while hot Jupiters are misaligned after protoplanetary disk
dispersal. We have also proposed several potential explanations for the
preferential misalignment of Saturn-mass planets, including a secular disk-
dispersal resonance, the Kozai-Lidov mechanism, or secular chaos.

Additional stellar obliquity measurements for warm Jupiters in binary star
systems are needed to evaluate whether the observed trends extend beyond
single-star systems. Further warm Jupiter spin-orbit angle measurements will
also help to elucidate any potential correlations between eccentricities and
stellar obliquities, which may provide improved insights into the key
mechanisms contributing to extrasolar system misalignments.

##  8 Acknowledgements

We thank our anonymous reviewer for their comments that have strengthened this
manuscript. Over the duration of this project, M.R. was supported by the
Heising-Simons Foundation 51 Pegasi b Fellowship and by the National Science
Foundation Graduate Research Fellowship Program under Grant Number
DGE-1752134. The data presented herein were obtained at the W. M. Keck
Observatory, which is operated as a scientific partnership among the
California Institute of Technology, the University of California and the
National Aeronautics and Space Administration. The Observatory was made
possible by the generous financial support of the W. M. Keck Foundation. This
work is supported by the Astronomical Big Data Joint Research Center, co-
founded by National Astronomical Observatories, Chinese Academy of Sciences
and Alibaba Cloud. This research has made use of the Keck Observatory Archive
(KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet
Science Institute (NExScI), under contract with the National Aeronautics and
Space Administration. 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. This paper contains data taken with the NEID
instrument, which was funded by the NASA-NSF Exoplanet Observational Research
(NN-EXPLORE) partnership and built by Pennsylvania State University. NEID is
installed on the WIYN 3.5m telescope at KPNO, which is managed by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the NSF. The NEID archive is operated by the NASA
Exoplanet Science Institute at the California Institute of Technology. We
thank the NEID Queue Observers and WIYN Observing Associates for their
skillful execution of our NEID observations. The authors are honored to be
permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a
mountain with particular significance to the Tohono O’odham.

Keck: I (HIRES), WIYN (NEID), Exoplanet Archive, Extrasolar Planets
Encyclopaedia

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Table 4: Parameters for the population of warm Jupiters, hot Jupiters, and Saturns studied in this work. Systems are ordered by $T_{\mathrm{eff},*}$. Uncertainties in all parameters except $\lambda$ were used only for display purposes. Where uncertainties were not available, we used the central value for the system. Upper limits are provided from the literature where possible.  | System | Planet |  $M_{\rm pl}$ ($M_{J}$) | $a/R_{*}$ | $e$ | Age (Gyr) |  $T_{\mathrm{eff},*}$ (K) | $\lambda\,(\arcdeg)$
---|---|---|---|---|---|---|---|---
Warm Jupiters: |  |  |  |  |  |  |  |
| WASP-80 | b | $0.538^{+0.035}_{-0.036}$ | $12.54^{+0.56}_{-0.58}$ | $0.002^{+0.010}_{-0.002}$ | $0.1^{+0.03}_{-0.02}$ | $4145\pm 100$ | $-14\pm 14$
| WASP-53 | b | $2.132^{+0.092}_{-0.094}$ | $11.02^{+0.35}_{-0.37}$ | $<0.03$ | - | $4950\pm 60$ | $-1\pm 12$
| WASP-84 | b | $0.694\pm 0.028$ | $22.11\pm 0.45$ | $0$ | $1$ | $5280\pm 80$ | $-0.3\pm 1.7$
| HAT-P-17 | b | $0.58\pm 0.06$ | $22.89\pm 1.56$ | $0.35\pm 0.01$ | 7.8 | $5322\pm 55$ | $19^{+14}_{-16}$
| TOI-1478 | b | $0.851^{+0.052}_{-0.047}$ | $18.54_{-0.6}^{+0.7}$ | $0.024^{+0.032}_{-0.017}$ | $9.1^{+3.1}_{-3.9}$ | $5597^{+83}_{-82}$ | $6.2_{-5.5}^{+5.9}$
| HD 63433 | b | $<5$ | $16.99^{+0.92}_{-1.18}$ | $0$ | $0.414\pm 0.023$ | $5640\pm 74$ | $8^{+33}_{-45}$
| K2-140 | b | $0.93\pm 0.04$ | $13.79^{+0.93}_{-0.81}$ | $0$ | $9.8^{+3.4}_{-4.6}$ | $5654\pm 55$ | $0.5\pm 9.7$
| HIP 67522 | b | $<5$ | $11.88\pm 0.54$ | $0.059^{+0.193}_{-0.046}$ | $0.017\pm 0.002$ | $5675\pm 75$ | $5.1^{+2.5}_{-3.7}$
| WASP-25 | b | $0.44\pm 0.10$ | $11.14\pm 1.65$ | $0$ | $0.1^{+5.7}_{-0.1}$ | $5736\pm 35$ | $14.6\pm 6.7$
| HD 17156 | b | $3.51\pm 0.21$ | $23.34\pm 0.90$ | $0.68$ | $3.38\pm 0.47$ | $6079\pm 56$ | $10\pm 5.1$
| WASP-38 | b | $3.44\pm 0.36$ | $12.09\pm 0.85$ | $0.03$ | $>5.0$ | $6436\pm 60$ | $7.5^{+4.7}_{-6.1}$
| KOI-12 | b | $1.1^{+3.5}_{-0.8}$ | $23.65^{+1.33}_{-1.11}$ | $0.34^{0.08}_{-0.07}$ | $1.5\pm 0.5$ | $6820\pm 120$ | $-7.1^{+4.2}_{-2.8}$
Hot Jupiters: |  |  |  |  |  |  |  |
| WASP-43 | b | $1.78\pm 0.1$ | $4.58\pm 0.15$ | $0$ | $0.4$ | $4520\pm 120$ | $3.5\pm 6.8$
| Qatar-2 | b | $2.494\pm 0.054$ | $5.935\pm 0.094$ | $0$ | $5.0$ | $4645\pm 50$ | $15\pm 20$
| Qatar-1 | b | $1.294^{+0.052}_{-0.049}$ | $6.27\pm 0.19$ | $0$ | $1.7^{+2.8}_{-1.1}$ | $4910\pm 100$ | $-8.4\pm 7.1$
| TOI-942 | b | $<2.6$ | $10.11^{+0.25}_{-0.24}$ | $0$ | $0.09\pm 0.07$ | $4928^{+125}_{-85}$ | $1^{+41}_{-33}$
| WASP-52 | b | $0.46\pm 0.02$ | $7.40\pm 0.20$ | $0$ | $0.4^{+0.3}_{-0.2}$ | $5000\pm 100$ | $1.1\pm 1.1$
| HATS-2 | b | $1.35\pm 0.15$ | $5.50\pm 0.14$ | $0$ | $9.7\pm 2.9$ | $5227\pm 95$ | $8\pm 8$
| HATS-14 | b | $1.07\pm 0.07$ | $8.82^{+0.20}_{-0.12}$ | $<0.142$ | $4.9\pm 1.7$ | $5346\pm 60$ | $76^{+4}_{-5}$
| CoRoT-18 | b | $3.47\pm 0.38$ | $6.34\pm 0.89$ | $<0.08$ | $0.10^{+0.80}_{-0.04}$ | $5440\pm 100$ | $-10\pm 20$
| WASP-19 | b | $1.154^{+0.078}_{-0.080}$ | $3.52^{+0.15}_{-0.16}$ | $0.0126^{+0.0140}_{-0.0089}$ | $6.4^{+4.1}_{-3.5}$ | $5460\pm 90$ | $-1.9\pm 1.1$
| WASP-4 | b | $1.186^{+0.090}_{-0.098}$ | $5.46^{+0.25}_{-0.27}$ | $0$ | $7.0$ | $5540\pm 55$ | $-1.0^{+14}_{-12}$
| WASP-41 | b | $0.85\pm 0.11$ | $9.95\pm 0.78$ | $0$ | $1.80\pm 0.27$ | $5546\pm 33$ | $29^{+10}_{-14}$
| WASP-47 | b | $1.142\pm 0.023$ | $9.68\pm 0.13$ | $0.0028\pm 0.0028$ | $6.7^{+1.5}_{-1.1}$ | $5576\pm 67$ | $0\pm 24$
| HAT-P-36 | b | $1.848\pm 0.087$ | $4.98\pm 0.11$ | $0.063\pm 0.032$ | $6.6\pm 1.8$ | $5620\pm 40$ | $-14\pm 18$
| WASP-16 | b | $1.24\pm 0.25$ | $9.55\pm 1.23$ | $0$ | $2.3\pm 2.2$ | $5630\pm 70$ | $-4.2^{+11.0}_{-13.9}$
| HAT-P-13 | b | $0.851\pm 0.038$ | $5.872$ | $0.0133\pm 0.0041$ | $5.0\pm 0.8$ | $5653\pm 90$ | $1.9\pm 8.6$
| WASP-5 | b | $1.58^{+0.13}_{-0.10}$ | $5.57\pm 0.41$ | $0.038^{+0.026}_{-0.018}$ | $5.4^{+4.4}_{-4.3}$ | $5770\pm 65$ | $12.1^{+8.0}_{-10.0}$
| Kepler-17 | b | $2.45\pm 0.11$ | $5.31\pm 0.17$ | $<0.011$ | $<1.78$ | $5781\pm 85$ | $0\pm 15$
| HAT-P-23 | b | $1.34\pm 0.59$ | $4.16\pm 1.26$ | $0.11\pm 0.04$ | $4\pm 1$ | $5885\pm 72$ | $15\pm 22$
| CoRoT-1 | b | $1.03\pm 0.12$ | $4.91\pm 0.34$ | $0$ | $0.5$ | $5950\pm 150$ | $77\pm 11$
| WASP-74 | b | $0.72\pm 0.12$ | $4.89\pm 0.53$ | $0$ | $2.0^{+1.0}_{-1.6}$ | $5984\pm 57$ | $0.77\pm 0.99$
| WASP-28 | b | $0.907\pm 0.043$ | $8.82\pm 0.29$ | $0$ | $5^{+3}_{-2}$ | $6084\pm 45$ | $6\pm 17$
| CoRoT-19 | b | $1.11\pm 0.06$ | $6.75\pm 0.19$ | $0.047\pm 0.045$ | $5\pm 1$ | $6090\pm 70$ | $-52^{+27}_{-22}$
| WASP-32 | b | $2.63\pm 0.82$ | $7.74\pm 1.73$ | $0$ | $2.42^{+0.53}_{-0.56}$ | $6100\pm 100$ | $-2^{+17}_{-19}$
| WASP-60 | b | $0.55\pm 0.19$ | $10.09\pm 2.16$ | $0$ | $3.6^{+2.1}_{-4.3}$ | $6105\pm 50$ | $-129\pm 17$
| WASP-103 | b | $1.490\pm 0.088$ | $2.96^{+0.11}_{-0.07}$ | $0$ | $4\pm 1$ | $6110\pm 160$ | $3\pm 33$
| HD 209458 | b | $0.73\pm 0.04$ | $8.78\pm 0.26$ | $0$ | $4\pm 2$ | $6117\pm 50$ | $1.58\pm 0.08$
| K2-34 | b | $1.78\pm 0.13$ | $6.69^{+0.41}_{-0.38}$ | $0$ | $4.24^{+0.39}_{-0.44}$ | $6131\pm 47$ | $-1^{+10}_{-9}$
| WASP-22 | b | $0.67\pm 0.19$ | $8.93\pm 1.59$ | $0.02\pm 0.01$ | $1.3^{+0.6}_{-1.3}$ | $6153\pm 46$ | $22\pm 16$
| WASP-71 | b | $1.39\pm 0.33$ | $4.30\pm 0.68$ | $0$ | $3.6^{+1.6}_{-1.0}$ | $6180\pm 52$ | $-1.9^{+7.1}_{-7.5}$
| Kepler-8 | b | $0.59^{+0.13}_{-0.12}$ | $6.84^{+0.35}_{-0.41}$ | $0$ | $3.84\pm 1.5$ | $6213\pm 150$ | $5\pm 7$
| WASP-62 | b | $0.52\pm 0.08$ | $9.53\pm 0.95$ | $0$ | $0.8\pm 0.6$ | $6230\pm 80$ | $19.4^{+5.1}_{-4.9}$
| KELT-6 | b | $0.52\pm 0.12$ | $10.80\pm 1.39$ | $0.22\pm 0.11$ | $4.90^{+0.66}_{-0.46}$ | $6246\pm 88$ | $-36\pm 11$
| WASP-61 | b | $2.68\pm 0.84$ | $8.15\pm 1.79$ | $0$ | $2.7^{+0.1}_{-0.6}$ | $6250\pm 150$ | $4.0^{+17.1}_{-18.4}$
| HAT-P-2 | b | $8.62\pm 0.17$ | $10.54\pm 0.69$ | $0.5172\pm 0.0019$ | $1.44\pm 0.47$ | $6290\pm 60$ | $9\pm 10$
| CoRoT-11 | b | $2.33\pm 0.34$ | $6.84\pm 0.80$ | $0$ | $2\pm 1$ | $6343\pm 72$ | $0.1\pm 2.6$
| HAT-P-9 | b | $0.749^{+0.064}_{-0.063}$ | $8.48^{+0.46}_{-0.40}$ | $0.084^{+0.052}_{-0.047}$ | $1.6\pm 1.4$ | $6350\pm 150$ | $-16\pm 8$
| HATS-3 | b | $1.071\pm 0.136$ | $7.45^{+0.17}_{-0.18}$ | $0$ | $3.2^{+0.6}_{-0.4}$ | $6351\pm 76$ | $3\pm 25$
| XO-4 | b | $1.42\pm 0.19$ | $7.71\pm 0.70$ | $0$ | $2.1\pm 0.6$ | $6397\pm 70$ | $-46.7^{+8.1}_{-6.1}$
| WASP-190 | b | $1.0\pm 0.1$ | $8.91\pm 0.57$ | $0$ | $2.8\pm 0.4$ | $6400\pm 100$ | $21\pm 6$
| XO-3 | b | $7.29\pm 1.19$ | $4.94\pm 0.53$ | $0.29$ | $2.82\pm 0.82$ | $6429\pm 75$ | $37.3\pm 3.0$
| HAT-P-34 | b | $3.33^{+0.21}_{-0.20}$ | $9.52^{+0.88}_{-0.63}$ | $0.432^{+0.029}_{-0.027}$ | $1.7^{+0.4}_{-0.5}$ | $6442\pm 88$ | $0\pm 14$
| WASP-87 | b | $2.18\pm 0.15$ | $3.89\pm 0.17$ | $0$ | $3.8\pm 0.8$ | $6450\pm 120$ | $-8\pm 11$
| NGTS-2 | b | $0.74^{+0.13}_{-0.12}$ | $7.97^{+0.38}_{-0.42}$ | $0$ | $2.17\pm 0.37$ | $6450\pm 50$ | $-11.3\pm 4.8$
| WASP-7 | b | $0.96\pm 0.13$ | $9.28\pm 0.61$ | $0$ | $2.4^{+0.9}_{-1.1}$ | $6520\pm 70$ | $86\pm 6$
| TOI-2109 | b | $5.02\pm 0.75$ | $2.27\pm 0.11$ | $0$ | $1.77^{+0.88}_{-0.68}$ | $6530^{+160}_{-150}$ | $1.7\pm 1.7$
| WASP-17 | b | $0.78\pm 0.23$ | $8.95\pm 1.73$ | $0$ | $3.0\pm 2.6$ | $6550\pm 100$ | $-148.7^{+7.7}_{-6.7}$
| HAT-P-56 | b | $2.31\pm 0.44$ | $6.38\pm 0.69$ | $<0.246$ | $2.01\pm 0.35$ | $6566\pm 50$ | $8\pm 2$
| HAT-P-6 | b | $1.32\pm 0.3$ | $7.76\pm 1.19$ | $0$ | $2.3\pm 0.7$ | $6570\pm 80$ | $165\pm 6$
| WASP-15 | b | $0.54\pm 0.29$ | $7.26\pm 2.67$ | $0$ | $3.9\pm 1.3$ | $6573\pm 70$ | $-139.6^{+4.3}_{-5.2}$
| WASP-121 | b | $1.16\pm 0.07$ | $3.82^{+0.10}_{-0.12}$ | $0$ | $1.5\pm 1.0$ | $6586\pm 59$ | $87.08^{+0.29}_{-0.27}$
| WASP-79 | b | $0.85\pm 0.18$ | $7.04\pm 0.94$ | $0$ | $1.4\pm 0.3$ | $6600\pm 100$ | $-99.1^{+4.1}_{-3.9}$
| WASP-66 | b | $2.35\pm 0.98$ | $6.72\pm 1.93$ | $0$ | $3.3^{+10.0}_{-2.7}$ | $6600\pm 150$ | $-4\pm 22$
| XO-6 | b | $<4.4$ | $9.08\pm 1.21$ | $0$ | $1.88^{+0.90}_{-0.20}$ | $6720\pm 100$ | $-20.7\pm 2.3$
| KELT-7 | b | $1.39\pm 0.22$ | $5.49\pm 0.34$ | $0$ | $1.3\pm 0.2$ | $6789^{+50}_{-49}$ | $2.7\pm 0.6$
| WASP-100 | b | $1.26\pm 0.45$ | $4.94\pm 1.03$ | $0$ | - | $6940\pm 120$ | $79^{+19}_{-10}$
| TOI-1518 | b | $<2.3$ | $4.29\pm 0.16$ | $<0.01$ | - | $7300\pm 100$ | $-119.66^{+0.93}_{-0.98}$
| KELT-17 | b | $1.31^{+0.28}_{-0.29}$ | $6.36^{+0.25}_{-0.24}$ | - | $0.65\pm 0.15$ | $7454\pm 49$ | $-115.9\pm 4.1$
| MASCARA-1 | b | $3.7\pm 0.9$ | $4.40\pm 0.66$ | $0$ | $1.0\pm 0.2$ | $7554\pm 150$ | $69.2^{+3.1}_{-3.4}$
| KELT-21 | b | $<3.91$ | $6.85\pm 0.13$ | $0$ | $1.6\pm 0.1$ | $7598^{+81}_{-84}$ | $-5.6^{+1.7}_{-1.9}$
| TOI-1431 | b | $3.12\pm 0.18$ | $5.15\pm 0.29$ | $0.0022^{+0.0030}_{-0.0016}$ | $0.29^{+0.32}_{-0.19}$ | $7690^{+400}_{-250}$ | $-155^{+20}_{-10}$
| HAT-P-69 | b | $3.58\pm 0.58$ | $7.30^{+0.24}_{-0.12}$ | $0$ | $1.27^{+0.28}_{-0.44}$ | $7724^{+250}_{-360}$ | $30.3^{+6.1}_{-7.3}$
| HATS-70 | b | $12.9^{+1.8}_{-1.6}$ | $4.15^{+0.16}_{-0.18}$ | $<0.18$ | $0.81^{+0.50}_{-0.33}$ | $7930^{+630}_{-820}$ | $8.9^{+5.6}_{-4.5}$
| HAT-P-70 | b | $<6.78$ | $5.48^{+0.36}_{-0.29}$ | $0$ | $0.60^{+0.38}_{-0.20}$ | $8450^{+540}_{-690}$ | $107.9^{+2.0}_{-1.7}$
| KELT-20 | b | $<3.382$ | $7.47^{+0.35}_{-0.41}$ | - | $0.6$ | $8730^{+250}_{-260}$ | $1.6\pm 3.1$
| KELT-9 | b | $2.88\pm 0.84$ | $3.15^{+0.14}_{-0.12}$ | $0$ | $0.3$ | $9600\pm 400$ | $-85.01\pm 0.23$
Saturns: |  |  |  |  |  |  |  |
| HAT-P-12 | b | $0.211\pm 0.012$ | $11.80^{+0.35}_{-0.19}$ | 0 | $2.5\pm 2.0$ | $4665\pm 45$ | $-54^{+41}_{-13}$
| WASP-69 | b | $0.29\pm 0.03$ | $11.98\pm 0.71$ | 0 | $2.0$ | $4700\pm 50$ | $0.4^{+2.0}_{-1.9}$
| TOI-1268 | b | $0.29$ | $17.20^{+0.58}_{-0.60}$ | $0.12^{+0.24}_{-0.12}$ | $0.39^{+0.28}_{-0.27}$ | $5257\pm 40$ | $40^{+7.2}_{-9.9}$
| WASP-6 | b | $0.37\pm 0.08$ | $10.60\pm 1.49$ | $0.05\pm 0.02$ | $11\pm 7$ | $5375\pm 65$ | $7.2\pm 3.7$
| WASP-148 | b | $0.291\pm 0.025$ | $17.64\pm 3.46$ | $0.202\pm 0.063$ | - | $5437\pm 21$ | $-8.2^{+8.7}_{-9.7}$
| WASP-39 | b | $0.28\pm 0.03$ | $11.24\pm 0.38$ | 0 | $8.5^{+3.5}_{-1.0}$ | $5485\pm 50$ | $0\pm 11$
| Kepler-63 | b | $0.378$ | $19.11^{+0.80}_{-0.64}$ | 0.45 | $0.210\pm 0.045$ | $5576\pm 50$ | $-110^{+22}_{-14}$
| WASP-21 | b | $0.300\pm 0.011$ | $10.55\pm 0.41$ | 0 | $12\pm 2$ | $5924\pm 55$ | $8^{+26}_{-27}$
| WASP-13 | b | $0.36\pm 0.09$ | $7.34\pm 1.17$ | 0 | $7.4\pm 0.4$ | $6025\pm 21$ | $8^{+13}_{-12}$
| WASP-117 | b | $0.30\pm 0.05$ | $17.46\pm 1.60$ | $0.30\pm 0.02$ | $4.6\pm 2.0$ | $6040\pm 90$ | $-46.9^{+5.5}_{-4.8}$
| HD 332231 | b | $0.244\pm 0.021$ | $24.12^{+0.93}_{-0.94}$ | $0.032^{+0.030}_{-0.022}$ | $4.3^{+2.5}_{-1.9}$ | $6089^{+97}_{-96}$ | $-1\pm 7$
| HD 149026 | b | $0.38\pm 0.06$ | $6.79\pm 0.55$ | 0 | $2.0\pm 0.8$ | $6147\pm 50$ | $12\pm 7$
| K2-232 | b | $0.398\pm 0.037$ | $19.20\pm 0.36$ | $0.258\pm 0.025$ | $1.43^{+0.82}_{-0.75}$ | $6154\pm 60$ | $-11.1\pm 6.6$
| WASP-174 | b | $0.330\pm 0.091$ | $8.77\pm 0.16$ | 0 | $2.20\pm 0.52$ | $6400\pm 100$ | $31\pm 1$