# A Testbed for Tidal Migration: the 3D Architecture of an Eccentric Hot
Jupiter HD 118203 b Accompanied by a Possibly Aligned Outer Giant Planet

Jingwen Zhang (张婧雯) NASA FINESST Fellow  Institute for Astronomy, University
of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Daniel Huber
Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu,
HI 96822, USA  Sydney Institute for Astronomy (SIfA), School of Physics,
University of Sydney, NSW 2006, Australia  Lauren M. Weiss Department of
Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA
Jerry W. Xuan NASA FINESST Fellow  Department of Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Jennifer A. Burt Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Fei Dai Institute for Astronomy, University of
Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Nicholas Saunders NSF
Graduate Research Fellow  Institute for Astronomy, University of Hawai‘i, 2680
Woodlawn Drive, Honolulu, HI 96822, USA  Erik A. Petigura Department of
Physics & Astronomy, University of California Los Angeles, Los Angeles, CA
90095, USA  Ryan A. Rubenzahl NSF Graduate Research Fellow  Department of
Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Joshua
N. Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy
Lane, Princeton, NJ 08540, USA  Sharon X. Wang Department of Astronomy,
Tsinghua University, Beijing 100084, People’s Republic of China  Judah Van
Zandt Department of Physics & Astronomy, University of California Los Angeles,
Los Angeles, CA 90095, USA  Max Brodheim W. M. Keck Observatory, 65-1120
Mamalahoa Hwy, Kamuela, HI 96743, USA  Zachary R. Claytor Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Department
of Astronomy, University of Florida, 211 Bryant Space Science Center,
Gainesville, FL 32611, USA  Ian Crossfield  Department of Physics and
Astronomy, University of Kansas, Lawrence, KS, USA  William Deich UC
Observatories, University of California, Santa Cruz, CA 95064, USA  Benjamin
J. Fulton Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Steven R. Gibson Caltech Optical Observatories,
California Institute of Technology, Pasadena, CA 91125, USA  Grant M. Hill W.
M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI 96743, USA  Bradford
Holden UC Observatories, University of California, Santa Cruz, CA 95064, USA
Aaron Householder Department of Earth, Atmospheric and Planetary Sciences,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Andrew W. Howard Department of
Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Howard
Isaacson Department of Astronomy, 501 Campbell Hall, University of California,
Berkeley, CA 94720, USA  Stephen Kaye  Caltech Optical Observatories,
California Institute of Technology, Pasadena, CA 91125, USA  Kyle Lanclos W.
M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI 96743, USA  Russ R.
Laher (良主嶺亞) NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E
California Blvd, Pasadena, CA 91125, USA  Jack Lubin Department of Physics &
Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA
Joel Payne W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI 96743,
USA  Arpita Roy Astrophysics & Space Institute, Schmidt Sciences, New York, NY
10011, USA  Christian Schwab School of Mathematical and Physical Sciences,
Macquarie University, Balaclava Road, North Ryde, NSW 2109, Australia  Abby P.
Shaum Department of Astronomy, California Institute of Technology, Pasadena,
CA 91125, USA  Josh Walawender W. M. Keck Observatory, 65-1120 Mamalahoa Hwy,
Kamuela, HI 96743, USA  Edward Wishnow  Space Sciences Laboratory, University
of California, Berkeley, CA 94720, USA  Sherry Yeh W. M. Keck Observatory,
65-1120 Mamalahoa Hwy, Kamuela, HI 96743, USA

(Received June 1, 2020; Revised June 1, 2020)

###### Abstract

Characterizing outer companions to hot Jupiters plays a crucial role in
deciphering their origins. We present the discovery of a long-period giant
planet, HD 118203 c ($m_{c}=11.9^{+0.69}_{-0.63}\ \mathrm{M_{J}}$,
$a_{c}=5.36^{+0.09}_{-0.10}$ AU, $e_{c}=0.26^{+0.03}_{-0.02}$) exterior to
previously known close-in eccentric hot Jupiter HD 118203 b ($P_{b}=6.135\
\mathrm{days}$, $m_{b}=2.14\pm{0.12}\ \mathrm{M_{J}}$, $r_{b}=1.14\pm{0.029}\
\mathrm{R_{J}}$, $e_{b}=0.31\pm{0.007}$) based on twenty-year radial velocity
observations. Using Rossiter-McLaughlin (RM) observations from the Keck Planet
Finder (KPF), we measured a low sky-projected obliquity
$\lambda_{b}=-11^{\circ}.7^{+7.6}_{-10}$ for HD 118203 b and detected stellar
oscillations in the host star, confirming its evolved status. Combining the RM
observation with the stellar inclination measurement, we constrained the true
obliquity of HD 118203 b as $\Psi_{b}<33^{\circ}.5\ (2\sigma)$, indicating the
orbit normal of the hot Jupiter nearly aligned with the stellar spin axis.
Furthermore, by combining radial velocities and Hipparcos-Gaia astrometric
acceleration, we constrained the line-of-sight mutual inclination between the
hot Jupiter and the outer planet to be $9^{\circ}.8^{+16.2}_{-9.3}$ at
$2\sigma$ level. HD 118203 is the first hot Jupiter system where both the true
obliquity of the hot Jupiter and the mutual inclination between inner and
outer planets have been determined. Our results are consistent with a system-
wide alignment, with low mutual inclinations between the outer giant planet,
the inner hot Jupiter, and the host star. This alignment, along with the
moderate eccentricity of HD 118203 c, implies that the system may have
undergone coplanar high-eccentricity tidal migration. Under this framework,
our dynamical analysis suggests an initial semi-major axis of 0.3 to 3.2 AU
for the proto-hot Jupiter.

editorials, notices — miscellaneous — catalogs — surveys

††facilities: Keck(HIRES), Keck(KPF), ELODIE, TESS, Gaia, Hipparcos††software:
allesfitter (Günther & Daylan, 2019, 2021) RadVel (Fulton et al., 2018) TESS-
SIP(Hedges et al., 2020)

##  1 Introduction

The discovery of Hot Jupiters, defined as giant gaseous planets with short
orbital periods ($P<10$ days), upended the traditional planet formation
theories based on our Solar System. Various theories have been proposed to
explain how these massive gaseous planets end up so close to their stars. The
in-situ formation theory suggests that hot Jupiters form directly in their
current, close orbits (Bodenheimer et al., 2000; Boley et al., 2016; Batygin
et al., 2016). However, the typical giant planet formation processes – either
core accretion or gravitational instability – encounter difficulties operating
at such close distances (Rafikov, 2005, 2006). Disk-driven migration is
another proposed mechanism in which the proto-hot Jupiters initially form in
the outer region of the protoplanetary disks and migrate inward due to the
torques from the disk (Goldreich & Tremaine, 1980; Lin et al., 1996; Ida &
Lin, 2008). Alternatively, in the high-eccentricity tidal migration scenario,
the orbit of a giant planet that formed beyond the ice line attains a high
eccentricity and then shrinks and circularizes through dissipative tidal
interactions with the host star (Fabrycky & Tremaine, 2007; Fabrycky & Winn,
2009; Naoz et al., 2011). The trigger for tidal migration is the eccentricity
excitation of the proto-hot Jupiters, which could be caused by dynamical
events such as planet-planet scattering, or secular interactions with another
planet such as von-Zeipel-Kozai-Lidov cycles (von Zeipel, 1910; Kozai, 1962;
Lidov, 1962)111The phenomenon is commonly referred to as Kozai-Lidov cycles or
Lidov-Kozai cycles, acknowledging that Kozai and Lidov independently
discovered it in the 1960s. However, recent investigations into the literature
have revealed that Swedish astronomer von Zeipel described a similar theory as
early as 1910 (Ito & Ohtsuka, 2019)..

The obliquities of stars hosting hot Jupiters are considered a crucial
indicator of their formation channel. Stellar obliquity is the angle between
the star’s spin axis and the planet’s orbital axis. In scenarios involving
tidal migration, where dynamic events cause eccentric orbits, the angular
momentum of the proto-hot Jupiters would be altered, leading to high stellar
obliquities. Additionally, disk-driven migration could lead to misalignment
between planetary orbits and their host stars if stellar companions are
present to disturb the proto-planetary disks (Batygin, 2013). However,
observations have suggested that the stellar obliquity distribution may not
only reflect the hot Jupiters’ origins but also sculpting by tidal
realignment. Schlaufman (2010) and Winn et al. (2010a) indicate a significant
dichotomy in the obliquities of stars hosting hot Jupiters based on the star’s
temperature. They discovered that cool stars hosting hot Jupiters generally
show low obliquities, likely because effective tidal realignment processes
erase their initial obliquities. In contrast, hotter stars exhibit a broader
range of obliquities, indicating that they may retain the initial obliquities
from when hot Jupiters first migrated to their close-in orbits. Furthermore,
Rice et al. (2022) found that the stellar obliquities of eccentric hot
Jupiters do not exhibit such a trend with stellar temperature, which is
consistent with the high-eccentricity migration scenario from their models.

The presence and properties of companions to hot Jupiters serve as another
clue to distinguish different origin theories. In-situ formation could produce
hot Jupiters with nearby planets that are not necessarily in orbital resonance
(Hansen & Murray, 2013; Boley et al., 2016). In contrast, in the disk
migration scenario, hot Jupiters tend to be accompanied by nearby planets in
orbital resonance (Malhotra, 1993; Lee & Peale, 2002). On the other hand,
high-eccentricity tidal migration will destroy the nearby companions during
process, but requires a distant companion outside the forming region of the
proto-hot Jupiter to trigger the eccentricity excitation. Several surveys have
been conducted to search for stellar or planetary companions to hot Jupiters
out to tens of AU using radial velocities (RVs) and direct imaging (Endl et
al., 2014; Knutson et al., 2014; Bryan et al., 2016; Ngo et al., 2016). Ngo et
al. (2016) revealed that fewer than $16\%$ of hot Jupiters possess stellar
companions capable of driving proto-hot Jupiters into highly eccentric orbits,
suggesting stellar companions are not typically the instigators of high-
eccentricity tidal migration. In contrast, Knutson et al. (2014) reported that
$51\pm{10}\%$ of hot Jupiters have an outer giant planet with masses between
$1-13\ M_{J}$ and orbital semi-major axes between 1 and 20 AU, which are
nearby and massive enough to raise the eccentricities of proto-hot Jupiters.
In a recent study, Zink & Howard (2023) found that outer companions of all
five hot Jupiters in their sample are at least three times more massive than
the inner hot Jupiters and have an average eccentricity of 0.34. They argue
that these features favor coplanar high-eccentricity migration as the dominant
mechanism for the formation of hot Jupiters (Petrovich, 2015).

In addition to obliquities and companion statistics, the mutual inclinations
between the hot Jupiters and their outer companions might provide clues about
the formation of hot Jupiters. For the secular von-Zeipel-Kozai-Lidov cycles
to be effective, there must be an initial mutual inclination of over 40
degrees between the orbits of the two objects. On the other hand, recent
studies propose that hot Jupiters originate through coplanar high-eccentricity
migration, where the outer planets have eccentric orbits and relatively low
mutual inclinations relative to the hot Jupiters (Petrovich, 2015; Espinoza-
Retamal et al., 2023). Dawson et al. (2014) constrained the mutual inclination
between a warm Jupiter and its outer giant planet companion in the Kepler-419
system to be $\sim 9^{\circ}$ using Transit Timing Variations (TTV). By
combining radial velocity data with Gaia-Hipparcos astrometric accelerations,
Xuan & Wyatt (2020) found evidence for misalignments between the orbits of an
outer giant planet and a smaller inner planet in the $\pi$ Men
($52.3^{\circ}<\Delta I<127.7^{\circ}$) and HAT-P-11 ($63.1^{\circ}<\Delta
I<117^{\circ}$) systems (see also De Rosa et al. 2020; Damasso et al. 2020).
However, direct measurements of mutual inclination between transiting planets,
their companions and their orientation to the host star have been limited to
only a handful systems.

HD 118203 (TOI-1271) is a sub-giant star ($M_{*}=1.25^{+0.028}_{-0.032}\
M_{\odot}$, $R_{*}=2.03\pm{0.03}\ R_{\odot}$) hosting a transiting hot Jupiter
with an orbital period of 6.135 days, a mass of $2.166^{+0.074}_{-0.079}\
M_{J}$ and an eccentricity of 0.3 (da Silva et al., 2006; Pepper et al.,
2020). In this paper, we first combined Rossiter-McLaughlin observations from
the Keck Planet Finder (KPF) and stellar rotation periods derived from
Transiting Exoplanet Survey Satellite (TESS) light curves to determine the
true stellar obliquity of HD 118203 b, which is the angle between stellar spin
axis and the orbital axis of the planet. Furthermore, we present the discovery
of HD 118203 c, a long-period giant planet orbiting outside HD 118203 b. We
characterized the three-dimensional orbital parameters of this outer planet by
combining RVs and Hipparcos and Gaia astrometric data. HD 118203 is one of the
first systems where the stellar obliquity for both the hot Jupiter and the
outer giant planet, and the mutual inclination between the hot Jupiter and
outer planet have been determined. Recently, Maciejewski et al. (2024)
presented an independent detection of HD 118203 c using long-baseline radial
velocity data from HARPS-N (Cosentino et al., 2012) and the High-Resolution
Spectrograph (HRS; Tull 1998) at the Hobby-Eberly Telescope (Ramsey et al.,
1998). We compare our results with their findings in Section 5.3.

##  2 Observations

###  2.1 ELODIE and HIRES radial velocities

We used 56 archival RVs of HD 118203 taken with the ELODIE spectrograph
installed on the 1.93m reflector at Observatoire de Haute-Provence in France
(Baranne et al., 1996) between 2004 May and 2006 June from da Silva et al.
(2006) and Pepper et al. (2020). We also include 18 RVs from High Accuracy RV
Planet Searcher in the northern hemisphere (HARPS-N) and 51 RVs from High-
Resolution Spectrograph (HRS) at the 9.2 m Hobby-Eberly Telescope published by
Maciejewski et al. (2024). The HRS observation were taken from January 2006
and June 2013, and HARP-N data were taken from December 2012 and August 2015.
Furthermore, we collected 26 RV data for HD 118203 from 2010 to 2024 using the
High Resolution Echelle Spectrometer (HIRES, Vogt et al. 1994) at the Keck I
10 m telescope on Maunakea. The observations are part of a survey aiming to
search for substellar/stellar companions to transiting planet hosts with
significant Hipparcos and Gaia astrometric accelerations (Zhang et al., 2024).
We used the standard California Planet Search (CPS) pipeline described in
Howard et al. (2010) to determine RVs. Spectra were obtained with an iodine
gas cell in the light path for wavelength calibration. An iodine-free template
spectrum bracketed by observations of rapidly rotating B-type stars was used
to deconvolve the stellar spectrum from the spectrograph PSF. We then forward-
model the spectra taken with the iodine cell using the deconvolved template
spectra (Butler et al., 1996). The wavelength scale, the instrumental profile,
and the RV in each of the $\sim 700$ segments of 80 pixels were solved
simultaneously (Howard et al., 2010). The RVs used in this work are presented
in Table 1.

Table 1: HD 118203 RVs

Time | RV | $\sigma_{\mathrm{RV}}$ | Inst
---|---|---|---
(BJD - 2450000) | (m/s) | (m/s) |
3151.4302000002936 | -29262.81 | 13.40 | ELODIE
3153.423499999568 | -29230.81 | 14.47 | ELODIE
3155.3946000002325 | -29639.81 | 10.79 | ELODIE
… | … | … | …
5370.815355828032 | -126.58 | 0.86 | HIRES
5371.83661200013 | 44.42 | 1.24 | HIRES
8885.015345999971 | -327.16 | 1.58 | HIRES
8906.171260000207 | 38.02 | 1.41 | HIRES
… | … | … | …
10043.80571338 | 42.93 | 0.96 | KPF
10043.808489768 | 43.79 | 0.96 | KPF
10043.811272037 | 47.59 | 1.05 | KPF
… | … | … | …

Note. — Times are in BJD - 2450000.0. The RV uncertainties do not include RV
jitter. ELODIE RVs are from Pepper et al. (2020), HIRES and KPF RVs are from
this work. All RV data utilized in this paper, including those sourced from
the literature, are available in a machine-readable format.

###  2.2 KPF Rossiter-McLaughlin observations

The Keck Planet Finder (KPF Gibson et al., 2016, 2018, 2020) is an echelle
spectrometer that was commissioned on the Keck 1 telescope at the W.M. Keck
Observatory in March 2023. KPF covers the wavelength range of 445–870 nm with
a resolving power of 98,000. The Rossiter-McLaughlin (RM, Rossiter 1924;
McLaughlin 1924 ) observations of HD 118203 b were conducted during one
transit on April 09 2023 UT. A total of 76 exposures were obtained, each with
an integration time of 180 s and a readout time of 60 s. We achieved a typical
SNR of 176 at 550 nm. The observation covered the ingress and egress phases of
the transit with additional 30 mins before and after the transit. The
observation was interrupted for $\sim 1.5$ hours in the middle of the transit
due to high humidity. The spectra were reduced with the KPF Data Reduction
Pipeline (DRP,Gibson et al. 2020), which is publicly available on
Github222https://github.com/Keck-DataReductionPipelines/KPF-Pipeline. The KPF
DRP performs quadrant stitching, flat-fielding, order tracing, and optimal
extraction. KPF’s wavelength calibration sources include Th-Ar and U-Ne lamps,
a Laser Frequency Comb, a Fabry-Pérot Etalon, and the Solar Calibrator
(Rubenzahl et al., 2023). Our observations were taken with simultaneous
wavelength calibration sourced from an etalon lamp fed through a dedicated
calibration fiber. In addition to simultaneous calibrations, we performed
hourly dedicated Etalon calibration exposures, known as ‘slew-cals,’ to
monitor the stability of RVs. We used the adapted version of publicly
available template-matching code SpEctrum Radial Velocity Analyser serval
(Zechmeister et al., 2018) to extract the RVs from our KPF spectra. The KPF
RVs used in this work are presented in Table 1.

###  2.3 TESS photometry

HD 118203 appears in the TESS Asteroseismic Target List (Schofield et al.,
2019) as a solar-like oscillator to be observed in 2-minute cadence. TESS data
were available as 2-minute cadence light curves for Sectors 15, 16, 22, 49 and
76 of the mission, spanning from 2019 Aug 15 to 2024 Mar 26. It was also
selected for TESS 20-second cadence observations in Sector 76. The light
curves were processed by the Science Processing Operations Center (SPOC) data
reduction pipeline (Jenkins et al., 2016). We downloaded all five sectors of
Pre-search Data Conditioning simple aperture photometry (PDC$\\_$SAP, Smith et
al. 2012; Stumpe et al. 2012, 2014) light curves, and then stitched and
normalized them using the lightkurve package (Lightkurve Collaboration et al.,
2018).

###  2.4 Hipparcos and Gaia astrometric acceleration333Also known as proper
motion anomaly.

HD 118203 was identified as a transiting-planet host with significant
Hipparcos and Gaia astrometric acceleration by Zhang et al. (2024), suggesting
the existence of an outer companion in the system. We used the astrometric
data for HD 118203 from the Hipparcos-Gaia Catalog of Accelerations (HGCA,
Brandt 2021). The HGCA catalog provides three proper motions in units of
$\mathrm{mas}\ \mathrm{yr}^{-1}$: (1) the Hipparcos proper motion
$\mu_{\rm{H}}$ measured at an epoch near 1991.25; (2) the Gaia EDR3 proper
motion $\mu_{\rm{G}}$ measured at an epoch near 2016.01; (3) the long-term
proper motion $\mu_{\rm{HG}}$ calculated as the difference in positions
between Hipparcos and Gaia divided by the $\sim 25$-year baseline. The long-
term proper motion $\mu_{\rm{HG}}$ can be approximately considered as the
velocity of stellar linear motion across the sky plane over nearly 25 years.
Following Kervella et al. (2019) and Brandt (2021), we computed the
astrometric acceleration by subtracting the long-term proper motion
$\mu_{\rm{HG}}$ from the Hipparcos or Gaia proper motion. Table 2 presents the
astrometric accelerations, which have a signal-to-noise ratio ($\rm{S/N}$) of
3.47 at the Gaia epoch, and 2.71 at the Hipparcos epoch.

Table 2: Hipparcos-Gaia astrometric acceleration in declination and right ascension for HD 118203 from Brandt (2021). The $\Delta\mu_{\delta*}$ components have the $\cos{\delta}$ factor included. $\sigma[\Delta\mu]$ represent the uncertainties. We assume that the uncertainties on the proper motions and $\mu_{\rm{HG}}$ are independent and add them in quadrature to calculate these uncertainties (see Zhang et al. 2024 for details). Data | $\Delta\mu_{\alpha}$ | $\sigma[\Delta\mu_{\alpha}]$ | $\Delta\mu_{\delta*}$ | $\sigma[\Delta\mu_{\delta*}]$ | S/N
---|---|---|---|---|---
epoch | $\mathrm{mas}\ \mathrm{yr}^{-1}$ | $\mathrm{mas}\ \mathrm{yr}^{-1}$ |
Gaia | -0.1038 | 0.0377 | 0.0905 | 0.0462 | 3.47
Hipparcos | 0.842 | 0.663 | -1.628 | 0.884 | 2.71

##  3 Host star Properties

###  3.1 Literature values

HD 118203 is a sub-giant G0V star at an age of $5.32^{+0.96}_{-0.73}$ Gyr and
a distance of $91.811\pm{0.236}$ pc (Pepper et al., 2020). The star has a mass
of $1.25\pm{0.06}\ M_{\odot}$, a radius of $2.10\pm{0.06}\ R_{\odot}$, and a
metallicity [Fe/H] of $0.223\pm{0.076}$ dex. Its spectroscopically determined
projected rotation velocity, $v\sin i_{*}$, is $5.32\pm{0.5}$ km s-1 (Luck,
2017).

###  3.2  High-resolution Spectroscopy

We obtained iodine-free spectra with Keck/HIRES for HD 118203 to characterize
stellar parameters. We used SpecMatch synthetic methodology (SpecMatch-Synth,
Petigura 2015) to compute stellar properties. We obtained a projected rotation
velocity of the star as $5.13\pm{1.0}~{}\rm{km}\ s^{-1}$. Other parameters are
listed in Table 3.

###  3.3 Isochrone fitting

As described in Section 4.2, we measured asteroseismic oscillations with a
frequency at maximum power of $\nu_{\rm{max}}=967.2^{+46.5}_{-51.7}$ $\mu$Hz.
We also employed isoclassify (Huber et al., 2017; Berger et al., 2020) to
derive the stellar parameters with following input observables: (1) the
frequency at maximum power $\mathrm{\nu_{max}}$, (2) the spectroscopic stellar
effective temperature and metallicity (Pepper et al., 2020), (3) the 2MASS
$K_{s}$ magnitude (Cutri et al., 2003), and (4) the Gaia DR3 parallax Gaia
Collaboration et al. (2022). Our measurement of $\nu_{max}$ helps to break the
mass and age degeneracy reported by Pepper et al. (2020). We derived a mass of
$1.27^{+0.028}_{-0.032}\ M_{\odot}$, a radius of $2.04\pm{0.03}\ R_{\odot}$,
and an age of $4.73\pm{0.39}$ Gyr, consistent with the classification of the
host star as a sub-giant. The stellar parameters are given in Table 3.

Table 3: Stellar Parameters of HD 118203

Parameters | SpecMatch | Isochrone | Reference
---|---|---|---
| Synth | Fitting | Value
$T_{\rm eff}\ (\rm{K})$ | $5802\pm{100}$ | $5742^{+93}_{-95}$ |  $5683^{+84}_{-85}$ (a)
$M_{\star}$ ($M_{\odot}$) | $1.32\pm{0.09}$ | $1.27^{+0.028}_{-0.032}$ |  $1.25\pm{0.06}$ (a)
$R_{\star}$ ($R_{\odot}$) | $2.08\pm{0.05}$ | $2.04\pm{0.03}$ |  $2.10\pm{0.06}$ (a)
$\rho_{\star}$ (cgs) | $0.206\pm{0.02}$ | $0.21\pm{0.01}$ |  $0.19\pm{0.01}$ (a)
log g (cgs) | $3.93\pm{0.1}$ | $3.92\pm{0.02}$ |  $3.89\pm{0.02}$(a)
$\rm[Fe/H]$ (dex) | $0.27\pm{0.06}$ | $0.287^{+0.069}_{-0.111}$ |  $0.223\pm 0.076$ (a)
$V_{mag}$ | – | – |  $8.135\pm 0.03$ (a)
Age (Gyr) | – | $4.73\pm{0.39}$ |  $5.12^{+0.64}_{-0.61}$ (a)
$v\sin i_{*}$ ($km\ s^{-1}$) | $5.13\pm{1}$ | – |  $5.32\pm{0.5}$ (b)
$\varpi$ (mas) | – | – |  $10.86\pm{0.018}$ (c)
$\nu_{\rm{max}}$ ($\rm{\mu Hz}$) | – | $967.2_{-51.7}^{+46.5}$ | –
$\mathrm{P_{rot}}$ (days)∗ | – | $23.1\pm{3.8}$ | –
$i_{*}$ (deg)∗ | – | $89.9_{-13.8}^{+13.7}$ | –

Note. — (a). Pepper et al. (2020); (b). Luck (2017); The reported velocity
($7\ \mathrm{km\ s^{-1}}$) in Luck (2017) represents a combination of rotation
and macroturbulence velocities, despite being referred to as Vr. After
deducting the macroturbulence velocities, the $v\sin i_{*}$ stands at $5.32\
\mathrm{km\ s^{-1}}$. (c). Gaia Collaboration et al. (2022). *
$\mathrm{P_{rot}}$ is measured from TESS light curves described in Section
4.3. $i_{*}$ is constrained with spectroscopic $v\sin i_{*}$ and stellar
rotation period described in Section 4.3.

##  4 The Stellar Obliquity

In this paper, we define the stellar obliquity as the angle between the
stellar spin axis and the orbital axis of a planet.

###  4.1 RM effect modelling without oscillations

We used the allesfitter (Günther & Daylan, 2019, 2021) package to determine
the sky-projected stellar obliquity for HD 118203 b. In the first step, we
performed a simultaneous global fit of 2-minute cadence light curves from TESS
sectors 15, 16, 22, 49, and 76 out-of-transit RVs from ELODIE (da Silva et
al., 2006; Pepper et al., 2020) and HIRES, and RM measurements from KPF. We
allowed 24 free parameters in the fitting (see Table LABEL:tab:rm_result). The
planetary properties and orbital elements of HD 118203 were described by the
following parameters: $r_{p}/R_{*}$, $(R_{*}+r_{p})/a_{b}$, $\cos{I_{b}}$,
$T_{0,b}$, $P_{b}$, $K_{b}$, $\sqrt{e_{b}}\cos{\omega_{b}}$, and
$\sqrt{e_{b}}\sin{\omega_{b}}$, where $R_{*}$ and $r_{p}$ are stellar and
planet radii, $\cos{I_{b}}$ is the cosine of the orbital inclination,
$T_{0,b}$ is the midtime of transit, $P_{b}$ is the orbital period, $K_{b}$ is
the RV semi-amplitude, $e_{b}$ is the eccentricity, and $\omega_{b}$ is the
argument of pericenter. The RM effect was modeled with two additional
parameters: $\lambda_{b}$, the projected stellar obliquity for HD 118203 b,
and $v\sin{i_{\star}}$, the projected stellar rotation velocity. In addition,
we fitted the transformed quadratic limb darkening coefficients for TESS
($q_{1,\rm{TESS}}$, $q_{2,\rm{TESS}}$) and KPF ($q_{1,\rm{KPF}}$,
$q_{2,\rm{KPF}}$). We also included constant zero point offsets for TESS light
curves and KPF RVs ($\gamma_{\rm{TESS}}$, $\gamma_{\rm{KPF}}$) and linear
slopes for ELODIE and HIRES RVs ($\gamma_{\rm{ELODIE}}$,
$\dot{\gamma}_{\rm{ELODIE}}$, $\gamma_{\rm{HIRES}}$,
$\dot{\gamma}_{\rm{HIRES}}$). Finally, we added four free parameters to
account for the jitter terms of the four instruments ($\ln\sigma_{\rm{TESS}}$,
$\ln\sigma_{\rm{ELODIE}}$, $\ln\sigma_{\rm{HIRES}}$, $\ln\sigma_{\rm{KPF}}$).
We adopted uniform priors for each of the parameters (see Table
LABEL:tab:rm_result). The initial guesses for the planetary and orbital
parameters were adopted from Pepper et al. (2020), while the initial guess for
$v\sin{i_{\star}}$ was the spectroscopically determined value from Luck
(2017).

Figure 1: Lomb-Scargle periodograms of the KPF RV residuals from RM fitting
without oscillation. The solid grey and orange lines represent the original
and smoothed periodograms (filter width=$340\rm{\ \mu Hz}$). The blue dashed
line indicates the predicted frequency at maximum power, $\nu_{\rm{max}}$,
derived using scaling relations
$\frac{g}{g_{\odot}}=\frac{\nu_{\rm{max}}}{\nu_{\odot}}\frac{T_{\rm{eff}}}{T_{\rm{eff,\odot}}}$.
The values for the star’s surface gravity $g$ and effective temperature
$T_{\rm{eff}}$ were taken from Pepper et al. (2020).  Figure 2: Panel a: TESS
SPOC light curves phase-folded to the best-fit orbital period of HD 118203 b.
The pink points present the TESS data binned every 15 minutes, and the red
line shows the best-fit transit model. Panel b: The out-of-transit RVs from
ELODIE (green) and HIRES (orange) phase-folded to the best-fit orbital period
of HD 118203 b. The red line depicts the best-fit RV model. The linear
baseline has been removed. Panel c: the in-transit RVs from KPF in blue and
the combined RM and GP model in red. Panel d: KPF data and RM model after
removing the contribution from oscillations. Panel e: KPF data and SHO GP
model after removing RM effect. Panel f: the residuals of the fit.

We performed a dynamic Nested Sampling fit with dynesty (Speagle, 2020) to
sample the posterior distributions of all model parameters. Our Nested
Sampling analysis used 500 live points and a convergence criterion that
required the uncertainty of the Bayesian evidence estimate to be $<0.01$. The
best-fit parameters and their $1\sigma$ uncertainties are reported in Table
LABEL:tab:rm_result. The fitting results in a sky-projected stellar obliquity
for HD 118203 b of $\lambda_{b}=-11_{-13}^{+11}\,{}^{\circ}$, and a projected
stellar rotation speed of $v\sin{i_{\star}}=5.85_{-0.32}^{+0.41}\ \mathrm{km\
s^{-1}}$ (see Appendix A.1 for the best-fit models and Appendix A.2 for the
joint posterior distributions of the 24 parameters). Our fitted
$v\sin{i_{\star}}$ is consistent with both the spectroscopic measurements
($5.13\pm{1}$ km s-1) from the HIRES spectra and the value ($5.32\pm{0.5}$ km
s-1) reported by Luck (2017) .

Additionally, we computed the power spectrum of the KPF RV residuals using a
Lomb-Scargle periodogram (Press & Rybicki, 1989). The power spectrum reveals a
clear pattern of peaks attributable to solar-like oscillations centered around
$950\ \mathrm{\mu Hz}$ (see Figure 1). The observed frequency at the peak is
consistent with the expected $\nu_{\rm{max}}$ calculated using the scaling
relation
$\frac{g}{g_{\odot}}=\frac{\nu_{\rm{max}}}{\nu_{\odot}}\frac{T_{\rm{eff}}}{T_{\rm{eff,\odot}}}$,
based on the stellar parameters from Pepper et al. (2020). Because the KPF RM
observation only spans about six hours, it is not feasible to distinguish
individual oscillation frequencies. On the other hand, we performed Lomb-
Scargle periodogram analysis on the combined TESS light curves spanning 5
sectors with a 2-minute cadence, as well as on each sector individually.
Furthermore, We examined the Lomb-Scargle periodogram of light curves with a
20-second cadence specifically from Sector 76. No oscillation modes were
identified in any of these analyses.

###  4.2 Gaussian process modelling of the RM effect with oscillations

Since the stellar oscillation period (17.5 min) is longer than the cadence of
the KPF RVs (4 min), we implemented a Gaussian process (GP) model alongside
the Rossiter-McLaughlin effect to account for any possible systematic errors
in the obliquity measurement introduced by the asteroseismic oscillations. The
oscillations can be modelled with a sum of stochastically driven damped simple
harmonic oscillators (SHO), as described by Foreman-Mackey et al. (2017). We
used a single SHO term to describe the envelope of the spectrum of
oscillations. The power spectrum of a SHO is

$S(\omega)=\sqrt{\frac{2}{\pi}}\frac{S_{0}\omega^{4}}{(\omega^{2}-\omega_{\mathrm{max}}^{2})^{2}+\omega^{2}\omega_{\mathrm{max}}^{2}/Q_{0}^{2}}$
(1)

where $\omega_{\mathrm{max}}$ is the angular frequency at the maximum power,
which can be converted to frequency at maximum power using
$\nu_{max}=\omega_{max}/2\pi$. $Q_{0}$ is the quality factor describing the
oscillation damping time-scale, and $S_{0}$ is proportional to the power at
$\omega=\omega_{\mathrm{max}}$:$S(\omega_{\mathrm{max}})=\sqrt{2/\pi}S_{0}Q_{0}^{2}$.

We used 27 free parameters in the Nested Sampling, with the same parameters as
described in section 4.1 and three additional GP parameters ($\ln S_{0}$, $\ln
Q_{0}$, and $\ln\omega_{\mathrm{max}}$). We adopted wide uniform priors on the
three GP parameters (see Table LABEL:tab:rm_result). The damping timescale of
sub-giant oscillations is typically a few days, which is significantly longer
than our observing baseline (Lund et al., 2017). Therefore, we set a uniform
prior of $\ln Q_{0}$ from 1 to 4, consistent with coherent (undamped)
oscillations. We used 500 live points, and required the Bayesian evidence
estimate to be $<0.01$ for convergence.

Figure 2 shows the results of modeling the RM effect alongside the stellar
oscillations. We obtained a sky-projected spin-orbit angle of
$\lambda_{b}=-11^{\circ}.7^{+7.6}_{-10}$ and a projected stellar rotation
speed of $v\sin i_{*}=5.89^{+0.31}_{-0.26}\ \mathrm{km\ s^{-1}}$, which agrees
with the results we obtained when fitting the RM effect without modeling the
stellar oscillations (section 4.1). The Bayes factor, with a value of
$\Delta\log(Z)=6.19>5$, indicates that the RM model with the GP is favored
over the RM model without the GP. Despite this, the fitting outcomes are
consistent, indicating that the stellar oscillations have a negligible effect
on the derived obliquity. This result aligned with our expectations, given
that the amplitude and timescale of the RM signal are both much larger than
those of the oscillations. The GP model with an SHO kernel yields a frequency
of maximum oscillation power $\nu_{max}=967.2_{-46.5}^{+51.7}\ \mathrm{\mu
Hz}$, which corresponds to an oscillation period of $\sim 17$ minutes.

Table 4: Priors and posterior distributions of RM modelling for HD 118203 b | RM without oscillation | RM+ oscillation
---|---|---
Parameter (Unit) | Prior | Value | Prior | Value
Planetary Parameters
Planet-to-star radius ratio $r_{b}/R_{\star}$ | $\mathcal{U}(0.01,0.1)$ | $0.05568_{-0.00021}^{+0.00024}$ | $\mathcal{U}(0.01,0.1)$ | $0.05573_{-0.00021}^{+0.00025}$
Sum of radii over semi-major axis $(R_{\star}+r_{b})/a_{b}$ | $\mathcal{U}(0.1,0.3)$ | $0.1443_{-0.0016}^{+0.0024}$ | $\mathcal{U}(0.1,0.3)$ | $0.1445_{-0.0016}^{+0.0025}$
Cosine of the orbital inclination $\cos{I_{b}}$ | $\mathcal{U}(0,0.2)$ | $0.022_{-0.014}^{+0.018}$ | $\mathcal{U}(0,0.2)$ | $0.027_{-0.013}^{+0.016}$
Orbital period $P_{b}$ (days) | $\mathcal{U}(6.12,6.15)$ | $6.1349787\pm 0.000002$ | $\mathcal{U}(6.12,6.15)$ | $6.1349779\pm 0.000002$
Transit epoch - 2459000 $T_{0;b}$ (BJD) | $\mathcal{U}(1711,1714)$ | $-262.80110\pm 0.0008^{*}$ | $\mathcal{U}(1711,1714)$ | $-268.93581\pm 0.0008^{*}$
RV semi-amplitude $K_{b}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(0.1,0.3)$ | $0.2165\pm 0.0031$ | $\mathcal{U}(0.1,0.3)$ | $0.2169\pm 0.0031$
Eccentricity term $\sqrt{e_{b}}\cos{\omega_{b}}$ | $\mathcal{U}(-0.7,0.7)$ | $-0.508\pm 0.012$ | $\mathcal{U}(-0.7,0.7)$ | $-0.510\pm 0.012$
Eccentricity term $\sqrt{e_{b}}\sin{\omega_{b}}$ | $\mathcal{U}(-0.7,0.7)$ | $0.235\pm 0.016$ | $\mathcal{U}(-0.7,0.7)$ | $0.232\pm 0.016$
Rossiter-McLaughlin Parameters
Sky-projected stellar obliquity $\lambda_{b}$ (deg) | $\mathcal{U}(-180,180)$ | $-11_{-13}^{+11}$ | $\mathcal{U}(-180,180)$ | $-11.7_{-10.}^{+7.6}$
Projected rotational velocity $v\sin{i_{*}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(1,10)$ | $5.85_{-0.32}^{+0.41}$ | $\mathcal{U}(1,10)$ | $5.89_{-0.26}^{+0.31}$
GP SHO parameter $lnS_{0}\ (\mathrm{km^{2}s^{-2}})$ | – | – | $\mathcal{U}(-35,15)$ | $-20.47_{-0.78}^{+0.54}$
GP SHO parameter $lnQ_{0}$ | – | – | $\mathcal{U}(1,4)$ | $1.70_{-0.46}^{+0.70}$
GP SHO parameter $ln\omega_{\rm{max}}\ (\mathrm{d^{-1}})$ | – | – | $\mathcal{U}(5,7)$ | $6.266\pm 0.055$
Other Parameters
Transformed limb darkening $q_{\mathrm{1;TESS}}$ | $\mathcal{U}(0,1)$ | $0.248_{-0.040}^{+0.043}$ | $\mathcal{U}(0,1)$ | $0.250\pm 0.038$
Transformed limb darkening $q_{\mathrm{2;TESS}}$ | $\mathcal{U}(0,1)$ | $0.341_{-0.069}^{+0.076}$ | $\mathcal{U}(0,1)$ | $0.333_{-0.061}^{+0.071}$
Transformed limb darkening $q_{\mathrm{1;KPF}}$ | $\mathcal{U}(0,1)$ | $0.69_{-0.11}^{+0.14}$ | $\mathcal{U}(0,1)$ | $0.658_{-0.079}^{+0.11}$
Transformed limb darkening $q_{\mathrm{2;KPF}}$ | $\mathcal{U}(0,1)$ | $0.81_{-0.18}^{+0.13}$ | $\mathcal{U}(0,1)$ | $0.864_{-0.15}^{+0.095}$
Jitter term $\ln{\sigma_{\mathrm{TESS}}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-8,-5)$ | $-7.697\pm 0.013$ | $\mathcal{U}(-8,-5)$ | $-7.697\pm 0.012$
Jitter term $\ln{\sigma_{\mathrm{ELODIE}}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-10,-1)$ | $-4.24\pm 0.20$ | $\mathcal{U}(-10,-1)$ | $-4.24_{-0.21}^{+0.19}$
Jitter term $\ln{\sigma_{\mathrm{HIRES}}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-10,-1)$ | $-3.63_{-0.14}^{+0.15}$ | $\mathcal{U}(-10,-1)$ | $-3.64_{-0.14}^{+0.16}$
Jitter term $\ln{\sigma_{\mathrm{KPF}}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-10,-1)$ | $-6.19\pm 0.11$ | $\mathcal{U}(-10,-1)$ | $-7.32_{-1.5}^{+0.61}$
Baseline offset $\gamma_{\mathrm{TESS}}$ | $\mathcal{U}(-0.003,0.003)$ | $0.00023\pm 0.000015$ | $\mathcal{U}(-0.003,0.003)$ | $0.00023\pm 0.000015$
Baseline offset $\gamma_{\mathrm{ELODIE}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-50,50)$ | $-29.3671\pm 0.0057$ | $\mathcal{U}(-50,50)$ | $-29.3670\pm 0.0057$
Baseline slope $\dot{\gamma}_{\mathrm{ELODIE}}$ ($\mathrm{km\ s^{-1}d^{-1}}$) | $\mathcal{U}(-0.1,0.1)$ | $0.0549\pm 0.0098$ | $\mathcal{U}(-0.1,0.1)$ | $0.0554\pm 0.0095$
Baseline offset $\gamma_{\mathrm{HIRES}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-50,50)$ | $-0.030\pm 0.015$ | $\mathcal{U}(-50,50)$ | $-0.029\pm 0.015$
Baseline slope $\dot{\gamma}_{\mathrm{HIRES}}$ ($\mathrm{km\ s^{-1}d^{-1}}$) | $\mathcal{U}(-0.1,0.1)$ | $0.021\pm 0.017$ | $\mathcal{U}(-0.1,0.1)$ | $0.020\pm 0.018$
Baseline offset $\gamma_{\mathrm{KPF}}$ ($\mathrm{km\ s^{-1}}$) | $\mathcal{U}(-50,50)$ | $0.0624\pm 0.0024$ | $\mathcal{U}(-50,50)$ | $0.0628\pm 0.0025$
Derived Parameters
Planet mass $m_{b}$ ($\mathrm{M_{J}}$) | – | $2.13\pm 0.12$ | – | $2.14\pm 0.12$
Planet radius $r_{b}$ ($\mathrm{R_{J}}$) | – | $1.139\pm 0.028$ | – | $1.140\pm 0.029$
Semi-major axis $a_{b}$ (AU) | – | $0.0714\pm 0.002$ | – | $0.0713\pm 0.002$
Inclination $I_{\mathrm{b}}$ (deg) | – | $88.71_{-1.02}^{+0.79}$ | – | $88.48_{-0.93}^{+0.73}$
Eccentricity $e_{\mathrm{b}}$ | – | $0.3135\pm 0.0078$ | – | $0.3136\pm 0.0075$
Argument of periastron $w_{\mathrm{b}}$ (deg) | – | $155.2_{-2.0}^{+1.9}$ | – | $155.5_{-2.0}^{+1.9}$
Impact parameter $b_{\mathrm{tra;b}}$ | – | $0.131_{-0.080}^{+0.10}$ | – | $0.155_{-0.074}^{+0.090}$
Total transit duration $T_{\mathrm{tot;b}}$ (h) | – | $5.640_{-0.014}^{+0.015}$ | – | $5.643_{-0.014}^{+0.016}$
Full-transit duration $T_{\mathrm{full;b}}$ (h) | – | $5.028_{-0.016}^{+0.014}$ | – | $5.026_{-0.017}^{+0.014}$
Planet Equilibrium temperature $T_{\mathrm{eq;b}}$ (K) | – | $1360\pm 23$ | – | $1361_{-22}^{+23}$
Planet density $\rho_{\mathrm{b}}$ (cgs) | – | $1.78_{-0.18}^{+0.21}$ | – | $1.79_{-0.18}^{+0.21}$
Planet surface gravity $g_{\mathrm{b}}$ (cgs) | – | $4198_{-168}^{+155}$ | – | $4192_{-179}^{+150}$
Limb darkening $u_{\mathrm{1;TESS}}$ | – | $0.339\pm 0.049$ | – | $0.333\pm 0.046$
Limb darkening $u_{\mathrm{2;TESS}}$ | – | $0.159\pm 0.087$ | – | $0.166\pm 0.078$
Limb darkening $u_{\mathrm{1;KPF}}$ | – | $1.33_{-0.23}^{+0.16}$ | – | $1.39_{-0.17}^{+0.11}$
Limb darkening $u_{\mathrm{2;KPF}}$ | – | $-0.51_{-0.18}^{+0.29}$ | – | $-0.59_{-0.13}^{+0.21}$
Jitter term $\sigma_{\mathrm{ELODIE}}(m\ s^{-1})$ | – | $14.38_{-2.6}^{+3.0}$ | – | $14.4_{-2.7}^{+3,0}$
Jitter term $\sigma_{\mathrm{HIRES}}(m\ s^{-1})^{\dagger}$ | – | $26.5_{-3.3}^{+4.3}$ | – | $26.4_{-3.4}^{+4.4}$
Jitter term $\sigma_{\mathrm{KPF}}(m\ s^{-1})$ | – | $2.0_{-0.21}^{+0.23}$ | – | $0.66_{-0.51}^{+0.55}$
Host density from all orbits $\rho_{\mathrm{\star}}$ (cgs) | – | $0.1968_{-0.0095}^{+0.0068}$ | – | $0.1960_{-0.0099}^{+0.0067}$
Frequency at maximum power $\nu_{\rm{max}}$ ($\mu\rm{Hz}$) | – | – | – | $967.2_{-46.5}^{+51.7}$
Model Comparison
Bayesian evidence $\rm{log(Z)}$ | – | $20089.45\pm{0.42}$ | – | $20095.64\pm{0.43}$
Figure 3: Panel a-d: TESS light curves of HD 118203 obtained in Sector 15, 16,
22, 49 and 76 with the modulation model at a period of 23.091 days. Panel e:
the Lomb-Scargle periodogram of the detrended TESS lightcurves of HD 118203
using SIP (Hedges et al., 2020). The highest peak in the periodogram is
located at 23.091 days. The red dashed line shows the Gaussian fit to the peak
to determine the uncertainty of rotation period.  Figure 4: (a): Posterior
distribution of projected stellar obliquity for HD 118203 b, obtained from RM
effect modeling with stellar oscillations. (b): Posterior distribution of
stellar inclination (in orange) versus an isotropic distribution. The stellar
inclination ranges from $0^{\circ}$ to $180^{\circ}$ to accounts for scenarios
where the stellar axis may point either toward ($<90^{\circ}$) or away
($>90^{\circ}$) from the observer. The red dashed lines correspond to
$1\sigma$ significance. (c): Posterior distribution of cosine of true stellar
obliquity for HD 118203 b. (d): True stellar obliquity distribution for HD
118203 b. The blue dashed lines correspond to $2\sigma$ significance.

###  4.3 True stellar obliquity distribution

The true stellar obliquity $\Psi_{p}$ of a planet is given by

$\cos{\Psi_{p}}=\cos{i_{*}}\cos{i_{p}}+\sin{i_{*}}\sin{i_{p}}\cos{\lambda_{p}},$
(2)

where $i_{p}$ is the planet’s orbital inclination, $i_{*}$ is the star’s spin-
axis inclination, and $\lambda_{p}$ is the sky-projected obliquity. For HD
118203 b, $i_{b}$ is tightly constrained by the TESS transit observations, and
$\lambda_{b}$ is constrained by modeling the RM effect. Therefore, it is
essential to determine the stellar inclination $i_{*}$ in order to derive the
true stellar obliquity of a planet. When the stellar radius $R_{*}$ and
rotational period $P_{\rm{rot}}$ are known, the equatorial rotational
velocities can be calculated as $v_{eq}=2\pi R_{*}/P_{\rm{rot}}$. If $R_{*}$,
$P_{\rm{rot}}$ and $v\sin{i_{*}}$ are independent, $i_{*}$ can be derived as
$\sin^{-1}(\frac{v\sin{i_{*}}}{v_{eq}})$. However, Masuda & Winn (2020)
pointed out that $v_{eq}$ and $v\sin{i_{*}}$ are not statistically
independent, and the consideration of their correlation is needed when
determining the probability distribution of $i_{*}$. Addressing this
correlation, Bowler et al. (2023) presents a Bayesian probabilistic framework
to constrain the stellar inclination as

${}P(i_{*}\mid P_{\mathrm{rot}},R_{*},v\sin i_{*})\propto\sin
i_{*}\times\frac{e^{-\frac{\big{(}v\sin i_{*}-\frac{2\pi
R_{*}}{P_{\mathrm{rot}}}\sin i_{*}\big{)}^{2}}{2\big{(}\sigma_{v\sin
i_{*}}^{2}+\sigma_{v_{\mathrm{eq}}}^{2}\sin^{2}i_{*}\big{)}}}}{\sqrt{\sigma_{v\sin
i_{*}}^{2}+\sigma_{v_{\mathrm{eq}}}^{2}\sin^{2}i_{*}}},$ (3)

where $\sigma_{v\sin i_{*}}$ and $\sigma_{v_{\mathrm{eq}}}$ are the
uncertainties in the projected velocity $v\sin{i_{*}}$ and the equatorial
velocity $v_{eq}$. We approximated $\sigma_{v_{\mathrm{eq}}}$ as

${}\sigma_{v_{\mathrm{eq}}}=\frac{2\pi
R_{*}}{P_{\mathrm{rot}}}\sqrt{\Big{(}\frac{\sigma_{R_{*}}}{R_{*}}\Big{)}^{2}+\Big{(}\frac{\sigma_{P_{\mathrm{rot}}}}{P_{\mathrm{rot}}}\Big{)}^{2}}.$
(4)

We applied a Systematics-insensitive Periodogram (SIP, Hedges et al. 2020) to
stitch and detrend systematics in the TESS SPOC light curves from sectors 15,
16, 22, 49, and 76. This process enabled us to generate a Lomb-Scargle
periodogram. We identified a rotation period $P_{\mathrm{rot}}$ of
$23.1\pm{3.8}$ days (see Figure 3). We derived the uncertainty from the width
of the periodogram peak, adding an extra $10\%$ to account for surface
differential rotation (Epstein & Pinsonneault, 2014; Claytor et al., 2022). We
adopted the stellar projected rotational velocity $v\sin i_{*}$ of
$5.32\pm{0.5}$ km s-1 reported from Luck (2017) and stellar radius $R_{*}$of
$2.04\pm{0.03}\ R_{\odot}$ (see Section 3 for details). Figure 4 b shows the
resulting distribution of the stellar inclination which ranges from
$0^{\circ}$ to $180^{\circ}$ to accommodate the possibility that the stellar
axis points either toward ($<90^{\circ}$) or away($>90^{\circ}$) from the
observer. We obtained a stellar inclination of
$i_{\star}=89^{\circ}.9^{+13.7}_{-13.8}$ at $1\sigma$ confidence, suggesting
that the stellar rotation axis is orthogonal to the line of sight. The
findings align with the calculation in Pepper et al. (2020), where a
rotational period of 22 days was estimated based on the $v\sin i_{*}$ and
$R_{*}$, assuming that the axis of stellar rotation is perpendicular to our
line of sight.

By combining this information with the sky-projected obliquity from our RM
measurements with Eqn. 2, we derived the cosine term of the true stellar
obliquity $\cos{\Psi_{b}}$, which is strongly peaked at 1 (see Figure 4 c).
This is consistent with expectations as we have projected stellar obliquity
$\lambda_{b}$ is close to 0 deg and stellar inclination peaks at 90 deg.
Figure 4 d shows the distribution of the true stellar obliquity $\Psi_{b}$ for
HD 118203 b after transforming from cosine terms to angles. However, this
distribution shows a low probability near $\Psi_{b}=0$. This does not
necessarily imply that the true obliquity is away from zero, but rather
reflects the effect of transforming from cosine terms to angles. Therefore, we
believe it is more appropriate to provide an upper limit for $\Psi_{b}$, which
is $\Psi_{b}<33^{\circ}.5$ at $95\%$ probability level. The low stellar
obliquity indicates HD 118203 b is nearly aligned with the host star.

##  5 The Discovery of an outer long-period planet

Figure 5:  a: The periodogram of RV residuals computed using RVsearch. The
signal from the hot Jupiter HD 118203 b has been subtracted. The blue line
corresponds to the periodogram using all RVs. The black line corresponds to
the periodogram using only HIRES RVs. b: The periodogram of HIRES
$S_{\rm{HK}}$ index. Figure 6:  Best-fit 2-planet Keplerian orbital model for
HD 118203 a: HD 118203 RVs with errors (black) and their best fit model (blue)
as a function of time. b: the residuals. c$\sim$d: RV data and models for each
planet phase-folded at the best-fit orbital period with all other planets’
signals removed.

Pepper et al. (2020) reported an RV trend in addition to HD 118203 b’s
Keplerian signal across two years of ELODIE observations. The Gemini-
North/Alopeke speckle imaging ruled out the possibility of a stellar companion
within 1.25 arcseconds. Therefore, the RV trend could possibly arise from
either a massive planet or a brown dwarf companion. Recently, Maciejewski et
al. (2024) used long-baseline RV data from HARPS and HRS to detect the outer
planet, HD 118203 c. In this work, we combine archival data with our new HIRES
RVs, which extended the observation baseline to nearly 20 years, to further
refine the orbit of the outer planet.

###  5.1 Lomb-Scargle Periodigram

Figure 5 a displays the radial velocity (RV) periodogram in which the vertical
axis shows the Bayesian Information Criterion (BIC) difference,
$\Delta\rm{BIC}$, when comparing single-planet and two-planet models. These
calculations were performed using the RVsearch package, as detailed in
Rosenthal et al. (2021). Specifically, RVsearch establishes an orbital period
grid ranging from two days to five times the span of observational data to
search for periodic signals. The algorithm iteratively fits a sinusoid at each
fixed period within this grid and computes $\Delta\rm{BIC}$ for each point.
The algorithm calculates an empirical false alarm probability (FAP) by fitting
a linear model to a log-scale histogram of periodogram power values. From
this, it extrapolates a detection threshold in terms of $\Delta\rm{BIC}$
corresponding to an empirical FAP of $0.1\%$, corresponding to that only
$0.1\%$ of periodogram values are anticipated to exceed this threshold (see
Howard & Fulton (2016) for details). We calculated two $\rm{\Delta\ BIC}$
periodograms: the first one using all available RVs and the second one only
using HIRES RVs. The two periodograms both reveal pronounced peaks at $\sim
4979$ days surpassing the detection threshold (a FAP of $0.1\%$). The lower
peaks observed at shorter periods are attributable to the harmonics of the
period. To investigate whether stellar activity could account for the long-
period signal, we also conducted a periodogram analysis of the S-values. We
computed S-values for HIRES data by measuring the core flux of Calcium H $\&$
K lines. Note that we only used the S-values from the HIRES observations. We
do not see the peak at 4979 days shown in the RV periodogram (see Figure 6 d).
Moreover, HD 118203 shows a Hipparcos-Gaia astrometric acceleration with
$>3\sigma$ significance, which can not be accounted by the stellar activity,
and suggests the star undergoes orbital motion due to gravitational
interaction with a long-period giant planet.

###  5.2 RV-only Keplerian Fit

We first performed a two-planet Keplerian orbital fit to the RVs using RadVel
(Fulton et al., 2018). The model includes five orbital elements for each
planet $K$, $P$, $T_{conj}$, $\sqrt{e}sin\omega$,$\sqrt{e}cos\omega$, where
$K$ is the RV semi-amplitude, $P$ is the orbital period, $T_{conj}$ is the
time of conjunction, $e$ is the eccentricity, and $\omega$ is the argument of
pericenter. For HD 118203 b, we used Gaussian priors informed by TESS
observations (Pepper et al., 2020) for the orbital period $P_{b}$ and
conjunction time $T_{conj,b}$. We set bounds on $0<e<1$, $K>0$ for all
planets.

Table 5:  MCMC Posteriors for RV-only fitting

Parameter | Credible Interval | Max Likelihood | Units
---|---|---|---
Fitted parameters
$P_{b}$ | $6.134982^{+2.2e-05}_{-2.3e-05}$ | $6.134983$ | days
$T\rm{conj}_{b}$ | $2459129.795\pm 0.022$ | $2459129.794$ | JD
$T\rm{peri}_{b}$ | $2459130.44^{+0.035}_{-0.037}$ | $2459130.444$ | JD
$e_{b}$ | $0.2851^{+0.0074}_{-0.0072}$ | $0.2841$ |
$\omega_{b}$ | $2.697^{+0.032}_{-0.034}$ | $2.702$ | radians
$K_{b}$ | $220.9\pm 1.8$ | $221.0$ | m s-1
$P_{c}$ | $5126^{+76}_{-72}$ | $5123$ | days
$T\rm{conj}_{c}$ | $2456857^{+54}_{-51}$ | $2456856$ | JD
$T\rm{peri}_{c}$ | $2457654^{+68}_{-71}$ | $2457662$ | JD
$e_{c}$ | $0.268^{+0.028}_{-0.026}$ | $0.267$ |
$\omega_{c}$ | $3.066^{+0.07}_{-0.073}$ | $3.078$ | radians
$K_{c}$ | $121.8^{+6.6}_{-5.7}$ | $122.1$ | m s-1
Other parameters
$\gamma_{\rm nea_{E}LODIE}$ | $\equiv-29290.0306$ | $\equiv-29290.0306$ | m s$-1$
$\gamma_{\rm hires_{j}}$ | $\equiv-95.6072$ | $\equiv-95.6072$ | m s$-1$
$\gamma_{\rm Harps}$ | $\equiv-29180.777$ | $\equiv-29180.777$ | m s$-1$
$\gamma_{\rm HRS}$ | $\equiv-22.9028$ | $\equiv-22.9028$ | m s$-1$
$\dot{\gamma}$ | $\equiv 0.0$ | $\equiv 0.0$ | m s-1 d-1
$\ddot{\gamma}$ | $\equiv 0.0$ | $\equiv 0.0$ | m s-1 d-2
$\sigma_{\rm nea_{E}LODIE}$ | $14.0^{+2.8}_{-2.5}$ | $13.3$ | $\rm m\ s^{-1}$
$\sigma_{\rm hires_{j}}$ | $11.7^{+2.2}_{-1.6}$ | $10.7$ | $\rm m\ s^{-1}$
$\sigma_{\rm Harps}$ | $6.9^{+2.1}_{-1.5}$ | $5.6$ | $\rm m\ s^{-1}$
$\sigma_{\rm HRS}$ | $15.2^{+2.4}_{-2.0}$ | $14.4$ | $\rm m\ s^{-1}$
Derived parameters
$M_{b}\sin i$ | $2.24\pm 0.12$ | $2.23$ | MJup
$a_{b}$ | $0.0711^{+0.0018}_{-0.0019}$ | $0.0709$ | AU
$M_{c}\sin i$ | $11.7^{+0.87}_{-0.83}$ | $11.64$ | MJup
$a_{c}$ | $6.32^{+0.17}_{-0.18}$ | $6.31$ | AU

Note. — The epoch has been shifted by -322 periods to put the epoch in the
middle of the dataset. $\dagger$ We only consider a single-planet Keplerian
model in the global fitting of RM effect. Therefore, the excess of HIRES
jitter is attributed to the presence of the outer planet, detail see section
5.

We performed the Markov Chain Monte Carlo (MCMC) exploration with emcee
(Foreman-Mackey et al., 2013) to estimate parameter credible levels. Our MCMC
analysis used 50 walkers and ran for $1.5\times 10^{6}$ steps per walker,
achieving a maximum Gelman-Rubin(GR) statistic of 1.004. Figure 6 shows the
best fit Keplerian solution. We derived a minimum mass for HD 118203 c from RV
amplitudes as $m_{c}\sin{c}=11.7^{+0.87}_{-0.83}\ \rm M_{J}$, semi-major axis
as $a_{c}=6.32^{+0.17}_{-0.18}$ AU and an eccentricity of
$e_{c}=0.268^{+0.028}_{-0.026}$. The derived planetary parameters are given in
Table 5.

Figure 7:  Three-dimensional orbit characterization of the outer giant planet
around HD 118203 using RVs and Hipparcos\- Gaia astrometric acceleration. (a)
$\&$ (b): Observed and fitted Hipparcos and Gaia proper motion of HD118203 in
right ascension and declination. (c): astrometric orbits of HD 118203 c
projected on the sky plane from our model. In all of above panels, the thicker
black lines represent the best-fit orbit in the MCMC chain while the other 100
blue lines represent random draws from the chain.

###  5.3 Three-dimensional orbit fitting

We then characterized the three-dimensional orbit of the outer companion by
conducting a joint fit to the Hipparcos-Gaia astrometric accelerations and RV
time series, following the methodology outlined in Xuan & Wyatt (2020). The
long-baseline RVs over 20 years provide crucial information about the period,
eccentricity, and companion mass. Meanwhile, the two astrometric accelerations
measured at the Hipparcos and Gaia epochs, with a baseline around 25 years,
offer the essential constraint on the inclination. We used the parallel-
tempering Markov chain Monte Carlo (PT-MCMC) ensemble sampler in emcee
(Foreman-Mackey et al., 2017). We use 40 different temperatures to sample the
parameter space, and our results are taken from the ‘coldest’ chain, which
corresponds to the original, unmodified likelihood function. For each of the
30 temperatures, we use 100 walkers to sample the 18-parameter model. Two of
these parameters define the mass of the host star $M_{\star}$ and the parallax
of the system $\varpi$. Seven parameters define the orbits and mass of the
outer companion: the true mass of companion $m_{c}$, the cosine of its orbital
inclination $\cos{I_{c}}$, the orbital period $P_{c}$, the longitude of the
ascending node $\Omega_{c}$, the eccentricity $e_{c}$, and the argument of
periastron $\omega_{c}$ (fitted as $\sqrt{e_{c}}\cos{\omega_{c}}$ and
$\sqrt{e_{c}}\sin{\omega_{c}}$)555Note that $\omega_{c}$ is the argument of
periastron of the planet, not the star. The argument of periastron of the
planet is offset by $\pi$ from that of the star., and the epoch of periastron
$\tau_{c}$ at a reference epoch ($t_{\rm{ref}}=2451544.5$ JD). $\tau$ is a
dimensionless quantity ranging from 0 to 1, calculated relative to a reference
epoch $t_{\mathrm{ref}}$, the time of pariastron $t_{\mathrm{p}}$ and the
planet orbital period $P$ : $\tau=(t_{p}-t_{\mathrm{ref}})/P$ (Blunt et al.,
2020). Determining the bounds of $\tau$ is straightforward, even if the
orbital period $P$ is uncertain for planets with long orbital periods. Five
parameters are for the orbit of the inner hot Jupiter HD 118203 b: RV semi-
amplitude $K_{b}$, orbital period $P_{b}$, $\sqrt{e_{b}}\cos{\omega_{b}}$,
$\sqrt{e_{c}}\sin{\omega_{b}}$ and mean longitude at a reference epoch
$\tau_{b}$. Given that the orbital period of HD 118203 b (6.13 days) is
significantly shorter than the observation periods of both Hipparcos and Gaia,
its astrometric signal would be canceled out. Therefore, we did not consider
it in the astrometric model and set its inclination to be
$I_{b}=88^{\circ}.48$, which is well-determined by the transiting
observations. Finally, we include four parameters to account for the RV zero
points ($\gamma_{\rm{ELODIE}}$, $\gamma_{\rm{hires}}$, $\gamma_{\rm{HRS}}$,
$\gamma_{\rm{harps}}$) and jitter terms ($\sigma_{\rm{ELODIE}}$,
$\sigma_{\rm{hires}}$, $\sigma_{\rm{HRS}}$, $\sigma_{\rm{harps}}$) for each
instrument. The final likelihood used in our MCMC is given by

$\ln\mathcal{L}=-\frac{1}{2}(\chi^{2}_{\Delta\mu}+\chi^{2}_{RV}),$ (5)

with

$\begin{split}\chi^{2}_{RV}=\sum_{j}^{N_{\rm{inst}}}\sum_{i}^{N_{\rm{RV}}}[\frac{(\mathcal{M}[{\rm
RV}_{i}]-{\rm RV}_{i}+{\rm
RV}_{\rm{offset,j}})^{2}}{\sigma_{i}^{2}+\sigma_{\rm{jit,j}}^{2}}\\\
+\ln{2\pi(\sigma_{i}^{2}+\sigma_{\rm{jit,j}}^{2})}],\end{split}$ (6)

$\begin{split}\chi^{2}_{\Delta\mu}=\sum_{j}^{G,H}\left[\left(\frac{\mathcal{M}[\Delta\mu_{j,\alpha}]-\Delta\mu_{j,\alpha}}{\sigma[\Delta\mu_{j,\alpha}]}\right)^{2}\right.\\\
\left.+\left(\frac{\mathcal{M}[\Delta\mu_{j,\delta}]-\Delta\mu_{j,\delta}}{\sigma[\Delta\mu_{j,\delta}]}\right)^{2}\right]\end{split}$
(7)

Equation 6 presents the chi-square of RV fitting $\chi^{2}_{RV}$, where
$\mathcal{M}[{\rm RV}_{i}]$, ${\rm RV}_{i}$, and $\sigma_{i}$ are the model,
data, and uncertainties of the radial velocity at time $i$. ${\rm RV}_{\rm
zero,j}$ and $\sigma_{\rm jit,j}$ are the RV zero-points and jitter terms for
the instrument $j$. The RV model is given by the third component of Eq. 4 in
Xuan & Wyatt (2020). Specifically, we consider a two-planet model to fit RV
data.

Table 6: MCMC Posteriors for combined fitting of RV and Hipparcos-Gaia
astrometric acceleration1

Parameter (Unit) | Median $\pm$1$\sigma$ | Prior
---|---|---
Fitted parameters
HD 118203 b
RV semi-amplitude of HD118203 b $K_{b}$ ($\mathrm{m\,s^{-1}}$) | $218.0\pm 2.2$ | $\mathcal{N}(217.7,5)$
Orbital periods $P_{b}$ (days) | $6.13499\pm{0.000025}$ | $\mathcal{N}(6.13,0.1)$
Eccentricity term$\sqrt{e}_{b}\sin{\omega_{b}}$ | $0.48\pm{0.01}$ | no prior
Eccentricity term$\sqrt{e_{b}}\cos{\omega_{b}}$ | $-0.23\pm{0.02}$ | no prior
Epoch of periastron at 2451544.5 JD, $\tau_{\rm b}$ | $0.51\pm{0.01}$ | no prior
HD 118203 c
Planet mass $m_{\rm c}$ ($\mathrm{M_{J}}$) | $11.9^{+0.69}_{-0.63}$ | no prior
Orbital periods $P_{c}$ (days) | $5103_{-77}^{+76}$ | no prior
Cosine term of inclination $\cos{I_{c}}$ | $-0.06_{-0.20}^{+0.24}$ | no prior
Longitude of ascending node $\Omega_{c}^{2}$ (deg) | $10_{+55}^{-43}$ | no prior
Eccentricity term$\sqrt{e}_{c}\sin{\omega_{c}}$ | $-0.04^{+0.04}_{-0.03}$ | no prior
Eccentricity term$\sqrt{e_{c}}\cos{\omega_{c}}$ | $0.51\pm{0.03}$ | no prior
Epoch of periastron at 2451544.5 JD, $\tau_{\rm c}$ | $0.19\pm{0.02}$ | no prior
Others
Host-star mass $M_{*}$ ($\mathrm{M_{\odot}}$) | $1.27\pm 0.05$ | $\mathcal{N}(1.27,0.09)$
Parallax $\varpi$ (mas) | $10.86\pm 0.02$ | $\mathcal{N}(\varpi_{\rm DR3},\sigma[\varpi_{\rm DR3}])^{3}$
RV zero point $\gamma_{\rm{ELODIE}}$($\mathrm{m\,s^{-1}}$ ) | $29620\pm{4}$ | no prior
RV zero point $\gamma_{\rm{HIRES}}$($\mathrm{m\,s^{-1}}$ ) | $-98.51\pm{4}$ | no prior
RV zero point $\gamma_{\rm{HARPS}}$($\mathrm{m\,s^{-1}}$ ) | $29180^{+3.6}_{-3.5}$ | no prior
RV zero point $\gamma_{\rm{HRS}}$($\mathrm{m\,s^{-1}}$ ) | $25.61^{+7.7}_{-7.9}$ | no prior
RV jitter term $\sigma_{\rm{ELODIE}}$($\mathrm{m\,s^{-1}}$) | $14.8^{+2.2}_{-1.9}$ | $\mathcal{U}(0,25)$
RV jitter term $\sigma_{\rm{HIRES}}$($\mathrm{m\,s^{-1}}$) | $6.5^{+1.8}_{-1.3}$ | $\mathcal{U}(0,25)$
RV jitter term $\sigma_{\rm{HARPS}}$($\mathrm{m\,s^{-1}}$) | $11.5^{+2.0}_{-1.6}$ | $\mathcal{U}(0,25)$
RV jitter term $\sigma_{\rm{HRS}}$($\mathrm{m\,s^{-1}}$) | $13.4^{+2.7}_{-2.5}$ | $\mathcal{U}(0,25)$
Derived parameters
$\rm{Inclination}$ $I_{c}$ (deg) | $93.7_{-13.8}^{+11.9}$ | no prior
Semi-major axis $a_{c}$ (AU) | $5.36\pm{0.09}$ | no prior
Eccentricity $e_{c}$ | $0.26^{+0.03}_{-0.02}$ | $\mathcal{U}(0,0.99)$
Argument of periastron $\omega_{c}$ (deg) | $175\pm{4}$ | no prior
Time of periastron $T_{p,c}$ (JD) | $2452527_{-76}^{+73}$ | no prior

Note. — 1.The $\chi^{2}$ of RVs is 148 for 151 measurements. 2. The
distribution of $\Omega_{c}$ is centered around $0^{\circ}$, which breaks down
into two halves. We shifted values of $\Omega_{c}$ greater than $\pi$ by
$-2\pi$ to combine the distribution. 3.$\varpi_{\rm DR3}$ and
$\sigma[\varpi_{\rm DR3}]$ present the parallax and parallax uncertainty of HD
118203 from Gaia DR3 observations. 3. The HIRES, HRS RVs are median-
subtracted, whereas the ELODIE and HARPS RVs are not.

Figure 8:  The joint posterior distributions for orbital inclination $I_{c}$,
mass $m_{c}$, and semi-major axis $a_{c}$ of HD 118203 c from combined fitting
of RV and Hipparcos-Gaia astrometric acceleration. Moving outward, the dashed
lines on the corner plots correspond to $1\sigma$, $2\sigma$ and $3\sigma$
contours. The orange dashed lines in the 1D distribution denote mean values
with $1\sigma$ interval.

Equation 7 calculates the chi-square of astrometric fitting
$\chi^{2}_{\Delta\mu}$ by aggregating data across Hipparcos and Gaia epochs in
the Right Ascension (RA,$\alpha$) and Declination(DEC,$\delta$) dimensions.
$\Delta\mu$ and $\sigma[\Delta\mu]$ are the astrometric accelerations and
uncertainties listed in Table 2. $\mathcal{M}[\Delta\mu]$ is the astrometric
model that depends on the orbital elements of the outer companion, including
$I_{c}$, $M_{c}$, $\Omega_{c}$ (for details, see Xuan & Wyatt 2020). We
neglect HD 118203 b and only consider the outer companion in the astrometric
model. We also take into account two important systematic corrections in the
astrometric model. First, Hipparcos and Gaia do not provide truly
instantaneous proper motions, but averaged values over multiple measurements
during the Hipparcos ($\delta_{\rm{H}}=1227$ days) and Gaia DR3
($\delta_{\rm{GDR3}}=1002$ days) observation windows. Consequently, the
astrometric acceleration from a companion whose orbital periods are comparable
to or shorter than the time spans of 1227 days or 1002 days will experience a
smearing effect throughout the observation period. Furthermore, even targets
with longer periods will suffer some amount of smearing. To account for the
smearing effect, we compute the tangential velocities at every individual
observation time. The Hipparcos observation times can be found from the
Hipparcos Epoch Photometry Annex (van Leeuwen et al., 1997) and the Gaia DR3
observation times can be downloaded in Gaia Observation Forecast Tool
666https://gaia.esac.esa.int/gost/ within the Gaia DR3 and Hipparcos observing
periods, and then average over them to mimic the smearing effect. Second, the
star was in different orbital phases during the Hipparcos and Gaia epochs, and
the star’s orbital motion contributed to the long-term mean motion vector
$\mu_{HG}$. To account for this effect, we modeled the orbital positions of
the star at each observation time of Hipparcos and Gaia DR3 and averaged these
positions to obtain the mean orbital positions at the two epochs.

Our PT-MCMC analysis used 100 walkers and stabilized in the mean and root mean
square (rms) of the posteriors of each of the model parameters after $1\times
10^{5}$ steps. We saved every 100th step of our chains and discarded the first
$75\%$ of the chain as the burn-in portion. Figure 7 shows the astrometric
models and data. Table 6 presents the posterior distributions of our fitted
parameters, as well as some derived parameters.

Figure 9 presents the joint posterior distributions for the orbital
inclination, mass and orbital period of HD 118203 c (see Appendix A.4 for
additional parameters). Our results show that HD 118203 c is a giant planet
with a mass of $m_{c}=11.9^{+0.69}_{-0.63}\ \mathrm{M_{J}}$ semi-major axis of
$a_{c}=5.36^{+0.09}_{-0.10}$ AU and eccentricity $e_{c}=0.26^{+0.03}_{-0.02}$.
We obtained the orbital inclination of the outer giant planet as
$I_{c}=93^{\circ}.7^{+11.9}_{-13.8}$, suggesting that HD 118203 c orbits the
star in an almost edge-on orbit. Our results are consistent with the values
reported from Maciejewski et al. (2024) within $1\sigma$.

Figure 9:  Panel a: sketch showing the orbital geometry for the mutual
inclination between HD 118203 b$\&$c. The observer is looking down from the
z-axis. The x-y plane is the plane of the sky, and the star is centered on the
origin. The orbital inclination of a planet is defined as the angle between
the planet orbital axis and line of sight direction (z-axis). The mutual
inclination is the angle between the orbital axes of the two planets. Panel b:
For a transiting planet with orbital inclination $I=90^{\circ}$ without
knowing the longitude of acceding node, its orbital axis could fall anywhere
along the sky plane (orange). For a planet with an orbital inclination $I\neq
90^{\circ}$ without knowing the longitude of acceding node, its orbital axis
falls on the surface of a cone (blue) with a vertex angle equals to the
orbital inclination (toward observer, $I<90^{\circ}$) or $\pi-$ orbital
inclination (away from observer, $I>90^{\circ}$). The dots $P_{1}$ and
$P^{\prime}_{1}$ represent two orbits that exhibit pro-grade and retrograde
degeneracy: they both lie within the same plane but orbit in opposite
directions. Figure 10: (a) Orbital inclination of HD 118203 c ($I_{c}$, blue)
, compared with that of HD 118203 b ($I_{b}$, red) and the line-of-sight
stellar inclination ($i_{*}$, orange). The zero degree corresponds to the line
of sight, same as z-axis in Figure 9. The stellar inclination ranges from
$0^{\circ}$ to $180^{\circ}$ to accounts for scenarios where the stellar axis
may point either toward ($<90^{\circ}$) or away ($>90^{\circ}$) from the
observer. (b): Distribution of HD 118203 b$\&$c mutual inclination projected
in line of sight direction. (c): Distribution of HD 118203 c obliquity
projected in line of sight direction.

###  5.4 Constraints on the planet-planet mutual inclination and stellar
obliquity for HD 118203 c

Figure 9 a$\&$b depict the fundamental orbital geometry illustrating the
mutual inclination between two planets, defined as the angle between the their
orbital axes. In the diagrams, the z-axis corresponds to the line of sight,
and is perpendicular to the x-y sky plane. The orientation of a planet’s orbit
is determined by two parameters: the orbital inclination $I$, which represents
the angle between a planet’s orbital axis and the line of sight (z-axis), and
the longitude of ascending node $\Omega$ defined as the angle of the line of
nodes from a reference direction (celestial north). In principle, the mutual
inclination between two planets can be calculated by measuring their
inclination and longitude of ascending node (Fabrycky & Winn, 2009):

$\cos{\Delta
I_{12}}=\cos{I_{1}}\cos{I_{2}}+\sin{I_{1}}\sin{I_{2}}\cos{(\Omega_{1}-\Omega_{2})}$
(8)

where $I_{i}$ and $\Omega_{i}$ is the inclination and longitude of ascending
node for planet $i$. Typically, using transit and radial velocity data, we can
infer the inclination of a transiting planet, but not its longitude of the
ascending node. In such scenarios, the orbital axis of a transiting planet
with an inclination close to $90^{\circ}$ might be oriented in any direction
within the sky plane. Similarly, for a non-transiting planet with an
inclination away from $90^{\circ}$, its orbital axis could lie on the surface
of a cone, which is either towards ($I_{2}<90^{\circ}$) or away from
($I_{2}>90^{\circ}$) the observer. Furthermore, when two orbits have an
inclination sum equal to $180^{\circ}$, as seen in $P_{1}$ and
$P^{\prime}_{1}$ in Figure 9b, their orbital axes align along the same line
but in opposite directions. This indicates that both orbits occupy the same
plane but the orbit directions are opposite. This phenomenon is known as the
retrograde-prograde degeneracy. Therefore, if we only have information about
the orbital inclinations of two planets and the longitude of ascending nodes
for one or both planets is unknown, we cannot determine the true mutual
inclinations. Instead, we can only derive the mutual inclinations as projected
along the line of sight.

Figure 10 a presents a comparison of the orbital inclinations of HD 118203 b,
c and stellar inclination using polar coordinates. By calculating the absolute
difference between $I_{b}$ and $I_{c}$, we can obtain their mutual inclination
in the line of sight direction, as shown in Figure 10 b. The resulting
distribution is compressed towards zero, yielding a line of sight $\Delta
I_{bc}$ as $9^{\circ}.8^{+8.6}_{-6.7}$ at $1\sigma$ and
$9^{\circ}.8^{+16.2}_{-9.3}$ at $2\sigma$. The peak of this distribution,
where the probability is highest, is at $1.5^{\circ}$. The measurement is
consistent with the scenario of a low mutual inclination between HD 118203 b
and c. Similarly, we can also constrain the stellar obliquity of HD 118203 c
in line of sight by calculating the absolute difference between stellar
inclination and $I_{c}$. Figure 10 c illustrates that the line of sight
$\Psi_{c}$ is $14^{\circ}.7^{+13.3}_{-10.2}$ at the $1\sigma$ confidence level
and $14^{\circ}.7^{+26.4}_{-13.9}$ at the $2\sigma$ level. The most probable
value is $4^{\circ}.5$. Our findings indicate that HD 118203 c is aligned with
the host star at the $2\sigma$ level. Therefore, the outer planet inclination
is similar to that of both the star and the hot Jupiter. While the line of
sight measurements of $\Delta I_{bc}$ and $\Psi_{c}$ only provide minimum
values because the angles in the sky plane are not determined, the similar
inclinations observed suggest a strong possibility of alignment across the
entire system. For instance, if the $\cos{i}$ values for the inclinations of
planets b and c and the stellar inclination were to be randomly selected from
a uniform distribution between -1 and 1, the likelihood that all of these
bodies would have inclinations within the range of $70^{\circ}$to
$110^{\circ}$ is merely $3.4\%$.

In summary, our analysis reveals that the inner hot Jupiter, HD 118203 b, is
nearly aligned with the host star’s spin axis. Our measurements are also
consistent with the outer giant planet, HD 118203 c, exhibiting a low mutual
inclination relative to both planet b and the host star spin axis.

##  6 Formation Scenarios for HD 118203 b

###  6.1 Disk-driven migration

The disk-driven migration scenario (e.g., Lin et al., 1996; Ida & Lin, 2008)
cannot explain the moderate eccentricity of HD 118203 b
($e_{b}=0.3136\pm{0.0075}$), because disk-driven migration would keep the
planet’s eccentricity low while shrinking its semi-major axis. Duffell &
Chiang (2015) finds that the eccentricity excitation through planet-disk
interaction is limited to small values (e $<0.1$) as the planet tends to
undergo collisions with the gap edge that dampen its eccentricity.

While the outer giant planet could potentially increase the eccentricity of
the hot Jupiter following disk-driven migration, the hot Jupiter would
experience apsidal precession due to general relativity and tidal forces at
its close-in orbit. This precession could counteract the gravitational
perturbations induced by the outer giant planet. Fabrycky (2010) present an
approximate criterion for the precession caused by the outer planet to
overcome the general-relativistic precession utilizing the masses and orbital
periods of both the inner and outer planets:

$\left(\frac{m_{out}}{M_{J}}\right)\left(\frac{37\
\rm{day}}{P_{out}}\right)^{2}\geq\left(\frac{M_{\odot}}{M_{*}}\right)^{5/3}\left(\frac{3\
\rm{day}}{P_{in}}\right)^{8/3}$ (9)

With a mass of $m_{c}=11.9^{+0.69}_{-0.63}\ \mathrm{M_{J}}$ and an orbital
period of $\sim 5103$ days, HD 118203 c does not meet the threshold required
to excite the eccentricity of HD 118203 b in its current close-in orbit.

Figure 11: Top: Mass versus semi-major axis of detected companions to hot
Jupiters. Red circles are stellar companions from Ngo et al. (2016), and
orange circles are giant planet companions from Knutson et al. (2014); Bryan
et al. (2016); Rosenthal et al. (2021). The orange line represents the limit
for a companion to overcoming GR precession at a close-in orbit of 6.13 days.
The blue lines represent the limit for a companion overcoming GR precession at
0.3 AU, 1 AU and 4 AU, respectively. Bottom: Eccentricity versus semi-major
axis of confirmed planets with mass larger tha $0.8\mathrm{M_{J}}$ within 1
AU, based on data from the NASA Exoplanet Archive as of June 2024. The orange
region indicates the trajectory associated with high-eccentricity tidal
migration, where tidal dissipation within the planet reduces its orbital
energy while keeping constant angular momentum. The final semi-major axis
range is set between 0.02 and 0.1 AU, representing the upper and lower limits.

###  6.2 Traditional high-eccentricity tidal migration vs. Coplanar high-
eccentricity tidal migration

The hierarchical star-planet-planet configuration in HD 118203 system fits the
high-eccentricity tidal migration scenario for explaining the origin of the
hot Jupiter. Tidal migration involves two steps: first, diminishing the
orbital angular momentum of the proto-hot Jupiter by exciting it into a highly
eccentric orbit, and second, tidally dissipating the orbital energy of the
planet through the interaction with the host star. We will examine how our
study of HD 118203 system sheds light on each of these steps below.

####  6.2.1 Eccentricity excitation

In the first step, gravitational interactions with a third body extract
orbital angular momentum from the proto-hot Jupiter and trigger the excitation
of its orbital eccentricity, although the specific mechanism responsible for
this process is still unknown.

Secular perturbation is a plausible mechanism in which a proto-hot Jupiter
swaps angular momentum with a third object gradually, on timescales much
longer than the orbital periods. Unlike short-term perturbations causing
noticeable changes in an orbit, secular perturbations manifest over a
timescale of many orbits, often affecting the orbit’s shape and orientation
gradually and cumulatively. Von-Zeipel-Kozai-Lidov cycles (von Zeipel, 1910;
Kozai, 1962; Lidov, 1962) is a specific type of secular perturbation that can
occur in hierarchical triple systems. Traditionally, the von Zeipel-Kozai-
Lidov mechanism describes the evolution of the inner orbit under secular
perturbation from a circular outer orbit, which can be analyzed using a
perturbing function truncated to quadrupole order in the semi-major axis ratio
($a_{1}/a_{2}$). This approximation provides a well-defined analytical
solution for the periodic exchange between the orbital eccentricity and
inclination of the inner orbit. For von-Zeipel-Kozai-Lidov cycles to occur,
there must be an initial misalignment of at least $40^{\circ}$ between the two
objects.

However, our findings that the outer planet has an eccentricity of
$e_{c}=0.26$, imply that we need to take into account the effects of the non-
circular outer orbit. Naoz et al. (2011) explored a more generalized mechanism
in which the outer orbit can be eccentric, called the eccentric Kozai-Lidov
(EKL) mechanism. They found that considering an eccentric outer orbit requires
an expansion of the perturbing function to octupole order, which is
proportional to $(a_{1}/a_{2})^{3}$. In the octupole level of approximation,
EKL mechanism can drive the eccentricity of the inner orbit to extremely high
levels, and can even induce a complete reversal in the orientation of the
inner orbit, flipping it from prograde to retrograde. The EKL mechanism is
also applicable across a broader range of parameters, including scenarios with
low initial mutual inclinations between the two objects. Li et al. (2014)
demonstrated that when the two planets have eccentric orbits, the inner
planet’s eccentricity can grow to large values even when the orbits are
initially close to being coplanar. Petrovich (2015) investigated the
consequence of this eccentricity excitation in the context of tides and found
that subsequent high-eccentricity migration kept the two planets nearly
coplanar. According to their study, hot Jupiters formed via this coplanar
high-eccentricity migration (CHEM) are likely to have distant ( $a\gtrsim
5\rm{AU}$) and massive (three times more massive than the hot Jupiter)
companions with relatively low mutual inclinations ( $\leq 20^{\circ}$) and
moderate eccentricities (0.2-0.5). Our measurements for HD 118203 c, with
properties of $a_{c}=5.36\pm{0.09}\ \rm{AU}$, $m_{c}=11.9^{+0.69}_{-0.63}\
M_{J}$, and $e_{c}=0.26^{+0.03}_{-0.02}$, align well with these predictions.
Furthermore, our analysis indicates that $\approx 95\%$ values from the
posterior distribution of the line-of-sight $\Delta I_{bc}$ are below
$25^{\circ}$, consistent with the outer giant planet having a low mutual
inclination relative to the inner hot Jupiter.

In reality, there are additional effects beyond secular perturbations from the
outer companion that can influence the evolution of inner orbit. For example,
general relativity (GR) effect from the host star could induce apsidal
precession of the inner orbit to the opposite direction than that induced by
the outer giant planet. If GR precession takes place at a faster rate, either
quadrupole-level von Zeipel-Kozai-Lidov cycles or octupole-level EKL would be
suppressed. Therefore, we need to compare the apsidal precession rate induced
by GR effect and the precession rate induced by the outer giant planet. Here,
we used the well-determined analytic solution for the quadrupole-level von
Zeipel-Kozai-Lidov cycles (Naoz et al., 2013) to provide a rough estimate of
the relevant time-scales.

$\tau_{\rm{quad}}=P_{b}\frac{m_{b}+M_{\star}}{m_{c}}(\frac{a_{c}}{a_{b}})^{3}(1-e_{c})^{3/2}$
(10)

In the other hand, the precession rate due to GR is given by Fabrycky &
Tremaine (2007),

$\tau_{\rm{GR}}=\frac{a_{b}c^{2}(1-e_{b}^{2})}{3G(M_{\star}+m_{b})}P_{b}$ (11)

where $c$ is the speed of light, and $G$ is the gravitational constant. In the
top panel of Figure 11, HD 118203 c is located to the right of the orange
line, indicating the general relativistic (GR) precession at the current
close-in orbit of HD 118203 b is significantly faster than the precession
induced by the distant secular perturber. Therefore, the hot Jupiter has
decoupled from the outer perturber as tides shrank its semi-major axis to
current close-in orbit. However, the companions to the left of blue lines are
massive and nearby enough to counter the GR effect at the initial orbital
distances of the proto-hot Jupiters. The exact boundary depends on the initial
orbital distances of the proto-hot Jupiter. Our results suggest that HD 118203
c is able of triggering the high-eccentricity migration as long as the proto-
hot Jupiter starts from a position beyond 0.3 AU.

Planet-planet scattering is another mechanism to create high eccentricity. In
systems where multiple giant planets are tightly packed, their gravitational
interactions can destabilize their orbits, causing them to cross paths and
have close encounters (Chatterjee et al., 2008). These encounters can lead to
a range of outcomes, including the ejection of one or more planets from the
system, collisions between planets or with the central star, and significant
alterations in the eccentricity and inclination of planetary orbits. The last
scenario could produce planets with highly eccentric orbits that can
subsequently undergo tidal orbital circularization. Numerical simulations have
found that planet-planet scattering would also alter the orientation of planet
orbits and increase their orbital inclination (Chatterjee et al., 2011; Li,
2021). Possibly, the HD 118203 system initially contained more than two giant
planets. A series of close encounters among these planets may have led to the
ejection of at least one planet, and the eccentricity excitation of HD 118203
b followed by tidal migration. However, it is highly unlikely that HD 118203 b
reached its close-in orbit solely through planet-planet scattering, as the
system’s total orbital energy is conserved. For instance, a Jupiter-mass
planet would need to eject 100 other planets of similar mass to shrink its
semi-major axis by a factor of 100 (Dawson & Johnson, 2018).

Figure 12: Eccentricity vs. stellar obliquity of HD 118203 b in comparison to
other hot Jupiters ($P<10\ \rm{days}$) from Rice et al. (2022). HD 118203 b is
one of few systems where the mutual inclinations between the hot Jupiters and
outer planets have been directly measured. The colors denote the effective
temperature of the host star.

####  6.2.2 Tidal dissipation and realignment

Once the proto-hot Jupiter’s orbit becomes sufficiently eccentric, tidal
dissipation in the planet initiates its inward migration. Specifically, during
each close approach to the host star, tides raised on the planet by the star
convert a small fraction of the planet’s orbital energy into heat, resulting
in the gradual shrinking and circularization of the planet’s orbit. During the
tidal dissipation process, the planet’s angular momentum remains constant.
Therefore, we can reverse the trajectory of the hot Jupiter’s semi-major axis
$a(t)$ and eccentricity $e(t)$ over time as
$a_{\rm{final}}=a(t)[1-e(t)^{2}]=\rm{constant}$, where $a_{\rm{final}}$ is the
final semi-major axis of the hot Jupiter when it’s fully circularized (Dawson
et al., 2014). The bottom panel of Figure 11 shows that HD 118203 b may have
experienced high-eccentricity tidal migration along constant angular momentum
tracks. Given its current semi-major axis of $a=0.07$ AU and eccentricity of
$e=0.32$, assuming an initial eccentricity of 0.99 would imply an upper limit
for the initial semi-major axis of 3.2 AU. In Section 6.2.1, we have shown
that the initial semi-major axis needs to exceed 0.3 AU for the outer
companion to overcome GR effects. Therefore, the initial position of the
proto-hot Jupiter could range from 0.3 to 3.2 AU.

Upon reaching its close-in orbit, the hot Jupiter might have been misaligned
relative to the host star due to tilting throughout the tidal migration.
However, previous studies have found that hot Jupiters orbiting cool stars
with temperatures below the Kraft break (6250 K, Kraft 1967) tend to be
aligned with their host stars, while hot stars show a range of obliquities. To
interpret this, Winn et al. (2010a) proposed that cool stars realign with the
hot Jupiters’ orbits due to tidal dissipation in their convective zones while
hot stars cannot realign because of their thinner convective zones. Lai (2012)
suggested that in the presence of a convective envelope, the dissipation of
inertial waves would cause efficient spin–orbit alignment for cool star
systems. Recently, Zanazzi et al. (2024) suggested that the gravity mode
pulsations in the radiative interior of cool stars can lock the hot Jupiters’
orbits and efficiently damp the system’s obliquity. Our analysis reveals that
HD 118203 b has a low stellar obliquity of $<33^{\circ}.5\ (95\%)$, indicating
the hot Jupiter orbits its host star in a nearly aligned orbit. This alignment
is consistent with the above trend as HD 118203 is a cool star
($T_{\rm{eff}}=5742^{+93}_{-95}\ K$). The stellar evolutionary models for HD
118203 show that it is at the sub-giant evolutionary stage and its effective
temperature has remained below the Kraft Break (6250K) throughout its entire
life (see Appendix A.5). It is possible that the orbit of HD 110233 b has
undergone realignment process. As a sub-giant system, our result is also
consistent with those of Saunders et al. (2024), who report on the aligned
orbits of three hot Jupiters orbiting sub-giant stars and suggest an increased
efficiency in obliquity damping after a star leaves the main sequence.

On the other hand, the orbital eccentricity of HD 118203 b is currently 0.32.
This suggests that if obliquity damping did occur, it was faster than
eccentricity damping. Based on the analytical model from Lai (2012), the
timescale of obliquity damping depends on tidal quality factor of the star
$Q_{*}$, which ranges over multiple orders of magnitude for sub-giants. With
an age of $4.73\ \mathrm{Gyr}$ determined by our isochrone model in Section 3,
we assume an upper limit for the timescale of obliquity damping
($\tau_{\lambda}<4.73\ \mathrm{Gyr}$) and constrain the tidal quality factor
of the star $Q_{*}<2.5\times 10^{5}$. On the other hand, we constrain a lower
limit for the timescale of eccentricity damping ($\tau_{e}>4.73\
\mathrm{Gyr}$). If we only consider the eccentricity damping driven by tidal
dissipation within the planet, the timescale $\tau_{e}$ could be derived with
the planet’s effective tidal quality factor $Q_{p}$, planet’s love number
$k_{2}$, star mass $M_{*}$ and the orbital parameters of the hot
Jupiter(Murray & Dermott, 1999; Rice et al., 2022). If we adopted a
$k_{2}=0.565$ and used the known stellar and planetary parameters, we can
obtain a constraint on the planet’s tidal quality factor $Q_{p}>1.1\times
10^{5}$.

Alternatively, the coplanar high-eccentricity migration provides a plausible
explanation for the aligned yet eccentric orbits of HD 118203 b. A hot Jupiter
undergoing coplanar high-eccentricity migration would maintain a low stellar
obliquity throughout the entire process. Recent observations of warm Jupiters
that have high eccentricities and aligned orbits support this scenario
(Espinoza-Retamal et al., 2023; Hu et al., 2024; Rubenzahl et al., 2024).

###  6.3 Comparison to other systems

Figure 12 compares the dynamical architecture of HD 118203 b to other
exoplanet systems with hot Jupiters ($P<10\ \rm{days}$). We can see that HD
118203 b exhibits an eccentric but aligned orbit, a trait shared by many other
hot Jupiters. This raises an intriguing question: Why are the orbits of these
planets aligned with their host star yet not fully circularized? One
possibility is that they experience obliquity damping more rapidly than
eccentricity damping. Alternatively, these planets might have a low stellar
obliquity upon reaching their current close-in orbits, potentially resulting
from a formation pathway such as coplanar high-eccentricity migration. Future
Gaia Data Release 4 is expected to offer high-precision epoch astrometry,
which will improve our ability to constrain the mutual inclination between hot
Jupiters and their distant giant companions and thereby investigate their
formation.

Figure 12 reveals that most hot Jupiters with eccentricity, either with low or
large obliquity, usually orbit around cool stars. The lack of eccentric hot
Jupiters around hot stars could be a result of detection bias because hot
stars typically rotate rapidly, making it challenging to obtain reliable RVs.
However, if the trend is true, that might suggest that the hot stars would
facilitate the circularization of the hot Jupiter orbits.

Although dozens of hot Jupiters have been discovered with outer giant planets,
direct measurements of mutual inclination between these planets are still
scarce. Here, we compare HD 118203 with similar systems that have been
characterized in three dimensions.

  * •

HAT-P 11 b is a transiting planet with a mass of $0.09\ M_{J}$ and an orbital
period of 4.88 days. While HAT-P 11 b doesn’t strictly fall into the category
of hot Jupiters with a mass below $0.1\ M_{J}$, its characteristics suggest it
likely resulted from high-eccentricity migration and serves as a good
comparison to HD 118203. Unlike the aligned orbit of HD 118203 b, HAT-P-11 b
has a large sky-projected stellar obliquity ($\sim 100^{\circ}$, Winn et al.
2010b; Hirano et al. 2011) although its host star has an effective temperature
below Kraft break. Yee et al. (2018) discovered a outer giant planet at $\sim
4AU$ and with a eccentricity of 0.6. Xuan & Wyatt (2020) combined RVs and
Hipparcos-Gaia astrometric acceleration to determine the mutual inclination
between HAT-P-11 b and the outer giant planet to be at least $50^{\circ}$.
This is consistent with previous studies that the nodal precession driven by
an inclined outer companions could cause the spin-orbit misalignment of inner
planets (Huber et al., 2013; Pu & Lai, 2019; Zhang et al., 2021). The N-body
simulation shows that HAT-P-11 b has undergone nodal precession driven by the
outer giant planet HAT-P-11 c, which surpassed the tidal realignment forces
exerted by the star (Yee et al., 2018; Xuan & Wyatt, 2020). To interpret the
difference between HD 118203 with HAT-P 11, we have estimated the nodal
precession rate of HD 118203 b driven by HD 118203 c using the analytical
model from Lai et al. (2018). It shows that HD 118203 c is not massive or
nearby enough to overcome the tidal realignment forces exerted by the star,
potentially explaining why we see HD 118203 b aligns with the host star, but
HAT-P-11 b does not.

  * •

Kepler-419 b is eccentric warm Jupiter with a mass of $2.5\ M_{J}$ and
eccentricity of 0.83 at 0.37 AU. Dawson et al. (2014) used RVs and TTV
observations to discover a non-transiting Kepler-419 c at 1.68 AU wth a mass
of $7.3\ M_{J}$. They also constrained that mutual inclination between the two
planets as low as $\sim 9^{\circ}$ from TTV observations. The system serves as
an example of undergoing co-planar high eccentricity migration.

  * •

WASP-148 b is a hot Jupiter with a nearby warm Jupiter companion (Hébrard et
al., 2020). The TTV observation show that the mutual inclnation between the
two planets to be below $35^{\circ}$.

##  7 Conclusions

We measured the true stellar obliquity of a hot Jupiter HD 118203 b and
presented the discovery of a long-period giant planet HD 118203 c in the same
system. HD118203 is the first system for which the true obliquity of the host
star relative to the hot Jupiter, and the mutual inclination between the hot
Jupiter and an outer giant planet, have both been determined. Our main
conclusions are as follows:

  * •

HD 118203 is a sub-giant ($M_{*}=1.27^{+0.028}_{-0.032}\ M_{\odot}$,
$R_{*}=2.04\pm{0.03}\ R_{\odot}$, $T_{\rm{eff}}=5742^{+93}_{-95}\ K$) hosting
a hot Jupiter HD 118203 b with a moderate eccentricity ($P_{b}=6.135$ days,
$e_{b}=0.31\pm{0.007}$, $m_{b}=2.14\pm{0.12}\ M_{\rm{J}}$,
$r_{b}=1.14\pm{0.029}\ R_{\rm{J}}$). We detected stellar oscillations using
KPF RVs and measured the frequency at maximum power to be
$\mu_{\rm{max}}=967.2^{+51.7}_{-46.5}\ \mu Hz$, which confirms that HD 118203
is a sub-giant.

  * •

We measured the sky-projected stellar obliquity of HD 118203 b to be
$\lambda_{b}=-11.7^{+7.6}_{-10}$ deg through RM observations using KPF. We
measured the stellar rotational period as $23\pm{4}$ days using TESS light
curves and constrained the stellar inclination to be
$i_{*}=89^{\circ}.9^{+13.7}_{-13.8}$. We obtained the true stellar obliquity
of HD 118203 b as $\Psi_{b}<33^{\circ}.5\ (2\sigma)$. Our results reveal that
HD 118203 is nearly aligned with the stellar spin axis.

  * •

HD 118203 c is a long-period giant planet ($m_{c}=11.9^{+0.69}_{-0.63}\
\mathrm{M_{J}}$, $a_{c}=5.36^{+0.09}_{-0.10}$ AU,
$e_{c}=0.26^{+0.03}_{-0.02}$) with moderate eccentricity outside the hot
Jupiter. We constrained the orbital inclination of HD 118203 c
($I_{c}=93^{\circ}.7^{+11.9}_{-13.8}$ ) by combing RV and Hipparcos-Gaia
astrometric accelerations. The line-of-sight mutual inclination between HD
118203 b $\&$ c is $9^{\circ}.8^{+8.6}_{-6.7}$ at $1\sigma$ and
$9^{\circ}.8^{+16.2}_{-9.3}$ at $2\sigma$ confidence level. The line-of-sight
stellar obliquity for HD 118203 c is $14^{\circ}.7^{+13.3}_{-10.2}$ at the
$1\sigma$ confidence level and $14^{\circ}.7^{+26.4}_{-13.9}$ at the $2\sigma$
level confidence level. Our measurements demonstrate that the orbit of HD
118203 c is most likely aligned with both the host star spin axis and the
orbit of the inner hot Jupiter.

  * •

The architecture of the HD 118203 system (star, hot Jupiter, distant giant
planet) is compatible with the high-eccentricity tidal migration scenario as a
possible explanation for the origin of the hot Jupiter. The observed low
mutual inclination between HD 118203 b and c, along with the moderate
eccentricity of HD 118203 c, suggests that the system underwent eccentric
Kozai-Lidov cycles (EKL) or another coplanar high eccentricity migration
scenario.

  * •

The low stellar obliquity of HD 118203 b indicates that either the obliquity
has always been low, for example through co-planar high-eccentricity
migration, or that tidal dissipation in the host star has realigned the
stellar spin and hot Jupiter’s orbit. If tidal obliquity damping did occur,
the moderate eccentricity of HD 118203 b suggests that obliquity damping was
more rapid than eccentricity damping.

  * •

Based on the constant angular momentum evolution during high-eccentricity
migration and the requirement of companion secular perturbation to counteract
general relativity effects from the host star, we constrain the initial
position of the proto-hot Jupiter to be between 0.3 and 3.2 AU.

The authors wish to recognize and acknowledge the very significant cultural
role and reverence that the summit of Mauna Kea has always had within the
indigenous Hawaiian community. We are most fortunate to have the opportunity
to conduct observations from this mountain. J.Z. would like to thank
Maximilian N. G$\ddot{u}$nther for helpful suggestion and advice regarding the
use of package allesfitter. J.Z. and D.H. acknowledge support from the Alfred
P. Sloan Foundation, the National Aeronautics and Space Administration
(80NSSC22K0303). D.H. also acknowledges support from the Alfred P. Sloan
Foundation, and the Australian Research Council (FT200100871). J.Z. and L.M.W.
acknowledge support from NASA-Keck Key Stragetic Mission Support Grant No.
80NSSC19K1475. L.M.W. also acknowledges support from the NASA Exoplanet
Research Program through grant 80NSSC23K0269. This research was carried out,
in part, at the Jet Propulsion Laboratory and the California Institute of
Technology under a contract with the National Aeronautics and Space
Administration and funded through the President’s and Director’s Research &
Development Fund Program. This research has made use of the NASA Exoplanet
Archive and ExoFOP, which are operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program.

##  Appendix A Supplementary figures

Figure A.1: The Rossiter-McLaughlin (RM) model for HD 118203 b without GP for
oscillation, integrating a global fitting of photometric data, non-transit
radial velocity (RV) measurements, and RM data during transit. Panel a: TESS
SPOC light curves phase-folded to the best-fit orbital period of HD 118203 b.
The pink points present the TESS data binned every 15 minutes, and the red
line shows the best-fit transit model. Panel b: The out-of-transit RVs from
ELODIE (green) and HIRES (orange) phase-folded to the best-fit orbital period
of HD 118203 b. The red line depicts the best-fit RV model. Panel c: the in-
transit RVs from KPF (blue) and best-fit RM model (red). Panel d: The
residuals of the fitting. The residuals show periodic patterns that could come
from the oscillation of the star.  Figure A.2: Joint posterior distributions
for 24 parameters used in the Nested Sampling fitting of the RM effect without
oscillation. The values and histogram distributions of all parameters are
shown, along with 1 $\sigma$ uncertainties.

Figure A.3: Joint posterior distributions for 27 parameters used in the Nested
Sampling fitting of the RM effect with SHO GP to account for stellar
oscillation. The values and histogram distributions of all parameters are
shown, along with 1 $\sigma$ uncertainties.

Figure A.4: Joint posterior distributions for 18 parameters used in the MCMC
fitting for the 3D orbits of the outer planet HD 118203 c. The values and
histogram distributions of all parameters are shown, along with 1 $\sigma$
uncertainties.

Figure A.5: Left: the stellar evolution models fit to HD 118203, and trace
back to their ZAMS location. Right: the mass fraction of hydrogen, helium and
carbon in the core of the star from the evolution models.

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