#  TESS Giants Transiting Giants VI: Newly Discovered Hot Jupiters Provide
Evidence for Efficient Obliquity Damping After the Main Sequence

Nicholas Saunders NSF Graduate Research Fellow  Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Samuel K. Grunblatt Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Ashley Chontos Henry Norris Russell Fellow  Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Fei Dai Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Daniel Huber Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Jingwen Zhang (张婧雯) NASA FINESST Fellow  Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Guðmundur Stefánsson Anton Pannekoek Institute for Astronomy, 904 Science Park, University of Amsterdam, Amsterdam, 1098 XH  Jennifer L. van Saders Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Joshua N. Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Daniel Hey Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Andrew W. Howard Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Benjamin Fulton NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E. California Boulevard, Pasadena, CA 91125, USA  Howard Isaacson Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA  Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD, Australia  Corey Beard NASA FINESST Fellow  Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA  Steven Giacalone NSF Astronomy and Astrophysics Postdoctoral Fellow  Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Judah van Zandt Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA  Joseph M. Akana Murphey NSF Graduate Research Fellow  Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA  Malena Rice Department of Astronomy, Yale University, New Haven, CT 06511, USA  Sarah Blunt Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  CIERA, Northwestern University, Evanston IL 60201, USA  Emma Turtelboom Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA  Paul A. Dalba Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA  Jack Lubin Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA  Casey Brinkman Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Emma M. Louden Department of Astronomy, Yale University, New Haven, CT 06511, USA  Emma Page Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA  Cristilyn N. Watkins Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Karen A. Collins Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Chris Stockdale Hazelwood Observatory, Australia  Thiam-Guan Tan Perth Exoplanet Survey Telescope, Perth, Western Australia  Richard P. Schwarz Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Bob Massey Villa ’39 Observatory, Landers, CA 92285, USA  Steve B. Howell NASA Ames Research Center, Moffett Field, CA 94035, USA  Andrew Vanderburg Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  George R. Ricker  Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA  Jon M. Jenkins NASA Ames Research Center, Moffett Field, CA, 94035  Sara Seager Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA  Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA  Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA  Jessie L. Christiansen NASA Exoplanet Science Institute, IPAC, MS 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Tansu Daylan Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA  Ben Falk  Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Max Brodheim W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743  Steven R. Gibson Caltech Optical Observatories, Pasadena, CA, 91125, USA  Grant M. Hill W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743  Bradford Holden UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California at Santa Cruz, 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  Stephen Kaye  Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA  Russ R. Laher NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA  Kyle Lanclos W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743  Erik A. Petigura Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA  Arpita Roy Astrophysics & Space Institute, Schmidt Sciences, New York, NY, 10011, USA  Ryan A. Rubenzahl NSF Graduate Research Fellow  Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, 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  Martin M. Sirk Space Sciences Laboratory, University of California, 7 Gauss Way, Berkeley, CA 94720, USA  Christopher L. Smith  Space Sciences Laboratory, University of California, 7 Gauss Way, Berkeley, CA 94720, USA  Josh Walawender W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743  Sherry Yeh W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

###### Abstract

The degree of alignment between a star’s spin axis and the orbital plane of
its planets (the stellar obliquity) is related to interesting and poorly
understood processes that occur during planet formation and evolution. Hot
Jupiters orbiting hot stars ($\gtrsim$6250 K) display a wide range of
obliquities, while similar planets orbiting cool stars are preferentially
aligned. Tidal dissipation is expected to be more rapid in stars with thick
convective envelopes, potentially explaining this trend. Evolved stars provide
an opportunity to test the damping hypothesis, particularly stars that were
hot on the main sequence and have since cooled and developed deep convective
envelopes. We present the first systematic study of the obliquities of hot
Jupiters orbiting subgiants that recently developed convective envelopes using
Rossiter-McLaughlin observations. Our sample includes two newly discovered
systems in the Giants Transiting Giants Survey (TOI-6029 b, TOI-4379 b). We
find that the orbits of hot Jupiters orbiting subgiants that have cooled below
$\sim$6250 K are aligned or nearly aligned with the spin-axis of their host
stars, indicating rapid tidal realignment after the emergence of a stellar
convective envelope. We place an upper limit for the timescale of realignment
for hot Jupiters orbiting subgiants at $\sim$500 Myr. Comparison with a
simplified tidal evolution model shows that obliquity damping needs to be
$\sim$4 orders of magnitude more efficient than orbital period decay to damp
the obliquity without destroying the planet, which is consistent with recent
predictions for tidal dissipation from inertial waves excited by hot Jupiters
on misaligned orbits.

††software: Lightkurve (Lightkurve Collaboration et al., 2018), Astropy
(Astropy Collaboration et al., 2013; Price-Whelan et al., 2018; Astropy
Collaboration et al., 2022), Astroquery (Ginsburg et al., 2019)

ʻ ‘

##  1 Introduction

Giant planets on close-in orbits (hot Jupiters) have been observed in a wide
variety of stellar environments, but it remains unclear how they arrive at
such small orbital separations from their host stars (Chatterjee et al., 2008;
Lee & Chiang, 2017). Two popular scenarios are disk-driven migration and high-
eccentricity migration (Dawson & Johnson, 2018). In both theories, giant
planets form at larger semi-major axes, before losing angular momentum to
either the protoplanetary disk (in disk-driven migration; e.g. Lin et al.
1996) or to another planet or star (in high-eccentricity migration; e.g. Rasio
& Ford 1996). In the latter case, the planet ends up on an eccentric orbit
that gradually shrinks and circularizes as the eccentricity is damped by tidal
interactions with the star.

The stellar obliquity—the angle between the spin axis of the star and the
planet’s orbital axis—is a key piece of evidence to distinguish between these
pathways as each scenario produces a distinct distribution of spin-orbit
angles. Disk migration results in aligned populations at early times, while
high-eccentricity migration predicts early scattering events that might alter
the obliquity (Dawson & Johnson, 2018). Observations of hot Jupiters around
main-sequence stars have shown that the stellar obliquity distribution has a
strong dependence on the star’s effective temperature (Winn et al., 2010; Rice
et al., 2022), with hot hosts displaying a wide range of obliquities, and cool
hosts being preferentially aligned. The critical separation is approximately
at the Kraft break (Kraft, 1967), around 6250 K, corresponding to the break
between stars with primarily radiative envelopes ($>$6250 K) and stars with
thick convective envelopes ($<$6250 K). The distinct distributions of
obliquities separated by the Kraft Break suggest two possible scenarios:

  1. 1.

Hot Jupiters have unique formation pathways around hot and cool stars,
resulting in aligned orbits around cool stars and a broad range of obliquities
around hot stars.

  2. 2.

Hot Jupiters are formed by the same mechanism around hot and cool stars, and
the spin-orbit angle is realigned during the main sequence by tidal
dissipation in stars with convective envelopes.

The correlation between spin-orbit alignment and the presence of a convective
envelope has driven theories for realignment through the tidal damping of
obliquity (Anderson et al., 2021). During their main sequence lifetimes, low
mass stars ($\lesssim$1.2 M⊙) have surface convection zones that deepen as the
star cools and evolves into a subgiant. However, it is challenging to
constrain the timescales of damping mechanisms using a main sequence
population in part due to uncertain age estimates. More massive stars
$\gtrsim$1.2 M⊙) have radiative envelopes on the main sequence, which cool and
become convective as the stars drop below the Kraft break during the subgiant
or red giant branch phase. As these stars only develop substantial convective
envelopes after leaving the main sequence, they provide a unique avenue to
test the timescales of tidal damping. If convection zones efficiently damp
obliquities, as required by high-eccentricity migration, subgiants that were
formerly ’hot’ and had a broad range of obliquities should be found to have
low obliquities due to rapid obliquity damping.

Studies have suggested that the presence of a convective envelope drives
efficient damping of orbital period, eccentricity, and obliquity (Winn et al.,
2010; Lai, 2012; Rogers & Lin, 2013; Albrecht et al., 2021; Patel & Penev,
2022). Diminishing orbital period has been unambiguously observed in only a
small handful of systems (Maciejewski et al., 2016, 2018; Yee et al., 2020;
Vissapragada et al., 2022), though these measurements provide useful
benchmarks for the timescales and efficiencies of mechanisms for orbital
evolution. Orbital decay is driven by the transfer of orbital angular momentum
to stellar rotation (Hut, 1980, 1981). As an orbiting planet raises a tidal
bulge in the outer envelope of its host star, turbulent viscosity (Lai, 2012)
and the excitation of waves through the stellar interior (Rogers & Lin, 2013)
sap energy from the orbit. This orbital evolution can take place over human
timescales (Yee et al., 2020; Vissapragada et al., 2022), and the changing
orbital period can be observed.

The sky-projected obliquity can be constrained by obtaining high-resolution
spectra during a planet’s transit and modeling the Rossiter-McLaughlin (RM)
effect (Rossiter, 1924; McLaughlin, 1924). As a planet crosses in front of its
host star, it blocks regions of the stellar surface that are red-shifted or
blue-shifted relative to the observer due to the stellar rotation. These red-
and blue-shifts imprint an additional signal onto the traditional radial
velocity curve caused by the planet’s orbit, the morphology and amplitude of
which depend on the relative alignment of the planet’s orbital path across the
rotating star, and can be used to constrain the sky-projected obliquity of the
system. The nearly all-sky coverage of NASA’s Transiting Exoplanet Survey
Satellite (TESS; Ricker et al. 2015) has enabled the discovery of a large
sample of hot Jupiters orbiting bright subgiants which are amenable to
measurements of the RM effect (Saunders et al., 2022; Grunblatt et al., 2022,
2023; Pereira et al., 2024), allowing us measure spin-orbit alignment and
understand late-stage planetary orbit evolution.

In this paper we present observations of the RM effect for three hot Jupiters
orbiting subgiants observed by TESS: TOI-1181 b ($R_{\rm p}=$ $1.519\pm 0.059$
$R_{\rm J}$, $P_{\rm orb}=$ $2.103189^{+0.000003}_{-0.000005}$ days,
$R_{\star}=1.89\pm 0.18R_{\odot}$, T${}_{\rm eff}=6106\pm 100$ K), TOI-6029 b
($R_{\rm p}=$ $1.284\pm 0.098$ $R_{\rm J}$, $P_{\rm orb}=$ $5.7987\pm 0.0004$
days, $R_{\star}=2.23\pm 0.05R_{\odot}$, T${}_{\rm eff}=6223\pm 100$ K), and
TOI-4379 b ($R_{\rm p}=$ $1.489\pm 0.079$ $R_{\rm J}$, $P_{\rm orb}=$
$3.25352\pm 0.00002$ days, $R_{\star}=1.82\pm 0.25R_{\odot}$, T${}_{\rm
eff}=6020\pm 100$ K). TOI-1181 b is a previously confirmed system (Kabáth et
al., 2022; Chontos et al., 2024), while TOI-6029 b and TOI-4379 b are newly-
identified planets confirmed by our work as part of the Giants Transiting
Giants Survey. We find that all three systems have a low sky-projected
obliquity ($|\lambda|<15^{\circ}$), indicating that tidal realignment is
efficient after the post-main-sequence emergence of a convective envelope. We
then test models for realignment, and find that enhanced efficiency of
obliquity damping after the main sequence can lead to realignment without
inspiral and engulfment.

##  2 Observations

###  2.1 TESS Photometry

TOI-6029 b and TOI-4379 b were discovered as part of our Giants Transiting
Giants survey (GTG; Saunders et al. 2022; Grunblatt et al. 2022, 2023; Pereira
et al. 2024). TESS data are available for TOI-6029 b in 30-minute cadence
Full-Frame Images (FFIs) for Sectors 17, 18, and 24, spanning from October 7,
2019 - May 13, 2020, and in 200-second cadence FFIs for Sector 58 from October
29 - November 26, 2022. It was flagged by our survey as a planet candidate in
2021, and was selected for TESS 2-minute cadence observations in Sector 58.
TESS data are available for TOI-4379 b in 30-minute FFIs in Sector 12 (April
22 - May 21, 2019), 10-minute cadence in Sector 39 (May 26 - June 24, 2021),
and 200-second cadence in Sector 66 (June 2 - July 1, 2023). We identified
TOI-4379 b as a planet candidate in 2021, and it received 2-minute cadence
observations in Sectors 39 and 66.

TOI-1181 b was first identified by the MIT Quick Look Pipeline (QLP; Huang et
al. 2020) in 2019, and TESS data are available in 30-minute cadence FFIs for
Sectors 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26, spanning from July
18, 2019 - July 4, 2020. TESS data are available in 10-minute cadence FFIs for
Sectors 40, 41, 47, 48, 49, 50, 51, 52, 53, 54, 55, spanning from June 24,
2021 - September 1, 2022. In 200-second cadence FFIs, it has so far been
observed in Sectors 56, 57, 58, 59, and 60, covering September 1, 2022 -
January 18, 2023. TOI-1181 b was scheduled for 2-minute cadence observations
beginning on April 16, 2020 with Sector 24, and was selected for 2-minute
cadence data through Sector 60. The planet was confirmed by Kabáth et al.
(2022), and was followed up as part of the TESS-Keck Survey (TKS; Chontos et
al. 2022, 2024).

All photometry data used are the Presearch Data Conditioning simple aperture
photometry (PDC_SAP; Smith et al. 2012; Stumpe et al. 2012, 2014) TESS
2-minute light curves produced by the Science Processing Operations Center
(SPOC; Jenkins et al. 2016a) at NASA Ames Research Center downloaded from the
Mikulski Archive for Space Telescopes (MAST).

We searched for additional transiting planets within these systems in the TESS
photometry by masking the known transits and performing a Box-fitting Least
Squares (BLS) periodogram analysis with the lightkurve implementation of the
astropy BLS tools. We did not identify additional transiting planet signals.

The Science Processing Operations Center (SPOC; Jenkins et al. 2016a)
performed multi-transiting planet search in the subsequent 2-min cadence
observations and detected the signatures of TOIs 1181 b, 6029 b and 4379 b
using a noise-compensating matched filter (Jenkins, 2002; Jenkins et al.,
2010, 2020). The transit signatures were fitted with an initial limb-darkened
transit model (Li et al., 2019) and were subjected to a number of diagnostic
tests (Twicken et al., 2018), including the difference image centroiding test,
which located the host stars within $0\arcsec.409$ $\pm$ $2\arcsec.6$,
$0\arcsec.78$ $\pm$ $2\arcsec.5$, and $2\arcsec.6$ $\pm$ $2\arcsec.6$. The
SPOC searches likewise failed to identify additional transiting planet
signals.

###  2.2 Radial Velocity Follow-up

High precision RVs were taken using the High Resolution Echelle Spectrograph
(HIRES) on the 10-meter Keck I telescope on Maunakea, Hawaiʻi (Vogt et al.,
1994). For TOI-1181 b, 87 RV measurements were taken between December 2, 2019
and May 16, 2022. For TOI-6029 b, 46 RV measurements were taken between
December 25, 2020 and October 6, 2022. For TOI-4379 b, 12 RV measurements were
taken between September 21, 2021 and July 16, 2022. RV observations were taken
in the HIRESr configuration with an iodine cell in the light path for
wavelength calibration. The C2 decker was used for all observations, providing
a spectral resolution of $R\approx 60,000$. RVs were reduced with the
California Planet Search (CPS; Howard et al. 2010) pipeline. Table
LABEL:tab:my_label in Appendix B contains all RV observations and
corresponding uncertainties used in this analysis.

###  2.3 Rossiter-McLaughlin Observations

In addition to RVs over the full phase of the planets’ orbits, we obtained RV
observations during each planet’s transit to measure the Rossiter-McLaughlin
(RM) effect. We obtained high resolution spectra during transit events with
HIRES for TOI-1181 b and TOI-6029 b, and with the Keck Planet Finder (KPF;
Gibson et al. 2016, 2018, 2020) for TOI-4379 b.

The HIRES observing configuration during the RM observations matches that
described in §2.2. We obtained 40 HIRES RV observations during the transit of
TOI-1181 b on September 8, 2020. The typical counts for each $\sim$10-minute
exposure were 20,000 e- with an average RV uncertainty of $\sim$3 m s-1. We
obtained full coverage of the transit of TOI-1181 b, with $\sim$1 hour before
ingress and $\sim$1.5 hours after egress to establish an RV baseline. For
TOI-6029 b, we obtained 34 HIRES RVs on September 18, 2022 with typical counts
of 5,000 e- over $\sim$13-minute exposures and average RV uncertainty of
$\sim$5 m s-1. Poor weather prevented observation of the full transit egress,
and we were unable to continue the observations after the transit event
concluded.

We obtained 30 observations of TOI-4379 b on June 6, 2023 using KPF, with
9-minute exposures. KPF is a fiber-fed spectrograph with a resolution of
$R\approx 98,000$. Observations were taken with simultaneous wavelength
calibration from an etalon lamp fed through a dedicated calibration fiber. In
addition to simultaneous calibrations, we took dedicated etalon calibration
exposures (so-called “slew-cals") each hour to monitor RV stability. Due to
its southerly declination ($-32^{\circ}$), the window to observe the transit
of TOI-4379 b was tight, and the star was not visible above the Keck telescope
dome during ingress. RVs were measured using two distinct approaches—first,
using a cross-correlation function (CCF) on a set of spectral lines produced
by the public KPF data reduction pipeline111https://github.com/Keck-
DataReductionPipelines/KPF-Pipeline/. This reduction method resulted in a
typical RV uncertainty of $\sim$7 m s-1. We also reduced the RVs by analyzing
the spectra using an adapted version of the SpEctrum Radial Velocity Analyser
code (SERVAL; Zechmeister et al. 2018), which measures shifts in the observed
spectra relative to a matched template spectrum, but found that the CCF
reduction resulted in RVs with higher precision and smaller scatter. The KPF
data reduction pipeline is being actively developed, and future improvements
to the measured RV precision are likely to be implemented.

###  2.4 Ground-based Imaging

We obtained high-contrast imaging of TOI-6029 and TOI-4379 to search for close
stellar companions which might have biased the measurement of the amplitude of
the transit signal observed by TESS. TOI-6029 was observed by the NIRC2 near-
infrared imager in the Kp bandpass on the 10-meter Keck II telescope on June
26, 2023.

We first pre-processed the data using the Vortex Imaging Processing (VIP)
software package (Gomez Gonzalez et al., 2017; Christiaens et al., 2023). We
performed flat-fielding, bad-pixel removal, and correct for geometric
distortions by applying the solution in Service et al. (2016). Since the data
were taken in vertical angle mode, we de-rotated each image according to the
parallactic angles and then stacked the individual images into a combined
image. To register the eight frames, we identify the star’s position by
fitting a 2D Gaussian to the stellar point spread function (PSF) in each
frame.

A companion to TOI-6029 was identified with a separation of $763\pm 20$
milliarcseconds at a position angle of $110.2\pm 0.5^{\circ}$. We measured a
difference in magnitude in the Ks bandpass of $\Delta m=4.44\pm 0.05$. The
reduced and derotated image can be found in Figure 13 in Appendix A.

After obtaining the pre-processed cubes, we extracted the astrometry of the
companion applying a least-squares minimization code. Specifically, we fit the
PSF of the companion using the unsaturated PSF of the primary star in the same
images (see details in Xuan et al. 2024).

TOI-4379 b was observed by the 8-meter Gemini South telescope on Cerro Pachon
using the speckle interferometric instrument Zorro (Scott et al., 2021; Howell
& Furlan, 2022) on May 18, 2022 UT. Three sets of 1000 $\times$ 60 ms images
were obtained simultaneously at 562 and 832 nm, the data was reduced using our
standard software pipeline (Howell et al., 2011) and the limiting magnitude
contrast was computed for each filter as a function of the angular separation
from TOI-4379. The 5-σ contrast curves reached depths of 5 to 6.5 magnitudes
in the two filters over the angular range of 0.1 to 1.2 arcsec. Within these
contrast levels and angular limits, no close companion was detected. The raw
and reduced data, including the contrast curves and reconstructed images at
each wavelength, are available on the EXOFOP-TESS
website222exofop.ipac.caltech.edu/tess/target.php?id=97766057.

###  2.5 Ground-based Photometry

The TESS pixel scale is $\sim 21\arcsec$ pixel-1, often causing multiple stars
to blend in the TESS photometric aperture. To confirm the true source of the
TESS detection, we acquired ground-based follow-up photometry of the fields
around TOI-6029 and TOI-4379 as part of the TESS Follow-up Observing Program
(TFOP; Collins, 2019)333tess.mit.edu/followup. We used the TESS Transit
Finder, which is a customized version of the Tapir software package (Jensen,
2013), to schedule our transit observations.

####  2.5.1 LCOGT

We observed an ingress window of TOI-6029 b in Sloan $i^{\prime}$ band on UTC
2023 August 03 from the Las Cumbres Observatory Global Telescope (LCOGT; Brown
et al., 2013) 1 m network node at McDonald Observatory near Fort Davis, Texas,
United States (McD). The 1 m telescopes are equipped with $4096\times 4096$
SINISTRO cameras having an image scale of $0\farcs 389$ per pixel, resulting
in a $26\arcmin\times 26\arcmin$ field of view. The images were calibrated by
the standard LCOGT BANZAI pipeline (McCully et al., 2018), and differential
photometric data were extracted using AstroImageJ (Collins et al., 2017). The
transit is detected within the $3\farcs 5$ follow-up photometric aperture
which excludes the flux from the nearest known neighbor in the Gaia DR3
catalog (Gaia DR3 430269136521931776) that is $\sim 17\arcsec$ north of
TOI-6029.

We also observed a full transit window of TOI-4379 b in Sloan $i^{\prime}$
band on May 15, 2022 (UTC) from the LCOGT 1 m network node at Cerro Tololo
Inter-American Observatory in Chile (CTIO). The transit is detected within the
$2\farcs 7$ follow-up photometric aperture, which excludes most of the flux
from the nearest known neighbor in the Gaia DR3 catalog (Gaia DR3
6027376203516633728), which is $\sim 3\arcsec$ west of TOI-4379.

####  2.5.2 PEST

We observed a full transit window of TOI-4379 b in Sloan $g^{\prime}$ band on
May 27, 2023 (UTC) using the Perth Exoplanet Survey Telescope (PEST), located
near Perth, Australia. The 0.3 m telescope is equipped with a $5544\times
3694$ QHY183M camera with an image scale of 0$\farcs$7 pixel-1, resulting in a
$32\arcmin\times 21\arcmin$ field of view. A custom pipeline based on
C-Munipack444c-munipack.sourceforge.net was used to calibrate the images and
extract the differential photometry. The transit is detected within the
$6\farcs 4$ follow-up photometric aperture, which includes the flux from the
nearest known neighbor in the Gaia DR3 catalog. Figure 14 shows the ground-
based transit observations of TOI-6029 b and TOI-4379 b with best-fit transit
models. All light curve data are available on the EXOFOP-TESS
website555exofop.ipac.caltech.edu/tess/target.php?id=419523962, but are not
included in the joint model of this work.

##  3 Host Star Characterization

###  3.1 High-resolution Spectroscopy

We obtained out-of-transit, iodine-free spectra with Keck/HIRES for each of
our host stars to extract precise stellar parameters. Stellar properties were
computed using SpecMatch empirical (SpecMatch-Emp; Yee et al. 2017) and
synthetic (SpecMatch-Synth; Petigura et al. 2015) methodologies. All of our
host stars have effective temperatures T${}_{\rm eff}>5000$ K, and we
therefore adopt values from SpecMatch-Synth. Our adopted values can be found
in Table 1.

###  3.2 Isochrone Fitting

We used the isoclassify666https://github.com/danxhuber/isoclassify Python
package (Huber, 2017; Berger et al., 2020) to compute additional stellar
properties. We ran the code in grid mode where we pass input observables taken
from SpecMatch-Synth and position and parallax distance from Gaia DR3 (Gaia
Collaboration et al., 2023) and K-band magnitude adopted from the TESS Input
Catalog (TIC; Stassun et al. 2019; Paegert et al. 2022 v8.2. isoclassify uses
the stellar model grid to compute stellar mass ($M_{\star}$), stellar radius
($R_{\star}$), surface gravity ($\log{g}$), distance ($d$), luminosity ($L$),
and stellar age. We used the Combined19 allsky dust map from mwdust (Bovy et
al., 2016).

We account for the companion to TOI-6029 described in §2.4 by running
isoclassify in binary mode, including the correction for the measured
companion magnitude. For the companion to TOI-6029, we obtain a mass of
$0.44\pm 0.02$ M⊙, radius of $0.45\pm 0.02$ R⊙, and temperature of $3528\pm
40$ K.

The isoclassify outputs and associated uncertainties can be found in Table 1.
All host stars are found to be evolved ($\sim$2 R⊙) stars with masses between
1.2 and 1.6 M⊙ and metallicities between 0.1 and 0.4 dex.

###  3.3 Asteroseismology

Finally, to improve the characterisation of the host stars, we searched for
signatures of stellar p-mode oscillations in the light curve. To do this, we
first remove transits and apply a simple high pass-filter to remove further
low-frequency variability. We use the PySYD pipeline (Chontos et al., 2021) to
search for variability in the host stars. We find no evidence of oscillations
in TOI-6029 and TOI-4379. In TOI-1181, however, we identify a weak signal
associated with the global oscillation power excess, commonly referred to as
$\nu_{\rm max}$, at 737$\pm$69 $\mu$Hz at an SNR of 4.55. This value is
slightly lower than the expected $\nu_{\rm max}$ of 930 $\mu$Hz based on
asteroseismic scaling relations (cf. Stello et al. 2009). As a result of this,
and non-detections in the other host stars, we do not use asteroseismology to
constrain the stellar parameters.

|  | TOI-1181 b | TOI-6029 b | TOI-4379 b | Source
---|---|---|---|---|---
TIC ID |  | 229510866 | 419523962 | 97766057 | a
RA |  | $19:48:51.8$ | $00:33:32.7$ | $16:48:11.49$ | a
dec |  | $+64:21:15.66$ | $+61:51:38.76$ | $-32:26:01.51$ | a
V Mag |  | $10.582\pm 0.006$ | $12.858\pm 0.038$ | $12.661\pm 0.092$ | a
Gaia Mag |  | $10.4776\pm 0.0003$ | $12.2681\pm 0.0002$ | $12.4083\pm 0.0002$ | a
TESS Mag |  | $10.079\pm 0.0086$ | $11.6593\pm 0.006$ | $11.8994\pm 0.0069$ | a
Teff | (K) | $6054\pm 81$ | $6223\pm 100$ | $6020\pm 100$ | b*
[Fe/H] | (dex) | $0.38\pm 0.06$ | $0.37\pm 0.06$ | $0.16\pm 0.06$ | b*
$v\sin{i_{\star}}$ |  (km s-1) | $10.6\pm 1.0$ | $10.4\pm 1.0$ | $5.9\pm 1.0$ | b*
M⋆ |  (M⊙) | $1.467\pm 0.024$ | $1.548^{+0.027}_{-0.085}$ | $1.372^{+0.037}_{-0.105}$ | c*
R⋆ |  (R⊙) | $1.961\pm 0.030$ | $2.338^{+0.027}_{-0.085}$ | $1.976\pm 0.040$ | c*
$\log{g}$ | (dex) | $4.02\pm 0.02$ | $3.886^{+0.016}_{-0.024}$ | $3.929^{+0.038}_{-0.023}$ | c*
Age | (Gyr) | $2.7\pm 0.2$ | $2.31^{+0.0.58}_{-0.19}$ | $4.51^{+0.40}_{-1.21}$ | c*
L |  (L⊙) | $4.39^{+0.23}_{-0.15}$ | $6.30^{+0.27}_{-0.30}$ | $4.34^{+0.28}_{-0.25}$ | c
$d$ | (pc) | $310.5^{+6.2}_{-4.8}$ | $596.7^{+11.5}_{-11.1}$ | $617.4^{+11.6}_{-11.9}$ | c
Table 1: Stellar properties derived for each of the host stars in our sample.
Sources: (a) TESS input catalog (Stassun et al., 2019), (b) SpecMatch-Synth
(this work; Petigura et al. 2015), (c) isoclassify (this work; Huber 2017;
Berger et al. 2020). *Stellar properties for TOI-1181 adopted from Chontos et
al. (in prep).

##  4 Planet Modeling

###  4.1 Joint Transit and RV Fit

We used the allesfitter Python package (Günther & Daylan, 2021, 2019) to
simultaneously model the planet transit, radial velocity curve, and RM effect.
All photometry data used are TESS 2-minute cadence observations reduced by the
Science Processing Operations Center (SPOC; Jenkins et al. 2016b) pipeline. We
used the 2-minute cadence data as it provides the best constraint on the
ingress and egress times, which are important for characterizing the shape of
the Rossiter-McLaughlin signal. These data are also reduced with a uniform
approach which accounts for contamination by nearby stars, providing the most
accurate constraint on the transit depth and planet radius. To enable longer
sampling within reasonable computation time ($\lesssim 24$ hours), we select
only one Sector of 2-minute cadence data for each target (Sectors 73, 58, 66
for TOI-1181 b, TOI-6029 b, and TOI-4379 b, respectively). Including
additional Sectors did not significantly impact the measured transit depth,
period, or epoch.

We fit for orbital period, $P$, transit epoch, $t_{0}$, ratio of planetary to
stellar radius, $R_{p}/R_{\star}$, cosine of orbital inclination, $\cos{i}$,
RV semi-amplitude, $K$, sky-projected spin-orbit angle, $\lambda$, and the
eccentricity and argument of periastron, parameterized by the expressions
$\sqrt{e}\cos{\omega}$ and $\sqrt{e}\sin{\omega}$. For each instrument, we
also fit for two quadratic limb darkening coefficients, described by
$q_{1,{\rm TESS}}$, $q_{2,{\rm TESS}}$, $q_{1,{\rm HIRES}}$, $q_{2,{\rm
HIRES}}$, $q_{1,{\rm KPF}}$, and $q_{2,{\rm KPF}}$. We include an offset
parameter for the RV instruments, which captures small global offsets for all
RVs taken by a given instrument.

For each target, we tested (a) constant instrumental offset shared between all
observations for a given instrument and (b) instrumental offset with a linear
slope to capture long term trends in the RV signal. The linear offset improved
the fit by reducing structure in the residuals for TOI-6029 b, while the fits
to TOI-1181 b and TOI-4379 b preferred the constant intrumental offset. We
predicted the RV signal due to the resolved companion for TOI-6029 b to test
whether it could cause the observed trend, assuming the companion was observed
at maximum separation on a circular orbit. We found that the maximum slope
caused by a companion of mass M$=0.44$ M⊙ at the observed angular separation
is $0.001$ m s-1 day-1, a factor of 10 smaller than the observed slope of
$\sim 0.01$ m s-1 day-1. We conclude that the outer companion is unlikely to
be the source of the RV slope and may indicate the presence of an unresolved
stellar or planetary-mass companion. Further monitoring of RV and astrometry
for this system is required to determine the source of the RV trend.

We also included the RM effect in our allesfitter model (see Günther & Daylan
2019 for details). The amplitude of the RM effect depends on the ratio of the
planet’s radius ($R_{\rm p}$) to the star’s radius ($R_{\star}$), the
projected rotation velocity of the star ($v\sin{i}$), and impact parameter
($b$), given by Triaud (2018) as:

$A_{\rm RM}\approx\frac{2}{3}\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}v\sin{i}\sqrt{1-b^{2}}.$

As stars evolve off of the main sequence, they become larger, more luminous,
and slower in their rotation. These changes have three primary effects that
make RM observations challenging—first, the larger radius causes the transit
duration to increase, limiting available observation windows; second, the
increased stellar radius also causes the ratio $R_{\rm p}/R_{\star}$ to
decrease, reducing the RM amplitude; finally, the slowed stellar rotation
($v\sin{i_{\star}}$) also reduces the RM amplitude. Additionally, granulation
for subgiants can introduce RV jitter with an amplitude of $\sim$5 m s-1 on
timescales of hours (Tayar et al., 2019).

By modeling the RM signal jointly with the photometric and RV observations,
the constraints on the obliquity are strengthened. The RVs provide additional
information about the out-of-transit RV baseline, and the deviation from that
baseline informs the spin-orbit angle. The photometric transit gives a
constraint on the planet radius, which plays a strong role in the amplitude of
the RM signal. In turn, the amplitude of the RM signal sheds additional light
on the planet radius. Because we have an independent constraint on the planet
radius, the projected stellar rotation velocity, $v\sin{i_{\star}}$, is also
well-constrained by the amplitude of the RM signal.

All priors and inferred values for our model parameters can be found in Table
2. For each planet, we additionally derive planet mass, $M_{p}$, planet
radius, $R_{p}$, and eccentricity, $e$ from the outputs of our model. Figures
1, 2, and 3 show the transit, RV, and RM fits, respectively. For TOI-6029 b, a
significant long-term RV trend was detected, shown in Figure 4.

For each target, we sample the posterior distribution using affine-invariant
Markov Chain Monte Carlo (MCMC). We initiate 100 walkers with 50,000 sampling
steps and 5,000 burn-in steps per walker, ensuring that the chains ran at
least 30 times the autocorrelation length and successfully converged. The
residual scatter in the RVs is consistent with what we would expect for a
subgiant ($10-15$ m s-1), and there is no remaining structure in the residual
TESS photometry.

Figure 1: Top: Normalized TESS photometry for each of our systems, phase-folded to the orbital periods of the planets. Raw TESS photometry is marked by gray points, and binned photometry is marked by blue points. The median of the posterior transit model is shown by the black line. Bottom: Residuals between observations and the best-fit transit model. Figure 2: Top: Radial velocity measurements for each of our systems, phase-folded to the orbital periods of the planets. Keck/HIRES observations are shown as red points and Keck/KPF observations are marked by blue points. The black line shows the best-fit orbital model to the system, including the Rossiter-McLaughlin signal. Bottom: Residuals between observations and the best-fit RV model. Figure 3: Top: Observations of the Rossiter-McLaughlin effect for each of our targets. Data from Keck/HIRES and Keck/KPF are shown as red and blue points, respectively, and the line shows our model fit. Bottom: Residuals between observations and the best-fit RM model. Figure 4: Top: RV observations for TOI-6029 b, shown in red, with the full RV model shown in black. Bottom: Residuals after removing the RV contribution from the transiting planet. |  | TOI-1181 b | TOI-6029 b | TOI-4379 b
---|---|---|---|---
Fitted parameters: |  |
$P$ | (days) | $2.103189^{+0.000003}_{-0.000005}$ | $5.7987\pm 0.0004$ | $3.25352\pm 0.00002$
$t_{0}$ | (BJD - 2457000) | $2481.51696^{+0.00055}_{-0.00057}$ | $2667.067^{+0.015}_{-0.017}$ | $2734.269^{+0.048}_{-0.003}$
$R_{\rm p}/R_{\star}$ |  | $0.07947^{+0.00042}_{-0.00045}$ | $0.056^{+0.00074}_{-0.00074}$ | $0.077^{+0.002}_{-0.002}$
$a/R_{\star}$ |  | $3.976^{+0.078}_{-0.049}$ | $7.17^{+0.39}_{-0.31}$ | $4.63^{+0.56}_{-0.21}$
$\cos{i}$ |  | $0.018^{+0.028}_{-0.013}$ | $0.023^{+0.020}_{-0.015}$ | $0.041^{+0.049}_{-0.029}$
$K$ |  (m s-1) | $146\pm 2$ | $137^{+9}_{-10}$ | $119^{+8}_{-5}$
$v\sin{i}$ |  (km s-1) | $12.96^{+0.88}_{-0.64}$ | $11.38^{+0.45}_{-0.76}$ | $6.41^{+0.42}_{-0.66}$
$\lambda$ | (deg) | $-1.5^{+15.9}_{-15.3}$ | $-14.4^{+16.7}_{-12.7}$ | $-1.0^{+22.1}_{-29.3}$
$\sqrt{e}\cos{\omega}$ |  | $-0.040^{+0.056}_{-0.046}$ | $-0.130^{+0.120}_{-0.083}$ | $-0.018^{+0.088}_{-0.083}$
$\sqrt{e}\sin{\omega}$ |  | $0.042^{+0.079}_{-0.086}$ | $-0.040^{+0.164}_{-0.148}$ | $0.096^{+0.159}_{-0.140}$
$q_{1,{\rm TESS}}$ |  | $0.119^{+0.032}_{-0.014}$ | $0.041^{+0.030}_{-0.019}$ | $0.114^{+0.197}_{-0.074}$
$q_{2,{\rm TESS}}$ |  | $0.156^{+0.078}_{-0.041}$ | $0.767^{+0.168}_{-0.269}$ | $0.404^{+0.376}_{-0.283}$
$q_{1,{\rm HIRES}}$ |  | $\ldots$ | $\ldots$ | $\ldots$
$q_{2,{\rm HIRES}}$ |  | $\ldots$ | $\ldots$ | $\ldots$
$q_{1,{\rm KPF}}$ |  | $\ldots$ | $\ldots$ | $0.620^{+0.263}_{-0.327}$
$q_{2,{\rm KPF}}$ |  | $\ldots$ | $\ldots$ | $0.589^{+0.287}_{-0.359}$
Derived parameters: |  |
$M_{p}$ |  ($M_{J}$) | $1.179\pm 0.023$ | $1.635\pm 0.032$ | $1.113\pm 0.071$
$R_{p}$ |  ($R_{J}$) | $1.519\pm 0.059$ | $1.284\pm 0.098$ | $1.489\pm 0.079$
$e$ |  | $0.011^{+0.007}_{-0.008}$ | $0.091^{+0.017}_{-0.015}$ | $0.018^{+0.029}_{-0.018}$
Allesfitter priors: |  |
$P$ | (days) | $\mathcal{U}[2.1032;2.0,2.2]$ | $\mathcal{U}[5.7998;5.75,5.85]$ | $\mathcal{U}[3.2535;3.2,3.3]$
$t_{0}$ | (BJD - 2457000) | $\mathcal{U}[1684.4058;1684.0,1685.0]$ | $\mathcal{U}[2307.4306;2307.0,2308.0]$ | $\mathcal{U}[1628.0701;1628.0,1628.1]$
$R_{\rm p}/R_{\star}$ |  | $\mathcal{U}[0.08;0.01,1.0]$ | $\mathcal{U}[0.057;0.0,1.0]$ | $\mathcal{U}[0.075;0.05,1.0]$
$\cos{i}$ |  | $\mathcal{U}[0.01;0.0,1.0]$ | $\mathcal{U}[0.01;0.0,1.0]$ | $\mathcal{U}[0.01;0.0,1.0]$
$K$ |  (m s-1) | $\mathcal{U}[150;50,400]$ | $\mathcal{U}[150;0,400]$ | $\mathcal{U}[120;50,400]$
$v\sin{i_{\star}}$ |  (km s-1) | $\mathcal{U}[10.71;5.0,15.0]$ | $\mathcal{U}[10.4;9.0,12.0]$ | $\mathcal{U}[5.9;4.0,7.0]$
$\lambda$ | (deg) | $\mathcal{U}[0.0;-180.0,180.0]$ | $\mathcal{U}[0.0;-180.0,180.0]$ | $\mathcal{U}[0.0;-180.0,180.0]$
$\sqrt{e}\cos{\omega}$ |  | $\mathcal{U}[0,0;0.0,1.0]$ | $\mathcal{U}[0.0;0.0,1.0]$ | $\mathcal{U}[0.0;0.0,1.0]$
$\sqrt{e}\sin{\omega}$ |  | $\mathcal{U}[0.0;0.0,1.0]$ | $\mathcal{U}[0.0;0.0,1.0]$ | $\mathcal{U}[0.0;0.0,1.0]$
$q_{1,{\rm TESS}}$ |  | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$
$q_{2,{\rm TESS}}$ |  | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$
$q_{1,{\rm HIRES}}$ |  | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$
$q_{2,{\rm HIRES}}$ |  | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$ | $\mathcal{U}[0.5;0.0,1.0]$
$q_{1,{\rm KPF}}$ |  | … | … | $\mathcal{U}[0.5;0.0,1.0]$
$q_{2,{\rm KPF}}$ |  | … | … | $\mathcal{U}[0.5;0.0,1.0]$
Table 2: Fit and derived planet parameters, and allesfitter model priors for
each of our systems. $\mathcal{U}[a;b,c]$ denotes a uniform distribution from
$b$ to $c$, starting with the test value $a$.

###  4.2 Doppler Shadow

As an independent measure of the obliquity, we analyzed the spectral
timeseries for each of our systems, and identified a Doppler shadow signal in
the HIRES observations for TOI-1181 b and TOI-6029 b (Figures 5 and 6,
respectively). We selected lines in the non-Iodine portion of the HIRES
spectra (4000-5000 Å) to search for variations in the line profile during
transit. We performed 5-$\sigma$ outlier clipping and removed the continuum
and blaze function with a polynomial fit. Each spectrum was then cross-
correlated with the SpecMatch template spectrum and the globabally-averaged
line profile was subtracted off to isolate line profile variations. Our
approach follows Dai et al. (2020). Analysis of the KPF data for TOI-4379 b
did not result in the Doppler shadow detection, due to low signal-to-noise
ratio of the observations, and the fact that the line profile is dominated by
turbulence rather than rotational modulation.

Figure 5: Measurement of the Doppler shadow signal for TOI-1181 b from line
profile residuals in the HIRES spectra. The horizontal dotted line marks the
ingress and egress of the transit, the vertical dotted line indicate
approximately the rotational broadening of the star ($v\sin{i}$). The
planetary shadow is a diagonal pattern moving from the blue-shifted to the
red-shifted part of the stellar disk. This is consistent with a well-aligned
orbit seen in the RM measurement. Figure 6: Same as Figure 5 but for TOI-6029
b.

##  5 Results

###  5.1 TOI-1181

TOI-1181 b is an intermediate-mass ($M_{p}=$ $1.179\pm 0.023$ $M_{J}$) hot
Jupiter on a circular orbit. The planet radius is typical for a hot Jupiter of
this mass ($R_{p}=$ $1.519\pm 0.059$ $R_{J}$). At a Teff of 6054 K, its
subgiant host star has cooled after its main sequence lifetime and is likely
to have developed a non-negligible surface convective zone. The RM effect of
TOI-1181 b has the highest amplitude in our sample, due in part to the high
projected rotation velocity of its host star ($v\sin{i_{\star}}=10.6\pm 1.0$
km s-1), providing the most precise measurement of the spin-orbit angle. The
posterior from our model gives the constraint on the sky-projected obliquity
of TOI-1181 of $\lambda=$ $-1.5^{+15.6}_{-15.3}$ ∘. For TOI-1181 b, we measure
an obliquity from the Doppler shadow analysis of $7.5\pm 17.2^{\circ}$, which
is consistent with the measurement from the RM analysis.

###  5.2 TOI-6029

TOI-6029 b is the most massive planet in our sample, with $M_{p}=$ $1.635\pm
0.032$ $M_{J}$, as well as the smallest ($R_{p}=$ $1.284\pm 0.098$ $R_{J}$),
indicating a high density relative to other hot Jupiters. Its host star is the
hottest, most massive, and most luminous in the sample. TOI-6029 sits near the
Kraft break at T${}_{\rm eff}=6223$ K, indicating that the system has had a
substantial surface convective envelope ($\gtrsim$10% of stellar radius) for
only a relatively brief window of its lifetime. The out-of-transit residuals
to our RV model have larger scatter than those of the other two systems, and
we searched for additional non-transiting planets. We performed a Lomb-Scargle
periodogram analysis on the residuals as well as attempting a two-planet fit
to the observations, neither of which returned conclusive evidence of
additional planetary companions. The lack of evidence for additional non-
transiting planets indicates that the residual scatter is likely caused by
stellar jitter. The spin-orbit alignment of the TOI-6029 system is similar to
that measured for TOI-1181, with $\lambda=$ $-14.4^{+16.7}_{-12.7}$ ∘. The
Doppler shadow measurement for TOI-6029 b provides an obliquity constraint of
$22.3\pm 8.6^{\circ}$, which is consistent with the degree of misalignment
provided by the RM measurement. We note that the sign of the obliquity from
the different methods is flipped for TOI-6029 b due to the degeneracy of the
orientation of the transit chord.

###  5.3 TOI-4379

TOI-4379 b is similar to Jupiter in mass ($M_{p}=$ $1.113\pm 0.071$ $M_{J}$)
but has a larger radius ($R_{p}=$ $1.489\pm 0.079$ $R_{J}$). Its host star has
the lowest projected stellar surface rotation velocity in our sample
($v\sin{i_{\star}}=5.9\pm 1.0$ km s-1). The lack of sufficient RV baseline
observations caused the measured sky-projected obliquity of the system to have
larger uncertainty than the previously discussed measurements. The spin-orbit
angle of the TOI-4379 system is consistent with being fully aligned
($\lambda=$ $-1.0^{+22.1}_{-29.3}$ ∘).

###  5.4 Population Comparison

We now compare the radii, masses, and incident flux of our planet sample to
the known population of exoplanets. Literature values were downloaded from the
NASA Exoplanet Science Institute (NExScI) Exoplanet
Archive777exoplanetarchive.ipac.caltech.edu on January 3, 2024. We limited the
sample to exoplanets with both measured radii and masses, and removed planets
for which the fractional uncertainty on either mass or radius exceeded 20%.
Figure 7 shows the well established relationship between incident flux, planet
radius, and planet mass. We estimate incident flux with the relation from
Seager (2010) given by

$\frac{F_{\rm
p}}{F_{\oplus}}=\left(\frac{R_{\star}}{R_{\oplus}}\right)^{2}\left(\frac{\rm
T_{\rm eff,\star}}{\rm T_{\rm eff,\odot}}\right)^{4}\left(\frac{a}{\rm
AU}\right)^{-2}.$

TOI-1181 b, TOI-6029 b, and TOI-4379 b show inflation that is consistent with
known planets of similar mass. The planets presented in this work constitute a
small representative sample of hot Jupiters. We refer to Chontos et al. (2024)
for a population-level comparison of masses and radii for planets orbiting
main-sequence and subgiant stars.

Figure 7: Planet radius versus incident flux received for planets drawn from
the literature (circles) and planets in our sample (stars). The colorbar
indicates planet mass, and the dotted vertical line shows the empirical
threshold above which planet inflation is observed.

##  6 Tidal Realignment

###  6.1 Comparison to Main Sequence Obliquities

Figure 8 shows a Hertzsprung-Russell (H-R) diagram of host stars for systems
with measured sky-projected obliquities. To populate this figure, we
downloaded obliquity measurements from the TEPCat catalog (Southworth, 2011)
and included additional measurements reported in Albrecht et al. (2021). We
then performed cuts to select hot Jupiters systems ($M_{P}>0.3M_{J}$, $P_{\rm
orb}<10$ days, $a/R_{\star}\leq 8$) and removed obliquity measurements with
uncertainties $>30$∘.

To estimate the evolution of our host stars, we fit evolutionary tracks from
MESA Isochrones & Stellar Tracks (MIST; Dotter 2016; Choi et al. 2016; Paxton
et al. 2011, 2013, 2015). We adopt the isoclassify mass and interpolate
between values for metallicity in order to identify isochrone models that pass
through our host stars near our stellar age estimates. The resulting tracks
are shown as dotted lines in Figure 8. The evolutionary tracks show that these
stars have indeed dropped below 6250 K during their post-main sequence
evolution. In Figure 8 we highlight two additional hot Jupiter systems for
which the host star is likely somewhat evolved: WASP-71 b at 6180 K (Smith et
al., 2013) and HAT-P-7 b at 6310 K (Winn et al., 2009).

Near the estimated main-sequence positions of our targets, hot Jupiter hosts
are observed with a wide range of obliquities. We test the likelihood that our
three aligned systems are randomly drawn from the previously-aligned
population in the observed distribution using a bootstrapping technique: we
select a range of effective temperature between 6300 K and 6700 K which
encompasses the zero-age main sequence (ZAMS) positions of our host stars.
Next, we perform 100,000 random draws of three measured obliquities and
calculate the fraction of draws in which all three systems were aligned
($<20$∘), and find that we retrieve the observed measurements $\sim$10% of the
time. If we include the literature measurement of WASP-71 b, the fraction
drops to $\sim$4%. This indicates that it is unlikely for our measured
obliquities to be randomly sampled, however a larger sample would improve the
confidence of this constraint.

Using a sample of subgiants which have decreased in temperature and crossed
the Kraft break, we can test whether systems that were part of a frequently
misaligned population become realigned after developing a deep surface
convective envelope. Main sequence stars hotter than the Kraft break—similar
to the H-R positions of our sample at the ZAMS—show a wide range of
obliquities. If spin-orbit misalignment persists over the full lifetime of
planetary systems, hot Jupiters orbiting stars which are hotter than the Kraft
break should still be misaligned. In contrast, if the presence of a convective
envelope causes spin-orbit realignment, the population of stars which gained
deep convective envelopes after the main sequence should host hot Jupiters
which are aligned or in the process of being aligned.

Figure 8: H-R diagram of stars hosting planets with measured sky-projected
obliquities. Main sequence host stars are marked by circles, while subgiant
hosts are shown as stars. The color of each point indicates the sky-projected
obliquity, $\lambda$. Dotted black lines show our stellar evolution models fit
to each of our host stars, and trace back to their ZAMS location.

###  6.2 Evolution of the Convective Envelope

Figure 8 qualitatively suggests that misaligned hot Jupiters can be realigned
as their host stars cross the Kraft break on the subgiant branch, implying
efficient obliquity damping due to a convective envelope. To quantify whether
obliquity damping can indeed act over subgiant evolution timescales, we
examine the evolution of the surface convective envelope. We performed stellar
modeling with the Modules for Experiments in Stellar Astrophysics code (MESA;
Paxton et al. 2010, 2013, 2015, 2018, 2019) to trace the emergence of surface
convective envelopes and estimate the time each star spent cooler than the
Kraft break. Our models used initial elemental abundances from Grevesse &
Sauval (1998) and an atmospheric temperature structure following an Eddington
$T(\uptau)$ relation with fixed opacity. We assume no diffusion and no core or
envelope overshoot.

We used the best-fit stellar properties for each star (given in Table 1) as
initial properties, and evolved our stellar models from the pre-main sequence
until the base of the red giant branch. In Figure 9, we show the depth of the
surface convection zone boundary in units of fractional stellar radius $R_{\rm
CZ}/R_{\star}$ (where 0 is the center and 1 is the surface) as a function of
stellar age.

Figure 9: Fractional stellar radius, $R_{\rm CZ}/R_{\star}$, as a function of
stellar age. On the y-axis, 0 represents the center of the star and 1 is its
surface. The dashed line shows when the star was above the Kraft break ($\geq
6250$ K) and the solid line shows when the star was below the Kraft break
($<6250$ K). The shaded regions show the range of estimated stellar ages from
isoclassify.

Near the ZAMS, the convective boundary is near $R_{\rm CZ}/R_{\star}=1$,
indicating a thin surface convection zone. As the stars evolve into subgiants,
the convective boundary moves deeper into the stellar interior. While the mass
fraction ($M_{\rm CZ}/M_{\star}$) contained in the convective envelope remains
small ($\lesssim$0.01%), the deepening surface convection zone may lead to
enhanced tidal damping. We note that at the age when these stars drop below
6250 K, the outer $\sim$10% of the stellar envelope is convective, but there
is no clear feature to indicate a substantial deepening of the convective
envelope over a short timescale. However, the observed change in the
distribution of obliquities corresponds to this temperature, and we therefore
select 6250 K as the threshold for enhanced damping. Realignment may require a
minimum threshold of material for efficient damping, or it may correspond to
the disappearance of the convective core. Further stellar modeling will be
required to examine this threshold with more detail in the future.

If we assume that efficient realignment begins when a star decreases in
temperature and drops below the Kraft break, we can roughly estimate the
timescale for realignment. Because all systems in our sample are already
aligned, we can only estimate an upper limit to this timescale by calculating
the time between when the host star crossed the Kraft break and the estimated
age at which it is observed today. This results in an upper limit on the
realignment timescale of $\sim$1.5 Gyr for TOI-1181 b, and $\sim$0.5 Gyr for
TOI-6029 b and TOI-4379 b. Using TOI-6029 b and TOI-4379 b as references, we
can place a global upper limit on the timescale for realignment at $\sim 500$
Myr.

###  6.3 Realignment Model

Stellar tides have been suggested to effectively damp out the obliquity of hot
Jupiter systems, however they also damp orbital periods on similar or shorter
timescales (Winn et al., 2010). This leads us to an important question—how can
a system realign without the planet facing runaway inspiral and engulfment?

To test this, we model the evolution of orbital period and obliquity under the
influence of tidal damping. We construct a toy model for orbital evolution,
using a single stellar evolution model that is characteristic of stars in our
sample, with $M_{\star}=1.4M_{\odot}$, [Fe/H]$=0.3$ dex, and $M_{p}=1.2M_{J}$.
For simplicity, we assume a circular orbit for the planet. We use our toy
model to explore the damping efficiencies required to reproduce a post-main
sequence distribution similar to what we observe.

We assume exponential decay prescribed by $\dot{a}=-a/\uptau_{a}$ for the
semi-major axis, $a$, and $\dot{\lambda}=-\sin{(\lambda)}/\uptau_{\lambda}$
for the obliquity, $\lambda$. The coefficients $\uptau_{a}$ and
$\uptau_{\lambda}$ are the damping timescales, defined by Lai (2012) as

$\uptau_{a}\approx
1.28\left(\frac{Q^{\prime}_{\star}}{10^{7}}\right)\left(\frac{M_{\star}}{10^{3}M_{\rm
p}}\right)\left(\frac{\bar{\rho}_{\star}}{\bar{\rho}_{\odot}}\right)^{5/3}\left(\frac{P_{\rm
orb}}{1{\rm d}}\right)^{13/3}{\rm Gyr}$ (1)

and

$\uptau_{\lambda}\approx
1.13\uptau_{a}\left(\frac{\kappa}{0.1}\right)\left(\frac{M_{\star}}{10^{3}M_{\rm
p}}\right)\left(\frac{\bar{\rho}_{\odot}}{\bar{\rho}_{\star}}\right)^{2/3}\left(\frac{10{\rm
d}}{P_{\rm s}}\right)\left(\frac{1{\rm d}}{P_{\rm orb}}\right)^{1/3}$ (2)

where $Q^{\prime}_{\star}$ is the tidal quality factor of the star,
$\bar{\rho}_{\star}$ is the mean stellar density, and $\kappa$ is related to
the stellar moment of inertia $I_{\star}$ by $I_{\star}=\kappa MR^{2}$.

The calculation of $\uptau_{\lambda}$ in Equation 2 requires knowledge of the
stellar rotation period. We therefore produced our stellar models with a
modified version of MESA which calculates and stores parameters necessary to
trace the rotational evolution of the star (see Saunders et al. 2024 for
details). We assume standard Skumanich-like spindown over the main sequence
lifetime. This is a reasonable assumption for stars in this mass range, as
they are not expected to undergo significant magnetic braking (van Saders et
al., 2016, 2019).

The choice of tidal quality factor $Q^{\prime}_{\star}$ has a significant
effect on the dynamical evolution. However, values for $Q^{\prime}_{\star}$
range over multiple orders of magnitude for subgiants (Patel & Penev, 2022).
In Figure 10, we show the impact of this choice on our dynamical models. If
$Q^{\prime}_{\star}$ is assumed to be the same for the damping of $P$ and
$\lambda$, the planet enters a phase of runaway inspiral and is engulfed
before it can realign (panel a). If we instead define the damping timescales
$\uptau_{a}$ and $\uptau_{\lambda}$ with different effective tidal quality
factors, it is possible to produce a model in which the planet realigns before
it can be engulfed. Because the obliquity distribution appears distinct on
either side of the Kraft break, we assume that $\uptau_{a}$ and
$\uptau_{\lambda}$ share the same tidal quality factor until the model drops
below 6250 K, at which point we invoke a deviation in $Q^{\prime}_{\star}$,
described by the order of magnitude difference
$\delta\log{Q^{\prime}_{\star}}$. Figure 10b shows that a difference in
effective tidal quality factor $\delta\log{Q^{\prime}_{\star}}\gtrsim 4$ is
required to efficiently realign hot Jupiters on orbits between $\sim$3-8 days
without entering a phase of runaway inspiral.

Figure 10: (a) Top: Damping timescales for our toy model, calculated using
Equations 1 & 2, as a function of stellar age, with $\uptau_{a}$ in black and
$\uptau_{\lambda}$ in blue. Middle: Model for the evolution of orbital period,
$P$ (shown by the black line). Bottom: Model for the evolution of obliquity,
$\lambda$ (shown by the blue line). These models were computed assuming the
damping efficiency is the same for both parameters. The red region indicates
the runaway inspiral regime. (b) Same as panel (a), but with different
efficiencies for the damping of $P$ and $\lambda$, prescribed by the tidal
quality factor $Q^{\prime}_{\star}$. This model has a difference in effective
tidal quality factor of $\delta\log{Q^{\prime}_{\star}}=4$.

Figure 11 shows the sample from Figure 8 but instead plotting obliquity versus
effective temperature, highlighting the correlation between alignment and
stellar effective temperature. To examine the change in $\lambda$ as the host
star evolves, we take the model in Figure 10 b and plot it in $\lambda$ versus
Teff space. We observe that the toy model indeed reproduces an evolutionary
track in which a deepening convective envelope during subgiant evolution
causes obliquity damping.

Figure 11: Distribution of hot Jupiter obliquities as a function of stellar
effective temperature, Teff. Blue points indicate cool ($<6250$ K) hosts, and
red points indicate hot ($>6250$ K) hosts. The vertical dashed black line
marks the Kraft break. Subgiant hosts are marked by large circles with black
outlines, and all other points are main sequence hosts. The distributions on
the far right of the plot show the kernel densities of each sample,
corresponding to color. The black line shows a model for the post-main
sequence realignment of a hot Jupiter. This model shows a possible pathway
from the hot, misaligned population to the observed cool, well-aligned
population.

The toy model shown in Figures 10 and 11 make several assumptions. First, we
assume that the tidal quality factor $Q^{\prime}_{\star}$ does not evolve
smoothly in time, and instead drops precipitously in the case of
$\lambda$-damping after the star crosses the Kraft break. It is unlikely that
this fully captures the behavior seen in nature, as $Q^{\prime}_{\star}$
depends on the interior stellar structure, which evolves smoothly over main
sequence lifetimes. However, main sequence internal structure changes are
small, and we choose to fix the values of $Q^{\prime}_{\star}$ on the main
sequence, as well as after the threshold is crossed. We also assume that the
value of $Q^{\prime}_{\star}$ applied to the damping of period is held fixed
for the entire evolution of the star. As the stellar structure evolves after
the star leaves the main sequence, its $Q^{\prime}_{\star}$ should change,
though the relationship between $Q^{\prime}_{\star}$ and stellar radius and
density is uncertain. We again choose the simple case of holding the value
fixed, though future studies of how $Q^{\prime}_{\star}$ relates to stellar
interior evolution may allow us to improve this model in the future. There are
promising pathways to better characterize $Q^{\prime}_{\star}$ through
eclipsing binary system evolution (Fabrycky & Tremaine, 2007) and
asteroseismology (Chontos et al., 2019).

We now consider what physical mechanisms could lead to a large difference in
the efficiencies of the orbital period and obliquity damping timescales. Winn
et al. (2010) invoke sustained core-envelope decoupling as a potential source
of enhanced obliquity damping. In this scenario, the planet is able to exert a
torque on the outer envelope of the star without rapidly losing orbital
angular momentum. They note, however, that this explanation is challenged by
evidence that the core and envelope of main sequence stars appear to be well-
coupled (Winn et al., 2010; Jianke & Collier Cameron, 1993). It is unclear
whether this expectation applies to subgiant stars, and the theory of
sustained core-envelope decoupling could be tested with asteroseismic
measurements of radial differential rotation or core-envelope misalignments in
evolved stars (Lund et al., 2014).

Another explanation, posed by Lai (2012), suggests that in the presence of a
convective envelope, a hot Jupiter on a misaligned orbit can excite inertial
waves which dissipate angular momentum as they propagate through the stellar
interior. Ogilvie & Lin (2007) examined tidal dissipation in stars with
convective envelopes in the context of hydrodynamical theory, and found that
these inertial waves can result in the value of $Q^{\prime}_{\star}$ being
reduced by up to 4 orders of magnitude. Because these inertial waves are
static in the inertial frame, they do not contribute to orbital decay (Lai,
2012), and only impact the efficiency of obliquity damping. This suggested
order of magnitude difference is consistent with our toy model in Figures 10
and 11 following the prescription for tidal damping laid out in Lai (2012).

A recent study by Zanazzi et al. (2024) showed that hot Jupiter orbits can
resonantly lock to gravity mode pulsations in the radiative interior of cool
stars and efficiently damp the system’s obliquity (similar to semi-major axis
damping described in Ma & Fuller 2021). This mechanism is able to damp the
stellar obliquity without causing the planet to inspiral into its host star.
The interior structure of our subgiant hosts is amenable to gravity mode
pulsations, providing another possible mechanism for the observed obliquity
damping.

###  6.4 Predicting Orbital Evolution of Main Sequence Systems

To investigate period and obliquity damping over a range of parameters, we
calculate our toy model over a grid of main sequence orbital period, $P_{\rm
ZAMS}$, planet mass, $M_{\rm p}$, and $\delta\log{Q^{\prime}_{\star}}$. We
consider a system “engulfed” if the orbital period is reduced to zero before
the star reaches the base of the red giant branch, “realigned” if the
obliquity drops below 10∘ and the orbital period is greater than zero for the
full track, and “not aligned” if the planet is not engulfed but retains an
obliquity greater than 10∘. This threshold corresponds to the typical
uncertainty on a sky-projected obliquity measurement (see Figure 11).

Figure 12 shows the parameter space where realignment is expected for a range
of values for $\delta\log{Q^{\prime}_{\star}}$ as a function of main sequence
orbital period, $P_{\rm ZAMS}$, and planet mass, $M_{\rm p}$. The regions
inside the contours are expected to realign without being engulfed before the
star reaches the base of the red giant branch. As the orbital period
increases, a larger $M_{\rm p}$ is required to efficiently damp the obliquity.
However, if the planet mass is too high, the damping of orbital period will be
efficient enough to catalyze runaway inspiral. Figure 12 can be used as a
rough guide to predict whether a hot Jupiter orbiting a hot main sequence star
will realign after the main sequence.

Figure 12: Regions of expected realignment given different values of
$\delta\log{Q^{\prime}_{\star}}$. The x-axis shows the initial orbital period
of the planet (at ZAMS), and the y-axis shows the planet mass. Inside the
shaded contours, realignment without engulfment is expected. Below the
contours, $\lambda$ is not damped efficiently enough to realign, and above the
contours the planet enters runaway inspiral.

##  7 Conclusions

The obliquities of hot Jupiter systems have been observed to show a strong
dependence on the effective temperature of the host star, with cool stars
($<6250$ K) hosting mostly aligned planets and hot stars ($\geq 6250$ K)
hosting mostly misaligned planets. Theories have suggested that a surface
convective envelope can damp the obliquity of a hot Jupiter’s orbit.

To test this theory, we have measured the sky-projected spin-orbit angles for
three subgiants hosting hot Jupiters. Each of the host stars in our sample
crossed the Kraft break and gained deep surface convective envelopes during
their post-main sequence evolution. Our evolved hot Jupiter systems have low
sky-projected obliquities, indicating efficient realignment as stars develop
convective envelopes.

We summarize our main conclusions as follows:

  1. 1.

We confirm two newly identified hot Jupiters orbiting subgiant stars: TOI-6029
b ($R_{\rm p}=$ $1.284\pm 0.098$ $R_{\rm J}$, $M_{\rm p}=$ $1.635\pm 0.032$
$M_{\rm J}$, $P_{\rm orb}=$ $5.7987\pm 0.0004$ days), and TOI-4379 b ($R_{\rm
p}=$ $1.489\pm 0.079$ $R_{\rm J}$, $M_{\rm p}=$ $1.113\pm 0.071$ $M_{\rm J}$,
$P_{\rm orb}=$ $3.25352\pm 0.00002$ days).

  2. 2.

We measured the RM effect for three hot Jupiters orbiting subgiants that have
recently developed convective envelopes. All three are aligned
($|\lambda|<15^{\circ}$), which is consistent with the picture that convective
envelopes can efficiently damp obliquities. We place an upper limit on the
timescale for tidal realignment of $\sim$500 Myr.

  3. 3.

A simple model incorporating tidal realignment to explain these observations
requires the effective tidal quality factor for obliquity damping to be at
least four orders of magnitude lower than that for orbital decay ($\delta\log
Q^{\prime}_{\star}\gtrsim 4$). This is consistent with angular momentum
dissipation through the driving and damping of inertial waves in the stellar
convective envelope or coupling to gravity modes in the radiative interior.

Further work is required to place tighter constraints on the timescales for
obliquity damping. A larger sample of obliquity measurements for planets
orbiting subgiant stars similar to those reported here would provide
additional benchmarks for the dynamical evolution, and potentially distinguish
between mechanisms for enhanced $\lambda$-damping efficiency. One approach to
better understand the relationship between $\lambda$ and Teff would be to
measure the stellar obliquity for massive subgiants which have not yet crossed
the Kraft break. By observing systems at a variety of evolutionary stages, we
will better constrain the efficiency of obliquity damping to test high-
eccentricity migration as the mechanism for hot Jupiter formation.

## Acknowledgements

N.S. acknowledges support by the National Science Foundation Graduate Research
Fellowship Program under Grant Numbers 1842402 & 2236415\. A.C. & D.H.
acknowledge support from the National Aeronautics and Space Administration
(80NSSC21K0652). D.H. acknowledges support from the Alfred P. Sloan Foundation
and the Australian Research Council (FT200100871). M.R. is supported by
Heising-Simons grant #2023-4478. K.A.C. and C.N.W. acknowledge support from
the TESS mission via subaward s3449 from MIT. T.D. acknowledges support by the
McDonnell Center for the Space Sciences at Washington University in St. Louis.
J.M.A.M. is supported by the National Science Foundation (NSF) Graduate
Research Fellowship Program (GRFP) under Grant No. DGE-1842400.

This paper made use of data collected by the TESS mission and are publicly
available from the Mikulski Archive for Space Telescopes (MAST) operated by
the Space Telescope Science Institute (STScI). Funding for the TESS mission is
provided by NASA’s Science Mission Directorate. We acknowledge the use of
public TESS data from pipelines at the TESS Science Office and at the TESS
Science Processing Operations Center. Resources supporting this work were
provided by the NASA High-End Computing (HEC) Program through the NASA
Advanced Supercomputing (NAS) Division at Ames Research Center for the
production of the SPOC data products. Funding for the TESS mission is provided
by NASA’s Science Mission Directorate.

Some of the data presented herein were obtained at Keck Observatory, which is
a private 501(c)3 non-profit organization operated as a scientific partnership
among the California Institute of Technology, the University of California,
and the National Aeronautics and Space Administration. The Observatory was
made possible by the generous financial support of the W. M. Keck Foundation.

The authors wish to recognize and acknowledge the very significant cultural
role and reverence that the summit of Maunakea has always had within the
Native Hawaiian community. We are most fortunate to have the opportunity to
conduct observations from this mountain.

This work makes use of observations from the LCOGT network. Part of the LCOGT
telescope time was granted by NOIRLab through the Mid-Scale Innovations
Program (MSIP). MSIP is funded by NSF.

Some of the observations in this paper made use of the High-Resolution Imaging
instrument Zorro and were obtained under Gemini LLP Proposal Number:
GN/S-2021A-LP-105. Zorro was funded by the NASA Exoplanet Exploration Program
and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott,
Elliott P. Horch, and Emmett Quigley. Zorro was mounted on the Gemini South
telescope of the international Gemini Observatory, a program of NSF’s OIR Lab,
which is managed by the Association of Universities for Research in Astronomy
(AURA) under a cooperative agreement with the National Science Foundation. on
behalf of the Gemini partnership: the National Science Foundation (United
States), National Research Council (Canada), Agencia Nacional de Investigación
y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación
(Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações
(Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).

This research has made use of the Exoplanet Follow-up Observation Program
(ExoFOP; DOI: 10.26134/ExoFOP5) website, which is operated by the California
Institute of Technology, under contract with the National Aeronautics and
Space Administration under the Exoplanet Exploration Program.

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##  Appendix A Ground-based Imaging Figures

Figure 13 shows the high-contrast image of TOI-6029 obtained with the
Keck/NIRC2 near-infrared imager (for details, see §2.4). The corrected and
derotated image, taken in the Kp bandpass, shows a companion with a separation
of $763\pm 20$ milliarcseconds at a position angle of $110.2\pm 0.5^{\circ}$.

Figure 14 shows photometric time-series observations of the transits of
TOI-6029 b and TOI-4379 b. These data were obtained with the ground-based
LCOGT and PEST telescopes (for details, see §2.5), and they display transit
depths and ephemerides consistent with our predictions.

Figure 13: High-contrast image of TOI-6029 taken with the Keck/NIRC2 near-
infrared imager in the Kp bandpass. The image has been corrected and
derotated. The colorbar indicates the measured flux. Figure 14: Ground-based
time-series photometry observations of the transits of TOI-6029 b and TOI-4379
b. The light green symbols show the unbinned data. The larger dark green
symbols show the data binned in 10 minute bins. The solid green line is the
best fit transit model. (a) Partial transit of TOI-6029 b observed by LCOGT on
August 3, 2023. (b) Transit of TOI-4379 b observed by LCOGT on May 15, 2022.
(c) Transit of TOI-4379 b observed by PEST on May 27, 2023.

##  Appendix B Radial Velocity Measurements

Table LABEL:tab:my_label lists RV measurements obtained with Keck/HIRES and
Keck/KPF for each of our targets. A machine-readable version of this table
containing all observations is available.

Table 3: Radial velocity measurements for TOI-1181, TOI-6029, and TOI-4379 used in our analysis. A full machine-readable table can be accessed online. Target | Instrument | Time (BJD) | RV (m/s) | RV Uncertainty (m/s)
---|---|---|---|---
TOI-1181 | Keck/HIRES | $2459000.040402$ | $-67.993$ | $3.735$
TOI-1181 | Keck/HIRES | $2459003.045446$ | $14.927$ | $3.907$
TOI-1181 | Keck/HIRES | $2459004.050001$ | $72.699$ | $3.539$
TOI-1181 | Keck/HIRES | $2459026.093511$ | $-50.794$ | $4.016$
TOI-1181 | Keck/HIRES | $2459029.024667$ | $118.585$ | $3.540$
⋮ |  |  |  |
TOI-6029 | Keck/HIRES | $2459423.093788$ | $84.436$ | $6.597$
TOI-6029 | Keck/HIRES | $2459442.102513$ | $-136.860$ | $6.772$
TOI-6029 | Keck/HIRES | $2459447.07665$ | $-26.309$ | $7.122$
TOI-6029 | Keck/HIRES | $2459449.108516$ | $-30.814$ | $6.492$
TOI-6029 | Keck/HIRES | $2459457.094269$ | $111.598$ | $6.563$
⋮ |  |  |  |
TOI-4379 | Keck/HIRES | $2459478.753368$ | $-38.193$ | $8.845$
TOI-4379 | Keck/HIRES | $2459482.716898$ | $104.904$ | $5.204$
TOI-4379 | Keck/HIRES | $2459633.119397$ | $63.349$ | $4.963$
TOI-4379 | Keck/HIRES | $2459694.041773$ | $79.915$ | $5.086$
TOI-4379 | Keck/HIRES | $2459710.004739$ | $24.236$ | $4.605$
⋮ |  |  |  |
TOI-4379 | Keck/KPF | $2460101.830527187$ | $68.210$ | $15.086$
TOI-4379 | Keck/KPF | $2460101.837867083$ | $49.998$ | $9.299$
TOI-4379 | Keck/KPF | $2460101.844402326$ | $37.009$ | $7.578$
TOI-4379 | Keck/KPF | $2460101.85159022$ | $40.815$ | $9.384$
TOI-4379 | Keck/KPF | $2460101.858644097$ | $48.221$ | $7.549$
⋮ |  |  |  |