# Evidence for Primordial Alignment: Insights from Stellar Obliquity
Measurements for Compact Sub-Saturn Systems

Brandon T. Radzom Department of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN 47405-7105, USA  Jiayin Dong Flatiron Research Fellow  Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA  Malena Rice Department of Astronomy, Yale University, 52 Hillhouse Ave, New Haven, CT 06511, USA  Xian-Yu Wang Department of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN 47405-7105, USA  Samuel W. Yee 51 Pegasi b Fellow  Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Tyler R Fairnington Centre for Astrophysics, University of Southern Queensland, West St, Darling Heights, Toowoomba, Queensland 4350, Australia  Cristobal Petrovich Department of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN 47405-7105, USA  Songhu Wang Department of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN 47405-7105, USA

###### Abstract

Despite decades of effort, the mechanisms by which the spin axis of a star and
the orbital axes of its planets become misaligned remain elusive.
Particularly, it is of great interest whether the large spin-orbit
misalignments observed are driven primarily by high-eccentricity migration —
expected to have occurred for short-period, isolated planets — or reflect a
more universal process that operates across systems with a variety of present-
day architectures. Compact multi-planet systems offer a unique opportunity to
differentiate between these competing hypotheses, as their tightly-packed
configurations preclude violent dynamical histories, including high-
eccentricity migration, allowing them to trace the primordial disk plane. In
this context, we report measurements of the sky-projected stellar obliquity
($\lambda$) via the Rossiter-McLaughlin effect for two sub-Saturns in
multiple-transiting systems: TOI-5126 b ($\lambda=1\pm 48\,^{\circ}$) and
TOI-5398 b ($\lambda=-24^{+14}_{-13}\,{}^{\circ}$). Both are spin-orbit
aligned, joining a fast-growing group of just three other compact sub-Saturn
systems, all of which exhibit spin-orbit alignment. Our results strongly
suggest that sub-Saturn systems are primordially aligned and become misaligned
largely in the post-disk phase through violent dynamical interactions inherent
to eccentric migration, as appears to be the case increasingly for other
exoplanet populations.

Exoplanet astronomy (486); Exoplanet dynamics (490); Exoplanet evolution
(491); Extrasolar gaseous giant planets (509); Radial velocity (1332);
Transits (1711)

††facilities: WIYN/NEID††software: NumPy (van der Walt et al., 2011; Harris et
al., 2020), SciPy (Virtanen et al., 2020), pandas (McKinney, 2010), matplotlib
(Hunter, 2007), allesfitter (Günther & Daylan, 2021), emcee (Foreman-Mackey et
al., 2013b), iSpec (Blanco-Cuaresma et al., 2014)

##  1 Introduction

Stellar obliquity—i.e., the angle between the spin axis of the host star and
the net orbit normal axis of the planets—is a powerful probe of a planetary
system’s formation history. While the precise mechanisms driving spin-orbit
misalignment are not yet clear, it is important to distinguish whether
misalignment is largely a product of high-eccentricity migration, which should
be confined to short-period isolated planets111Planets that are single or have
only distant planetary companion(s) (e.g., hot Jupiters; see Dawson & Johnson
2018 and references therein), or some universal process that operates across
planetary systems with a variety of architectures, e.g., magnetic warping (Lai
et al., 2011; Romanova et al., 2013), interactions with stellar companions
(Lubow & Ogilvie, 2000; Batygin, 2012), stellar flybys (Hao et al., 2013), or
stellar gravity waves (Rogers et al., 2012, 2013).

Compact multi-planet systems may be critical to differentiate between these
two scenarios, as their tight orbital configurations are inconsistent with
high-eccentricity migration induced by planet-planet scattering (Rasio & Ford,
1996; Chatterjee et al., 2008), Lidov-Kozai cycling (Wu & Murray, 2003;
Fabrycky & Tremaine, 2007; Naoz, 2016), or secular interactions (Naoz et al.,
2011; Wu & Lithwick, 2011; Petrovich, 2015) — all of which tend to disrupt the
orbits of nearby planetary neighbors (Mustill et al., 2015). As such, if high-
eccentricity migration is the primary driver of misalignment, compact multi-
planet systems will retain their primordially aligned configurations. On the
other hand, if universal misalignment processes dominate, compact systems may
be misaligned at a rate comparable to isolated systems.

The great majority of spin-orbit angle measurements have been made for close-
in Jupiters, particularly hot Jupiters (see recent review by Albrecht et al.
2022), which rarely host nearby companions (Steffen et al., 2012; Huang et
al., 2016; Hord et al., 2021; Wu et al., 2023). Therefore, it is useful to
expand the census of stellar obliquities to other types of exoplanets. Sub-
Saturns, broadly defined as giant planets with masses ranging between that of
Neptune ($\sim$17 M${\oplus}$) and Saturn ($\sim$95 M${\oplus}$), represent a
particularly interesting population for such studies. The core accretion model
implies that they are ‘failed gas giants’ (Pollack et al., 1996), suggesting
they have analogous origins to Jupiters (Dong et al., 2018; Lee, 2019; Hallatt
& Lee, 2022). However, sub-Saturns are less amenable to detection than
Jupiters and are relatively understudied as an exoplanet population, so few
constraints on their formation histories exist. As of this writing, only about
two dozen sub-Saturns have had their spin-orbit angle measured, demonstrating
that these planets can take on a range of values from near-perfect alignment
to almost completely retrograde. Yet, only _three_ of these measurements have
been made for sub-Saturns in compact multi-planet systems: Kepler-9 b (Wang et
al., 2018), AU Mic b (Hirano et al., 2020a), and WASP-148 b (Wang et al.,
2022). While this sample is limited in number, all three systems show evidence
of spin-orbit alignment and thus provide tentative support for primordial
alignment.

The Transiting Exoplanets Survey Satellite (TESS) mission (Ricker et al.,
2015), now well into its Extended Mission phase, has produced a growing sample
of sub-Saturns around bright stars. Of particular interest to this work, TESS
is responsible for the discovery of at least two sub-Saturns in compact multi-
planet systems: TOI-5126 b (Fairnington et al., 2023), and TOI-5398 b
(Mantovan et al., 2022, 2024). Given the brightness of their host stars and
advances in Extreme Precision Radial Velocity (EPRV), these systems are
amenable to stellar obliquity measurements.

In this study, we present Rossiter-McLaughlin (RM) effect (Holt, 1893;
Rossiter, 1924; McLaughlin, 1924; Queloz et al., 2000) measurements for both
confirmed compact TESS sub-Saturn systems, TOI-5126 and TOI-5398, using the
high-precision NEID spectrograph (Schwab et al., 2016) on the WIYN
$3.5\,\mathrm{m}$ telescope at Kitt Peak, AZ. TOI-5126, a relatively bright
FV-type star ($V=10.1\,\mathrm{mag}$), hosts a hot sub-Saturn with a 5.5-day
period and an outer warm sub-Neptune with a 17.9-day period (Fairnington et
al., 2023). TOI-5398, a GV-type star ($V=10.1\,\mathrm{mag}$), harbors a warm
sub-Saturn with a 10.6-day period and an inner hot sub-Neptune with a 4.8-day
period (Mantovan et al., 2022, 2024). Our spin-orbit measurements represent a
significant contribution to the limited sample of sub-Saturns in compact
multi-planet systems and are part of the 11 published outcome from the Stellar
Obliquities in Long-period Exoplanet Systems (SOLES) survey, as detailed in a
series of publications (Rice et al. 2021; Wang et al. 2022; Rice et al. 2022;
Rice et al. 2023a; Hixenbaugh et al. 2023; Dong et al. 2023; Wright et al.
2023; Rice et al. 2023b; Lubin et al. 2023; Hu et al. 2024).

This paper is organized as follows. In Section 2, we outline our observational
methods. In Section 3, we describe our determination of stellar parameters. In
Section 4, we detail our modeling of the RM signals and, subsequently, the
stellar obliquities. In Section 5, we quantify the significance of our results
and discuss their implications for the origins of spin-orbit misalignment and
the formation histories of sub-Saturns.

##  2 Observations

The RM effect of TOI-5126 and TOI-5398 were observed using the WIYN/NEID
spectrograph in the High Resolution (HR) mode ($R\sim 110,000$, Halverson et
al. 2016; Schwab et al. 2016) on February 28, 2023 and April 16, 2023,
respectively. NEID is a fiber-fed (Kanodia et al., 2018), actively
environmentally stabilized spectrograph (Stefansson et al., 2016; Robertson et
al., 2019) with a wavelength coverage of 380 nm to 930 nm. For TOI-5126 b, we
obtain 20 radial velocity (RV) measurements in HR mode with 1000-second
exposures from 02:20-07:48 UT, covering its full transit. These observations
occurred under atmospheric conditions with a seeing range of
1.2$\arcsec$-2.8$\arcsec$ (median 1.75$\arcsec$) and an airmass range of
$\mathrm{z}=1.03-2.31$. At a wavelength of $5500\text{\AA}$, the NEID
spectrograph achieved a signal-to-noise ratio (SNR) of 56 pixel-1. For
TOI-5398 b, nearly the full transit (excluding ingress) was covered by 14 RV
measurements obtained in HR mode with 1250-second exposures from 02:55-09:04
UT. The conditions featured a seeing range of 0.7$\arcsec$-1.5$\arcsec$
(median 1.0$\arcsec$) and an airmass range of $\mathrm{z}=1.08-1.71$. At the
same wavelength, the spectrograph’s SNR was 60 pixel-1. An additional 14 out-
of-transit NEID observations of TOI-5398 b were obtained between April 8,
2022, and December 27, 2023, with 300-second exposure times.

The NEID spectra were analyzed using version 1.3.0 of the NEID Data Reduction
Pipeline (NEID-DRP)222Detailed information is available at:
https://neid.ipac.caltech.edu/docs/NEID-DRP/, and radial velocities were
derived via the cross-correlation function (CCF) method within NEID-DRP. We
extracted the resulting barycentric-corrected radial velocities for re-
weighted orders (CCFRVMOD) from the NExScI NEID
Archive333https://neid.ipac.caltech.edu/. The RV data gathered from NEID for
this study are illustrated in Figure 1 (in baseline-subtracted form) and
available through the Data Behind the Figure program (see figure caption).

##  3 Stellar parameters

###  3.1 Synthetic Spectral Fit By iSpec

We adopt the co-added NEID spectra of TOI-5126 and TOI-5398 to determine their
respective stellar atmospheric parameters, including stellar effective
temperature ($T_{\rm eff}$ ), surface gravity ($\log g_{*}$ ), metallicity
($\left[{\rm Fe}/{\rm H}\right]$ ), and projected rotational velocity
($v\sin{i_{*}}$ ). We use the synthetic spectral fitting technique provided by
the Python package iSpec (Blanco-Cuaresma et al., 2014; Blanco-Cuaresma, 2019)
to measure these parameters.

We employ the SPECTRUM radiative transfer code (Gray & Corbally, 1994), the
MARCS atmosphere model (Gustafsson et al., 2008), and the sixth version of the
GES atomic line list (Heiter et al., 2021), all integrated within iSpec, to
create a synthetic model for the combined NEID spectra (SNR of 206 for
TOI-5126; SNR of 176 for TOI-5398). In our fitting process, we treat micro-
turbulent velocities as a variable, allowing us to accurately represent the
small-scale turbulent motions in the stellar atmosphere. Conversely, we
determine macro-turbulent velocities using an empirical relationship (Doyle et
al., 2014) that leverages established correlations with various stellar
attributes. We select specific spectral regions to streamline the fitting
process, focusing on the wings of the H$\alpha$, H$\beta$, and Mg I triplet
lines, which are indicative of $T_{\rm eff}\,$ and $\log g_{*}\,$, as well as
Fe I and Fe II lines, which are crucial for constraining $\left[{\rm Fe}/{\rm
H}\right]\,$ and $v\sin{i_{*}}\,$. We then utilize the Levenberg-Marquardt
nonlinear least-squares fitting algorithm (Markwardt, 2009) to refine our
spectroscopic parameters by iteratively minimizing the $\chi^{2}$ value
between the synthetic and observed spectra. The final spectroscopic parameters
are detailed in Table 1.

###  3.2 SED+MIST Fit By EXOFASTv2

To ascertain additional stellar parameters, such as stellar mass ($M_{*}$) and
radius ($R_{*}$), we utilize the MESA Isochrones & Stellar Tracks (MIST) model
(Choi et al., 2016; Dotter, 2016) in combination with a spectral energy
distribution (SED) fit. Photometry for the SED fit was compiled from various
catalogs, including 2MASS (Cutri et al., 2003), WISE (Cutri et al., 2021),
TESS (Ricker et al., 2015), and Gaia DR3 (Gaia Collaboration et al., 2023). We
apply Gaussian priors based on our synthetic spectral fit to $T_{\rm eff}$ and
$\left[{\rm Fe}/{\rm H}\right]$ , along with the parallax from Gaia DR3, and
an upper limit for the $V$-band extinction from Schlafly & Finkbeiner (2011).
To accommodate the approximate 2.4% systematic uncertainty floor in $T_{\rm
eff}$ , as suggested by Tayar et al. (2022), we increase the uncertainties of
$T_{\rm eff}$ to 150 K for both TOI-5126 and TOI-5398.

As mentioned in Fairnington et al. (2023) and Mantovan et al. (2024)
respectively, both TOI-5126 and TOI-5398 exhibit strong stellar rotation
signals, providing additional stringent constraints on their stellar ages. We
adopt Equation 7 from Barnes (2007) to calculate the gyrochronological ages
for these stars based on the stellar rotation periods derived from our work
(see Section 4). The resulting gyrochronological ages for TOI-5126 and
TOI-5398 are $0.13\pm 0.02$ Gyr and $0.42^{+0.07}_{-0.05}$ Gyr, respectively.
Subsequently, we adopt these ages and their 3$\sigma$ uncertainties as priors
for the stellar ages in our SED fit.

We perform the SED fitting using the Differential Evolution Markov Chain Monte
Carlo (DEMCMC) technique, integrated within EXOFASTv2 (Eastman, 2017; Eastman
et al., 2019), from which we obtain uncertainty estimates. The DEMCMC
procedure was considered converged when the Gelman-Rubin diagnostic
($\hat{R}$; Gelman & Rubin, 1992) fell below 1.01 and the count of independent
draws surpassed 1000. Our final stellar parameters are listed in Table 1.

Table 1: Stellar and Planetary Parameters of TOI-5126 and TOI-5398

| Description | Priorsa | TOI-5126b | Priorsa | TOI-5398
---|---|---|---|---|---
Stellar Parameters:
Synthetic spectral fit:
$T_{\rm eff}\,$ | Effective temperature (K) | - | $6166^{+95}_{-97}$ | - | $6267\pm 90$
$\left[{\rm Fe}/{\rm H}\right]\,$ | Metallicity (dex) | - | $0.10\pm 0.06$ | - | $0.01\pm 0.06$
$\log g_{*}\,$ | Surface gravity (log10(cm/s2)) | - | $4.37\pm 0.20$ | - | $4.44\pm 0.22$
SED+MIST fit (adopted):
$M_{*}$ | Stellar mass ($M_{\odot}$) | - | $1.260^{+0.035}_{-0.034}$ | - | $1.105^{+0.031}_{-0.029}$
$R_{*}$ | Stellar radius ($R_{\odot}$) | - | $1.208^{+0.033}_{-0.032}$ | - | $1.038^{+0.029}_{-0.028}$
$T_{\rm eff}\,$ | Effective temperature (K) | $\mathcal{N}(6166|6166,150)$ | $6253\pm 84$ | $\mathcal{N}(6267|6267,150)$ | $6028^{+82}_{-81}$
$\left[{\rm Fe}/{\rm H}\right]\,$ | Metallicity (dex) | $\mathcal{N}(0.1|0.1,0.06)$ | $0.116^{+0.059}_{-0.058}$ | $\mathcal{N}(0.01|0.01,0.06)$ | $0.018^{+0.059}_{-0.060}$
$\log g_{*}\,$ | Surface gravity (log10(cm/s2)) | - | $4.374\pm 0.019$ | - | $4.449\pm 0.019$
Age | Age (Gyr) | $\mathcal{N}(0.13|0.13,0.06)$ | $0.169^{+0.056}_{-0.051}$ | $\mathcal{N}(0.42|0.42,0.21)$ | $0.55^{+0.19}_{-0.17}$
$\varpi$ | Parallax (mas) | $\mathcal{N}(6.257|6.257,0.024)$ | $6.257\pm 0.025$ | $\mathcal{N}(7.648|7.648,0.017)$ | $7.650\pm 0.018$
d | Distance (pc) | - | $159.82\pm 0.65$ | - | $130.73\pm 0.30$
$v\sin i_{*}$ (iSpec) | Projected stellar | - | $13.84\pm 0.80$ | - | $7.65\pm 0.54$
rotational velocity (km/s)
Rossiter-McLaughlin Parameters:
$\lambda$ | Sky-projected spin-orbit angle (deg) | $\mathcal{U}(0|-180,180)$ | $1\pm 48$ | $\mathcal{U}(0|-180,180)$ | $-24_{-13}^{+14}$
$v\sin{i_{\star}}$ | Projected stellar | $\mathcal{N}(13.84|13.84,0.80)$ | $13.83\pm 0.79$ | $\mathcal{N}(7.65|7.65,0.54)$ | $7.50_{-0.38}^{+0.43}$
rotational velocity (km/s)
$\xi$ | Micro-turbulent velocity (km/s) | $\mathcal{T}(1.23|1.23,1,0,10)$ | $1.38_{-0.81}^{+0.95}$ | $\mathcal{T}(1.14|1.14,1,0,10)$ | $1.36_{-0.80}^{+0.92}$
$\zeta$ | Macro-turbulent velocity (km/s) | $\mathcal{T}(4.52|4.52,1,0,10)$ | $4.62\pm 0.99$ | $\mathcal{T}(3.77|3.77,1,0,10)$ | $3.96\pm 0.95$
Planetary Parameters:
$R_{b}/R_{\star}$ | Planet-to-star radius ratio | $\mathcal{U}(0.0348|0,1)$ | $0.03612\pm 0.00042$ | $\mathcal{U}(0.0898|0,1)$ | $0.09092_{-0.00059}^{+0.00062}$
$(R_{\star}+R_{b})/a_{b}$ | Ratio of the sum of star and | $\mathcal{U}(0.0916|0,1)$ | $0.08951_{-0.00078}^{+0.0015}$ | $\mathcal{U}(0.0543|0,1)$ | $0.05259_{-0.00057}^{+0.0011}$
planet radii to semi-major axis
$\cos{i_{b}}$ | Cosine of inclination | $\mathcal{U}(0.0|0,1)$ | $0.0097_{-0.0067}^{+0.0091}$ | $\mathcal{U}(0.0|0,1)$ | $0.0079_{-0.0042}^{+0.0052}$
$T_{0;b}$ | Mid-transit time - 2459600 ($\rm BJD_{\rm TDB}$) | $\mathcal{U}(27.039|26.939,27.139)$ | $27.03719_{-0.00043}^{+0.00053}$ | $\mathcal{U}(16.492|16.392,16.592)$ | $16.49202\pm 0.00024$
$P_{b}$ | Orbital period (days) | $\mathcal{U}(5.458|5.358,5.558)$ | $5.4588627_{-0.0000085}^{+0.0000077}$ | $\mathcal{U}(10.591|10.491,10.691)$ | $10.590530\pm 0.000037$
$K_{b}$ | Radial velocity semi-amplitude (m/s) | $\mathcal{N}(6.6|6.6,2.5)$ | $6.6\pm 2.5$ | $\mathcal{N}(15.7|15.7,1.5)$ | $15.2\pm 1.3$
$q_{1;\rm NEID}$ | Linear limb-darkening coefficient for NEID | $\mathcal{U}(0.5|0,1)$ | $0.51\pm 0.34$ | $\mathcal{U}(0.5|0,1)$ | $0.38_{-0.26}^{+0.34}$
$q_{2;\rm NEID}$ | Quadratic limb-darkening coefficient for NEID | $\mathcal{U}(0.5|0,1)$ | $0.46\pm 0.31$ | $\mathcal{U}(0.5|0,1)$ | $0.35_{-0.25}^{+0.35}$
Derived Parameters:
$M_{b}$ | Planetary mass ($\mathrm{M_{J}}$) | - | $0.067\pm 0.026$ | - | $0.171\pm 0.017$
$R_{b}$ | Planetary radius ($\mathrm{R_{J}}$) | - | $0.425\pm 0.012$ | - | $0.919\pm 0.026$
$a_{b}/R_{\star}$ | Semi-major axis scaled by stellar radius | - | $11.57_{-0.19}^{+0.10}$ | - | $20.74_{-0.43}^{+0.23}$
$a_{b}$ | Semi-major axis (AU) | - | $0.0649\pm 0.0020$ | - | $0.0998\pm 0.0032$
$i_{b}$ | Inclination (deg) | - | $89.45_{-0.53}^{+0.39}$ | - | $89.55_{-0.30}^{+0.24}$
$T_{14;b}$ | Total transit duration (hours) | - | $3.711_{-0.020}^{+0.022}$ | - | $4.208_{-0.019}^{+0.020}$
$u_{1;\rm NEID}$ | Linear limb-darkening coefficient for NEID | - | $0.56_{-0.38}^{+0.57}$ | - | $0.34_{-0.22}^{+0.48}$
$u_{2;\rm NEID}$ | Quadratic limb-darkening coefficient for NEID | - | $0.04\pm 0.42$ | - | $0.14_{-0.34}^{+0.41}$
GP Parameters:
$ln\rm B_{NEID}$ | Amplitude scaling factor | - | - | $\mathcal{U}(0|-10,10)$ | $-6.67_{-1.1}^{+0.59}$
$ln\rm C_{NEID}$ | Balance factor | - | - | $\mathcal{U}(0|-10,10)$ | $-0.7_{-7.6}^{+8.2}$
$ln\rm L_{NEID}$ | Coherence timescale | - | - | $\mathcal{U}(0|-10,10)$ | $-1.3_{-7.1}^{+5.2}$
$ln\rm P_{rot,NEID}$ | Stellar rotation period | - | - | $\mathcal{N}(2|2,0.2)$ | $2.03_{-0.18}^{+0.17}$
$ln\rm B_{HARPS-N}$ | Amplitude scaling factor | - | - | $\mathcal{U}(0|-10,10)$ | $-7.16\pm 0.16$
$ln\rm C_{HARPS-N}$ | Balance factor | - | - | $\mathcal{U}(0|-10,10)$ | $-0.9_{-7.5}^{+8.6}$
$ln\rm L_{HARPS-N}$ | Coherence timescale | - | - | $\mathcal{U}(0|-10,10)$ | $-1.19_{-1.4}^{+0.69}$
$ln\rm P_{rot,HARPS-N}$ | Stellar rotation period | - | - | $\mathcal{N}(2|2,0.2)$ | $2.01\pm 0.19$

aafootnotetext: The notation $\mathcal{U}(p_{0}|a,b)$,
$\mathcal{N}(p_{0}|\mu,\sigma)$, and $\mathcal{T}(p_{0}|\mu,\sigma,a,b)$
represent the uniform, Gaussian, and truncated Gaussian distributions,
respectively. Here, $p_{0}$ denotes the initial guess, $a$ and $b$ the lower
and upper bounds, $\mu$ the median value, and $\sigma$ the standard deviation.

bbfootnotetext: RV amplitude and Planetary mass of TOI-5126 b are derived from
Fairnington et al. (2023).

##  4 Obliquity Modeling

We determine the sky-projected spin-orbit angle $\lambda$ for TOI-5126 b and
TOI-5398 b by jointly fitting their two-minute cadence TESS lightcurves and
radial velocities from NEID using a modified version of allesfitter package
(Günther & Daylan, 2021), along with ancillary ground-based data available
from their discovery papers. The modified version of allesfitter incorporates
the Hirano et al. (2011) RM model implemented in tracit (Hjorth et al., 2021;
Knudstrup & Albrecht, 2022). We exclude the transit data for the non-sub-
Saturn planet in both systems but model their RVs (TOI-5126 c at
$P=17.9\,\mathrm{d}$ and TOI-5398 c at $P=4.77\,\mathrm{d}$).

For TOI-5126 b, we modeled eight full transits and one partial transit from
TESS Sector 45, four full transits and one partial transit from Sector 46, and
four full transits from Sector 48. Within allesfitter, we simultaneously fit
the TESS and NEID data together with all photometric data utilized in TOI-5126
b’s discovery paper (Fairnington et al., 2023), including observations from
the _CHaracterising_ _ExOplanets_ _Satellite_ (CHEOPS, Benz et al. 2021) as
well as ground-based $1.0\,\mathrm{m}$ telescopes at McDonald Observatory in
Fort Davis, TX and at the Cerro Tololo Inter-American Observatory (CTIO) in
Cerro Tololo, Chile, which are both part of the Las Cumbres Observatory Global
Network (LCOGT; Brown et al. 2013). We do not make use of the out-of-transit
radial velocities presented in Fairnington et al. (2023) due to their
relatively large scatter.

For TOI-5398 b, we model two full transits from TESS Sector 48, as well as
photometry from the LCOGT $1.0\,\mathrm{m}$ telescopes at McDonald Observatory
and Teide Observatory and from the $1.2\,\mathrm{m}$ KeplerCam at the Fred
Lawrence Whipple Observatory (FLWO) on Mount Hopkins, Arizona (all sourced
from Mantovan et al. 2024). We additionally incorporate the out-of-transit
radial velocities originally presented in Mantovan et al. (2024), which were
obtained from the High Accuracy Radial velocity Planet Searcher (HARPS-N,
Cosentino et al. 2012) at the Italian Telescopio Nazionale Galileo. To account
for strong stellar activity in the RVs, we apply a rotational Gaussian Process
(GP) regression kernel formulated by Foreman-Mackey et al. (2017),

$\displaystyle k(\tau)$ $\displaystyle=$
$\displaystyle\frac{B}{2+C}e^{-\tau/L}\left[\cos\left(\frac{2\pi\tau}{P_{\mathrm{rot}}}\right)+(1+C)\right]$
(1)

where $P_{\mathrm{rot}}$ represents the period of stellar rotation, $L$
denotes the coherence timescale, $\tau$ signifies the interval between two
successive data points, $B$ is a scaling hyperparameter that adjusts the
amplitude, and $C$ serves as the balance parameter for the periodic and
nonperiodic components of the GP kernel.

For both TOI-5126 b and TOI-5398 b, we adopt priors but utilize the results
reported in the literature (Fairnington et al. 2023 and Mantovan et al. 2024,
respectively) as our initial guesses for $R_{\mathrm{p}}/R_{*}$,
$(R_{*}+R_{\mathrm{p}})/a$, $\cos(i)$, $T_{0}$, $P$, and $K$ (see Table 1 for
parameter definitions). We fixed eccentricity ($e$) and the argument of
periastron ($\omega$) to 0 due to the near-zero eccentricity reported by the
discovery papers of TOI-5126 b and TOI-5398 b. The two transformed quadratic
limb-darkening coefficients $q_{1}$ and $q_{2}$ are initialized at 0.5 for
each instrument. As an initial guess for $v\sin{i_{*}}$ , for which we apply
Gaussian priors, we adopt the values derived from our synthetic spectral fit.
Finally, we initialize $\lambda$ at $0\,^{\circ}$ for both systems. All
parameters (except $e$ and $\omega$) are allowed to vary during the fitting
process. Note that, to account for the short-term overnight instrumental
systematics and stellar variability in the RM fit, we employ a quadratic
function.

For the fits of both systems, we sample the posterior distributions for all
modeled parameters using an affine-invariant Markov Chain Monte Carlo (MCMC)
grounded in emcee (Foreman-Mackey et al., 2013a) with 100 walkers. Best-fit
parameters were obtained after 200,000 accepted steps per walker were reached
(note that each ran 10,000 burner steps); our final MCMC results and
associated $1\sigma$ uncertainties are summarized in Table 1. Additionally,
all Markov Chains reached $>50\times$ their autocorrelation lengths,
indicating convergence.

We find good agreement ($<2\sigma$ difference) with all stellar, planetary,
and derived parameters reported in the discovery papers of both systems
(Fairnington et al. 2023 for TOI-5126 b and Mantovan et al. 2024 for TOI-5398
b), except $T_{0;b}$ (2.5$\sigma$ discrepant) for TOI-5126, which may be due
to tentative transit timing variations (see Fairnington et al. 2023).
Critically, our RM fits show that both sub-Saturns are consistent with spin-
orbit alignment, with TOI-5126 b having a best-fit $\lambda=1\pm 48\,^{\circ}$
and TOI-5398 b having a best-fit $\lambda=-24_{-13}^{+14}\,{}^{\circ}$. Figure
1 reveals the best-fit joint models and associated residuals for both systems,
while Table 1 contains all fitted and derived parameters.

Figure 1: Keplerian signal-subtracted radial velocities (top panels) and RM
model residuals (bottom panels) for TOI-5126 (left) and TOI-5398 (right). In
the top panels, the best-fitted RM models are shown with red dashed lines, and
their uncertainties are indicated by grey shadows, while the planets’ spin-
orbit configurations are depicted in the bottom left portion of the plot. Note
that the first data point of TOI 5398 was taken on April 10, 2022. The data
used are available here.

It is useful to additionally measure the 3D stellar obliquity $\psi$, which
yields a system’s true spin-orbit configuration. To estimate $\psi$, we first
derive the stellar inclination $i_{*}$ and stellar equatorial velocity
$v_{eq}$ for both TOI-5126 and TOI-5398 by applying the Bayesian inference
method presented by Masuda & Winn (2020) and Hjorth et al. (2021) to our
values for stellar radius $R_{*}$, stellar rotation period $\,{P_{\rm
rot}}\,$, and $\cos i_{*}$. We compute the stellar rotation periods via the
autocorrelation function (ACF) implemented in SpinSpotter (Holcomb et al.,
2022), finding $P_{\rm rot}=4.05\pm 0.21$ days and $P_{\rm rot}=7.443\pm
0.041$ days for TOI-5126 and TOI-5398, respectively. The resulting stellar
rotation periods are consistent with those reported by Fairnington et al.
(2023) for TOI-5126 ($P_{\mathrm{rot}}=4.602^{+0.071}_{-0.067}$ days) and
Mantovan et al. (2024) ($P_{\mathrm{rot}}=7.34\pm 0.05$ days) for TOI-5398.
From our Bayesian analysis, we find $i_{*}=90\pm 15\,^{\circ}$ and
$v_{eq}=14.36\pm 1.02\,\mathrm{km/s}$ for TOI-5126 and $i_{*}=91\pm
17\,^{\circ}$ and $v_{eq}=8.09\pm 0.87\,\mathrm{km/s}$ for TOI-5398, and
compute the true 3D stellar obliquities as (see Eqn. 9 of Fabrycky & Winn
2009):

$\cos{\psi}=\cos{i_{*}}\cos{i}+\sin{i_{*}}\sin{i}\cos{\lambda}$ (2)

where $i$ is the inclination angle of the planet. For TOI-5126 b, we find a
true obliquity of $\psi=36.06^{+19.63}_{-31.42}\,{}^{\circ}$. Similarly, we
derive $\psi=30.06^{+12.29}_{-11.48}\,{}^{\circ}$ for TOI-5398 b,
demonstrating that both systems are consistent with alignment or near-
alignment. Note that, in our derivation of $\psi$, we adopt an uncertainty
floor of 10% in the stellar rotation periods due to the latitudinal
differential rotation as suggested in Epstein & Pinsonneault (2014) and
Aigrain et al. (2015) and a systematic uncertainty floor of $\approx$ 4.2% in
stellar radii reported by Tayar et al. (2022)

##  5 Discussion

While sub-Saturns broadly appear to boast a wide range of stellar obliquities,
the few previously confirmed to be in compact multi-planet systems are
exclusively aligned. Our RM measurements for two of such recently-confirmed
sub-Saturns, TOI-5126 b and TOI-5398 b, continue this trend, as both are
consistent with alignment. We show the updated stellar obliquity distribution
for the sub-Saturn population as a function of stellar effective temperature
in Figure 2. To construct this sub-Saturn sample, we combine the catalogs of
Albrecht et al. (2022) and
TEPCat444https://www.astro.keele.ac.uk/jkt/tepcat/obliquity.html (Southworth,
2011) accessed on March 7, 2024. We consider only RM effect measurements,
which exhibit near-uniform sensitivity to the full range of possible $\lambda$
values (compared to nonuniform methods such as starspot tracking or gravity
darkening, Siegel et al. 2023; Dong & Foreman-Mackey 2023) and constitute the
majority of obliquity measurements (Triaud, 2018; Albrecht et al., 2022),
prioritizing those selected in Albrecht et al. (2022) (we otherwise adopt the
“preferred” measurements from TEPCat). Finally, we apply a mass cut of
$17\,\,M_{\oplus}<M_{pl}<95\,\,M_{\oplus}$, and select only single-star
systems by cross-matching our sample with the multi-star catalog of Rice et
al. (2024), which is based on the most recent data from Gaia DR3. We
distinguish between sub-Saturns that are isolated and those in compact multi-
planet systems, which we define as having a nearby companion with a period
ratio of $<4$. All sub-Saturns considered in this work are presented in Table
2 (note that we supplement any missing parameters using the NASA Exoplanet
Archive555https://exoplanetarchive.ipac.caltech.edu and values adopted by
their respective $\lambda$ reference papers). Our RM measurements allow for
more reliable statistical analyses to be performed on the relationship between
compact configurations and spin-orbit alignment for sub-Saturns (Section 5.1)
and thus more robust constraints on the origins of spin-orbit misalignment as
well as the formation histories of the sub-Saturn population (Sections 5.2 and
5.3).

Table 2: Sub-Saturns Around Single Stars With a Stellar Obliquity Measurement

System | Planet |  $T_{\mathrm{eff}}$ (K) |  $M_{*}$ [$\mathrm{M}_{\odot}$] |  $P$ (d) |  $M_{pl}$ ($\mathrm{M}_{\oplus}$) |  $\lambda$ ($\,{}^{\circ}$) |  $\lambda$ Reference
---|---|---|---|---|---|---|---
Compact Multi-planet Systems:
AU Mic | b | 3665$\pm 100$ | 0.5$\pm 0.03$ | 8.46$\pm 0.0000052$ | 20.12${}^{+1.57}_{-1.72}$ | $-4.7^{+6.4}_{-6.8}$ | Hirano et al. (2020a)
Kepler-9 | b | 5774$\pm 60$ | 1.02${}^{+0.03}_{-0.04}$ | 19.24$\pm 0.00006$ | 43.41${}^{+1.6}_{-2.0}$ | 13$\pm 16$ | Wang et al. (2018)
WASP-148 | b | 5555$\pm 21$ | 0.95${}^{+0.03}_{-0.05}$ | 8.80$\pm 0.000043$ | 91.24${}^{+6.99}_{-5.09}$ | $-8.2^{+9.7}_{-8.7}$ | Wang et al. (2022)
TOI-5126 | b | 6253$\pm 84$ | 1.26${}^{+0.035}_{-0.034}$ | 5.46$\pm 0.0000062$ | 21.3$\pm 8.3$ | 1$\pm 48$ | this work
TOI-5398 | b | 6028${}^{+82}_{-81}$ | 1.105${}^{+0.031}_{-0.029}$ | 10.59${}^{+0.000041}_{-0.000046}$ | 54.4$\pm 5.4$ | $-24^{+14}_{-13}$ | this work
Isolated Systems: |  |  |  |  |  |  |
GJ 436 | b | 3586$\pm 54$ | 0.47$\pm 0.07$ | 2.64$\pm 0.00000057$ | 22.25$\pm 2.23$ | 72${}^{+33}_{-24}$ | Bourrier et al. (2018)
HAT-P-11 | b | 4780$\pm 50$ | 0.81${}^{+0.02}_{-0.03}$ | 4.89$\pm 0.0000068$ | 26.7$\pm 2.23$ | 103${}^{+26}_{-10}$ | Winn et al. (2010b)
HAT-P-12 | b | 4650$\pm 45$ | 0.73$\pm 0.02$ | 3.21$\pm 0.0000021$ | 67.08$\pm 3.81$ | $-54^{+41}_{-13}$ | Mancini et al. (2018)
HD 89345 | b | 5576${}^{+70}_{-71}$ | 1.16${}^{+0.04}_{-0.05}$ | 11.81$\pm 0.0002$ | 34.97${}^{+5.4}_{-5.72}$ | 74.2${}^{+33.6}_{-32.5}$ | Bourrier et al. (2023)
HD 332231 | b | 6089${}^{+97}_{-96}$ | 1.13$\pm 0.08$ | 18.71$\pm 0.00043$ | 77.57$\pm 6.68$ | $-2\pm 6$ | Knudstrup & Albrecht (2022)
HIP 67522 | b | 5675$\pm 75$ | 1.22$\pm 0.05$ | 6.96${}^{+0.000016}_{-0.000015}$ | $79^{a}$ | $-5.8^{+2.8}_{-5.7}$ | Heitzmann et al. (2021)
K2-25 | b | 3207$\pm 58$ | 0.26$\pm 0.01$ | 3.48${}^{+0.0000006}_{-0.0000005}$ | 24.51${}^{+5.7}_{-5.2}$ | 3$\pm 16$ | Stefansson et al. (2020)
K2-105 | b | 5636$\pm 70$ | 1.05$\pm 0.02$ | 8.27${}^{+0.000007}_{-0.0000067}$ | 30.01$\pm 19.0$ | $-81^{+50}_{-47}$ | Bourrier et al. (2023)
KELT-11 | b | 5375$\pm 25$ | 1.44$\pm 0.07$ | 4.74$\pm 0.00003$ | 54.36$\pm 4.77$ | $-77.86^{+2.36}_{-2.26}$ | Mounzer et al. (2022)
TOI-1842 | b | 6230$\pm 50$ | 1.46$\pm 0.03$ | 9.57${}^{+0.0002}_{-0.0001}$ | 68.03${}^{+12.72}_{-12.08}$ | $-68.1^{+21.2}_{-14.7}$ | Hixenbaugh et al. (2023)
TOI-1859 | b | 6341${}^{+68}_{-70}$ | 1.29$\pm 0.06$ | 63.48$\pm 0.0001$ | $68.67^{a}$ | 38.9${}^{+2.8}_{-2.7}$ | Dong et al. (2023)
WASP-39 | b | 5400$\pm 50$ | 0.93$\pm 0.03$ | 4.06$\pm 0.000009$ | 89.01$\pm 9.54$ | 0$\pm 11$ | Mancini et al. (2018)
WASP-69 | b | 4700$\pm 50$ | 0.98$\pm 0.14$ | 3.87$\pm 0.000002$ | 92.19$\pm 9.54$ | 0.4${}^{+2.0}_{-1.9}$ | Casasayas-Barris et al. (2017)
WASP-107 | b | 4425$\pm 70$ | 0.68$\pm 0.02$ | 5.72$\pm 0.000002$ | 30.52$\pm 1.59$ | 118.1${}^{+37.8}_{-19.1}$ | Rubenzahl et al. (2021)
WASP-131 | b | 6030$\pm 50$ | 1.06$\pm 0.06$ | 5.32$\pm 0.000005$ | 85.83$\pm 6.36$ | 162.4${}^{+1.3}_{-1.2}$ | Doyle et al. (2023)
WASP-156 | b | 4910$\pm 61$ | 0.84$\pm 0.05$ | 3.84$\pm 0.000003$ | 40.69${}^{+3.18}_{-2.86}$ | $105.7^{+14.0}_{-14.4}$ | Bourrier et al. (2023)
WASP-166 | b | 6050$\pm 50$ | 1.19$\pm 0.06$ | 5.44$\pm 0.000004$ | 32.11$\pm 1.59$ | 3$\pm 5$ | Hellier et al. (2019)

aafootnotetext: Calculated using the Chen & Kipping (2017) mass-radius
relationship.

Figure 2: Sky-projected stellar obliquity $\lambda$ versus stellar effective
temperature $T_{\rm eff}$ for all sub-Saturns considered in this work, with
$1\sigma$ error bars shown. The sub-panel in the top right displays the
resultant distributions of the number of misaligned sub-Saturns found in
compact multi-planet systems (dashed line) and the number of misaligned
isolated sub-Saturns after performing 100,000 random draws (bars), with
statistical significance ($2.6\sigma$) indicated. In both panels, orange
corresponds to sub-Saturns in compact multi-planet systems while blue
corresponds to isolated sub-Saturns.

###  5.1 Sub-Saturns in Compact Multi-planet Systems are Significantly Aligned

It is clear from Figure 2 that spin-orbit alignment is much more common for
sub-Saturns in compact multi-planet systems than for isolated sub-Saturns,
while no clear relation exists for the host star’s effective temperature (as
seen in previous works, e.g., Albrecht et al. 2022). Our two stellar obliquity
measurements increase the sample of sub-Saturns in compact configurations to
five, enabling the first statistical tests of this trend. We investigate the
statistical significance of preferential alignment in compact systems by
comparing the number of misaligned systems found in the compact multi-planet
sub-Saturn sample (zero) with that found in a series of random draws from the
isolated sub-Saturn sample.

Namely, of the 22 sub-Saturns with $\lambda$ measurements considered in this
work, five are in compact multi-planet systems, while the other 17 are
isolated. Of the five in compact configurations, none are misaligned, while
9/17 (53%) isolated sub-Saturns are misaligned. We define misalignment as
$|\lambda|>10\,^{\circ}$ at the $1\sigma$ level (i.e.,
$|\lambda|-\sigma_{\lambda}>10\,^{\circ}$) and $\lambda>0\,^{\circ}$ at the
$2\sigma$ level (i.e., $|\lambda|-2\sigma_{\lambda}>0\,^{\circ}$), where we
adopt the $1\sigma$ uncertainties ($\sigma_{\lambda}$) reported for each
$\lambda$ measurement (see Table 2). In order to perform a fair comparison
between the compact multi-planet sample and the isolated sample, we randomly
select and classify the alignment of five isolated sub-Saturns. We perform
100,000 iterations of these random draws, counting the number of misaligned
systems for each, and fit a Gaussian function over the resulting distribution.
The upper-right sub-panel of Figure 2 reveals the result, illustrating that
sub-Saturns in compact multi-planet systems are more often aligned than
isolated sub-Saturns to a significance level of $2.6\sigma$.

We verified that this significance is robust to the precise misalignment
criteria adopted by re-running the above procedure with the following
alternative cuts: $\lambda>15\,^{\circ}$ and $\lambda>0\,^{\circ}$ at the
$2\sigma$ level ($2.6\sigma$ significance), (ii) $\lambda>20\,^{\circ}$ and
$\lambda>0\,^{\circ}$ at the $2\sigma$ level ($2.6\sigma$ significance), (iii)
$\lambda>15\,^{\circ}$ ($2.3\sigma$ significance), and (iv)
$\lambda>20\,^{\circ}$ ($2.3\sigma$ significance). Assuming our original
alignment criteria, we additionally vary the Neptune/sub-Saturn mass boundary
and re-run our procedure: (i) $15\,\mathrm{M}_{\oplus}$ ($3.0\sigma$
significance), and (ii) $0.1\,\mathrm{M}_{\mathrm{J}}$ ($1.8\sigma$
significance). We note that while the significance appears to degrade with an
increasing mass bound, this trend is likely not astrophysical, as the
significance is directly proportional to the number of sub-Saturns in compact
systems considered, which drops from six assuming a $15\,\mathrm{M}_{\oplus}$
bound to only three assuming a $0.1\,\mathrm{M}_{\mathrm{J}}$ ($\approx
32\,\mathrm{M}_{\oplus}$) bound. Finally, we note that the young ($\sim
20\,\mathrm{Myr}$), aligned, compact sub-Saturn system AU Mic still hosts a
dusty debris disk (Plavchan et al., 2020; Hirano et al., 2020a), meaning that
its spin-orbit angle and compactness has likely not yet had the opportunity to
be affected by various pathways to trigger post-disk eccentric migration (Wu
et al., 2023). Therefore, while this system provides strong support for
primordial alignment, it may bias our statistical test; we re-run our original
procedure excluding AU Mic (such that there are only four compact sub-Saturn
systems) and find that the significance is maintained, but reduced to
$2.2\sigma$.

###  5.2 Evidence for Primordial Alignment and Misalignment Via Eccentric
Migration

In this work, we have demonstrated that sub-Saturns in compact planetary
systems are significantly more aligned than those without nearby planetary
companions (at the $2.6\sigma$ level). This trend of spin-orbit alignment in
compact multi-planet systems around single stars is not limited to sub-
Saturns, however. Short-period ($P\lesssim 100\,\mathrm{d}$) Jupiters provided
the first robust evidence for this phenomenon. Specifically, all measured
spin-orbit angles for short-period Jupiters with nearby companions, even hot
Jupiters, are consistent with alignment, e.g., Kepler-30 (Sanchis-Ojeda et
al., 2012), Kepler-89 (Albrecht et al., 2013), WASP-47 (Sanchis-Ojeda et al.,
2015), WASP-148 (Wang et al., 2022), TOI-1478 Rice et al. (2022), TOI-2202
(Rice et al., 2023b), TOI-1670 (Lubin et al., 2023). Conversely, isolated
short-period Jupiters display a wide range of obliquities and are frequently
misaligned (Albrecht et al. 2012, 2022). While there is a paucity of spin-
orbit measurements for sub-Neptunes and super-Earths, most of these lower-mass
planets reside in compact multi-planet systems, and those with secure
obliquity measurements are aligned, e.g., Kepler-50, Kepler-65 (Chaplin et
al., 2013), Kepler-25 (Albrecht et al., 2013), HD 106315 (Zhou et al., 2018;
Bourrier et al., 2023), TRAPPIST-1 (Hirano et al., 2020b), TOI-1726 (Dai et
al., 2020), HD 63433 (Mann et al., 2020), TOI-1136 (Dai et al., 2023), and
TOI-2076 (Frazier et al., 2023).

As the spin-orbit angles of compact multi-planet systems should remain largely
unaltered following the dispersal of the disk, the low stellar obliquities
observed for sub-Saturns and other types of exoplanets in compact systems
suggest that planetary systems are initially formed spin-orbit aligned.
Furthermore, the large misalignments observed for isolated hot Jupiters and
sub-Saturns are generally consistent with violent eccentric migration pathways
(Dawson & Johnson, 2018; Wu et al., 2023). This scenario is additionally
supported by observational evidence showing that i) both young stellar systems
($\lesssim 100\,\mathrm{Myr}$ old) and planets still embedded within their
debris disks are aligned (Kraus et al., 2020; Plavchan et al., 2020; Hirano et
al., 2020b; Martioli et al., 2020; Palle et al., 2020; Carmichael et al.,
2021; Albrecht et al., 2022; Johnson et al., 2022), and ii) that misaligned
hot Jupiter systems tend to be older (Hamer & Schlaufman, 2022), reinforcing
the notion that misalignments are acquired well into in the post-disk phase
rather than near the onset of system formation.

###  5.3 On the Temperature-Obliquity Relation

There is an interesting discrepancy between the obliquity distributions of
isolated hot Jupiters and isolated hot sub-Saturns: hot Jupiters are commonly
misaligned around hot stars but exclusively aligned around cool stars (i.e.,
the $T_{\rm eff}-\lambda$ relation, see Winn et al. 2010a; Albrecht et al.
2012, 2022), while hot sub-Saturns are commonly misaligned around cool stars
(and no data exist for those orbiting hot stars, see Figure 2). Realignment
via tidal dissipation offers a plausible explanation for this phenomenon; cool
stars are susceptible to realignment by massive hot Jupiters, while lower-mass
hot sub-Saturns are less likely to realign them on timescales shorter than the
system’s age (e.g., see Eqn. 2 of Albrecht et al. 2012). Hence, both hot
Jupiters and hot sub-Saturns may indeed become misaligned through high-
eccentricity migration, but only cool stars with hot Jupiters can be
realigned. This tidal realignment mechanism, however, requires finely-tuned
parameters in order to completely realign all hot Jupiters around cool stars
without leaving any on polar or retrograde orbits (Rogers et al., 2013; Li &
Winn, 2016), which are not seen in observations.

Alternatively or in addition to this tidal realignment scenario, it is
possible that sub-Saturns are subject to misalignment mechanisms that may not
operate for Jupiters. For instance, Petrovich et al. (2020) illustrates that
secular resonance between an outer gas giant companion and an inner sub-
Saturn, both embedded in a decaying disk, can drive the sub-Saturn to a polar
orbit. The level of obliquity excitation depends on the ratio of the angular
momentum between the inner and outer orbits, with nearly polar orbits
preferentially excited for inner sub-Saturn-mass planets over Jupiter-mass
planets. This resonance sweeping offers a natural explanation for the excess
of sub-Saturns observed to be on polar orbits, regardless of stellar effective
temperature (Bourrier et al., 2018; Hixenbaugh et al., 2023), as well as a
potential mechanism to stunt the runaway growth of sub-Saturns since they are
assumed to be embedded within an actively dispersing disk. However, this
mechanism predicts similar misalignment rates for compact multi-planet and
isolated systems, holding all other factors equal. This would imply that
isolated sub-Saturns, which are more often misaligned, host outer companions
at a higher rate than sub-Saturns in compact systems, but no statistical
studies on these relative rates have yet been completed.

To explain the $T_{\rm eff}-\lambda$ relation and spin-orbit misalignment in
single-star exoplanetary systems more broadly, Hixenbaugh et al. (2023)
outline a unified model wherein planets gain misalignments if other planets of
comparable (or higher) mass form within their disk. Specifically, hotter, more
massive stars tend to host more massive protoplanetary disks (Williams &
Cieza, 2011; Andrews et al., 2013; Andrews, 2020), which are more likely to
form multiple Jupiter-mass planets (Johnson et al., 2010; Ghezzi et al., 2018;
Yang et al., 2020). These Jupiters may interact with each other to produce
spin-orbit misalignments after the disk dissipates. By contrast, the limited
disk mass around cooler stars is unlikely to produce more than one Jupiter-
mass planet, if any (Andrews et al., 2013; Ansdell et al., 2016; Pascucci et
al., 2016; Dawson & Johnson, 2018; Yang et al., 2020). As a result, these
giants, which have no comparable-mass planetary perturbers, should largely
retain their primordial alignments throughout the post-disk phase.
Nevertheless, despite their less massive disks, cool stars may still form
several sub-Saturn planets that undergo an array of dynamical interactions in
the post-disk phase that are capable of exciting their stellar obliquities,
analogously to multiple Jupiters around hot stars. As present-day compact
systems may have avoided violent dynamical interactions, they preserve their
primordial spin-orbit angle, allowing them to serve as a window into the
early, undisturbed state of planetary systems.

## Acknowledgments

We acknowledge the independent work on the Rossiter-McLaughlin effect
measurement of TOI-5398 b by Mantovan et al. Two manuscripts were prepared
concurrently but independently. We appreciate the collegiality of the team
regarding the manuscript submission process.

We express our gratitude for the insightful discussions with George Zhou and
Chelsea Huang regarding the stellar activity of young planetary systems. We
would also like to thank Armaan Goyal for insightful discussions regarding
statistical analyses.

Data presented were obtained by the NEID spectrograph built by Penn State
University and operated at the WIYN Observatory by NOIRLab, under the NN-
EXPLORE partnership of the National Aeronautics and Space Administration and
the National Science Foundation. These results are based on observations
obtained with NEID on the WIYN 3.5m Telescope at Kitt Peak National
Observatory (PI: Songhu Wang, IU TAC). Additional NEID data were obtained
through time allocated by NSF’s NOIRLab (NOIRLab Prop. IDs 2022A-543544,
2023A-904842, 2023B-738235; PI: S. Yee), under the NN-EXPLORE partnership of
the National Aeronautics and Space Administration and the National Science
Foundation. WIYN is a joint facility of the University of Wisconsin–Madison,
Indiana University, NSF’s NOIRLab, the Pennsylvania State University, Purdue
University, University of California, Irvine, and the University of Missouri.
The authors are honored to be permitted to conduct astronomical research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham.

M.R. acknowledges support from Heising-Simons Foundation grant $\\#$2023-4478.
S.W. acknowledges support from Heising-Simons Foundation grant $\\#$2023-
4050. We acknowledge support from the NASA Exoplanets Research Program
NNH23ZDA001N-XRP (grant $\\#$80NSSC24K0153). This research was supported in
part by Lilly Endowment, Inc., through its support for the Indiana University
Pervasive Technology Institute.

All the TESS data used in this paper can be found in MAST:
doi:10.17909/t9-nmc8-f686 (catalog 10.17909/t9-nmc8-f686). This research has
made use of the NASA Exoplanet Archive (NASA Exoplanet Archive, 2024), 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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