# Revisiting the Full Sets of Orbital Parameters for the XO-3 System:
No evidence for Temporal Variation of the Spin-Orbit Angle

Keduse Worku  Department of Astronomy, Yale University, New Haven, CT 06511,
USA  Songhu Wang Department of Astronomy, Indiana University, Bloomington, IN
47405, USA  Jennifer Burt Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA  Malena Rice
Department of Astronomy, Yale University, New Haven, CT 06511, USA  Xian-Yu
Wang National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  University of Chinese Academy of Sciences, Beijing, 100049,
China  Yong-Hao Wang School of Physics and Astronomy, Sun Yat-Sen University,
Zhuhai, 519082, China  Steven S. Vogt  UCO/Lick Observatory, Department of
Astronomy and Astrophysics, University of California at Santa Cruz, Santa
Cruz, CA 95064, USA  R. Paul Butler Earth and Planets Laboratory, Carnegie
Institution for Science, 5241 Broad Branch Road NW, Washington, DC 20015, USA
Brett Addison University of Southern Queensland, Centre for Astrophysics, USQ
Toowoomba, West Street, QLD 4350 Australia  Brad Holden  UCO/Lick Observatory,
Department of Astronomy and Astrophysics, University of California at Santa
Cruz, Santa Cruz, CA 95064, USA  Xi-Yan Peng  Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China  Zhen-Yu Wu
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  University of Chinese Academy of Sciences, Beijing, 100049,
China  Xu Zhou  National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China  Hui-Gen Liu School of Astronomy and Space
Science and Key Laboratory of Modern Astronomy and Astrophysics in Ministry of
Education, Nanjing University, Nanjing 210093, China  Hui Zhang Shanghai
Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
Ji-Lin Zhou School of Astronomy and Space Science and Key Laboratory of Modern
Astronomy and Astrophysics in Ministry of Education, Nanjing University,
Nanjing 210093, China  Gregory Laughlin Department of Astronomy, Yale
University, New Haven, CT 06511, USA

###### Abstract

We present 12 new transit light curves and 16 new out-of-transit radial
velocity measurements for the XO-3 system. By modelling our newly collected
measurements together with archival photometric and Doppler velocimetric data,
we confirmed the unusual configuration of the XO-3 system, which contains a
massive planet ($M_{\rm P}=11.92^{+0.59}_{-0.63}{\,\rm\,{\rm M_{\rm J}}}$) on
a relatively eccentric ($e=0.2853^{+0.0027}_{-0.0026}$) and short-period
($3.19152\pm 0.00145\,$day) orbit around a massive star
($M_{*}=1.219^{+0.090}_{-0.095}M_{\odot}$). Furthermore, we find no strong
evidence for a temporal change of either $V\sin i_{*}$ (and by extension, the
stellar spin vector of XO-3), or the transit profile (and thus orbital angular
momentum vector of XO-3b). We conclude that the discrepancy in previous
Rossiter-McLaughlin measurements ($70.0\,^{\circ}\pm 15.0\,^{\circ}$ (Hébrard
et al., 2008); $37.3\,^{\circ}\pm 3.7\,^{\circ}$ (Winn et al., 2009);
$37.3\,^{\circ}\pm 3.0\,^{\circ}$ (Hirano et al., 2011)) may have stemmed from
systematic noise sources.

##  1 Introduction

The existence of hot Jupiters, giant planets orbiting perilously close to
their parent stars, was wholly unpredicted; as a consequence, their initial
discoveries twenty-five years ago occurred with very high signal-to-noise. In
retrospect, this historical development was largely a consequence of the
emptiness of the inner reaches of the Solar System.

Although it has been suggested that hot Jupiters may form in situ (Batygin et
al., 2016), conventional wisdom still holds that they form at larger distances
– where cold, ice-based materials are plentiful – and then migrate inward
(Bodenheimer et al., 2000). Over the past two decades, two distinct and
competing long-distance migration mechanisms have been established: namely,
quiescent disk migration (Lin et al., 1996) and violent dynamical migration
(Lidov-Kozai cycling with tidal friction (Wu et al., 2007; Fabrycky &
Tremaine, 2007; Naoz, 2016); planet-planet scattering (Rasio & Ford, 1996;
Nagasawa et al., 2008); or secular interactions (Wu & Lithwick, 2011;
Petrovich, 2015)). Nonetheless, the most workable process for delivering a
“normal Jupiter” to its final location remains controversial.

Measurements of stellar obliquity (i.e., the sky-projected angle, $\lambda$,
between the orbital angular momentum vector of a transiting planet and its
host star’s spin vector) through the Rossiter-Mclaughlin effect (R-M effect;
Rossiter, 1924; McLaughlin, 1924; Queloz et al., 2000) were initially thought
to provide a zeroth-order discriminating test between quiescent disk-driven
migration and violent dynamical migration. Hot Jupiters with low spin-orbit
angles were thought to have migrated through the disk, while those with high
spin-orbit angles were believed to owe their orbits to high-eccentricity
dynamical migration.

The origin and evolution of spin-orbit misalignment has since been extensively
studied (see Winn & Fabrycky 2015 and references therein), however, and the
connection between the spin-orbit misalignment and the migration process may
be more complicated than was initially thought. These misalignments can either
be primordial (chaotic star formation (Bate et al., 2010; Fielding et al.,
2015); magnetic star-disk interactions (Lai et al., 2011; Spalding & Batygin,
2014); torques from the stellar companions (Batygin, 2012)), with planets born
in tilted disks, or they can be modified later by post-migration evolution
(tidal and magnetic realignment (Winn et al., 2010; Dawson, 2014; Li & Winn,
2016); gravitational perturbation from the companions (Innanen et al., 1997;
Li et al., 2014; Storch et al., 2014; Lai, 2016; Gomes et al., 2017); internal
gravity waves (Rogers et al., 2012)). The spin-orbit angle generally evolves
on very long timescales. Under certain circumstances, however, it can vary on
an observable timescale (See Rogers et al. 2012, for example).

In this light, the XO-3 system (Johns-Krull et al., 2008) has a special
importance as one of few transiting planet systems that displays a discrepancy
between multiple R-M measurements ($70.0\,^{\circ}\pm 15.0\,^{\circ}$, Hébrard
et al. 2008; $37.3\,^{\circ}\pm 3.7\,^{\circ}$, Winn et al. 2009;
$37.3\,^{\circ}\pm 3.0\,^{\circ}$, Hirano et al. 2011). This discrepancy, in
combination with the planet’s unusual mass ($M_{\rm P}=11.7\,M_{\rm J}$,
Bonomo et al. 2017) – lying just at the mass limit between giant planets and
low-mass stars – its eccentricity ($e=0.28$, Wong et al. 2014), and its short
period ($P=3.19\,{\rm day}$, Winn et al. 2008), suggests that XO-3 merits
further scrutiny.

In this paper, we present new transit light curves and Doppler velocimetric
measurements for XO-3 to address the source of the discrepancy in previous R-M
measurements, as well as implications for the origins of hot Jupiters.

The paper is organized as follows. §2 presents 33 photometric transit
observations from the literature and 12 new transits, as well as 16 new
Doppler velocity measurements of XO-3 that were used in this study. §3 focuses
on the characterization of the stellar atmospheric parameters using the APF
data. §4 describes the joint analysis of the in-transit photometric and out-
of-transit radial velocity measurements. §5 compares these findings with
previous results and outlines the potential implications of this study.

##  2 Observation and Data Reduction

###  2.1 Photometry

Our new photometric dataset, comprised of twelve light curves, was collected
using the Xinglong Schmidt and Xinglong $60\,{\rm cm}$ telescopes operated by
the National Astronomical Observatories of China (NAOC). These observations
span roughly four years, from 2014 March to 2017 December.

Four of these light curves were obtained using the Xinglong Schmidt telescope
(Zhou et al., 1999, 2001), which utilizes a $4{\rm K}\times 4{\rm K}$ CCD.
This CCD has a field of view (FOV) of $94^{\prime}\times 94^{\prime}$ and a
pixel scale of $1.38^{\prime\prime}\,{\rm pixel^{-1}}$. To reduce the initial
readout time ($93\,{\rm s}$), we windowed the frames down to $512\times 512$
pixels, which results in a readout time of $12\,{\rm s}$. A Johnson/Cousins
$R$-band filter was used during these observations.

The remaining eight light curves were obtained with the Xinglong $60\,{\rm
cm}$ telescope. The observation conducted on UT 2014-03-02 used a $512\times
512$ CCD, giving an FOV of $17^{\prime}\times 17^{\prime}$, a pixel scale of
$1.95^{\prime\prime}\,{\rm pixel^{-1}}$, and a readout time of $3\,{\rm s}$.
The observations conducted on UT 2015-02-16, UT 2017-11-11, and UT 2017-11-14
used a $1{\rm K}\times 1{\rm K}$ CCD, giving an FOV of $17^{\prime}\times
17^{\prime}$, a pixel scale of $0.99^{\prime\prime}\,{\rm pixel^{-1}}$, and a
readout time of $23\,{\rm s}$. The observations conducted on UT 2016-01-14, UT
2016-02-18, UT 2016-03-05 used a $2{\rm K}\times 2{\rm K}$ CCD, giving an FOV
of $36^{\prime}\times 36^{\prime}$, a pixel scale of
$1.06^{\prime\prime}\,{\rm pixel^{-1}}$, and a readout time of $6\,{\rm s}$.
All of the observations for this telescope utilized a Johnson/Cousins $R$-band
filter, except the one from UT 2017-11-11 which alternated between the
Johnson/Cousins $B$-band and $V$-band filters.

As XO-3 is bright (Vmag$=9.86$), we defocused the telescopes to avoid non-
linear effects on the CCD. The defocusing method usually requires longer
exposure times, which is helpful for increasing the duty cycle of our
observations and reducing the scintillation and Poisson noise (Southworth et
al., 2009). The focus was kept unchanged during our observation. The exposure
time was changed only if required by weather conditions. The telescope time
was synchronized with online GPS time servers. The beginning time of each
exposure was recorded in the frame header using the UTC time standard, and it
was then converted to ${\rm BJD_{\rm TDB}}$ as described in Eastman et al.
(2010). All times reported from previous works have been converted to ${\rm
BJD_{\rm TDB}}$ for congruency with our time standard.

We conducted standard bias and flat-field corrections on all the frames
following the procedures described in Wang X. et al. (2018, 2021a); Wang et
al. (2018a, c); Wang Y. et al. (2017, 2019). We then performed aperture
photometry using SExtractor (Bertin & Arnouts, 1996). We identified the best
aperture for both the target and reference stars as the one that minimized the
root mean square (RMS) of the final differential light curves, which are
obtained by comparing XO-3 with three reference stars in the field. Highly
discrepant points and/or linear trends presented in these light curves were
removed. A summary of the observations and the data reduction procedures are
listed in Table 1. The final light curves are presented in Table 2 and plotted
in Figures 1 and 2.

Table 1: Overview of Observations and Data Reduction for XO-3 Date | Time | Telescope | Band | Frames | Exposure | Read | Airmass | Moon | Aperture aaThis column indicates the aperture diameter used in SExtractor. | Scatter bbThis column presents the RMS scatter of residuals from the best-fitting model.
---|---|---|---|---|---|---|---|---|---|---
(UTC) | (UTC) |  |  |  | (second) | (second) |  | illum. | (pixels) | (mmag)
2014 Mar 02 | $10:43:00\rightarrow 16:01:40$ | Xinglong $60\,{\rm cm}$ | $R$ | 306 | 60 | 3 | $1.06\rightarrow 1.88$ | 0.03 | 18 | 1.7
2014 Dec 11 | $12:05:58\rightarrow 16:27:51$ | Xinglong Schmidt | $R$ | 464 | 20 | 12 | $1.21\rightarrow 1.05\rightarrow 1.07$ | 0.75 | 16 | 2.5
2014 Dec 27 | $11:15:39\rightarrow 15:32:22$ | Xinglong Schmidt | $R$ | 515 | 18 | 12 | $1.18\rightarrow 1.05\rightarrow 1.08$ | 0.38 | 17 | 2.2
2015 Feb 16 | $10:37:34\rightarrow 16:45:33$ | Xinglong $60\,{\rm cm}$ | $R$ | 269 | 60 | 23 | $1.05\rightarrow 1.82$ | 0.08 | 30 | 1.6
2016 Jan 14 | $10:15:34\rightarrow 15:31:46$ | Xinglong $60\,{\rm cm}$ | $R$ | 336 | 10-25 | 6 | $1.17\rightarrow 1.05\rightarrow 1.15$ | 0.25 | 23 | 1.7
2016 Feb 02 | $11:17:33\rightarrow 15:47:20$ | Xinglong Schmidt | $R$ | 460 | 25 | 12 | $1.05\rightarrow 1.35$ | 0.35 | 18 | 1.5
2016 Feb 15 | $10:43:10\rightarrow 13:28:28$ | Xinglong Schmidt | $R$ | 198 | 25-55 | 12 | $1.05\rightarrow 1.15$ | 0.54 | 18 | 2.4
2016 Feb 18 | $10:35:44\rightarrow 14:54:19$ | Xinglong $60\,{\rm cm}$ | $R$ | 550 | 10-25 | 6 | $1.05\rightarrow 1.38$ | 0.84 | 18 | 2.7
2016 Mar 05 | $11:33:32\rightarrow 16:02:43$ | Xinglong $60\,{\rm cm}$ | $R$ | 506 | 25-45 | 6 | $1.11\rightarrow 2.00$ | 0.15 | 20 | 2.2
2017 Nov 11 | $11:07:08\rightarrow 14:59:15$ | Xinglong $60\,{\rm cm}$ | $B$ | 97 | 20-30 | 23 | $1.82\rightarrow 1.12$ | 0.41 | 23 | 4.8
2017 Nov 11 | $11:08:05\rightarrow 14:57:34$ | Xinglong $60\,{\rm cm}$ | $V$ | 96 | 20-30 | 23 | $1.82\rightarrow 1.12$ | 0.41 | 23 | 3.6
2017 Nov 14 | $15:31:09\rightarrow 21:57:23$ | Xinglong $60\,{\rm cm}$ | $R$ | 336 | 20 | 23 | $1.08\rightarrow 1.05\rightarrow 1.55$ | 0.14 | 25 | 2.3
Figure 1: Phased light curve of XO-3b including 33 literature transits
provided by previous works (Johns-Krull et al., 2008; Winn et al., 2008, 2009;
Bonomo et al., 2017) and 12 new transits collected in this work. To estimate
the system parameters, these light curves were simultaneously fitted with the
radial velocity measurements (Figure 3) as described in detail in §4. The
orange solid line represents the best-fitting model to all of the data and the
residuals are plotted below. Figure 2: Light curves of XO-3b transits,
obtained from the literature and as part of this work. The best-fitting models
are overplotted as solid red lines in the left panel. The residuals are
displayed in the right panel. Both light curves and residuals are sorted by
observation date for clarity. Table 2: Photometry for XO-3

${\rm BJD}$ | Flux | $\sigma_{\rm Flux}$ | Instrument | Filter
---|---|---|---|---
2456719.055590 | 0.9950 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.056318 | 0.9943 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.057046 | 0.9948 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.057773 | 0.9965 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.058501 | 0.9956 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.059229 | 0.9940 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.059957 | 0.9947 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.060685 | 0.9935 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.061412 | 0.9962 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.062140 | 0.9943 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.062868 | 0.9944 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
2456719.063597 | 0.9935 | 0.0017 | Xinglong $60\,{\rm cm}$ | $R$
… | … | … | … | …

Note. — The complete table is available in the machine readable format. We put
a portion here just for guidance concerning the form and content.

###  2.2 Velocimetry

XO-3 has been the subject of a variety of radial velocity (RV) observing
campaigns over the past two decades. In this work, we combine all previously
published RV data sets with a new set of velocity measurements obtained using
the Automated Planet Finder (APF) telescope located at Lick Observatory. The
APF couples a 2.4m primary mirror with the slit-fed, Levy echelle
spectrograph, which works at a typical spectral resolution of R $\sim$ 110,000
and delivers a peak overall system efficiency (fraction of photons striking
the telescope primary that are detected by the CCD) of 15% (Vogt et al.,
2014). The telescope was designed to search for planets in the liquid water
habitable zone of nearby stars. The APF is driven by a dynamic scheduling
software system that can make minute-to-minute decisions on what target to
observe based on the ambient atmospheric transparency, atmospheric seeing, and
lunar phase (Burt et al., 2015). This allows the telescope to operate
efficiently throughout the year without the need for human supervision.

Like its predecessors on the Keck and Magellan telescopes (HIRES: Vogt et al.
1994, and PFS: Crane et al. 2010, respectively), the APF uses a gaseous I2
cell to imprint a forest of narrow absorption lines on the stellar spectrum
before its incidence on the spectrograph slit (Butler et al., 1996). These I2
absorption lines create a stable wavelength calibration source and permit the
measurement of the spectrometer’s point spread function (PSF). For each
stellar spectrum, the 5000 - 6200Å region (which contains the highest density
of I2 lines) is subdivided into $\sim$700 individual 2Å segments, with each
segment providing an independent measure of the wavelength, the PSF, and the
Doppler shift. Our reported overall stellar velocity from a given spectrum is
a weighted mean of the individual segments’ velocity measurements. The
uncertainty for each velocity is the RMS of the individual segment velocity
values about the mean divided by the square root of the number of segments.
This “internal” uncertainty primarily represents errors in the fitting
process, which are dominated by Poisson statistics. The velocities are
expressed relative to the solar system barycenter, but are not referenced to
any absolute fiducial point. Since it began scientific operations in Q2 2013,
the APF has contributed to a number of planet detections (e.g., Burt et al.
2014, 2021; Fulton et al. 2015; Vogt et al. 2015; Christiansen et al. 2017)
and has showcased its ability to reach internal precisions of $\sim$ 1m/s on
bright, quiet stars. Indeed, the APF has consistently achieved internal
velocity precision of order $\sigma\lesssim$ 2 m/s on bright (e.g. V
$\lesssim$ 8) stars (Vogt et al., 2015).

Table 3 presents our newly collected RV measurements for XO-3b, with 16
individual exposures. The median internal uncertainty for our observations is
$\sigma_{i}\approx$ 15.9 m/s with an exposure time of $45\,{\rm mins}$. These
large internal uncertainties are driven by (1) the star’s high rotational
velocity ($V\sin i_{*}=17.3\pm 0.9{\rm\,km\,s^{-1}}$, see Section 3 for more
details), which rotationally broadens the stellar absorption lines, thereby
reducing their Doppler information content, and (2) by the star’s high
effective temperature ($T_{\rm eff}$ = 6471${}^{+83}_{-82}$K), which reduces
the overall number of absorption lines (Torres et al., 2012; Bouchy et al.,
2011; Beatty, & Gaudi, 2015).

Table 3:  APF RVs for XO-3 ${\rm BJD}$ |  | RV |  | $\sigma_{\rm RV}$
---|---|---|---|---
|  | m s-1 |  | m s-1
2457683.767 |  | -918.53333 |  | 14.410
2457683.798 |  | -951.22600 |  | 15.857
2457683.830 |  | -946.57400 |  | 15.773
2457683.861 |  | -951.54200 |  | 14.680
2457683.892 |  | -1031.46600 |  | 15.292
2457683.923 |  | -994.45200 |  | 14.566
2457683.955 |  | -1016.47200 |  | 15.046
2457683.986 |  | -986.96500 |  | 14.303
2457684.017 |  | -1077.11600 |  | 15.232
2457684.048 |  | -1050.91000 |  | 16.088
2457809.638 |  | 383.11333 |  | 25.398
2457809.669 |  | 422.80429 |  | 23.685
2457809.700 |  | 596.86000 |  | 23.145
2457809.732 |  | 720.67714 |  | 18.881
2457809.763 |  | 824.05000 |  | 18.534
2457809.794 |  | 852.69000 |  | 22.612

These APF velocities are combined with previously published RV data sets from
the following instruments (Figure 3): the HIgh Resolution Echelle Spectrometer
(HIRES) on Keck I (Vogt et al., 2014); the northern High Accuracy Radial
velocity Planet Searcher (HARPS-N) on the Telescopio Nazionale Galileo
(Cosentino et al., 2012, 2014); the Spectrographe pour l’Observation des
Ph$\mathrm{\acute{e}}$nom$\mathrm{\grave{e}}$nes des
Int$\mathrm{\acute{e}}$rieurs stellaires et des Exoplan$\mathrm{\grave{e}}$tes
(SOPHIE) on the 1.93m reflector telescope at the Haute-Provence Observatory
(Perruchot et al., 2008); the High Dispersion Spectrograph (HDS) on the Subaru
telescope (Noguchi et al., 2002); the High Resolution Spectrograph (HRS) on
the Hobby-Eberly Telescope and the Tull Coude spectrograph on the HJS
telescope (though, in this case, it was coupled to the Hobby-Eberly Telescope
via a fiber optic cable) (Tull et al., 1995; Tull, 1998); and the eShel
spectrograph at Stara Lesna Observatory (SLO) (Eversberg, 2016). Detailed
information on each RV dataset can be found in the references list in Table 4.

Like the APF and Keck/HIRES, HET’s HRS makes use of an iodine cell for its
wavelength calibration efforts and applies a forward modeling approach for
determining each observation’s RV measurement. The other instruments listed
here handle wavelength calibration with reference spectra taken using a ThAr
calibration lamp, either via simultaneous reference spectra in the case of
fiber-fed instruments or via separate calibration frames for the slit-fed
instruments.

For the instruments with a ThAr-based wavelength calibration, a 2-D spectrum
is extracted from the FITS file once an observation is complete. The stellar
spectrum is cross-correlated with a reference (in the case of HARPS-N, SOPHIE,
and HDS this is a numerical mask corresponding to the appropriate spectral
type, while for the Tull Coude spectrograph it is a particular spectrum of
XO-3 taken on BJD 2454137.8215). The resulting cross-correlation function
(CCF) is fit with a Gaussian curve to produce a radial velocity measurement,
and it is calibrated to determine the RV photon-noise uncertainty
$\sigma_{RV}$ (e.g. Baranne et al., 1996; Pepe et al., 2002).

Table 4: RV data sources Reference | Facility | Calibration |  | Nobs |  | RMS (m s-1 )
---|---|---|---|---|---|---
This work | Levy (APF) | I2 cell |  | 16 |  | 29.95
Bonomo et al. (2017) | HARPS-N (TNG) | ThAr lamp |  | 19 |  | 44.13
Hebrard et al. (2008) |  SOPHIE (1.93-m, HPO) | ThAr lamp |  | 34 |  | 66.37
Hirano et al. (2011) | HDS (Subaru) | ThAr lamp |  | 10 |  | 47.64
Johns-Krull et al. (2008) | HRS (HET) | I2 cell |  | 11 |  | 151.41
Johns-Krull et al. (2008) | Tull Coude (HJS) | I2 cell |  | 10 |  | 141.73
Knutson et al. (2014) | HIRES (Keck I) | I2 cell |  | 11 |  | 37.20
Winn et al. (2009) | HIRES (Keck I) | ThAr lamp |  | 11 |  | 11.45
Garai et al. (2017) | eShel (0.6-m, SLO) | ThAr lamp |  | 20 |  | 269.30
Figure 3: The radial velocity measurements of XO-3 from previous works (Hirano
et al., 2011; Johns-Krull et al., 2008; Hébrard et al., 2008; Winn et al.,
2009; Knutson et al., 2014; Bonomo et al., 2017; Garai et al., 2017) and from
this work, marked as different colors and shown as a function of orbital
phase. The best-fitting Keplerian orbit model (solid red line) is determined
from the joint fitting of RVs and light curves. The residuals of the best fit,
with an RMS scatter $\sigma=267.90\,\rm{m\,s^{-1}}$, is shown in the bottom
panel.

##  3 Stellar Atmospheric Parameters from APF

We determined the values of the stellar atmospheric parameters, including
$V\sin i_{*}$, $T_{\rm eff}$, and $\,{\rm log}\,{g}$, for XO-3 directly from
our APF spectrum using The Cannon (Ness et al., 2015; Casey et al., 2016), a
data-driven approach using generative modeling to determine stellar
parameters. The Cannon trains on an input set of stellar spectra with
previously determined parameter values in order to “learn” the characteristics
of this dataset. This trained model can then be applied to obtain the
corresponding labels for a new set of input spectra. The Cannon requires
overlapping wavelength coverage and assumes a similar set of systematics
between the training set of spectra and the data to which it is applied.

Following the methods of Rice & Brewer (2020), we ran The Cannon to determine
the properties of XO-3 directly from our APF spectrum. We trained our model
using Keck/HIRES spectra included in the Spectral Properties of Cool Stars
(SPOCS) dataset (Valenti $\&$ Fischer, 2005; Brewer et al., 2016), with
stellar parameters determined using the program Spectroscopy Made Easy (SME;
Valenti $\&$ Piskunov 1996).

From the full SPOCS sample, we removed all spectra flagged as ‘bad’ or labeled
with ‘NGC’, indicating that the target was not an individual star. We also cut
all spectra from our input sample with SNR $<$ 100, and we used only the
highest-SNR spectrum for each star in the sample. This resulted in a total of
1202 spectra in our training/validation sample. We used 86 overlap stars
observed by both Keck/HIRES (as part of the SPOCS sample) and by the APF as a
validation set to test our model performance, leaving 1116 stars in our
training sample.

We interpolated the HIRES and APF spectra onto an overlapping wavelength grid
to directly compare each spectrum. Then, we trained our model using only HIRES
spectra and applied it to our overlapping validation set of APF spectra to
determine uncertainties in each parameter. Our final values for XO-3 are
included in Table 5.

Because we trained our model using HIRES spectra, the differing systematics
between the Keck and APF instrumentation setups may affect the precision of
our label transfer process. Indeed, we find that the scatter when transferring
labels across datasets, quantified using our validation set, is higher than
when using only Keck/HIRES spectra as in Rice & Brewer (2020). This results in
higher uncertainties for our reported values. From our validation set, we also
find a systematic offset in the values of $V\sin i_{*}$ derived from our APF
spectra relative to the nominal SPOCS values; that is, the $V\sin i_{*}$
values obtained for APF spectra with The Cannon are on average -1.87 km/s
lower than the associated SPOCS values. This offset is likely attributable to
the difference in the line spread-function of the APF as compared to that of
Keck/HIRES. We correct for this offset in Table 5.

XO-3 is also included in the SPOCS sample with previously determined stellar
parameters obtained based on the Keck spectra alone. We compared these pre-
existing parameters with our results as an additional test to check the
fidelity of our model, focusing on $\,{\rm log}\,{g}$ and $T_{\rm eff}$, two
other dominant global stellar properties that should show no temporal changes.
As shown in Table 5, we find that all global stellar parameters obtained for
XO-3 from the APF data are in agreement with previously reported values
derived from the HIRES dataset.

Our results are detailed in Table 5. We report our uncertainties as the
scatter in our test set results when testing and training with the SPOCS
sample; these uncertainties may be underestimated due to the differing
systematics across instruments, which are not encompassed by this scatter.
Although our stellar atmospheric parameters from APF data show good agreement
with previous estimates derived from a wide range of independent datasets, we
adopt $\,{\rm log}\,{g}$ and $T_{\rm eff}$ from Torres et al. (2012) as priors
in our global fitting (see §4 for details) because their parameters benefit
from using combined high-SNR data from both Keck/HIRES and TRES/FIES.

Our primary parameter of interest is $V\sin i_{*}$ (see §5 for details), which
we compare with previous estimates in Table 5. Ultimately, we find that the
value of $V\sin i_{*}$ that we obtain directly from the APF spectrum is
consistent with previous estimates derived from a wide range of independent
datasets.

##  4 Planetary Parameters from Global Fitting

Table 5:  Stellar parameters for XO-3

Telescope | HJS | Keck | Keck | Subaru | Subaru+Keck | Keck+TRES | APF
---|---|---|---|---|---|---|---
Instrument | Tull Coude | HIRES | HIRES | HDS | HDS+HIRES | HIRES+FIES | Levy
Observation Date | 10/2006-02/2007 | 02/2009 | 02/2009 | 11/2009-02/2010 | 02/2009-02/2010 | 02/2009 | 10/2016
Method | SME | SME | R-M Fitting | R-M Fitting | R-M Fitting | SPC+SME+MOOG | Cannon
References | 1 | 2 | 3 | 4 | 4 | 5 | This work
$T_{\rm eff}$ (K) | $6429\pm 50$ | $6673\pm 25$ | … | … | … | $6759\pm 79$ | $6430\pm 69$
$[M/H]$ | $-0.204\pm 0.023$ | $-0.054\pm 0.01$ | … | … | … | $-0.05\pm 0.08$ | …
$\mathrm{log}g$ ($\rm{cm\,s^{-2}}$) | $3.95\pm 0.062$ | $4.15\pm 0.028$ | … | … | … | $4.24\pm 0.03^{\rm a}$ | $4.3\pm 0.1$
$V\sin i_{*}$ ($\rm{km\,s^{-1}}$) | $18.54\pm 0.17$ | $16.19\pm 0.5$ | $18.31\pm 1.3$ | $17.0\pm 1.2$ | $18.4\pm 0.8$ | $20.3\pm 2.0$ | $17.3\pm 0.9$

††footnotetext: aExternal constraint on $\mathrm{log}g$ was obtained from the
light curves (Winn et al., 2008).

††footnotetext: 1Johns-Krull et al. (2008); 2Brewer et al. (2016); 3Winn et
al. (2009); 4Hirano et al. (2011); 5Torres et al. (2012).

System Parameters. To determine the XO-3 system parameters, we used EXOFASTv2
(Eastman et al., 2013; Eastman, 2017) to simultaneously fit the transit light
curves and out-of-transit radial velocity data from the literature (Johns-
Krull et al., 2008; Winn et al., 2008; Hébrard et al., 2008; Winn et al.,
2009; Bonomo et al., 2017; Hirano et al., 2011; Knutson et al., 2014; Bonomo
et al., 2017; Garai et al., 2017), as well as our newly collected photometric
and RV data.

EXOFASTv2 performs a global analysis of exoplanetary and stellar parameters
using a differential evolution Markov Chain Monte Carlo (DE-MCMC, ter Braak
2006) simulation to simultaneously fit, for an arbitrary number of planets,
the Spectral Energy Distribution (SED), transit data, and RV data taken from
multiple instruments.

To constrain the stellar parameters, we used the MESA Isochrones and Stellar
Tracks (MIST) model (Choi et al., 2016; Dotter, 2016) included in EXOFASTv2.
The Gaussian priors were applied to the $T_{\rm eff}$ and $\left[{\rm Fe}/{\rm
H}\right]$ of the star derived in Torres et al. (2012). The limb-darkening
coefficients were assumed to be a quadratic function. We imposed wavelength-
dependent priors on limb-darkening coefficients from Claret & Bloemen (2011)
based on the $T_{\rm eff}$, $\,{\rm log}\,{g}$, and $\left[{\rm Fe}/{\rm
H}\right]$ from Torres et al. (2012). The priors for the orbital parameters,
including all transit and RV parameters, were adopted from the results of Winn
et al. (2009).

To minimize the convergence time, the fitting process required multiple short
runs before longer ones. We derived a new set of Gaussian priors after each
run, allowing us to begin subsequent fits at the most likely model. We
continually refined the fit until our criteria – both the number of
independent draws being greater than 1000 and a Gelman-Rubin statistic of less
than 1.01 for all parameters – were satisfied six consecutive times,
indicating that the chains were considered to be well-mixed (Eastman et al.,
2013).

The system parameters derived from global fit are listed in Table 6. The
fitting results are shown in Figure 1, 2 and 3.

Transit Timing Variations $\&$ Orbital Ephemeris. We modelled each available
transit light curve for XO-3 using the JKTEBOP (Southworth, 2008) code. We
fixed all global parameters to the results derived from the global fitting,
and we allowed only the transit mid-time ($T_{0}$) and baseline flux ($F_{0}$)
to vary as free parameters in the fit. We utilized the bootstrapping
technique, Monte Carlo simulations, and the residual-shift method to estimate
the errors of mid-transit times separately. The largest errors were selected
as the final errors to provide a conservative estimate. The result is shown in
Figure 4, which is consistent with a constant period. No significant transit
timing variations were detected.

To update the linear ephemeris ($T_{\rm C}+N\times P$), we performed a
weighted least squares fit to the derived mid-transit times ($T_{0}$). During
the fit, we followed the approaches described in Southworth et al. (2017), and
we re-scaled the uncertainties of each transit mid-time such that
$\chi^{2}_{\rm reduced}=1$. This choice was made to provide conservative
errors for the transit mid-time at the reference epoch ($T_{\rm C}$) and
orbital period ($P$) for future scheduling purposes. The result agrees with
the values from Winn et al. (2008) within 1$\sigma$.

Table 6: System Parameters for XO-3

Parameter | Units | This work | Previous work | Agreement($\sigma$) | Ref
---|---|---|---|---|---
Stellar Parameters:
$M_{*}$ | Mass ($\,{\rm M_{\sun}}$) | $1.219^{+0.090}_{-0.095}$ | $1.213\pm 0.066$ | 0.05 | Winn2008
$R_{*}$ | Radius ($\,{\rm R_{\sun}}$) | $1.371^{+0.041}_{-0.042}$ | $1.377\pm 0.083$ | 0.06 | Winn2008
$L_{*}$ | Luminosity ($\,{\rm L_{\sun}}$) | $2.97^{+0.27}_{-0.26}$ | $2.92^{+0.59}_{-0.48}$ | 0.09 | Winn2008
$\rho_{*}$ | Density (cgs) | $0.665^{+0.033}_{-0.030}$ | $0.650\pm 0.086$ | 0.16 | Winn2008
$\log{g}$ | Surface gravity (cgs) | $4.249^{+0.017}_{-0.018}$ | $4.244\pm 0.041$ | 0.11 | Winn2008
$T_{\rm eff}$ | Effective Temperature (K) | $6471^{+83}_{-82}$ | $6429\pm 100$ | 0.32 | Winn2008
$[{\rm Fe/H}]$ | Metallicity | $-0.176^{+0.080}_{-0.081}$ | $-0.177\pm 0.080$ | 0.01 | Winn2008
$[{\rm Fe/H}]_{0}$ | Initial Metallicity | $-0.025^{+0.072}_{-0.071}$ | … | … | …
$Age$ | Age (Gyr) | $2.6^{+1.6}_{-1.1}$ | $2.82^{+0.58}_{-0.82}$ | 0.18 | Winn2008
$EEP$ | Equal Evolutionary Point | $361^{+39}_{-21}$ | … | … | …
$A_{v}$ | V-band extinction | $0.069^{+0.091}_{-0.049}$ | … | … | …
$\sigma_{\rm SED}$ | SED photometry error scaling | $2.8^{+2.8}_{-1.1}$ | … | … | …
$d$ | Distance (pc) | $178.6^{+6.9}_{-7.4}$ | $174\pm 18$ | 0.24 | Winn2008
$\pi$ | Parallax (mas) | $5.60^{+0.24}_{-0.21}$ | … | … | …
Planetary Parameters:
$P$ | Period (days) | $3.19152\pm 0.00145^{a}$ | $3.1915239\pm 0.0000068$ | 0.00 | Winn2008
$R_{P}$ | Radius ($\,{\rm R_{\rm J}}$) | $1.219^{+0.039}_{-0.040}$ | … | … | …
$T_{C}$ | Time of Transit ($\rm{BJD_{TDB}}$) | $2454449.86969\pm 0.00073^{a}$ | $2454449.868937\pm 0.00023$ | 0.99 | Winn2008
$T_{0}$ | Optimal Transit Time ($\rm{BJD_{TDB}}$) | $2455314.77290\pm 0.00015$ | … | … | …
$a$ | Semi-major axis (AU) | $0.0455^{+0.0011}_{-0.0012}$ | $0.04539^{+0.00081}_{-0.00084}$ | 0.08 | Bonomo2017
$i$ | Inclination (Degrees) | $84.26\pm 0.19$ | $84.20\pm 0.54$ | 0.1 | win2008
$e$ | Eccentricity | $0.2853^{+0.0027}_{-0.0026}$ | $0.27587^{+0.00071}_{-0.00067}$ | 3.39 | Bonomo2017
$\omega_{*}$ | Argument of Periastron (Degrees) | $349.6\pm 1.2$ | $349.35^{+0.67}_{-0.68}$ | 1.78 | Bonomo2017
$T_{\mathrm{eq}}$ | Equilibrium temperature (K) | $1714\pm 26$ | $1710\pm 46$ | 0.08 | Winn2008
$M_{P}$ | Mass ($\,{\rm M_{\rm J}}$) | $11.92^{+0.59}_{-0.63}$ | $11.7^{+0.43}_{-0.43}$ | 0.3 | Bonomo2017
$K$ | RV semi-amplitude (m/s) | $1488.0^{+8.7}_{-9.1}$ | $1468.9^{+4.6}_{-4.5}$ | 1.95 | Bonomo2017
$\mathrm{log}K$ | Log of RV semi-amplitude | $3.1726^{+0.0025}_{-0.0026}$ | … | … | …
$R_{P}/R_{*}$ | Radius of planet in stellar radii | $0.09139^{+0.00039}_{-0.00040}$ | $0.09057\pm 0.00057$ | 1.19 | Winn2008
$a/R_{*}$ | Semi-major axis in stellar radii | $7.12^{+0.12}_{-0.11}$ | $7.07\pm 0.31$ | 0.15 | Winn2008
$\delta$ | Transit depth (fraction) | $0.008352^{+0.000071}_{-0.000072}$ | … | … | …
Depth | Flux decrement at mid transit | $0.008352^{+0.000071}_{-0.000072}$ | … | … | …
$\tau$ | Ingress/egress transit duration (days) | $0.01904\pm 0.00067$ | $0.466\pm 0.033$ | 0.25 | Winn2008
$T_{14}$ | Total transit duration (days) | $0.12334\pm 0.00060$ | … | … | …
$T_{\mathrm{FWHM}}$ | FWHM transit duration (days) | $0.10430\pm 0.00033$ | … | … | …
$b$ | Transit Impact parameter | $0.700^{+0.011}_{-0.012}$ | $0.705\pm 0.023$ | 0.19 | Winn2008
$b_{S}$ | Eclipse impact parameter | $0.614^{+0.012}_{-0.013}$ | … | … | …
$\tau_{S}$ | Ingress/egress eclipse duration (days) | $0.01509^{+0.00053}_{-0.00052}$ | … | … | …
$T_{S,14}$ | Total eclipse duration (days) | $0.1167\pm 0.0012$ | … | … | …
$T_{S,\mathrm{FWHM}}$ | FWHM eclipse duration (days) | $0.10159^{+0.00069}_{-0.00075}$ | … | … | …
$\delta_{S,3.6\mu m}$ | Blackbody eclipse depth at 3.6$\mu$m (ppm) | $754^{+24}_{-25}$ | … | … | …
$\delta_{S,4.5\mu m}$ | Blackbody eclipse depth at 4.5$\mu$m (ppm) | $975\pm 27$ | … | … | …
$\rho_{P}$ | Density (cgs) | $8.15^{+0.54}_{-0.50}$ | $8.1^{+1.6}_{-1.3}$ | 0.04 | Bonomo2017
$\mathrm{log}g_{P}$ | Surface gravity | $4.298^{+0.017}_{-0.016}$ | $4.293^{+0.055}_{-0.053}$ | 0.09 | Bonomo2017
$\Theta$ | Safronov Number | $0.729^{+0.025}_{-0.023}$ | … | … | …
$\langle F\rangle$ | Incident Flux (109 erg s-1 cm-2) | $1.81\pm 0.11$ | … | … | …
$T_{P}$ | Time of Periastron ($\rm{BJD_{TDB}}$) | $2454449.2483^{+0.0086}_{-0.0089}$ | … | … | …
$T_{S}$ | Time of eclipse ($\rm{BJD_{TDB}}$) | $2454448.8324^{+0.0036}_{-0.0038}$ | … | … | …
$T_{A}$ | Time of Ascending Node ($\rm{BJD_{TDB}}$) | $2454449.3105^{+0.0035}_{-0.0037}$ | … | … | …
$T_{D}$ | Time of Descending Node ($\rm{BJD_{TDB}}$) | $2454447.852\pm 0.010$ | … | … | …
$e\cos{\omega_{*}}$ |  | $0.2778^{+0.0018}_{-0.0019}$ | $0.27111_{-0.00033}^{+0.00034}$ | 3.65 | Bonomo2017
$e\sin{\omega_{*}}$ |  | $-0.0648^{+0.0061}_{-0.0063}$ | $-0.0510_{-0.0034}^{+0.0033}$ | 1.93 | Bonomo2017
$M_{P}\sin i$ | Minimum mass ($\,{\rm M_{\rm J}}$) | $11.86^{+0.58}_{-0.63}$ | … | … | …
$M_{P}/M_{*}$ | Mass ratio | $0.00934^{+0.00026}_{-0.00023}$ | $0.00927\pm 0.00036$ | 0.16 | Winn2008
$d/R_{*}$ | Separation at mid transit | $7.00\pm 0.13$ | … | … | …
$P_{T}$ | A priori non-grazing transit prob | $0.1298^{+0.0023}_{-0.0024}$ | … | … | …
$P_{T,G}$ | A priori transit prob | $0.1559\pm 0.0029$ | … | … | …
$P_{S}$ | A priori non-grazing eclipse prob | $0.1478^{+0.0021}_{-0.0022}$ | … | … | …
$P_{S,G}$ | A priori eclipse prob | $0.1776\pm 0.0027$ | … | … | …
Wavelength Parameters:
$u_{1B}$ | linear limb-darkening coeff | $0.497\pm 0.028$ | … | … | …
$u_{2B}$ | quadratic limb-darkening coeff | $0.277\pm 0.026$ | … | … | …
$u_{1I}$ | linear limb-darkening coeff | $0.197\pm 0.017$ | $0.06\pm 0.15$ | 0.91 | Winn2008
$u_{2I}$ | quadratic limb-darkening coeff | $0.301\pm 0.016$ | 0.35b | … | Winn2008
$u_{1R}$ | linear limb-darkening coeff | $0.260^{+0.013}_{-0.014}$ | $0.16\pm 0.14$ | 0.71 | Winn2008
$u_{2R}$ | quadratic limb-darkening coeff | $0.317\pm 0.012$ | 0.37b | … | Winn2008
$u_{1\mathrm{Sloan}r}$ | linear limb-darkening coeff | $0.293\pm 0.044$ | … | … | …
$u_{2\mathrm{Sloan}r}$ | quadratic limb-darkening coeff | $0.336^{+0.046}_{-0.045}$ | … | … | …
$u_{1\mathrm{Sloan}z}$ | linear limb-darkening coeff | $0.174\pm 0.019$ | $0.11\pm 0.07$ | 0.88 | Winn2008
$u_{2\mathrm{Sloan}z}$ | quadratic limb-darkening coeff | $0.301\pm 0.019$ | 0.353b | … | Winn2008
$u_{1V}$ | linear limb-darkening coeff | $0.366^{+0.020}_{-0.019}$ | $0.47\pm 0.14$ | 0.74 | Winn2008
$u_{2V}$ | quadratic limb-darkening coeff | $0.327\pm 0.019$ | 0.36b | … | Winn2008

††footnotetext: a Conservative values from JKTEBOP fit.

††footnotetext: b In the fit of Winn et al. (2008), the quadratic coefficient
is fixed.

Table 7: Comparison of the System Parameters before Hébrard et al. (2008) and after Winn et al. (2009) Parameter | Units | Before Hébrard et al. (2008) | After Winn et al. (2009) | Agreement($\sigma$)
---|---|---|---|---
Stellar Parameters:
$M_{*}$ | Mass ($\,{\rm M_{\sun}}$) | $1.225^{+0.090}_{-0.097}$ | $1.216^{+0.087}_{-0.092}$ | 0.07
$R_{*}$ | Radius ($\,{\rm R_{\sun}}$) | $1.395\pm 0.060$ | $1.325\pm 0.050$ | 0.9
$L_{*}$ | Luminosity ($\,{\rm L_{\sun}}$) | $3.07^{+0.34}_{-0.32}$ | $2.81^{+0.29}_{-0.27}$ | 0.6
$\rho_{*}$ | Density (cgs) | $0.634^{+0.067}_{-0.059}$ | $0.734^{+0.061}_{-0.055}$ | 1.18
$\log{g}$ | Surface gravity (cgs) | $4.236\pm 0.030$ | $4.277\pm 0.024$ | 1.07
$T_{\rm eff}$ | Effective Temperature (K) | $6467^{+83}_{-81}$ | $6490^{+83}_{-79}$ | 0.2
$[{\rm Fe/H}]$ | Metallicity | $-0.174^{+0.079}_{-0.078}$ | $-0.179^{+0.081}_{-0.082}$ | 0.04
$[{\rm Fe/H}]_{0}$ | Initial Metallicity | $-0.021^{+0.072}_{-0.071}$ | $-0.041\pm 0.070$ | 0.2
$Age$ | Age (Gyr) | $2.7^{+1.6}_{-1.1}$ | $2.2^{+1.6}_{-1.0}$ | 0.26
$EEP$ | Equal Evolutionary Point | $365^{+38}_{-23}$ | $349^{+38}_{-18}$ | 0.38
$A_{v}$ | V-band extinction | $0.068^{+0.092}_{-0.048}$ | $0.072^{+0.090}_{-0.051}$ | 0.04
$\sigma_{\mathrm{SED}}$ | SED photometry error scaling | $2.8^{+2.8}_{-1.1}$ | $2.6^{+2.5}_{-1.0}$ | 0.07
$d$ | Distance (pc) | $181.6^{+8.8}_{-9.0}$ | $173.4^{+7.6}_{-7.8}$ | 0.7
$\pi$ | Parallax (mas) | $5.51^{+0.29}_{-0.25}$ | $5.77^{+0.27}_{-0.24}$ | 0.71
Planetary Parameters:
$P$ | Period (days) | $3.1915219\pm 0.0000035$ | $3.1915264^{+0.0000020}_{-0.0000019}$ | 1.12
$R_{P}$ | Radius ($\,{\rm R_{\rm J}}$) | $1.229^{+0.055}_{-0.056}$ | $1.183^{+0.050}_{-0.049}$ | 0.62
$T_{C}$ | Time of Transit ($\rm{BJD_{TDB}}$) | $2454449.86996^{+0.00025}_{-0.00026}$ | $2454449.8685\pm 0.0016$ | 0.9
$T_{0}$ | Optimal Transit Time ($\rm{BJD_{TDB}}$) | $2454357.31583\pm 0.00023$ | $2454858.3839\pm 0.0016$ | .
$a$ | Semi-major axis (AU) | $0.0455^{+0.0011}_{-0.0012}$ | $0.0454^{+0.0011}_{-0.0012}$ | 0.06
$i$ | Inclination (Degrees) | $84.08^{+0.38}_{-0.40}$ | $84.67^{+0.31}_{-0.30}$ | 1.17
$e$ | Eccentricity | $0.261\pm 0.015$ | $0.2887^{+0.0061}_{-0.0059}$ | 1.71
$\omega_{*}$ | Argument of Periastron (Degrees) | $-11.9^{+6.3}_{-6.0}$ | $-15.4\pm 2.3$ | 0.52
$T_{\rm eq}$ | Equilibrium temperature (K) | $1727\pm 35$ | $1691\pm 31$ | 0.77
$M_{P}$ | Mass ($\,{\rm M_{\rm J}}$) | $11.93^{+0.69}_{-0.71}$ | $11.56^{+0.59}_{-0.63}$ | 0.40
$K$ | RV semi-amplitude (m/s) | $1474\pm 45$ | $1449^{+26}_{-27}$ | 0.48
$\mathrm{log}K$ | Log of RV semi-amplitude | $3.169\pm 0.013$ | $3.1612^{+0.0077}_{-0.0082}$ | 0.51
$R_{P}/R_{*}$ | Radius of planet in stellar radii | $0.09050^{+0.00052}_{-0.00054}$ | $0.09174^{+0.00063}_{-0.00064}$ | 1.49
$a/R_{*}$ | Semi-major axis in stellar radii | $7.01^{+0.24}_{-0.23}$ | $7.36^{+0.20}_{-0.19}$ | 1.15
$\delta$ | Transit depth (fraction) | $0.008191^{+0.000095}_{-0.000097}$ | $0.00842\pm 0.00012$ | 1.48
$Depth$ | Flux decrement at mid transit | $0.008191^{+0.000095}_{-0.000097}$ | $0.00842\pm 0.00012$ | 1.48
$\tau$ | Ingress/egress transit duration (days) | $0.01948^{+0.00089}_{-0.00090}$ | $0.0182\pm 0.0010$ | 0.96
$T_{14}$ | Total transit duration (days) | $0.12298^{+0.00083}_{-0.00085}$ | $0.12315^{+0.00094}_{-0.00093}$ | 0.13
$T_{\rm FWHM}$ | FWHM transit duration (days) | $0.10350\pm 0.00046$ | $0.10495^{+0.00053}_{-0.00054}$ | 2.07
$b$ | Transit Impact parameter | $0.713^{+0.013}_{-0.015}$ | $0.679^{+0.019}_{-0.021}$ | 1.38
$b_{S}$ | Eclipse impact parameter | $0.640^{+0.039}_{-0.037}$ | $0.582^{+0.022}_{-0.023}$ | 1.28
$\tau_{S}$ | Ingress/egress eclipse duration (days) | $0.0159^{+0.0018}_{-0.0015}$ | $0.01404^{+0.00084}_{-0.00080}$ | 0.94
$T_{S,14}$ | Total eclipse duration (days) | $0.1182^{+0.0031}_{-0.0036}$ | $0.1142^{+0.0022}_{-0.0023}$ | 1.04
$T_{S,\mathrm{FWHM}}$ | FWHM eclipse duration (days) | $0.1023^{+0.0012}_{-0.0021}$ | $0.1001^{+0.0014}_{-0.0016}$ | 1.1
$\delta_{S,3.6\mu m}$ | Blackbody eclipse depth at 3.6$\mu$m (ppm) | $754\pm 38$ | $731^{+36}_{-35}$ | 0.45
$\delta_{S,4.5\mu m}$ | Blackbody eclipse depth at 4.5$\mu$m (ppm) | $973\pm 42$ | $951\pm 40$ | 0.38
$\rho_{P}$ | Density (cgs) | $7.97^{+1.0}_{-0.90}$ | $8.64^{+0.93}_{-0.83}$ | 0.52
$\mathrm{log}g_{P}$ | Surface gravity | $4.292^{+0.035}_{-0.036}$ | $4.310^{+0.029}_{-0.028}$ | 0.39
$\Theta$ | Safronov Number | $0.722^{+0.043}_{-0.041}$ | $0.730^{+0.034}_{-0.033}$ | 0.15
$\langle F\rangle$ | Incident Flux (109 erg s-1 cm-2) | $1.89^{+0.16}_{-0.15}$ | $1.71^{+0.13}_{-0.12}$ | 0.9
$T_{P}$ | Time of Periastron ($\rm{BJD_{TDB}}$) | $2454449.234^{+0.050}_{-0.051}$ | $2454449.230\pm 0.016$ | 0.08
$T_{S}$ | Time of eclipse ($\rm{BJD_{TDB}}$) | $2454448.785^{+0.026}_{-0.028}$ | $2454448.8326^{+0.0074}_{-0.0078}$ | 1.64
$T_{A}$ | Time of Ascending Node ($\rm{BJD_{TDB}}$) | $2454449.293^{+0.021}_{-0.023}$ | $2454449.3025^{+0.0066}_{-0.0067}$ | 0.40
$T_{D}$ | Time of Descending Node ($\rm{BJD_{TDB}}$) | $2454447.809^{+0.045}_{-0.046}$ | $2454447.869^{+0.021}_{-0.020}$ | 1.19
$ecos{\omega_{*}}$ |  | $0.254^{+0.013}_{-0.014}$ | $0.2784^{+0.0038}_{-0.0041}$ | 1.68
$esin{\omega_{*}}$ |  | $-0.053\pm 0.029$ | $-0.077\pm 0.012$ | 0.76
$M_{P}\sin i$ | Minimum mass ($\,{\rm M_{\rm J}}$) | $11.87^{+0.68}_{-0.71}$ | $11.51^{+0.59}_{-0.62}$ | 0.39
$M_{P}/M_{*}$ | Mass ratio | $0.00932^{+0.00038}_{-0.00036}$ | $0.00909^{+0.00029}_{-0.00027}$ | 0.49
$d/R_{*}$ | Separation at mid transit | $6.90^{+0.40}_{-0.38}$ | $7.31^{+0.24}_{-0.23}$ | 0.91
$P_{T}$ | A priori non-grazing transit prob | $0.1317^{+0.0077}_{-0.0073}$ | $0.1243\pm 0.0039$ | 0.86
$P_{T,G}$ | A priori transit prob | $0.1579^{+0.0093}_{-0.0088}$ | $0.1494^{+0.0049}_{-0.0048}$ | 0.81
$P_{S}$ | A priori non-grazing eclipse prob | $0.1466^{+0.0033}_{-0.0034}$ | $0.1450^{+0.0034}_{-0.0035}$ | 0.33
$P_{S,G}$ | A priori eclipse prob | $0.1758\pm 0.0041$ | $0.1743^{+0.0043}_{-0.0044}$ | 0.25
Wavelength Parameters:
$u_{1B}$ | linear limb-darkening coeff | $0.500\pm 0.032$ | $0.488\pm 0.050$ | 0.2
$u_{2B}$ | quadratic limb-darkening coeff | $0.277\pm 0.030$ | $0.277\pm 0.050$ | 0.0
$u_{1I}$ | linear limb-darkening coeff | $0.190\pm 0.018$ | $0.221\pm 0.042$ | 0.68
$u_{2I}$ | quadratic limb-darkening coeff | $0.299^{+0.017}_{-0.018}$ | $0.300\pm 0.044$ | 0.02
$u_{1R}$ | linear limb-darkening coeff | $0.272^{+0.019}_{-0.020}$ | $0.258\pm 0.016$ | 0.56
$u_{2R}$ | quadratic limb-darkening coeff | $0.324\pm 0.018$ | $0.319\pm 0.015$ | 0.21
$u_{1\mathrm{Sloan}z}$ | linear limb-darkening coeff | $0.166\pm 0.021$ | … | …
$u_{2\mathrm{Sloan}z}$ | quadratic limb-darkening coeff | $0.294\pm 0.021$ | … | …
$u_{1\mathrm{Sloan}i}$ | linear limb-darkening coeff | … | $0.306\pm 0.044$ | …
$u_{2\mathrm{Sloan}i}$ | quadratic limb-darkening coeff | … | $0.340\pm 0.045$ | …
$u_{1V}$ | linear limb-darkening coeff | $0.364\pm 0.023$ | $0.341^{+0.049}_{-0.050}$ | 0.42
$u_{2V}$ | quadratic limb-darkening coeff | $0.324^{+0.022}_{-0.021}$ | $0.310\pm 0.049$ | 0.26

##  5 Discussion

Our global analysis of 45 transit light curves (including 12 collected in this
work) and 142 Doppler velocities (including 16 collected in this work) spans
more than 10 years, making XO-3b one of the best-studied exoplanets.

The results from our global analysis (Table 6) show good agreement with
previous work (Winn et al., 2009; Wong et al., 2014; Bonomo et al., 2017),
except that we find a slightly higher eccentricity ($e$) by $0.01$.

Our result confirms that the XO-3 system is unique, containing a massive
planet ($M_{\rm P}=11.92^{+0.59}_{-0.63}\,{\rm\,{\rm M_{\rm J}}}$) in a
relatively eccentric ($e=0.2853^{+0.0027}_{-0.0026}$) and short-period
($3.19152\pm 0.00145\,$day) orbit around a massive star
($M_{*}=1.219^{+0.090}_{-0.095}M_{\odot}$).

Our result also confirms the relatively high Safronov number, $\Theta=0.73$,
of XO-3b (Safronov, 1972). The Safronov number is defined as

$\Theta={{1}\over{2}}[{{V_{\rm esc}}\over{V_{\rm orb}}}]^{2}={M_{\rm
P}\over{M_{*}}}{{a}\over{R_{\rm P}}},$ (1)

where $V_{\rm esc}=\sqrt{2GM_{\rm P}/R_{\rm P}}$ is the planetary escape
velocity, and $V_{\rm orb}=\sqrt{GM_{*}/a}$ is the planet’s circular orbital
velocity. $\Theta$ connects to the outcome of instability in an N-body system
(for example, a system that hosts a hot Jupiter and close-in test particles).
If $\Theta>>1$, the ejection of test particles is very likely when instability
occurs, whereas for most hot Jupiters, $\Theta$ is substantially smaller than
1, and collisions (either between test particles and the planet, or between
test particles and the star) constitute a more likely outcome. Although XO-3b
is on a 3-day orbit, its high mass drives its Safronov number close to unity.

We explored the dynamical behavior of test particles on both interior and
exterior orbits between $1.3$ to $7.5$ mutual Hill radii (Gladman, 1993) with
XO-3b. As shown in Figure 6, the stability of test particles increases with
increasing distance from XO-3b. All of the unstable particles initially lay
within 3.5 mutual Hill radii of the planet, in concordance with the rule of
thumb found by Chambers et al. 1996. In terms of the channels for instability,
$80\%$ collided with XO-3, while $20\%$ were ejected from system. The
integrations indicate that XO-3 is approaching (but has not reached) a regime
in which small bodies in the vicinity of the planet are predominantly ejected
from the system rather than incorporated into the planet. This suggests the
existence of a cross-over mass slightly above that of XO-3 where further
growth in mass would be restricted, and it is consistent with the observed
near-absence of such objects in the short-period planet population (Marcy &
Butler, 2000), despite their ready detectability through either transit
photometry or Doppler velocity.

Figure 4: Observed minus calculated mid-transit times (O-C) for XO-3b,
according to the updated linear ephemeris. The $(O-C)=0$ reference is
represented by a red solid line, and the red dashed line indicates the
propagation of $\pm$ 1 $\sigma$ errors of the updated linear ephemeris. Figure
5: Spectroscopic radial velocities of XO-3 from Hébrard et al. (2008) (left),
Winn et al. (2009) (center), and Hirano et al. (2011) (right) as a function of
orbital phase (minutes from mid-transit), coupled with the best-fitting R-M
models (red dashed line). The RMS of the residuals from the three works, from
left to right, is 40 $\rm m\,s^{-1}$, 18 $\rm m\,s^{-1}$, and 23 $\rm
m\,s^{-1}$. Figure 6: A top-down view of the XO-3 system. Initial orbits of
300 test particles are evenly distributed over $1.3-7.5$ mutual Hill radii
with XO-3b. Orbits remaining after a $10^{6}$ year integration are shown in
blue. All orbits lying within $3.5$ mutual Hill radii of XO-3b have been
destabilized. Among the unstable particles, $80\%$ collide with XO-3 (red),
and, as a consequence of XO-3b’s large mass, $20\%$ are ejected (green).

XO-3b is also known as the first exoplanet measured to have a large spin-orbit
misalignment ($70.0\,^{\circ}\pm 15.0\,^{\circ}$, Hébrard et al. 2008).
Follow-up studies on the stellar obliquity of the XO-3 system based on radial
velocities collected with Keck/HIRES and Subaru/HDS found a smaller angle
($37.3\,^{\circ}\pm 3.7\,^{\circ}$, Winn et al. 2009; and $37.3\,^{\circ}\pm
3.0\,^{\circ}$, Hirano et al. 2011). Although there are indications of
systematic effects in previous datasets (see Hirano et al. 2011 for detailed
discussion), the discrepancy could also be caused by true astrophysical
reasons that warrant investigation.

Astrophysical Origin. The true angle between the orbital angular momentum
vector of XO-3b and its stellar spin vector can be determined from three
independent angles: The sky-projected spin-orbit angle ($\lambda$), which can
be measured through the Rossiter-McLaughlin effect; the orbital angular
momentum vector of the transiting planet along the line of sight (e.g.,
transit inclination, $i$), which can be determined by modelling the transit
light curve; and the stellar spin vector of the star along the line of sight
($i_{*}$), which can be estimated by measuring $V_{*}\sin{i_{*}}$ from
spectroscopic observation and $V_{*}$ by analyzing periodic photometric
variations in the light curve caused by the stellar spots.

If the discrepancy between previous $\lambda$ measurements is truly caused by
a change in the spin-orbit angle of XO-3 system, then it is statistically
improbable that it only changed within the projected plane (perpendicular to
the line of sight, e.g., $\lambda$). In the following analyses, we examine
whether the stellar spin vector of XO-3 along the line of sight ($i_{*}$) or
the orbital angular momentum vector of XO-3b along the line of sight (e.g.,
transit inclination, $i$) also changed over time. We do not find strong
evidence in support of these changes.

Stellar Spin Vector along the line of sight ($i_{*}$). Internal gravity waves
within hot (and thus massive) stars (Rogers et al., 2012) can induce time-
dependent variations in the direction of the stellar surface spin. Given that
XO-3 is a fairly hot and fairly massive star, we tested whether this type of
stellar polar wander occurs in this system.

The combination of estimates of $V_{*}\sin{i_{*}}$ (from spectroscopic
measurements) and $V_{*}$ (from light curve measurements) can provide the
stellar spin vector of a star along the line of sight ($i_{*}$). Although the
spectroscopic rotation velocity obtained from Johns-Krull et al. (2008)
($V_{\rm*before}\sin{i_{\rm*before}}=18.54\pm 0.17\rm\ km\ s^{-1}$) agrees
with the average of Winn et al. (2009), Hirano et al. (2011), Torres et al.
(2012), and Brewer et al. (2016) ($V_{\rm*after}\sin{i_{\rm*after}}=17.9\pm
0.5\rm\ km\ s^{-1}$) within $1.2\sigma$, we cannot constrain the possible
variation of the stellar spin vector ($\Delta i_{*}$) that might occur, since
XO-3 did not leave a sufficient signal (which would typically manifest as
periodic photometric variations from star spots) in its light curve to measure
its true rotation velocity ($V_{*}$).

However, the difference between the stellar spin vector of XO-3 along the line
of sight before Hébrard et al. (2008) and after Winn et al. (2009) ($\Delta
i_{*}=i_{\rm*after}-i_{\rm*before}=\arcsin(\frac{V_{\rm*after}\sin{i_{\rm*after}}}{V_{*}})-\arcsin(\frac{V_{\rm*before}\sin{i_{\rm*before}}}{V_{*}})$)
only depends on the true rotational velocity of the star $V_{*}$, if we assume
that the rotational velocity of the star, $V_{*}$, does not change over time,
this means that the rotational velocity before Hébrard et al. (2008) and after
Winn et al. (2009) $V_{\rm*before}=V_{\rm*after}=V_{*}$. The maximum possible
variation is then $\Delta i_{*}=15.26\,^{\circ}\pm 5.82\,^{\circ}$, when
$V_{*}=\mathrm{max}(V_{\rm*}\sin{i_{\rm*\ before}},V_{\rm*}\sin{i_{\rm*\
after}})=V_{\rm*\ before}=18.54\pm 0.17\rm\ km\ s^{-1}$. This agrees with no
change (e.g., $\Delta i_{*}=0^{\circ}$) within $2.6\sigma$. This is the
maximum change that is consistent with current data, since here we assume the
minimum possible value of $V_{*}$ corresponds to the largest possible $\Delta
i_{*}$.

Orbital Angular Momentum Vector. Precession of nodes due to an additional
perturber could also cause the temporal variation of the direction of the
orbital angular momentum vector (Innanen et al., 1997). Precession can
manifest as temporal variation of the transit profile, with evolution
occurring in the impact parameter ($b$), the transit inclinations ($i$), the
transit duration ($T_{14}$), and the transit depth ($\delta$). We thus
separately fit the transit and radial velocity data before the R-M measurement
conducted by Hébrard et al. (2008), and after the R-M measurement conducted by
Winn et al. (2009). As shown in Table 7 and Figure 7, we found that the
transit inclinations derived from the data before Hébrard et al. (2008)) and
after Winn et al. (2009) are in excellent agreement with each other with only
$\Delta i=0.59^{\circ}\pm 0.5^{\circ}$ difference, which means that the
orbital angular momentum vector of XO-3b along the line of sight agrees with
no change within $1.2\sigma$.

In conclusion, we found no evidence for temporal changes of either the stellar
spin vector or the orbital angular momentum vector of XO-3b along the line of
sight.

Systematic noise. Winn et al. (2009) and Hirano et al. (2011) suspected that
the discrepancy in multiple R-M measurements for XO-3b is most likely due to
systematic errors in the datasets (Hébrard et al., 2008), which will lead to
underestimated uncertainties in $\lambda$.

We bin the residuals of each of three R-M measurements (Hébrard et al., 2008;
Winn et al., 2009; Hirano et al., 2011) into bin sizes $N=1-15$ and evaluate
the RMS of the data. We found that the RMS of the time-binned residuals for
all three datasets decreases more slowly than $N^{1/2}$ ($N^{1/3}=$ for
Hébrard et al. 2008, $N^{1/5}=$ for Winn et al. 2009, $N^{1/3}=$ for Hirano et
al. 2011), suggesting that strong correlated noise is present.

This is also clear in the residual panels of Figure 5. In this figure, we took
the data directly from Hébrard et al. (2008), Winn et al. (2009), and Hirano
et al. (2011), and we subtracted away the baseline RV trend from each dataset
using the parameters reported by each respective paper. We then plotted the
reported model (using the previous authors’ results) in Figure 5, showing the
residual amplitudes below.

The data from all three datasets (Hébrard et al., 2008; Winn et al., 2009;
Hirano et al., 2011) suffers from strong systematic errors: that is, the
residuals show clear structure that remains after the best-fitting model has
been subtracted from the data. This is not surprising for a hot and massive
star like XO-3, and it leads to an underestimate of uncertainties in
$\lambda$.

Although current observations still allow for quite a bit of temporal
variation of the spin-orbit angle of XO-3 system, we have demonstrated that
the disagreement between sky-projected spin-orbit angles measured from
previous studies very likely results from the underestimate of the
uncertainties of $\lambda$ due to the presence of systematic noise in the
data, with no requirement for a true temporal variation in the spin-orbit
angle. We also find no strong evidence that two other angles have changed.

Figure 7: Left: Phased transit light curves before the R-M measurement
conducted by Hébrard et al. (2008). Right: Phased transit light curves after
the R-M measurement conducted by Winn et al. (2009). The best-fitting model
for each dataset is overplotted as a solid orange line. The two transit
profiles are in good agreement.

The origin and evolution of spin-orbit misalignments remains one of the most
interesting unsolved problems posed by the observed properties of the
exoplanets (Winn & Fabrycky, 2015; Albrecht et al., 2021). A definitive
assessment would be easier to formulate if measurements of spin-orbit angles
could be made for a variety of system types. The R-M effect, however, is much
more easily measured when transits are frequent and deep. Therefore, while R-M
observations of small planets and/or long-period planets play a critical role
in understanding the origin of spin-orbit misalignment (Albrecht et al., 2013;
Rice et al., 2021; Wang et al., 2018b, 2021; Wang X. et al., 2021b; Zhou et
al., 2018), they are difficult to successfully carry out.

Although the R-M effect was first established through the observation of an
eclipsing binary more than a century ago (Schlesinger, 1910), such
measurements of low-mass eclipsing binaries are surprisingly rare (see BANANA
Project, Albrecht et al. 2007). NASA’s TESS mission has been steadily
discovering suitable targets (e.g., Huang et al. 2018; Wang et al. 2019; Jones
et al. 2019; Cañas et al. 2019; Günther et al. 2019) and will detect a large
number of low-mass eclipsing binaries orbiting bright stars that are suitable
for R-M follow-up. This new population may shed light on not only planet
formation, but also on the genesis of low-mass stars.

K.W. thanks Yale College for supporting this work through the First-Year
Research Fellowship in the Sciences and the Edward A Bouchet Fellowship. S.W.
thanks the Heising-Simons Foundation for their generous support as a 51 Pegasi
b fellow. J.A.B. thanks MIT’s Kavli Institute for its support as a Torres
postdoctoral fellow. M.R. is supported by the National Science Foundation
Graduate Research Fellowship Program under Grant Number DGE-1752134. The
research was carried out in part at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (80NM0018D0004). This work is supported by Astronomical
Big Data Joint Research Center, co-founded by National Astronomical
Observatories, Chinese Academy of Sciences and Alibaba Cloud. We thank Jason
Eastman, John Michael Brewer, and Pia Cortes-Zuleta for useful discussions.

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