# Evidence that the Hot Jupiter WASP-77 A b Formed Beyond Its Parent
Protoplanetary Disk’s H2O Ice Line

Henrique Reggiani Carnegie Fellow  The Observatories of the Carnegie
Institution for Science, 813 Santa Barbara St, Pasadena, CA 91101, USA  Kevin
C. Schlaufman William H. Miller III Department of Physics and Astronomy, Johns
Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Brian F.
Healy William H. Miller III Department of Physics and Astronomy, Johns Hopkins
University, 3400 N Charles St, Baltimore, MD 21218, USA  Joshua D. Lothringer
Department of Physics, Utah Valley University, College of Science - MS 179,
800 W University Pkwy, Orem, UT 84058, USA  David K. Sing William H. Miller
III Department of Physics and Astronomy, Johns Hopkins University, 3400 N
Charles St, Baltimore, MD 21218, USA  Department of Earth and Planetary
Sciences, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218,
USA

(Received December 28, 2021; Accepted January 18, 2022)

###### Abstract

Idealized protoplanetary disk and giant planet formation models have been
interpreted to suggest that a giant planet’s atmospheric abundances can be
used to infer its formation location in its parent protoplanetary disk. It has
recently been reported that the hot Jupiter WASP-77 A b has sub-solar
atmospheric carbon and oxygen abundances with a solar C/O abundance ratio.
Assuming solar carbon and oxygen abundances for its host star WASP-77 A,
WASP-77 A b’s atmospheric carbon and oxygen abundances possibly indicate that
it accreted its envelope interior to its parent protoplanetary disk’s H2O ice
line from carbon-depleted gas with little subsequent planetesimal accretion or
core erosion. We comprehensively model WASP-77 A and use our results to better
characterize WASP-77 A b. We show that the photospheric abundances of carbon
and oxygen in WASP-77 A are super-solar with a sub-solar C/O abundance ratio,
implying that WASP-77 A b’s atmosphere has significantly sub-stellar carbon
and oxygen abundances with a super-stellar C/O ratio. Our result possibly
indicates that WASP-77 A b’s envelope was accreted by the planet beyond its
parent protoplanetary disk’s H2O ice line. While numerous theoretical
complications to these idealized models have now been identified, the
possibility of non-solar protoplanetary disk abundance ratios confound even
the most sophisticated protoplanetary disk and giant planet formation models.
We therefore argue that giant planet atmospheric abundance ratios can only be
meaningfully interpreted relative to the possibly non-solar mean compositions
of their parent protoplanetary disks as recorded in the photospheric
abundances of their solar-type dwarf host stars.

Exoplanet astronomy(486) — Exoplanet atmospheric composition(2021) — Exoplanet
formation(492) — Exoplanet migration(2205) — Exoplanet systems(484) —
Exoplanet tides(497) — Exoplanets(498) — Hot Jupiters(753) — Planet hosting
stars(1242) — Star-planet interactions(2177) — Stellar abundances(1577)

††facilities: ARC, CDS, Exoplanet Archive, Gaia, IRSA, Skymapper, Sloan,
TESS††software: astropy (Astropy Collaboration et al., 2013, 2018), CERES
(Brahm et al., 2017), colte (Casagrande et al., 2021), eleanor (Feinstein et
al., 2019), IRAF (Tody, 1986, 1993), isochrones (Morton, 2015), iSpec (Blanco-
Cuaresma et al., 2014; Blanco-Cuaresma, 2019), lightkurve (Lightkurve
Collaboration et al., 2018), MOOG (Sneden, 1973), MultiNest (Feroz & Hobson,
2008; Feroz et al., 2009, 2019), numpy (Harris et al., 2020), pandas
(McKinney, 2010; pandas Development Team, 2020), PyMultinest (Buchner et al.,
2014), q2 (Ramírez et al., 2014), RadVel (Fulton et al., 2017, 2018), scipy
(Virtanen et al., 2020)

Astronomical Journal

##  1 Introduction

It has been proposed that a giant planet’s formation location in its parent
protoplanetary disk can be discerned by studying the abundances of the
elements in the planet’s atmosphere. While acknowledging the importance of
planetesimal accretion after runaway envelope accretion in determining giant
planet envelope abundances, based on static disk models Öberg et al. (2011)
suggested that stellar C/O abundance ratios should be realized for giant
planets forming inside both their parent protoplanetary disks’ H2O ice and
carbon-grain sublimation lines where all volatiles are in the gas phase. Sub-
stellar or stellar C/O abundance ratios combined with super-stellar carbon
abundances were argued to result from the accretion of large amounts of icy
planetesimals after envelope accretion. Super-stellar C/O abundance ratios and
carbon abundances could be attributed either to formation close to the CO2 or
CO ice lines or by the accretion of carbon-rich grains in the narrow range
inside the H2O ice line but outside the carbon-grain sublimation line. Super-
stellar C/O abundance ratios and sub-stellar carbon and oxygen abundances were
put forward as a unique signature of formation beyond the H2O ice line.

This straightforward scenario outlined for the case of static disks is
significantly complicated in more realistic models that include disk chemical
& structural evolution, the radial migration of solids, detailed models for
planetesimal accretion, and/or post envelope-accretion migration. The
locations of the H2O, CO2, and CO ice lines depend on stellar luminosity and
move inward with time as a viscously evolving disk’s mass accretion rate
declines (e.g., Harsono et al., 2015). The inclusion of radially migrating
solids in viscously evolving disks can also cause the H2O ice line to move in
by a factor of two (Piso et al., 2015). In this case, the C/O abundance ratio
in disk gas is most likely to be super-stellar between the CO2 and CO ice
lines (Piso et al., 2015). Gas drag, sublimation, vapor diffusion,
condensation, and coagulation on major carbon- and oxygen-bearing volatiles
indicate that giant planets that accreted their envelopes inside the H2O ice
line should have super-stellar C/O (Ali-Dib et al., 2014; Ali-Dib, 2017). In
the absence of core erosion, pebble accretion is thought to produce sub-
stellar carbon and oxygen abundances but super-stellar C/O abundance ratios
(e.g., Madhusudhan et al., 2017; Booth et al., 2017). The point in time during
a disk’s evolution at which a giant planet forms also affects the C/O
abundance ratio of its envelope (Cridland et al., 2016). Chemical evolution in
disks can mask the influence of formation location on envelope composition
though (Eistrup et al., 2018), with super-stellar C/O only realized between
CO2 and CH4 ice lines and greater giant planet envelope abundance dispersions
expected in more metal-rich disks (Cridland et al., 2020; Notsu et al., 2020).
High-eccentricity migration after disk dissipation can lead to sub-stellar
carbon and oxygen abundances with stellar or super-stellar C/O abundance
ratios (Madhusudhan et al., 2014). For ultra-hot Jupiters, atmospheric
abundances of refractory elements like magnesium, silicon, and iron also can
constrain formation location (e.g., Lothringer et al., 2021).

It has also been argued that the abundances in giant planet envelopes depend
critically on the assumptions made regarding the refractory composition of the
inner disk (Mordasini et al., 2016; Cridland et al., 2019a, 2020).
Compositions inherited from the interstellar medium but including the
destruction of carbon grains in the inner disk by oxidizing reactions on grain
surfaces require $\text{C/O}<1$, both inside and outside the H2O ice line. The
reason is that the accretion of oxygen in the form of silicate planetesimals
in the former case and H2O-rich material in the latter case drive C/O below
its abundance in disk gas at a giant planet’s formation location. In contrast,
the assumptions that solids form along the condensation sequence as a disk
cools can still produce giant planet envelopes with $\text{C/O}>1$.

All of these analyses rely on two assumptions: (1) that the envelope of a
young giant planet stays well mixed during its formation even though most
metals are accreted before most gas and (2) that a mature giant planet’s
atmosphere has a similar composition to the average composition of its
envelope at the end of the planet formation process. While fully convective
and therefore well-mixed envelopes are the usual outcome of giant planet
formation models, compositional gradients can quench convection (e.g., Leconte
& Chabrier, 2012). In this case, the abundances in a giant planet’s envelope
will only reflect the abundances of the gas it accreted during runaway
accretion (Thiabaud et al., 2015). Despite all of these complications, one
robust prediction of giant planet formation models in dynamically evolving
disks is that the metal abundances of giant planets with $M_{\text{p}}\lesssim
2~{}M_{\text{Jup}}$ are dominated by the accretion of planetesimals after
envelope accretion. On the other hand, the metal abundances of giant planets
with $M_{\text{p}}\gtrsim 2~{}M_{\text{Jup}}$ are dominated by envelope
accretion itself (Mordasini et al., 2014, 2016; Cridland et al., 2019b).

In short, the interpretation of giant planet atmospheric carbon, oxygen, and
C/O abundances is far from simple. Moreover, giant planet atmospheric
abundance ratios can only be meaningfully interpreted relative to the mean
compositions of their parent protoplanetary disk. Because the protoplanetary
disks that formed the observed giant planets disappeared long ago, the only
way to reveal the mean compositions of those disks is to use the photospheric
abundances of their host stars. During the era of giant planet formation, the
star growing at the center of a protoplanetary disk has already accreted 99%
of the material that ever passed through its disk. As a result, the
photospheric abundances of solar-type planet-host stars are an excellent proxy
for mean protoplanetary disk abundances. The implication is that accurate and
precise host star elemental abundances for the same elements observed in giant
planet atmospheres are critically needed to achieve the full potential of
giant planet atmospheric abundance inferences as planet formation constraints.

In this article, we infer photospheric and fundamental stellar parameters as
well as individual elemental abundances—including carbon and oxygen—for the
hot Jupiter host star WASP-77 A. Atmospheric carbon and oxygen abundances for
the hot Jupiter WASP-77 A b were recently published by Line et al. (2021), and
the carbon and oxygen abundances we infer for WASP-77 A qualitatively change
the interpretation of WASP-77 A b’s atmospheric abundances. We describe in
Section 2 the high-resolution optical spectrum we collected for WASP-77 A. We
then infer stellar parameters from that spectrum and all available astrometric
and photometric data in Section 3. We derive the individual elemental
abundance in WASP-77 A’s photosphere in Section 4. We infer updated parameters
for the planet WASP-77 A b that are self consistent with the stellar
parameters we inferred for its host star in Section 5. We review our results
and their implications in Section 6. We conclude by summarizing our findings
in Section 7.

##  2 Data

The hot Jupiter host star WASP-77 A has right ascension $\alpha$ = 02 28
37.23, declination $\delta$ = –07 03 38.4, and is also known as BD-07 436 A,
TYC 4697-201-1, TOI-398, and Gaia EDR3 5178405479961698048. We observed it
from Apache Point Observatory (APO) with the Astrophysical Research Consortium
(ARC) 3.5-m Telescope and its ARC Echelle Spectrograph (ARCES; Wang et al.,
2003) for 750 seconds starting at UTC 2021 February 23 01:42:55. We used the
standard 1$\farcs$6 x 3$\farcs$2 slit, yielding a spectrum between 320 nm and
1000 nm with spectral resolution $R\approx 31,\\!500$. We collected all
calibration data (e.g., bias, quartz flat field, and ThAr lamp frames) in the
late afternoon before the start of our observations. We reduced the raw
spectra and calibration frames using a customized version of the CERES echelle
spectrograph data reduction package111https://github.com/rabrahm/ceres (Brahm
et al., 2017). We cross-correlated each order with a theoretical solar
spectrum and averaged the result to calculate the barycentric radial velocity
$v_{r}=2\pm 1$ km s-1 consistent with the High Accuracy Radial velocity Planet
Searcher (HARPS)-based radial velocity presented in Maxted et al. (2013). We
then used our radial velocity measurement to place the normalized spectrum in
its rest frame. Because WASP-77 A is a solar-type star with close-to-solar
photospheric stellar parameters, we used
iSpec222https://www.blancocuaresma.com/s/iSpec (Blanco-Cuaresma et al., 2014;
Blanco-Cuaresma, 2019) and a solar template spectrum to continuum normalize
the individual orders. Our reduced, continuum-normalized one-dimensional (1D)
spectrum has a signal-to-noise ratio $\text{S/N}\gtrsim 100$ pixel-1 at 500 nm
and $\text{S/N}\gtrsim 150$ pixel-1 at 620 nm.

##  3 Stellar Parameters

We derive photospheric and fundamental stellar parameters for WASP-77 A using
the algorithm described in Reggiani et al. (2020, 2021) that makes use of both
the classical spectroscopy-only approach333The classical spectroscopy-only
approach to photospheric stellar parameter estimation involves simultaneously
minimizing for individual line-based iron abundance inferences the difference
between Fe I & Fe II-based abundances as well as their dependencies on
transition excitation potential and measured reduced equivalent width. and
isochrones to infer accurate, precise, and self-consistent photospheric and
fundamental stellar parameters. Isochrones are especially useful for effective
temperature $T_{\text{eff}}$ inferences in this case, as high-quality
multiwavelength photometry from the ultraviolet to the red optical are
available from Data Release (DR2) of the SkyMapper Southern Survey (SMSS) and
Gaia DR2. Similarly, the Gaia EDR3 parallax-based distance of WASP-77 A makes
the calculation of surface gravity $\log{g}$ via isochrones straightforward.
With both $T_{\text{eff}}$ and $\log{g}$ available via isochrones, the
equivalent widths of iron lines can be used to self-consistently determine
metallicity $[\text{Fe/H}]$ and microturbulence $\xi$ by minimizing the
dependence of individual line-based iron abundance inferences on reduced
equivalent width.

The inputs to our photospheric and fundamental stellar parameter inference
include the equivalent widths of Fe I and Fe II atomic absorption lines,
multiwavelength photometry, a Gaia parallax, and an extinction estimate. Using
atomic absorption line data from Yana Galarza et al. (2019) for lines from
Meléndez et al. (2014) found to be insensitive to stellar activity, we first
measure these equivalent widths by fitting Gaussian profiles with the splot
task in IRAF to our continuum-normalized spectrum. Whenever necessary, we use
the deblend task to disentangle absorption lines from adjacent spectral
features. We gather $u$, $v$, $g$, $r$, and $z$ photometry and their
uncertainties from SMSS DR2 (Onken et al., 2019) as well as $G$ photometry and
its uncertainty from Gaia DR2 (Gaia Collaboration et al., 2016, 2018; Arenou
et al., 2018; Evans et al., 2018; Hambly et al., 2018; Riello et al., 2018).
We use a Gaia EDR3 parallax and its uncertainty (Gaia Collaboration et al.,
2021; Fabricius et al., 2021; Lindegren et al., 2021a, b; Torra et al., 2021)
as well as an extinction $A_{V}$ inference based on three-dimensional (3D)
maps of extinction in the solar neighborhood from the STructuring by Inversion
the Local Interstellar Medium (Stilism) program (Lallement et al., 2014, 2018;
Capitanio et al., 2017).

We assume Asplund et al. (2021) solar abundances and use these inputs to infer
photospheric and fundamental stellar parameters using the following steps.

  1. 1.

We use 1D plane-parallel solar-composition ATLAS9 model atmospheres (Castelli
& Kurucz, 2003), the 2019 version of the MOOG radiative transfer code (Sneden,
1973), and the q2 MOOG wrapper444https://github.com/astroChasqui/q2 (Ramírez
et al., 2014) to derive an initial set of photospheric stellar parameters
$T_{\text{eff}}$, $\log{g}$, $[\text{Fe/H}]$, and $\xi$ using the classical
spectroscopy-only approach.

  2. 2.

We then use the isochrones
package555https://github.com/timothydmorton/isochrones (Morton, 2015) to fit
the MESA Isochrones and Stellar Tracks (MIST; Dotter, 2016; Choi et al., 2016;
Paxton et al., 2011, 2013, 2015, 2018, 2019) library to our photospheric
stellar parameters as well as our input multiwavelength photometry, parallax,
and extinction data using
MultiNest666https://ccpforge.cse.rl.ac.uk/gf/project/multinest/ (Feroz &
Hobson, 2008; Feroz et al., 2009, 2019) via PyMultinest (Buchner et al.,
2014). We restricted the MIST library to extinctions in the range
$0~{}\text{mag}\leq A_{V}\leq 0.05~{}\text{mag}$ based on the maximum
plausible $A_{V}$ value suggested by Stilism. This produces a new set of
photospheric and fundamental stellar parameter posteriors that are both self-
consistent and physically consistent with stellar evolution.

  3. 3.

We next impose the posterior-median $T_{\text{eff}}$ and $\log{g}$ inferred in
step 2 on our grid of model atmospheres and minimize the dependence of
individual line-based iron abundance inferences on reduced equivalent width to
derive model atmosphere $[\text{Fe/H}]_{\text{atmo}}$ and $\xi$ values
consistent with our measured Fe I & Fe II equivalent widths and our isochrones
inferred $T_{\text{eff}}$ & $\log{g}$.

  4. 4.

We then use the model atmosphere selected in step 3 to calculate
$[\text{Fe/H}]$ as the average of all
$n_{\text{Fe}}=n_{\text{{Fe~{}I}}}+n_{\text{{Fe~{}II}}}$ equivalent width-
based iron abundance inferences for individual Fe I & Fe II lines. We take the
uncertainty of our $[\text{Fe/H}]$ inference as the standard deviation of the
individual line-based abundance inferences $\sigma_{[\text{Fe/H}]}^{\prime}$
divided by $\sqrt{n_{\text{Fe}}}$.

  5. 5.

We next check if the $[\text{Fe/H}]$ inferred in step 4 agrees to two decimal
places with $[\text{Fe/H}]_{\text{atmo}}$. If so, we proceed to step 6. If
not, we replace $[\text{Fe/H}]_{\text{atmo}}$ with $[\text{Fe/H}]$ and repeat
steps 3 to 5 until agreement is achieved.

  6. 6.

We then repeat steps 2 to 5 until the metallicities inferred from both the
isochrones analysis and the reduced equivalent width balance approach are
consistent within their uncertainties (typically a few iterations).

We use a Monte Carlo simulation to derive the final values and uncertainties
in our adopted $[\text{Fe/H}]$ and $\xi$ values due to the uncertainties in
our adopted $T_{\text{eff}}$ and $\log{g}$.

  1. 1.

We randomly sample a self-consistent pair of $T_{\text{eff}}$ and $\log{g}$
from our converged isochrones posteriors described above and calculate the
values of $[\text{Fe/H}]_{\text{atmo}}$ and $\xi$ that produce the best
reduced equivalent width balance given our Fe I & Fe II equivalent width
measurements.

  2. 2.

We use the model atmosphere selected in step 1 to calculate the average of all
$n_{\text{Fe}}=n_{\text{{Fe~{}I}}}+n_{\text{{Fe~{}II}}}$ individual equivalent
width-based iron abundance inferences and save the resulting metallicity of
each iteration.

  3. 3.

We repeat steps 1 and 2 200 times and adopt as our final photospheric stellar
parameters the (16,50,84) percentiles of the 200 self-consistent sets of
$T_{\text{eff}}$, $\log{g}$, $[\text{Fe/H}]$, and $\xi$ produced in this way.

We find good agreement between these photospheric stellar parameter derived
from our Monte Carlo simulation and those resulting from a single iteration of
reduced equivalent width balance using the median $T_{\text{eff}}$ and
$\log{g}$ from the posteriors produced by our converged analysis. We report
our adopted photospheric and fundamental stellar parameters in Table 1 and
plot samples from their posteriors in Figure 1. All of the uncertainties
quoted in Table 1 include random uncertainties only. That is, they are
uncertainties derived under the unlikely assumption that the MIST isochrone
grid we use in our analyses perfectly reproduces all stellar properties.

Figure 1: Samples from photospheric and fundamental stellar parameter
posteriors from our Bayesian analyses of astrometric, photometric, and
spectroscopic data for WASP-77 A. We emphasize that our adopted
$[\text{Fe/H}]$ comes from averaging individual equivalent width-based iron
abundance inferences as described above and not directly from the isochrones-
inferred metallicity plotted here.

We use Transiting Exoplanet Survey Satellite (TESS \- Ricker et al., 2015)
data from Sectors 4 and 31 to measure the rotational period of WASP-77 A. We
used the eleanor software framework777https://adina.feinste.in/eleanor/
(Feinstein et al., 2019) to generate light curves for WASP-77 A based on
Sector 4 and 31 TESS full-frame image data. We masked data points coincident
with transits of WASP-77 A b or that had non-zero data quality flags. We used
the lightkurve package888https://docs.lightkurve.org/ (Lightkurve
Collaboration et al., 2018) to linearly flatten the light curves across a
single-orbit window to preserve variability while removing orbit-to-orbit
trends. After normalizing each light curve and eliminating 4-$\sigma$
outliers, we followed Healy & McCullough (2020) and Healy et al. (2021) to use
both a stellar light curve’s autocorrelation function (ACF) and its
periodogram to infer stellar rotation period. For both the combined light
curve and each sector individually, we used the first peak of the ACF and its
half-width at half-maximum to estimate the rotation period of WASP-77 A and
its uncertainty. In all cases, the ACFs indicate statistically consistent
periods. The ACF-inferred rotation periods are consistent with the periodogram
peaks for both the Sector 4 and the Sectors 4+31 analyses, though for the
Sector 31 analysis the periodogram peak suggests a rotation period a factor of
two higher than the ACF. Based on the combined two-sector light curve, the
first peak of the ACF produces our final estimate of WASP-77 A’s rotation
period $P_{\text{rot}}=15.6\pm 3.5$ reported in Table 1. The periodogram of
the combined two-sector light curve is consistent with this period
determination. We use the the rotational evolution models presented in van
Saders et al. (2016) to roughly infer an age $\tau_{\text{gyro}}\approx 1.5$
Gyr for WASP-77 A based on this rotation period inference.

Table 1: Adopted Stellar Parameters Property | Value | Unit
---|---|---
SkyMapper DR2 $u$ | $12.084\pm 0.008$ | AB mag
SkyMapper DR2 $v$ | $11.756\pm 0.008$ | AB mag
SkyMapper DR2 $g$ | $10.443\pm 0.034$ | AB mag
SkyMapper DR2 $r$ | $10.088\pm 0.04$ | AB mag
SkyMapper DR2 $z$ | $9.881\pm 0.046$ | AB mag
Gaia DR2 DR2 $G$ | $10.096\pm 0.002$ | Vega mag
Gaia EDR3 parallax | $9.459\pm 0.056$ | mas
Isochrone-inferred parameters
Effective temperature $T_{\text{eff}}$ | $5525^{+23}_{-25}$ | K
Surface gravity $\log{g}$ | $4.44\pm 0.02$ | cm s-2
Stellar mass $M_{\ast}$ | $0.92\pm 0.02$ | $M_{\odot}$
Stellar radius $R_{\ast}$ | $0.96^{+0.01}_{-0.02}$ | $R_{\odot}$
Luminosity $L_{\ast}$ | $0.77\pm 0.01$ | $L_{\odot}$
Isochrone-based age $\tau_{\text{iso}}$ | $7.99^{+1.90}_{-1.71}$ | Gyr
Spectroscopically inferred parameters
$[\text{Fe/H}]$ | $-0.05^{+0.02}_{-0.01}$ |
$\xi$ | $1.26\pm 0.02$ | km s-1
Light-curve inferred parameters
Rotational period $P_{\text{rot}}$ | $15.6\pm 3.5$ | day
Gyrochronology-based age $\tau_{\text{gyro}}$ | 1.5 | Gyr

To evaluate the impact any possible systematic uncertainties resulting from
our analysis, we compare the photospheric and fundamental stellar parameters
we infer for WASP-77 A with those reported by other groups for the same star.
WASP-77 A was initially studied by Maxted et al. (2013) in the paper
announcing the discovery of WASP-77 A b. Photospheric and/or fundamental
parameters for the star based on high-resolution spectroscopy have since been
published by Mortier et al. (2013) (updated in Sousa et al., 2021) and Kolecki
& Wang (2021). Maxted et al. (2013) found $T_{\text{eff}}=5500\pm 80$ K,
$\log{g}=4.33\pm 0.08$, $[\text{Fe/H}]=0.00\pm 0.11$, $M_{\ast}=1.00\pm
0.07~{}M_{\odot}$, $\tau_{\text{iso}}=8$ Gyr, and $P_{\text{rot}}=15.4\pm 0.4$
days. Mortier et al. (2013) found $T_{\text{eff}}=5605\pm 41$ K,
$\log{g}=4.37\pm 0.09$, $[\text{Fe/H}]=0.07\pm 0.03$, and $M_{\ast}=0.96\pm
0.07~{}M_{\odot}$. Updated parameters from the same group presented in Sousa
et al. (2021) found $T_{\text{eff}}=5595\pm 16$ K, $\log{g}=4.41\pm 0.03$,
$[\text{Fe/H}]=0.01\pm 0.01$, and $M_{\ast}=0.915\pm 0.004~{}M_{\odot}$.
Kolecki & Wang (2021) found $T_{\text{eff}}=5660\pm 30$ K, $\log{g}=4.49\pm
0.01$, and $[\text{Fe/H}]=-0.15\pm 0.06$. Our photospheric and fundamental
stellar parameters agree well with those presented in Maxted et al. (2013) and
Mortier et al. (2013) as updated by Sousa et al. (2021). There is some tension
between our parameters and those reported by Kolecki & Wang (2021) that we
attribute to their neglect of extinction and their use of Two Micron All Sky
Survey (2MASS) and Wide-field Infrared Survey Explorer (WISE) photometry for
WASP-77 A that were contaminated by its binary companion WASP-77 B. The
excellent agreement between our photospheric and fundamental stellar
parameters for WASP-77 A and those inferred by other groups suggests that any
systematic uncertainties present must be small.

As an additional check we inferred $T_{\text{eff}}$ for WASP-77 A using the
colte code999https://github.com/casaluca/colte (Casagrande et al., 2021) that
estimates $T_{\text{eff}}$ using a combination of color–$T_{\text{eff}}$
relations obtained by implementing the InfraRed Flux Method for Gaia and 2MASS
photometry. As required by colte, we used Gaia EDR3 $G$, $G_{\text{BP}}$, and
$G_{\text{RP}}$ plus 2MASS $J$, $H$, and $K_{s}$ photometry as input. We used
these data despite the fact that 2MASS $J$ and $H$ are affected by confusion
and 2MASS $K_{s}$ is affected by a diffraction spike. We find a colte-based
$T_{\text{eff}}=5354\pm 203$ K. While we suspect its large uncertainty is due
to the poor quality of the required input 2MASS photometry, it is consistent
with our adopted $T_{\text{eff}}=5525^{+23}_{-25}$ K.

There are significant differences between the ages inferred for WASP-77 A from
isochrones and from gyrochronology. We find $\tau_{\text{iso}}$ =
$7.99^{+1.90}_{-1.71}$ Gyr in full agreement with the $\tau_{\text{iso}}=8$
Gyr inference from the full MCMC fit to all data from Maxted et al. (2013).
These inferences are also consistent with the fits produced by five additional
isochrones grids presented in Maxted et al. (2013) using the methodology
described in Southworth (2012). Similarly, if we fit Yonsei-Yale isochrones
(Yi et al., 2001, 2003; Kim et al., 2002; Demarque et al., 2004) to our
$T_{\text{eff}}$ and $\log{g}$ inferences using q2 we find
$\tau_{\text{iso}}=8.3\pm 3.5$ Gyr. On the other hand, Maxted et al. (2013)
used their $P_{\text{rot}}=15.4\pm 0.4$ days to infer a gyrochronology-based
age $\tau_{\text{gyro}}=1.0^{+0.5}_{-0.3}$ Gyr using the gyrochronology
relation from Barnes (2007). Using our TESS-based $P_{\text{rot}}=15.6\pm 3.5$
days we find $\tau_{\text{gyro}}\approx 1.5$ Gyr using the updated van Saders
et al. (2016) models.

We argue that $\tau_{\text{iso}}$ should be preferred to $\tau_{\text{gyro}}$
for WASP-77 A. We assume that WASP-77 B has the same metallicity and
extinction we inferred for WASP-77 A and use the same procedure described
above to fit the MIST isochrone grid to Sloan Digital Sky Survey (SDSS;
Fukugita et al., 1996; Gunn et al., 1998, 2006; York et al., 2000; Doi et al.,
2010; Abdurro’uf et al., 2021) $u$ and Gaia DR2 $G$ photometry as well as a
Gaia EDR3 parallax for WASP-77 B. We find consistent ages for WASP-77 A
$\tau_{\text{iso}}$ = $7.99^{+1.90}_{-1.71}$ and WASP-77 B $\tau_{\text{iso}}$
= $7.3^{+4.4}_{-4.6}$. We also note that Salz et al. (2015) studied the X-ray
properties of the WASP-77 system and found X-ray-based ages consistent with
our isochrone-based age inferences for both stars. As we will show in Section
6, the difference between the isochrone- and gyrochronology-based age
inferences results from tidally mediated angular momentum exchange between the
WASP-77 A system’s orbital angular momentum and the rotational angular
momentum of WASP-77 A.

It has been proposed that the chemical abundances of solar neighborhood “solar
twin” stars can be used as “chemical clocks” to complement other age inference
techniques. WASP-77 A is a thin disk star, so its elemental abundances are the
result of thin disk Galactic chemical evolution. We use our inferred elemental
abundances and the empirical chemical clocks derived from thin disk solar
twins by Spina et al. (2018) to infer two more, independent, age estimates
based on $[\text{Y/Mg}]$ and $[\text{Y/Al}]$ abundance ratios. According to
Equations (4) and (5) from Spina et al. (2018), WASP-77 A has a
$[\text{Y/Mg}]$ chemical clock-based age $\tau_{\text{cc}}=8.0\pm 1.0$ Gyr and
a $[\text{Y/Al}]$ chemical clock-based age $\tau_{\text{cc}}=6.5\pm 0.9$ Gyr.
The consistency of $\tau_{\text{iso}}$ and these $\tau_{\text{cc}}$ estimates
supports our interpretation of the gyrochronology-based age inferences for
WASP-77 A.

##  4 Elemental Abundances

To infer the elemental abundances of several $\alpha$, light odd-$Z$, iron-
peak, and neutron-capture elements we first measure the equivalent widths of
atomic absorption lines of C I, O I, Na I, Mg I, Al I, Si I, K I, Ca I, Sc II,
Ti I, Ti II, V I, Cr I, Fe I, Fe II, Ni I, and Y II in our continuum-
normalized spectrum by fitting Gaussian profiles with the splot task in IRAF.
We use the deblend task to disentangle absorption lines from adjacent spectral
features whenever necessary. We measure an equivalent width for every
absorption line in our line list that could be recognized, taking into
consideration the quality of the spectrum in the vicinity of a line and the
availability of alternative transitions of the same species. We assume Asplund
et al. (2021) solar abundances and local thermodynamic equilibrium (LTE) and
use the 1D plane-parallel solar-composition ATLAS9 model atmospheres and the
2019 version of MOOG to infer elemental abundances based on each equivalent
width measurement. We report our adopted atomic data, equivalent width
measurements, and individual line-based abundance inferences in Table 2.

Table 2: Atomic Data, Equivalent-width Measurements, and Individual-line
Abundance Inferences

Wavelength | Species | Excitation Potential | log($gf$) | EW | $\log_{\epsilon}(\rm{X})$
---|---|---|---|---|---
(Å) |  | (eV) |  | (mÅ) |
$6154.225$ | Na I | $2.102$ | $-1.547$ | $47.8$ | $6.302$
$6160.747$ | Na I | $2.104$ | $-1.246$ | $59.4$ | $6.169$
$4571.095$ | Mg I | $0.000$ | $-5.623$ | $118.3$ | $7.374$
$4730.040$ | Mg I | $4.340$ | $-2.389$ | $71.6$ | $7.595$
$6319.236$ | Mg I | $5.108$ | $-2.165$ | $67.0$ | $7.986$
$6318.717$ | Mg I | $5.108$ | $-1.945$ | $54.4$ | $7.603$
$6696.018$ | Al I | $3.143$ | $-1.481$ | $49.4$ | $6.453$
$6698.667$ | Al I | $3.143$ | $-1.782$ | $28.3$ | $6.391$
$7835.309$ | Al I | $4.021$ | $-0.680$ | $46.7$ | $6.371$
$7836.134$ | Al I | $4.021$ | $-0.450$ | $66.1$ | $6.376$

Note. — This table is published in its entirety in the machine-readable
format. A portion is shown here for guidance regarding its form and content.

We report in Table 3 our abundance inferences in three common systems:
$A(\text{X})$, $[\text{X/H}]$, and $[\text{X/Fe}]$. The abundance
$A(\text{X})$ is defined $A(\text{X})=\log{N_{\text{X}}/N_{\text{H}}}+12$, the
abundance ratio $[\text{X/H}]$ is defined as
$[\text{X/H}]=A(\text{X})-A(\text{X})_{\odot}$, and the abundance ratio
$[\text{X/Fe}]$ is defined as $[\text{X/Fe}]=[\text{X/H}]-[\text{Fe/H}]$. We
define the uncertainty in the abundance ratio $\sigma_{[\text{X/H}]}$ as the
standard deviation of the individual line-based abundance inferences
$\sigma_{[\text{X/H}]}^{\prime}$ divided by $\sqrt{n_{\text{X}}}$. We define
the uncertainty $\sigma_{[\text{X/Fe}]}$ as the square root of the sum of
squares of $\sigma_{[\text{X/H}]}$ and $\sigma_{[\text{Fe/H}]}$.

Table 3: Elemental Abundances Species | $A(\text{X})$ | [X/H] | $\sigma_{\text{[X/H]}}$ | [X/Fe] | $\sigma_{[\text{X/Fe}]}$ | $n$
---|---|---|---|---|---|---
LTE abundances |
C I | $8.57$ | $0.11$ | $0.09$ | $0.16$ | $0.11$ | $5$
O I | $9.03$ | $0.34$ | $0.02$ | $0.39$ | $0.04$ | $3$
Na I | $6.24$ | $0.02$ | $0.05$ | $0.07$ | $0.07$ | $2$
Mg I | $7.64$ | $0.09$ | $0.11$ | $0.14$ | $0.13$ | $4$
Al I | $6.46$ | $0.03$ | $0.04$ | $0.08$ | $0.05$ | $6$
Si I | $7.58$ | $0.07$ | $0.02$ | $0.12$ | $0.03$ | $13$
K I | $5.20$ | $0.13$ | $\cdots$ | $0.18$ | $0.06$ | $1$
Ca I | $6.29$ | $-0.01$ | $0.03$ | $0.04$ | $0.05$ | $11$
Sc II | $3.04$ | $-0.11$ | $0.03$ | $-0.05$ | $0.06$ | $4$
Ti I | $4.92$ | $-0.06$ | $0.03$ | $0.00$ | $0.05$ | $14$
Ti II | $5.00$ | $0.03$ | $0.04$ | $0.08$ | $0.06$ | $10$
V I | $3.84$ | $-0.06$ | $0.03$ | $-0.01$ | $0.06$ | $8$
Cr I | $5.51$ | $-0.11$ | $0.05$ | $-0.06$ | $0.06$ | $10$
Fe I | $7.39$ | $-0.07$ | $0.01$ | $\cdots$ | $\cdots$ | $80$
Fe II | $7.50$ | $0.04$ | $0.03$ | $\cdots$ | $\cdots$ | $17$
Ni I | $6.23$ | $0.03$ | $0.02$ | $0.08$ | $0.03$ | $17$
Y II | $2.14$ | $-0.08$ | $0.05$ | $-0.02$ | $0.08$ | $5$
1D non-LTE abundances
C I | $8.42$ | $-0.04$ | $\cdots$ | $-0.03$ | $\cdots$ | $1$
O I | $8.93$ | $0.24$ | $\cdots$ | $0.25$ | $\cdots$ | 3aa1D non-LTE corrections from Amarsi et al. (2020) for the three lines of the oxygen triplet are limited to a mean correction.
Na I | $6.16$ | $-0.06$ | $0.10$ | $-0.05$ | $0.10$ | $2$
Al I | $6.35$ | $-0.08$ | $0.01$ | $-0.07$ | $0.01$ | $2$
Si I | $7.56$ | $0.05$ | $0.03$ | $0.06$ | $0.03$ | $13$
K I | $4.89$ | $-0.18$ | $\cdots$ | $-0.17$ | $\cdots$ | $1$
Ca I | $6.07$ | $-0.23$ | $0.01$ | $-0.22$ | $0.01$ | $2$
Fe I | $7.42$ | $-0.04$ | $0.01$ | $\cdots$ | $\cdots$ | $80$
Fe II | $7.53$ | $0.07$ | $0.03$ | $\cdots$ | $\cdots$ | $17$
3D non-LTE abundances |
C I | $8.56$ | $0.10$ | $0.09$ | $0.15$ | $0.09$ | $5$
O I | $8.92$ | $0.23$ | $0.02$ | $0.28$ | $0.02$ | $3$
Additional abundance ratios of interest
$[\text{Fe/H}]_{\text{1Dnon-LTE}}=-0.01\pm 0.01$
$[\text{(C+O)/H}]_{\text{3Dnon-LTE}}=0.33\pm 0.09$
$[\text{C/O}]_{\text{3Dnon-LTE}}=-0.13\pm 0.09$
$\text{C/O}_{\text{3Dnon-LTE}}=0.44^{+0.07}_{-0.08}$

When possible, we update our elemental abundances derived under the
assumptions of LTE to account for departures from LTE (i.e., non-LTE
corrections) by linearly interpolating published grids of non-LTE corrections
using scipy (Virtanen et al., 2020). We make use of 1D non-LTE corrections for
carbon (Amarsi et al., 2020), oxygen (Amarsi et al., 2020), sodium (Lind et
al., 2011), aluminum (Amarsi et al., 2020), silicon (Amarsi & Asplund, 2017),
potassium (Reggiani et al., 2019), calcium (Amarsi et al., 2020), and iron
(Amarsi et al., 2016). We also make use of 3D non-LTE corrections for carbon
and oxygen (Amarsi et al., 2019). We present these abundances corrected for
departures from LTE in Table 3.

Our 1D non-LTE abundance corrections for carbon and oxygen come from Amarsi et
al. (2020) and are limited to one carbon line and a mean correction for the
three lines of the oxygen triplet. We therefore cannot calculate individual
line-based abundance dispersions for our 1D non-LTE corrected carbon and
oxygen abundance inferences, so our uncertainties might be underestimated. On
the other hand, our 3D non-LTE carbon and oxygen abundance corrections come
from Amarsi et al. (2019) and are available for all the lines in our line
list. We do find good agreement between our 1D non-LTE and 3D non-LTE
corrected carbon abundances for the one line with corrections in both sources.
Because both our 1D non-LTE and 3D non-LTE corrected carbon and oxygen
abundance inferences for WASP-77 A indicate that it has a sub-solar C/O ratio,
our preference for 3D non-LTE corrected abundances does not affect the
constraints we derive in Section 6 on the formation of WASP-77 A b.

Elemental abundances for WASP-77 A have also been published in Maxted et al.
(2013) and Kolecki & Wang (2021). All of our elemental abundance inferences
are consistent with those presented in Maxted et al. (2013). Our abundance
inferences for carbon, sodium, magnesium, and silicon are consistent with
those presented in Kolecki & Wang (2021), though our oxygen and potassium
abundances are significantly higher than those presented in that paper. We are
able to reproduce the oxygen abundance reported in Kolecki & Wang (2021) using
their photospheric stellar parameters, our equivalent width measurements, and
3D non-LTE abundance corrections from Amarsi et al. (2019). We are therefore
confident that the tension between our oxygen abundances results from the
photospheric stellar parameters Kolecki & Wang (2021) derived for WASP-77 A
that are hotter and more metal-poor than those derived by us, Maxted et al.
(2013), and Mortier et al. (2013) as updated by Sousa et al. (2021).

##  5 Updated Planetary Parameters

We make use of our accurate, precise, and self-consistent stellar mass and
radius inferences for WASP-77 A to update the parameters of the planet WASP-77
A b. Taking the transit depth $\delta=(R_{\text{p}}/R_{\ast})^{2}=0.01693\pm
0.00017$ from Maxted et al. (2013), our stellar radius
$R_{\ast}=0.96^{+0.01}_{-0.02}~{}R_{\odot}$ implies a planet radius
$R_{\text{p}}=0.125^{+0.001}_{-0.004}~{}R_{\odot}=1.22^{+0.01}_{-0.04}~{}R_{\text{Jup}}$
in agreement with the $R_{\text{p}}=1.21\pm 0.02~{}R_{\text{Jup}}$ value
inferred by Maxted et al. (2013). We fit the HARPS radial velocities from
Maxted et al. (2013) as updated in Trifonov et al. (2020) using the RadVel
package101010https://radvel.readthedocs.io/en/latest/ (Fulton et al., 2017,
2018). We fix the orbital period $P=1.3600306\pm 0.0000012$ days to the value
from Turner et al. (2016). We fix the eccentricity $e=0$ as expected for such
a short-period hot Jupiter and find a planet mass $M_{\text{p}}=1.66\pm
0.03~{}M_{\text{Jup}}$, marginally lower than those presented in Maxted et al.
(2013) and Bonomo et al. (2017). We plot our fit to the Doppler data in Figure
2 and summarize our updated planet parameters in Table 4.

Figure 2: RadVel fit of a Keplerian orbit to the WASP-77 A differential radial velocities from Maxted et al. (2013) as updated in Trifonov et al. (2020). We fix the orbital period $P=1.3600306\pm 0.0000012$ to the value from Turner et al. (2016) and the eccentricity to zero as expected for short-period hot Jupiter orbiting a mature solar-type star. Using the stellar mass we inferred from our self-consistent analysis of astrometric, photometric, and spectroscopic data for WASP-77 A $M_{\ast}=0.92\pm 0.02~{}M_{\odot}$ we find that WASP-77 A b’s mass is $M_{\text{p}}=1.66\pm 0.03~{}M_{\text{Jup}}$. Table 4: Updated Planetary Parameters Parameter | Credible Interval | Units
---|---|---
Fixed values
Orbital period $P_{\text{orb}}$ | $1.3600306\pm 0.0000012$ | days
Eccentricity $e$ | 0.00 |
Fit values
Argument of periapse $\omega$ | $177\pm 120$ | degrees
Doppler semiamplitude $K_{\ast}$ | $321.4^{+3.9}_{-3.8}$ | m s-1
Planet mass $M_{\text{p}}$ | $1.66\pm 0.03$ | $M_{\text{Jup}}$
Planet radius $R_{\text{p}}$ | $1.22^{+0.01}_{-0.04}$ | $R_{\text{Jup}}$
Planet density $\rho_{\text{p}}$ | $1.14^{+0.12}_{-0.10}$ | g cm-3
Semimajor axis $a$ | $0.02338\pm 0.00017$ | AU

##  6 Discussion

###  6.1 The Atmospheric Abundances of WASP-77 A b Indicate it Formed Beyond
the H2O Ice Line

For the photosphere of WASP-77 A and therefore the protoplanetary disk in
which WASP-77 A b formed, we find super-solar carbon and oxygen abundances
with a sub-solar C/O abundance ratio111111According to Asplund et al. (2021),
the solar C/O abundance ratio is $\text{C/O}=0.59^{+0.08}_{-0.06}$.. We infer
$[\text{C/H}]_{\text{3Dnon-LTE}}=0.10\pm 0.09$ and $[\text{O/H}]_{\text{3Dnon-
LTE}}=0.23\pm 0.02$, implying $[\text{(C+O)/H}]_{\text{3Dnon-
LTE}}=\left[A(\text{C})_{\ast}-A(\text{C})_{\odot}\right]+\left[A(\text{O})_{\ast}-A(\text{O})_{\odot}\right]=0.33\pm
0.09$ and C/O abundance ratios $[\text{C/O}]_{\text{3Dnon-
LTE}}=[\text{C/H}]-[\text{O/H}]=-0.13\pm 0.09$ and $\text{C/O}_{\text{3Dnon-
LTE}}=10^{A(\text{C})-A(\text{O)}}=0.44^{+0.07}_{-0.08}$.

For the atmosphere of the hot Jupiter WASP-77 A b, Line et al. (2021) found
$[\text{C/H}]=-0.46^{+0.17}_{-0.16}$, $[\text{O/H}]=-0.49^{+0.14}_{-0.12}$,
$[\text{(C+O)/H}]=-0.48^{+0.15}_{-0.13}$, and $\text{C/O}=0.59\pm 0.08$. Those
authors also considered the possibility that cloud condensates like MgSiO3 and
Mg2SiO4 “rain out” on WASP-77 A b’s night side and remove oxygen from its
atmosphere. That process would enhance the gas-phase C/O ratio of WASP-77 A b.
When accounting for this possibility, Line et al. (2021) found
$[\text{(C+O)/H}]=-0.41\pm 0.14$ and $\text{C/O}=0.46\pm 0.08$ for the
atmosphere of WASP-77 A b. Line et al. (2021) assumed that because of its near
solar metallicity the star WASP-77 A has a solar C/O abundance ratio. Based on
that assumption, those authors asserted that the atmosphere of the planet
WASP-77 A b has sub-stellar $[\text{C/H}]$ and $[\text{O/H}]$ plus a stellar
C/O abundance ratio. They suggested that WASP-77 A b accreted its envelope
interior to its parent protoplanetary disk’s H2O ice line from carbon-depleted
gas with little subsequent planetesimal accretion or core erosion.

We find that carbon and oxygen are 3.6 times and 5.2 times less abundant in
the atmosphere of WASP-77 A b than in the photosphere of WASP-77 A. Our
abundance inferences for WASP-77 A increase the statistical significance of
the Line et al. (2021) observation that WASP-77 A b’s atmosphere has sub-
stellar carbon and oxygen abundances, from 2.6 $\sigma$ to 2.9 $\sigma$ for
carbon and from 3.4 $\sigma$ to 5.1 $\sigma$ for oxygen. Our C/O abundance
ratio inference for WASP-77 A qualitatively changes the status of WASP-77 A
b’s atmospheric C/O abundance ratio from stellar to super-stellar by a factor
of 1.4 at the 1.4-$\sigma$ level.

As we argued in Section 1, it is not trivial to connect a giant planet’s
atmospheric carbon and oxygen abundances to its formation location in its
parent protoplanetary disk. Nevertheless, WASP-77 A b’s significantly sub-
stellar carbon and oxygen abundances combined with a super-stellar C/O
abundance ratio is consistent with formation beyond its parent protoplanetary
disk’s H2O ice line (e.g., Öberg et al., 2011), possibly from pebble accretion
followed by high-eccentricity migration after disk dissipation (e.g.,
Madhusudhan et al., 2014, 2017; Booth et al., 2017). The possibility that some
of the oxygen in WASP-77 A b’s atmosphere has rained-out in the form of
condensates means that the bulk C/O ratio of the planet may indeed match the
stellar abundance.

The qualitative change in the interpretation of WASP-77 A b’s atmospheric
carbon and oxygen abundances brought about by our inference of WASP-77 A’s
non-solar $[\text{C/H}]$, $[\text{O/H}]$, and C/O abundance ratio emphasizes
the importance of stellar abundance inferences for the interpretation of giant
planet atmospheric abundances. Indeed, even though WASP-77 A has a solar
metallicity $[\text{Fe/H}]_{\text{1Dnon-LTE}}=-0.01\pm 0.01$ it has non-solar
carbon and oxygen abundances. To illustrate this point, we plot in Figure 3
the 1D non-LTE carbon and oxygen abundances of solar neighborhood dwarf stars
(i.e., $\log{g}\geq 4$) over the 1D non-LTE correct metallicity range
characteristic of virtually all known planet host stars
$-0.5\leq[\text{Fe/H}]\leq 0.5$ from the third data release of the Galactic
Archaeology with HERMES (GALAH) survey (Buder et al., 2021). Figure 3 shows
that WASP-77 A is fully consistent with the solar neighborhood carbon and
oxygen abundance distributions. It also shows that the solar neighborhood
carbon, oxygen, and C/O abundance ratios are much broader than usually
considered in planet formation models, with 95% of solar neighborhood dwarf
stars in the ranges $-0.15\leq[\text{C/Fe}]\leq 0.32$,
$-0.19\leq[\text{O/Fe}]\leq 0.43$, $-0.26\leq[\text{C/O}]\leq 0.29$, and
$0.27\leq\text{C/O}\leq 0.94$.

Figure 3: 1D non-LTE carbon and oxygen abundances of WASP-77 A compared to the
same 1D non-LTE abundances for solar neighborhood dwarfs from GALAH DR3 (Buder
et al., 2021). In each kernel density estimator-smoothed scatter plot, darker
blue regions correspond to more densely populated regions of abundance space.
We plot outliers as individual dots. We plot our abundance inferences for
WASP-77 A’s as black-bordered white points. Left: $[\text{C/H}]$,
$[\text{C/Fe}]$, $[\text{O/H}]$, and $[\text{O/Fe}]$. Right:
$[\text{(C+O)/H}]$, $[\text{(C+O)/Fe}]$, $[\text{C/O}]$, and C/O. In the range
$-0.1\leq[\text{Fe/H}]\leq 0$, WASP-77 A is consistent with the GALAH-inferred
carbon ($[\text{C/Fe}]=0.05^{+0.12}_{-0.10}$), oxygen
($[\text{O/Fe}]=0.06^{+0.14}_{-0.10}$), and C/O abundance ratio distributions
($[\text{C/O}]=0.01^{+0.11}_{-0.12}$ and $\text{C/O}=0.49^{+0.15}_{-0.12}$).
In the metallicity range $-0.5\leq[\text{Fe/H}]\leq 0.5$ appropriate for
virtually all known exoplanet host stars, GALAH DR3 finds that 95% of solar
neighborhood dwarf stars have $-0.15\leq[\text{C/Fe}]\leq 0.32$,
$-0.19\leq[\text{O/Fe}]\leq 0.43$, $-0.26\leq[\text{C/O}]\leq 0.29$, and
$0.27\leq\text{C/O}\leq 0.94$. These broad distributions emphasize that the
assumption of solar abundance ratios of carbon and oxygen for exoplanet host
stars is not always appropriate.

If a planet host star’s carbon and oxygen abundances are unknown, then a
planet with a sub-stellar but super-solar C/O abundance ratio would be
mischaracterized as carbon rich. Likewise, a planet with a super-stellar but
sub-solar C/O abundance ratio would be mischaracterized as carbon poor. We
therefore recommend that all future studies of exoplanet atmospheric
abundances also infer stellar photospheric and fundamental parameters that are
self-consistent and physically consistent with stellar evolution. Those
photospheric stellar parameters should be used in all elemental abundance
inferences, and those fundamental stellar parameters should be used to
recalculate exoplanet parameters (e.g, $M_{\text{p}}$, $R_{\text{p}}$,
$\log{g}_{\text{p}}$, etc) using observables like Doppler velocities and light
curves. In that way, all of the parameters necessary to characterize an
exoplanet system can be made self-consistent to eliminate the small but
possibly important systematic uncertainties that can effect an exoplanet
atmosphere abundance inference. If the photospheric stellar parameters that
produce carbon and oxygen abundances are not used to self consistently infer
(1) stellar masses and radii and (2) planet parameters using observables, then
those photospheric abundances are not suitable for exoplanet atmospheric
characterization (e.g., Kolecki & Wang, 2021).

###  6.2 Angular Momentum Exchange Explains the Tension Between Isochrone- and
Gyrochronology-based Ages for WASP-77 A

The utility of gyrochronology-based age inferences for hot Jupiter host stars
has been questioned both theoretically and empirically (e.g., Barnes, 2007;
Brown, 2014; Tejada Arevalo et al., 2021). Similarily, we argue that
gyrochronology-based age inferences for WASP-77 A are unreliable because of
the possibility of tidally mediated angular momentum exchange between the
system’s orbital angular momentum $L_{\text{orb}}$ and the rotational angular
momentum of the host star WASP-77 A $L_{\text{rot}}$ (the rotational angular
momentum of WASP-77 A b is negligible). The total angular momentum
$L_{\text{tot}}$ of the WASP-77 A system in its current configuration is

$\displaystyle L_{\text{tot},2}$ $\displaystyle=$ $\displaystyle
L_{\text{orb},2}+L_{\text{rot},2},$ (1) $\displaystyle L_{\text{orb},2}$
$\displaystyle=$
$\displaystyle\left(\frac{M_{\ast}M_{\text{p}}}{M_{\ast}+M_{\text{p}}}\right)\left(\frac{2\pi}{P_{\text{orb},2}}\right)a_{2}^{2}\sqrt{1-e^{2}},$
(2) $\displaystyle L_{\text{rot},2}$ $\displaystyle=$ $\displaystyle
I_{\ast}P_{\text{rot},2}.$ (3)

Given the currently observed stellar rotation period $P_{\text{rot},2}=15.6\pm
3.5$ days, the system’s orbital period $P_{\text{orb},2}=1.3600306\pm
0.0000012$ days, and the system’s semimajor axis $a_{2}=0.0234\pm 0.0002$ AU,
if we assume a constant stellar momentum of inertia $I_{\ast}$ with the solar
value $I/(M_{\odot}R_{\odot}^{2})=0.070$, the stellar mass and radius we
inferred $M_{\ast}=0.92\pm 0.02~{}M_{\odot}$ and
$R_{\ast}=0.96^{+0.01}_{-0.02}~{}R_{\odot}$, and the planet mass we inferred
$M_{\text{p}}=1.66\pm 0.03~{}M_{\text{Jup}}$, we find
$L_{\text{orb},2}=2.05\times 10^{49}$ g cm2 s-1, $L_{\text{rot},2}=3.2\times
10^{48}$ g cm2 s-1, and $L_{\text{tot},2}=2.4\times 10^{49}$ g cm2 s-1.

For our isochrone-inferred age of the system $\tau_{\text{iso}}$ =
$7.99^{+1.90}_{-1.71}$ Gyr, the rotational evolution models of van Saders et
al. (2016) predict $P_{\text{rot}}\approx 30$ days for a system with the zero-
age main sequence $T_{\text{eff}}$ expected for WASP-77 A. Assuming the
conservation of angular momentum over the system’s evolution and taking
$P_{\text{rot},1}\approx 30$ days as the rotation period WASP-77 A would have
had in the absence of tidal evolution, we find that the current rapid rotation
of WASP-77 A given the system’s age can be explained if the system had
$P_{\text{orb},1}\approx 1.7$ days and $a_{1}\approx 0.027$ AU before tidal
evolution. That is, the disagreement between the isochrone- and
gyrochronology-based ages for WASP-77 A can easily be explained by tidal
evolution mediated angular momentum exchange.

Making the assumptions that the orbit of the WASP-77 A system had circularized
early in its evolution and that the spin-up of WASP-77 A results from tidally
mediated angular momentum exchange after orbit circularization, one can
analytically solve the ordinary differential equation for semimajor axis
evolution from Jackson et al. (2008) for WASP-77 A’s tidal dissipation
parameter $Q_{\ast}^{\prime}$ (i.e., following the convention of Goldreich &
Soter, 1966). Taking $a_{1}\approx 0.027$ AU, $a_{2}\approx 0.023$ AU, and
$\Delta t\approx 8$ Gyr we find $Q_{\ast}^{\prime}\sim 10^{8}$. In other
words, even in the limit of inefficient dissipation tides can easily explain
the spin-up of WASP-77 A due to tidal evolution over the life of the system.
These results provide further evidence that gyrochronology-based ages should
not be relied on for hot Jupiter systems.

##  7 Conclusion

We find that the hot Jupiter host star WASP-77 A has super-solar carbon
$[\text{C/H}]_{\text{3Dnon-LTE}}=0.10\pm 0.09$ and oxygen
$[\text{O/H}]_{\text{3Dnon-LTE}}=0.23\pm 0.02$ abundances plus sub-solar C/O
abundance ratios $[\text{C/O}]_{\text{3Dnon-LTE}}=-0.13\pm 0.09$ and
$\text{C/O}_{\text{3Dnon-LTE}}=0.44^{+0.07}_{-0.08}$. As reported by Line et
al. (2021), the atmosphere of the hot Jupiter WASP-77 A b has
$[\text{C/H}]=-0.46^{+0.17}_{-0.16}$, $[\text{O/H}]=-0.49^{+0.14}_{-0.12}$,
and $\text{C/O}=0.59\pm 0.08$. Though Line et al. (2021) assumed that WASP-77
A has solar carbon and oxygen abundances and therefore asserted that WASP-77 A
b has a stellar C/O abundance ratio, we find that WASP-77 A b has a
significantly super-stellar C/O abundance ratio. The non-solar abundance
ratios of WASP-77 A qualitatively change the interpretation of WASP-77 A b’s
atmospheric abundances put forward in Line et al. (2021). While Line et al.
(2021) suggested that WASP-77 A b formed its envelope interior to its parent
protoplanetary disk’s H2O ice line from carbon-depleted gas with little
subsequent planetesimal accretion or core erosion, we find that its super-
stellar C/O abundance ratio implies formation outside its parent
protoplanetary disk’s H2O ice line.

While the use of exoplanet atmospheric abundances to inform planet formation
is not always straightforward, this qualitative change in the interpretation
of the atmospheric abundances of WASP-77 A b emphasizes the importance of
exoplanet host star elemental abundances inferences. For the most powerful
constraints, these carbon and oxygen abundances cannot be inferred
independently of stellar and exoplanet parameters. Instead, photospheric and
fundamental stellar parameters that are self-consistent and physically
consistent with stellar evolution should be used both in elemental abundance
inferences and to re-derive planet parameters from observables like Doppler
velocities and light curves. This approach results in self-consistent
photospheric and fundamental stellar parameters, elemental abundances, and
exoplanet parameters that can be used to explore planet formation and
evolution. Accounting for the possibility of tidally mediated angular momentum
exchange in the WASP-77 A system resolves the tension between its isochrone-
and gyrochronology-based ages. This result is further evidence that in general
gyrochronology-based ages should not be relied on for hot Jupiter systems.

## Acknowledgments

We thank the referee for a prompt and helpful review. Henrique Reggiani
acknowledges support from a Carnegie Fellowship. Based on observations
obtained with the Apache Point Observatory 3.5-meter telescope, which is owned
and operated by the Astrophysical Research Consortium. The national facility
capability for SkyMapper has been funded through ARC LIEF grant LE130100104
from the Australian Research Council, awarded to the University of Sydney, the
Australian National University, Swinburne University of Technology, the
University of Queensland, the University of Western Australia, the University
of Melbourne, Curtin University of Technology, Monash University and the
Australian Astronomical Observatory. SkyMapper is owned and operated by The
Australian National University’s Research School of Astronomy and
Astrophysics. The survey data were processed and provided by the SkyMapper
Team at ANU. The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is
hosted at the National Computational Infrastructure (NCI). Development and
support of the SkyMapper node of the ASVO has been funded in part by Astronomy
Australia Limited (AAL) and the Australian Government through the
Commonwealth’s Education Investment Fund (EIF) and National Collaborative
Research Infrastructure Strategy (NCRIS), particularly the National eResearch
Collaboration Tools and Resources (NeCTAR) and the Australian National Data
Service Projects (ANDS). This work has made use of data from the European
Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This publication makes use
of data products from the Two Micron All Sky Survey, which is a joint project
of the University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation. This research
has made use of the NASA/IPAC Infrared Science Archive, which is funded by the
National Aeronautics and Space Administration and operated by the California
Institute of Technology. Funding for the Sloan Digital Sky Survey IV has been
provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy
Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High Performance Computing at the
University of Utah. The SDSS website is www.sdss.org. SDSS-IV is managed by
the Astrophysical Research Consortium for the Participating Institutions of
the SDSS Collaboration including the Brazilian Participation Group, the
Carnegie Institution for Science, Carnegie Mellon University, Center for
Astrophysics — Harvard & Smithsonian, the Chilean Participation Group, the
French Participation Group, Instituto de Astrofísica de Canarias, The Johns
Hopkins University, Kavli Institute for the Physics and Mathematics of the
Universe (IPMU) / University of Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik
Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-
Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für
Extraterrestrische Physik (MPE), National Astronomical Observatories of China,
New Mexico State University, New York University, University of Notre Dame,
Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State
University, Shanghai Astronomical Observatory, United Kingdom Participation
Group, Universidad Nacional Autónoma de México, University of Arizona,
University of Colorado Boulder, University of Oxford, University of
Portsmouth, University of Utah, University of Virginia, University of
Washington, University of Wisconsin, Vanderbilt University, and Yale
University. This research has made use of NASA’s Astrophysics Data System
Bibliographic Services. This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France (Wenger et al., 2000). This research has
made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI:
10.26093/cds/vizier). The original description of the VizieR service was
published in 2000, A&AS 143, 23 (Ochsenbein et al., 2000). This research has
made use of the NASA Exoplanet Archive, which is operated by the California
Institute of Technology, under contract with the National Aeronautics and
Space Administration under the Exoplanet Exploration Program.

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