# Retrieving C and O Abundance of HR 8799 c by Combining High- and Low-
Resolution Data

Ji Wang (王吉) Department of Astronomy, The Ohio State University, 100 W 18th
Ave, Columbus, OH 43210 USA  Jason J. Wang (王劲飞) 51 Pegasi b Fellow  Center
for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics and Astronomy, Northwestern University, Evanston, IL
60208, USA  Jean-Baptiste Ruffio Department of Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Geoffrey A. Blake  Division of
Geological & Planetary Sciences, California Institute of Technology, Pasadena,
CA 91125, USA  Dimitri Mawet  Department of Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA  Ashley
Baker  Department of Astronomy, California Institute of Technology, Pasadena,
CA 91125, USA  Randall Bartos  Jet Propulsion Laboratory, California Institute
of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA  Charlotte Z. Bond
UK Astronomy Technology Centre, Royal Observatory, Edinburgh EH9 3HJ, United
Kingdom  Benjamin Calvin  Department of Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA  Department of Physics & Astronomy, 430
Portola Plaza, University of California, Los Angeles, CA 90095, USA  Sylvain
Cetre  W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
Jacques-Robert Delorme W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela,
HI, USA  Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Greg Doppmann  W. M. Keck Observatory, 65-1120
Mamalahoa Hwy, Kamuela, HI, USA  Daniel Echeverri  Department of Astronomy,
California Institute of Technology, Pasadena, CA 91125, USA  Luke Finnerty
Department of Physics & Astronomy, 430 Portola Plaza, University of
California, Los Angeles, CA 90095, USA  Michael P. Fitzgerald Department of
Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles,
CA 90095, USA  Nemanja Jovanovic Department of Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Ronald Lopez  Department of Physics &
Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095,
USA  Emily C. Martin Department of Astronomy & Astrophysics, University of
California, Santa Cruz, CA95064, USA  Evan Morris  Department of Astronomy &
Astrophysics, University of California, Santa Cruz, CA95064, USA  Jacklyn
Pezzato  Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Sam Ragland  W. M. Keck Observatory, 65-1120
Mamalahoa Hwy, Kamuela, HI, USA  Garreth Ruane Department of Astronomy,
California Institute of Technology, Pasadena, CA 91125, USA  Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,Pasadena,
CA 91109, USA  Ben Sappey  Center for Astrophysics and Space Sciences,
University of California, San Diego, La Jolla, CA 92093  Tobias Schofield
Department of Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Andrew Skemer  Department of Astronomy & Astrophysics, University
of California, Santa Cruz, CA95064, USA  Taylor Venenciano  Physics and
Astronomy Department, Pomona College, 333 N. College Way, Claremont, CA 91711,
USA  J. Kent Wallace Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA  Peter Wizinowich  W.
M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA  Jerry W. Xuan
Department of Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Marta L. Bryan  Department of Astronomy, 501 Campbell Hall,
University of California at Berkeley, CA 94720, USA  Arpita Roy Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles
St, Baltimore, MD 21218, USA  Nicole L. Wallack  Division of Geological &
Planetary Sciences, California Institute of Technology, Pasadena, CA 91125,
USA

###### Abstract

The formation and evolution pathway for the directly-imaged multi-planetary
system HR 8799 remains mysterious. Accurate constraints on the chemical
composition of the planetary atmosphere(s) are key to solving the mystery. We
perform a detailed atmospheric retrieval on HR 8799 c to infer the chemical
abundances and abundance ratios using a combination of photometric data along
with low- and high-resolution spectroscopic data (R$\sim$20-35,000). We
specifically retrieve [C/H], [O/H], and C/O and find them to be
0.55${}^{+0.36}_{-0.39}$, 0.47${}^{+0.31}_{-0.32}$, and
0.67${}^{+0.12}_{-0.15}$ at 68% confidence. The super-stellar C and O
abundances, yet a stellar C/O ratio, reveal a potential formation pathway for
HR 8799 c. Planet c, and likely the other gas giant planets in the system,
formed early on (likely within $\sim$1 Myr), followed by further atmospheric
enrichment in C and O through the accretion of solids beyond the CO iceline.
The enrichment either preceded or took place during the early phase of the
inward migration to the planet current locations.

††journal: ApJ

.

##  1 Introduction

The HR 8799 multiple planetary system (Marois et al., 2008, 2010) remains an
interesting and challenging case to test planet formation scenarios. Despite
numerous studies, it is still uncertain where and how the four planets formed
and how they arrived at their current locations. Chemical composition (e.g.,
Öberg et al., 2011; Madhusudhan et al., 2017) and orbital configuration (e.g.,
Konopacky et al., 2016; Goździewski & Migaszewski, 2020) may reveal key
information as to the origin and evolutionary history of the system. A
comprehensive review can be found in Wang et al. (2020).

In Wang et al. (2020), we found that retrieval analyses from different groups
led to drastically different abundances and C/O ratios. Even with the same
retrieval code, the results were sensitive to the choice of priors and data
sets. The analyses in Wang et al. (2020) painted a bleak picture of spectral
retrievals for directly-imaged exoplanets. Similar conclusion was also drawn
for transiting planets using the James Webb Space Telescopes (JWST), but with
an emphasis on the systematics due to the opacity data (Niraula et al., 2022).
However, some of the issues in Wang et al. (2020) had been addressed with e.g.
a more flexible temperature profile and a more realistic cloud treatment (Wang
et al., 2022a). Consequently, promising results had been shown when analyzing
data from a bench-mark brown dwarfs: consistent chemical abundances and C/O
ratios can be retrieved for bench-mark brown dwarfs and their host stars (Wang
et al., 2022a; Xuan et al., 2022). The promising results motivate a new
retrieval analysis for the HR 8799 system.

In addition to the progress of spectral analysis, recent observational
progress has been made on the study of the HR 8799 planetary system since Wang
et al. (2020). Doelman et al. (2022) used LMIRCam/ALES observations to further
constrain the planetary properties and cloud conditions, while Sepulveda &
Bowler (2022) and Brandt et al. (2021) further pinned down the dynamical
masses for HR 8799 and planets bcde using Gaia data. Ruffio et al. (2021)
employed Keck/OSIRIS data to obtain consistent C/O ratios for planets b,c, and
d.

Notably, Wang et al. (2021) used the Keck Planet Imager and Characterizer
(KPIC, Mawet et al., 2018; Jovanovic et al., 2019; Delorme et al., 2020, 2021)
to obtain high-resolution (R$\sim$35,000) K-band spectra of the four planets
orbiting HR 8799. With high-spectral resolution data, they measured previously
inaccessible properties such as planet projected rotational velocities.
Furthermore, higher spectral resolution can better resolve molecular lines and
provide key constraints on chemical compositions even in cloudy conditions
(Xuan et al., 2022).

With the addition of the KPIC data for HR 8799 planets, we will again use HR
8799 c to stress test our retrieval analysis and the influence of different
data sets on the retrieval results, similar to Wang et al. (2020, 2022a). A
full analysis of all four planets including the KPIC data and other newly
available data will be presented in a forthcoming manuscript. This paper is
organized as follows. §2 summarizes the observational data. §3 presents a test
of our retrieval code using a PHOENIX synthetic spectrum as a known input.
Main results are in §4. Discussions on the results are provided in §5. A
summary of the paper can be found in §6.

##  2 Data

Our data for HR 8799 c are detailed in Wang et al. (2020) and we provide a
summary of the data sets in Table 1. The data sets include (from lower to
higher spectral resolution): $L$\- and $M$-band photometric data points
(Bonnefoy et al., 2016; Skemer et al., 2012, 2014; Currie et al., 2014;
Galicher et al., 2011), integral field unit (IFU) data from CHARIS (R$\sim$20,
PI: Wang), IFU data from GPI (R$\sim$45-80, Greenbaum et al., 2018), and IFU
data from OSIRIS (R=5000, Konopacky et al., 2013). In addition, we utilize
high-resolution spectral data (R$\sim$35,000) for HR 8799 c from KPIC (Wang et
al., 2021).

##  3 Testing with a Phoenix BT-Settl Spectrum

We test our retrieval framework using a synthetic spectrum for which we know
the C and O abundance. We refer to Wang et al. (2022a) for details of the
retrieval framework because we use the same methodology to perform retrieval
analyses. In summary, we model exoplanet atmospheres based on petitRADTRANS
and consider both low and high resolution modes (R=1,000 and R=1,000,000). We
include MgSiO3 clouds (Mollière et al., 2020) and adopt a flexible P-T profile
(Petit dit de la Roche et al., 2020). To sample the posterior distribution in
a Bayesian framework, we used PyMultiNest (Buchner et al., 2014).

The synthetic spectrum is obtained from the PHOENIX BT-Settl model (Baraffe et
al., 2015). We choose an effective temperature (Teff) of 1200 K, a surface
gravity ($\log g$) of 3.5, and solar abundances111The fits file is available
at: https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011_2015/SPECTRA/. The
effective temperature and surface gravity of the synthetic spectrum are
similar to those of HR 8799 c and the other planets in this system (e.g.,
Doelman et al., 2022; Mollière et al., 2020). To calculate the luminosity, we
set the radius to be 1.20 RJupiter. This corresponds to a mass of 1.84
MJupiter for the assumed $\log g$ of 3.5, and a bolometric luminosity of
$\log(L/L_{\odot})$=-4.62. The luminosity is consistent with the previously
measured values of $\log(L/L_{\odot})$ from -4.58 to -4.83 (Greenbaum et al.,
2018)

###  3.1 Simulating the Data

For low-resolution and photometric data, we resample the synthetic spectrum to
the wavelength grid of existing data. We assume a top-hat response function
for the photometric filters. To describe the top-hat function for each $L$\-
and $M$-band photometric filter, we use the central wavelength and the full
width at half maximum (FWHM) in Table 1 of Bonnefoy et al. (2016). Since the
spectral resolution of low-resolution data is not uniform across the
wavelength range (as shown in Table 1), sampling the synthetic spectrum to the
wavelength grid of existing data ensures that the synthetic spectrum has the
same varying spectral resolution as the original spectrum.

To simulate high-resolution spectroscopic data, we use a wavelength range from
2.29 to 2.48 $\mu$m. We apply a rotational broadening of 10 km$\cdot$s-1,
which is consistent with the upper limit of 14 km$\cdot$s-1 as measured in
Wang et al. (2021). We then convolve the spectrum with a Gaussian kernel that
corresponds to spectral resolution of R=35,000, and resample the spectrum to
the wavelength grid of the KPIC data. We add random gaussian errors to all
simulated data points based on the reported measurement uncertainties.

We simulate three data sets. The first simulated data set (SDS I) contains
only the medium-to-low-resolution and photometric data, i.e., data set #1, #2,
#3, and #4 in Table 1. This is the same data set that is used in Wang et al.
(2020). The second simulated data set (SDS II) contains the high-resolution
data (data set #5). Since the high-resolution data are normalized, we also
include simulated CHARIS data (data set #1) and photometric data (data set #4)
so as to provide flux information for $J$ through $M$ band. The third
simulated data set (SDS III) contains simulated data for all observations as
listed in §2 and Table 1. The data set includes high- and low-resolution
spectral data along with photometric data.

###  3.2 Retrieval Analyses For the Simulated Data

The retrieved C and O abundances and C/O ratios are broadly consistent with
the solar values, which are the input of the PHOENIX spectrum that we use to
simulate the data (Fig. 1). Retrieved values of model parameters can be found
in Table 3.

For SDS I, the [C/H], [O/H], and the C/O ratio are $0.31^{+0.12}_{-0.12}$,
$0.31^{+0.08}_{-0.08}$, and $0.56^{+0.06}_{-0.06}$. The C and O abundances are
2.6-$\sigma$ and 3.9-$\sigma$ higher than solar values and the C/O ratio is
consistent with the solar value (see Table 2 for solar values).

For SDS II, the [C/H], [O/H], and the C/O ratio are $0.14^{+0.33}_{-0.32}$,
$0.20^{+0.23}_{-0.18}$, and $0.48^{+0.14}_{-0.15}$. While the uncertainties
are larger, the C and O abundances and the C/O ratio are consistent with the
solar values.

For SDS III, the [C/H], [O/H], and the C/O ratio are $0.21^{+0.20}_{-0.20}$,
$0.24^{+0.13}_{-0.12}$, and $0.52^{+0.10}_{-0.10}$. The C and O abundances are
1.0-$\sigma$ and 2.0-$\sigma$ higher than solar values and the C/O ratio is
consistent with the solar value within 1-$\sigma$.

---
Figure 1: Retrieved C and O abundances and C/O based on the PHOENIX spectrum.
Top SDS I: Low-resolution and photometric data only. C and O abundances from
posterior samples (blue histograms) tend to be overestimated by 2.6-3.9
$\sigma$ when compared with the solar values (vertical dashed lines, Asplund
et al., 2009). Vertical blue lines mark 16% and 84% percentile values of
posterior distributions. Middle SDS II: High-resolution data plus CHARIS and
photometric data. C/O is underestimated and C and O abundance uncertainties
are the largest. Bottom SDS III: High and low-resolution along with
photometric data. C and O abundances and C/O agree with the solar values
within $\sim$1-2 $\sigma$.

Upon further consideration, we decide that combining high- and low-resolution
spectral and photometric data offers a balance between accuracy and precision
(as demonstrated with SDS III, bottom panel of Fig. 1). In comparison,
retrieval on the data set without high-resolution data (SDS I, top panel of
Fig. 1) significantly overestimates the C and O abundances by 2.6-3.9
$\sigma$. On the other hand, while unbiased, the retrieval analysis on the
data set with only high-resolution data and photometric data points (i.e., SDS
II) returns the largest uncertainty (middle panel in Fig. 1).

Although the analyses work reasonably well in retrieving C and O abundances,
we find that we cannot satisfactorily retrieve the input $\log g$. The input
value for $\log g$ is 3.5, but we consistently retrieve a $\log g$ that is
higher than 4.2, which is $>$3-$\sigma$ higher than the input value. In
comparison, the retrieved planet radius is generally overestimated by
$\sim$2-$\sigma$ when compared to the input value of 1.2 RJupiter. The
retrieved Teff is consistent with the input value (1200 K) within
$\sim$1-$\sigma$: the retrieved values of Teff from SDS I, II, and III are
$1228^{+126}_{-40}$ K, $1215^{+69}_{-42}$ K, and $1300^{+253}_{-95}$ K,
respectively. While the retrieved $\log g$ is inconsistent at $>$3-$\sigma$
level, this does not affect the main conclusion of the paper, which rests on
the C and O abundances and the C/O ratio of HR 8799 c (see §5.3 for more
discussion).

---
Figure 2: Retrieved results for HR 8799 c. Top five panels are the observed
spectroscopic and photometric data (black) and the 1-$\sigma$ (16 to 84
percentile, darker red) and 2-$\sigma$ (2.5 to 97.5 percentile, lighter red)
distribution of modeled spectra. OSIRIS and KPIC spectral data are normalized
so the flux on the y-axis is arbitrary. The bottom panel is a residual plot
with data minus model and divided by the individual errors.  Figure 3: Carbon
and oxygen abundance for HR 8799 (blue) and planet c (black and red data
points). The dashed line represents solar C/O (0.55, Asplund et al., 2009) and
the red shaded region is the 1-$\sigma$ uncertainty region for HR 8799 stellar
C/O (0.54${}^{+0.12}_{-0.09}$). The quoted values for HR 8799 c are obtained
from the retrieval analysis on DS III which include low- and high-resolution
spectral data as well as photometric data. See §5.1 for more details for DS
III and other comparison data sets (DS I and DS II).

##  4 Key Results From the Retrieval Analysis for HR 8799 c

We focus on the retrieval analysis on all available data for HR 8799 c (DS III
in Table 3) because our retrieval results on simulated data sets in §3 shows
that combining high- and low-res spectral and photometric data points leads to
an accurate and precise measurement of C and O abundances.

Fig. 2 shows that our model fits reasonably well for all available data: our
retrieved model spectra fit data within 3-$\sigma$ (bottom panel in Fig. 2).
Noticeable mismatches include the short-wavelength end of the $J$-band CHARIS
data, shorter wavelength to 2 $\mu$m in the $K$-band GPI data, and the
$M$-band photometric data. These mismatches were also seen in Wang et al.
(2020).

We retrieve C and O abundances and the C/O ratio that are higher than (at
$\sim$1-$\sigma$ level) the host star values (Table 2). For HR 8799 c, the
[C/H], [O/H], and C/O are 0.55${}^{+0.36}_{-0.39}$, 0.47${}^{+0.31}_{-0.32}$,
and 0.67${}^{+0.12}_{-0.15}$ at 68% credible interval. The 68% credible
interval applies to all quoted uncertainties in this paper. In comparison, the
host star HR 8799 [C/H], [O/H], and C/O are 0.04$\pm$0.12, 0.08$\pm$0.14, and
0.54${}^{+0.12}_{-0.09}$ (Wang et al., 2020).

It is important to note that our test on SDS III in §3 suggests that the
retrieved [C/H] and [O/H] may be overestimated by 0.21 and 0.24 dex and the
C/O ratio underestimated by 0.03. Assuming the same bias exist for the
analysis on DS III and after correcting for the bias, the [C/H], [O/H], and
C/O are 0.34${}^{+0.36}_{-0.39}$, 0.23${}^{+0.31}_{-0.32}$, and
0.70${}^{+0.12}_{-0.15}$ for HR 8799 c. As a result, the conclusion of super-
stellar abundance is weakened and super-stellar C/O is strengthened. However,
the super-stellar abundances and C/O ratio remain at $\sim$1-$\sigma$ level.

###  4.1 Formation Pathway

####  4.1.1 A Brief Introduction

Fig. 3 shows the 2-D parameter space of [C/H] vs. [O/H], which may be
informative as to the planet formation pathway (Madhusudhan et al., 2017). For
example, if the planet and the host star share similar chemical abundances,
then it is likely that they form in a similar fashion to a binary star via
gravitational instability that leads to fragmentation in a molecular cloud.
Alternatively, a significant difference in chemical abundances and elemental
ratios would indicate formation scenarios that are more akin to core accretion
pathway.

As summarized in Öberg et al. (2011), when migration is minimal a stellar C/O
ratio is expected for planets forming interior to both the water iceline and
the carbon-grain evaporation line, where all volatiles are in the gas phase.
Sub-stellar C/O and sub-stellar abundance indicate formation beyond the water
iceline but inside the CO iceline. Substellar or stellar C/O and super-stellar
absolute abundances (relative to hydrogen) indicate that a large amount of
solids “polluted” the atmosphere following gas accretion. Super-stellar C/O
and abundances may be due either to gas accretion close to the CO or CO2
icelines, or indicate that the atmosphere was substantially polluted by pebble
(early on) or planetesimal (above the pebble isolation mass) accretion and/or
core erosion. When migration is significant, the various environments along
the path can contribute, leading to less stringent constraints on planet
formation (Pacetti et al., 2022).

Results from Konopacky et al. (2013) and Wang et al. (2020) are consistent
with a scenario in which the planet forms between the H2O and CO2 icelines,
accreting gas that is deficient in H2O, and core erosion plays no role in
enriching the planet atmosphere. In comparison, the result from Lavie et al.
(2017) points to a scenario in which the planet atmosphere is significant
affected by pebble/planetesimal accretion and/or core erosion post formation.

####  4.1.2 One Likely Scenario

Our new result—super-stellar/stellar C/O and super-stellar
metallicity—suggests the following formation scenario: the HR 8799 Jovian
planets formed well before $\sim$1 Myr, with their atmospheres contaminated by
C and O through accretion of solids, and the planets migrated inward to their
current locations.

First, we address the formation time scale. The formation of the planets need
to be very early to be consistent with the observed level of metal
contamination. It is estimated that 65-360 M⊕ of solids have been accreted to
account for the observed $\sim$0.5 dex metal enrichment (Mollière et al.,
2020). The large solid budget is only possible to be available during the
Class I phase (Najita & Kenyon, 2014), which is much younger than 1 Myr. The
early planet formation is also supported by evidence of planet formation in
embedded phases for young stars (Harsono et al., 2018).

Forming multiple planets early on can be challenging for both gravitational
instability and core accretion. While gravitational instability can form
companions on a time scale that is less than $\sim$0.1 Myr (Pineda et al.,
2015), there is little evidence that the instability can form planets in
orbital resonance, which is a sign of orbital migration for HR 8799 planets
(Konopacky et al., 2016; Wang et al., 2018a). Early work also disfavors the
core-accretion scenario given the insufficient mass budget and short formation
time scale (Dodson-Robinson et al., 2009). However, recent studies show that
gas giant planets can form within $\sim$1 Myr via streaming instability
(Youdin & Goodman, 2005; Li et al., 2019) and pebble accretion (Ormel & Klahr,
2010; Johansen & Lambrechts, 2017).

Second, we address the accretion of solids. The accreted solids can be in the
form of dust, pebbles, and/or planetesimals. In the core accretion scenario
via streaming instability and pebble accretion, the accreted solids can be
dominated by pebbles. The key difference between pebbles and other smaller
dust or larger planetesimals is that pebbles feel the gas drag and move toward
locations of pressure maxima. In the case of forming planets via gravitational
instability, as planets form early and open gaps, pebbles can no longer be
accreted onto the growing planets. In comparison, accretion of planetesimals
and small dust can ensue through diffusion and dynamical scattering.

Lastly, we address the inward orbital migration and atmospheric contamination.
In accretion models, only solid accretion outside the CO iceline, where solids
have stellar C/O, would result in a stellar C/O in the planet atmosphere. At
other radial locations, the planet C/O can vary greatly. For example, the C/O
ratio be significantly lower than stellar C/O, e.g., accretion of solids that
are enriched by O but not C (e.g., H2O) within the CO iceline but outside the
water ice line. Mollière et al. (2020) shows that the location for CO iceline
($\sim$45 AU) is much further away than the current location of HR 8799 e
($\sim$15 AU). Therefore, under this scenario, HR 8799 e would have to migrate
inward and other planets (including planet c) would have to migrate inward too
in order to maintain the resonant chain.

Moreover, the C and O abundances and C/O for HR 8799 e (Mollière et al., 2020)
and 8799 c in this work have comparable C and O abundances and C/O. Both
planets have super-solar metallicities at $\sim$0.5 dex and a C/O that is
consistent with the stellar value. For 8799 e, it is estimated that 65-360 M⊕
of solids have been accreted (Mollière et al., 2020). The amount of accreted
solids for 8799 c should be comparable.

The timing of the atmospheric pollution should precede the inward migration or
take place during the early phase of the migration when the planets are
outside the CO iceline. This is to ensure that the majority of the accreted
solids have stellar C/O ratio. However, accretion of gas and solids from the
infalling protostellar envelope may complicate the interpretation. Depending
on the original formation location, the gas/dust ratio for the infalling
material can vary significantly (e.g., Visser et al., 2009). Thus, detailed
modeling works of the HR 8799 exoplanetary atmospheric compositions that
involve infalling gas+dust, accretion, and migration should provide further
constraints on their formation and migration history.

####  4.1.3 Unlikely Scenarios

In principle, gas accretion close to the CO iceline can also lead to
stellar/super-stellar CO and super-stellar metallicity (Öberg et al., 2011),
as the evaporation front close to the CO iceline results in an overabundance
of CO. Therefore, the accretion of CO gas would lead to stellar C/O and super-
stellar abundances that we measured. From Öberg et al. (2011), CO/H2 gas
enrichment by a factor of two would lead to a stellar C/O and an enrichment of
2 times (0.3 dex). A higher level of CO enrichment would result in a higher
level of metal-enrichment. However, in multiple-planet systems such as the HR
8799 system, it is unlikely that both planet c and e form in the same region
where CO abundance is enhanced.

In a different scenario in which planets form inside the water ice line but
outside the refractory carbon stability front, the C/O ratio can approach
unity from the accreted solids (Chan et al., submitted). This would require
outward migration for the 8799 planets. However, we consider this scenario
unlikely given the insufficient mass budget in forming multiple planets
between the iceline and the soot line.

We also discuss a core-erosion scenario (Madhusudhan et al., 2017), but
conclude that the scenario is unlikely given the high-level of metal-
enrichment of the atmosphere.

In the core-erosion scenario, the post-formation atmospheric contamination is
achieved by core erosion. The level of enrichment depends on how much metal
can be provided by the core. In the core-accretion planet formation scenario,
the mass threshold for the core is $\sim$10 M⊕, so the core can provide up to
$\sim$10 M⊕ metals. This is insufficient when compared to the amount of solids
(65-360 M⊕, Mollière et al., 2020) that is needed to enrich the atmosphere.

In the disk-fragmentation scenario, the core mass is estimated to be $\sim$15
M⊕ assuming a planet mass of 5 MJupiter and a dust-to-gas ratio of 1 to 100.
While the dust-to-gas ratio can be enhanced in a proto-planetary disk, the
core mass seems insufficient to enrich the atmosphere at the observed level.

###  4.2 Temperature Structure and Cloudiness

The retrieved P-T profile is shown in Fig. 4. The contribution function (black
solid line in Fig. 4) shows that the majority of the thermal flux comes from
pressures between 0.1 and 1 bar, at a temperature of $\sim$1200 K. This is
consistent with the effective temperature of the planet (Wang et al., 2020).
Below $\sim$10 bar, the P-T profile has a thermal gradient ($d\log T/d\log P$)
about 0.25, indicating a convective layer. Between 0.1 and 10 bar, the thermal
gradient decreases, implying that the dominant energy transporting mechanism
shifts from convection to radiation.

At lower pressure levels (P$<$0.1 bar), the P-T profile is less constrained,
but the increased thermal gradient may indicate some detached convective
regions as suggested in Marley & Robinson (2015). The upper convective regions
can facilitate vertical mixing that has been suggested to cause chemical
disequilibrium for many brown dwarfs and exoplanets, including HR 8799 c
(e.g., Skemer et al., 2014).

The contribution function partially overlaps with the cloud opacity (dashed
line in Fig. 4). Below we provide a calculation of cloud optical depth and
discuss the implication of the cloudiness of HR 8799 c. The retrieved cloud
opacity is $\sim$10-4 cm2/g with a long tail towards to lower values. At the
cloud base, the pressure is $\sim$10 bar and the temperature is $\sim$1600 K.
The density $\rho$ is 2$\times 10^{-4}$ g/cm3 assuming an ideal gas law with a
mean molecular weight of 2.3. The cloud spans $\sim$1.5 dex of pressure change
(from 10 bar to 0.3 bar as shown in Fig. 4. This corresponds to $\sim$6 scale
heights. At a temperature of $\sim$1600 K and a $\log g$ of 4.2 with a mean
molecular weight of 2.3, each scale height is 36 km. Therefore, 6 scale
heights are 216 km. Based on $\tau=\kappa\rho l$, where $\tau$ is optical
depth, $\kappa$ is opacity, and $l$ is the cloud thickness, the optical depth
is 0.4.

The optical depth is comparable with unity. Note that the retrieval model is
one-dimensional and retrieves the average of a 3D object. This means that
certain regions would have cloud optical depth higher than 1 given the
patchiness of clouds. Therefore, it is likely that clouds play a role in
affecting the emerging thermal flux even though the cloud contribution
function is slightly below the flux contribution function. Our inference of
the cloudiness is consistent with previous observational and modeling work for
HR 8799 planets (e.g., Skemer et al., 2014; Mollière et al., 2020; Skemer et
al., 2012; Currie et al., 2014).

Figure 4: Retrieved P-T profile (1-$\sigma$ region in red shaded region) for
HR 8799 c using DS III. The data set (DS III) include low- and high-resolution
spectral data as well as photometric data. See §5.1 for more details for this
data set and other data sets (DS I and DS II). The spectrally-averaged
contribution function is shown as the black solid line, which partially
overlaps with retrieved cloud layer (black dashed line). The pressure level of
the retrieved cloud layer is consistent with that of a physical MgSiO3 cloud
(blue dotted line).

##  5 Discussion

###  5.1 Comparing C and O Abundances From Different Data Sets

In addition to the retrieval analysis using all available data for HR 8799 c,
which include high- and low-spectral resolution along with photometric data
points (i.e., DS III, similar to the nomenclature in §3), we also run separate
analyses using different combination of data sets. The results are shown in
Table 3. The choices to construct a data set for retrieval are similar to
those in §3. One data set (DS I hereafter) uses all data but the high spectral
resolution data from KPIC. The other data set (DS II hereafter) uses high
spectral resolution data from KPIC, CHARIS data that covers $J$ through
$M$-band, and photometric data points.

The most interesting case is DS II. The analysis on DS II returns much higher
C and O abundances and C/O than that for DS I and DS III (Fig. 5). In
addition, the retrieved uncertainties are the smallest among all DS data sets.
This is not seen in our analysis on the simulated data set SDS II.

Moreover, DS II is the data set with the most constrained fblur term at
$12.08^{+4.23}_{-2.89}$ km$\cdot$s-1. The term consists of both the spectral
blurring due to a finite resolving power and the rotational broadening.
Assuming the spectral blurring term and rotational broadening term sum up in
quadrature to equal to the measured fblur term, if the spectral blurring is
$\sim$8.5 km$\cdot$s-1 at R=35,000, the rotational broadening is $\sim$8.7
km$\cdot$s-1 with a sizable uncertainty of 3.0-4.5 km$\cdot$s-1. This is
consistent with the upper limit of 14 km$\cdot$s-1 given in Wang et al.
(2021). DS III constrains fblur to be $\sim$20 km$\cdot$s-1, which is also
consistent with the the upper limit in Wang et al. (2021) given the large
uncertainty (Table 3). DS I does not constrain fblur because the low-
resolution data have no constraining power on fblur: OSIRIS data with the
highest spectral resolution (R=5000) among low-resolution data can only
resolve fblur as low as $\sim$60 km$\cdot$s-1, which is not sufficient for the
$\sim$10 km$\cdot$s-1 level for HR 8799 c.

Overall, we find that DS III and DS I return similar C and O abundances and
C/O. This is not surprising given that the most constraining power comes from
the lower-resolution data sets. We follow Xuan et al. (2022) to quantify the
relative constraining power between the high-resolution KPIC and the lower-
resolution IFU data. The detection significance using KPIC high-resolution
data is $\sim$10 (Wang et al., 2021). This detection significance is
quantified by the significance of the peak of a cross correlation function
(CCF) when cross-correlating the KPIC spectrum with a template. The detection
significance using CCF is also commonly referred to as CCF signal-to-noise-
ratio (SNR). In comparison, median SNR per wavelength bin is typically higher
than 10 for lower-resolution date sets (Table 1). Therefore, the lower-
resolution data sets dominate the constraining power, DS I and DS III return
similar results because lower-resolution date sets are used in these two data
sets.

Based on our tests on the synthetic PHOENIX spectrum, we adopt the retrieval
result using DS III. Our key findings in §4 remain unchanged for DS I-III even
with the high C and O abundances and C/O retrieved for DS II (Fig. 3).

---
Figure 5: Retrieved C and O abundances and C/O based on the HR 8799 c data: C
and O abundances from posterior samples are blue histograms, and stellar
values are the black data points. Top: DS I: Low-resolution and photometric
data only. Middle: DS II: High-resolution data plus CHARIS and photometric
data. Bottom: DS III: High and low-resolution along with photometric data. The
difference between DS and SDS in Fig. 1 is that DS indicates real data and SDS
indicates simulated data.

###  5.2 Comparing Retrieved Uncertainties

Properly estimated uncertainties enable accurate interpretation of the
retrieval results. Below we compare uncertainties in our retrieval analyses.

The uncertainties for the HR 8799 c data analyses (DS I and III) are generally
$\sim$2-3 time higher than those for the PHOENIX analyses (SDS I and III). We
attribute the smaller uncertainties for SDS I and III to the similarities of
underlying physics of radiative transfer in atmospheres between petitRADTRANS
and PHOENIX. The larger uncertainties in DS I and III can be interpreted as
accumulative differences between modeling and real atmospheres as marginalized
over many modeling parameters.

DS I was used for retrieval analysis in Wang et al. (2020), and is re-analyzed
using our updated retrieval code (Wang et al., 2022a). The uncertainties for
[C/H], [O/H], and C/O are 0.1 dex, 0.06 dex, and 0.05 in Wang et al. (2020).
In our new analysis on DS I, the uncertainties for [C/H], [O/H], and C/O are
$\sim$0.35 dex, 0.3 dex, and 0.1. The new uncertainties are 2-5 times larger
than those from the previous analysis. We attribute the increased
uncertainties, and likely more realistic, to the updated and more flexible
model, e.g., cloud treatment and P-T profile, as detailed in §4.2 in Wang et
al. (2022a).

DS II represents an interesting case as discussed in §5.1. A relevant case is
the analysis on HR 7672 B (Wang et al., 2022a) where the data set is similarly
a combination of high-resolution spectral data and photometric data points.
The uncertainties for [C/H] and [O/H] are $\sim$0.05 dex and C/O uncertainty
is 0.02. These values are much lower than those for DS II in this work
($\sim$0.2 dex for C and O abundances and 0.06 for C/O). The higher
uncertainties for DS II are due to the much lower SNR (SNR$\sim$1 per pixel)
of the HR 8799 c high-resolution data than the HR 7672 B data (SNR$\sim$10-20
per pixel).

DS II analysis also returns the lowest uncertainties among DS I-III. One
possible explanation is the more homogeneous data used in DS II. More data
from inhomogeneous sources can introduce difference systematics and therefore
broadens the posterior distribution (Wang et al., 2020).

An alternative explanation is that high spectral resolution data are more
effective in constraining chemical abundances and atmospheric properties (for
HD 4747 B, Xuan et al., 2022). However, in the case of HR 8799 c, we prefer
results using DS III over DS II for the following reasons. First, our test
using PHOENIX spectrum shows that combining high- and low-resolution spectral
data as well as photometric data points leads to the most accurate results
(§3). Second, our low-resolution spectral and photometric data covers a large
wavelength from $J$ to $M$-band. The large wavelength coverage includes
spectral range from 2.2 to 2.5 $\mu$m and the $L$ and $M$-band, which are
thought to be crucial in constraining chemical abundances (Xuan et al., 2022)
and cloud properties (Wang et al., 2020). This is a key difference between the
HD 4747 B data and the HR 8799 c data. Lastly, the major conclusions of the
paper would remain the same even if we adopt the retrieval result from DS II.

###  5.3 The Influence of Radius and Gravity on C and O Abundances

As noted in §3 for the synthetic data, the retrieved radius ($\sim$1.3
RJupiter) and surface gravity ($\log g\sim$4.2) are off by more than
3-$\sigma$ from the input values of the synthetic PHOENIX spectrum, which are
1.2 RJupiter for radius and 3.5 for $\log g$.

We run a separate retrieval analysis to investigate the influence of radius
and surface gravity on C and O abundances (see results in Table 3). We compare
the results from the analysis in §3 to those for which we fix the radius and
$\log g$ at 1.2 RJupiter and 3.5. We find that the retrieved C and O
abundances and C/O are consistent between these retrieval analyses. Changing
the radius and surface gravity from 1.2 and 3.5 to 1.3 and 4.2 shifts the C
and O abundance upward by $\sim$0.2 dex and the C/O downward by 0.03.

These changes are smaller than, but not negligible when compared to, the
uncertainties in the C and O abundances and the C/O ratio. If accounting for
the effect of surface gravity on C and O abundances, [C/H] and [O/H] for HR
8799 c would shift downward by $\sim$0.2 dex if $\log g$ drops from 4.2 to
3.5, but the major conclusions in §4 remain unchanged. We also note that the
retrieved radius at $1.01^{+0.09}_{-0.08}$ RJupiter and $\log g$ at
$4.17^{+0.41}_{-0.47}$ for HR 8799 c (DS III in Table 3) correspond the a
planetary mass of $\sim$6 MJupiter. These values agree well with most recent
literature values (Doelman et al., 2022) and the mass is unlikely cause the
long-term dynamical instability of the system (Wang et al., 2018b) .

###  5.4 Chemical Disequilibrium

The atmosphere of HR 8799 c is known to be in chemical disequilibrium (e.g.,
Skemer et al., 2012; Konopacky et al., 2013). Here, we investigate (1) if our
results are consistent with the chemical disequilibrium case; and (2) to what
extent we can constrain the vertical mixing which is thought to disrupt
equilibrium chemistry.

We use poor_mans_nonequ_chem to interpolate a pre-calculated chemical grid
from easyCHEM (Mollière et al., 2017). The grid spans multiple dimensions
including temperature (60 - 4000 K), pressure (10-8 \- 1000 bar), C/O (0.1 -
1.6) and [Fe/H] (-2 - 3). To calculate the equilibrium abundance, we use the
median of the retrieved P-T profile and stellar values of C/O = 0.67 and
[Fe/H] = 0.50. The value for the metallicity [Fe/H] is chosen to be consistent
with the retrieved [C/H] and [O/H] at 0.55 dex and 0.47 dex, respectively (see
Table 3 for DS III results).

Fig. 6 shows the abundances (in mass mixing ratio) for four species (CO, H2O,
CH4, and CO2) assuming two conditions: chemical equilibrium, and a quenched
case of chemical disequilibrium, where vertical mixing homogenizes abundances
above a quench pressure which we set at 50 bar.

The retrieved abundances are inconsistent with the predicted values assuming
equilibrium chemistry. At pressures between 0.1 and 1 bar, which is the peak
of the contribution function (see Fig. 4), the abundances as predicted from
equilibrium chemistry are too high for CH4 and too low for H2O, when compared
to retrieved values. That is, CH4 needs to be quenched and H2O needs to be
enhanced to match with the retrieval result.

This can be achieved by invoking vertical mixing at a quench pressure. Indeed,
the retrieved abundances are consistent with the predicted values assuming a
perfect vertical mixing for pressures lower than 50 bar (vertical solid lines
in Fig. 6). This indicates the case of chemical disequilibrium and a quench
pressure that is higher than $\sim$10 bar.

###  5.5 CO2 vs. no CO2

Similar to Wang et al. (2020), the peak of the retrieved CO2 mass mixing ratio
is $\sim$10-4. While the posterior distribution has a long tail toward much
lower abundance, the retrieved median of CO2 mass mixing ratio is only two
orders of magnitude lower than the retrieved median of CO mass mixing ratio.
This is at odds with Sorahana & Yamamura (2012), who predicted that CO2 should
be $\sim$4 orders of magnitude less abundant than CO for a brown dwarf with
similar effective temperature although CO2 may be produced through
photochemistry (Moses et al., 2016). We therefore perform another retrieval
analysis where we fix the CO2 mass mixing ratio to be 10-10. The retrieved C
and O abundances and C/O are consistent between the two models with varying
and fixed CO2 abundance. Neither model is preferred given the difference of
$\ln$(Z) is only 1.3, where is Z is the Bayesian evidence. We therefore
conclude that including/excluding CO2 does not affect the retrieved C and O
abundances and C/O.

Figure 6: Abundances assuming equilibrium chemistry (dashed lines) and a
quenched pressure at 50 bar (the horizontal blue dashed line). 1-$\sigma$
ranges of retrieved abundances are shown in shaded regions. The retrieved
abundances are inconsistent with the predicted values assuming equilibrium
chemistry, at pressures between 0.1 and 1 bar which is the peak of the
contribution function (see Fig. 4). Instead, the retrieved abundances are
consistent with the predicted values assuming a perfect vertical mixing for
pressures lower than 50 bar (vertical solid lines). This indicates a case of
chemical disequilibrium and a quench pressure that is higher than $\sim$10
bar.

##  6 Summary

  * •

Using photometric and spectroscopic data with resolving powers up to 35,000,
we retrieve both elemental abundances and abundance ratios for HR 8799 c. We
find that [C/H], [O/H], and C/O are 0.55${}^{+0.36}_{-0.39}$,
0.47${}^{+0.31}_{-0.32}$, and 0.67${}^{+0.12}_{-0.15}$ at 68% credible
interval. Retrieval results are summarized in Table 3 while a detailed
discussion can be found in §5.

  * •

The super-stellar C and O abundances and the stellar C/O ratio reveal
potential formation pathways for HR 8799 c. Planet c, and likely the other
planets in the system, formed within $\sim$1 Myr when there were sufficient
solids to enhance the planetary atmosphere C and O abundances. The metal
enrichment likely preceded or took place during the early phase of the inward
migration to the planet current locations. More details can be found in §4.1.

  * •

The retrieval code is tested using a physically- and chemically-motivated
PHOENIX synthetic spectrum with similar properties as those for HR 8799 c
(§3). We find that combining all available data, i.e., photometric data, and
low- and high-resolution spectroscopic data, result in the most accurate
inference of C and O abundances. Further discussions on retrieval
uncertainties and systematics can be found in §5.1, §5.2, and §5.3.

In the future, we will extend this work to other planets in HR 8799 and test
the consistency of formation pathway for the planets in the same system. We
will take a similar approach to analyze other directly-imaged exoplanets with
high-resolution spectroscopic data (e.g., KPIC) and lower resolution data that
cover larger range of wavelength (e.g., the Jame Webb Space Telescope).

Acknowledgments This work is supported by the National Science Foundation
under Grant No. 2143400. We thank Andrew Youdin, Joan Najita, and Kaitlin
Kratter for insightful discussions on gravitational instability vs. pebble
accretion. We thank the anonymous referee for constructive comments and
suggestions that significantly improve the paper.

Funding for KPIC has been provided by the California Institute of Technology,
the Jet Propulsion Laboratory, the Heising-Simons Foundation (grants
#2015-129, #2017-318 and #2019-1312), the Simons Foundation (through the
Caltech Center for Comparative Planetary Evolution), and NSF under grant
AST-1611623.

The data presented herein were obtained at the W. M. Keck Observatory, which
is operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Mauna Kea has always had within the indigenous Hawaiian community.
We are most fortunate to have the opportunity to conduct observations from
this mountain.

We thank the Heising-Simons Foundation for supporting the workshop on
combining high-resolution spectroscopy and high-contrast imaging for exoplanet
characterization, where the idea originated on combining photometric data and
spectral data of different resolutions.

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Table 1: Summary of data sets

# | Telescope/ | Wavelength | Resolution | Mean | Reference
---|---|---|---|---|---
| Instrument | Range ($\mu$m) |  | SNR |
1 | Subaru/CHARIS | 1.16 - 2.37 |  $\sim$20 | 13.8 | Wang et al. (2022b)
2 | Gemini/GPI | 1.50 - 2.37 | 45 - 80 | 15.3 | Greenbaum et al. (2018)
3 | Keck/OSIRIS | 1.96 - 2.38 | 5,000 | 42.2 | Konopacky et al. (2013)
4∗ |  $L$-$M$ photometry | 3.04 - 4.67 | $\cdots$ | 7.2 | Bonnefoy et al. (2016)
5 | Keck/KPIC | 2.291 - 2.482 |  $\sim$35,000 | 0.6∗∗ | Wang et al. (2021)
HR 8799 c

Note. — $\ast$: A collection of data points from Keck/NIRC2, VLT/NaCo,
Subaru/IRCS, and LBTI/LMIRCam (Galicher et al., 2011; Skemer et al., 2012,
2014; Currie et al., 2014). $\ast\ast$: CCF SNR$\sim$10 (Wang et al., 2021).

Table 2: C and O abundances for HR 8799 and planet c

| $\log{\epsilon_{\rm C}}$ | [C/H] | $\log{\epsilon_{\rm O}}$ | [O/H] | C/O
---|---|---|---|---|---
HR 8799
Wang et al. (2020) | 8.54$\pm$0.12 | 0.04$\pm$0.12 | 8.81$\pm$0.14 | 0.08$\pm$0.14 | 0.54${}^{+0.12}_{-0.09}$
Solar∗ | 8.43 | 0.00 | 8.69 | 0.00 | 0.55
HR 8799 c
This work | 8.98${}^{+0.36}_{-0.39}$ | 0.55${}^{+0.36}_{-0.39}$ | 9.16${}^{+0.31}_{-0.32}$ | 0.47${}^{+0.31}_{-0.32}$ | 0.67${}^{+0.12}_{-0.15}$

Note. — $\ast$: We use solar elemental abundances from Asplund et al. (2009).
We note that solar C and O abundances have been revised upward significantly
in a recent work (Magg et al., 2022). The revision affects values of [C/H] and
[O/H] and the solar C/O, but will not affect results in this paper since we
compare planet atmospheric abundances and abundance ratios to those of the
host star. This highlights the importance of using stellar abundances instead
of solar abundances when interpreting retrieval results.

Table 3: Summary of Retrieval Results.

Parameter | Unit | PHOENIX | HR 8799 c
---|---|---|---
|  | SDS I | SDS II | SDS III | Fixed $\log g$ | DS I | DS II | DS III | Fixed CO2
$\log$(g) | cgs | $4.33^{+0.13}_{-0.14}$ | $4.53^{+0.31}_{-0.33}$ | $4.20^{+0.21}_{-0.25}$ | $3.50^{+0.00}_{-0.00}$ | $4.33^{+0.39}_{-0.42}$ | $4.32^{+0.44}_{-0.60}$ | $4.17^{+0.41}_{-0.47}$ | $4.30^{+0.33}_{-0.36}$
RP | RJupiter | $1.30^{+0.06}_{-0.05}$ | $1.26^{+0.07}_{-0.06}$ | $1.36^{+0.07}_{-0.07}$ | $1.20^{+0.00}_{-0.00}$ | $1.06^{+0.06}_{-0.07}$ | $0.95^{+0.05}_{-0.04}$ | $1.01^{+0.09}_{-0.08}$ | $0.96^{+0.06}_{-0.05}$
$\Delta_{y}$ |  $\cdots$ | $-0.01^{+0.31}_{-0.31}$ | $-0.04^{+0.06}_{-0.06}$ | $-0.08^{+0.10}_{-0.09}$ | $-0.09^{+0.09}_{-0.08}$ | $0.01^{+0.30}_{-0.31}$ | $0.24^{+0.03}_{-0.03}$ | $0.14^{+0.06}_{-0.06}$ | $0.16^{+0.05}_{-0.05}$
$\log$(mr${}_{\rm{H}_{2}\rm{O}}$) |  $\cdots$ | $-2.24^{+0.05}_{-0.05}$ | $-2.28^{+0.13}_{-0.11}$ | $-2.27^{+0.08}_{-0.07}$ | $-2.50^{+0.04}_{-0.03}$ | $-2.18^{+0.21}_{-0.22}$ | $-2.07^{+0.19}_{-0.25}$ | $-2.22^{+0.24}_{-0.25}$ | $-2.14^{+0.20}_{-0.20}$
$\log$(mrCO) |  $\cdots$ | $-1.95^{+0.12}_{-0.12}$ | $-2.14^{+0.33}_{-0.33}$ | $-2.05^{+0.20}_{-0.21}$ | $-2.17^{+0.18}_{-0.22}$ | $-1.75^{+0.34}_{-0.37}$ | $-1.30^{+0.19}_{-0.28}$ | $-1.72^{+0.37}_{-0.39}$ | $-1.60^{+0.31}_{-0.33}$
$\log$(mr${}_{\rm{C}\rm{O}_{2}}$) |  $\cdots$ | $-3.79^{+0.51}_{-3.49}$ | $-3.51^{+0.75}_{-3.34}$ | $-5.45^{+1.84}_{-2.85}$ | $-7.24^{+1.83}_{-1.71}$ | $-2.85^{+0.34}_{-0.50}$ | $-6.20^{+1.95}_{-2.35}$ | $-3.50^{+0.67}_{-3.44}$ | $-10.00^{+0.00}_{-0.00}$
$\log$(mr${}_{\rm{C}\rm{H}_{4}}$) |  $\cdots$ | $-4.24^{+0.13}_{-0.12}$ | $-4.23^{+0.27}_{-0.24}$ | $-4.31^{+0.17}_{-0.17}$ | $-4.67^{+0.09}_{-0.10}$ | $-4.20^{+0.33}_{-0.36}$ | $-6.66^{+1.65}_{-2.07}$ | $-4.39^{+0.38}_{-0.44}$ | $-4.34^{+0.33}_{-0.43}$
tint | K | $1850^{+67}_{-66}$ | $1642^{+95}_{-113}$ | $1809^{+101}_{-92}$ | $1983^{+150}_{-86}$ | $1380^{+99}_{-79}$ | $1494^{+180}_{-92}$ | $1421^{+92}_{-72}$ | $1443^{+100}_{-79}$
$\Delta\lambda_{\rm{KPIC}}$ | $\AA$ | $-0.54^{+62.52}_{-62.42}$ | $-0.13^{+0.33}_{-0.33}$ | $0.49^{+54.02}_{-27.10}$ | $0.16^{+37.63}_{-5.30}$ | $-1.32^{+61.58}_{-61.40}$ | $3.04^{+0.11}_{-0.12}$ | $3.04^{+0.52}_{-0.54}$ | $3.03^{+0.27}_{-0.28}$
$\Delta\lambda_{\rm{OSIRIS}}$ | $\AA$ | $-0.48^{+0.25}_{-0.37}$ | $1.00^{+62.41}_{-63.05}$ | $-0.61^{+0.44}_{-0.50}$ | $-0.81^{+0.30}_{-0.42}$ | $-4.85^{+0.70}_{-0.76}$ | $-1.63^{+63.16}_{-60.56}$ | $-4.80^{+0.86}_{-1.17}$ | $-4.58^{+0.57}_{-0.57}$
$\Delta T$ between 100 and 32 bar | K | $736^{+780}_{-509}$ | $644^{+783}_{-462}$ | $1019^{+852}_{-666}$ | $1242^{+767}_{-756}$ | $1102^{+822}_{-719}$ | $1193^{+792}_{-748}$ | $1126^{+811}_{-728}$ | $1163^{+806}_{-744}$
$\Delta T$ between 32 and 10 bar | K | $126^{+172}_{-92}$ | $150^{+212}_{-109}$ | $265^{+368}_{-193}$ | $457^{+499}_{-316}$ | $685^{+799}_{-538}$ | $740^{+680}_{-506}$ | $557^{+721}_{-402}$ | $728^{+753}_{-530}$
$\Delta T$ between 10 and 3.2 bar | K | $62^{+75}_{-44}$ | $105^{+125}_{-75}$ | $112^{+132}_{-80}$ | $93^{+106}_{-68}$ | $360^{+653}_{-297}$ | $289^{+432}_{-207}$ | $216^{+513}_{-156}$ | $383^{+564}_{-295}$
$\Delta T$ between 3.2 and 1 bar | K | $806^{+99}_{-116}$ | $497^{+156}_{-146}$ | $659^{+146}_{-165}$ | $280^{+148}_{-98}$ | $112^{+127}_{-72}$ | $111^{+158}_{-76}$ | $104^{+115}_{-68}$ | $128^{+137}_{-81}$
$\Delta T$ between 1 and 0.1 bar | K | $248^{+160}_{-137}$ | $565^{+271}_{-290}$ | $413^{+264}_{-213}$ | $969^{+22}_{-36}$ | $115^{+102}_{-75}$ | $338^{+86}_{-90}$ | $152^{+79}_{-82}$ | $119^{+86}_{-73}$
$\Delta T$ between 0.1 bar and 1 mbar | K | $380^{+343}_{-258}$ | $477^{+321}_{-306}$ | $413^{+351}_{-275}$ | $247^{+252}_{-164}$ | $659^{+226}_{-288}$ | $640^{+231}_{-289}$ | $592^{+257}_{-305}$ | $578^{+275}_{-320}$
$\Delta T$ between 1 mbar and 10 nbar | K | $507^{+306}_{-307}$ | $496^{+318}_{-310}$ | $495^{+315}_{-316}$ | $509^{+304}_{-309}$ | $496^{+308}_{-310}$ | $497^{+312}_{-307}$ | $496^{+316}_{-301}$ | $496^{+315}_{-309}$
fblur | km$\cdot$s-1 | $50.45^{+31.35}_{-29.89}$ | $22.58^{+13.18}_{-9.51}$ | $67.53^{+21.82}_{-27.48}$ | $68.51^{+20.77}_{-26.34}$ | $50.45^{+30.50}_{-30.58}$ | $12.08^{+4.23}_{-2.89}$ | $20.35^{+28.09}_{-10.59}$ | $14.01^{+11.13}_{-6.13}$
$\log$(mr${}_{\rm{MgSiO}_{3}}$) |  $\cdots$ | $-4.47^{+1.24}_{-0.42}$ | $-4.85^{+1.17}_{-0.98}$ | $-3.69^{+1.02}_{-1.02}$ | $-3.11^{+0.37}_{-0.40}$ | $-3.89^{+1.06}_{-3.57}$ | $-3.68^{+0.97}_{-1.34}$ | $-4.76^{+1.67}_{-3.06}$ | $-3.80^{+1.07}_{-2.76}$
$\log$(Kzz) | cm$2\cdot$s-1 | $5.75^{+2.60}_{-0.57}$ | $7.30^{+1.78}_{-1.67}$ | $7.33^{+1.85}_{-1.82}$ | $5.27^{+0.40}_{-0.20}$ | $7.81^{+1.25}_{-1.33}$ | $7.73^{+1.37}_{-1.46}$ | $7.78^{+1.38}_{-1.50}$ | $7.79^{+1.34}_{-1.42}$
$f_{sed}$ |  $\cdots$ | $4.65^{+0.25}_{-0.41}$ | $4.16^{+0.59}_{-0.88}$ | $4.47^{+0.38}_{-0.59}$ | $4.55^{+0.31}_{-0.51}$ | $2.41^{+1.19}_{-1.05}$ | $2.69^{+1.26}_{-1.16}$ | $2.60^{+1.33}_{-1.29}$ | $2.65^{+1.25}_{-1.18}$
$\sigma_{g}$ |  $\cdots$ | $2.40^{+0.44}_{-0.65}$ | $2.10^{+0.60}_{-0.65}$ | $2.18^{+0.58}_{-0.66}$ | $2.44^{+0.39}_{-0.57}$ | $2.03^{+0.60}_{-0.59}$ | $2.03^{+0.61}_{-0.60}$ | $2.03^{+0.62}_{-0.59}$ | $2.02^{+0.63}_{-0.60}$
$[$C/H$]$ | dex | $0.31^{+0.12}_{-0.12}$ | $0.14^{+0.33}_{-0.32}$ | $0.21^{+0.20}_{-0.20}$ | $0.09^{+0.18}_{-0.22}$ | $0.54^{+0.34}_{-0.36}$ | $0.95^{+0.19}_{-0.28}$ | $0.55^{+0.36}_{-0.39}$ | $0.65^{+0.31}_{-0.33}$
$[$O/H$]$ | dex | $0.31^{+0.08}_{-0.08}$ | $0.20^{+0.23}_{-0.18}$ | $0.24^{+0.13}_{-0.12}$ | $0.06^{+0.12}_{-0.12}$ | $0.49^{+0.28}_{-0.29}$ | $0.80^{+0.18}_{-0.27}$ | $0.47^{+0.31}_{-0.32}$ | $0.56^{+0.27}_{-0.27}$
C/O |  $\cdots$ | $0.56^{+0.06}_{-0.06}$ | $0.48^{+0.14}_{-0.15}$ | $0.52^{+0.10}_{-0.10}$ | $0.58^{+0.09}_{-0.12}$ | $0.63^{+0.11}_{-0.13}$ | $0.78^{+0.06}_{-0.07}$ | $0.67^{+0.12}_{-0.15}$ | $0.69^{+0.10}_{-0.12}$

Note. — a: We report the median of posterior distribution and error bars
correspond to the difference of the median and the 68% credible interval. b:
We adopt solar elemental abundances from Asplund et al. (2009)