# Brown Dwarf Retrievals on FIRE!: Atmospheric Constraints and Lessons Learned
from High Signal-to-Noise Medium Resolution Spectroscopy of a T9 Dwarf

Callie E. Hood Department of Astronomy & Astrophysics, University of
California, Santa Cruz, CA 95064, USA  Jonathan J. Fortney Department of
Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064, USA
Michael R. Line School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287, USA  Jacqueline K. Faherty Department of
Astrophysics, American Museum of Natural History, Central Park West at 79th
Street, NY 10024, USA

###### Abstract

Brown dwarf spectra offer vital testbeds for our understanding of the chemical
and physical processes that sculpt substellar atmospheres. Recently,
atmospheric retrieval approaches have been successfully applied to low-
resolution (R$\sim$100) spectra of L, T, and Y dwarfs, yielding constraints on
the chemical abundances and temperature structures of these atmospheres.
Medium-resolution (R$\sim 10^{3}$) spectra of brown dwarfs offer additional
insight, as molecular features are more easily disentangled and the thermal
structure of the upper atmosphere is better probed. We present results from a
GPU-based retrieval analysis of a high signal-to-noise, medium-resolution
(R$\sim$6000) FIRE spectrum from 0.85-2.5 $\mu$m of the T9 dwarf UGPS
J072227.51-054031.2. At 60$\times$ higher spectral resolution than previous
brown dwarf retrievals, a number of novel challenges arise. We examine the
effect of different opacity sources, in particular for CH4. Furthermore, we
find flaws in the data like errors from order stitching can bias our
constraints. We compare these retrieval results to those for a R$\sim$100
spectrum of the same object, revealing how constraints on atmospheric
abundances and temperatures improve by an order of magnitude or more with
increased spectral resolution. In particular, we can constrain the abundance
of H2S which is undetectable at lower spectral resolution. While these medium-
resolution retrievals offer the potential of precise, stellar-like constraints
on atmospheric abundances ($\sim$0.02 dex), our retrieved radius is
unphysically small ($R=0.50_{-0.01}^{+0.01}$RJup), indicating shortcomings
with our modeling framework. This work is an initial investigation into brown
dwarf retrievals at medium spectral resolution, offering guidance for future
ground-based studies and JWST observations.

##  1 Introduction

Brown dwarfs, objects more massive than gas giant planets but yet not massive
enough to sustain hydrogen fusion like a star (13 MJup $\lesssim$ M $\lesssim$
73 MJup, Burrows et al., 2001), provide essential testbeds of our
understanding of the physics and chemical processes that sculpt substellar
atmospheres. Without a sustained central energy source from fusion, brown
dwarfs instead cool over time, leading to the formation of molecules and
condensates in their atmospheres which dramatically affect their emitted
spectra across the M, L, T, and Y spectral types (e.g. Kirkpatrick, 2005;
Cushing et al., 2011). The chemical and physical processes shaping these
spectra are expected to be similar to those of gas giant exoplanets due to
their similar effective temperatures (Faherty et al., 2016). Thus, the often
more easily-observable spectra of brown dwarfs can inform our predictions for
and interpretations of spectra of directly imaged planets.

Traditionally, brown dwarf spectra have been compared to theoretical “grid
models” which use our current understanding of substellar atmospheres and
evolution to produce model spectra for a small number of fundamental
parameters, such as composition, effective temperature, and surface gravity
(see Marley & Robinson (2015) for a review). The cost of a small number of
parameters is the number of chemical and physical assumptions, for example
radiative-convective and thermochemical equilibrium, that are required. These
grid models are an important resource for connecting observed properties with
physical parameters of brown dwarfs as well as predicting signatures to be
tested with future observations. However, while advancements in molecular
opacities and the increasing model complexity have led to improved fits to
observed spectra (Phillips et al., 2020; Marley et al., 2021), notable
discrepancies remain (Leggett et al., 2021) indicating there is still much to
be learned about modeling these cool atmospheres.

An alternative way to glean information from brown dwarf spectra is
atmospheric retrieval, a data-driven Bayesian inverse method where minimal
assumptions are made for the cost of far more free parameters. First developed
for Earth and Solar System sciences (e.g. Rodgers, 2000; Fletcher et al.,
2007) and then adapted for exoplanets (e.g. Madhusudhan & Seager, 2009;
Benneke & Seager, 2012; Line et al., 2013), atmospheric retrievals have been
applied successfully to brown dwarf spectra of various spectral types (Line et
al., 2014, 2015, 2017; Burningham et al., 2017, 2021; Zalesky et al., 2019,
2022; Gonzales et al., 2020, 2021, 2022; Kitzmann et al., 2020; Piette &
Madhusudhan, 2020; Howe et al., 2022; Lueber et al., 2022; Wang et al., 2022;
Xuan et al., 2022; Calamari et al., 2022). Retrievals provide a way to test
the assumptions included in grid models; for example, Line et al. (2017) and
Zalesky et al. (2019) used retrievals of T and Y dwarfs to show a decrease of
Na and K abundances with effective temperature, validating the rainout
chemistry paradigm over pure equilibrium. However, while atmospheric retrieval
can explore a wider range of possible atmospheres, unphysical combinations of
parameters can still provide good fits to the data and therefore spuriously be
preferred in retrieval frameworks. In particular, a number of brown dwarf
retrieval studies have yielded unphysically small radius constraints (e.g.
Burningham et al., 2021; Lueber et al., 2022) or very high surface gravities
(Zalesky et al., 2019). Therefore, comparison to theoretical expectations from
grid models are still needed to ensure retrieval results are fully
contextualized.

A vast majority of brown dwarf retrieval studies have been conducted on low-
resolution (R $\sim$ 100) spectra. At medium-resolution, R $\gtrsim$ 1000,
molecular bandheads are resolved into unique groups of densely-packed lines,
allowing for more robust detections of molecules. Furthermore, the cores of
strong lines are formed at lower pressures than can be sensed at low spectral
resolutions, providing better probes of the upper end of the atmosphere’s
temperature-pressure profile. Comparisons of medium-resolution spectra of
brown dwarfs to grid models have provided validations of certain line lists
(Canty et al., 2015) and constraints on the brown dwarf’s fundamental
properties (e.g. Bochanski et al., 2011; Petrus et al., 2022; Hoch et al.,
2022). Spectroscopy at medium-to-high spectral resolutions of brown dwarfs
have been analyzed in retrieval frameworks, but often over a narrow wavelength
range and with relatively low signal-to-noise, though they can also be
combined with low-resolution observations for better constraints (Wang et al.,
2022; Xuan et al., 2022).

The aim of this work is to test how the atmospheric retrieval framework works
at medium spectral resolution (R $\sim$ 6000), in terms of both new insights
and novel challenges. We use the same framework successfully applied at low-
resolution (e.g. Line et al., 2017; Zalesky et al., 2022) for a spectrum with
roughly 60$\times$ higher spectral resolution. This work is structured as
follows. In Section 2, we describe our dataset, retrieval framework, and the
modifications necessary at this spectral resolution. In Section 3, we give an
overview of the tests performed and changes made throughout this project, and
the accompanying results and lessons learned. In Section 4, we put our results
in context with constraints from low-resolution spectra of the same object,
previous analysis of this dataset, and grid models. Finally, our conclusions
are summarized in Section 5.

##  2 Methods

###  2.1 Spectra of UGPS 0722

We perform our analysis on the medium-resolution spectrum of UGPS 0722
presented by Bochanski et al. (2011, hereafter B11) obtained with the Folded-
port InfraRed Echellette Simcoe et al. (2013, FIRE) at the Magellan
Telescopes. This spectrum covers 0.85 to 2.5 $\mu$m over 21 orders with R
$\sim$ 6000\. We scaled the reduced and order-stitched spectrum of B11 to the
observed H-band photometry on the Mauna Kea Observatories (MKO) photometric
system following Line et al. (2017). We will explore potential issues with
which regions of this spectrum to include in our analysis (for example due to
telluric absorption or order stitching problems) in later sections.

We also use the low-resolution spectrum of UGPS 0722 from the SpeX Prism
Library (Burgasser, 2014), which covers a similar wavelength range of 0.8 to
2.5 $\mu$m with a wavelength-dependent resolution of $\sim$ 87-300. This
spectrum was also calibrated to flux units using the same H-band photometry as
for the FIRE spectrum. The flux-calibrated FIRE and SpeX spectra are shown in
Figure 1.

Figure 1: The FIRE (R $\sim$ 6000 B11) and SpeX (Burgasser, 2014, R $\sim$ 87
- 300) spectra of UGPS 0722. The FIRE fluxes are in blue, FIRE errors in gray,
and SpeX datapoints in orange. The FIRE spectrum is available for download
online as supplementary data.

###  2.2 GPU Retrieval Framework

We use the CHIMERA retrieval framework successfully applied previously to low-
resolution brown dwarf spectra (Line et al., 2015, 2017; Zalesky et al.,
2019). However, generating a forward model emission spectrum at a resolution
of $\sim$ 60,000 (which is then binned to R $\sim$ 6,000) is quite
computationally expensive. Thus, to make this study feasible, we must use a
modified version for use with graphical processing units (GPUs), that builds
upon the code described in Zalesky et al. (2022). Specifically, we modify the
radiative transfer to solve the two stream multiple scattering problem using
the methods described in Toon et al. (1989). However, as we are not in a
particularly cloud regime, the effects of multiple scattering are neglible. As
in the previous studies done with this framework, we use the affine-invariant
MCMC ensemble sampler package emcee (Foreman-Mackey et al., 2013). We include
uniform-with-altitude volume mixing ratios of H2O, CH4, CO, NH3, H2S, Na, and
K, the surface gravity, a radius-to-distance scaling factor, the temperature-
pressure (TP) profile, and three cloud parameters: cloud volume mixing ratio,
the cloud pressure base, and the sedimentation efficiency (Ackerman & Marley,
2001). As in Line et al. (2017), the TP profile is parameterized by 15
independent temperature-pressure points subject to two smoothing
hyperparameters. These 15 temperature-pressure points are interpolated onto a
finer 70 layer pressure grid for the radiative transfer using a cubic Hermite
spline.

Table 1: Free Parameters in Our Retrieval Model

Parameter | Description
---|---
log(fi) | log of the uniform-with-altitude volume
| mixing ratios of H2O, CH4, CO, NH3,
| H2S, Na, and K
log(g) | log surface gravity [cm s-2]
(R/D)2 | radius-to-distance scale [RJup/pc]
T(P) | temperature at 15 pressure levels [K]
$b$ | errorbar inflation exponent (Line et al., 2015)
$\gamma,\beta$ | TP profile smoothing hyperparameters
| (Line et al., 2015)
log(Cloud VMR) | log of the cloud volume mixing ratio
log(Pc) | log of the cloud base pressure
fsed | sedimentation efficiency
RV | radial velocity [km s-1]
$v$ sin $i$ | rotational velocity [km s-1]

Table 2: Opacity Sources for Our Retrieval Model

Species | Opacity Sources
---|---
H2-H2, | Richard et al. (2012)
H2-He CIA |
H2O | Polyansky et al. (2018)
CH4 | (1) Yurchenko & Tennyson (2014)
| (2) Hargreaves et al. (2020) (Section 3.3.2)
CO |  Rothman et al. (2010),
| isotopologues Li et al. (2015)
NH3 | (1) Yurchenko et al. (2011)
| (2) Coles et al. (2019) (Section 3.3.2)
H2S |  Tennyson & Yurchenko (2012),
|  Azzam et al. (2015),
| isotopologues Rothman et al. (2013)
K | (1) see Marley et al. (2021)
| (2) Allard et al. (2016) (Section 3.6)
Na | (1) see Marley et al. (2021)
| (2) Allard et al. (2019) (Section 3.6)

The chemical species and associated opacity sources used in this work are
listed in Table 2.2. We began with the set of absorption cross-sections
presented in Freedman et al. (2008) and subsequently updated as detailed in
Freedman et al. (2014), Lupu et al. (2014), and Marley et al. (2021). However,
we use a set of H2O opacities calculated based on the POKAZATEL line list
(Polyansky et al., 2018). In Section 3, we explore the effect of switching our
opacity source for a number of species, including NH3 (Coles et al., 2019),
CH4 (Hargreaves et al., 2020), K (Allard et al., 2016), and Na (Allard et al.,
2019). For the cloud opacity, we used Mie scattering theory assuming a Mg2SiO4
cloud with optical properties from Wakeford & Sing (2015). However, the exact
cloud species assumed should not particularly matter as cloud optical
properties tend to be gray over these near-infrared wavelengths and the
cloud’s placement and extent in the atmosphere are parameterized independent
of composition.

At these moderate resolutions, the radial and rotational velocities present in
the spectrum, thus we add these as two additional parameters in the retrieval
forward model. Both properties were measured in B11 by comparison to grid
models. We use the dopplerShift function from PyAstronomy (Czesla et al.,
2019) to shift the forward modeled emission spectrum by a given radial
velocity and interpolate it back onto the input wavelength grid.

For the rotational velocity, we first tested the rotBroad function from
PyAstronomy which implements rotational broadening as described by Gray (2008)
for a given $v$ sin $i$ and linear limb-darkening coefficient (we chose 0).
This function convolves the modeled spectral lines with a wavelength-dependent
line profile representing the Doppler line broadening from rotation, called
the “broadening kernel.” However, this method proved infeasibly slow,
requiring 89 seconds to generate one broadened forward model spectrum. We then
tested the fastRotBroad function from PyAstronomy, which uses a single
broadening kernel that only depends on the median wavelength of the input
data, leading to a much faster forward model generating time of 0.19 seconds
but differences from the slower version that were larger than the error bars
of the FIRE spectrum as shown in Figure 2. As a compromise, we split our
spectrum into two and used fastRotBroad for each half, taking 0.2 seconds but
leading to differences from the more accurate function that were smaller than
the error bars.

Figure 2: Comparison of different rotational broadening methods. The top panel
shows modeled emission spectra using different rotational broadening methods
compared to a snippet of the FIRE spectrum of UGPS 0722 shown in the grey data
points. The bottom panel shows the difference between the model broadened with
a wavelength-dependent kernel (“Slow”) and those using either 1 or 2
broadening kernels for the spectrum (“Fast”), and the data error bars in this
region. At least two broadening kernels, using the median wavelengths of the
blue and red halves of the spectrum, are required to reduce the difference
with the slower, more accurate method to smaller than the error bars on the
FIRE spectrum.

Figure 3: Retrieved TP profiles and posteriors of certain parameters for a
“fake” FIRE spectrum based on the Sonora Bobcat model with an effective
temperature of 600 K and surface gravity log(g)=4.0.

To test how our retrieval framework might perform on the FIRE data of UGPS
0722, we generated a fake test data set. We used CHIMERA to create one forward
model based on the TP profile from the Sonora Bobcat grid (Marley et al.,
2021) for an object with Teff = 600 K and log(g)= 4.0, giving it a radial
velocity of 47 km s-1 and $v$ sin $i$ of 40 km $s^{-1}$. We assume constant
with altitude chemical abundances with log volume mixing ratios of H2O =
-3.09, CH4 = -3.33, CO = -10.37, H2S = -4.59, NH3 =-5.00, K = -6.80, and Na =
-4.90. This forward model was convolved to the FIRE instrument resolution,
interpolated onto the wavelength grid of the FIRE spectrum, and given the same
error bars as the FIRE spectrum of UGPS 0722. As such, we assume a $(R/D)^{2}$
value of 0.0158 (corresponding to a radius of $\sim$0.528 RJup for an object
at the distance of UGPS 0722) which is needed to give the same peak flux to
error ratio as the FIRE spectrum of UGPS 0722. The results from a CHIMERA
retrieval on this test dataset compared to the input values are shown in
Figure 3. We see that we are able to recover the input values with high
accuracy and precision (with the exception of CO as it is unconstrained due to
the low input mixing ratio), particularly with an unprecedented uncertainty of
$\sim$ 0.01 dex on the chemical abundances and surface gravity,
$\sim$10$\times$ more precise than constraints from spectra at R$\sim$100
(Line et al., 2017; Zalesky et al., 2022).

##  3 Results

###  3.1 Initial Fire Retrieval vs. SpeX

Initially, we used the entire order-stitched and flux-calibrated FIRE spectrum
UGPS 0722, with a few spurious data points effectively ignored by greatly
inflating their error bars. However, in regions of high telluric absorption,
the error bars on the spectrum were artificially underestimated (almost 14
orders of magnitude more precise than elsewhere). To prevent these data points
from incorrectly driving our results, the error bars in these regions are
inflated to a high enough value that these data points functionally do not
contribute to our retrieval analysis. We also mask out those regions of strong
telluric absorption in the SpeX spectrum as well to allow for a direct
comparison.

Figure 4 shows the retrieved TP profiles and posterior distributions for
selected parameters from our initial analysis on the UGPS FIRE spectrum
compared to results from the SpeX spectrum. Though the FIRE spectrum
potentially offers more precision than the SpeX results, we find our MCMC
chains have trouble converging, with bimodal posteriors, unphysical values for
certain parameters like the surface gravity, and very jagged TP profiles.

Figure 4: Retrieved TP profiles and posterior distributions for selected
parameters for initial retrievals on the SpeX and FIRE spectra of U0722.

We suspected our forward model was too flexible, which was leading to
overfitting and unphysical results. We reduced the number of parameters in our
model by fixing the second TP profile smoothing hyperparameter $\beta$ and the
cloud parameters to specific values. Line et al. (2017) found letting $\beta$
vary had a negligible effect compared to the nominal fixed value of
$\beta=5\times 10^{-5}$ used by Line et al. (2015). As previous work (Line et
al., 2015, 2017) showed little evidence for optically thick clouds in T
dwarfs, we set the cloud opacity parameters log(Cloud VMR), log(Pc), fsed =
[-15, 2, 10], consistent with an optically thin cloud deck.

The effect of fixing $\beta$ and the cloud parameters is shown in Figure 5 in
orange. The TP profiles are smoother and more precisely constrained, while we
no longer see bimodal posteriors for the plotted parameters. We note a
potential detection of H2S in this object, although the posterior on the H2S
abundance has a long lower tail. We will explore letting the cloud parameters
vary again in Section 3.8.

Figure 5: Effect on the retrieved TP profiles and selected posteriors of
fixing the smoothing hyperparameter $\beta$ and cloud parameters to set values
(orange). In Section 3.2, we then limited the input spectrum to 0.9-2.35
$\mu$m and took every fourth pixel to limit the analysis to one data point per
resolution element (grey).

###  3.2 Effect of Resolution Element and Wavelength Limits

Next, we decided to more mimic the analysis of B11 and limit the FIRE spectrum
to 0.9 - 2.35 $\mu$m when comparing to models, effectively getting rid of the
noisiest regions of the spectrum at the beginning and end. Additionally, like
SpeX, the FIRE spectrum is oversampled compared to a spectral resolution
element, so we take every 4th pixel of the spectrum to ensure independent data
points (e.g. Line et al., 2017; Kitzmann et al., 2020).

Figure 5 shows the effect of these two changes in grey. While the retrieved TP
profiles are very similar to the previous results at pressures less than
$\sim$ 15 bars, the deep atmosphere is much warmer. The molecular abundances
often shift slightly, but with the exception of H2O the new posteriors are
within 1$\sigma$. Our precision on almost all parameters also decreases, due
to the substantial decrease in the number of included data points.

###  3.3 Effect of Updated Line Lists

In an effort to improve our best model fit to the observed spectrum, we
investigated the effect of changing the sources of line lists used for CH4 and
NH3. For NH3, we upgraded to the more recent CoYuTe (Coles et al., 2019)
ExoMol line list instead of the older BYTe (Yurchenko et al., 2011) list we
were using previously. For CH4, we replaced the ExoMol 10to10 list (Yurchenko
& Tennyson, 2014) with the recent HITEMP line list published by Hargreaves et
al. (2020) which combined the more accurate ab initio line lists of Rey et al.
(2017) with HITRAN2016 data (Gordon et al., 2017).

Comparisons of these “old” and “new” line lists for CH4 and NH3 at a specific
pressure and temperature are shown in Figure 6. The two CH4 opacities on the
left show clear deviations beyond just shifts to line positions, especially
blueward of $\sim$ 1.62 $\mu$m. Hargreaves et al. (2020) demonstrate the
better match of their CH4 line list to experimental data for this wavelength
region than the ExoMol CH4 line list. In contrast, while minor differences are
evident between the two NH3 line lists in the right panel, they have similar
overall values across the wavelength region of the spectrum. The effect of
these changed line lists on our retrieved parameters is discussed below.

Figure 6: Comparison of old and new molecular cross sections at 725 K and 1
bar, and smoothed to R $\sim$ 6000\. Left: Comparison of CH4 cross sections
from the ExoMol 10to10 (Yurchenko & Tennyson, 2014) and HITEMP (Hargreaves et
al., 2020) line lists. Right: Comparison of NH3 cross sections from the older
BYTe (Yurchenko et al., 2011) and newer CoYuTe (Coles et al., 2019) ExoMol
line lists.

####  3.3.1 Low Resolution Retrievals

Figure 7 shows the effect of these updated line lists on the retrieved TP
profile and selected posteriors for the SpeX spectrum. The retrieved TP
profiles are consistently warmer than those obtained when using the Yurchenko
& Tennyson (2014) CH4 and Yurchenko et al. (2011) NH3 line lists. Constraints
on the abundances of H2O, CH4, and NH3 all shift to lower values, while the
retrieved abundance of K increases by $\sim$ 0.5 dex. The retrieved surface
gravity also decreases, from log(g)=5.19${}_{{-0.40}}^{{+0.27}}$ to
log(g)=4.43${}_{{-0.17}}^{{+0.27}}$ (cgs). Furthermore, the retrieved radius,
assuming the parallax distance of 4.12 $\pm$ 0.04 pc (Leggett et al., 2012),
decreases from 0.76${}_{{-0.04}}^{{+0.06}}$ to 0.5${}_{{-0.02}}^{{+0.03}}$
RJup, an unphysically small value (see Section 4.1 for further discussion).
Notably, the dramatic decrease in both surface gravity and radius are driven
by updating the CH4 line list, as this effect occurs even when the NH3 line
list is kept the same.

Figure 7: Effect on the retrieved TP profiles and selected posteriors of
updating the CH4 and NH3 line lists to those of Hargreaves et al. (2020) and
Coles et al. (2019), respectively, for the SpeX spectrum (orange). Results for
the FIRE spectrum smoothed to the resolution of SpeX (grey) agree well for all
parameters with those from the SpeX spectrum.

To assess if the FIRE dataset is consistent with the SpeX spectrum, we
smoothed and sampled the FIRE data down to the SpeX resolution. Our retrieval
results on this smoothed FIRE spectrum are compared to the original SpeX
results (both with the updated cross sections) in Figure 7. The retrieved
constraints arising from the smoothed FIRE spectrum and the SpeX spectrum show
remarkable agreement. This suggests that at low resolutions, observations at
different times with completely different instruments produce extremely
consistent results, given our retrieval model assumptions.

####  3.3.2 Medium Resolution Retrievals

With the increased spectral resolution of FIRE, we can better assess the
accuracy of the line positions in various line lists. Figure 8 shows two
narrow regions of the FIRE spectrum of U0722 where CH4 and NH3 are dominant in
the left and right panels, respectively. For each molecule, we show model
spectra generated with the updated line lists. In both cases, our newer line
lists are better able to replicate the line positions of their respective
molecules. Both the Hargreaves et al. (2020) and Coles et al. (2019) line
lists incorporated empirical energy levels where available, as opposed to
solely computed ones, leading to improved line position accuracy.

Figure 8: Comparisons of median model spectra for retrieval results using old
and new molecular line lists, compared to the FIRE spectrum of U0722 in narrow
regions of the spectrum where CH4 and NH3 are expected to dominate on the left
and right, respectively. Left: Models with Yurchenko & Tennyson (2014) and
Hargreaves et al. (2020) CH4 line lists; the Hargreaves et al. (2020) line
list does a significantly better job at matching the CH4 lines in this region.
Right: Models with Yurchenko et al. (2011) and Coles et al. (2019) NH3 line
lists; the Coles et al. (2019) line list improves the fit to NH3 lines in this
region.

Unsurprisingly, improving the fit to line positions in our model does affect
our retrieved atmospheric parameters. Figure 9 shows how updating the line
lists of CH4 and NH3 affects our retrieved TP profiles and posteriors in
orange. While the TP profiles are relatively similar deeper than $\sim$ 1 bar,
the scenario with updated line lists prefers the atmosphere to be as cold as
possible around 0.2 bars. Our retrieved posteriors for all molecular
abundances as well as potassium do shift to higher values. The surface gravity
also increases, from a median log(g) = 3.66 ${}_{-0.02}^{+0.02}$ to log(g) =
4.08 ${}_{-0.03}^{+0.03}$(cgs), a more plausible value. However, the radius
decreases just like for the SpeX retrieval in Section 3.3.1, from
0.76${}_{-0.01}^{+0.01}$ to 0.53${}_{-0.01}^{+0.01}$ RJup, an unphysically
small size, which we discuss more in Section 4.1. The retrieved radial
velocity and $v$ sin $i$ are relatively unaffected by our choice of CH4 and
NH3 line lists, indicating these measurements may be primarily driven by H2O
lines in the spectrum.

Figure 9: Effect on the retrieved TP profiles and selected posteriors from the
FIRE spectrum when updating the CH4 and NH3 line lists to those of Hargreaves
et al. (2020) and Coles et al. (2019), respectively (orange). The retrieved TP
profiles and selected posteriors resulting from removing regions of the data
where order stitching was not successful (Section 3.4) are shown in grey.

Notably, we retrieve bounded constraints on the H2S and CO abundances; CO
constraints were unbounded when using the older line lists. Figure 10 shows
model spectra for the median retrieved parameters, with and without H2S, where
we expect H2S opacity to have an effect. Both H2S and CO have the greatest
effect on our model spectra between $\sim 1.56-1.60\mu$m where there is a
window in the combined opacity of H2O and CH4. Including H2S does improve the
model fit in this region of the spectrum, but mostly as an overall shift in
strength of features. However, around 1.59 $\mu$m there is what appears to be
an H2S line that is blended with another feature but clear in the residuals
shown on the bottom. Tannock et al. (2022) reported an H2S detection in a high
signal-to-noise, R $\sim$ 45,000 spectrum of a T6 dwarf, where they show a
strong H2S feature at this exact location around 1.59 $\mu$m. Thus, our
constraint on the H2S abundance is consistent with this feature, though it is
dependent on a relatively small number data points ($\sim$3 points for the
1.59 $\mu$m feature and $\sim$ 20 others). Figure 11 shows models with and
without CO, for the region of the spectrum where the model spectra are most
different from each other. While the inclusion of CO does improve our model
fit in this region, there is not a clear CO feature to point to as the source
of our CO constraint. Thus, while we do have a bounded posterior for the CO
abundance, it is perhaps a less trustworthy detection.

Figure 10: Model spectrum from the median retrieved parameters generated with
(orange) and without (blue) H2S, compared to a snipppet of the FIRE spectrum
where H2S opacity is expected. Some features are better fit with the inclusion
of H2S. The black arrow indicates a blended H2S line consistent with the H2S
detection of Tannock et al. (2022). Figure 11: Model spectrum from the median
retrieved parameters generated with and without CO, compared to a snipppet of
the FIRE spectrum where some CO opacity is expected. While including CO does
slightly improve the fit to the data, the effect is slight.

###  3.4 Issues with Order Stitching

FIRE observations in cross-dispersed mode are spread over 21 different orders.
These orders have some overlap in wavelength coverage on either end. The final
stitched spectrum published by Bochanski et al. (2011) combined these orders
into a single spectrum by averaging the regions where the orders overlapped.
However, in some cases when one order was noisier than the other, this
averaging can lead to data artifacts. We first noticed this issue when
inspecting the spectrum around 2.1 $\mu$m, where a jump or step in the data
appears that is not possible to reproduce with our forward models, as shown in
the top panel of Figure 12. Additional examples of overlap regions potentially
subject to order stitching problems are shown in the lower two panels. While
neither panel shows a step similar to the one seen at 2.09 $\mu$m, both cover
regions where there is substantial disagreement between the two component
orders leading to perhaps unphysical features in the stitched spectrum,
particularly towards the middle of the overlapping areas.

To avoid these potential order stitching issues from biasing our retrieved
results, we artificially inflated the error bars in order overlap regions
where the final stitched product differed from either input order spectrum by
more than 10%, effectively removing these data points from our analysis. 9
shows the result of removing these order stitching artifacts on our retrieval
results in grey. Though the region of the atmosphere shown in Figure 12 is
near the peak of K band and thus probes hotter parts of the atmosphere, other
problematic order overlap regions masked by this process were on the edges of
peaks in the spectrum and therefore contributed spuriously to constraints on
the TP profile of the upper atmosphere. Our molecular and alkali abundances
change slightly, with a less precise constraint on the amount of CO. While our
median retrieved surface gravity increases to a slightly more plausible value
of log(g)= 4.19 ${}_{-0.03}^{+0.04}$ from log(g) = 4.08
${}_{-0.03}^{+0.03}$(cgs), our median radius shrinks even more to
0.50${}_{-0.01}^{+0.01}$ RJup. Finally, our retrieved radial velocity and $v$
sin $i$ posteriors also slightly shift as well.

Figure 12: Example order overlap regions of the stitched FIRE spectrum (grey)
that were removed due to issues with order stitching, compared to the
component orders (blue and orange).

###  3.5 Retrieving on Subsections of the Spectrum

While our stitched FIRE spectrum of U0722 covers y \- K bands, we wanted to
investigate the constraining power of different sets of spectral bands. In
particular, Hargreaves et al. (2020) retains the completeness limits of the
input line lists from Rey et al. (2017), giving a temperature-dependent
maximum wavenumber limit for which the CH4 line list can be considered
complete, or including all lines strong enough to affect the resulting
opacity. This limit is 10700 cm-1 or 0.93 $\mu$m for 1300 K, and 9500 cm-1 or
1.05 $\mu$m for 1400 and 1500 K. Thus, the Hargreaves et al. (2020) CH4 line
list is not complete for the deeper, hotter temperatures probed by the y band.
Motivated by this potential completeness problem, we retrieved on the J \- K
band data only, followed by a retrieval on solely H \- K bands.

Figure 13: Effect on the retrieved TP profiles and selected posteriors when
performing the retrieval on the full wavelength range, J-K, or just H and K
bands.

Comparisons of the retrieved TP profiles and select posteriors are shown in
Figure 13. The TP profiles get cooler as the shorter wavelength data is
progressively removed from the analysis. As the flux in y and J bands comes
from deeper in the atmosphere, we do not constrain the temperatures to as deep
of pressures without these data, as indicated by the “fanning out” of the TP
profiles in these deeper layers. The H2O and CH4 abundances shift to higher
values when just looking at H and K bands. While our constraint on the CO
abundance becomes more precise for the J-K retrieval, there is still a long
tail towards low values. The H2S and NH3 posteriors shift slightly with the
exclusion of the y band data. While we lose all constraints on the K abundance
for just H & K, we have a very precise and high constraint on the Na abundance
when looking at J-K bands. Our retrieved surface gravity is also higher for
the J-K retrieval than the other two, with a median of
log(g)=4.31${}^{+0.06}_{-0.04}$. We retrieve a larger radius as we remove each
bluer band of data, perhaps reflecting the cooler TP profiles retrieved for
these cases as well. Finally, removing the J band data causes quite a shift in
the retrieved radial velocity and $v$ sin $i$ values, pointing to the
importance of the strong water lines in J band to our constraints on these
values for the full spectrum.

Given that the Hargreaves et al. (2020) line list is not complete for y band
at the temperatures found in this object, the J-K retrieval is perhaps a
compromise between CH4 line completeness while retaining some flux from deeper
in the atmosphere. However, the anomalously high Na abundance does call into
question the physical plausibility of these results.

###  3.6 Effect of Alkali Opacities

Another change we made to our framework was to update the opacities used for
the alkali metals Na and K. In particular, their strong lines at $\sim$ 0.59
and 0.77 $\mu$m can be very broadened and significantly impact the near
infrared spectrum of a brown dwarf. Prescriptions for these line profiles can
vary quite a bit. We first used older alkali opacities based on the unified
line-shape theory (Allard et al., 2007b, a), as used in the Sonora Bobcat grid
(Marley et al., 2021); example cross-sections of Na and K for 725 K and 1 bar
are shown in Figure 14, which basically become flat continuum opacity sources
after a certain wing cutoff point. The high retrieved Na abundance for the J-K
retrieval in Section 3.5 above can then be understood as an additional source
of continuum opacity used to reduce J band flux which is not penalized when
the y band is excluded. These alkali opacities were shown by Gonzales et al.
(2020) to produce more physically reasonable alkali abundances in their
retrieval study of a d/sdL7+T7.5p binary than those from Burrows & Volobuyev
(2003). In contrast, newer cross sections based on the recent theoretical
advancements of Allard et al. (2016, 2019) have more complicated shapes and
significantly lower cross sections redward of $\sim$ 1.2 $\mu$m.

Figure 14: Comparison of old (dash-dot) and new (solid) alkali cross sections
at 725 K and 1 bar, and smoothed to R $\sim$ 6000\. The older alkali cross
sections are from Freedman et al. (2014), while the newer ones are based on
Allard et al. (2016) and Allard et al. (2019) for K and Na, respectively.

These differences demonstrated in Figure 14 between the older and newer line
profiles can greatly affect our retrieved parameters. To isolate the
differences, we retrieved on the SpeX data with just updating the Na cross
sections, just updating K, and then updating both simultaneously. The
retrieved TP profiles and selected posteriors for the different alkali
treatments for the SpeX data are shown in Figure 15. Updating the Na opacity
on its own does not particularly affect our retrieved results, perhaps
unsurprisingly since the older K cross sections have so much more opacity.
Updating to the new K causes a large change to our retrieved TP profiles,
causing a very cold upper atmosphere with an inversion and a hotter, more
precisely-constrained profile from $\sim$1 - 10 bars. The retrieved surface
gravity and radius also decrease. Finally, updating Na and K at the same time
is mostly similar to when just using the new K, except for a very high amount
of retrieved Na.

Figure 15: Effect of updating alkali cross sections on retrieved TP profiles
and selected posteriors using the SpeX spectrum.

Figure 16 shows the effect of updating the alkali cross sections on our
retrieved TP profiles and posteriors of selected parameters for the FIRE data
of U0722. Similarly to the SpeX retrieval, the retrieved TP profile is hotter
from $\sim$1 - 10 bars and goes to 0 K (unphysical) at $\sim$0.1 bars before
inverting back to higher temperatures. The abundances of many species shift to
higher values, particularly Na which again has an extremely high abundance.
Additionally, the CO posterior loses its long low tail, leading to a higher
median abundance. We also retrieve a lower surface gravity, concordant with
the SpeX results, while the radius increases slightly.

Figure 16: Effect on the retrieved TP profiles and selected posteriors for the
FIRE spectrum when updating our alkali cross sections (orange). As our Na
constraint appears to be only from a small snippet on the blue end of the FIRE
spectrum, removing any data $<1\mu$m leads to an unconstrained Na abundance
(grey).

Looking at a spectrum generated with the median retrieved parameters compared
to one with significantly less Na, we see that the only discernible
differences occur from $\sim$0.9 - 0.98 $\mu$m shown in Figure 17. The gaps in
the FIRE data shown were regions where order stitching was found to be an
issue as discussed in Section 3.4. Although the high Na model does get closer
to fitting the data in this region, it is still not a particularly good fit by
eye, completely missing many of the data points. Furthermore, this is one of
the noisiest regions of the data. Thus, the high retrieved Na abundance is
likely a spurious result driven by low signal-to-noise data in these regions.

Figure 17: Model spectra from the median retrieved parameters, both using the
new alkali cross sections, generated with and without the retrieved high Na
abundance. This wavelength region is the only one where the increased Na
abundance leads to a notable difference in the model spectra.

We ran another retrieval on the FIRE data with the new alkali cross sections,
but without any data blueward of 1 $\mu$m. The results are shown in Figure 16.
The previous high Na constraint disappears as expected. While other posteriors
shift as well, the constraints on the other species change only slightly.
However, the inverted TP profile and even lower surface gravity compared to
the retrieval with the old alkali cross sections are a cause for concern.
Therefore, the old alkali cross sections are preferred as they give us more
physical results. Further studies of the alkali line profiles and their effect
on retrievals in particular would help provide context for these results and
guidance for future medium resolution retrievals.

###  3.7 Setting the Radius to 1 Jupiter Radius

Given the unphysically small retrieved radius for U0722, we explored the
impact of fixing the retrieved radius to 1 Jupiter radius on the TP profile
and abundance constaints. Figure 18 shows the retrieved TP profiles and
selected posteriors with this fixed radius in orange compared to when the
radius is allowed to vary for the retrieval described in Section 3.4. The
retrieved TP profiles overlap with the previous results for $\sim$2-4 bars but
with a slightly different slope, as they are hotter above and cooler below
this region. All abundance posteriors shift, including the disappearance of
the long low tail for CO, except for H2S which remains relatively unchanged.
In particular all abundance shifts are to lower values, perhaps balancing out
the cooler temperatures in the deep atmosphere where the flux in y and J bands
originates. The radial velocity posterior is strangely trimodal; however, one
peak does correspond to the previous value. The $v$ sin $i$ posterior does
shift as well. Importantly, the surface gravity decreases to an unphysically
small median value of log(g)= 3.45. Figure 28 in the Appendix shows the
correlations between parameters for the retrieval discussed in Section 3.4.
Gravity is notably positively correlated with the abundances of major
absorbers (H2O, CH4, NH3) as well negatively correlated with the radius
(through the scaling factor (R/D)2). As gravity increases, the column optical
depth decreases, leading to more flux particularly in the peak of each band
(see Figure 2 of Line et al., 2015). Therefore, when the radius is fixed to
the larger value of 1 RJup, the retrieval converges on a smaller gravity value
to balance out the increase in flux (which is also consistent with the lowered
abundances). Due to this questionably low surface gravity, we do not use this
fixed radius retrieval to compare to SpeX and grid models in Section 4.

We next considered fixing the surface gravity simultaneously with the radius,
choosing log(g)=4.5 consistent with a recent grid model study of U0722
(Leggett et al., 2021). The resulting TP profile and posteriors are shown in
Figure 18 in grey. Since the gravity can no longer be decreased to account for
the increased flux from a 1 RJup object, the retrieved TP profile is
significantly colder than the other two plotted retrievals for the entire
region of the atmosphere where it is well-constrained, about 0.5 to 50 bars.
We will compare this TP profile to grid model predictions in Section 4. All
abundance posteriors shift significantly compared to when the radius and
surface gravity are allowed to vary, though the direction of this shift varies
for different species. The radial velocity and $v$ sin $i$ posteriors change
slightly as well, though the former does not exhibit the same trimodal
behavior as when the radius alone is fixed. While these constraints seem more
plausible than those obtained when only the radius was fixed to a certain
value, the fit to the data does suffer compared to when the radius and gravity
are allowed to vary, as shown in Figure 19. Furthermore, our choice of fixed
radius and surface gravity, while guided by previous studies of this object,
are still relatively arbitrarily chosen. As the benefit of a retrieval
analysis is to obtain constraints with fewer a priori assumptions than fitting
the data to grid models, we do not use this fixed radius and gravity retrieval
as our “preferred” FIRE retrieval in Section 4 for comparing to SpeX and grid
model results.

Figure 18: Effect on the retrieved TP profiles and selected posteriors of
fixing the radius to 1 RJup (orange), and then setting the surface gravity to
log(g)=4.5 as well. Almost all parameters are significantly affected. Figure
19: Select regions of the model spectrum from the median retrieved parameters
when the radius and log(g) are fixed to 1 R RJup and 4.5, respectively,
compared to the median model when both are allowed to vary (from Section 3.4)
and the FIRE spectrum of U0722. The fit to the data is worse when the radius
and gravity are not allowed to vary in the retrieval.

###  3.8 Cloudy Retrieval

Figure 20: Effect on the retrieved TP profiles and selected posteriors of
allowing the cloud parameters to vary (orange), instead of being fixed to
values corresponding to optically thin clouds (blue). We find little evidence
for optically thick clouds in the atmosphere of U0722.

In Section 3.1 we fixed our cloud parameters to be consistent with an
optically thin cloud that would not affect the emission spectrum when it
appeared our model was perhaps too flexible. After incorporating the preceding
changes we found improved our results (fixing $\beta$, changing the wavelength
limits to match B11, sampling one data point per resolution element, updating
the CH4 and NH3 opacities, and ignoring regions victim to poor order
stitching), we allowed the cloud parameters to vary once again. Figure 20
shows the resulting TP profiles and posteriors in grey. The cloud parameters
are unconstrained, except for a potential upper limit on the cloud volume
mixing ratio. Other parameters are relatively unaffected, suggesting the
inclusion of cloud opacity has little impact. This lack of evidence for
optically thick clouds in the atmosphere of U0722 are consistent with
expectations for late T dwarfs (e.g. Kirkpatrick, 2005) as well as other T
dwarf retrieval studies at lower spectral resolution (e.g. Line et al., 2017;
Zalesky et al., 2022).

##  4 Discussion and Conclusions

###  4.1 SpeX vs FIRE

Figure 21: Comparison of the FIRE spectrum (light blue) with the median model
of our initial retrieval (navy) and the median model of our “final” preferred
FIRE retrieval from Section 3.4 (orange). The FIRE errors are shown in gray.
Red lines at the top of each panel indicate sections of the FIRE spectrum that
were removed from our final analysis as described in Sections 3.1, 3.2, and
3.4. Our final preferred median model spectrum does a better job fitting the
FIRE data, particularly the peak of y band and the CH4 features in H band.

We choose the FIRE retrieval from Section 3.4, with pieces of the spectrum
affected by order stitching removed and all the preceding changes, as our
“preferred” FIRE retrieval. While Sections 3.5 - 3.8 explore various other
tests we performed, the retrieval from Section 3.4 uses the entire wavelength
range while providing more physical constraints than those gained when using
the new alkali opacities or fixing the radius for example. Table 3 lists each
change to our retrieval framework we tested, the section of the paper where it
is discussed, and whether or not it was applied for our preferred retrieval.
Figure 21 shows the FIRE spectrum compared to the median model spectrum from
the initial retrieval described in 3.1 and the median model spectrum from this
final preferred FIRE retrieval. Red lines at the top of each panel indicate
data points removed from our analysis due to spurious flux values and high
telluric absorption (Section 3.1) or suspected problems with order stitching
(Section 3.4). In Figure 21, the model from our final retrieval fits the FIRE
spectrum much better than for our initial retrieval, particularly the peak of
y band and the CH4 features in H band, as discussed in Section 3.3. However,
though the posteriors of the retrieved parameters are quite different, there
are large sections of the spectrum where both models reproduce the observed
spectrum relatively well, as should be expected for a data-driven model
fitting method. Most of the tests of our framework outlined in Section 3,
other than updating the line lists, were motivated by intuition or unphysical
results rather than poor fits to the data. As such, we do not plot the median
models from all our different retrieval runs in the interest of brevity and
clarity.

Figure 22: Retrieved TP profiles and selected posterior distribution for
selected parameters for SpeX and FIRE spectra of U0722 after making the
changes discussed in Section 3.

Figure 22 shows how much the constraints on our retrieved posteriors improve
when using the R $\sim$ 6000 FIRE spectrum compared to the R $\sim$ 100 SpeX
spectrum. The “preferred” SpeX retrieval here uses the same updated line lists
as described in Section 3.3. The H2O, CH4, and NH3 abundance posteriors with
FIRE are $\sim$3-6$\times$ more precise than those from SpeX. Similarly, we
can constrain the surface gravity to within about 0.04 dex, about 5$\times$
more precise than SpeX. Furthermore, between $\sim$0.5 to 20 bars, the TP
profile is constrained within 100 K compared to the much wider 500 K spread
for SpeX. Finally, our retrieval on the medium resolution FIRE spectrum allows
constraints on parameters such as H2S abundance, CO abundance, radial
velocity, and $v$ sin $i$ which are not constrained with the SpeX spectrum.
However, while the preferred FIRE retrieval yields very precise constraints,
they may not be accurate descriptions of the characteristics of U0722 and the
uncertainties do not include any systematic effects, which we will discuss
further below.

Table 3: Changes to our retrieval framework tested for application to the FIRE spectrum of U0722. Change | Section of Paper | In preferred retrieval?
---|---|---
Add radial velocity and $v$ sin $i$ | 2.2 | yes
Mask regions of total telluric absorption | 3.1 | yes
Remove second T-P smoothing hyperparameter | 3.1 | yes
Fix cloud parameters to optically thin values | 3.1 | yes
Limit data to 0.9-2.35 $\mu$m | 3.2 | yes
Take every 4th data point to not oversample the FIRE resolution element | 3.2 | yes
Update NH3 and CH4 opacities | 3.3 | yes
Mask data points affected by issues with orders stitching | 3.4 | yes
Limit analysis to subsections of the spectrum, J-K and H-K | 3.5 | no
Update alkali opacities | 3.6 | no
Fix radius to 1 RJup | 3.7 | no
Fix log(g)=4.5 and R = 1 RJup | 3.7 | no
Allow cloud parameters to vary | 3.8 | no

Table 4 lists physical parameters of U0722 for both retrievals, as well as
from previous studies which will be discussed below. To calculate a number of
these quantities and their uncertainties from the parameters in our retrieval
framework, we take 5000 random samples of our posterior. With these precise
constraints on the molecular abundances, we can consider the metallicity and
C/O ratio as most of the metal content in these cool brown dwarf atmospheres
is contained in H2O and CH4, with some contribution from CO and NH3. We
calculate these quantities from our retrieved abundances following Equations 1
and 2 from Zalesky et al. (2022) and assuming solar abundances from Lodders
(2010). To account for condensation processes that can deplete atmospheric
oxygen, we multiply our H2O abundance by 1.3 to better approximate the
intrinsic metallicity and C/O as in Zalesky et al. (2022). Our retrieved
abundances yield [M/H] = -0.10${}^{+0.02}_{-0.02}$ dex and C/O =
0.54${}^{+0.01}_{-0.02}$ from the FIRE retrieval, with $\sim$3-4$\times$ the
precision than from the SpeX retrieval. For comparison, Zalesky et al. (2022)
on average constrain [M/H] and C/O to within 0.2 dex with 50 T dwarf spectra
at R$\sim$100, although the precision varies among the objects. Though we have
achieved precise constraints on these bulk properties, Calamari et al. (2022)
carried out a retrieval study of GI 229B and showed the 1.3 oxygen scaling
factor lead to a calculated C/O that was unexpectedly inconsistent with
measurements of the primary star. They suggest better understanding of the
potential oxygen sinks in brown dwarf interiors could refine the best way to
connect measured atmospheric C/O to the bulk value.

For each sample of our posterior, we also generate a low-resolution spectrum
over 0.3 to 250 $\mu$m, which we integrate and use to compute the bolometric
luminosity LBol and effective temperature Teff. Leggett et al. (2012) report a
luminosity range of log(LBol/LSun) = -6.05-6.17 from observed spectra of U0722
over $\sim$ 0.7 - 4 $\mu$m. The calculated log(LBol/LSun) from the FIRE
results of -6.23${}^{+0.20}_{-0.09}$ is consistent with this literature value,
but the luminosity from our SpeX results log(LBol/LSun) =
-6.42${}^{+0.08}_{-0.05}$ is significantly lower. We also note that the
luminosity constraints from the SpeX retrieval are more precise than those
from the FIRE retrieval. While between 0.5 - 20 bars the TP profile is much
better constrained by the FIRE retrieval, outside of this range the
temperatures vary considerably and are hotter than the corresponding SpeX
values, leading to a calculated luminosity that is both higher and less well-
constrained. The effective temperatures calculated from the luminosities and
radius values unsurprisingly show the same behavior.

Table 4: Parameters of U0722 calculated from this work and previous studies. Parameter | FIRE Retrieval | SpeX Retrieval | BT-Settl Model Grid Fit | Tuned ATMO Model Grid Fit
---|---|---|---|---
| (this work) | (this work) | Bochanski et al. (2011) | Leggett et al. (2021)
Wavelength Range ($\mu$m) | 0.9 - 2.35 | 0.9 - 2.35 | 0.9 - 2.35 | 0.6 - 5.1
Spectral Resolution | 6000 | 100 | 6000 | 180-500ffUsed flux-calibrated SED with datasets from multiple instruments published by Lucas et al. (2010); Leggett et al. (2012); Miles et al. (2020)
log(g) (cgs) | 4.19${}^{+0.04}_{-0.03}$ | 4.43${}^{+0.28}_{-0.17}$ | 4.0${}^{+0.3}_{-0.3}$ | 4.5
Radius (RJup) | 0.5${}^{+0.01}_{-0.01}$ | 0.5${}^{+0.03}_{-0.02}$ | 1.14${}^{+0.46}_{-0.33}$ aaCalculated from reported log(g) and mass | 1.12ggDetermined from evolutionary models of Phillips et al. (2020)
Distance (pc) | Fixed at 4.12 | Fixed at 4.12 | 4.6 (restricted)bbBochanski et al. (2011) perform two fits, one with distance as a free parameter and one where the model-derived distance must be within 5$\sigma$ of the parallax measurement from Lucas et al. (2010). | Fixed at 4.12
|  |  | 49 (unrestricted) |
Mass (MJup) | 1.5${}^{+0.1}_{-0.1}$ | 1.9${}^{1.2}_{-0.6}$ | 5.24ccDetermined from evolutionary models of Baraffe et al. (2003) and the best fit Teff and log(g) | 15ggDetermined from evolutionary models of Phillips et al. (2020)
C/O | 0.54${}^{+0.01}_{-0.02}$ | 0.55${}^{+0.08}_{-0.06}$ | 0.55ddConsidered models from the BT-Settl model grid (Allard & Freytag, 2010) assuming solar metallicity and C/O using solar abundances from Asplund et al. (2009) | 0.55hhConsidered models from the ATMO grid (Phillips et al., 2020) assuming solar C/O and with [M/H] = -0.5, 0.0, and +0.3 dex using solar abundances reported by Asplund et al. (2009) and updated by Caffau et al. (2011)
[M/H] (dex) | -0.10${}^{+0.02}_{-0.02}$ | -0.02${}^{+0.07}_{-0.06}$ | 0.0ddConsidered models from the BT-Settl model grid (Allard & Freytag, 2010) assuming solar metallicity and C/O using solar abundances from Asplund et al. (2009) | 0.0hhConsidered models from the ATMO grid (Phillips et al., 2020) assuming solar C/O and with [M/H] = -0.5, 0.0, and +0.3 dex using solar abundances reported by Asplund et al. (2009) and updated by Caffau et al. (2011)
log(LBol/LSun) | -6.23${}^{+0.20}_{-0.09}$ | -6.42${}^{+0.08}_{-0.05}$ | -6.11 (restricted)eeCalculated from radius and Teff. | -5.99eeCalculated from radius and Teff.
|  |  | -5.52 (unrestricted) |
Teff (K) | 711${}^{+84}_{-35}$ | 638${}^{+32}_{-15}$ | 500${}^{+50}_{-50}$ (restricted) | 540
|  |  | 700${}^{+50}_{-50}$ (unrestricted) |

However, both our SpeX and FIRE retrievals do result in an unphysically small
radius of about 0.5 RJup, potentially calling the accuracy of our other
constraints into question. The parameter actually constrained by CHIMERA is
the radius-to-distance scaling factor (R/D)2; however, U0722 has a well-
constrained distance from parallax measurements (4.12 $\pm$ 0.04 pc, Leggett
et al. 2012) that is most likely not the source of our impossibly small
radius. From the Sonora Bobcat evolutionary tracks (Marley et al., 2021), the
minimum possible radius for even a 10 Gyr object at subsolar metallicity is
0.75 RJup. Furthermore, this small radius combined with our median retrieved
$v$ sin$i$ would imply an unreasonably fast rotation period of about 1.55
hours. Figures 7 and 9 show this radius problem particularly gets worse when
updating the CH4 and NH3 cross sections which are necessary to accurately
reproduce the line positions seen in the FIRE spectrum. One potential source
of error could be the completeness of the Hargreaves et al. (2020) CH4 line
list in y band as discussed in Section 3.5. Furthermore, even with the
previously used Yurchenko & Tennyson (2014) line list we were retrieving a
smaller radius than expected (0.76 RJup), indicating additional issues that
are perhaps only exacerbated by the change to the newer CH4 line list.

Further work is needed to assess whether our small radius could be attributed
to completeness issues with the CH4 line list from Hargreaves et al. (2020),
uncertainties on how to treat alkali opacities, poor photometric calibration,
or some other unseen flaw in our modeling framework. Recently, multiple
retrieval studies have found radii smaller than expected from evolutionary
models for both L dwarfs (Gonzales et al., 2020; Burningham et al., 2021; Xuan
et al., 2022) and T dwarfs (Kitzmann et al., 2020; Lueber et al., 2022) across
different retrieval frameworks and instruments with varying spectral
resolutions. Furthermore, Zhang et al. (2021) performed a uniform comparison
of 55 late T dwarf spectra with the Sonora Bobcat grid (Marley et al., 2021)
of forward models using the Bayesian framework Starfish, finding small radii
for a number of the studied objects indicating this issue is not solely found
in retrieval analyses. Notably, Zhang et al. (2021) fit a SpeX spectrum of
U0722, getting Teff= 680 $\pm$ 26 K, log(g)=3.6 $\pm$ 0.3 dex, [M/H]= -0.06
$\pm$ 0.2 dex, and R = 0.43 $\pm$ 0.04 RJup, yielding even more unphysical
values of the surface gravity and radius than in this work. More retrieval
studies for large samples of brown dwarf spectra, particularly including mid-
infrared data for the coolest objects and high-quality observations from JWST,
will hopefully illuminate the source of this small radius problem. In
addition, the growing sample of transiting brown dwarfs from the TESS mission
(e.g. Šubjak et al., 2020; Carmichael et al., 2022) can provide independent
tests of the radii predicted by evolutionary models, though their irradiated
nature may cause difficulty in making comparisons.

###  4.2 Comparison to Bochanski et al. Results

We can compare the results from our preferred retrieval on the FIRE spectrum
of U0722 to the original analysis of the dataset presented by B11. By
comparing the spectrum to line lists, they were able to identify features of
H2O, CH4, and NH3, as well as broad absorption from K. Similarly, we are able
to place constraints on the abundances of all of these species to within $\pm$
0.02 dex, indicating the impact of these species on the spectrum even if the
retrieved abundances are not exactly accurate. In addition, we are also able
to constrain the H2S abundance due to a few distinct H2S features in the
spectrum (see Section 3.3.2) unidentified in the previous study.

B11 fit the FIRE spectrum of U0722 with BT-Settl models of Allard et al.
(2011) based on the fitting procedure outlined by Cushing et al. (2008). The
large differences we see in retrieved posteriors when looking at subsections
of the spectrum in Section 3.5 reflect similar variance across the model fits
of B11 of data in different bandpasses. However, the authors find significant
differences between the best fitting models and the data in many places across
the spectrum. Furthermore, there are notable discrepancies in our retrieved
physical values from those of B11 when considering their full spectrum fit
(most analogous to our results) as shown in Table 4. While their log(g)
4.0${}^{+0.3}_{-0.3}$ is consistent, they report a larger mass of 5.24 MJup
from evolutionary models, indicating a much larger radius value of
1.14${}^{+0.46}_{-0.33}$ RJup. Due to the large uncertainty on U0722’s
parallax at the time, B11 allow distance instead of the radius to vary when
fitting the (R/d)2 scaling factor. When allowing this distance to freely vary,
their best fit to the entire spectrum also prefers a hotter object with
effective temperature 700 $\pm$ 50 K, consistent with the calcualted Teff from
our FIRE retrieval but with a far larger distance than the literature value
(49 pc vs. 4.12 pc). Our retrieval functionally achieves the same effect with
a small (R/d)2 scaling factor instead attributed to a small radius given the
reliable parallax measurement. However, when the distance is required to be
within 5$\sigma$ of the parallax measurement from Lucas et al. (2010), the
authors get a much colder Teff of 500 $\pm$ 50 K.

Finally, while their measured $v$ sin$i$ of 40 $\pm$ 10 km s-1 agrees with our
results, their radial velocity measurement of 46.9 $\pm$ 2.5 km s-1 is
significantly higher than our result of 35.80 $\pm$ 0.15 km s-1. Given that
the radial velocity was measured in part with models using significantly older
line lists for basically all molecules present in this object’s spectrum, such
an inconsistency is perhaps expected. Future work to calculate if an updated
RV measurement would change the original determination by B11 of a thin disk
Galactic orbit could provide interesting context to U0722’s potential age and
evolutionary trajectory, though the change in radial velocity may not be
enough to matter. We use the tool Bayesian Analysis for Nearby Young
AssociatioNs $\Sigma$ (BANYAN$\Sigma$, Gagné et al., 2018) to determine the
probability of U722 being a member of any well-characterized young
associations within 150 pc, inputting the previously measured proper motion
(Faherty et al., 2009) and parallax (Faherty et al., 2012) as well as the
radial velocity measured in this work. BANYAN$\Sigma$ gives a 99.9% likelihood
that U0722 is a field brown dwarf. Furthermore, for both velocity parameters
our reported precision from emcee is significantly higher than those reported
for the measurements by B11. Both the radial velocity and $v$ sin $i$ were
calculated by cross-correlating the 1.27 - 1.31 $\mu$m section of FIRE
spectrum with a template, either a grid model spectrum or another observation
of a T dwarf, which may or may not be a good fit to the data in question. In
contrast, the log-likelihood based MCMC approach reported here both considers
significantly more data points across the whole spectrum and has been shown to
produce smaller error bars than cross-correlation methods (Brogi & Line,
2019).

###  4.3 Comparison to Grid Models

Figure 23: TP profiles from our preferred FIRE retrieval (blue) and the fixed
radius and surface gravity retrieval (grey) to the tuned ATMO model (pink) and
two disequilibrium forward models from Mukherjee et al. (2022). The tuned ATMO
model is significantly colder than the retrieved TP profiles from our
preferred FIRE retrieval, and has a different gradient in accordance with the
adjusted adiabat discussed by Leggett et al. (2021). However, the retrieved TP
profiles when fixing the radius and surface gravity to set values is in
excellent agreement with the tuned ATMO model.  Figure 24: Molecular volume
mixing ratios from the tuned ATMO model (dotted lines) as a function of
atmospheric pressure compared to the median and 1$\sigma$ posteriors from our
FIRE retrieval (solid lines and shaded regions). With the exception of NH3,
the abundances are quite similar.  Figure 25: Comparison of the FIRE (blue)
and SpeX (orange) spectra of U0722 analyzed in this work to the spectrum
(yellow) and tuned ATMO model (pink) used in Leggett et al. (2021). There are
notable discrepancies in the peak of the y band.

Given the many advances in line lists over the past decade, it is instructive
to look at more recent comparisons of U0722 spectra with grid models. Miles et
al. (2020) find that the low-resolution (R $\sim$ 300) M band spectrum of
U0722 indicates disequilibrium amounts of CO from vertical mixing, consistent
with our tentative detection of CO in the FIRE spectrum. Leggett et al. (2021)
compare the flux-calibrated 0.6-5.1 $\mu$m spectrum of U0722 created from
multiple published observations (Lucas et al., 2010; Leggett et al., 2012;
Miles et al., 2020) with the ATMO 2020 grid of models, finding that the models
with and without disequilibrium chemistry still have trouble fitting the data
well. The authors improve this fit by “tuning” the gradient of the TP profile
away from the standard adiabat. As such, we compare our retrieved results to
this adiabat-adjusted model in Table 4, which has an effective temperature of
540 K, log surface gravity (cgs) of 4.5, solar metallicity, log Kzz (cm2
s-1)=7, and an adiabatic index of 1.27.

Figure 23 shows our retrieved TP profiles from the FIRE spectrum of U0722,
compared to the abundances of the tuned ATMO model (Tremblin 2021, private
communication), as well as two forward grid models with disequilibrium
chemistry from Mukherjee et al. (2022). The tuned ATMO TP profile has a
steeper gradient than either of the other grid models, reflecting the adjusted
adiabat. Our retrieved TP profiles from the preferred FIRE retrieval are
significantly hotter than that of the tuned ATMO model, more closely
resembling that of a hotter 700 K object from the Mukherjee et al. (2022)
grid, consistent with our calculated Teff. The difference in temperature is
consistent with the differences in radii- the tuned ATMO model assumed a
radius of 1.12 RJup from the evolutionary models of Phillips et al. (2020),
more than double that of our retrieved value. Thus, a cooler object would be
necessary to achieve a similar amount of flux. In fact, when we fixed the
radius to 1 RJup and surface gravity to log(g)=4.5, as discussed in Section
3.7, the retrieved TP profiles are consistent with the tuned ATMO model as
shown by the grey curves in Figure 23. This agreement is reasonable, as
Leggett et al. (2021) similarly set R and log(g) to values from evolutionary
model before tuning the TP profile to match the observed data.

Figure 24 compares our retrieved molecular abundances, shown by the solid
lines and shaded 1$\sigma$ regions, to the mixing ratios in the tuned ATMO
model as a function of pressure in the atmosphere. Unlike the tuned ATMO model
which assumes disequilibrium chemistry governed by the eddy diffusion
parameter log(Kzz), our volume mixing ratios for each species are allowed to
vary independently and are not tied to one another by any assumptions or
parameterizations of the chemical timescales involved. Our retrieved values
for CO, CH4, and H2S are all in agreement with the ATMO abundances above 100
bars, and our H2O value posterior is only 0.003 dex away from the ATMO model
value. However, our NH3 posterior is $\sim$0.5 dex lower, perhaps reflecting
the differences in effective temperature and surface gravity between our
retrieval results and the tuned ATMO model. Given a measured CO abundance, we
can estimate log(Kzz) by comparing to chemical equilibrium abundances from the
Sonora Bobcat structure models (Marley et al., 2021) of similar Teff and
log(g) as described by Miles et al. (2020). Since our CO abundance has large
uncertainties, our results could imply no vertical mixing for the lowest
amount of CO up to log(Kzz) $\sim$ 3.3 cm2 s-1 for the maximum amount of CO.
The comparatively larger log(Kzz) = 7 cm2 $s^{-1}$ from the tuned ATMO model
is needed to get a similar CO abundance for a much colder object with higher
surface gravity. While our retrieved surface gravity and radius are
unphysical, our molecular abundances are plausible and mostly consistent with
grid model predictions. The similarity in molecular abundances is also
reflected in the consistency of the C/O ratios and metallicities reported in
Table 4.

Figure 26: Comparison of the median model of our preferred FIRE retrieval and
the ATMO tuned model to the L and M band spectra included in the analysis of
Leggett et al. (2021). The FIRE retrieval clearly does not reproduce the
measured flux of U0722 at longer wavelengths, whereas the ATMO tuned model
gets closer at matching the observed slopes of these regions of the spectrum.

Differences in the data considered perhaps help explain why our retrieval
prefers a hotter, smaller object than the tuned ATMO model when the retrieval
is solely driven by the data (rather than any assumed radius or gravity
value). We note that the NIR portion of the U0722 spectrum used in Leggett et
al. (2021) is slightly inconsistent with that of the FIRE and SpeX spectra
examined in this work. Figure 25 shows all three datasets, as well as the
tuned ATMO model; the peak y band flux in particular of the Leggett et al.
(2021) is offset in both strength and wavelength. While the source of this
discrepancy is unclear, it undoubtedly contributes to differences in our
fitted or retrieved values for U0722. However, a more important distinction is
the inclusion of longer wavelength data for U0722 in the analysis of Leggett
et al. (2021).

Figure 27: Normalized contribution functions for a model generated from the
median results of preferred FIRE retrieval (which covered 0.9-2.5 $\mu$m)
extrapolated out to longer wavelengths.

The authors included L and M band spectra from Leggett et al. (2012) and Miles
et al. (2020), respectively. Figure 26 shows these datasets compared to the
median model from our preferred FIRE retrieval extrapolated out to these
wavelengths, as well as the tuned ATMO model. Our model clearly underpredicts
the observed flux of U0722 and does not reproduce the observed slope in the M
band data. Figure 27 shows the normalized contribution functions for a model
using the median parameter values from our preferred FIRE retrieval, generated
at lower resolution and out to $\sim$ 15 $\mu$m. The L and M bands probe the
upper atmosphere, and we note that if we were to perform a retrieval on these
datasets we would need to extend the atmosphere in our radiative transfer
model to pressures below 10-3 bars, the current top of the atmosphere in this
framework. Notably, the TP profile from our preferred FIRE retrieval is not
well constrained at these low pressures in our retrieval, perhaps contributing
to the model’s failure to reproduce these data points. Furthermore, the
discrepancy in M band could be due to less CO present in the median model than
is needed to fit the data, as it is a major opacity source in this wavelength
region. Figure 24 shows that the median CO abundance from our preferred FIRE
retrieval is more than a dex less than the amount of CO in the tuned ATMO
model. In combination with the unphysically small radius, the inability of our
preferred FIRE retrieval to produce fluxes in accordance with longer
wavelength observations points to our precise constraints on the atmosphere of
U0722 being inaccurate and not necessarily representative of the true physical
properties of this object. Future work on combining medium-resolution spectra
with broader wavelength coverage data at lower resolution may provide further
insight into the best way to obtain more plausible results from an atmospheric
retrieval analysis.

##  5 Conclusions

In this work, we have applied the CHIMERA atmospheric retrieval framework to a
high signal-to-noise, medium-resolution (R $\sim$ 6000) FIRE spectrum of a T9
dwarf. Key takeaways from this work are as follows:

  1. 1.

Limitations of the dataset must be taken into account: In Section 3.2, we show
that ensuring only one data point per resolution element is sampled as well as
cutting out very noisy regions of data at either end of the spectrum can
affect our retrieved posteriors. More dramatically, areas of the spectrum
where orders were stitched poorly can negatively bias the retrieval analysis
as discussed in Section 3.4. Future improvements to order stitching methods,
or new ways to account for potential stitching issues within a retrieval
framework, could help alleviate this problem in future analysis of data from
echelle spectrographs.

  2. 2.

Using different opacity sources may lead to very different results: Updating
the line lists for CH4 and NH3 to those from Hargreaves et al. (2020) and
Coles et al. (2019), respectively, greatly improved the ability of the forward
models to match the line positions in the data, as shown in Figure 8. Almost
all retrieved posteriors were affected by this change, even for the retrieval
on the R $\sim$ 100 SpeX spectrum as shown in Figures 7 and 9. For example for
the FIRE retrieval, changing the line lists lead the median abundances of H2O
and CH4 to increase by $\sim$ 0.3 dex and the surface gravity log(g) to
increase by $\sim$0.4 dex. However, the updated line lists also resulted in a
radius decreased by about 30% to an unphysical 0.5 RJup. We also tested
different treatments of Na and K opacities in Section 3.6, again finding
largely disparate results depending on which cross sections were used. While
we do recommend the Hargreaves et al. (2020) CH4 line list due to its ability
to match the line positions in our FIRE spectrum, more comparison with other
line lists with regards to completeness as well as treatment of the alkali
wings are needed to fully utilize this kind of high-quality data.

  3. 3.

Medium-resolution retrievals offer very precise constraints, but they may not
be accurate: As shown in Figure 22, the constraints on the temperature-
pressure profile and abundances from our FIRE retrieval of U0722 are
significantly more precise than those from the SpeX spectrum. In particular,
we are able to retrieve the abundances of H2S and tentatively CO which is not
possible with the lower resolution spectrum. However, while we do get these
precise, stellar-like constraints on atmospheric abundances ($\sim$0.02 dex),
the radius is far too small to be physically plausible. This small size is in
accordance with our retrieved TP profile being hotter than previous analyses
of this object, yielding a similar overall observed flux. Thus, this study
joins a growing number of modeling analyses of brown dwarf spectra that have
yielded smaller radii than allowed by our understanding of brown dwarf
evolution. Furthermore, extrapolating a median model from our retrieval cannot
to L and M band cannot reproduce observations of U0722 at these wavelengths,
indicating the constraints from the medium-resolution FIRE spectrum alone do
not accurately describe the conditions of this object.

This work is a first foray into the challenges and benefits of applying
atmospheric retrieval tools to medium-resolution spectra of brown dwarfs. With
the launch of JWST and future ground-based studies, more work is needed to
assess how we can improve our current modeling frameworks to address these
challenges and unlock the potential for trustworthy, precise constraints on
substellar atmospheres from retrievals of medium-resolution spectra.

We thank the anonymous reviewer for their thoughtful comments and suggestions
that have improved the quality of this work. We acknowledge use of the lux
supercomputer at UC Santa Cruz, funded by NSF MRI grant AST 1828315. This work
benefited from the 2022 Exoplanet Summer Program in the Other Worlds
Laboratory (OWL) at the University of California, Santa Cruz, a program funded
by the Heising-Simons Foundation. MRL acknowledges support from NASA XRP grant
80NSSC19K0293. JJF acknowledges support from NSF Award Number:1909776 and NASA
Award Number: 80NSSC22K0142 as well as the Simons Foundation.

Figure 28: Corner plot summarizing the posterior of our preferred FIRE
retrieval for the listed parameters. Correlations between parameters are shown
in the two-dimensional histograms. Marginalized posteriors for each parameter
are shown along the diagonal, with the dashed lines indicating the 16th, 50th,
and 84th percentiles. The median and $\pm 1\sigma$ values for each parameter
are shown above the histograms.

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