# The Hawaii Infrared Parallax Program. VI. The Fundamental Properties of
1000+ Ultracool Dwarfs and Planetary-mass Objects Using Optical to Mid-IR SEDs
and Comparison to BT-Settl and ATMO 2020 Model Atmospheres

Aniket Sanghi NSF Graduate Research Fellow.  Department of Astronomy,
California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA
91125, USA  Institute for Astronomy, University of Hawaii, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA  Michael C. Liu Institute for Astronomy,
University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  William M.
Best The University of Texas at Austin, Department of Astronomy, 2515
Speedway, C1400, Austin, TX 78712, USA  Trent J. Dupuy Institute for
Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill,
Edinburgh, EH9 3HJ, UK  Robert J. Siverd Institute for Astronomy, University
of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Zhoujian Zhang
University of California, Santa Cruz, 1156 High St. Santa Cruz, CA 95064, USA
Spencer A. Hurt Department of Physics and Astronomy, University of Wyoming,
Laramie, WY 82071, USA  Eugene A. Magnier Institute for Astronomy, University
of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Kimberly M. Aller
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu,
HI 96822, USA  Niall R. Deacon Max Planck Institute for Astronomy, Konigstuhl
17, D-69117 Heidelberg, Germany

###### Abstract

We derive the bolometric luminosities ($L_{\mathrm{bol}}$) of 865 field-age
and 189 young ultracool dwarfs (spectral types M6–T9, including 40 new
discoveries presented here) by directly integrating flux-calibrated optical to
mid-IR spectral energy distributions (SEDs). The SEDs consist of low-
resolution ($R\sim$ 150) near-IR (0.8–2.5 $\mu$m) spectra (including new
spectra for 97 objects), optical photometry from the Pan-STARRS1 survey, and
mid-IR photometry from the CatWISE2020 survey and Spitzer/IRAC. Our
$L_{\mathrm{bol}}$ calculations benefit from recent advances in parallaxes
from Gaia, Spitzer, and UKIRT, as well as new parallaxes for 19 objects from
CFHT and Pan-STARRS1 presented here. Coupling our $L_{\mathrm{bol}}$
measurements with a new uniform age analysis for all objects, we estimate
substellar masses, radii, surface gravities, and effective temperatures
($T_{\mathrm{eff}}$) using evolutionary models. We construct empirical
relationships for $L_{\mathrm{bol}}$ and $T_{\mathrm{eff}}$ as functions of
spectral type and absolute magnitude, determine bolometric corrections in
optical and infrared bandpasses, and study the correlation between
evolutionary model-derived surface gravities and near-IR gravity classes. Our
sample enables a detailed characterization of BT-Settl and ATMO 2020
atmospheric model systematics as a function of spectral type and position in
the near-IR color-magnitude diagram. We find the greatest discrepancies
between atmospheric and evolutionary model-derived $T_{\mathrm{eff}}$ (up to
800 K) and radii (up to 2.0 $R_{\mathrm{Jup}}$) at the M/L transition
boundary. With 1054 objects, this work constitutes the largest sample to date
of ultracool dwarfs with determinations of their fundamental parameters.

Fundamental parameters of stars (555); Astrometry (80); Photometry (1234);
Brown dwarfs (185); M dwarf stars (982); L dwarfs (894); T dwarfs (1679);
Spectral energy distribution (2129); Bolometric correction (173); Stellar
evolutionary models (2046); Stellar atmospheres (1584); Exoplanet atmospheres
(487)

##  1 Introduction

Brown dwarfs are substellar objects more massive than gas-giant planets
($\gtrsim$ 13 $M_{\mathrm{Jup}}$; Spiegel et al., 2011) but less massive than
stars ($\lesssim$ 70 $M_{\mathrm{Jup}}$; Dupuy & Liu, 2017). All brown dwarfs
fuse deuterium (e.g. Saumon et al., 1996) and those between 60–75
$M_{\mathrm{Jup}}$ will also burn lithium (Basri, 1998). Brown dwarfs cool
through their lifetimes, progressing through a wide temperature range spanning
the M, L, T and Y spectral types (e.g. Burrows et al., 1997; Kirkpatrick,
2005; Cushing et al., 2011). This continuous cooling means that for a given
luminosity and effective temperature, and hence spectral type, the age and
mass of a brown dwarf are not uniquely determined. Such an observational
degeneracy makes it challenging to use colors, magnitudes, and spectral types
to determine the physical properties of brown dwarfs, such as mass, age, and
radius. However, the luminosities and temperatures for a population of brown
dwarfs do characterize their evolutionary history and provide unique
information about the star-formation history of our Galaxy (Best et al., 2021;
Kirkpatrick et al., 2021). Additionally, young brown dwarfs, identified by
their unusually red near- to mid-infrared colors as well as lower surface
gravities, serve as crucial templates for understanding the atmospheric
properties (e.g., clouds) of giant exoplanets given their similar effective
temperatures and ages (e.g. Liu et al., 2013; Faherty et al., 2016; Liu et
al., 2016). A large, well-defined sample of brown dwarfs with accurate and
precise luminosity measurements would enhance our ability to test models of
substellar formation, evolution, and atmospheres.

A proven approach to determining the physical properties of ultracool dwarfs
($\gtrsim$ M6) is to assemble and integrate broadband spectral energy
distributions (SEDs) consisting of flux-calibrated optical, near-infrared, and
mid-infrared spectra and/or photometry to obtain bolometric fluxes (e.g.
Filippazzo et al., 2015). Given parallax measurements, we can then derive the
corresponding bolometric luminosities ($L_{\mathrm{bol}}$). The bolometric
luminosities can be used together with age estimates and evolutionary models
to compute the masses and radii of ultracool dwarfs. Finally, combining the
bolometric luminosities with evolutionary model-derived radii and applying the
Stefan-Boltzmann Law enables the calculation of effective temperatures.

This approach should yield less model-dependent results than deriving the
bolometric luminosities, effective temperatures, and surface gravities
simultaneously from atmospheric model spectra fits, and then obtaining the
masses and radii. Bolometric luminosites derived by atmospheric model fits can
introduce systematic uncertainties since such measurements are dependent upon
the fidelity of the fitting procedures adopted, the sampling of the model
grid, and the differing input physics between various atmospheric models (e.g.
Stephens et al., 2009; Zhang et al., 2021b). Filippazzo et al. (2015) adopted
the SED integration strategy, combining optical, near-infrared, and mid-
infrared spectra and photometry to construct spectral energy distributions for
145 field-age and 53 young ultracool dwarfs and thereby derive their
fundamental physical properties. The uncertainties in the distances were the
largest contributor to the uncertainty in their bolometric luminosities. Thus,
high-precision parallax measurements are a necessity for accurate and precise
determinations of bolometric luminosities. Similarly, since the uncertainty in
age is the dominant contributor to the uncertainty in ultracool dwarf masses,
radii, surface gravities, and effective temperatures, well-constrained ages
are a necessity for accurate and precise determinations of these physical
properties. The limited availability of such measurements has constrained the
sample size of objects in previous work.

Table 1: Comprehensive Reference List for Table of Ultracool Fundamental
Properties Data

References
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Journals, submitted); Hurt et al. (AAS Journals, submitted); _The
UltracoolSheet_ (version 2.0.0, in preparation); This Work.

Note. — References for ultracool dwarf discoveries, spectral types
(optical/IR), gravity classifications (optical/IR), parallaxes, SpeX prism
spectra, ages, Pan-STARRS 1 $grizy$ photometry, 2MASS $JHK_{S}$ photometry,
MKO $JHK$ photometry, WISE _W1_ , _W2_ , _W3_ , and _W4_ photometry, and
Spitzer Channel 1 and Channel 2 photometry presented in the Table of Ultracool
Fundamental Properties associated with this work (see §1) are included above.

The first generation of ultracool dwarf parallax programs focused on obtaining
measurements for small samples of objects across the spectral type range M, L,
and T as they were newly discovered (Dahn et al., 2002; Tinney et al., 2003;
Vrba et al., 2004). With the rapid acceleration in the number of ultracool
dwarf discoveries catalyzed by the advent of large digital sky surveys, the
latest generation of ultracool parallax programs have focused their efforts on
distinct classes of objects and large volume-limited samples. The Hawaii
Infrared Parallax Program has observed L/T transition dwarfs (Dupuy & Liu,
2012), young ultracool dwarfs (Liu et al., 2016), and ultracool binaries
(Dupuy & Liu, 2012, 2017) with WIRCam on the Canada–France–Hawaii Telescope
(CFHT), and most recently obtained infrared parallax measurements for 348 L
and T dwarfs with WFCAM on the United Kingdom Infrared Telescope (Best et al.,
2020). Several programs have targeted late-T and Y dwarfs, the coldest objects
in the ultracool dwarf temperature sequence (Dupuy & Kraus, 2013; Beichman et
al., 2014; Tinney et al., 2014; Martin et al., 2018; Kirkpatrick et al., 2019,
2021). Therefore, a large-scale analysis aimed at deriving ultracool dwarf
properties by taking advantage of the tremendous advances in precision and
availability of trigonometric parallax measurements, particularly for later-
type objects, is timely.

Age dating stars and brown dwarfs is a notoriously difficult task, but steady
progress has been made over the past decade. The simultaneous study of a large
ensemble of stars has proven to be a successful avenue to obtaining new age
measurements. The discovery of coeval, kinematically comoving associations of
young stars and brown dwarfs in the solar neighborhood called young moving
groups (YMGs) has been instrumental to the effort (e.g. Zuckerman & Song,
2004; Gagné et al., 2014c; Aller et al., 2016; Faherty et al., 2016; Liu et
al., 2016; Gagné et al., 2018, 2022). Group members generally span a wide
range of masses and, thus, can be used to age date each group using lithium
depletion boundaries and isochrone fitting (e.g. Bell et al., 2015; Herczeg &
Hillenbrand, 2015). For the field population of ultracool dwarfs, Dupuy & Liu
(2017) empirically determined an age distribution consistent with the Besançon
model of the solar neighborhood (Robin et al., 2003) based on cooling ages of
20 ultracool dwarfs in binaries derived from high-precision dynamical mass
measurements. In addition, various techniques have been developed geared
towards identifying signs of youth in individual objects. Activity-based
indicators such as chromospheric H$\alpha$ emission (West et al., 2008; Kiman
et al., 2021), coronal X-ray emission (Jackson et al., 2012; Booth et al.,
2017), and UV emission (Shkolnik et al., 2011; Rodriguez et al., 2013) are
found to be well-correlated with stellar ages. Tracing lithium absorption
features in ultracool dwarf spectra and perfoming a comparison with
predictions of lithium-burning timescales from evolutionary models can provide
valuable insights into the age of young dwarfs (Kirkpatrick et al., 2008;
Binks & Jeffries, 2016; Zhang et al., 2022). For mid-M to L dwarfs, there are
a number of gravity-sensitive features in the near-infrared such as FeH (0.99,
1.20, 1.55 $\mu$m), VO (1.06 $\mu$m), Na I (1.138 $\mu$m), K I (1.169, 1.177,
1.244, 1.253 $\mu$m), and the shape of the _H_ -band continuum (1.46–1.68
$\mu$m) that can be used to uncover low gravity (young) objects (Allers & Liu,
2013; Cruz et al., 2018). Similarly, at optical wavelengths, weak Na I (8183,
8195 Å), Cs I (8521, 8943 Å), Rb I doublet lines (7800, 7948 Å), weak
pressure-broadened K I wings (7665, 7699 Å), weak FeH (8692 Å), and strong VO
(7300–7550 Å and 7850–8000 Å) in early type dwarfs act as spectroscopic
signatures of low-gravity (youth) (Martín et al., 1998; Cruz et al., 2009;
Kirkpatrick et al., 2010).

The large-scale effort to leverage advances in parallax and age measurements
to empirically determine the fundamental properties of ultracool dwarfs has
been greatly facilitated by the development of _The UltracoolSheet_ (version
2.0.0, in preparation). _The UltracoolSheet_ is a catalog of 3000+ ultracool
dwarfs and directly imaged exoplanets, including optical to mid-infrared
photometry from Pan-STARRS1 (Chambers et al., 2016), 2MASS (Skrutskie et al.,
2006), UKIDSS/UHS (Lawrence et al., 2007; Dye et al., 2018), and CatWISE
(Eisenhardt et al., 2020; Marocco et al., 2021), absolute astrometry, proper
motions, parallaxes, multiplicity (including orbit determinations), and
spectroscopic classifications. A major benefit offered by _The UltracoolSheet_
for deriving bolometric luminosities is that it not only compiles high-
precision parallax values from the latest surveys (e.g., _Gaia DR3_) and
literature measurements (e.g. Best et al., 2020; Kirkpatrick et al., 2021),
but also compares the various sources to readily make available the most
precise measurement available to-date for each listed object. Additionally, it
provides photometric distance estimates for objects without parallax
measurements. In short, _The UltracoolSheet_ is a valuable resource well
matched to the goals of this work.

Figure 1: Near-infrared (0.8–2.5 $\mu$m) SpeX prism spectra for the 57 known
objects presented in Table 2. From top to bottom and left to right, plotting
order follows the table order for objects, namely by category (which is
colorized in the plot) and then by right ascension. Light blue corresponds to
T dwarfs, red corresponds to red L dwarfs, purple corresponds to candidate
low-gravity objects, dark blue corresponds to Hyades members, green
corresponds to wide substellar/planetary-mass companions, yellow corresponds
to solar neighborhood candidates, and gray corresponds to M/L transition
dwarfs. The horizontal line to the left of each spectrum shows the zero flux
level.

In this paper, we derive the fundamental physical properties of 1054 ultracool
dwarfs by directly integrating flux-calibrated SEDs consisting of low-
resolution near-infrared SpeX prism spectra and optical and mid-infrared
photometry. This work increases the sample of ultracool dwarfs with semi-
empirical physical parameter measurements by nearly five-fold, and takes
advantage of significant growth in parallax data for ultracool dwarfs.
Additionally, we construct and analyze empirical relationships for bolometric
luminosity and effective temperature as functions of spectral type and
absolute magnitude, determine bolometric corrections across the optical to
mid-infrared filters, benchmark the near-infrared gravity classifications
against evolutionary model-derived surface gravity measurements, and study
systematic offsets between fundamental parameters derived with evolutionary
(BHAC15 and SM08) and atmospheric (BT-Settl and ATMO 2020) models. We prepare
a Table of Ultracool Fundamental Properties111Zenodo Data Publication:
https://doi.org/10.5281/zenodo.8315643 that is a subset of _The
UltracoolSheet_ (version 2.0.0, in preparation) tied uniquely to this work. It
is a comprehensive compilation of the astrometric, photometric, spectroscopic,
and age properties of all objects in our sample. Importantly, it contains the
fundamental parameters calculated in this work for all objects in our sample.
References for all data used in this work, compiled in the Table of Ultracool
Fundamental Properties, are included in Table 1.

Figure 2: Near-infrared (0.8–2.5 $\mu$m) SpeX prism spectra for the 40 new
discoveries presented in Table 3. From top to bottom and left to right,
plotting order follows the table order for objects, namely by category (which
is colorized in the plot) and then by right ascension. Light blue corersponds
to T dwarfs, magenta corresponds to L dwarfs, red corresponds to red L dwarfs,
purple corresponds to low-gravity objects, and gray corresponds to M/L
transitions dwarfs. The horizontal line to the left of each spectrum shows the
zero flux level.

This paper is organized as follows. Section 2 details the construction of our
ultracool dwarf sample and presents newly discovered objects, new object
spectra, and new parallax measurements. Section 3 describes the spectra and
photometry used to assemble and flux-calibrate multiwavelength ultracool dwarf
spectral energy distributions (SEDs). Section 4 details our procedures to
estimate flux in regions where no spectra are available using atmospheric
model fitting. Sections 5 and 6 discuss the calculation of bolometric
luminosities and assignment of ages for the 1054 ultracool dwarfs in our
sample. Section 7 summarizes the techniques used to derive the masses, radii,
surface gravities, and effective temperatures for objects in our ultracool
dwarf sample. Section 8 presents calculations of bolometric corrections,
analysis of the relationship between gravity classes and derived surface
gravities, and a detailed look at atmospheric model-derived parameters in
comparison with evolutionary model-derived parameters. Finally, Section 9
presents our conclusions. Appendix A presents revised absolute magnitude-
spectral type relations for young ultracool dwarfs based on all available
objects in _The UltracoolSheet_.

Figure 3: Parallax solutions for 11 objects with new CFHT/WIRCam astrometry.
For each object, three panels are shown: Decl. measurements with the proper
motion and zero point subtracted (top); R.A. measurements with the proper
motion and zero point subtracted (middle); and the residuals of both R.A.
(open symbols) and Decl. (solid symbols) plotted with the best-fit parallax
and proper motion solution subtracted (bottom). The standard deviation of the
residuals of both R.A. and Decl. combined is quoted as the rms in the bottom
panel. The top and middle subpanels show the motion expected from the best-fit
parallax (solid line). Objects have varying observational time baselines,
which is reflected in the varying x-axis ranges.

##  2 Sample & Additional Observations

###  2.1 Construction of Ultracool Dwarf Sample

Our sample is constructed using _The UltracoolSheet_ (version 2.0.0, in
preparation), selecting all objects with parallaxes or photometric distances
and SpeX prism spectra readily available to us. We exclude unresolved binaries
and multiples, subdwarfs, and members of active star-forming regions (e.g.
Taurus, Chamaeleon, Lupus, IC348, etc.). _The UltracoolSheet_ columns relevant
to the above criteria are multiple_unresolved_in_this_table, flag, and
age_category, respectively. In total, 1054 objects in _The UltracoolSheet_
satisfy the above criteria (including the newly discovered ultracool dwarfs
discussed below).

###  2.2 New Objects and Spectra

We obtained low-resolution near-IR (0.8–2.5 µm) spectroscopy for 97 objects in
our sample using the NASA Infrared Telescope Facility (IRTF) on the summit of
Mauna Kea, Hawaii. We used the facility spectrograph SpeX (Rayner et al.,
2003) in prism mode with either the 0.5″ or 0.8″ slit, which provided an
average spectral resolution ($R\equiv\Delta\lambda/\lambda$) of $\approx$150
or $\approx$100, respectively. We dithered in an ABBA pattern on the science
target to enable sky subtraction during the reduction process. For each epoch,
we obtained calibration frames (arcs and flats) and observed an A0 V star
contemporaneously for telluric calibration. All spectra were reduced using the
Spextool software package (Vacca et al., 2003; Cushing et al., 2004).

These spectra were obtained for a variety of programs conducted by us over the
past decade, either for (1) spectroscopic characterization of known objects
(Figure 1 and Table 2) or (2) discovery of new ultracool dwarfs (Figure 2 and
Table 3). For (1), we obtained spectra of known objects that had potential
indications of low-gravity spectra from previous work (e.g. Allers & Liu,
2013) or objects that could serve as benchmark objects by virtue of being
members of known stellar associations (the Hyades and Pleiades) or being wide
substellar/planetary-mass companions to stars (e.g. Metchev & Hillenbrand,
2006; Dupuy et al., 2018). Finally, as a bad-weather program, we obtained
spectra of objects with pre-Gaia parallaxes and with spectral types at the M/L
transition. For (2), the discoveries result from various multi-catalog
photometric searches relying on the Pan-STARRS 1 optical survey as a staring
point, including searches for T dwarfs (e.g. Deacon et al., 2011; Liu et al.,
2011), very red (low-gravity) L dwarfs (e.g. Liu et al., 2013), young moving
group members (e.g. Aller et al., 2016; Aller, 2016), L/T transition dwarfs
and objects in the solar neighborhood (e.g. Best et al., 2015). For spectral
typing and gravity classification, we use the near-IR classification system
described by Allers & Liu (2013) and Aller et al. (2016) for late-M and L
dwarfs. For T dwarfs, we follow the near-IR system described by Burgasser et
al. (2006b). Among the new discoveries, we find that PSO J080.3940+23.0999 is
a new candidate AB Doradus Moving Group member with an 82.5% BANYAN $\Sigma$
probability (Gagné et al., 2018).

###  2.3 New Parallaxes

For eleven objects — WISEA J004403.39+022810.6, 2MASS J03264225-2102057, PSO
J057.2893+15.2433, PSO J077.1034+24.3810, PSO J127.4696+10.5777, SDSS
J161731.65+401859.7, SDSS J213240.36+102949.4, WISEA J235422.31-081129.7, PSO
J178.1434-23.8603, PSO J349.3359+32.0532, and PSO J243.9421+67.2075 — we
present new parallax measurements (Figure 3). They are available in the Table
of Ultracool Fundamental Properties associated with this paper (see §1). We
monitored these objects with the facility infrared camera WIRCam (Puget et
al., 2004) on the Canada-France-Hawaii Telescope (CFHT). In the $J$ band, we
measured their $(x,y)$ positions and those of many reference stars. Using our
custom pipeline (Dupuy & Liu, 2012; Dupuy et al., 2015), we reduced these
individual measurements into high-precision multi-epoch relative astrometry,
with the absolute calibration provided by low-proper-motion 2MASS stars (Cutri
et al., 2003). We derived the relative parallax and proper motion for the
targets using our standard MCMC approach and then, in order to be consistent
with our many previously published CFHT parallaxes, converted to an absolute
reference frame using the Besançon galaxy model to simulate the distances of
the reference stars (Robin et al., 2003).

We also present new Pan-STARRS 1 (PS1) parallaxes for eight objects in our
sample: 2MASS J03001631+2130205, 2MASSW J0309088-194938, SDSSp
J032817.38+003257.2, 2MASSW J0355419+225702, 2MASSW J0918382+213406, 2MASS
J13120707+3937445, SDSS J151603.03+025928.9, and SDSS J154849.02+172235.4.
They are available in the Table of Ultracool Fundamental Properties associated
with this paper (see §1). These objects were observed by the PS1 telescope
from 2009–2015 as part of the Pan-STARRS $3\pi$ Steradian Survey (Chambers et
al., 2016) in the $grizy$ filters. The $3\pi$ Survey was well-suited to
parallaxes as every survey region was observed at opposition as well as
evening and morning twilight. Astrometric calibration and automated
calculation of parallaxes and proper motions are described by Magnier et al.
(2020). The resulting astrometry is tied to the Gaia DR1 inertial system, with
a correction for the proper motion bias introduced by Galactic rotation and
solar motion.

Table 2: IRTF/SpeX Observing Log: Objects from the Literature

Object | UT Date | Slit (″) |  $<$Airmass$>$ |  $T_{\mathrm{int}}$ (s) |  $<$S/N$>(Y,J,H,H_{\mathrm{blue}},K,K_{\mathrm{blue}})$
---|---|---|---|---|---
T dwarfs
PSO J005.6302-06.8669 | 2012-01-20 | 0.8″ | 1.49 | 120.0 | 16, 37, 20, 20, 16, 24
PSO J052.2746+13.3754 | 2012-09-24 | 0.8″ | 1.24 | 420.0 | 18, 45, 23, 25, 9, 18
PSO J088.5709-00.1430 | 2012-09-24 | 0.8″ | 1.08 | 1440.0 | 34, 103, 33, 46, 11, 35
PSO J089.1751-09.4513 | 2011-04-20 | 0.5″ | 2.10 | 240.0 | 7, 18, 3, 5, 1, 3
PSO J117.0600-01.6779 | 2012-04-20 | 0.8″ | 1.16 | 240.0 | 15, 28, 6, 12, 2, 10
PSO J127.5648-11.1861 | 2012-04-20 | 0.8″ | 1.24 | 240.0 | 30, 66, 30, 34, 11, 27
PSO J136.5380-14.3267 | 2012-04-20 | 0.8″ | 1.35 | 240.0 | 17, 34, 15, 17, 6, 13
ULAS J114340.47+061358.9 | 2015-05-26 | 0.8″ | 1.27 | 1554.1 | 16, 27, 7, 10, 3, 7
ULAS J131610.13+031205.5 | 2012-04-20 | 0.8″ | 1.05 | 240.0 | 23, 50, 31, 32, 30, 36
WISE J133750.46+263648.6 | 2012-01-19 | 0.8″ | 1.03 | 120.0 | 28, 68, 15, 23, 4, 14
WISE J163236.47+032927.3 | 2012-08-10 | 0.8″ | 1.29 | 60.0 | 19, 42, 10, 19, 3, 9
WISE J163645.56-074325.1 | 2012-04-20 | 0.8″ | 1.13 | 240.0 | 28, 62, 26, 31, 9, 21
WISE J174303.71+421150.0 | 2012-07-05 | 0.8″ | 1.62 | 240.0 | 15, 37, 10, 14, 4, 10
WISE J230356.79+191432.9 | 2012-01-22 | 0.8″ | 1.64 | 120.0 | 12, 26, 8, 11, 3, 8
2MASS J23322678+1234530 | 2012-02-01 | 0.8″ | 1.95 | 120.0 | 14, 31, 18, 16, 18, 20
Red L Dwarfs
WISEA J004403.39+022810.6 | 2012-10-14 | 0.8″ | 1.06 | 1320.0 | 11, 31, 34, 29, 46, 45
2MASS J00550564+0134365 | 2012-10-25 | 0.8″ | 1.06 | 840.0 | 25, 62, 48, 43, 61, 61
WISEA J010202.11+035541.4 | 2012-10-25 | 0.8″ | 1.04 | 840.0 | 27, 52, 46, 41, 44, 50
WISE J023038.90-022554.0 | 2012-10-28 | 0.8″ | 1.22 | 1080.0 | 15, 34, 31, 28, 34, 37
SDSSp J030321.24-000938.2 | 2012-11-08 | 0.8″ | 1.06 | 840.0 | 51, 85, 48, 43, 42, 43
SDSS J084016.42+543002.1 | 2012-04-30 | 0.8″ | 1.34 | 120.0 | 12, 20, 12, 11, 14, 14
WISEA J095729.41+462413.5 | 2012-11-08 | 0.8″ | 1.31 | 840.0 | 25, 61, 57, 51, 69, 69
2MASS J09593276+4523309 | 2012-04-20 | 0.8″ | 1.18 | 180.0 | 25, 68, 67, 58, 86, 87
SDSS J102947.68+483412.2 | 2012-04-30 | 0.8″ | 1.63 | 120.0 | 8, 14, 7, 7, 8, 7
WISEA J114724.10-204021.3 | 2014-01-17 | 0.8″ | 1.33 | 1080.0 | 9, 28, 37, 32, 46, 48
2MASS J12594167+1001380 | 2012-01-20 | 0.8″ | 1.02 | 120.0 | 13, 42, 43, 37, 51, 52
Candidate Low-Gravity Objects
SIMP J01205253+1518277 | 2015-09-25 | 0.5″ | 1.01 | 1075.9 | 49, 64, 49, 47, 43, 43
SERC 296A | 2015-11-24 | 0.5″ | 1.94 | 1912.7 | 118, 133, 108, 104, 71, 80
2MASS J04362788-4114465 | 2016-02-03 | 0.5″ | 2.06 | 418.4 | 102, 117, 83, 76, 65, 71
2MASS J06431685-1843375 | 2015-01-20 | 0.5″ | 1.28 | 418.4 | 110, 158, 130, 153, 110, 117
2MASS J07410404-0359495 | 2016-03-25 | 0.5″ | 1.39 | 836.8 | 83, 113, 79, 71, 72, 73
WISEA J090258.99+670833.1 | 2016-02-05 | 0.5″ | 1.47 | 2271.3 | 24, 52, 62, 50, 75, 75
2MASS J21512797+3547206 | 2015-08-07 | 0.5″ | 1.07 | 266.9 | 101, 135, 118, 111, 99, 100
Hyades Members
Hya03 | 2014-11-17 | 0.8″ | 1.03 | 87.6 | 19, 35, 31, 33, 28, 27
Cl Melotte 25 DKJ 14784e | 2016-03-21 | 0.5″ | 1.25 | 1554.1 | 41, 52, 33, 29, 44, 41
Hya09 | 2016-02-03 | 0.5″ | 1.41 | 1315.0 | 12, 25, 20, 17, 28, 28
Wide Substellar/Planetary-Mass Companions
CFBDS J022644.65-062522.0 | 2011-12-02 | 0.8″ | 1.13 | 120.0 | 9, 20, 8, 9, 3, 5
2MASS J0249-0557 c | 2015-09-25 | 0.5″ | 1.11 | 1554.1 | 20, 35, 33, 28, 38, 37
HD 203030B | 2012-07-07 | 0.5″ | 1.08 | 240.0 | 8, 19, 22, 20, 20, 23
Solar Neighborhood Candidates
WISE J040137.21+284951.7 | 2013-12-11 | 0.8″ | 1.04 | 420.0 | 117, 261, 242, 234, 189, 198
2MASS J06143818+3950357 | 2011-10-14 | 0.8″ | 1.11 | 120.0 | 22, 44, 39, 38, 32, 38
DENIS-P J104731.1-181558 | 2012-04-30 | 0.8″ | 1.38 | 15.0 | 49, 98, 65, 58, 62, 63
2MASS J12022564-0629026 | 2015-05-19 | 0.8″ | 1.12 | 133.4 | 67, 89, 68, 62, 58, 59
WISEA J130729.56-055815.4 | 2011-03-31 | 0.5″ | 1.18 | 120.0 | 23, 53, 48, 45, 45, 48
M/L Transition Dwarfs
RG 0050-2722 | 2014-11-17 | 0.8″ | 1.47 | 107.0 | 48, 76, 77, 75, 52, 53
BR B0246-1703 | 2014-01-19 | 0.8″ | 1.50 | 35.0 | 37, 79, 68, 68, 40, 43
TVLM 831-165166 | 2014-10-15 | 0.8″ | 1.07 | 87.6 | 20, 26, 23, 24, 15, 16
LP 412-31 | 2013-11-06 | 0.8″ | 1.00 | 50.0 | 45, 104, 95, 94, 58, 61
2MASSI J0445538-304820 | 2016-02-03 | 0.8″ | 1.57 | 418.4 | 66, 103, 83, 75, 74, 76
LSR J0510+2713 | 2014-01-19 | 0.8″ | 1.42 | 11.0 | 24, 55, 52, 52, 28, 30
LHS 2065 | 2014-01-19 | 0.8″ | 1.11 | 21.0 | 50, 127, 124, 124, 77, 81
TVLM 262-111511 | 2014-01-18 | 0.8″ | 1.21 | 105.0 | 68, 125, 99, 97, 62, 65
TVLM 262-70502 | 2014-01-18 | 0.8″ | 1.22 | 70.0 | 46, 85, 71, 71, 42, 44
TVLM 263-71765 | 2014-01-18 | 0.8″ | 1.19 | 35.0 | 32, 70, 62, 62, 35, 37
TVLM 213-2005 | 2014-01-19 | 0.8″ | 1.20 | 35.0 | 17, 38, 32, 32, 18, 19
TVLM 513-42404A | 2014-07-29 | 0.8″ | 1.02 | 836.8 | 128, 131, 108, 93, 101, 102
TVLM 513-42404B | 2014-07-29 | 0.8″ | 1.03 | 1315.0 | 76, 100, 67, 55, 66, 69

Note. — The last column gives the median S/N of the spectrum for the standard
IR bandpasses and also for two wavelength regions free of methane absorption
($H_{\mathrm{blue}}$ = 1.49–1.63 µm and $K_{\mathrm{blue}}$ = 2.03–2.20 µm),
which are relevant for T-dwarf spectra.

Table 3: IRTF/SpeX Observing Log: New Discoveries

Object | UT Date | Slit (″) |  $<$Airmass$>$ |  $T_{\mathrm{int}}$ (s) |  $<$S/N$>(Y,J,H,H_{\mathrm{blue}},K,K_{\mathrm{blue}})$
---|---|---|---|---|---
T dwarfs
PSO J002.0878+52.0687 | 2012-01-22 | 0.8″ | 1.52 | 120.0 | 12, 32, 9, 13, 3, 11
PSO J013.7740+38.2804 | 2011-10-14 | 0.8″ | 1.13 | 120.0 | 16, 36, 23, 23, 15, 21
PSO J033.2936+20.4493 | 2011-10-14 | 0.8″ | 1.11 | 120.0 | 11, 24, 20, 21, 16, 21
PSO J098.2822-23.2845 | 2011-10-14 | 0.8″ | 1.37 | 120.0 | 17, 36, 23, 23, 18, 21
PSO J216.4707-06.2849 | 2012-04-20 | 0.8″ | 1.12 | 240.0 | 18, 45, 18, 22, 6, 14
PSO J227.1576-07.9608 | 2012-04-20 | 0.8″ | 1.18 | 240.0 | 12, 23, 15, 15, 7, 12
L Dwarfs
PSO J009.8334+58.5781 | 2015-11-28 | 0.5″ | 1.33 | 836.8 | 27, 56, 42, 39, 48, 53
PSO J036.9069+08.5025 | 2012-01-20 | 0.8″ | 1.16 | 120.0 | 9, 31, 23, 21, 19, 21
PSO J063.1519-02.8904 | 2012-01-22 | 0.8″ | 1.08 | 120.0 | 40, 79, 53, 47, 46, 49
PSO J139.5493+39.0380 | 2012-01-19 | 0.8″ | 1.08 | 120.0 | 20, 50, 40, 38, 34, 37
PSO J154.9622+04.8279 | 2012-01-19 | 0.8″ | 1.09 | 120.0 | 22, 52, 35, 33, 29, 32
PSO J162.0614+13.9756 | 2012-01-19 | 0.8″ | 1.01 | 120.0 | 27, 59, 46, 45, 36, 39
PSO J277.3873+26.0116 | 2015-07-15 | 0.8″ | 1.01 | 207.6 | 41, 69, 57, 54, 47, 49
PSO J348.5125+69.6598 | 2015-12-08 | 0.5″ | 1.55 | 1793.2 | 50, 87, 76, 69, 86, 92
Red L Dwarfs
PSO J179.2489+41.5121 | 2014-01-17 | 0.8″ | 1.08 | 1080.0 | 10, 25, 28, 25, 29, 33
PSO J255.0913+10.3263 | 2013-09-23 | 0.8″ | 1.59 | 240.0 | 7, 20, 19, 16, 30, 31
PSO J319.7576-13.0982 | 2013-09-22 | 0.8″ | 1.38 | 240.0 | 11, 38, 40, 33, 52, 54
PSO J349.3359+32.0532 | 2012-10-14 | 0.8″ | 1.03 | 2280.0 | 25, 53, 38, 34, 41, 40
Low-Gravity Objects
PSO J000.2794+16.6237 | 2015-11-03 | 0.5″ | 1.26 | 418.4 | 58, 60, 42, 41, 34, 34
PSO J032.2211+67.4179 | 2015-12-24 | 0.5″ | 1.48 | 2032.3 | 65, 75, 56, 51, 47, 49
PSO J037.4616+17.2761 | 2015-09-30 | 0.5″ | 1.05 | 537.9 | 35, 52, 53, 49, 43, 43
PSO J042.8405+35.3726 | 2013-09-23 | 0.5″ | 1.08 | 120.0 | 73, 132, 93, 86, 77, 77
PSO J069.4987+19.6346 | 2015-01-21 | 0.5″ | 1.29 | 629.2 | 83, 111, 93, 89, 77, 74
PSO J075.4182-17.8327 | 2015-11-02 | 0.5″ | 1.32 | 418.4 | 47, 58, 41, 38, 38, 38
PSO J080.3940+23.0999 | 2015-10-14 | 0.5″ | 1.03 | 207.6 | 42, 61, 51, 47, 42, 42
PSO J138.1887-06.9862 | 2015-12-24 | 0.5″ | 1.17 | 418.4 | 55, 73, 58, 54, 52, 52
PSO J153.7969-11.3631 | 2015-05-20 | 0.5″ | 1.28 | 2510.4 | 28, 39, 26, 23, 23, 24
PSO J178.1434-23.8603 | 2013-04-04 | 0.8″ | 1.39 | 2400.0 | 22, 39, 22, 20, 22, 22
PSO J210.6211-26.9300 | 2016-05-29 | 0.5″ | 1.47 | 1554.1 | 44, 54, 40, 36, 37, 37
PSO J235.3149+61.5071 | 2015-07-06 | 0.5″ | 1.34 | 1434.5 | 43, 59, 47, 42, 40, 41
PSO J260.1789+15.3091 | 2013-04-19 | 0.5″ | 1.01 | 180.0 | 24, 50, 32, 28, 36, 34
PSO J271.8353+46.9326 | 2014-08-01 | 0.5″ | 1.17 | 836.8 | 72, 127, 89, 83, 68, 78
PSO J279.6187+11.6036 | 2015-09-25 | 0.5″ | 1.02 | 418.4 | 87, 111, 87, 78, 75, 79
PSO J281.4296+04.2296 | 2015-07-15 | 0.8″ | 1.07 | 207.6 | 77, 107, 88, 81, 75, 75
PSO J299.5032+55.0722 | 2015-09-25 | 0.5″ | 1.23 | 537.9 | 55, 73, 57, 52, 47, 50
PSO J307.7981-06.6854 | 2015-07-04 | 0.5″ | 1.14 | 1075.9 | 19, 29, 26, 24, 22, 22
PSO J313.1067+32.9117 | 2015-12-08 | 0.5″ | 1.44 | 2032.3 | 36, 57, 41, 34, 43, 44
PSO J323.4300-07.5353 | 2015-08-07 | 0.5″ | 1.17 | 266.9 | 91, 108, 88, 83, 71, 72
PSO J345.2949+07.7665 | 2015-07-04 | 0.5″ | 1.06 | 1315.0 | 43, 70, 68, 62, 53, 55
M Dwarfs
PSO J028.0715-15.5717 | 2012-10-15 | 0.8″ | 1.28 | 2520.0 | 13, 30, 17, 16, 14, 14

Note. — The last column gives the median S/N of the spectrum for the standard
IR bandpasses and also for two wavelength regions free of methane absorption
($H_{\mathrm{blue}}$ = 1.49–1.63 µm and $K_{\mathrm{blue}}$ = 2.03–2.20 µm).

Figure 4: Example flux-calibrated SpeX prism spectrum generated in this work.
_Top:_ Filter transmission profiles for the PS1, 2MASS, MKO, UKIDSS, WISE, and
Spitzer/IRAC filters. _Bottom:_ Flux-calibrated SpeX spectrum (slit size =
0.5″) of 2MASSI J0335020+234235 in gray with the corresponding photometry in
the same colors as the filter transmission profiles, if available. Photometry
marked with triangles represent upper limits. The gray points represents model
synthesized photometry. The black curve correponds to the best-fit atmospheric
model. The inset figure shows the $\chi^{2}$ surface for the atmospheric model
fits to the SpeX spectrum in $T_{\mathrm{eff}}$-log $g$ space. The white star
marks the location of the model-fit with the smallest $\chi^{2}$. Yellow plus
signs mark the $T_{\mathrm{eff}}$-log $g$ values at which ATMO 2020 model
spectra were preferred over the BT-Settl model spectra based on their lower
$\chi^{2}$.

##  3 Ultracool Dwarf Spectral Energy Distributions

The central goal of the following analysis is to construct and directly
integrate the spectral energy distributions of ultracool dwarfs to obtain
their bolometric fluxes. Combining this calculated quantity with a distance
measurement yields the bolometric luminosity. This then enables the derivation
of other fundamental parameters such as masses, radii, effective temperatures,
and surface gravities using evolutionary models. In this section, we discuss
the spectra used in this work and our spectral flux-calibration procedures.

###  3.1 SpeX Prism Spectra

Since most ultracool dwarfs have faint apparent magnitudes, SpeX is often used
in its low-resolution ($R\sim$ 75–200) prism mode to acquire 0.8–2.5 $\mu$m
spectra using one of the three different slit sizes — $0\farcs 3\times
15\farcs 0$, $0\farcs 5\times 15\farcs 0$, and $0\farcs 8\times 15\farcs 0$.
This wavelength range is sensitive to several well-known absorption features
in ultracool dwarf spectra, e.g., $\mathrm{CH_{4}}$, $\mathrm{H_{2}O}$, FeH,
and VO. SpeX prism spectra are ideal for SED measurements as its broad
wavelength coverage captures the NIR region in a single continguous spectrum
thereby minimizing systematic uncertainties associated with combining multiple
spectra from differing spectral orders or instruments. Filippazzo et al.
(2015) find that bolometric luminosities derived from spectral integration are
relatively independent of the spectral resolution. Thus, the accuracy of our
results will not be hindered by the use of low-resolution spectra.

The large majority of spectra in our study were obtained from the SpeX prism
Library (Burgasser, 2014). A comprehensive list of references for our object
spectra is provided in the Table of Ultracool Fundamental Properties
associated with this paper (see §1). Flux measurements made at the wavelength
extremes of the spectra generally have low S/N and can be unreliable. To
mitigate this, we apply cuts to the spectra on the blue end at $0.85\;\mu$m
and on the red end at $2.45\;\mu$m. The values are chosen to avoid exclusion
of wavelengths included in the filter transmission profiles of the PS1 $y$
band on the blue end and the UKIDSS $K$-band on the red end.

###  3.2 Absolute Flux Calibration of SpeX Prism Spectra

To derive the bolometric flux from SED integration, we need to first calibrate
each object’s SpeX spectrum to its observed photometry. We begin with the
highest available signal-to-noise ratio (S/N) SpeX spectrum of each object
($f_{\lambda,\mathrm{obs}}$) and the photometry with similar wavelength
coverage (PS1 $y$, 2MASS $JH$$K_{S}$ , MKO $JHK$, and UKIDSS $JHK$). Each
photometric point is converted to a flux density measurement
($f_{\lambda,\mathrm{phot}}$) using the corresponding filter’s zero point flux
density. For several objects, we find that photometric measurements for some
filters have formal uncertainties below the instrumental noise floor (PS1:
Magnier et al., 2020). To avoid underestimating the uncertainties associated
with flux calibration of the spectra, we adopt a uniform noise floor of 0.01
mag for all photometric measurements.

Figure 5: Normalized and vertically offset log $g$ = 5.0 dex BT-Settl and ATMO
2020 model atmospheric spectra, degraded to the wavelength-dependent
resolution of the 0.3″SpeX prism slit. This sequence spans the effective
temperature range from 3000 K to 600 K and approximately corresponds to the
ultracool dwarf spectral types M (top), L (middle), and T (bottom).

Next, we synthesize photometry from the SpeX spectrum of the object for all
the filters discussed previously. The filter transmission profile
$T_{\lambda}$ is retrieved from the SVO Filter Profile
Service222http://svo2.cab.inta-csic.es/theory/fps/index.php and the SpeX
spectrum is interpolated using cubic spline interpolation and mapped to the
wavelength grid of the transmission profile. We then calculate the synthetic
photometry $f_{\lambda,\mathrm{syn}}$ using,

$f_{\lambda,\mathrm{syn}}=\frac{\int
f_{\lambda,\mathrm{obs}}\;T_{\lambda}\;d\lambda}{\int T_{\lambda}\;d\lambda},$
(1)

where the limits of the integration are $\lambda_{\mathrm{min}}$ and
$\lambda_{\mathrm{max}}$ of the wavelength grid of the filter transmission
profile.

Combining the photometric flux densities and the synthetic flux densities
derived previously, we can compute a scale factor $k_{i}$ corresponding to
each filter ($i$) as,

$k_{i}=\left(\frac{f_{\lambda,\mathrm{phot}}}{f_{\lambda,\mathrm{syn}}}\right)_{i},$
(2) $\delta k_{i}=\left(\frac{\delta
f_{\lambda,\mathrm{phot}}}{f_{\lambda,\mathrm{syn}}}\right)_{i},$ (3)

where $\delta k_{i}$ is the associated uncertainty. We note here that since
the synthetic flux density is an integration over the spectrum’s many
wavelength points corresponding to the filter transmission profile, its
uncertainty can be neglected in comparison to the uncertainty in the
photometric flux density. We derive the scale factor $k$ for the calibration
of the SpeX spectrum by obtaining the weighted average of the individual scale
factors $k_{i}$. A 1% uncertainty floor is adopted for the weighted average
scale factor $k$ to avoid downweighting the photometric noise floor applied
earlier. Finally, the calibrated spectrum $f_{\lambda,\mathrm{cal}}$ (Figure
4) is found as $f_{\lambda,\mathrm{cal}}=f_{\lambda,\mathrm{obs}}\cdot k$.

##  4 Atmospheric Model Fitting

One of the challenges associated with deriving the bolometric luminosity by
SED integration is the lack of optical and mid-infrared (MIR) spectra for most
known ultracool dwarfs. However, optical and MIR photometry is well-measured
for many of these objects by the Pan-STARRS1 and WISE surveys, respectively.
In a few cases, individual objects also have MIR photometry from Spitzer/IRAC
Channels 1 and 2 (Fazio et al., 2004). We fit the SpeX prism spectrum and the
optical and/or MIR photometry with synthetic atmospheric grids produced by the
BT-Settl (CIFIST2011/2015; Allard et al., 2012; Baraffe et al., 2015) and ATMO
2020 (Phillips et al., 2020) models (Figure 5). Directly integrating the area
under the best-fit atmospheric model spectrum in wavelength regions not
covered by the SpeX prism spectrum then yields the associated optical and MIR
contributions to the bolometric luminosity.

BT-Settl (CIFIST2011/2015)333http://svo2.cab.inta-
csic.es/theory/newov2/index.php?models=bt-settl-cifist incorporates the role
of clouds and dust in its atmospheres, and thus resulting spectra, by self-
consistently modeling their formation and sedimentation (Allard et al., 2012)
and accounting for the condensation of nearly 200 liquid and solid species in
the atmospheric equation of state. This makes BT-Settl appropriate for late-M
and L dwarfs, which have cloudy photospheres. Additionally, it includes
linelists for several important molecules such as $\mathrm{H_{2}O}$, CO2, CaH,
FeH, CrH, and TiH. Each model is calculated at solar metallicity ($Z=0$ dex),
as defined by Caffau et al. (2011). To ensure a uniformly spaced grid for
subsequent uncertainty analysis (Section 5), we omit the 50 K interval model
spectra. Given that all our objects have spectral types $\geq$M6, we only
consider model spectra upto a temperature of 3500 K to reduce computation time
(the grid extends to 7000 K). Thus, the final BT-Settl grid adopted in our
fitting procedure spans effective temperatures ($T_{\mathrm{eff}}$) of
1200–3500 K in steps of 100 K and surface gravities ($\log g$) of 2.5–5.5 dex
in steps of 0.5 dex.

ATMO 2020 is one of the latest cloudless atmospheric model grid geared towards
T and Y spectral type ultracool dwarfs (Phillips et al., 2020). It brings
numerous improvements to the evolutionary and spectroscopic modeling of this
coldest subset of brown dwarfs. ATMO 2020 makes use of the new hydrogen and
helium equation of state from Chabrier et al. (2019) in its interior structure
model. It includes updated line lists for the most prominent molecular opacity
sources such CH4 and NH3 from the ExoMol group (Tennyson & Yurchenko, 2018)
and significantly more transitions required to accurately model the
atmospheres of brown dwarfs. Furthermore, ATMO 2020 provides a more detailed
and careful treatment of the Na and K resonant lines, which play an important
role in shaping the visible and red-optical spectra of ultracool dwarfs.
Similar to BT-Settl, each grid point in the model is calculated at solar
metallicity ($Z=0$ dex). In this work, we make use of the weak non-equilibrium
chemistry model set (NEQ Weak). Numerous observations have highlighted the
importance of non-equilibrium chemistry in the spectra of cool brown dwarfs
due to vertical mixing in their atmospheres (e.g. Noll et al., 1997; Saumon et
al., 2000; Geballe et al., 2009; Leggett et al., 2015). Phillips et al. (2020)
model non-equilibrium chemistry by consistently coupling the relaxation scheme
of Tsai et al. (2018) to their 1D atmosphere code ATMO while considering the
non-equilibrium abundances of H2O, CO, CO2, CH4, N2, and NH3. Similar to the
approach with BT-Settl, to ensure a uniformly spaced rectangular grid for
subsequent uncertainty analysis (Section 5), we omit the 50 K interval model
spectra. The final ATMO 2020 grid in our fitting procedure spans
$T_{\mathrm{eff}}$ of 500–1800 K in steps of 100 K and $\log g$ of 2.5–5.5 dex
in steps of 0.5 dex. We note that the NEQ Weak ATMO 2020 grid does not provide
a spectrum for the $T_{\mathrm{eff}}=$ 1100 K and $\log g=$ 3.5 dex grid point
due to lack of convergence in the model generation. As a substitute, we use an
interpolated model spectrum at this grid point (Mark Phillips; private
communication). We refer the reader to Section 2 of Lueber et al. (2023) for a
summary of brown dwarf atmospheric model grids not used in this work.

Figure 6: Summary of the bolometric luminosity uncertainty analysis in this
work. The model-included total uncertainty (scale factor uncertainty +
spectral data points uncertainties + model-contributed uncertainty + distance
uncertainty) is plotted on the y-axis against the model-excluded total
uncertainty (scale factor uncertainty + spectral data points uncertainties +
distance uncertainty). Each point represents one of the 1054 objects in our
sample and is colored based on the fractional uncertainty in the distance
measurement for that object (color bar at right). Four dashed lines of slopes
1, 2, 5, and 10 plot differing multiplicative relations between the model-
included total uncertainty and the model-excluded total uncertainty. The inset
figure zooms in near the tail end of the points close to the origin. A noise
floor of $\approx$0.015 dex is identified and marked with a dashed gray line.
The model-contributed uncertainty component dominates for low fractional
distance uncertainty objects, thereby limiting the gains possible with our
high precision parallaxes.

We begin by converting the PS1 optical (excluding $y$ band since it is
included in the wavelength range of the SpeX prism spectra) and WISE and
Spitzer/IRAC MIR photometry to flux density measurements using the zero points
for the corresponding filters. Model spectra are typically provided at high
spectral resolution and are first degraded to the non-linear spectral
resolution of the SpeX prism data (R$\sim$75–200) corresponding to the three
different slit sizes of the SpeX spectrograph (Hurt et al., AAS Journals,
submitted). Such a resolution degradation is, however, not required for
fitting the optical and/or MIR photometry. Keeping this in mind, we apply the
following method to each model spectrum.

First, we map the degraded model spectra to the wavelength grid of the SpeX
prism data using cubic spline interpolation. Second, we synthesize the PS1
$griz$, WISE $W1$, $W2$, $W3$, and $W4$, and Spitzer/IRAC Channel 1 and
Channel 2 photometric flux densities using the full resolution model spectra,
following Equation 1. Third, we determine the scale factor $k_{\mathrm{fit}}$
for a given model spectrum that would minimize the combined $\chi^{2}$ of the
degraded model spectrum and the synthesized photometriy fit to the observed
SED. $k_{\mathrm{fit}}$ is given by

$k_{\mathrm{fit}}=\frac{\sum(f_{i}F_{k,i}/\sigma_{f,i})}{\sum(f_{i}^{2}/\sigma_{f,i})},$
(4)

where $f_{i}$ represent the individual flux-calibrated SpeX prism flux
densities or the individual PS1/WISE/IRAC bandpass photometric flux densities,
$F_{k,i}$ represents the model spectrum or the synthesized PS1/WISE/IRAC
photometry, and $\sigma_{f,i}$ represent the uncertainty in the observed SpeX
spectrum or the PS1/WISE/IRAC photometry. To mitigate telluric contamination
due to low atmospheric transmission in the water bands, we exclude spectral
data points in the 1.355–1.415 $\mu$m and the 1.830–1.930 $\mu$m wavelength
ranges from the above fitting procedure. Additionally, to avoid biasing the
model fits, we exclude any spectral data points with unphysical S/N $<-3$.
This process is repeated for all available model spectra and the model
spectrum that yields the smallest $\chi^{2}$ value is chosen as the best-fit
model spectrum. Figure 4 shows an example flux-calibrated SED with the best-
fit atmospheric model spectrum overlaid, as well as the associated $\chi^{2}$
surface obtained from the fitting procedures.

Figure 7: Percentage of bolometric flux contained in different wavelength
regions as a function of effective temperature, computed by integrating BT-
Settl (1200–3000 K) and ATMO 2020 (600–1400 K) model spectra with
$\mathrm{log}\;g=$ 4.0–5.5. The shaded background areas roughly mark the
temperature range corresponding to M, L, and T dwarfs. The NIR contribution
dominates for the vast majority of spectral types. The SED tail flux (20–2000
$\mu$m) is a very small percentage of the bolometric flux across all spectral
types. The MIR contribution is most important for objects cooler than
$\sim$1800 K. Figure 8: _Left:_ Normalized distributions of the best-fit
$\chi^{2}_{\nu}$ values from the model atmosphere fitting for different
spectral types (top) and different youth categories (bottom). Young moving
group members and low gravity sources are designated as young objects. The bin
size is 12.5. Both distributions are skewed significantly towards smaller
$\chi^{2}_{\nu}$ values, indicating good quality model fits for the vast
majority of objects in the sample. Model fits to T dwarfs are better than
those to L dwarfs, which are in turn better than those to M dwarfs. Field
objects yield slightly better fits than young objects. We flag objects with
$\chi^{2}_{\nu}>$ 50 in the Table of Ultracool Fundamental Properties (see
§5.2). _Right:_ Example SpeX prism spectra and the corresponding best-fit
model spectra for six objects in the sample with gradually increasing
$\chi^{2}_{\nu}$ values. The S/N of the spectra are similar ($\approx$70–100).
This is a visual demonstration of the quality of the fits for different
$\chi^{2}_{\nu}$ values in the distribution. Figure 9: _Left:_ Comparison of
bolometric luminosities calculated in this work with those in Filippazzo et
al. (2015) for objects common to the two samples. The dashed black line
represents a 1:1 relation. Objects with disagreements ($>$2$\sigma$
discrepancy) in the two values are marked with a black-bordered star. _Right:_
Logarithmic difference in parallax measurements used in the bolometric
luminosity calculation vs logarithmic difference in bolometric luminosity
measurements between this work and Filippazzo et al. (2015) for the
_discrepant objects_ in the left panel. The two gray dashed lines correspond
to zero luminosity and parallax difference. The black dashed line corresponds
to the expected relation between $\Delta$log $\pi$ and $\Delta$log
($L_{\mathrm{bol}}/L_{\odot}$). Objects marked with a square correspond to the
discrepant objects whose luminosity difference is consistent with the
theoretically expected parallax difference within 1.5$\sigma$. Objects marked
with a triangle correspond to the discrepant objects whose luminosity
difference is _not_ consistent with the theoretically expected parallax
difference within 1.5$\sigma$. Observed differences in bolometric luminosity
can thus be explained by changes in parallax measurements for the majority of
discrepant objects.

##  5 Bolometric Luminosities

Given a flux-calibrated spectrum and best-fit atmospheric model spectrum, we
computed the bolometric flux of each object by integration of the SED. We take
care to correctly treat correlated and uncorrelated uncertainty contributions
to the calculated bolometric flux.

###  5.1 Calculation and Uncertainty Analysis

We integrate the area under the SpeX spectrum to obtain the NIR contribution
to the bolometric flux. The sum of the optical and MIR contribution is
obtained by integrating the full resolution best-fit model spectrum at
wavelengths outside those of the SpeX spectrum. The full wavelength range of
our SED integration is $\approx$ 0.1–2000 $\mu$m. Adding the two components,
we obtain a bolometric flux measurement ($f_{\mathrm{bol}}$). The uncertainty
in $f_{\mathrm{bol}}$ is a combination of the uncertainty from the NIR
contribution and the uncertainty from the summed optical and MIR contributions
to the bolometric flux.

The uncertainty in the NIR flux contribution is defined as the calibration
uncertainty and consists of the uncertainty in the scale factor
$k_{\mathrm{fit}}$ for the entire SpeX spectrum and the measurement
uncertainties for each individual spectral data point. We note that it only
comes from the NIR contribution since we use the zero uncertainty model
spectrum to estimate the optical and MIR contributions. The calibration
uncertainty is determined using Monte Carlo techniques. We generate $10^{4}$
different SpeX spectra by drawing the scaling factor and the value for each
spectral data point from their corresponding posterior distributions. Each
generated spectrum is integrated as described previously to obtain the NIR
flux contribution. The standard deviation of these measurements yields the
calibration uncertainty.

The uncertainty in the MIR and optical flux contribution is defined as the
model-contributed uncertainty ($\sigma_{\mathrm{mod}}$), which arises because
the atmospheric model fit to the SpeX spectrum and optical/MIR photometry is
not perfect. To calculate the model-contributed uncertainty, we re-compute the
sum of the optical and MIR flux contributions to the bolometric flux using the
four model spectra adjacent on the model grid in $T_{\mathrm{eff}}$ and
$\mathrm{log}\;g$ values to the best-fit model spectrum. The standard
deviation of this set of values gives the model-contributed uncertainty
(Cushing et al., 2008; Stephens et al., 2009).

The total uncertainty in the bolometric flux ($\delta f_{\mathrm{bol}}$) is
then obtained as the quadrature sum of the calibration and model-contributed
uncertainty. Given a bolometric flux $f_{\mathrm{bol}}\pm\delta
f_{\mathrm{bol}}$ and distance measurement $d\pm\delta d$, we can obtain the
bolometric luminosity $L_{\mathrm{bol}}$ and the associated uncertainty
$\delta L_{\mathrm{bol}}$ of the object as follows,

$L_{\mathrm{bol}}=f_{\mathrm{bol}}\cdot 4\pi d^{2},$ (5) $\frac{\delta
L_{\mathrm{bol}}}{L_{\mathrm{bol}}}=\sqrt{\left(\frac{\delta
f_{\mathrm{bol}}}{f_{\mathrm{bol}}}\right)^{2}+\left(2\cdot\frac{\delta
d}{d}\right)^{2}}.$ (6)

For 839 objects in our sample, the distance is given by a parallax
measurement. For the remaining 215 objects, we use photometric distances
computed using absolute magnitude-spectral type relations for 2MASS $J$$K_{S}$
, MKO $JK$, and WISE $W2$ bands. We use the polynomials from Dupuy & Liu
(2012) for field objects with 2MASS $J$$K_{S}$ , MKO $JK$, and ALLWISE $W2$
photometry; Feeser et al. (2022) for field objects with CatWISE $W2$
photometry; or this work (Appendix A) for young objects with 2MASS $J$$K_{S}$
, MKO $JK$, and ALLWISE/CatWISE $W2$ photometry. The luminosities are reported
as $\mathrm{log}(L_{\mathrm{bol}}/L_{\odot})$ where $L_{\odot}=3.828\times
10^{26}$ W (Mamajek et al., 2015).

We examine the effect of the model-contributed uncertainty on the total
uncertainty in our $L_{\mathrm{bol}}$ measurements by comparing this total
$L_{\mathrm{bol}}$ uncertainty against the total $L_{\mathrm{bol}}$
uncertainty computed assuming zero model-contributed uncertainty. This is
presented in Figure 6, where we have the defined the former quantity as the
model-included total uncertainty and the latter quantity as the model-excluded
total uncertainty. We find that the model-included total uncertainty is
significantly larger than the model-excluded total uncertainty for low
fractional distance uncertainty objects. A reduction in the fractional
distance uncertainty from high precision parallax programs should have
significantly lowered the total uncertainty in the bolometric luminosity (as
seen for the $\sigma_{\mathrm{mod}}=0$ assumption case), but a non-zero
$\sigma_{\mathrm{mod}}$ component results in an uncertainty floor at
$\approx$0.015 dex. Thus, the central limiting factor for these cases is the
presence of the model-contributed uncertainty. Obtaining high S/N spectra of
ultracool dwarfs in the optical and MIR wavelengths will be crucial to
lowering the identified noise floor.

Table 4: Objects with $>$2$\sigma$ Disagreement between This Work’s and
Filippazzo et al. (2015)’s Bolometric Luminosity Measurements

UCS Name | F+15 Name | $\pi$ |  $\pi$ (F+15) | log($L_{bol}$/$L_{\odot}$) | log($L_{bol}$/$L_{\odot}$) (F+15)
---|---|---|---|---|---
|  | (mas) | (mas) | (dex) | (dex)
2MASS J00332386-1521309 | 0033-1521 | $43.50\pm 0.80$ | $24.8\pm 2.5$ | $-4.20\pm 0.03$ | $-3.61\pm 0.09$
2MASS J00345157+0523050 | 0034+0523 | $118.80\pm 2.70$ | $105.4\pm 7.5$ | $-5.41\pm 0.06$ | $-5.27\pm 0.06$
2MASS J03231002-4631237 | 0323-4631 | $23.40\pm 0.70$ | $17.0\pm 3.0$ | $-3.66\pm 0.03$ | $-3.34\pm 0.15$
2MASS J06085283-2753583 | 0608-2753 | $22.62\pm 0.16$ | $32.0\pm 3.6$ | $-3.06\pm 0.02$ | $-3.34\pm 0.10$
2MASS J09490860-1545485 | 0949-1545 | $42.20\pm 2.70$ | $55.3\pm 6.6$ | $-4.62\pm 0.10$ | $-4.86\pm 0.10$
2MASS J13595510-4034582 | 1359-4034 | $47.35\pm 0.20$ | $64.2\pm 5.5$ | $-3.77\pm 0.02$ | $-4.0\pm 0.07$
2MASSI J0439010-235308 | 0439-2353 | $80.70\pm 0.40$ | $110.4\pm 4.0$ | $-4.37\pm 0.03$ | $-4.61\pm 0.03$
2MASSI J0445538-304820 | 0445-3048 | $61.94\pm 0.15$ | $78.5\pm 4.9$ | $-3.79\pm 0.02$ | $-3.99\pm 0.05$
2MASSI J0825196+211552∗ | 0825+2115 | $92.60\pm 0.80$ | $93.8\pm 1.0$ | $-4.63\pm 0.02$ | $-4.53\pm 0.01$
2MASSI J0835425-081923 | 0835-0819 | $138.31\pm 0.21$ | $117.3\pm 11.2$ | $-4.24\pm 0.02$ | $-4.05\pm 0.08$
2MASSI J0847287-153237 | 0847-1532 | $57.51\pm 0.21$ | $76.5\pm 3.5$ | $-3.75\pm 0.02$ | $-4.01\pm 0.04$
2MASSI J1546291-332511∗ | 1546-3325 | $88.00\pm 1.90$ | $88.0\pm 1.9$ | $-5.25\pm 0.04$ | $-5.07\pm 0.02$
2MASSI J2104149-103736 | 2104-1037 | $58.02\pm 0.27$ | $53.0\pm 1.71$ | $-3.91\pm 0.02$ | $-3.81\pm 0.03$
2MASSW J1155395-372735 | 1155-3727 | $84.73\pm 0.15$ | $104.4\pm 4.7$ | $-3.85\pm 0.02$ | $-4.01\pm 0.04$
2MASSW J1506544+132106∗ | 1506+1321 | $85.43\pm 0.19$ | $70.92\pm 2.5$ | $-4.02\pm 0.02$ | $-3.8\pm 0.03$
2MASSW J1515008+484742∗ | 1515+4847 | $101.96\pm 0.26$ | $95.24\pm 2.0$ | $-4.47\pm 0.02$ | $-4.32\pm 0.02$
2MASSW J1552591+294849 | 1552+2948 | $48.90\pm 0.16$ | $47.7\pm 0.9$ | $-3.60\pm 0.02$ | $-3.55\pm 0.02$
2MASSW J2208136+292121 | 2208+2921 | $25.10\pm 1.60$ | $21.2\pm 0.7$ | $-3.93\pm 0.06$ | $-3.7\pm 0.03$
BR B0246-1703 | 0248-1651 | $44.62\pm 0.13$ | $61.6\pm 5.4$ | $-3.22\pm 0.02$ | $-3.46\pm 0.08$
DENIS-P J0751164-253043 | 0751-2530 | $56.42\pm 0.12$ | $59.15\pm 0.84$ | $-3.68\pm 0.02$ | $-3.73\pm 0.01$
DENIS-P J142527.97-365023.4∗ | 1425-3650 | $84.40\pm 0.30$ | $86.45\pm 0.83$ | $-4.08\pm 0.02$ | $-4.04\pm 0.01$
ESO 207-61 | 0707-4900 | $41.69\pm 0.11$ | $54.1\pm 4.5$ | $-3.45\pm 0.02$ | $-3.66\pm 0.07$
G 196-3B | 1004+5022 | $46.20\pm 0.50$ | $41.0\pm 4.1$ | $-3.95\pm 0.02$ | $-3.75\pm 0.09$
LHS 1604∗ | 0351-0052 | $68.12\pm 0.07$ | $68.1\pm 1.9$ | $-3.08\pm 0.02$ | $-3.02\pm 0.02$
LHS 2924∗ | 1428+3310 | $91.16\pm 0.10$ | $90.8\pm 1.3$ | $-3.60\pm 0.02$ | $-3.54\pm 0.01$
LHS 3003 | 1456-2809 | $141.79\pm 0.06$ | $159.2\pm 5.1$ | $-3.21\pm 0.02$ | $-3.32\pm 0.03$
LP 412-31∗ | 0320+1854 | $68.27\pm 0.08$ | $68.9\pm 0.6$ | $-3.29\pm 0.02$ | $-3.25\pm 0.01$
SDSS J000013.54+255418.6∗ | 0000+2554 | $70.80\pm 1.90$ | $70.8\pm 1.9$ | $-4.78\pm 0.04$ | $-4.68\pm 0.02$
TVLM 831-161058 | 0251+0047 | $28.16\pm 0.18$ | $20.5\pm 2.2$ | $-3.32\pm 0.02$ | $-3.02\pm 0.09$
UGPS J072227.51-054031.2∗ | 0722-0540 | $242.80\pm 2.40$ | $242.8\pm 2.4$ | $-6.38\pm 0.11$ | $-6.02\pm 0.02$
WISEPA J164715.59+563208.2 | 1647+5632 | $42.70\pm 2.10$ | $116.0\pm 29.0$ | $-4.56\pm 0.05$ | $-5.31\pm 0.22$
Wolf 359∗ | 1056+0700 | $415.18\pm 0.07$ | $419.1\pm 2.1$ | $-2.97\pm 0.02$ | $-3.03\pm 0.00$

Note. — For objects marked with an asterisk (∗), the difference in parallax
measurements is not consistent with the theoretical $\Delta$log
($L_{\mathrm{bol}}/L_{\odot}$) = $-2\cdot\Delta$log $\pi$ scaling relation
within 1.5$\sigma$. For the remaining objects, the difference in parallax
measurements is consistent with the theoretical scaling relation within
1.5$\sigma$. See §5.3 for more details.

###  5.2 Quality and Reliability of Atmospheric Model Fits

In this work, we estimate the flux contribution from the optical (0.20–0.85
$\mu$m) and MIR (2.5–2000.0 $\mu$m) wavelengths using atmospheric model
spectra. Thus, it is important to characterize the quality and reliability of
atmospheric model spectrum fits to the objects’ NIR SpeX prism spectra and
associated PS1 optical and WISE/Spitzer MIR photometry.

The NIR flux contribution to an object’s bolometric flux is obtained by
directly integrating the observed SpeX prism spectrum across a wavelength
range of $\approx$0.85–2.5 $\mu$m. To gauge the percentage of flux missing
after this step, we plot the average percentage of the bolometric flux
contained in three defined regions — the optical region (0.20–0.85 $\mu$m),
the MIR region (2.5–20.0 $\mu$m), and the SED tail (20.0–2000.0 $\mu$m) — as a
function of effective temperature in Figure 7. These values are obtained by
integrating the BT-Settl and ATMO 2020 model spectra for each temperature in
the defined region. We average flux percentage values computed across 4.0–5.5
dex $\mathrm{log}\;g$ spectra (most typical range of surface gravities for
ultracool dwarfs) at a fixed temperature. It is observed that the SED tail
contribution to the bolometric flux is negligible ($<$4%) across all spectral
types and does not affect our luminosity calculations. This quantity is
important to check since the longest wavelength WISE photometry ($W4$) point
that constrains our fits lies near 20 $\mu$m. The MIR flux contribution to the
bolometric flux is between $\approx$14–17% for M dwarfs, $\approx$16–45% for L
dwarfs, and $\approx$25–60% for T dwarfs. The optical flux contribution is
only significant for M dwarfs ($\approx$3–10%). Thus, while the NIR
contribution dominates for the majority of objects, the MIR contribution does
comprise a significant percentage of the bolometric flux, especially for the
cooler $\gtrsim$L5 ultracool dwarfs. These are the objects for which the model
fits must be examined in greater detail.

We use the reduced chi-square $\chi^{2}_{\nu}$ as our figure of merit to
quantify the goodness of our atmospheric model spectrum fit to the observed
SED (SpeX spectrum + optical and MIR photometry). Figure 8 presents the
distribution of our best-fit $\chi^{2}_{\nu}$ values for different spectral
classes and youth categories. The distributions are significantly skewed
towards smaller $\chi^{2}_{\nu}$ indicating good model fits to the data. The
model fits have better (lower) $\chi^{2}_{\nu}$ values for T dwarfs, followed
by L and M dwarfs respectively. This is reassuring given that good quality
model fits to the SEDs of the cooler ultracool dwarfs in our sample are
important for accurate estimation of a signficant percentage of their total
bolometric flux. The distributions for field and young (young moving group
members/low gravity sources) objects are roughly similar with slightly better
(lower) $\chi^{2}_{\nu}$ values for field objects. Figure 8 provides examples
of atmospheric model spectra fits to the observed SpeX prism spectra for six
objects in our sample at different $\chi^{2}_{\nu}$ values.

Based on the shape of the $\chi^{2}_{\nu}$ distributions and visual inspection
of the atmospheric model fits, we flag objects with $\chi^{2}_{\nu}$ values
greater than 50 in our Table of Ultracool Fundamental Properties (see §1) to
indicate possibly lower reliability than the other fits. A total of 171
objects ($\sim$16% of the sample) are flagged: 80 M dwarfs (39% of M dwarfs in
our sample), 83 L dwarfs (13.4% of the L dwarfs in our sample), and 8 T dwarfs
(3.5% of the T dwarfs in our sample). For these objects, the relatively poorer
quality of the model spectrum fit may have impacted the estimated optical and
MIR flux contribution, and hence the calculated bolometric luminosity. We note
that the discrepancy is likely to be small for the majority of objects in this
subset since the MIR contribution to the bolometric flux is significant,
compared to the NIR contribution, primarily for $\gtrsim$L5 dwarfs (only 13
objects in the flagged subset).

###  5.3 Comparison with Filippazzo et al. (2015)

Figure 9 compares our $L_{\mathrm{bol}}$ values with those in Filippazzo et
al. (2015) for objects common to the two samples (N = 130) and characterizes
measurement differences as a function of the difference in parallaxes used in
the calculations. We find that our results are consistent ($<$2$\sigma$
difference) with Filippazzo et al. (2015) for the majority of objects (N =
98). $>$2$\sigma$ discrepancy in the bolometric luminosity is observed for 32
objects (Figure 9, left panel). The logarithmic difference in bolometric
luminosity ($\Delta$log [$L_{\mathrm{bol}}/L_{\odot}$]) is related to the
logarithmic difference in parallax ($\Delta$log $\pi$) as $\Delta$log
($L_{\mathrm{bol}}/L_{\odot}$) = $-2\cdot\Delta$log $\pi$. We can use this
scaling relationship to determine if the source of the discrepancy in
bolometric luminosity is a revised parallax measurement. The differences in
parallax measurements between our work and Filippazzo et al. (2015) can
explain the differences in bolometric luminosity measurements (within
1.5$\sigma$) for 21 of the 32 discrepant objects (Figure 9, right panel). For
the 11 discrepant objects (2MASSI J0825196+211552, 2MASSI J1546291-332511,
2MASSW J1506544+132106, 2MASSW J1515008+484742, DENIS-P J142527.97-365023.4,
LHS 1604, LHS 2924, LP 412-31, SDSS J000013.54+255418.6, UGPS
J072227.51-054031.2, and Wolf 359) where the difference in parallax
measurements is not consistent with the theoretical scaling relation within
1.5$\sigma$, the minor differences observed in $L_{\mathrm{bol}}$ values may
be attributed to our use of atmospheric models to evaluate the optical and MIR
flux contributions to the bolometric flux in contrast to the use of the
object’s optical and (in some cases) MIR spectra (combined with linear
interpolation across SED gaps), as in Filippazzo et al. (2015). A complete
list of the 32 discrepant objects is provided in Table 4. Overall, our work
increases the sample size of $L_{\mathrm{bol}}$ measurements by a factor of
$\sim$5 (Figure 10).

###  5.4 Polynomial Relations

We derive empirical relationships between the bolometric luminosity and
spectral type/absolute magnitude for field and young objects by performing
inverse variance-weighted polynomial fits to the data. The polynomial fit is
carried out for increasing orders and a lack-of-fit test (_F_ -test) is
conducted for the best-fit coefficients at each order to determine if a
higher-order fit is necessary to explain the data. We reject the null-
hypothesis (higher-order fit required) for $p>0.05$. A root-mean-square (rms)
value is determined to quantitatively characterize the scatter about the best-
fit relation. We note here that the _F_ -test does not always yield a
definitive answer for the order of the polynomial based on the $p>0.05$
rejection criterion. For such cases, we adopt the lowest order after which $p$
does not change significantly (factor of $\sim$5–10) as the best-fit
polynomial and validate it by visual inspection.

Figure 11 summarizes our $L_{\mathrm{bol}}$ measurements as a function of
spectral type and presents the corresponding sixth-order polynomial relation
(Table 5). For M dwarfs (SpT $=6-9.5$), young objects show considerable
scatter from the relation, as was observed by Filippazzo et al. (2015).
However, for later types, young objects do not scatter significantly different
than the field-age objects, an observation not previously confirmed given the
small sample size of ultracool dwarf measurements available. While the
relation does not exhibit significant differences compared to Filippazzo et
al. (2015), our larger sample size enables a robust determination of the rms
scatter in the polynomial relation.

Figure 10: A comparison of the number of SED-integrated $L_{\mathrm{bol}}$
measurements available as a function of spectral type (SpT) before (Filippazzo
et al., 2015) and after this work. The bin size is 1 SpT. We increase the
sample size of $L_{\mathrm{bol}}$ values by a factor of $\sim$5 and contribute
a significant number of measurements in the L/T transition region. Figure 11:
Bolometric luminosities derived for our sample of 1054 ultracool dwarfs as a
function of their spectral type (SpT). Field objects, low gravity ultracool
dwarfs, and young moving group members are marked by gray, cyan, and blue
points respectively. Random scatter in the x direction with amplitude 0.3 SpT
is applied to avoid overlapping points. Due to the large number of points,
uncertainties in individual measurements are not shown for figure readability.
The black line is the best sixth-order weighted polynomial fit to the data and
the shaded blue region represents the rms scatter about the fit. The
polynomial relation from Zhang et al. (2020, Z+20) derived using bolometric
luminosities of 198 field-age ultracool dwarfs from Filippazzo et al. (2015,
F+15) is plotted as a green solid line for comparison. Figure 12: Bolometric
luminosity vs absolute magnitude in the MKO $JHK$ bands. Symbols are the same
as in Figure 11. The black solid and dashed lines are the best-fit polynomial
relations for the field and young objects, respectively. The shaded blue and
orange regions represent the rms scatter about the field and young object
fits, respectively. Figure 13: Same as Figure 12 for PS1 $zy$ and WISE $W1$
and $W2$ bands. Figure 14: Age properties of the ultracool dwarf sample.
_Left:_ Number of objects in the sample in each age category as described in
§6. The corresponding count is listed to the right of each bar. _Right:_ Mean
age distribution of objects in the sample based on Monte Carlo simulations.
The bin size is 0.4 Gyr. Uncertainty in the density for each bin is marked
with a red errorbar. The median age for the sample is 1.9 Gyr.

Figures 12 and 13 summarize our $L_{\mathrm{bol}}$ measurements as a function
of absolute magnitude in the MKO $JHK$, PS1 $zy$, and WISE $W1$ and $W2$
bands. The full set of polynomial relations (including those for 2MASS
$JH$$K_{S}$ ) are presented in Table 5. We find that the MKO $H$-band
polynomial relation is the most reliable choice for $L_{\mathrm{bol}}$
determinations using absolute magnitudes, based on its linearity, low rms
scatter about the fit, and insensitivity to age and spectral type information.

Figure 15: Comparison of masses, radii, effective temperatures, and surface
gravities derived with BHAC15 and SM08 evolutionary models. The dashed black
line in each panel represents a 1:1 relation between the compared parameters.
While the parameters obtained are statistically consistent, we find that
BHAC15 generally predicts higher temperatures and smaller radii than SM08.

##  6 Ages

In order to compute properties of ultracool dwarfs from evolutionary models,
we must first estimate ages for all the objects. There is no single definitive
method for doing this, so we employ multiple methods, making best use of the
objects’ available measurements and previous assessments in the literature. We
follow a tiered approach to assigning ages, with the following priority order:

  1. 1.

Membership in a stellar association or moving group: The BANYAN $\Sigma$
algorithm (Gagné et al., 2018) provides a means for uniform probability
assessment of objects’ memberships in stellar associations or young moving
groups (YMGs) within 150 pc based on kinematics ($UVW$) and position ($XYZ$),
including handling objects that are missing parallax and/or radial velocity
(RV) measurements. We process our entire sample using this algorithm, using
the highest S/N astrometry for each object as compiled in The UltracoolSheet.
An object is designated as a YMG member if the BANYAN $\Sigma$ membership
probability is $\geq$90% and then assigned the corresponding YMG age tabulated
in Gagné et al. (2018), assuming a normal age distribution for ages
originating from Bell et al. (2015) and a uniform age distribution for all
other ages. (We note that BANYAN $\Sigma$ is a probabilistic assessment based
on an assumed spatial-kinematic model of the solar neighborhood, and
individual objects of interest might benefit from greater scrutiny using
additional methods.) In order to retain information about objects that might
be members, especially with the addition of missing measurements (typically
RVs), we assign potential membership for objects with probabilities of 80–90%
and tag these with a “?” suffix (e.g. “ABDMG?” indicates potential membership
in the AB Dor moving group). This suffix is also used for objects whose BANYAN
$\Sigma$ analysis was done with a photometric distance rather than a
parallactic one. Finally, for objects where the BANYAN $\Sigma$ calculation
was done with the full 6-d information, we append a “!” suffix (e.g.,
“ABDMG!”) to indicate the greater robustness of this membership assignment as
compared to most objects, where only 5-d information is available since RV
measurements are much less common than astrometry and proper motions. In
total, 61 objects are designated as YMG members based on the $\geq$90%
probability + parallactic distance criterion (for 22 out of these 61 objects,
the full 6-d information is available). 30 additional objects are designated
as potential YMG members based on the 80–90% probability criterion. 3 objects
with a $\geq$90% BANYAN $\Sigma$ probability are also designated as potential
YMG members based on the use of a photometric distance.

  2. 2.

Companionship to a star: For ultracool companions with spectral types L6 and
later, we use the literature compilation of ages for the primary stars
(Kirkpatrick et al., 2001; Seifahrt et al., 2005; Valenti & Fischer, 2005;
Reid & Walkowicz, 2006; Luhman et al., 2007; Burgasser et al., 2010b; Deacon
et al., 2012b, a; Pinfield et al., 2012; Deacon et al., 2014, 2017; Miles-Páez
et al., 2017; Gauza et al., 2019; Zhang et al., 2020, 2021a, 2021c). For 7
companions in our sample, The UltracoolSheet does not yet include the
literature host star ages, so these objects are not included in the
evolutionary model analysis that follows. In total, 78 objects are assigned
ages based on the age of the primary star.

Figure 16: Effective temperatures derived for our sample of 1054 ultracool
dwarfs as a function of their spectral type (SpT). Symbols are the same as in
Figure 11. Random scatter in the x direction with amplitude 0.3 SpT is applied
to avoid overlapping points. The black solid and dashed lines are the weighted
best-fit polynomial relations for field and young objects, respectively. The
shaded blue and orange regions represent the rms scatter about the field and
young object fits, respectively. The field polynomial relation from Filippazzo
et al. (2015, F+15) is plotted as a green solid line for comparison.

  3. 3.

Spectroscopic gravity: Spectroscopic signatures of low-gravity in either
optical (e.g. Cruz et al., 2009) or near-IR (e.g. Allers & Liu, 2013) spectra
indicate that an object is young enough ($\lesssim$300 Myr) such that its
radius is inflated compared to older (higher-gravity) field objects. We use
the set of objects with low-gravity spectra that reside in young moving groups
and stellar associations (as determined above with BANYAN $\Sigma$) to examine
the age range corresponding to the different gravity classes. We find objects
with vl-g  near-IR spectra (18 objects) are present in regions with ages of
10–150 Myr (bounded by the TW Hya Association and the AB Dor moving
group444Younger sources in star-forming regions, e.g., Taurus and Upper Sco,
also show vl-g  spectra (e.g. Zhang et al., 2018) but since our sample does
not include such regions, we take 10 Myr as the lower age bound for this
gravity class.) while objects with int-g  near-IR spectra (42 objects) span a
range of 20–250 Myr (bounded by the $\beta$ Pic moving group and the cooling
age for the int-g  binary system LP 349-25 [Dupuy & Liu, 2017]). We assume
uniformly distributed age values within each range. Each gravity class does
not correspond to a unique age range, as a given moving group may have sources
with both gravity classes (e.g. Aller et al., 2016). For objects with optical
gravity classifications, we treat the $\beta$ classification (8 objects) as
equivalent to int-g  and the $\delta$ (1 object) and $\gamma$ (13 objects)
classifications as equivalent to vl-g . For cases where the optical and near-
IR gravity classes are both young but do not agree, or the optical gravity
class is uncertain, we adopt an age range that encompasses both gravity
classifications: 10–250 Myr for $\beta$+vl-g  (2 objects), $\gamma$+int-g  (3
objects), and $\beta$+$\gamma$ (1 object), except for the few objects with
$\beta$+fld-g  (4 objects) where we assume 20–250 Myr. For objects with low-
gravity spectral signatures but no formal gravity class in the literature
(designated as low _g_), we adopt a uniform distribution of 10–300 Myr (1
object). Finally for objects with fld-g  gravity (148 objects), we assume the
same field-age distribution from Dupuy & Liu (2017) described below but with
an elevated lower bound of 300 Myr.

  4. 4.

Field age: For all remaining objects, i.e., those lacking evidence of youth or
ages based on group membership or companionship (634 objects), we assign an
age drawn from the distribution of Dupuy & Liu (2017, hereafter DL17). The
DL17 age distribution was empirically determined based on high precision
dynamical mass measurements of 10 ultracool dwarf binaries and is found to be
consistent with the Besançon model of the solar neighborhood (Robin et al.,
2003). The Besançon model predicts a field age distribution skewed towards
younger ages since dynamical heating preferentially excites older objects out
of the midplane of the Galaxy. According to DL17 age distribution, the ages
are uniformly distributed in the following Besançon model bins, where each bin
contains a fraction of the total population as follows: 8.1% for 0.01–0.15
Gyr, 20.0% for 0.15–1 Gyr, 16.1% for 1–2 Gyr, 11.9% for 2–3 Gyr, 16.6% for 3–5
Gyr, 12.9% for 5–7 Gyr, and 14.4% for 7–10 Gyr. We adopt this as an age
probability density function for each field object.

The number of ages obtained with each method is summarized in the left panel
of Figure 14. We derive the age distribution of our ultracool dwarf sample
using Monte Carlo simulations. $10^{5}$ ages are drawn for each object from
their assigned age distributions. We construct a sample age distribution
spanning 0–10 Gyr with a bin size of 0.4 Gyr from the drawn ages of all
objects in each trial. The mean age distribution is derived by averaging the
density values across the $10^{5}$ sample age distributions in each bin.
Uncertainty in the density value of each bin is obtained using the standard
deviation. The right panel of Figure 14 shows the mean age distribution with
uncertainties. The median age of the sample is 1.9 Gyr.

Figure 17: MKO $M_{J}$ vs $J-K$ color-magnitude diagram for ultracool dwarfs
in our sample with individual points colored by their derived effective
temperatures. Spectral blends and overluminous objects have been excluded. Due
to the large number of points, uncertainties in individual measurements are
not shown for figure readability. The median uncertainty in
$M_{J_{\mathrm{MKO}}}$ is 0.06 mag and in $J-K$ is 0.05 mag.

##  7 Evolutionary Model-Derived Parameters

In this section, we use the bolometric luminosities (Section 5) derived for
all objects in our sample along with their age estimates (Section 6) as inputs
to the Saumon & Marley (2008, SM08) hybrid and Baraffe et al. (2015, BHAC15)
evolutionary model grids to estimate their masses, radii, surface gravities,
and effective temperatures. Ultracool dwarfs in our sample span a wide range
of luminosities and thus, in some cases, are too bright for the SM08 models
($\mathrm{log}(L_{\mathrm{bol}}/L_{\odot})\gtrsim-3$) or too faint for the
BHAC15 models ($\mathrm{log}(L_{\mathrm{bol}}/L_{\odot})\lesssim-4.5$). We use
each evolutionary model grid in its respective non-overlapping luminosity-age
space. In the region of model overlap, we derive the fundamental properties
with both sets of models.

###  7.1 Bayesian Rejection Sampling

To estimate ultracool dwarf masses, radii, and surface gravities by
interpolation of evolutionary model grids, we implement the Bayesian rejection
sampling method described in Dupuy & Liu (2017). The first step is to randomly
draw $10^{6}$ log $L_{\mathrm{bol,model}}$ samples from a uniform distribution
spanning the bolometric luminosity range of the evolutionary model grid. The
second step involves randomly sampling the object’s age distribution,
calculating $\chi^{2}$ values for the sampled ages, and computing a
probability for each sample. The exact implementation of this step depends on
the adopted age distribution.

_Normal Age Distribution:_ $10^{6}$ $\mathrm{log}\;t_{\mathrm{model}}$ samples
are randomly drawn from a uniform distribution spanning the age range of the
evolutionary model grid. We compute $\chi^{2}$ for each sample,

$\chi^{2}=\frac{(\mathrm{log}\;L_{\mathrm{\mathrm{bol,model}}}-\mathrm{log}\;L_{\mathrm{bol}})^{2}}{\sigma_{\mathrm{log}\;L_{\mathrm{bol}}}^{2}}+\frac{(\mathrm{log}\;t_{\mathrm{model}}-\mathrm{log}\;t)^{2}}{\sigma_{\mathrm{log}\;t}^{2}},$
(7)

where
$\mathrm{log}\;L_{\mathrm{bol}}\pm\sigma_{\mathrm{log}\;L_{\mathrm{bol}}}$ is
the bolometric luminosity measurement and
$\mathrm{log}\;t\pm\sigma_{\mathrm{log}\;t}$ is the age measurement for the
object. The $\chi^{2}$ for each sample is converted to a probability $p$,
normalizing by the sample with the minimum $\chi^{2}$, as follows,

$p=\mathrm{exp}\left[-\frac{1}{2}\left(\chi^{2}-\chi^{2}_{\mathrm{min}}\right)\right].$
(8)

_Uniform Age Distribution:_ $10^{6}$ $\mathrm{log}\;t_{\mathrm{range}}$
samples are randomly drawn from a uniform distribution spanning the
intersection of the quoted age range and the evolutionary model grid age
range. We compute the $\chi^{2}$ for each sample, in this case, as follows,

$\chi^{2}=\frac{(\mathrm{log}\;L_{\mathrm{bol,model}}-\mathrm{log}\;L_{\mathrm{bol}})^{2}}{\sigma_{L_{\mathrm{bol}}}^{2}}.$
(9)

The $\chi^{2}$ for each sample is converted to a probability $p$, normalizing
by the sample with the minimum $\chi^{2}$ using Equation 8.

Figure 18: Comparison of mass, radius, effective temperature, and surface
gravity derived in this work with those in Filippazzo et al. (2015, F+15) for
objects common to the two samples. The dashed black line in each panel
represents a 1:1 relation between the compared parameters. The parameters
obtained are statistically consistent for the large majority of objects.
Objects with disagreements ($>$2$\sigma$ discrepancy) in the estimated values
are prominently marked with a black-bordered star. Figure 19: Bolometric
corrections in the 2MASS and MKO near-IR bands for ultracool dwarfs in our
sample as a function of spectral type (SpT). Random scatter in the x direction
with amplitude 0.3 SpT is applied to avoid overlapping points. Symbols are the
same as in Figure 11. Due to the large number of points, uncertainties in
individual measurements are not shown for figure readability. The median
uncertainty of measurements in each panel presented is 0.1 mag. The black
solid and dashed lines are the best-fit polynomial relations for field and
young objects, respectively. The blue and orange shaded regions are the rms
scatter about the field and young object relations, respectively. The field
and young object relations from Filippazzo et al. (2015) are plotted as green
solid and dashed lines, respectively, for comparison. Figure 20: Bolometric
corrections in the PS1 optical and WISE mid-infrared bands for ultracool
dwarfs in our sample as a function of spectral type (SpT). Figure elements are
same as in Figure 19. The median uncertainty of measurements in each panel
presented is 0.1 mag.

_DL17 Age Distribution:_ $10^{6}$ $\mathrm{log}\;t_{\mathrm{DL17}}$ samples
are randomly drawn from a uniform distribution spanning the intersection of
the DL17 age range and the evolutionary model grid age range. In each case,
the $\chi^{2}$ for each sample (Equation 9) is converted to a probability
$p_{*}$, normalizing by the sample with the minimum $\chi^{2}$ using Equation
8. We then compute the likelihood of drawing the sampled age from the DL17
distribution (truncated at 300 Myr) for field-age (fld-g ) objects
$p_{\mathrm{DL17}}=p(\mathrm{log}\;t_{\mathrm{DL17}}|\mathrm{DL17})$. The
final probability for each sample, normalized by the sample with the highest
probability, is,

$p=\frac{p_{*}\cdot p_{\mathrm{DL17}}}{\mathrm{max}(p_{*}\cdot
p_{\mathrm{DL17}})}$ (10)

The third step is to randomly draw $10^{6}$ uniform variates ($u$) distributed
in the range from 0 to 1 and reject any samples where $p<u$. The fourth and
final step is to linearly interpolate the evolutionary models at each accepted
luminosity-age point to calculate the corresponding mass ($\mathrm{log}\;M$),
radius ($\mathrm{log}\;R$), and surface gravity (log $g$). The final
mass/radius/gravity measurement and uncertainty are obtained as the median and
standard deviation of all the sample’s mass/radius/gravity values. Our
measurements are summarized in the Table of Ultracool Fundamental Properties
associated with this paper (see §1). A total of 243 sources’ parameters were
estimated using both BHAC15 and SM08, 339 sources’ parameters were estimated
using only BHAC15, and 464 sources’ parameters were estimated using only SM08.

Figure 21: Evolutionary model-derived surface gravity distribution for
different near-infrared gravity classifications. The smaller size markers
represent individual object measurements and the larger size markers represent
the mean surface gravity for a given gravity classification. DENIS-P
J1058.7-1548 (DPJ10-15) and EROS-MP J0032-4405 (EMPJ00-44) are significant
outliers. The median uncertainty of surface gravity measurements presented
here is 0.04 dex. The black dashed line connects the mean surface gravities
and serves as a visual aid to identify the trend. While individual groups
exhibit varying levels of dispersion in their log $g$ values, the index-based
NIR gravity classification scheme accurately groups objects with systematic
differences in their evolutionary model-derived surface gravities. Figure 22:
Atmospheric model-derived effective temperatures for objects in our sample as
a function of spectral type. The symbols are similar to those described in
Figure 16. A pile-up of objects at $\approx$1800 K creates two gaps at higher
and lower effective temperatures.

The choice of a given model, SM08 or BHAC15, for the adopted measurements is
based on the following conditions. BHAC15 is used if the $1\sigma$ lower limit
of the measured bolometric luminosity is greater than the luminosity on the
lowest mass luminosity-age model track interpolated at the $1\sigma$ lower
limit of the measured age (or lower limit of the quoted age range). SM08 is
used if the $1\sigma$ upper limit of the measured bolometric luminosity is
less than the luminosity on the highest mass luminosity-age model track
interpolated at the $1\sigma$ upper limit of the measured age (or upper limit
of the quoted age range). When both conditions are satisfied, calculations are
made using both evolutionary models. In the Table of Ultracool Fundamental
Properties associated with this paper (see §1), we present the lower
uncertainty measurement in such cases. Figure 15 presents a comparison of the
fundamental parameters derived between BHAC15 and SM08. While the parameters
obtained with BHAC15 and SM08 are statistically consistent, we find that
BHAC15 generally predicts higher temperatures (by $\approx$50 K) and smaller
radii (by $\approx$ 0.06 $R_{\mathrm{Jup}}$) than SM08.

This method significantly differs from the process implemented in Filippazzo
et al. (2015). To estimate an object’s radius (which is used to derive
effective temperature) and mass, Filippazzo et al. (2015) determine a
parameter range for each source spanning the minimum and maximum values of all
evolutionary model predictions given an object’s bolometric luminosity and
age. They quote the average of the minimum and maximum value as the final
radius/mass and half the radius/mass range as the uncertainty in Table 9.
However, this method does not take into account the shape of the underlying
radius/mass distribution when estimating the object’s properties. Except for
the case of a uniform distribution, the average parameter value does not
correspond to the distribution’s true expectation value. This is further
complicated by the use of parameter estimates from several sets of
evolutionary models to construct a single range in Filippazzo et al. (2015).
Our Bayesian rejection sampling technique addresses the limitations of the
above method by deriving each object’s underlying radius and mass distribution
independently for each of the two evolutionary model sets in this work. The
median value computed from the parameter’s cumulative distribution function is
then quoted as our final value.

###  7.2 Effective Temperature

Effective temperatures are calculated using the evolutionary model-inferred
radius $R$ and SED-integrated bolometric luminosity measurement
$L_{\mathrm{bol}}$ using the Stefan-Boltzmann Law,

$T_{\mathrm{eff}}=\left(\frac{L_{\mathrm{bol}}}{4\pi
R^{2}\sigma_{\mathrm{SB}}}\right)^{\frac{1}{4}},$ (11)

where $\sigma_{\mathrm{SB}}$ is the Stefan-Boltzmann constant. Figure 16
presents the calculated effective temperatures as a function of spectral type
along with our best-fit polynomial relations for field and young (low gravity
and YMG member) objects separately (Table 5). Our field relation is in good
agreement with the corresponding field polynomial relation derived in
Filippazzo et al. (2015). Figure 17 demonstrates the temperature evolution of
ultracool dwarfs as a MKO NIR color-magnitude diagram for objects in our
sample with derived fundamental parameters.

###  7.3 Comparison with Filippazzo et al. (2015)

Figure 18 compares the masses, radii, effective temperatures, and surface
gravities calculated in this work with those in Filippazzo et al. (2015). We
find that our results are consistent ($<$2$\sigma$ difference) with Filippazzo
et al. (2015) for the majority of objects (N = 91). $>$2$\sigma$ discrepancy
in at least one of the above fundamental properties is observed for 37 objects
(Figure 18). A complete list of the above objects is provided in Table 6.

The primary reason for the observed disagreements is a revision in the age
distribution adopted for the objects. For field (fld-g ) objects, Filippazzo
et al. (2015) adopt a uniform age distribution in the range 0.5–10.0 Gyr. We
adopt the (truncated) DL17 age distribution which skews towards younger ages:
the greatest probability density in this age distribution lies in the 0.15–1
Gyr bin and the median age is $\sim$2.5 Gyr. Consequently, our derived radii
for these objects are generally larger, and our derived masses, temperatures,
and gravities are generally smaller. This is the case for 22 out of the 37
discrepant objects. For 11 objects assigned a field age (0.5–10 Gyr) in
Filippazzo et al. (2015), there have been updates to their YMG membership or
spectroscopic gravity class. In this work, 2MASS J06244595-4521548, 2MASSW
J0030300-145033, 2MASSI J0451009-340214, and DENIS-P J1058.7-1548 are
designated as Argus members; 2MASSI J1010148-040649 is designated as a Carina-
Near member; and LHS 3003 is designated as an AB Doradus member based on
results from BANYAN $\Sigma$. 2MASS J13595510-4034582, DENIS-P
J0652197-253450, and LHS 132 are assigned a $\beta$ optical gravity
classification, and 2MASSI J2057540-025230 and Teegarden’s Star are assigned
an int-g  NIR gravity classification. Consequently, our derived masses,
temperatures, and gravities for these objects are generally smaller, and our
derived radii are generally larger. For 2 objects, 2MASS J03264225-2102057 and
2MASSW J2244316+204343, the YMG membership (AB Doradus) does not change but
the source of our adopted ages differ. Filippazzo et al. (2015) use an age
range of 50–120 Myr from Malo et al. (2013), whereas we use an older age of
$149^{+51}_{-19}$ Myr from Bell et al. (2015). Consequently, our derived radii
for these objects are generally smaller, and our derived masses, temperatures,
and gravities for these objects are generally larger.

Two objects remain to be accounted for: 2MASSW J2208136+292121 and 2MASS
J00332386-1521309. 2MASSW J2208136+292121 meets the consistency threshold for
mass, radius, and gravity. We derive a lower effective temperature that is
marginally inconsistent at the 2.1$\sigma$ level. The age range adopted for
this object is the same between Filippazzo et al. (2015) and this work. 2MASS
J00332386-1521309 meets the consistency threshold for mass, radius, and
gravity. The effective temperature is inconsistent at the 2.8$\sigma$ level.
Filippazzo et al. (2015) adopt an age range 100–150 Myr whereas we adopt a
truncated DL17 age distribution (lower bound of 300 Myr) based on its fld-g
classification. This corresponds to an older age assignment compared to
Filippazzo et al. (2015). However, our temperature estimate is lower than that
of Filippazzo et al. (2015), not higher as it would be expected. These two
disagreements arise because of the difference in bolometric luminosity
measurements (due to changes in parallax measurement). Both objects can be
found in Table 4. We measure lower bolometric luminosities for 2MASSW
J2208136+292121 and 2MASS J00332386-1521309, which explain the lower effective
temperatures derived for both objects.

More generally, the differences observed between our evolutionary model-
derived parameter estimates and those in Filippazzo et al. (2015) may also be
a result of the different estimation methods employed, as described in §7.1.

##  8 Discussion

###  8.1 Bolometric Corrections

We calculate bolometric corrections spanning the optical to MIR filters for
both field and young (low gravity and young moving group members) objects and
determine empirical relationships as a function of spectral type following the
process described in Section 5. For our calculations, we use
$M_{bol,\odot}=4.74$. We note here that the _F_ -test does not always yield a
definitive answer for the order of the polynomial based on the $p>0.05$
rejection criterion. For such cases, we adopt the lowest order after which $p$
does not change significantly (factor of $\sim$5-10) as the best-fit
polynomial, which is then validated by visual inspection. Figure 19 presents
our bolometric corrections for the 2MASS and MKO NIR filters as a function of
spectral type. Figure 20 presents our bolometric corrections for the PS1 $z$
and $y$ optical filters and the WISE $W1$ and $W2$ MIR filters as a function
of spectral type. The derived polynomial relations are presented in Table 5.

Next, we discuss some key observations. First, based on the rms about the fit,
the NIR relations generally show less scatter than the optical and MIR
relations and thus are preferable for bolometric magnitude calculations.
Second, within the NIR relations, the bolometric corrections generally show
smaller rms in the MKO filters than the 2MASS filters, likely due to the
higher precision MKO photometry for T-dwarfs in the sample. Finally, we find
that both the 2MASS $K_{S}$ and MKO $K$-band bolometric corrections are
relatively insensitive to the age of the ultracool dwarf and are thus most
appropriate for cases when no age information is available.

We find a few significant differences in the 2MASS $J$ and $K_{S}$ polynomial
relations on comparison with Filippazzo et al. (2015). Our field relation in
the 2MASS $J$ band predicts a higher bolometric correction (by $\approx
0.1-0.3$ mag) in the spectral range $\sim$L5–T2. The change can be attributed
to the larger number of L/T transition objects in our work compared to
Filippazzo et al. (2015). This enables a better constraint on the polynomial
relation. The difference in the young object relations in the 2MASS $J$ band
is even more striking. Our best-fit polynomial predicts a bolometric
correction up to $\approx$0.7 mag higher than that in Filippazzo et al. (2015)
across spectral types $\sim$L2–T6. A careful comparison of Figure 12 in
Filippazzo et al. (2015) with the top left panel of Figure 19 leads us to
conclude that the larger number of young L2–L5 dwarfs with bolometric
corrections $>1.5$ mag serves to anchor our fit and is responsible for the
differing solutions. However, in our measurements, we observe that the
bolometric corrections of young objects in the spectral range L5–L7 steeply
drop to $\approx$1 mag, a feature not reproduced in our best-fit relation.
This is also the case for our MKO $J$ band young object relation. Our results
suggest that a more appropriate empirical relationship may lie between the
solutions found in Filippazzo et al. (2015) and this work or by adopting the
direct averages as a function of spectral type. We note that the key to
determining this will be adding more young objects in the range L7–T2, where,
presently, there is a noticeable gap in our measurements. Until then, caution
should be exercised in using the 2MASS and MKO $J$ band young object relations
at the L/T transition. The 2MASS $K_{S}$ and MKO $K$-band young relations are
more tightly constrained and are the recommended alternatives. Finally, in the
2MASS $K_{S}$ band, we observe slight discrepancies between the field
relations, within the calculated rms scatter, in the spectral range L5–T2.
This can be similarly explained by the larger number of L/T transition object
measurements in this work. The Filippazzo et al. (2015) 2MASS $K_{S}$ band
field relation diverges for T9 dwarfs compared to the one in this work and is
likely an edge effect due to the small number of T9 objects in our sample.

Figure 23: _Left:_ Difference between the the atmospheric model-derived
effective temperatures and the evolutionary model-derived effective
temperatures ($\Delta T_{\mathrm{eff}}$) as a function of spectral type (SpT).
Random scatter in the x direction with amplitude 0.3 SpT is applied to avoid
overlapping points. Objects are colored based on their spectral type where the
darkest shade corresponds to M-dwarfs, the intermediate shade corresponds to
L-dwarfs, and the lightest shade corresponds to T-dwarfs. Objects using the
atmospheric-evolutionary model pairings of BT-Settl—BHAC15, BT-Settl—SM08, and
ATMO 2020–BHAC15/SM08 are marked with a star, triangle, and circle
respectively. BT-Settl—BHAC15 objects are presented with a higher color
opacity than BT-Settl—SM08 and ATMO 2020–BHAC15/SM08 objects to emphasize the
greater reliability of $\Delta T_{\mathrm{eff}}$ trends for the former (self-
consistently computed models). Symbols with black outlines mark young objects
based on signatures of low surface gravity. _Right:_ MKO $M_{J}$ vs $J-K$
color-magnitude diagram for ultracool dwarfs in our sample with each object
colored by its corresponding $\Delta T_{\mathrm{eff}}$ value. Due to the large
number of points, uncertainties in individual measurements are not shown for
figure readability. The median uncertainty in $M_{J_{\mathrm{MKO}}}$ is 0.06
mag and in $J-K$ is 0.05 mag. Figure 24: Same as Figure 23 but for difference
between atmospheric and evolutionary model-derived radius measurements.

###  8.2 Evolutionary Model-Derived Surface Gravity vs Ultracool Dwarf NIR
Gravity Classifications

Allers & Liu (2013) defined a set of near-infrared gravity-sensitive indices
based on FeH, VO, K I, Na I, and H-band continuum shape and created a gravity
classification scheme consisting of three classes: fld-g , int-g , and vl-g .
Here, we examine the empirical distribution of the evolutionary model-derived
surface gravity of ultracool dwarfs as a function of their NIR gravity
classifications. For an accurate analysis, we exclude ultracool dwarfs with
spectra $<$50 S/N in any of the following six bandpasses: $YJHK$ and two blue-
half regions in the $H$ and $K$ bands (provided in the Table of Ultracool
Fundamental Properties associated with this paper; see §1). We take care to
only include low gravity objects for which the ages were determined based on
the ages of their host stars or their membership to a YMG, independent of
their NIR gravity classification. This was done to avoid circular logic in the
analysis, since the assigned ages directly impact the estimation of surface
gravities in our evolutionary model interpolation procedures. This results in
6 fld-g , 7 int-g , and 12 vl-g objects with evolutionary model-derived
parameters from this work. These objects are plotted in Figure 21. We observe
varying levels of scatter in the evolutionary model-derived surface gravity
values for each gravity class — lowest for int-g , followed by fld-g , and
highest for vl-g . The fld-g outlier at log $g=4.27$ dex is DENIS-P
J1058.7-1548, and the vl-g outlier at log $g=4.89$ dex is EROS-MP J0032-4405.
DENIS-P J1058.7-1548 is a candidate Argus member (95% BANYAN-$\Sigma$
probability) and was thus assigned an age of $45\pm 5$ Myr. Such a young age
is uncharacteristic for a fld-g object and would explain the large discrepancy
in log $g$ from the fld-g mean value. A comparison of DENIS-P J1058.7-1548’s
spectrum with L-dwarf fld-g standards does not reveal any discrepancies that
could suggest a gravity misclassification and thus it is probable that the YMG
assignment is incorrect. EROS-MP J0032-4405 is an AB Doradus (AB Dor) Moving
Group member. The int-g classification is characteristic of AB Dor members
(all 7 int-g objects in our analysis are AB Dor members). Given that the
derived surface gravity for this object is consistent with the mean surface
gravity in the int-g class, it may be possible that int-g is a more suitable
gravity classification for the object. However, the object also possesses an
optical gravity classification of $\gamma$ (Cruz et al., 2009), reducing the
probability that a NIR gravity misclassification has occurred. We investigated
the possibility that this object is a spectral blend using the spectral
indices of Burgasser et al. (2010a) and Bardalez Gagliuffi et al. (2014), but
found that none of the criteria for binarity were met. The relatively higher
scatter in the fld-g and vl-g classes may be an edge case artifact due to the
limited number of classification groups and possibly highlights the need to
add new gravity classes. This can be more reliably determined with a similar
analysis consisting of a larger sample of objects. The primary factor limiting
the sample size in this work is the lack of high S/N spectra (indicative of
reliabilty of the index-based gravity classification) across all the six
wavelength regions defined previously. Nevertheless, the mean surface
gravities exhibit a trend consistent with the gravity classification scheme —
objects with a fld-g classification, on average, have higher surface gravities
than those with an int-g classification which, in turn, have higher surface
gravities, on average, than objects with a vl-g classification.

###  8.3 Differences between Atmospheric Model-derived and Evolutionary Model-
derived Parameters

Numerous works in the past have documented discrepant log $g$,
$T_{\mathrm{eff}}$, and thus radii values between those derived by fitting
atmospheric models to substellar SEDs and those derived by interpolating
evolutionary models (e.g. Liu et al., 2008; Dupuy et al., 2009; Bowler et al.,
2010; Barman et al., 2011; Marley et al., 2012; Dupuy & Kraus, 2013; Liu et
al., 2013; Zhang et al., 2021c; Carter et al., 2022). Here, we take advantage
of our large sample to study the noted differences as a function of spectral
type and in the near-infrared color-magnitude diagram for ultracool dwarf
effective temperatures, radii, and surface gravities.

####  8.3.1 Effective Temperature

First, we investigate the atmospheric model-derived effective temperatures
($T_{\mathrm{eff,atmo}}$) vs spectral type relation and then systematically
characterize the differences with evolutionary model-derived effective
temperatures ($T_{\mathrm{eff,evo}}$).

Figure 22 shows $T_{\mathrm{eff,atmo}}$ as a function of spectral type. A
pile-up of objects in the temperature range $\approx$1700-1900 K is observed.
This pile-up carves out gaps in the temperature sequence at $\approx$2000-2200
K and, less prominently, $\approx$1500-1600 K. Figure 23 investigates the
effects of these features in greater detail by presenting the difference in
$T_{\mathrm{eff,atmo}}$ and $T_{\mathrm{eff,evo}}$ measurements as a function
of spectral type. A comparison between self-consistently computed atmospheric
and evolutionary models is the most accurate choice for such an analysis.
However, the availability of models only allows such a comparison for objects
where we simultaneously employed the BT-Settl and BHAC15 models (BT-
Settl—BHAC15). We find that late-M and early L-dwarfs ($\sim$M6–L2) primarily
constitute our sample for this analysis (Figure 23). Given that we do not
observe a significant difference in the temperatures derived with BHAC15 vs
SM08 (median difference $\approx$50 K; Figure 15), we are also able to perform
plausible comparisons for the BT-Settl–SM08 and ATMO 2020–BHAC15/SM08
pairings. $\sim$L3–T2 dwarfs primarily form the sample for the former and
$>$T2 dwarfs form the sample for the latter.

  1. 1.

BT-Settl—BHAC15 objects: In Figure 23, we find that BT-Settl generally
underestimates the effective temperature compared to the BHAC15 evolutionary
model, with the largest discrepancy (up to $\approx$800 K) at the M/L
transition boundary. This discrepancy reflects the $\approx$1800 K pile-up of
M/L transition objects in Figure 22 and the $\approx$2000-2200 K gap in the
temperature sequence. Since cloud opacity increases as the effective
temperature decreases across the M/L transition ultracool dwarfs, an
underestimated effective temperature highlights that BT-Settl does not
incorporate sufficient dust in its $\approx$2000-2200 K models. Our results
are in agreement with Hurt et al. (AAS Journals, submitted), where a
$\approx$1800 K pile-up of objects was also identified in an analysis of the
BT-Settl model fitting results for 89 late-M and L benchmark dwarfs. They find
that M/L transition objects are poorly fit by the BT-Settl atmospheric models,
which consistently overestimate the $J$ and $H$ band flux. By including an
extinction component that follows the interstellar medium (ISM) reddening law,
Hurt et al. (AAS Journals, submitted) find a significant improvement in the
fits, similarly suggesting a lack of dust opacity in the BT-Settl models in
this regime.

  2. 2.

BT-Settl–SM08 objects: In Figure 23, BT-Settl generally yields consistent
results with the SM08 evolutionary model for spectral types $\sim$L3–L5 but
tends to overestimate the effective temperatures of late-L dwarfs. This
discrepancy reflects the $\approx$1800 K pile-up of late-L dwarfs in Figure 22
and possibly the less prominent $\approx$1500–1600 K gap in the temperature
sequence. Since cloud opacity increases from early-L to late-L dwarfs, an
overestimated effective temperature for late-L dwarfs may mean that BT-Settl
over-predicts dust opacity in its $\approx$1700–1800 K models.

  3. 3.

ATMO 2020–BHAC15/SM08 objects: In Figure 23, we observe that ATMO 2020 tends
to generally overestimate temperatures at a level of 100–200 K. This may
indicate possible incompleteness in incorporating cloud-free opacity sources
in the atmospheres of T-dwarfs.

It is important to note that a significant number of objects discussed in the
second and third categories are located at the L/T transition. This is a
relatively less understood phase of brown dwarf evolution. Since the
comparison is not being conducted for self-consistently computed models, the
discrepancies seen could be attributed to reasons outside the scope of our
investigation. As such, one should be cautious in definitively interpreting
BT-Settl–SM08 and ATMO 2020–BHAC15/SM08 object results. Combining SED-
integrated bolometric luminosities with direct radius measurements of
ultracool dwarfs would allow strictly empirical tests of the systematics in
both atmospheric and evolutionary models. A summary of our results is
presented as a MKO $M_{J}$ vs $J-K$ color-magnitude diagram in the right panel
of Figure 23.

Figure 25: Same as right panel of Figure 23 but for difference between
atmospheric and evolutionary model-derived surface gravity measurements.
Figure 26: Average difference between the atmospheric and evolutionary model-
derived effective temperatures (top), radii (middle), and surface gravities
(bottom) as a function of spectral type for field (triangles) and young
(squares) objects. Spectral types with $<$10 field objects (M6, T4, T9) and
$<$5 young objects (M6, M7, L6, $\geq$L8) in our sample have been skipped.
There was no significant difference between the mean and the median parameter
difference distributions as a function of spectral type. Such systematic
offsets may be adopted when combining fundamental parameter estimates obtained
with the two methods. Figure 27: Effective temperatures derived for objects in
our sample as a function of infrared (top) and optical (bottom) spectral types
(SpTs). Random scatter in the x direction with amplitude 0.3 SpT is applied to
avoid overlapping points. The figure elements are similar to those described
in Figure 16. In each panel, the best-fit polynomial relation for field and
young objects from the other panel is presented as a dashed line of the same
color as in the other panel.

####  8.3.2 Radius and Surface Gravity

Here, we note the observed differences between atmospheric and evolutionary
model-derived radii and surface gravities for our sample of ultracool dwarfs.
The analysis is conducted similar to §8.3.1.

The atmospheric model-derived radius $R_{\mathrm{atmo}}$ is obtained from the
scale factor $\Omega$ of the best-fit model spectrum as,

$R_{\mathrm{atmo}}=\sqrt{\Omega\cdot d^{2}},$ (12)

where $d$ is the object’s distance. Figure 24 shows the difference between the
two sets of radii measurements as a function of spectral type. For BT-
Settl—BHAC15 objects, which lie at the M/L transition boundary, radius is
significantly overestimated by BT-Settl (up to 1.5 $R_{\mathrm{Jup}}$). For
BT-Settl–SM08 and ATMO 2020–BHAC15/SM08 objects ($\gtrsim$ L3), atmospheric
models underestimate the radius by $\approx$0.1–0.7 $R_{\mathrm{Jup}}$, with
maximum discrepancy in the late-L spectral types. A summary of these results
is presented as a MKO $M_{J}$ vs $J-K$ color-magnitude diagram in the right
panel of Figure 24. Finally, Figure 25 presents the trend in log $g$
differences as a MKO $M_{J}$ vs $J-K$ color-magnitude diagram. Atmospheric
models underestimate the gravities of bluer M/L transition dwarfs by up to 2
dex but overestimate the gravities of redder M/L transition dwarfs by
$\approx$0.5 dex. No systematic trend is observed for later type objects in
the color-magnitude diagram.

###  8.4 Characterization of Systematic Offsets

The significant discrepancies observed in the fundamental parameters obtained
from evolutionary and atmospheric models raise questions about the reliability
of either method to accurately estimate ultracool dwarf fundamental
parameters. Past work on this topic has indicated that atmospheric models are
more likely to be the source of the discrepancy: they are sensitive to
boundary conditions unlike evolutionary models, involve many uncertain
parameters in the modeling inputs and prescriptions, and often yield
unphysical parameter estimates (e.g., Chabrier et al., 2000; Dupuy et al.,
2009, 2010; Liu et al., 2010; Barman et al., 2011; Dupuy & Liu, 2017). It has
been suggested that a systematic offset may exist between atmospheric and
evolutionary model-derived parameters. Previously, the characterization of
such systematic offsets has been limited to small samples of objects. Here, we
take advantage of our large sample size to derive the average systematic
offset in effective temperature, radius, and surface gravity as a function of
spectral type for field and young objects. We compute the mean of the
differences between atmospheric and evolutionary model-derived parameter
values ($\Delta P=P_{\mathrm{atmo}}-P_{\mathrm{evo}}$, where $P$ is the
relevant fundamental parameter) for each spectral type. We skip spectral types
with $<$10 field objects (M6, T4, T9) and spectral types with $<$5 young
objects (M6, M7, L6, $\geq$L8) in our sample. Figure 26 presents our results.
The average systematic offsets are as large as 400 K in effective temperature,
0.4 $R_{\mathrm{Jup}}$ in radius, and 1.2 dex in log $g$. Larger systematic
offsets are generally found near the M/L and L/T transition regions. Such
systematic offsets might be adopted when combining fundamental parameter
estimates obtained with the two methods.

###  8.5 Optical and Near-Infrared Spectral Type-$T_{\mathrm{eff}}$ Sequences
and Comparison

We present $T_{\mathrm{eff}}$ sequences as a function of optical and IR
spectral types in Figure 27 and the corresponding best-fit polynomial
relations for field and young objects in Table 5. Nearly two times as many
objects have IR spectral types in our sample as optical spectral types. We
observe that the $T_{\mathrm{eff}}$ sequence has a marginally lower rms about
the best-fit as a function of IR spectral type, for field objects, compared to
optical spectral types. While the relations for both spectral classifications
are consistent, the optical spectral type is observed to predict slightly
lower temperatures in the L/T transition for field objects compared to the IR
spectral type and marginally higher temperatures for late-T field-age objects.
We conclude that both spectral-typing schemes are good indicators of
$T_{\mathrm{eff}}$ across the complete ultracool dwarf spectral sequence,
given the availability of age information.

##  9 Conclusion

In this work, we performed a uniform calculation of the fundamental physical
properties of 1054 ultracool dwarfs using the most precise photometry and
astrometry available to date from The UltracoolSheet (version 2.0.0, in
preparation). Our results are summarized below:

  1. 1.

We obtained new low-resolution near-IR spectroscopy for 97 objects in our
sample using NASA/IRTF. These spectra were collected for the spectroscopic
characterization of known objects (57 potential low-gravity sources, benchmark
objects, and wide companions) or the discovery of new ultracool dwarfs (40
objects). Additionally, we obtained new parallaxes for 19 objects using
CFHT/WIRCam (11 sources) and Pan-STARRS 1 (8 sources).

  2. 2.

We have derived the bolometric luminosities ($L_{\mathrm{bol}}$) of 865 field-
age and 189 young ultracool dwarfs by assembling and directly integrating
flux-calibrated optical to mid-IR SEDs consisting of low-resolution ($R\sim$
150) near-IR (0.85–2.5 $\mu$m) NASA/IRTF SpeX prism spectra, optical
photometry from the Pan-STARRS1 survey, and mid-IR photometry from the
CatWISE2020 survey and Spitzer/IRAC. Flux contributions from the optical
(0.2–0.85 $\mu$m) and mid-IR (2.5–2000.0 $\mu$m) wavelengths were estimated
using the best-fit BT-Settl or ATMO 2020 atmospheric model spectrum to the
SED.

  3. 3.

A careful treatment of uncertainties revealed the presence of a noise floor in
our $L_{\mathrm{bol}}$ measurements at the $\approx$0.015 dex level due to the
use of atmospheric models to estimate flux contribution from wavelengths
outside our SpeX spectra. This highlighted that high precision parallaxes must
be paired with improved spectral coverage in the optical and mid-IR
wavelengths to further drive the uncertainty in $L_{\mathrm{bol}}$ lower.
Overall, the median uncertainty in our $L_{\mathrm{bol}}$ measurements was
0.05 dex.

  4. 4.

Using the derived $L_{\mathrm{bol}}$ values and a new uniform age analysis for
all objects in our sample, combined with the BHAC15 and SM08 evolutionary
models, we estimated ultracool dwarf masses, radii, and surface gravities by
employing Bayesian rejection sampling. Our $L_{\mathrm{bol}}$ measurements and
evolutionary model-derived radii enabled the determination of semi-empirical
effective temperatures ($T_{\mathrm{eff}}$), with a median uncertainty of 98
K.

  5. 5.

We showed that our $L_{\mathrm{bol}}$ measurements and evolutionary model-
derived fundamental parameter estimates are largely consistent with those from
Filippazzo et al. (2015), for objects common to the two samples (N = 130). For
a small subset of these objects, revisions to their parallaxes and age
distributions primarily contributed to the discrepancies observed in their
$L_{\mathrm{bol}}$ measurements and the evolutionary model-derived fundamental
parameter estimates, respectively.

  6. 6.

Our large sample enabled us to establish empirical relationships for
$L_{\mathrm{bol}}$ as a function of absolute magnitude, and
$L_{\mathrm{bol}}$, $T_{\mathrm{eff}}$, and bolometric correction in various
photometric bands as a function of spectral type, for both field-age and young
objects. Based on the low rms scatter about the fit and relative insensitivity
to age, we recommended the use of the MKO $H$-band polynomial relation for
$L_{\mathrm{bol}}$ calculations from absolute magnitude. Further, we found
that the 2MASS $K_{S}$-band or MKO $K$-band polynomial relations are the
preferred choice for bolometric correction calculations from spectral type.

  7. 7.

We performed a comparison between the evolutionary model-derived surface
gravities and near-infrared gravity classifications of ultracool dwarfs for a
carefully constructed sample of young moving group members and companion
objects. While the surface gravity measurements exhibited scatter in each of
the three categories (fld-g , int-g , and vl-g ), we observed a systematic
difference in the mean surface gravities between the three gravity classes.

  8. 8.

A detailed characterization of atmospheric model systematics as a function of
spectral type and in the near-infrared color magnitude diagram, based on
comparisons with evolutionary model-derived values, revealed significantly
large discrepancies in the effective temperatures (up to 800 K), radii (up to
2.0 $R_{\mathrm{Jup}}$), and surface gravities (up to 2 dex) of M/L transition
dwarfs. Our results suggested that BT-Settl lacks sufficient dust opacity in
its models, in this regime. Based on this comparative analysis, we provided
average systematic offsets between atmospheric and evolutionary model-derived
$T_{\mathrm{eff}}$, radii, and log $g$ as a function of spectral type for
field and young objects. These systematic offsets might be adopted when
combining fundamental parameter estimates obtained with the two methods.

The large-scale compilation of ultracool dwarf fundamental physical parameters
presented in this work represents a factor of $\sim$5 increase in the sample
size of such semi-empirical measurements compared to previous work.
Determining the ages of ultracool dwarf binaries (e.g. Dupuy et al., 2023),
calculating the radii of candidate transiting planets (e.g. Tamburo et al.,
2022), and constructing luminosity distributions to constrain the substellar
initial mass function (e.g. Kirkpatrick et al., 2021) are just a few of the
many exciting applications of this work as we continue to explore the
fascinating diversity of substellar objects and companions.

## Acknowledgments

The authors would like to thank the anonymous referee for many helpful
comments which improved the manuscript. We also thank Aaron Meisner and Adam
Schneider for helpful discussions on cross-matching ALLWISE sources with
CatWISE2020 sources. This work has benefitted from _The UltracoolSheet_ ,
maintained by Will Best, Trent Dupuy, Michael Liu, Aniket Sanghi, Rob Siverd,
and Zhoujian Zhang, and developed from compilations by Dupuy & Liu (2012),
Dupuy & Kraus (2013), Liu et al. (2016), Best et al. (2018), Best et al.
(2020), and Schneider et al. (2023). 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 (NASA). 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 Maunakea has always had within
the indigenous Hawaiian community. We are most fortunate to have the
opportunity to conduct observations from this mountain.

A. Sanghi acknowledges support from Research Experience for Undergraduate
program at the Institute for Astronomy, University of Hawaii-Manoa funded
through National Science Foundation (NSF) grant #2050710. A. Sanghi would like
to thank the Institute for Astronomy for their hospitality during the course
of this project. This research was funded in part by the Gordon and Betty
Moore Foundation through grant GBMF8550 to M. Liu. W.M.J. Best acknowledges
support from grant HST-GO-15238 provided by STScI and AURA. T. Dupuy
acknowledges support from UKRI STFC AGP grant ST/W001209/1. For the purpose of
open access, the author has applied a Creative Commons Attribution (CC BY)
licence to any Author Accepted Manuscript version arising from this
submission.

This publication makes use of data products from: the _Two Micron All Sky
Survey_ (2MASS), which is a joint project of the University of Massachusetts
and the Infrared Processing and Analysis Center/California Institute of
Technology, funded by the National Aeronautics and Space Administration and
the National Science Foundation; the UKIRT Infrared Deep Sky Survey (UKIDSS)
project, which is defined in Lawrence et al. (2007), uses the UKIRT Wide Field