Uniform Forward-Modeling Analysis of Ultracool Dwarfs. II. Atmospheric Properties of 55 Late-T Dwarfs

DRAFTVERSION MAY 13, 2021 Typeset using LATEX twocolumn style in AASTeX61

ZHOUJIAN ZHANG (张周健), 1, 2 MICHAEL C.LIU,1 MARK S.MARLEY,3, 4 MICHAEL R.LINE,5 AND WILLIAM M.J.BEST6

1Institute for Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA 2Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under contract 80HQTR19D0030 with the National Aeronautics and Space Administration. 3NASA Ames Research Center, Mail Stop 245-3, Moffett Field, CA 94035, USA 4The University of Arizona, Tuscon AZ 85721, USA 5School of Earth & Space Exploration, Arizona State University, Tempe AZ 85287, USA 6Department of Astronomy, University of Texas at Austin, Austin, Texas 78712, USA

(Received August 8, 2020; Revised April 29, 2021; Accepted May 5, 2021) Submitted to ApJ

ABSTRACT We present a large uniform forward-modeling analysis for 55 late-T (T7−T9) dwarfs, using low-resolution (R ≈ 50−250) near- infrared (1.0 − 2.5 µm) spectra and cloudless Sonora-Bobcat model atmospheres. We derive the objects’ effective temperatures, surface gravities, metallicities, radii, masses, and bolometric luminosities using our newly developed Bayesian framework, and use the resulting population properties to test the model atmospheres. We find (1) our objects’ fitted metallicities are 0.3−0.4 dex lower than those of nearby stars; (2) their ages derived from spectroscopic parameters are implausibly young (10 Myr−0.4 Gyr); (3) their fitted effective temperatures show a similar spread as empirical temperature scales at a given spectral type but are ∼ 50 − 200 K hotter for >T8 dwarfs; and (4) their spectroscopically inferred masses are unphysically small (mostly 1 − 8 MJup). These suggest the Sonora-Bobcat assumptions of cloudless and chemical-equilibrium atmospheres do not adequately reproduce late-T dwarf spectra. We also find a gravity- and a metallicity-dependence of effective temperatures as a function of spectral type. Combining the resulting parameter posteriors of our sample, we quantify the degeneracy between fitted surface gravity and metallicity such that an increase in Z combined with a 3.4× increase in logg results in a spectrum that has similar fitted parameters. We note the systematic difference between late-T dwarf spectra and Sonora-Bobcat models is on average ≈ 2% − 4% of the objects’ peak J-band fluxes over the 1.0 − 2.5 µm range, implying modeling systematics will exceed measurement uncertainties when analyzing data with J-band S/N & 50. Using our large, high-quality sample, we examine the spectral-fitting residuals as a function of wavelength and atmospheric properties to discern how to improve the model assumptions. Our work constitutes the largest analysis of brown dwarf spectra using multi-metallicity models and the most systematic examination of ultracool model atmospheres to date.

1. INTRODUCTION As brown dwarfs evolve to cooler effective temperatures, arXiv:2105.05256v1 [astro-ph.SR] 11 May 2021 Characterization of ultracool dwarf spectra is essential to substantial changes in atmospheric chemistry drive the emer- understand the physical and chemical processes in the atmo- gence and disappearance of various atomic and molecular spheres of brown dwarfs and exoplanets. Such processes spectral features. The transition from M to L spectral type govern these objects’ appearance and evolution, and the is marked by the increasing strengths of alkali lines (e.g., emergent spectra encode signatures of their formation path- Na and K) and metal hydride bands (e.g., FeH and CrH), ways (e.g., Burrows et al. 2001; Fortney et al. 2008; Marley and the disappearance of TiO and VO due to the forma- & Robinson 2015). However, spectroscopic analysis of sub- tion of titanium-bearing condensates (e.g., CaTiO3; Burrows stellar objects can be challenging given the numerous com- & Sharp 1999; Lodders 2002). During the M/L transi- plex processes and interactions in ultracool atmospheres. tion, clouds of iron and silicate grains (e.g., MgSiO3 and Mg2SiO4) form and cause the objects’ emergent spectra to 2 become redder toward later L subtypes (Burrows & Sharp limited to the physical assumptions made by grid models and 1999; Marley et al. 2002). The hallmark of the transition has the freedom to explore more physical (and unphysical) from L to T spectral type is the emergence of CH4 absorp- conditions of ultracool atmospheres by using many more free tion, along with the strengthening of H2O bands. The cool parameters (e.g., Line et al. 2015, 2017; Burningham et al. temperatures of the L/T transition (Teff ≈ 1200 − 1400 K) 2017; Zalesky et al. 2019; Gonzales et al. 2020). Aiming to apparently lead to cloud patchiness, condensation below the find models that almost exactly explain the observed spectra, photosphere, and dissipation (e.g., Saumon & Marley 2008; retrieval can robustly test whether the physical assumptions Marley et al. 2010), resulting in brighter J-band emission and made within grid models are reasonable. However, when an bluer J − K colors of early-T dwarfs as compared to late- object has a peculiar spectrum, this data-driven method might L dwarfs (e.g., Knapp et al. 2004; Liu et al. 2006; Dupuy converge to atmospheric compositions and thermal structures & Liu 2012). The transition from T to Y spectral type is that are not physically self-consistent (e.g., Zalesky et al. defined by the emergence of NH3 absorption in the near- 2019). infrared, along with the formation of new condensates (e.g., Studying large samples of brown dwarfs can help surpass Na2S, KCl, and H2O) given the cold atmosphere temperature the aforementioned limitations of both forward modeling and (. 600 K; Cushing et al. 2011; Morley et al. 2012). An alter- retrieval methods, so that fundamental properties of the ul- native interpretation of the L-T-Y evolution does not rely on tracool population can be robustly investigated. In addition, clouds but rather a thermo-chemical instability (e.g., Trem- forward-modeling analysis of an ensemble of brown dwarfs blin et al. 2015, 2016, 2017, 2019). Such an instability can can uncover the discrepancies between data and models, pro- trigger local compositional convection, drive chemical abun- viding feedback about the strength and weakness of current dances (e.g., CO/CH4 and N2/NH3) out of equilibrium, and modeling. Thanks to the wide coverage and depth of imag- reduce the vertical temperature gradient in the atmospheres. ing sky surveys, including Pan-STARRS1 (PS1; Chambers Efforts have been devoted to describe the above compli- et al. 2016), 2MASS (Skrutskie et al. 2006), the UKIRT In- cated processes via models of ultracool atmospheres (e.g., frared Deep Sky Survey (UKIDSS; Lawrence et al. 2007), Baraffe et al. 1998, 2003a; Chabrier et al. 2000; Burrows and the Wide-field Infrared Survey Explorer (WISE, Wright et al. 2002; Marley et al. 2002, 2010, 2017; Burrows et al. et al. 2010; CatWISE, Eisenhardt et al. 2020; Marocco et al. 2006; Saumon & Marley 2008; Allard et al. 2012; Mor- 2020), we now have a large census of brown dwarfs in the ley et al. 2012; Tremblin et al. 2015, 2016; Mollière et al. field and in young associations (e.g., Kirkpatrick et al. 1999, 2019; Phillips et al. 2020). Characterization of brown dwarfs 2011; Luhman 2006; Luhman et al. 2009; Luhman & Mama- is commonly conducted by comparing grids of such pre- jek 2012; Burningham et al. 2008, 2010a, 2013; Best et al. computed, forward models to observations. Synthetic model 2015, 2017, 2018; Best et al. 2020b; Lodieu et al. 2007; spectra have been quite successful in predicting the spectral Lodieu 2013; Gagné et al. 2018; Zhang et al. 2018a; Meisner morphology of brown dwarfs and giant planets, but inconsis- et al. 2020). Most notably, the sample of ultracool dwarfs tencies between data and grid models have been long noticed with near-infrared spectra is abundant (> 20 objects in each (e.g., Cushing et al. 2008; Stephens et al. 2009; Leggett et al. spectral subtype from M6 to T8; e.g., Burgasser 2014; Fil- 2017; Zhang et al. 2020a), indicating that the assumptions of ippazzo et al. 2016; Manjavacas et al. 2019), enabling en- these models (e.g., cloud properties, equilibrium chemistry, semble analyses instead of the more common single-object and radiative convective equilibrium) should be improved. studies. Recently, we have conducted a forward-modeling analysis In this paper, we apply our forward-modeling framework using cloudless Sonora-Bobcat models (Marley et al. 2017; constructed and validated in Paper I to 55 T7–T9 dwarfs Marley et al. submitted) for three benchmark late-T dwarfs, (Teff ≈ 600 − 1200 K). Late-T dwarfs likely lack optically HD 3651B, GJ 570D, and Ross 458C (Zhang et al. 2020b; thick clouds in their near-infrared photospheres (though op- referred to as “Paper I” hereinafter). Comparing our spectral- tically thin, sulfide clouds might exist; e.g., Morley et al. fitting results to those derived from these objects’ bolometric 2012). Therefore, choosing late-T dwarfs helps simplify the luminosities, their primary stars’ ages and metallicities, and model parameter space and can validate the cloud-free mod- the Sonora-Bobcat evolutionary models, we identified poten- els, which serve as the starting point for the more complex tial shortcomings of model predictions of these three objects. cloudy models. We compare our objects’ low-resolution However, such analysis is hampered by the small census of near-infrared spectra to cloudless Sonora-Bobcat models known benchmarks with high-quality spectroscopy, meaning with both solar and non-solar metallicities. Our forward- an insufficient diversity in surface gravity and metallicity at modeling analysis uses the Bayesian framework Starfish a given effective temperature (especially for . 1000 K). (developed by Czekala et al. 2015 and updated by Gully- The atmospheric retrieval technique is an alternative ap- Santiago et al. 2017; see Paper I), which accounts for un- proach for inferring brown dwarf properties, which is not certainties from model interpolation, as well as correlated 3

(data−model) residuals due to instrumental effects and mod- to +70◦1. Information about our sample’s spectroscopy, as- eling systematics, thereby providing robust physical param- trometry, and photometry is listed in Tables1,2, and3, re- eters and more realistic error estimates than the traditional spectively. (χ2-based) methods adopted in most previous work. Our work is the largest analysis of brown dwarf spectra using 2.2. IRTF/SpeX Spectroscopy multi-metallicity models and altogether the most systematic We obtained near-infrared spectra of 16 T7–T9 dwarfs in test of any set of ultracool model atmospheres to date. our sample using IRTF (Table1). We used SpeX in prism We aim to investigate the physical properties of late-T mode with either the 0.500 slit (R ≈ 80 − 250) or the 0.800 dwarfs and examine the model atmospheres through detailed slit (R ≈ 50 − 160). For each target, we took at least six ex- comparisons between data and models. We start with a de- posures in a standard ABBA pattern and contemporaneously scription of our sample (Section2) and present our method- observed a nearby A0V standard star within 0.1 airmass for ology and results of the forward-modeling analysis (Sec- telluric correction. We reduced the data using version 4.1 of tion3). We identify known and candidate binaries in our the Spextool software package (Cushing et al. 2004). Our sample and exclude them from our subsequent analysis (Sec- resulting spectra are reported with vacuum wavelengths and tion4). We then study the inferred atmospheric properties of have typical S/N ≈ 50 per pixel in J band. late-T dwarfs (Section5) and use them to investigate the per- Combining all available IRTF/SpeX spectra, we find 42 formance of the cloudless Sonora-Bobcat models (Section6). objects in our sample have spectra observed using the 0.500 Finally, we provide a summary and a brief discussion of fur- slit and 15 objects using the 0.800 slit, with 2 objects hav- ther work (Section7). ing spectra observed using both slit widths. Therefore, we have in total 57 near-infrared spectra for 55 objects. We flux- calibrate all spectra using the objects’ HMKO magnitudes, which are either observed (e.g., Knapp et al. 2004; Lawrence 2. OBSERVATIONS et al. 2012; Kirkpatrick et al. 2019) or synthesized from other 2.1. Sample of T7–T9 Dwarfs bands by Best et al.(2021). The WFCAM H-band filter We construct our sample of late-T dwarfs by using the cat- and the corresponding zero-point flux are from Hewett et al. alog of ultracool dwarfs by Best et al.(2018) and selecting (2006) and Lawrence et al.(2007), respectively. all T7–T9 objects with prism-mode (R ≈ 50 − 250) spec- 3. FORWARD-MODELING ANALYSIS tra taken from the SpeX spectrograph (Rayner et al. 2003) mounted on the NASA Infrared Telescope Facility (IRTF). 3.1. Methodology These low-resolution spectra are from our own observations In Paper I, we constructed and validated a forward- (Section 2.2), the SpeX Prism Library (Burgasser 2014; Bur- modeling framework using the Bayesian inference tool gasser & Splat Development Team 2017), and the litera- Starfish (Czekala et al. 2015; Gully-Santiago et al. 2017) ture (e.g., Burgasser et al. 2004, 2006a, 2010a; Best et al. and the Sonora-Bobcat models (Marley et al. 2017; Marley 2015), leading to a total of 55 late-T dwarfs. In this sam- et al. submitted), which assume cloudless and chemical- ple, 54 objects have parallaxes (van Leeuwen 2007; Bur- equilibrium atmospheres. Our framework is customized for gasser et al. 2008b; Dupuy & Liu 2012; Leggett et al. 2012; the parameter space relevant to late-T dwarfs: [600,1200] K Gaia Collaboration et al. 2016a,b, 2018; Kirkpatrick et al. in effective temperature Teff, [3.25,5.5] dex in logarithmic 2019; Best et al. 2020a) and the only remaining object, surface gravity logg, and [−0.5,+0.5] dex in bulk metallic- WISE J024512.62−345047.8 (WISE 0245−3450, T8; Mace ity Z (before removal of any species by condensation), and et al. 2013a), is part of our astrometric monitoring program. for near-infrared (0.8−2.5 µm) spectra taken using the prism Four objects in our sample are co-moving companions to ei- mode of IRTF/SpeX. Here we summarize our methods and ther stars or brown dwarfs: HD 3651B (T7.5; Mugrauer et al. refer readers to Paper I and Czekala et al.(2015) for more 2006; Luhman et al. 2007), GJ 570D (T7.5; Burgasser et al. details. 2000), Ross 458C (T8; Goldman et al. 2010; Scholz 2010b), We start our analysis by training Starfish’s spectral emula- and ULAS J141623.94+134836.3 (ULAS 1416+1348; T7.5 tor to generate a probability distribution of model spectra for Scholz 2010a; Burningham et al. 2010b), for which we adopt an arbitrary set of grid parameters. Starfish propagates the the parallaxes of their primary hosts from Gaia DR2 (Gaia Collaboration et al. 2016a, 2018). The first three bench- 1 mark objects have been analyzed in Paper I and here we sim- The five late-T dwarfs with such properties not included in our sample are: Gl 229B (T7; Nakajima et al. 1995), WISEPA J052844.51−330823.9 ply keep them in our sample. Our sample contains 90% of (T7; Kirkpatrick et al. 2011), WISE J214706.78−102924.0 (T7.5; Mace the known T7−T9 dwarfs that have distances 6 25 pc, J- et al. 2013a), WISEPA J085716.25+560407.6 (T8; Kirkpatrick et al. 2011), ◦ band magnitudes 6 17.5 mag, and declinations from −40 and WISEPA J143602.19−181421.8 (T8; Kirkpatrick et al. 2011). 4

0.45

0.40

0.35

0.30

0.25 1 MJup

0.20 10 MJup

Oblateness 70 MJup 0.15 10 Myr 0.10 100 Myr

0.05 1 Gyr 10 Gyr 0.00 0 50 100 150 200 250 300 Equatorial Rotational Velocity (km s 1)

Figure 1. Rotation-induced oblateness as a function of equatorial rotational velocity vrot with different ages and masses, based on the solar- 2 metallicity cloudless Sonora-Bobcat evolutionary models. We compute oblateness as f = 2Cvrot/3gR (e.g., Barnes & Fortney 2003), where {g, R} are derived from the evolutionary models and C = 0.9669 corresponds to the n = 1.5 polytrope representative of fully convective brown dwarfs (e.g., Burrows & Liebert 1993). We use the black horizontal line to mark the critical oblateness ( fcrit = 0.385) below which objects are −1 rotationally stable and have vrot 6 300 km s . These theoretical relations are only slightly changed using the Sonora-Bobcat models with non- solar metallicities (i.e., Z = −0.5 and +0.5). A 10 Gyr-old 70 MJup object can induce the same oblateness if it has a 0.5 dex lower metallicity −1 but ≈ 20 km s higher vrot. Such an effect due to non-solar metallicity diminishes with younger ages and lower masses. resulting interpolation uncertainties into the inferred posteri- fitting process by assuming the rotationally induced oblate- ors and is thereby fundamentally different from linear inter- ness is within the stability limit. For the only object in our polation adopted by traditional (χ2-based) forward-modeling sample (WISE 0245 − 3450) without a parallax, we use the analyses (e.g., Rice et al. 2010; Zhang et al. 2020a). We theoretical relation between the oblateness and the equato- first trim the cloudless Sonora-Bobcat models over the afore- rial rotational velocity based on the cloudless Sonora-Bobcat −1 mentioned grid parameter space and wavelength range, and models (see Figure1) and adopt vmax = 300 km s . then downgrade their spectral resolution using two differ- We also determine three hyper-parameters {aN , aG, `} as ent Gaussian kernels, which correspond to the 0.500 slit and part of our spectral-fitting process. These characterize the the 0.800 slit of SpeX. The convolution also accounts for the final covariance matrix which contains both diagonal and wavelength-dependent spectral resolution of the SpeX prism off-diagonal components. The latter accounts for correlated mode (Rayner et al. 2003). We conduct principal component data−model residuals due to instrumental effects and mod- analysis and train Gaussian processes on these processed eling systematics, leading to more realistic error estimates models, leading to spectral emulators tailored for spectra ob- than a covariance matrix with only diagonal components. We tained with the 0.500 and 0.800 slits. adopt the same hyper-parameter priors as in Paper I, but we We determine six physical parameters: effective tempera- assume a uniform prior of [425,1115 × 5] km s−1 in ` for 00 ture Teff, logarithmic surface gravity logg, metallicity Z, ra- the 0.8 spectra in our sample. This is slightly different from −1 00 dial velocity vr, projected rotational velocity vsini, and log- the [820,1840×5] km s we use for 0.5 spectra, given the arithmic solid angle logΩ = log (R/d)2, where i, R, and different instrumental line spread functions for these two slit d is the inclination of the rotation axis, the radius, and widths (see Appendix of Paper I). the distance, respectively. As in Paper I, we assume uni- We use emcee (Foreman-Mackey et al. 2013) to fit our ob- form priors of [600,1200] K in Teff, [3.25,5.5] dex in logg, jects’ 1.0−2.5 µm spectra with 48 walkers. We terminate the 5 [−0.5,+0.5] dex in Z, (−∞,+∞) for both vr and logΩ, and MCMC fitting process after 10 iterations, since such number [0,vmax] for vsini. We determine vmax using the objects’ dis- of iterations exceeds 50 times the autocorrelation length of tances and fitted {logg, logΩ} in each step of the spectral- all the fitted parameters. To complete our forward-modeling 5

Figure 2. Starfish-based forward-modeling results of our late-T dwarf sample. Left: The upper panel shows the observed spectrum (black) of each object and the median Sonora-Bobcat model spectra of those interpolated at parameters drawn from the MCMC chains (blue). The object’s name, spectral type, the slit width and J-band S/N (median S/N in 1.2 − 1.3 µm) of its spectrum, and inferred physical parameters are in the upper right corner. The lower panel shows the residual (data−model; black), with the blue shadows being 1σ and 2σ dispersions of 5 × 104 draws from the Starfish’s full covariance matrix. Right: Posteriors of the six physical parameters {Teff, logg, Z, vr, vsini, logΩ} derived from the Starfish-based forward-modeling analysis. We use grey vertical and horizontal lines to mark the {Teff, logg, Z} grids points of the cloudless Sonora-Bobcat models. Figures of spectral-fitting results for our entire sample (55 late-T dwarfs with 57 spectra) are accessible online. analysis, we add the following systematic uncertainties into We do not incorporate these small shifts (which are mostly the resulting parameter posteriors: 20 K in Teff, 0.2 dex in smaller than the uncertainties from the spectral fitting) into −1 −1 logg, 0.12 dex in Z, 180 km s in vr, 40 km s in vsini, the systematic errors, so our spectroscopically inferred prop- 2 2 1/2 and [0.05 + (0.4σHMKO ) ] dex in logΩ, where σHMKO is erties are all tied to the objects’ H-band photometry. the H-band magnitude uncertainty. These systematic errors were determined in Paper I by (1) fitting the original Sonora- 3.2. Results Bobcat model atmospheres themselves using our forward- Figure2 presents the resulting parameter posteriors of our modeling framework to quantify the uncertainties introduced late-T dwarfs and compares the observed data with Sonora- by Starfish’s spectral emulator, (2) incorporating the uncer- Bobcat model spectra interpolated at the physical parameters tainty in the wavelength calibration of the SpeX prism data drawn from the MCMC samples. These observed data and into the systematics of vr, and (3) incorporating the uncer- fitted model spectra of our entire sample are summarized in tainty in the flux calibration of spectra (due to H-band mag- Figure3. We list the fitted physical parameters and uncer- nitude errors) into the systematics of logΩ. We note all these tainties in Table4. 2 We further use the objects’ parallaxes added uncertainties are smaller than or comparable to the for- and their fitted logg and logΩ to derive their radii (R) and mal fitting errors of physical parameters (see Table6). masses (M). We also compute the bolometric luminosity In addition, we flux-calibrate the objects’ spectra using H- (Lbol) of each object by integrating its observed 1.0−2.5 µm band magnitudes in this work, and conducting this process SpeX spectrum combined with the fitted model spectra at using photometry in different bands will alter our resulting shorter and longer wavelengths spanning 0.4 − 50 µm. We logΩ posteriors (e.g., Figure 8 of Line et al. 2015). Each have incorporated uncertainties in observed spectral fluxes, late-T dwarf in our sample has similar photometric uncer- parallaxes, and fitted physical parameters into our resulting tainties among J, H, and K bands. Therefore, the objects’ Lbol values in a Monte Carlo fashion. Such computed Lbol’s logΩ posteriors, derived using spectra calibrated by JMKO or are not necessarily equal to those derived from the objects’ KMKO, will have different median values but similar uncer- Teff and R posteriors via the Stefan-Boltzmann law, given tainties compared to those inferred in this work using HMKO. the mismatch between observed spectra and fitted models in As a reference, Table4 lists the shifts ( ∆logΩ) that our fitted logΩ values would be changed by if our objects’ spectra were instead flux-calibrated using J or K band. Our derived 2 The spectroscopically inferred radial velocities and projected rotational spectroscopic radii, masses, and bolometric luminosities (see velocities in our analysis cannot be well constrained due to the low spectral resolution of the data. These physical parameters (vr and vsini) are in Section 3.2) would be accordingly increased by 0.5∆logΩ, fact coupled with the model systematics but have no correlations with, and ∆logΩ, and ∆logΩ in the logarithmic scale, respectively. thereby no impact on, the other physical parameters (Figure5; also see Sec- tion 4.2 of Paper I). 6

WISE 0040 + 0900 (T7.0) WISE 2340 0745 (T7.0) 2MASS 0729 3954 (T8.0pec)

2MASS 0050 3322 (T7.0) WISE 2348 1028 (T7.0) 2MASS 0939 2448 (T8.0)

WISE 0123 + 4142 (T7.0) HD 3651B (T7.5) ULAS 1029 + 0935 (T8.0)

WISE 0241 3653 (T7.0) WISE 0223 2932 (T7.5) Ross 458C (T8.0) 15 WISE 0325 + 0831 (T7.0) WISE 0521 + 1025 (T7.5) WISE 1322 2340 (T8.0)

WISE 0614 + 0951 (T7.0) WISE 1039 1600 (T7.5) WISE 1653 + 4444 (T8.0)

2MASS 0727 + 1710 (T7.0) WISE 1052 1942 (T7.5) WISE 1711 + 3500 (0.500; T8.0)

WISE 1124 0421 (T7.0) 2MASS 1114 2618 (T7.5) WISE 1711 + 3500 (0.800; T8.0)

WISE 1254 0728 (T7.0) 2MASS 1217 0311 (T7.5) WISE 1813 + 2835 (T8.0) 10 WISE 1257 + 4008 (T7.0) ULAS 1416 + 1348 (sd T7.5) WISE 1959 3338 (T8.0)

WISE 1457 + 5815 (T7.0) GJ 570D (T7.5) WISE 2000 + 3629 (T8.0)

PSO J224 + 47 (T7.0) WISE 1809 + 3838 (T7.5) WISE 2226 + 0440 (T8.0)

SDSS 1504 + 1027 (T7.0) WISE 2319 1844 (T7.5) WISE 2255 3118 (T8.0)

2MASS 1553 + 1532 (T7.0) ULAS 2321 + 1354 (T7.5) WISE 0049 + 2151 (0.500; T8.5)

Scaled Flux + constant 5

SDSS 1628 + 2308 (T7.0) WISE 0245 3450 (T8.0) WISE 0049 + 2151 (0.800; T8.5)

WISE 1852 + 3537 (T7.0) PSO J043 + 02 (T8.0) WISE 0458 + 6434 (T8.5)

WISE 2157 + 2659 (T7.0) 2MASS 0415 0935 (T8.0) UGPS 0521 + 3640 (T8.5)

WISE 2209 2734 (T7.0) WISE 0500 1223 (T8.0) UGPS 0722 0540 (T9.0)

WISE 2213 + 0911 (T7.0) WISE 0623 0456 (T8.0) WISE 1741 + 2553 (T9.0) 0 1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 wavelength ( m)

Figure 3. Observed spectra (black) of our entire late-T dwarf sample and the median Sonora-Bobcat model spectra interpolated at parameters drawn from the MCMC chains (blue).

1.0 − 2.5 µm wavelengths. We confirm our objects’ bolo- and the Sonora-Bobcat evolutionary models. Assuming the metric luminosities computed from these two approaches are evolutionary-based parameters are more robust, we found the all consistent within uncertainties. For HD 3651B, GJ 570D, accuracy of our forward-modeling results exhibited two out- and Ross 458C, we adopt the more accurate Lbol derived in comes. For HD 3651B and GJ 570D, our spectral fits pro- our Paper I using their primary stars’ ages and metallici- duced reliable Teff and R, but underestimated logg and Z ties and the Sonora-Bobcat evolutionary models. We com- by ≈ 1.1 − 1.3 dex and ≈ 0.3 − 0.4 dex, respectively. For pare our computed Lbol with literature in AppendixA and Ross 458C, our spectral fit produced reliable logg and Z, calculate bolometric corrections for T7−T9 dwarfs in Ap- but overestimated Teff by ≈ 120 K and underestimated R by pendixB. The derived physical properties ( R, M, Lbol) of our ≈ 0.4 RJup (or a factor of ≈ 1.6). Underestimation of the objects are summarized in Table5. spectroscopically inferred logg and/or R further led to un- We summarize the typical parameter uncertainties in Ta- physically small masses of these objects. The late-T dwarfs ble6 and find the resulting {Teff, logg, Z} uncertainties of in our sample might have their fitted parameters biased by our sample are about 1/3 − 1/2 of the Sonora-Bobcat model similar amount as these benchmarks, and in Sections5 and grid spacing, the same as our finding in Paper I for the three 6, we demonstrate that the spectroscopically inferred param- late-T benchmarks, HD 3651B, GJ 570D, and Ross 458C. eters of this ensemble of late-T dwarfs are useful to assess The fitted models generally match well the observed spectra, the model atmospheres. and we discuss the data-model comparison in Section6. We also conducted a forward-modeling analysis follow- Some of our spectroscopically inferred physical parame- ing the traditional approach, where we use linear interpola- ters of late-T dwarfs can be under- or over-estimated rel- tion to generate the model spectrum in between grid points ative to their true values, given that our set of models as- and adopt a diagonal covariance matrix (defined by observed sume cloudless and chemical-equilibrium atmospheres. In flux uncertainties) to evaluate free parameters. We describe Paper I, we analyzed three late-T benchmarks and compared details in AppendixC and present results in Figure 20 and the fitted parameters to those derived from their bolomet- Table9. Our Starfish analysis produces generally consis- ric luminosities, their primary stars’ metallicities and ages, tent physical parameters but with more realistic error esti- 7

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R 10 M Traditional ( ( / < 5% (0.500/0.800) 0.05 5 / 5% (0.500/0.800)

0.00 0 0 25 50 75 100 125 150 175 0 25 50 75 100 125 150 175 J-band S/N J-band S/N

Figure 4. Parameter uncertainties from Starfish (blue) and traditional (purple; AppendixC) forward-modeling analysis as a function of the J-band S/N, i.e., the median S/N in 1.2 − 1.3 µm. We use solid circles for results using spectra taken with the 0.500 slit and open circles for 00 the 0.8 slit. The uncertainties of {Teff, logg, Z, vr, vsini, logΩ} have a significant spread at low S/N, but then shrink to lower values at S/N & 50. Such trends also hold for R and M uncertainties of the objects with high-precision parallaxes (e.g., σπ/π < 5%). The vr uncertainties derived from the lower-resolution spectra (0.800 slit) are systematically larger than those from the higher-resolution ones as expected. Also, −1 the vr uncertainties from both Starfish and traditional spectral-fitting methods are larger than the 180 km s uncertainty in the wavelength calibration of the SpeX prism data. mates than the traditional approach. We therefore adopt the Our near-infrared spectra span a wide range in J-band Starfish-based properties as final results of our atmospheric S/N3 from 15 − 180 (Table1), and here we examine the im- model analysis and use them for subsequent discussions. pact of the S/N on the precision of our fitting results. Fig- In the rest of this section, we study the impact of the data ure4 plots uncertainties of the eight physical parameters S/N and spectral resolution on our forward-modeling analy- {Teff, logg, Z, vr, v sini, logΩ, R, M} as a function of J- sis (Section 3.2.1) and investigate the correlations among the band S/N. The uncertainties of the first six parameters have a inferred physical parameters (Section 3.2.2).

3 The median S/N around the J-band peak, i.e., 1.2 − 1.3 µm, of the 3.2.1. Impact of S/N and Spectral Resolution object’s spectra. 8

Figure 5. Stacked posteriors of the eight physical parameters {Teff, logg, Z, vr, vsini, logΩ, R, M} from all the MCMC chains of our late-T dwarf sample. These chains have been subtracted by their medians, so they represent the deviations of parameter posteriors from median values, which we denote as ∆Teff, ∆logg, ∆Z, etc. We stack all the modified chains, generate stacked posteriors, and compute the Pearson correlation coefficients between each pair of parameters and give the coefficient values in the lower left corner of each panel. Most parameter pairs have very weak correlations (grey) with the absolute values of their coefficients below 0.2. We find four parameter pairs (blue) with large correlation coefficients (absolute values above 0.4): (∆Teff, ∆logΩ), (∆R, ∆M), (∆logg, ∆vsini), and (∆logg, ∆Z). The strong correlations within the first three pairs are as expected, while the fourth one suggests a logg − Z degeneracy (Equation1) in our forward-modeling analysis. significant spread at low S/N, but then shrink to small value the parallax uncertainties. The R and M uncertainties of the at S/N& 50. We note the Starfish-derived parameter errors at objects with higher-precision parallaxes (e.g., relative uncer- high S/N (& 50) are in fact dominated by modeling systemat- tainties . 5%) have the same dependence on S/N as the first ics (instead of measurement uncertainties), which constitute six physical parameters. ≈ 2% − 4% of the peak J-band flux of late-T dwarf spectra We also investigate the impact of spectral resolution, since (see Section 6.1) and cause the parameter precision to be in- our sample contains spectra observed with both the 0.500 dependent of the data S/N in this regime. The precision of (R ≈ 80 − 250) and 0.800 (R ≈ 50 − 160) slits. As shown in R and M depends not only on the S/N of spectra but also on Figure4, the radial velocity uncertainties derived from the 9 lower-resolution spectra are systematically larger than those are naturally correlated due to the Stefan-Boltzmann law; (2) from the higher-resolution ones, as expected, but such dis- the mass is computed from radii and surface gravity and has a tinction between the two resolutions is not seen for the other stronger dependence on radii; and (3) the inferred vsini value parameters. of each object is primarily constrained by the prior, which is Two objects in our sample, WISEPA J171104.60 + determined by surface gravity, solid angle, and distance (Sec- 350036.8 (WISE 1711 + 3500; Kirkpatrick et al. 2011) and tion 3.1 and Equation 2 of Paper I). WISE J004945.61 + 215120.0 (WISE 0049 + 2151; Mace The fourth correlation between logg and Z is interesting. et al. 2013a), have spectra taken with both slits. Comparing Conducting an orthogonal distance regression to the modi- results from each object with the two resolutions, we find fied chains of ∆logg and ∆Z using a straight line across the almost all parameters are consistent within 0.6σ. The only origin, we derive the following relation: exception is the radial velocity of WISE 1711 + 3500, where the inferred values between the two slits differ by 2.1σ. ∆ logg = 3.42 × ∆Z (1) WISE 1711 + 3500 is a resolved 0.7800 T8+T/Y binary (Liu et al. 2012; also see Section4), so the two slits will exclude The qualitative form of such logg−Z relation has been noted different amounts of the integrated light from the system and from previous spectroscopic analyses using other model at- may cause the discrepant radial velocities. Also, the J-band mospheres (e.g., Burgasser et al. 2006a; Leggett et al. 2007; S/N of this object’s 0.500 spectrum is quite low (≈ 20) com- Liu et al. 2007; Burningham et al. 2009). The cause of this pared to the 0.800 spectrum (S/N ≈ 35), so the radial velocity degeneracy is that these two parameters have similar effect inferred from the 0.500 spectrum might be less accurate.4 on the spectral morphology of cloudless model atmospheres, as either a high (low) logg or a low (high) Z leads to the same 3.2.2. Parameter Correlations via Stacked Posteriors suppressed (enhanced) K-band flux in late-T dwarf spectra. In addition, we find that Z has the noticeable correlation only We now investigate the correlations between the physical with the logg parameter in our analysis, suggesting that ap- parameters {T , logg, Z, v , vsini, logΩ, R, M}. The eff r plying the solar-metallicity cloudless Sonora-Bobcat models correlations can shed light on how parameters change when to late-T dwarfs whose true metallicities are non-solar can one is under- or over-estimated. This information is use- bias logg but not the other parameters. ful when we examine the results from previous forward- modeling analyses (e.g., Cushing et al. 2008; Stephens et al. 4. BINARY SYSTEMS 2009), which relied on only solar-metallicity model atmospheres to analyze ultracool dwarfs which might have sub- Now we examine binary systems in our sample, for which solar or super-solar metallicities. our derived atmospheric model properties will be not reli- To investigate the parameter correlations, we stack the pa- able. This is because some of the integrated light from these rameter posteriors of all our late-T dwarfs which are directly systems might be missing from our prism spectra (depend- 00 00 from the formal spectral fitting and have no systematic errors ing on their angular separations relative to the 0.5 and 0.8 incorporated.5 For each object’s chain, we first subtract the slit widths), and our spectral-fitting process assumes all ob- median of each posterior so that the chain then represents the jects are single. We first discuss known resolved binaries deviations of the posteriors from the median values, which (Section 4.1) and then identify candidate binaries based on their spectrophotometric properties (Section 4.2). In the end, we denote as ∆Teff, ∆logg, ∆Z, etc. Then we concatenate all the modified chains to generate the stacked parameter pos- we exclude six known or likely binaries from our subsequent teriors (Figure5). We use these stacked posteriors to com- discussions. pute the Pearson correlation coefficients between all pairs of parameters. We find that four pairs exhibit significant cor- 4.1. Resolved Binaries relations: (∆Teff, ∆logΩ), (∆R, ∆M), (∆logg, ∆vsini), Three late-T dwarfs in our sample are known to be resolved and (∆logg, ∆Z). The correlations within the first three binaries: 2MASSI J1553022 + 153236 (2MASS 1553 + pairs are expected: (1) effective temperature and solid angle 1532), WISE 1711 + 3500, and WISEPA J045853.89 + 643452.9 (WISE 0458 + 6434). 2MASS 1553 + 1532 (T7) is a resolved 0.3500 T6.5+T7 binary (Burgasser et al. 2006c), 4 The J-band S/N of WISE 0049 + 2151 (which does not show the discrepant atmospheric parameters between two slits) is ≈ 80 for its 0.500 spec- with a bolometric luminosity ratio of 0.31 ± 0.12 mag and 6 trum and ≈ 35 for its 0.800 spectrum. a mass ratio of 0.90 ± 0.02 . Our fitted model spectra 5 We note the parameter correlation involves the change of parameters’ median values (due to the intrinsic parameter degeneracy). The systematic errors incorporated to our results only slightly inflate the parameter uncer- 6 Burgasser et al.(2006c) estimated the bolometric luminosity of binary tainties (instead of their median values; Section 3.1) and thus have no impact components using their HST/NICMOS resolved photometry and a bolomet- to the parameter correlations that we investigate here. ric correction, which they derived as a function of near-infrared colors using 10

10 T7+T7.5 Dwarfs ±0.07 RJup T8+T8.5 Dwarfs T9 Dwarfs 8

Number WISE 2340 0745 4 2MASS 1553 + 1532 WISE 1039 1600

0 0.4 0.6 0.8 1.0 1.2 R (RJup)

Figure 6. Distribution of our derived atmospheric radii for late-T dwarfs (violet for T7 and T7.5, orange for T8 and T8.5, and green for T9). We label the three objects whose radii stand out with large values, among which 2MASS 1553 + 1532 is a resolved binary and the other two are candidate binaries (Section4). of this object match the observed spectrum, but the fit- on their retrieval analysis which also did not account for +0.10 ted R = 1.21−0.09 RJup stands out from our sample (Fig- binarity. ure6) and is larger than the evolutionary model predictions WISE 1711+3500 (T8) is a resolved 0.7800 T8+T/Y binary (≈ 0.75 − 1.20 RJup; e.g., Burrows et al. 1997; Saumon (Liu et al. 2012). The binary has near-infrared (YJHK bands) & Marley 2008; Marley et al. submitted), as well as the flux ratios of 2.6 − 3.1 mag, indicating the integrated-light directly measured radii of transiting brown dwarfs, e.g., spectra of this system is dominated by the T8 primary. This +0.038 +0.16 KELT-1b (1.116−0.029 RJup; Siverd et al. 2012), KOI-205b might explain why its fitted radii R = 0.93−0.12 RJup is not as (0.81 ± 0.02 RJup; Díaz et al. 2013), and LHS 6343 C large as the nearly equal-brightness binary 2MASS 1553 + (0.783 ± 0.011 RJup; Montet et al. 2015). The large radii 1532. of 2MASS 1553 + 1532 is in accord with its nearly equal- WISE 0458 + 6434 (T8.5) is a resolved 0.5100 T8.5+T9 brightness binarity. A similar large spectroscopic radii binary (Gelino et al. 2011; Burgasser et al. 2012; Leggett +0.14 (R = 1.59−0.09 RJup) was inferred by Line et al.(2017) based et al. 2019) with near-infrared (JHK bands) flux ratios of 1.0 − 1.1 mag. The fitted model spectra from our analysis do not match the observed spectrum and predict too faint H- the photometry of unresolved T dwarfs in their sample and the K-band bolometric correction from Golimowski et al.(2004). Burgasser et al. estimated band fluxes. Also, this object’s parameter posteriors are bi- the mass of the binary components using the objects’ effective temperatures modal based on the traditional method (AppendixC), with based on the Golimowski et al.(2004) Teff-SpT relation, an assumed age of one peak consistent with the Starfish results and the other 0.5 − 5.0 Gyr, and the Burrows et al.(1997) evolutionary models. peak being 160 K cooler in Teff, 1.5 dex higher in logg, and 0.2 dex higher in Z. While the bimodal posteriors might 11 Spectral Type T1 T3 T5 T7 T9 Field Ultracool Dwarfs Single Resolved Binary

Our Late-T Dwarfs Single Likely Binary Resolved Binary

WISE 2340–0745 SDSS 1504+1027 2MASS 1553+1532AB WISE 1039–1600

WISE 1711+3500AB ULAS 1416+1348 2MASS 0939–2448 WISE 0458+6434AB

WISE 0500–1223

Figure 7. W2-band absolute magnitudes as a function of W1 −W2 colors for our late-T dwarfs and field ultracool dwarfs (Best et al. 2020b) whose absolute magnitudes and colors both have S/N> 5. We use red solid circles for resolved late-T binaries in our sample (Section 4.1), red open circles for likely binaries (Section 4.2), and blue solid circles for the remaining. For the field objects, we use white squares for resolved ultracool binaries and solid squares for those which are single or have unknown binarity. The upper x-axis shows corresponding T spectral types based on median W1−W2 colors of field dwarfs from Best et al.(2018). We draw a line in the diagram to help distinguish the T-type resolved binaries from the others, with the boundary defined by points of (W1−W2, M(W2)) = (0.72,9.4), (0.72,11.9), (2,12.3), (3,13.3), (4.5,13.3). T-type binaries tend to have brighter magnitudes and redder colors than such boundary in this diagram, but we caution that low-metallicity, high-gravity single objects might occupy the same area as the resolved binaries (Section 4.2). be due to the object’s binarity, they are likely caused by the (2) The parameter posteriors derived from either Starfish low J-band S/N= 24 of the data, which prevents the spectral- or traditional forward-modeling analysis are bimodal, fitting process from converging on a unique set of solutions. or We exclude all these three known binaries from our subse- (3) The AllWISE photometry is brighter and redder than quent analysis (Sections5 and6). most other objects in our sample, with critical values shown in Figure7. 4.2. Candidate Binaries Figure7 compares the integrated-light photometry of our Here we choose three criteria to identify candidate bina- late-T dwarfs to known field dwarfs from Best et al.(2020a), ries in our sample based on their fitted model parameters and which shows that T-type resolved binaries have distinctly observed photometry: brighter W2-band absolute magnitudes and redder W1 −W2 colors than the majority of field dwarfs which are single (1) The fitted R are outliers in the derived radii distribution (or not known to be binaries). We draw a line in Figure7 of our sample (Figure6) with very large values (e.g., to help distinguish the resolved binaries from the others. & 1.2 RJup), We caution that this line can also select single objects with 12 high-gravity, metal-poor atmospheres, as demonstrated by also cause this anomaly. We thus suggest a re-analysis of this 2MASS 0939−2448 and ULAS 1416+1348 (discussed fur- system using higher-quality data. ther below). Either high gravity or low metallicity can cause The remaining three candidates are all selected by the objects’ photospheres to reside at higher pressures, favor- the third criterion, i.e., anomalous AllWISE photometry. ing the formation of CH4 (with strong absorption in W1) over 2MASS 0939 − 2448 (T8) has a metal-poor atmosphere CO (with strong absorption in W2; e.g., Liebert & Burgasser given its broad Y-band and faint K-band spectra. We derive a +0.15 2007; Zahnle & Marley 2014; Leggett et al. 2017). This leads low metallicity, Z = −0.41−0.09 dex, which is consistent with to redder W1−W2 colors and might make such objects have the past forward-modeling (Burgasser et al. 2006a, 2008b; similar AllWISE photometry as binaries. Leggett et al. 2009) and retrieval analyses (Line et al. 2017). We select candidate binaries whose properties satisfy Combining near-infrared and mid-infrared spectra, both Bur- at least one of our criteria and end up with six ob- gasser et al.(2008b) and Leggett et al.(2009) have suggested jects: WISE J103907.73 − 160002.9 (WISE 1039 − 1600), 2MASS 0939 − 2448 is an unresolved binary, primarily due WISEPC J234026.62 − 074507.2 (WISE 2340 − 0745), to its uncommon brightness in the mid-infrared. However, WISEPA J050003.05 − 122343.2 (WISE 0500 − 1223), Dupuy & Liu(2012) suggested that 2MASS 0939 − 2448 2MASS J09393548 − 2448279 (2MASS 0939 − 2448), might be a single object with a very red W1 − W2 color ULAS 1416 + 1348, and SDSS J150411.63 + 102718.4 due to its low metallicity and/or high gravity, given that its (SDSS 1504 + 1027). Based on the recent near-infrared near-infrared absolute magnitudes are not brighter than field adaptive optics imaging (M. Liu, private communication), all dwarfs with similar spectral types. these candidates are unresolved down to 0.100, making their ULAS 1416 + 1348 (T7.5) is a 900 (85 au) companion potential binarity intriguing. (Burningham et al. 2010b; Scholz 2010a) to a L6 dwarf WISE 1039−1600 (T7.5) and WISE 2340−0745 (T7) are SDSS J141624.08+134826.7 (Bowler et al. 2010a; Schmidt selected by both the first and third criteria, with their spec- et al. 2010).7 Burningham et al.(2010b) noted the unusu- troscopically inferred radii among the largest in our sample ally red H −W2 and W1 −W2 colors of this object and con- (Figure6) and their AllWISE photometry being anomalous nected these properties to its metal-poor atmosphere. The (Figure7). The fitted model spectra of these two objects very blue Y − K color of this companion suggests high grav- match the observed spectra, so their large radii are unlikely ity and low metallicity, in accord with our derived logg = +0.26 +0.14 caused by poorly fitted models. Also, their kinematics do 5.21−0.25 dex and Z = −0.39−0.11 dex, which are consis- not support membership in any young associations (based tent with past spectral analyses (Burgasser et al. 2010b; Line on BANYAN Σ [Gagné et al. 2018] or LACEwING [Riedel et al. 2017; Gonzales et al. 2020).8 Kirkpatrick et al.(2019) et al. 2017]; see Zhang et al. 2021) and their fitted logg are suggested ULAS 1416 + 1348 is an unresolved binary, given consistent with the other late-T dwarfs in our sample. There- that three known late-T subdwarfs (BD +01◦ 2920B [Pin- fore, we find no evidence of youth that would lead to their field et al. 2012], Wolf 1130C [Mace et al. 2013b], and large radii (e.g., Burrows et al. 2001; Kirkpatrick et al. 2008; WISE J083337.82+005214.1 [Pinfield et al. 2014]) have sim- Allers & Liu 2013). In addition, their near-infrared spectra ilar positions as field dwarfs in the MH vs. H −W2 diagram do not exhibit any peculiarities that might result from high- whereas ULAS 1416 + 1348 is an apparent outlier. gravity, metal-poor atmospheres. Therefore, we conclude SDSS 1504 + 1027 (T7) has a normal near-infrared spec- their large atmospheric radii and atypical AllWISE photom- trum given its spectral type, and our fitted model spec- etry are likely caused by binarity. tra match the data (J-band S/N of 50 per pixel). Aberas- WISE 0500 − 1223 (T7) is selected by the second crite- turi et al.(2014) obtained high spatial-resolution images us- rion, i.e., bimodal posteriors from atmospheric model analy- ing HST/WFC3 and did not resolve this object down to a sis. Similar to WISE 0458 + 6434, the parameter posteriors of this object based on the traditional forward-modeling anal- 7 The primary has weak TiO and CaH absorption features in opti- ysis have two peaks, with one consistent with the Starfish cal, an unusually blue J − K color, enhanced FeH 0.99 µm absorption, results, and the other being 120 K cooler in Teff, 1.5 dex deep H2O absorption features, and thin-disk kinematics (e.g., Bowler et al. higher in logg, and 0.4 dex higher in Z. Its W2-band ab- 2010a; Schmidt et al. 2010). Bowler et al.(2010a) suggested these fea- solute magnitude is fainter than our critical line in Fig- tures are caused by the object’s low metallicity and its less-opaque conden- sate clouds than the majority of L6 dwarfs (similar to the blue L4.5 dwarf ure7, although this line is not well-established around the 2MASS J11263991 − 5003550 as discussed by Burgasser et al. 2008a). The W1 −W2 = 3.47 ± 0.16 mag of WISE 0500 − 1223, given metallicity and age of the system are not well constrained as compared to that there are too few known binaries with such red colors. the three benchmark companions (HD 3651B, GJ 570D, and Ross 458C) studied in Paper I. While the bimodal posteriors of this object could be caused 8 The best-fit parameters derived by Burgasser et al.(2010b) using the by binarity, we note the low J-band S/N≈ 13 of data might Saumon & Marley(2008) spectral models also favors a weak-mixing atmo- 4 2 −1 sphere with the vertical diffusion coefficients of Kzz = 10 cm s . 13

1.4 Our Late-T Dwarfs 1.2 Hypatia Catalog

1.0

0.8

0.6

0.4 Normalized Number 0.2

0.0 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 1.4 25pc Sample 1.2

1.0

0.8

0.6

0.4 Normalized Number 0.2

0.0 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 10 T7+T7.5 Dwarfs T8+T8.5 Dwarfs 8 T9 Dwarfs Hypatia Catalog

±0.16 dex 4 Number

0 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Z (dex)

Figure 8. Top: Metallicity distributions for late-T dwarfs (binaries removed) from our atmospheric model analysis and for thin-disk stars from the Hypatia catalog (Hinkel et al. 2014, 2016, 2017). For our late-T dwarfs, we generate 104 metallicity distributions with each constructed by a random draw from each object’s Z posterior. We then plot the median and the 1σ conﬁdence interval of these distributions using the blue line and shadow, respectively. Distributions for Hypatia stars (black) are produced in the same approach except that we assume the metallicity of each star follows a Gaussian distribution. We normalize these two types of distributions so that their peaks correspond to one. Middle: Similar format as the top but with late-T and stellar samples constrained to 6 25 pc. Bottom: The distribution of our late-T dwarfs, with the violet color for T7 and T7.5, orange for T8 and T8.5, and green for T9. Our typical Z uncertainty is shown as the black error bar. We use the vertical black line and grey shadow to mark the median (0.04 dex) and the 1σ spread (−0.21 to 0.22 dex) of the Hypatia stars. resolution of 0.09600 − 0.14200 and a magnitude contrast of etry is likely caused by low-S/N data or abnormal atmo- 1.5 − 2.0 mag. spheres. However, we do not rule out the possibility that To conclude, we ﬂag WISE 1039 − 1600, WISE 2340 − these objects are tight unresolved binaries. 0745, and SDSS 1504 + 1027 as likely binaries, and exclude them from our subsequent analysis (Sections5 and 5. ATMOSPHERIC PROPERTIES OF LATE-T DWARFS 6) along with the three known binaries discussed in Sec- Here we examine our forward-modeling results on the dis- tion 4.1. For the remaining candidates, WISE 0500 − 1223, tributions of metallicities (Section 5.1) and ages (Section 5.2) 2MASS 0939−2448, and ULAS 1416+1348, we keep them of late-T dwarfs, as well as their effective temperature– in our analysis, given that their anomalous spectrophotom- spectral type relation (Section 5.3). We demonstrate spectral- 14

fitting results for an ensemble of objects can provide use- et al. 2017). Metallicity estimates are viable using model at- ful diagnostics about the physical assumptions made within mosphere analyses. While there has been previous such stud- model atmospheres, by virtue of having a large sample of ies for one to a few objects (e.g., Saumon et al. 2007; Bowler spectra over a focused spectral type range (as opposed to et al. 2010b; Leggett et al. 2017), many analyses have used a smaller sample of objects spanning a wide spectral type models with only the solar metallicity (e.g., Cushing et al. range). 2008; Stephens et al. 2009; Del Burgo et al. 2009; Liu et al. 2011a; Schneider et al. 2015; Deacon et al. 2017). 5.1. Metallicity Distribution Recently, some studies have begun characterizing metallicities for large samples of ultracool dwarfs. Zhang et al. We investigate the metallicity of 49 late-T dwarfs9 in our (2017; 2018b) studied 28 M7−L7 subdwarfs with thick-disk sample (with the other 6 objects excluded due to their known or halo membership. They determined the objects’ physi- or likely binarity; Section4) and compare them to metallic- cal properties by visually comparing their optical and near- ities of nearby stars. This analysis can in principle assess infrared spectra to BT-Settl model atmospheres (Allard et al. whether these two populations have similar formation his- 2011; Allard 2014) and found low [Fe/H] values, spanning tories but can also shed light on the accuracy of our fitted −2.5 to −0.5 dex. Leggett et al.(2017) visually compared atmospheric properties. near-infrared spectra of 20 Y dwarfs to the Tremblin et al. Spectroscopic analysis of stellar metallicities and elemen- (2015) models, which assume cloudless atmospheres with tal abundances involves (1) measuring equivalent widths of dis-equilibrium chemistry and reduced vertical temperature individual absorption lines and then converting them into gradient, and found solar-like metallicities with a spread of abundances using the curve of growth and theoretical line ±0.3 dex. Line et al.(2017) conducted a retrieval analysis lists (e.g., Edvardsson et al. 1993; Santos et al. 2004; Bond for 11 late-T dwarfs and found notably lower metallicities et al. 2006; Ramírez et al. 2013; Soto & Jenkins 2018), or (2) (spanning −0.4 to 0.1 dex) and higher carbon-to-oxygen ra- fitting the entire observed spectra using stellar model atmo- tios (C/O) than nearby FGK stars. These results are likely spheres with a range of abundances (e.g., Valenti & Fischer related to the sequestration of oxygen into condensates in 2005; Jenkins et al. 2008; Petigura & Marcy 2011; Brewer late-T atmospheres, which then rain out of the photosphere et al. 2016). The abundances of several thousands of nearby (e.g., Fegley & Lodders 1994; Marley & Robinson 2015), ( 100 pc) stars have been studied via high-resolution spec- . resulting in retrieved metallicities and C/O that are system- troscopy (R 3 × 104). The vast majority of these stars have & atically too low and high, respectively. Applying the Line similar iron content ([Fe/H]) as our Sun (e.g., Figure 4 of et al.(2017) methodology to 14 late-T and Y dwarfs, Zalesky Hinkel et al. 2014). While it has been known that different et al.(2019) derived metallicities that span −0.2 to 0.6 dex, methods, line lists, and spectral resolutions can lead to no- which are consistent with (if not slightly higher than) those table spreads in the derived metallicities and elemental abun- of nearby stars. dances (e.g., Torres et al. 2012; Smiljanic et al. 2014; Hinkel Our work analyzes 49 late-T dwarfs and thus constitutes et al. 2016; Ivanyuk et al. 2017), much effort has been made the largest homogeneous analysis of brown dwarf metallic- to construct large, homogeneous stellar abundance catalogs ities to date. Figure8 compares the derived metallicities of (e.g., Hinkel et al. 2014, 2017; Brewer et al. 2016). our late-T dwarfs to those of F0–M4 stars from the Hypatia In contrast, systematic metallicity analysis of substellar ob- catalog10 (Hinkel et al. 2014, 2016, 2017). The latest version jects has been lacking in most previous spectroscopic studies. of this catalog (as of June 2019, compiling nearly 200 litera- For late-type ultracool dwarfs, direct metallicity measure- ture studies) contains 6196 stars within 150 pc that have spec- ments from atomic/molecular absorption features are partic- troscopically determined [Fe/H] and at least one other ele- ularly challenging, given that (1) the spectral continuum re- ment. This compilation of stellar abundance is homogeneous quired for computing absorption depths is hard to determine in a sense that Hinkel et al. carefully examined the results due to the presence of very broad molecular bands (e.g., H O 2 reported by different studies for the same element within the and CH ) and (2) elemental abundances cannot be entirely 4 same star and normalized all abundances to the same solar tracked by the observed absorption features, as some por- scale by Lodders et al.(2009). We only include thin-disk tions of species might have been perturbed by vertical mix- Hypatia stars (5021 objects) in our analysis and find these ing and/or sequestered into condensates and thereby sinking objects have a median metallicity of Z = 0.04 dex with a 1σ to below the photosphere (e.g., Fegley & Lodders 1994; Line confidence interval (i.e., 16th-to-84th percentiles) of −0.21 to 0.22 dex. In comparison, our 49 late-T dwarfs have much 9 Among these objects, WISE 0049 + 2151 has two sets of atmospheric model parameters, derived from 0.500 and 0.800 spectra (Section 3.2.1). These results are consistent, and we use those determined from the higher- 10 https://www.hypatiacatalog.com/hypatia. S/N 0.500 spectrum. 15

1.1 Ross 458C

1.0 p u J R

7 0 0.9 . 0

GJ 570D ±

) ±0.16 dex p u

J 0.8 HD 3651B R (

R 0.7

0.6

T7+T7.5 Dwarfs 0.5 T8+T8.5 Dwarfs T9 Dwarfs 0.4 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Z (dex)

Figure 9. Our spectroscopically inferred R and Z for late-T dwarfs (binaries removed; violet for T7 and T7.5, orange for T8 and T8.5, and green for T9), with typical uncertainties shown as black error bars. We use two dashed lines for Z = −0.21 dex and R = 0.75 RJup to divide the space into four parts. Objects located in the upper left area (i.e., Z < −0.21 dex and R > 0.75 RJup) have metallicities that are > 1σ lower than Hypatia stars but their radii appear reasonable. They might be in similar situations as GJ 570D and HD 3651B, whose fitted Z are underestimated by ≈ 0.35 dex but their R are reliable, as noted in Paper I. Objects located in the lower right area (i.e., Z > −0.21 dex and R < 0.75 RJup) have solar-like metallicities but unphysically small radii. Some of them might be in a similar situation as Ross 458C whose fitted Z is reliable but R is underestimated by a factor of ≈ 1.6. The evolutionary model parameters of three benchmark companions (derived from Paper I) are shown as stars and connected to their spectroscopically inferred parameters by arrows. lower metallicities, spanning −0.43 to 0.23 dex and with a unchanged when we limit our comparison to 6 25 pc for our median and mode of −0.24 dex and −0.40 dex, respectively. late-T dwarfs (47 objects) and the Hypatia stars (590 objects; We run the Kolmogorov-Smirnov (K-S) test for metallicities Figure8), with the K-S test provideing similarly small p- of Hypatia stars and our late-T dwarfs in a Monte Carlo fash- values of < 0.02 (with a median of 2 × 10−7). ion11 and obtain p-values that are all smaller than 2 × 10−3, To examine whether the metallicity discrepancy between thereby indicating they are not drawn from the same distribu- late-T and stellar populations is physical, we first analyze the tion. These two samples’ metallicity distributions are largely accuracy of our derived Z values. As shown in Figure9, 30 late-T dwarfs have Z of −0.43 to −0.21 dex and are thus > 1σ lower than the Hypatia stellar metallicity. Most of these 11 We generate 104 distributions of Hypatia stellar metallicities, with each distribution constructed by a random draw from each object’s measured Z objects have R = 0.75−1.2 RJup from our atmospheric mod- and errors by assuming a Gaussian distribution. We generate 104 distribu- eling, consistent with the evolutionary model predictions and tions of our late-T dwarfs’ metallicities in a same approach except that we directly measured radii of transiting brown dwarfs (see Sec- draw each object’s metallicities from its atmospheric-based Z posterior. We tion 4.1). The remaining 19 (= 49 − 30) late-T dwarfs have then run the K-S test for each pair of Hypatia and late-T distributions and compute the corresponding p-values. higher Z of −0.15 to 0.23 dex that are consistent with those 16

1.6 Our Late-T Dwarfs 1.4 Dupuy & Liu (2017)

1.2

1.0

0.8

0.6

0.4 Normalized Number

0.2

0.0 10 Myr 100 Myr 1 Gyr 10 Gyr

0.3 dex +1.0 dex 14 T7+T7.5 Dwarfs T8+T8.5 Dwarfs 12 T9 Dwarfs

10 Dupuy & Liu (2017)

Number 6

0 10 Myr 100 Myr 1 Gyr 10 Gyr Age

Figure 10. Top: Age distributions for our late-T dwarfs (binaries removed) and for benchmark binaries with dynamical masses from Dupuy & Liu(2017). We compute age posteriors for our late-T dwarfs using the spectroscopically inferred {Teff, logg, Z} values at each step of the MCMC chains and the interpolated Sonora-Bobcat evolutionary models. We then generate 104 distributions with each constructed by a random draw from each object’s age posterior. We plot the median and the 1σ confidence interval of these distributions using the blue line and shadow. Distributions for the Dupuy & Liu(2017) sample are produced in the same approach using the age posteriors of the mass benchmarks (T. Dupuy, private communication). We normalize these two types of distributions so that their peaks correspond to one. Bottom: The distribution of our late-T dwarfs, with the violet color for T7 and T7.5, orange for T8 and T8.5, and green for T9. Our typical logarithmic age uncertainty is shown as the black error bar. We use the vertical black line and shadow to mark the median (1.3 Gyr) and the 1σ spread (0.8 to 3.0 Gyr) of the Dupuy & Liu(2017) age distribution. of Hypatia stars, but most of these objects have unphysically given mass and age (also see Burrows et al. 2011). Also, this small radii of < 0.75 RJup. Overall, we find a correlation be- correlation is not found in stacked posteriors of atmospheric tween Z and R, where the objects with higher Z in our sample parameters (Figure5). Therefore, it is likely not due to the tend to have smaller R from our atmospheric model analysis. parameter degeneracy within the Sonora-Bobcat models or This Z − R correlation is in contradiction to the cloudless our spectral-fitting machinery. Sonora-Bobcat evolutionary models, which predict that ob- This Z − R trend might be because the spectroscopically jects with higher metallicities should have increased atmo- inferred Z or R of our late-T dwarfs are underestimated. As spheric opacity, slower cooling, and thereby larger radii at a shown in Paper I, the fitted metallicities of the two bench- 17 mark companions, HD 3651B and GJ 570D, are underesti- Schmidt et al. 2007; Zapatero Osorio et al. 2007). However, mated by ≈ 0.35 dex but their fitted radii are reliable, when based on tangential velocities of a large, volume-limited sam- compared to the radii derived from the Sonora-Bobcat evo- ple of M7−T8 objects within 20 pc, Faherty et al.(2009) lutionary models (using these companions’ bolometric lumi- suggested the kinematic age of ultracool dwarfs (3 − 8 Gyr) nosities and their primary stars’ ages and metallicities). For is indistinguishable from earlier-type stars, although the for- another benchmark, Ross 458C, its fitted metallicity is con- mer indeed becomes younger (2 − 4 Gyr) after excluding sistent with evolutionary-based results but its fitted radii is thick-disk or halo-like objects with high tangential velocities underestimated by a factor of ≈ 1.6. Among our sample, (> 100 km s−1; also see Schmidt et al. 2007). some objects with fitted Z < −0.21 dex and R > 0.75 RJup Thanks to the long-term astrometric monitoring, many might be in similar situations as HD 3651B and GJ 570D, brown dwarf binaries now have measured dynamical masses. while those with fitted Z > −0.21 dex and R < 0.75 RJup These “mass benchmarks” (e.g., Liu et al. 2008) can dis- might be in a similar situation as Ross 458C. Altogether, entangle the age-mass degeneracy of substellar objects and correcting the inaccurate atmospheric parameters of these thus yield some of the most robust ages. Recently, Dupuy & late-T dwarfs would be the equivalent of shifting their Z Liu(2017; DL17) studied the largest sample of such bench- toward higher values and/or their R toward larger values, mark binaries to date (31 systems) and derived ages from which would alter the Z − R correlation seen in Figure9. their measured dynamical masses and the Saumon & Mar- The underestimation of these fitted parameters in our sample ley(2008) evolutionary models. They derived a substellar suggests the model assumptions of cloudless and chemical- age distribution from a high-quality subset of 10 systems equilibrium atmospheres are not adequate to fully interpret (M8−T5 within 30 pc) and found a median and mean of late-T dwarf spectra and should be further improved. 1.3 Gyr and 2.3 Gyr, respectively. This distribution is sys- To conclude, many of our late-T dwarfs with low atmo- tematically younger than the past population synthesis anal- spheric Z might have values underestimated by as large as yses of Burgasser(2004) and Allen et al.(2005), which found 0.3 − 0.4 dex, and this can explain the discrepant metallicity that mid-L to early-T dwarfs have ages of ≈ 2 − 3 Gyr (al- distributions between late-T dwarfs and Hypatia stars. Thus, though the modeled age distribution at a given spectral type we cannot say more about whether the metallicities of nearby has a large spread, e.g., see Figure 8 of Burgasser 2004). The late-T dwarfs and stars are similar. More comprehensive DL17 age distribution is in accord with Galactic dynamic understanding of substellar metallicities would benefit from heating preferentially scattering older substellar objects out the continuing discoveries of brown dwarfs as companions of the immediate region (. 50 pc) around the Earth, which to stars or members of nearby associations. These objects has been the main focus of brown dwarf searches. can provide independent metallicities from their hosts and Here we compare the DL17 age distribution to that of our are thus “metallicity benchmarks” for calibrating forward- late-T dwarf sample, for which we compute ages using the modeling analyses and improving our understanding of ul- fitted {Teff, logg, Z} values at each step of the MCMC chains tracool atmospheres. and the interpolated cloudless Sonora-Bobcat evolutionary models (Table5). Both our analysis and that of DL17 are 5.2. Age Distribution subject to any systematics of the evolutionary models (e.g., Dupuy et al. 2009, 2014; Beatty et al. 2018; Bowler et al. We proceed to study ages of our late-T dwarfs. The age 2018; Dieterich et al. 2018; Brandt et al. 2020), but ours are distribution of field brown dwarfs is essential to constrain also impacted by those of the atmospheric models, which are their formation history and initial mass function (e.g., Kirk- likely more uncertain. Therefore, our derived ages are ex- patrick et al. 2012; Day-Jones et al. 2013). However, it is pected to be less accurate than the DL17 results. challenging to age-date individual field brown dwarfs, given The age distribution of our late-T dwarfs is shown in Fig- that their ages and masses are degenerate and thus the former ure 10, with a median of 50 Myr and 16th-to-84th percentiles cannot be well-constrained unless the latter is known inde- spanning 10 Myr to 0.4 Gyr. This distribution is signifi- pendently. Previous work has estimated low-precision ages cantly younger than DL17 (with a median of 1.3 Gyr and for large groups of ultracool dwarfs based on their kinemat- 16th-to-84th percentiles spanning 0.8 to 3.0 Gyr) and past ics. Such analysis is conducted in a statistical fashion and kinematic studies. Such a young distribution is implausible, assumes the objects’ space motions exhibit a larger spread especially since the majority of our sample are not associated with time due to dynamical evolution (e.g., Wielen 1977). to any stars or clusters with such young age.12 The same ef- Comparing tangential or three-dimensional space velocities of late-M, L, and early-T dwarfs to those of earlier-type main-sequence stars, several studies have found that local ul- 12 Based on the available astrometry and radial velocities of 694 T and tracool dwarfs have slightly smaller velocity dispersions and Y dwarfs, Zhang et al.(2021) recently identified a number of T dwarfs as candidate members of nearby young moving groups, with their final perhaps younger ages (0.5 − 4 Gyr; e.g., Dahn et al. 2002; 18

5.5 T7+T7.5 Dwarfs 10 Gyr HD 3651B T8+T8.5 Dwarfs T9 Dwarfs 4 Gyr 5.0 GJ 570D 2 Gyr x e d

1 Gyr 2 . ) 0 ± x

e 4.5 ±28 K d ( Ross 458C g g o

l 100 Myr 4.0

3.5 Z = 0.5 10 Myr Z = 0 Z = + 0.5

650 700 750 800 850 900 950 1000 1050 Teff (K)

Figure 11. Spectroscopically inferred Teff and logg of our late-T dwarfs (binaries removed; violet for T7 and T7.5, orange for T8 and T8.5, and green for T9) and the cloudless Sonora-Bobcat isochrones (grey lines) with different ages (10 Myr, 100 Myr, 1 Gyr, 2 Gyr, 4 Gyr, 10 Gyr) and metallicities (−0.5 dex, 0 dex, +0.5 dex). Typical uncertainties for our fitted Teff and logg are shown as black error bars. The evolutionary model parameters of the three benchmark companions (derived from Paper I) are shown as stars and connected to their spectroscopically inferred parameters by arrows. Our forward-modeling analysis underestimates these benchmarks’ logg and Z or overestimates their Teff, likely due to systematics of the cloudless Sonora-Bobcat models, and consequently, their ages derived from atmospheric model parameters are implausibly younger than the ages of their primary stars. Many of our late-T dwarfs might be in similar situations as these three companions, and thus their true ages should be systematically older. fect is seen for the 3 benchmarks, HD 3651B, GJ 570D, and troduce more atmospheric processes into models, e.g., clouds Ross 458C, whose primary stars have ages of 4.5 − 8.3 Gyr, and/or dis-equilibrium chemistry (see Section6). 1.4 − 5.2 Gyr, and 0.15 − 0.8 Gyr determined by various 5.3. Effective Temperature vs. Spectral Type age-dating techniques (as summarized in Paper I). In contrast, our spectroscopically inferred ages of these objects Now we study the relation between the atmospheric-based, +0.077 +0.052 spectroscopically inferred T and the spectral types of late-T are much younger, 0.038−0.025 Gyr, 0.031−0.020 Gyr, and eff +0.314 dwarfs. First, we briefly review empirical Teff–SpT relations 0.077−0.056 Gyr, respectively (Figure 11 and Table5). Our implausibly young age estimates represent another il- that have been established using evolutionary models. Simi- lustration that the assumptions made within our model atmo- lar to earlier work by Leggett et al.(2002), Golimowski et al. spheres should be further improved. To derive more reliable (2004) compiled parallaxes and infrared spectrophotometry ages of late-T dwarfs from near-infrared spectra, we could in- of 51 M1−T9 dwarfs. They measured bolometric luminosities from the objects’ spectral energy distributions (SEDs) and then converted these into Teff assuming an age range of membership assessment awaiting radial velocity measurements. Only 3 0.1 − 10 Gyr and using evolutionary models (Burrows et al. objects in our sample here are candidate young moving group members: WISE J024124.73 − 365328.0 (T7; Argus; 40 − 50 Myr, Zuckerman 2019), 1997; Baraffe et al. 1998; Chabrier et al. 2000). Looper et al. 2MASS 1553 + 1532 (T7; Carina-Near; 200 ± 50 Myr, Zuckerman et al. (2008) and Stephens et al.(2009) later modified this sam- 2006), and WISEPC J225540.74 − 311841.8 (T8; β Pictoris; 24 ± 3 Myr, ple by removing binaries and/or adding companions discov- Bell et al. 2015). ered since the Golimowski et al.(2004) work and obtained 19

Stephens et al. (2009) 1000 Dupuy & Kraus (2013) [1 Gyr] Dupuy & Kraus (2013) [5 Gyr] Filippazzo et al. (2015)

900 3.5 4.0 4.5 5.0 logg (dex) ) K

( 800

f ±28K f e T

700

600

T7 T7.5 T8 T8.5 T9 Spectral Type

1000 T7 T7.5 T8

) 900 K

( HD 3651B

f f 800 e

T GJ 570D 700 Ross 458C

600 3.5 4.0 4.5 5.0 3.5 4.0 4.5 5.0 3.5 4.0 4.5 5.0 logg (dex) logg (dex) logg (dex)

Figure 12. Top plot: Teff−SpT relation of our late-T dwarfs (binaries removed), with spectroscopically inferred logg encoded with blue shading. The median Teff uncertainty is shown as the black error bar. We overlay evolutionary-based effective temperature scales from Stephens et al. (2009; dashed) and Filippazzo et al.(2015; solid). We also plot the Dupuy & Kraus(2013) Teff−SpT relations corresponding to assumed ages of 1 Gyr (dotted) and 5 Gyr (dash-dotted). The gray shadow shows the rms of each empirical Teff−SpT relation. Bottom plots: Teff of our T7, T7.5, and T8 dwarfs as a function of logg. Median Teff and logg uncertainties are shown as black error bars. The evolutionary model parameters of three benchmarks (derived from Paper I) are shown as stars and connected to their spectroscopically inferred parameters by arrows. tighter temperature scales as a function of spectral types. (2015) temperature scale of field dwarfs is consistent with Focusing on the coolest substellar objects, Dupuy & Kraus results of Stephens et al.(2009) and Dupuy & Kraus(2013). (2013) computed Lbol for 21 T8–Y0 field dwarfs using par- Figures 12 and 13 compare our atmospheric-based Teff of allaxes, broadband photometry, and bolometric corrections, T7−T9 dwarfs to the evolutionary-based temperature scales and then derived Teff using the Baraffe et al.(2003b) evolu- from Stephens et al.(2009), Dupuy & Kraus(2013), and Fil- tionary models with age assumptions of 1 Gyr and 5 Gyr. ippazzo et al.(2015). At a given spectral type, our spectro- The most recent Teff−SpT relation has been derived by Fil- scopically inferred Teff show a spread of ±100 K, similar to ippazzo et al.(2015) using a much larger sample of nearly the empirical relations. Our results are consistent with empir- 200 M6−T9 dwarfs. They measured Lbol based on a ho- ical ones for T7 and T7.5 dwarfs, but appear 50−200 K hot- mogeneous SED analysis and then estimated Teff using the ter at later types, likely because our derived Teff for T8−T9 Saumon & Marley(2008) hybrid models with an assumed dwarfs are less reliable than those of T7 dwarfs. As found age of 0.5−10 Gyr for most objects and narrower ranges for in Paper I and summarized in Section 3.2 in this work, our association members and companions. The Filippazzo et al. spectral fits of Ross 458C (T8) predict ≈ 120 K higher Teff 20

Stephens et al. (2009) 1000 Dupuy & Kraus (2013) [1 Gyr] Dupuy & Kraus (2013) [5 Gyr] Filippazzo et al. (2015)

900 0.4 0.2 0 +0.2 Z (dex) ) K

( 800

f ±28K f e T

700

600

T7 T7.5 T8 T8.5 T9 Spectral Type

1000 T7 T7.5 T8

) 900 K (

f f 800 e HD 3651B T GJ 570D 700 Ross 458C

600 0.4 0.2 0 0.2 0.4 0.4 0.2 0 0.2 0.4 0.4 0.2 0 0.2 0.4 Z (dex) Z (dex) Z (dex)

Figure 13. The effect of metallicity on the Teff−SpT relation of our late-T dwarfs (binaries removed), with the same format as Figure 12. Spectroscopically inferred Z are encoded with green shading. Bottom plots show the objects’ Teff as a function of Z. than evolutionary model analysis, while the two T7.5 bench- Figure 12. For T7−T9 spectral types, there is only one low- marks (HD 3651B and GJ 570D) have no such significant gravity benchmark with direct spectroscopy, i.e., Ross 458C, discrepancy. for which our evolutionary model analysis derives Teff = +16 Figure 12 explores whether the derived Teff−SpT relation 682−17 K (Paper I). This temperature is consistent with those of our sample is gravity-dependent. We focus on the T7, of higher-gravity field T8 dwarfs (678 ± 113 K based on the T7.5, and T8 objects in our sample, since we have only four Teff−SpT relation of Filippazzo et al. 2015) and thus not in T8.5 and T9 dwarfs. Studying the objects’ Teff as a func- accord with the Teff − logg relation from our results, which tion of logg, we find lower-gravity T7.5 and T8 dwarfs have predicts its Teff to be much hotter. More discoveries of such on-average hotter effective temperatures at a given spectral young, late-type benchmarks will help investigate the gravity type. Intriguingly, this phenomenon is not found in earlier dependence of late-T dwarfs’ effective temperatures. T dwarfs. Based on a sample of 25 young (6 300 Myr) and In fact, the possible relation between the fitted Teff and logg old (> 300 Myr) T0–T6 benchmarks, Zhang et al.(2020a) of our sample is likely because these parameters are over- demonstrated that lower-gravity objects tend to have ≈ 100 K and/or under-estimated. Some of the T7.5 dwarfs in our sam- cooler Teff than their high-gravity counterparts at same spec- ple might be similar to HD 3651B and GJ 570D, whose fitted tral types (also see Metchev & Hillenbrand 2006; Filippazzo Teff are reliable but logg are underestimated by ≈ 1.2 dex et al. 2015; Faherty et al. 2016; Liu et al. 2016), which is when compared to the more robust evolutionary model pre- opposite of the Teff − logg relation for late-T dwarfs seen in dictions. Also, some of the T8 dwarfs might be similar to 21

) 10000 SpT Teff logg Z 1 0.500 s

m 8000 k (

e c 6000 0.800 n a i r a

v 4000 o C

l a

b 2000 o l G

T9 T8.5 T8 T7.5 T7 600 700 800 900 1000 3.5 4.0 4.5 5.0 5.5 0.50 0.25 0.00 0.25 0.50 Spectral Type Teff (K) logg (dex) Z (dex)

Figure 14. The covariance hyper-parameter ` of our 49 late-T dwarfs (binaries removed) as a function of spectral type and ﬁtted {Teff, logg, Z}, with median uncertainties shown as black error bars. We use green and violet for ` inferred from spectra taken from the 0.500 and 0.800 slits, respectively. For a given slit, we use the shaded area to mark the ` range expected by the instrumental line spread function (820 − 1840 km s−1 for the 0.500 slit and 425 − 1115 km s−1 for the 0.800 slit) and use dashed lines to mark upper boundaries of our adopted ` prior.

6% SpT Teff logg Z

J 4%

T9 T8.5 T8 T7.5 T7 600 700 800 900 1000 3.5 4.0 4.5 5.0 5.5 0.50 0.25 0.00 0.25 0.50 Spectral Type Teff (K) logg (dex) Z (dex)

Figure 15. The J values of our 49 late-T dwarfs (binaries removed) computed from their covariance hyper-parameters aG and aN and the observed spectra (Section 6.1). The J describes the fraction that model uncertainties (i.e., systematic ﬂux difference between the cloudless Sonora-Bobcat models and late-T dwarf spectra) comprise of the objects’ observed peak J-band ﬂuxes. Median uncertainties are shown as black error bars. The object with the highest J among our sample is WISE 1322 − 2340. The J values, namely model systematics, tend to become larger at later spectral types, cooler spectroscopically inferred Teff, and possibly higher logg and Z.

Ross 458C, whose fitted logg is reliable but Teff is overesti- seen in our results. Similar to the aforementioned Teff − logg mated by ≈ 120 K. Correcting their inaccurate atmospheric- trend, this Teff − Z relation for late-T dwarfs in Figure 13 based parameters would be the equivalent of shifting their might be related to modeling systematics. To further validate logg toward higher values and/or their Teff toward cooler val- this potential metallicity dependence for late-T dwarfs, we ues, which could alter the Teff − logg relation in Figure 12. need models with modified physical assumptions and more Figure 13 explores whether the derived Teff−SpT relation benchmarks over a wide range of metallicities. of our sample is metallicity-dependent. Studying the objects’ 6. DISCUSSION OF THE CLOUDLESS Teff as a function of Z, we find lower-metallicity T7, T7.5, and T8 dwarfs have on-average cooler effective temperatures at a SONORA-BOBCAT MODELS given spectral type. The validity of this metallicity depen- Our forward-modeling analysis shows the cloudless dence is uncertain. For the two late-T subdwarf benchmarks, Sonora-Bobcat models match the general spectroscopic ap- BD +01◦ 2920B (T8; Z = −0.38 ± 0.06 dex by Valenti & pearance of late-T dwarfs, but the fitted physical parameters Fischer 2005) and Wolf 1130C (T8; Z = −0.70 ± 0.12 by can be under- or over-estimated, suggesting the models’ as- Mace et al. 2018), previous work found their effective tem- sumptions might be inadequate to fully interpret late-T dwarf peratures are similar to those of the field population, which spectra (Section5). Here we explore specific shortcomings, has on average higher metallicities (Pinfield et al. 2012; Mace by focusing on 49 late-T dwarfs in our sample, as the other 6 et al. 2013b), and thus not in accord with the Teff −Z relation objects are resolved or likely binaries (Section4). 22

+40% Data Model Scaled Residual = +30% J-band Peak

20%

+10%

10% Scaled Residual 20%

30%

GJ 570D 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Wavelength ( m)

Figure 16. Stacked spectral-fitting residuals of our 49 late-T dwarfs (binaries removed), with each residual (i.e., data − model; black) normalized by the object’s observed peak J-band flux. The 1σ and 2σ dispersions of 5 × 104 draws from Starfish’s full covariance matrix are shown as deep and light blue shadows, respectively, and they are normalized by the same peak flux for each object. The resulting stack has a tight distribution at each wavelength and exhibits prominent features in YJHK bands (grey shadow, with specific wavelength ranges shown in Equation2). The spectrum of GJ 570D is plotted at the bottom as a reference.

6.1. Starfish Covariance Hyper-Parameters Given that the modeling systematics dominate the corre- As discussed in Paper I, we can assess the systematic dif- lated residuals, we can use the other two hyper-parameters (aN and aG) to quantify the modeling systematics. Specif- ference between data and models using the fitted covariance p ically, we can compute aG/aN and regard it as an equiv- hyper-parameters {aN , aG, `} of our late-T dwarfs. The parameter ` characterizes the autocorrelation wavelength of alent flux that describes the average “model uncertainties” data−model residuals, caused by (1) the oversampled instru- (as opposed to measurement uncertainties). Normalizing this mental line spread function (LSF), and (2) the systematics of value by an object’s peak J-band flux (Equation 7 of Pa- per I), we define the normalized model uncertainty as J = model assumptions. If the correlated residuals of an object p are solely due to the instrumental LSF, then its fitted ` should aG/aN max( fobs,J), with higher values suggesting more be within [820,1840] km s−1 and [425,1115] km s−1 for significant data-model discrepancies. spectra taken by the SpeX 0.500 and 0.800 slits, respectively We summarize the derived J of our sample in Table5 (see Appendix of Paper I). However, the spectroscopically and find the systematic difference between the cloudless inferred ` of almost all our late-T dwarfs are significantly Sonora-Bobcat models and late-T dwarf spectra is on av- higher than the range expected from the LSF, suggesting the erage ≈ 2% − 4% of the observed peak J-band fluxes over 13 correlated residuals of our sample do not arise from the in- 1.0 − 2.5 µm wavelengths , equivalent to a S/N (the ratio strument’s spectral resolution but rather shortcomings of the of signal to model uncertainty) of 50 − 25. This is in ac- model predictions (Figure 14). Figure 14 also shows the inferred ` of our sample has no correlations with the objects’ 13 We note these values describe an average systematic difference be- spectral types or fitted {Teff, logg, Z}. tween observed spectra and Sonora-Bobcat models over the 1.0 − 2.5 µm range. As shown in Figure 16, such systematic difference can be as high 23

SpT WISE 2000 + 3629 SpT = T8 WISE 0040 + 0900 SpT = T7

7 WISE 2226 + 0440 SpT = T8 2MASS 0050 3322 SpT = T7

UGPS 0722 0540 SpT = T9 WISE 2255 3118 SpT = T8 WISE 0123 + 4142 SpT = T7

6 WISE 1741 + 2553 SpT = T9 2MASS 0939 2448 SpT = T8 WISE 2213 + 0911 SpT = T7

WISE 0049 + 2151 SpT = T8.5 ULAS 1029 + 0935 SpT = T8 WISE 2209 2734 SpT = T7

UGPS 0521 + 3640 SpT = T8.5 ULAS 1416 + 1348 SpT = (sd) T7.5 WISE 2157 + 2659 SpT = T7 5

Ross 458C SpT = T8 HD 3651B SpT = T7.5 WISE 0241 3653 SpT = T7

2MASS 0729 3954 SpT = T8 pec WISE 2319 1844 SpT = T7.5 WISE 0325 + 0831 SpT = T7 4 WISE 1653 + 4444 SpT = T8 GJ 570D SpT = T7.5 WISE 1254 0728 SpT = T7

WISE 0623 0456 SpT = T8 WISE 1809 + 3838 SpT = T7.5 WISE 0614 + 0951 SpT = T7

3 WISE 1322 2340 SpT = T8 2MASS 1217 0311 SpT = T7.5 2MASS 0727 + 1710 SpT = T7

WISE 0500 1223 SpT = T8 ULAS 2321 + 1354 SpT = T7.5 SDSS 1628 + 2308 SpT = T7 Scaled Residuals + Constant

2 2MASS 0415 0935 SpT = T8 2MASS 1114 2618 SpT = T7.5 PSO J224 + 47 SpT = T7

WISE 1813 + 2835 SpT = T8 WISE 1052 1942 SpT = T7.5 WISE 1457 + 5815 SpT = T7

1 PSO J043 + 02 SpT = T8 WISE 0521 + 1025 SpT = T7.5 WISE 1124 0421 SpT = T7

WISE 0245 3450 SpT = T8 WISE 0223 2932 SpT = T7.5 WISE 1852 + 3537 SpT = T7

WISE 1959 3338 SpT = T8 WISE 1257 + 4008 SpT = T7 WISE 2348 1028 SpT = T7 0

1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 wavelength ( m)

Figure 17. Residuals of our 49 late-T dwarfs (binaries removed) shown in order of spectral types, and spectroscopically inferred Teff, logg, and Z. We use each object’s observed peak J-band flux to normalize its residual (black), as well as 1σ and 2σ dispersions of 5 × 104 draws from Starfish’s full covariance matrix (blue shadows). The J-band and H-band residuals tend to be more significant with later spectral types and cooler effective temperatures. The H-band residual also becomes more prominent with increasing metallicities. cord with our conclusion in Paper I based on the three late-T tral types, cooler effective temperatures, and possibly higher benchmarks, which are also included in the analysis here. logg and Z. These trends indicate some important atmo- This result implies that model uncertainties exceed measure- spheric processes are likely missing in the model assump- ment uncertainties when fitting the cloudless Sonora-Bobcat tions, which we discuss in the following section. models to late-T dwarf spectra with S/N & 50 per pixel in J band. As a consequence, increasing the S/N of observations does not necessarily improve the precision of the fitted 6.2. Spectral-Fitting Residuals physical parameters, as also seen in Figure4 and discussed Missing physical processes in model assumptions should in Section 3.2.1. leave footprints in our objects’ spectral-fitting residuals. Figure 15 plots our objects’ J as a function of their Thanks to our large sample of late-T dwarfs, we can in- spectral types and fitted physical parameters. We find the vestigate these residuals and assess how models deviate from data-model difference tends to be larger toward later spec- the observations as a function of wavelength and atmospheric parameters. In Figure 16, we normalize the fitting residuals as 10% − 20% of the objects’ peak J-band fluxes over narrow wavelength for each object by its observed peak J-band flux and then ranges in J and H bands. stack them together. The resulting stack has a tight distri- 24

WISE 1653 + 4444 Teff = 800 K WISE 0325 + 0831 Teff = 873 K Teff 7 WISE 2209 2734 Teff = 801 K WISE 1124 0421 Teff = 879 K

2MASS 0939 2448 Teff = 656 K Ross 458C Teff = 804 K WISE 1457 + 5815 Teff = 882 K

6 UGPS 0722 0540 Teff = 680 K WISE 0245 3450 Teff = 804 K WISE 0040 + 0900 Teff = 883 K

ULAS 1416 + 1348 Teff = 695 K 2MASS 0729 3954 Teff = 806 K 2MASS 1217 0311 Teff = 886 K

WISE 1741 + 2553 Teff = 719 K WISE 2000 + 3629 Teff = 810 K 2MASS 0727 + 1710 Teff = 896 K 5

WISE 0500 1223 Teff = 727 K WISE 1813 + 2835 Teff = 811 K WISE 1852 + 3537 Teff = 899 K

2MASS 1114 2618 Teff = 729 K WISE 1959 3338 Teff = 812 K WISE 2348 1028 Teff = 900 K 4

WISE 0623 0456 Teff = 743 K HD 3651B Teff = 818 K WISE 2213 + 0911 Teff = 908 K

WISE 0049 + 2151 Teff = 753 K WISE 1052 1942 Teff = 821 K WISE 0241 3653 Teff = 909 K

3 UGPS 0521 + 3640 Teff = 781 K ULAS 2321 + 1354 Teff = 822 K SDSS 1628 + 2308 Teff = 910 K

WISE 2319 1844 Teff = 782 K WISE 1322 2340 Teff = 822 K PSO J224 + 47 Teff = 930 K Scaled Residuals + Constant

2 WISE 0223 2932 Teff = 787 K GJ 570D Teff = 828 K WISE 1257 + 4008 Teff = 933 K

WISE 2255 3118 Teff = 790 K WISE 2226 + 0440 Teff = 829 K WISE 1254 0728 Teff = 936 K

1 2MASS 0415 0935 Teff = 793 K WISE 1809 + 3838 Teff = 838 K 2MASS 0050 3322 Teff = 947 K

ULAS 1029 + 0935 Teff = 794 K WISE 0521 + 1025 Teff = 846 K WISE 0614 + 0951 Teff = 956 K

PSO J043 + 02 Teff = 798 K WISE 2157 + 2659 Teff = 860 K WISE 0123 + 4142 Teff = 1023 K 0

1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 wavelength ( m)

Figure 17. Continued bution as a function of wavelength and exhibits prominent Here λλ corresponds to the wavelength range of prominent features in YJHK bands. YJHK-band features seen in Figure 16 and does not follow In order to study how the residuals are correlated with at- the standard definition of the filters. In addition, fmodel,λ mospheric properties, we plot residuals sorted by the objects’ and fobs,λ are the fitted model and the observed spectrum of spectral types and fitted {Teff, logg, Z} in Figure 17. We the object, respectively. The sign of q indicates whether the further define the following quantities qY , qJ, qH , and qK to models under-predict (positive) or over-predict (negative) the evaluate the data-model difference in YJHK bands for each data, with larger absolute values of q indicating larger data- object (Table5), model discrepancies. By definition, model atmospheres that perfectly match the data will have q = 0 for all wavelength ranges. Z In Y band (≈ 1.0 − 1.1 µm), our fitted models slightly fmodel,λ dλ λλ under-predict the spectra of most late-T dwarfs (Figure 16 q(λλ) = 1 − Z and 17). This is likely related to the potassium resonance fobs,λ dλ λλ doublet at 0.77 µm (Allard et al. 1999; Burrows et al. 2000), (2) such that qY ≡ q([1.00µm,1.10µm]) whose pressure-broadened wings can extend to Y and J bands. The K line profile depends on the treatment of colli- qJ ≡ q([1.18µm,1.35µm]) sions between H molecules and K atoms (e.g., Baudino et al. q ≡ q([1.50µm,1.65µm]) 2 H 2017; Phillips et al. 2020). The cloudless Sonora-Bobcat qK ≡ q([2.03µm,2.20µm]) 25

logg WISE 1959 3338 logg = 3.92 dex PSO J043 + 02 logg = 4.15 dex

7 WISE 0241 3653 logg = 3.93 dex WISE 1124 0421 logg = 4.18 dex

WISE 0521 + 1025 logg = 3.36 dex 2MASS 0727 + 1710 logg = 3.93 dex WISE 2213 + 0911 logg = 4.19 dex

6 WISE 0325 + 0831 logg = 3.42 dex HD 3651B logg = 3.94 dex WISE 1813 + 2835 logg = 4.20 dex

WISE 0245 3450 logg = 3.43 dex WISE 1852 + 3537 logg = 3.97 dex ULAS 2321 + 1354 logg = 4.20 dex

WISE 0049 + 2151 logg = 3.47 dex 2MASS 1217 0311 logg = 3.98 dex SDSS 1628 + 2308 logg = 4.21 dex 5

UGPS 0722 0540 logg = 3.60 dex WISE 1322 2340 logg = 3.98 dex WISE 0614 + 0951 logg = 4.21 dex

ULAS 1029 + 0935 logg = 3.61 dex WISE 1254 0728 logg = 3.98 dex WISE 1457 + 5815 logg = 4.27 dex 4 PSO J224 + 47 logg = 3.62 dex 2MASS 0050 3322 logg = 4.06 dex WISE 2319 1844 logg = 4.46 dex

WISE 1809 + 3838 logg = 3.63 dex 2MASS 0415 0935 logg = 4.07 dex UGPS 0521 + 3640 logg = 4.50 dex

3 WISE 2255 3118 logg = 3.66 dex WISE 0040 + 0900 logg = 4.07 dex 2MASS 0729 3954 logg = 4.51 dex

WISE 2000 + 3629 logg = 3.67 dex WISE 2348 1028 logg = 4.09 dex 2MASS 1114 2618 logg = 4.52 dex Scaled Residuals + Constant

2 WISE 2226 + 0440 logg = 3.78 dex Ross 458C logg = 4.09 dex WISE 2209 2734 logg = 4.54 dex

WISE 2157 + 2659 logg = 3.79 dex WISE 1653 + 4444 logg = 4.10 dex WISE 0623 0456 logg = 4.70 dex

WISE 1052 1942 logg = 3.83 dex WISE 0123 + 4142 logg = 4.13 dex 2MASS 0939 2448 logg = 4.74 dex 1

GJ 570D logg = 3.90 dex WISE 0500 1223 logg = 4.13 dex WISE 1257 + 4008 logg = 4.83 dex

WISE 1741 + 2553 logg = 3.91 dex WISE 0223 2932 logg = 4.15 dex ULAS 1416 + 1348 logg = 5.21 dex 0

1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 wavelength ( m)

Figure 17. Continued

models adopt the K line shape theory by Allard et al.(2007). less models. This is consistent with the qJ − Teff correlation Allard et al.(2016) have improved calculations of the K −H2 in Figure 18. potential, which might predict more accurate shapes for the Alternatively, the over-predicted J-band flux of the cloud- K I doublet and thereby the Y-band fluxes (e.g., Phillips et al. less Sonora-Bobcat models might be related to the assump- 2020). tion that the deep temperature gradient in convective regions In J band (≈ 1.18 − 1.35 µm), our fitted models over- lies along an adiabatic. As demonstrated by Tremblin et al. predict spectra of all late-T dwarfs, and we find qJ is corre- (2015, 2019), cloudless dis-equilibrium models with (1) at- lated with spectral type and Teff (Figure 18), as the J-band mospheric mixing (described by the eddy diffusion coeffi- residuals increase for later-type and cooler objects. This cients Kzz) and (2) reduced vertical temperature gradient (as residual plausibly arises from clouds (e.g., Morley et al. compared to the adiabatic lapse rate) are more appropriate for 2012), given that fluxes at near-infrared spectral peaks are spectra of T/Y dwarfs than cloudless, chemical-equilibrium emitted from the deep atmosphere and thus more sensitive to models. While these models can explain the J-band residu- cloud opacity. Including clouds into Sonora-Bobcat models als of our sample, they will need to explain the qJ−SpT and might therefore produce spectra that better match the obser- qJ − Teff correlations as seen in Figure 18. One implication vations. Also, with lower effective temperatures, conden- from these correlations is that the thermo-chemical instability sates of various species are expected to form (e.g., Na2S, in the Tremblin et al.(2015) models should be more signifi- KCl, MnS, ZnS; Lodders 1999; Morley et al. 2012), leading cant with later spectral types and cooler Teff for late-T dwarfs. to a larger J-band discrepancy between data and the cloud- To examine this hypothesis, further investigation is needed to 26

Z WISE 0223 2932 Z = 0.33 dex WISE 0325 + 0831 Z = 0.13 dex

7 WISE 0623 0456 Z = 0.32 dex 2MASS 0415 0935 Z = 0.13 dex

WISE 2209 2734 Z = 0.43 dex WISE 0123 + 4142 Z = 0.31 dex PSO J043 + 02 Z = 0.11 dex

6 2MASS 0939 2448 Z = 0.41 dex WISE 2213 + 0911 Z = 0.31 dex 2MASS 0050 3322 Z = 0.11 dex

2MASS 0729 3954 Z = 0.40 dex WISE 1457 + 5815 Z = 0.31 dex WISE 1959 3338 Z = 0.10 dex

2MASS 1114 2618 Z = 0.40 dex WISE 1809 + 3838 Z = 0.31 dex UGPS 0722 0540 Z = 0.06 dex 5

ULAS 1416 + 1348 Z = 0.39 dex WISE 0521 + 1025 Z = 0.29 dex WISE 1257 + 4008 Z = 0.06 dex

WISE 1052 1942 Z = 0.39 dex WISE 2157 + 2659 Z = 0.28 dex WISE 1741 + 2553 Z = 0.04 dex 4 WISE 1322 2340 Z = 0.38 dex WISE 2226 + 0440 Z = 0.26 dex WISE 1653 + 4444 Z = 0.01 dex

WISE 1124 0421 Z = 0.38 dex WISE 0049 + 2151 Z = 0.24 dex WISE 0614 + 0951 Z = 0.01 dex

3 WISE 0500 1223 Z = 0.38 dex ULAS 1029 + 0935 Z = 0.24 dex 2MASS 1217 0311 Z = 0.00 dex

WISE 2319 1844 Z = 0.37 dex HD 3651B Z = 0.22 dex WISE 1813 + 2835 Z = 0.05 dex Scaled Residuals + Constant

2 WISE 0245 3450 Z = 0.36 dex 2MASS 0727 + 1710 Z = 0.22 dex SDSS 1628 + 2308 Z = 0.06 dex

WISE 2000 + 3629 Z = 0.36 dex WISE 1852 + 3537 Z = 0.21 dex PSO J224 + 47 Z = 0.12 dex

WISE 0241 3653 Z = 0.34 dex WISE 2255 3118 Z = 0.21 dex ULAS 2321 + 1354 Z = 0.18 dex 1

WISE 0040 + 0900 Z = 0.34 dex WISE 1254 0728 Z = 0.15 dex UGPS 0521 + 3640 Z = 0.23 dex

GJ 570D Z = 0.34 dex WISE 2348 1028 Z = 0.15 dex Ross 458C Z = 0.23 dex 0

1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 1.2 1.6 2.0 2.4 wavelength ( m)

Figure 17. Continued study such instability in brown dwarf atmospheres as a func- over-predicted J-band flux by choosing models that under- tion of physical properties. predict the H-band flux. Studying our late-T dwarf spectra In H band (≈ 1.50 − 1.65 µm), our fitted models under- using (1) cloudless models with dis-equilibrium chemistry or predict spectra of all late-T dwarfs, and we find the resid- (2) cloudy models with equilibrium chemistry will help bet- uals (qH ) become more significant with later spectral types, ter understand this H-band residual. cooler effective temperatures, and higher metallicities (Fig- In K band (≈ 2.03 − 2.20 µm), our fitted models only ures 18). These can be related to the dis-equilibrium abun- slightly under-predict spectra of most late-T dwarfs. Simi- dance of NH3, which would be less than the amount assumed lar to J band, such K-band residuals may arise from clouds by the equilibrium chemistry within the cloudless Sonora- and/or reductions in the vertical temperature gradient. As Bobcat models and thus lead to weaker absorption in the shown by Morley et al.(2012; their Figure 11) and Tremblin blue wing of H band (Saumon et al. 2006, 2012; Cushing et al.(2015; their Figure 1), either of these two processes can et al. 2011; Zahnle & Marley 2014). However, the NH3 reduce fluxes in YJ bands and increase the flux in K band. cross-section is peaked around 1.5 µm (or wavenumbers of 6600 − 6700 cm−1; e.g., Yurchenko et al. 2011; Coles et al. 7. SUMMARY AND FUTURE WORK 2019), slightly offset from the H-band residuals of our late- We have conducted a forward-modeling analysis for 55 T dwarfs which are peaked near 1.58 µm. Alternatively, late-T (T7−T9) dwarfs using low-resolution (R ≈ 50 − 250) the under-predicted H-band spectra could well be a conse- near-infrared (1.0 − 2.5 µm) spectra and state-of-art, cloud- quence of the spectral-fitting procedure responding to the less Sonora-Bobcat model atmospheres with Teff = 600 − 27

SpT Teff logg Z +12%

+8% Y

q +4%

4% T9 T8.5 T8 T7.5 T7 600 800 1000 3.5 4.0 4.5 5.0 5.5 0.5 0.0 0.5 Spectral Type Teff (K) logg (dex) Z (dex)

SpT T logg Z 0% eff

5% J q 10%

15%

T9 T8.5 T8 T7.5 T7 600 800 1000 3.5 4.0 4.5 5.0 5.5 0.5 0.0 0.5 Spectral Type Teff (K) logg (dex) Z (dex)

+40% SpT Teff logg Z

+30%

H +20% q

+10%

0% T9 T8.5 T8 T7.5 T7 600 800 1000 3.5 4.0 4.5 5.0 5.5 0.5 0.0 0.5 Spectral Type Teff (K) logg (dex) Z (dex)

+50% SpT Teff logg Z

+40%

+30%

K +20% q +10%

10% T9 T8.5 T8 T7.5 T7 600 800 1000 3.5 4.0 4.5 5.0 5.5 0.5 0.0 0.5 Spectral Type Teff (K) logg (dex) Z (dex)

Figure 18. The qY (purple), qJ (blue), qH (green), and qK (orange) of our 49 late-T dwarfs (binaries removed) as a function of spectral type and atmospheric {Teff, logg, Z}, with median uncertainties shown by black error bars. The q values (Equation2) quantify the ﬁtted residuals of the object in YJHK bands. The sign of q indicates whether the models under-predict (positive) or over-predict (negative) the observed spectra over a given band pass, with larger absolute values of q indicating larger data-model discrepancies. By deﬁnition, perfect models should have q = 0 at all wavelengths (dashed horizontal lines).

1200 K, logg = 3.25−5.5 dex, and Z = {−0.5,0,+0.5} dex. pared to traditional forward-modeling studies, our analysis Our sample contains 90% of the nearby T7−T9 population produces more realistic error estimates since we account for with distances 6 25 pc, J-band magnitudes 6 17.5 mag, and uncertainties from model interpolation and correlated resid- declinations from −40◦ to +70◦. Our work is the largest uals due to instrumental effects and systematics of model as- analysis of brown dwarf spectra using multi-metallicity mod- sumptions. els to date, as well as the most systematic test of any set of We have inferred effective temperatures (Teff), surface ultracool model atmospheres. Our forward-modeling frame- gravities (logg), metallicities (Z), radii (R), masses (M), work was constructed and validated in Paper I, which uses the and bolometric luminosities (Lbol) for our late-T dwarfs, Bayesian inference tool Starﬁsh (Czekala et al. 2015). Com- with the typical resulting {Teff, logg, Z} uncertainties being 28

≈ 1/3 − 1/2 of the Sonora-Bobcat model grid spacing. We using more late-T benchmarks with diverse ages and found no difference in the precision of these physical pa- metallicities. rameters for spectra with two different spectral resolutions 4. The spectroscopically inferred masses of our sample (R ≈ 80 − 250 and 50 − 160). Combining the resulting pa- are unphysically small (mostly 1 − 8 MJup), due to the rameter posteriors of our entire sample, we found some fitted underestimate of their fitted logg and/or R. parameters are correlated, including Teff and R, R and M, and logg and Z. Correlations within the first two pairs are ex- Using the hyper-parameters from our spectral-fitting re- pected by the Stephen-Boltzmann law and the calculation of sults, we quantified that the systematic difference between mass from spectroscopic parameters, respectively. The third the observed late-T dwarf spectra and the Sonora-Bobcat correlation illustrates that logg and Z are degenerate in the models is on average ≈ 2%−4% of the objects’ peak J-band cloudless Sonora-Bobcat models, and we provide a quantifi- fluxes over the 1.0−2.5 µm range (as high as 10−20% of the cation of this logg − Z dependence. For late-T dwarfs, the objects’ peak J-band fluxes over narrow wavelength ranges degeneracy acts such that an increase in Z, combined with a in J and H bands), equivalent to a S/N of 50 − 25. There- 3.4× larger increase in logg, results in a spectrum that has fore, model uncertainties exceed measurement uncertainties similar fitted atmospheric parameters. Consequently, using when fitting the Sonora-Bobcat models to late-T dwarf spec- solar-metallicity model atmospheres to study late-T dwarfs tra with S/N higher than these values. This can also explain whose metallicities are in fact non-solar will bias the inferred why the fitted parameter precision of our sample does not im- logg, but likely not other parameters. prove with increasing S/N once it is above ≈ 50 per pixel in By virtue of having a large sample of spectra within a fo- J band. cused spectral type range (as opposed to a smaller sample of In order to investigate how to improve model assumptions, objects over a wide spectral type range), we can study pop- we stacked the spectral-fitting residuals of our entire sample ulation properties of T7–T9 dwarfs to provide useful diag- and investigated these as a function of wavelength and the nostics about our set of grid models, which assume cloudless inferred atmospheric properties. We found common, promi- and chemical-equilibrium atmospheres: nent residual features in YJHK bands: 1. The spectroscopically inferred metallicities of our en- 1. In Y band, the cloudless Sonora-Bobcat models tend tire sample and a volume-limited subset within 25 pc to under-predict the observed fluxes, which is likely are 0.3 − 0.4 dex lower than those of nearby FGKM related to the potassium line profiles. Further improve- stars (e.g., Hinkel et al. 2014). This significant discrep- ments of the alkali opacities might help reduce this ancy is unlikely to be a real difference between the sub- residual (e.g., Allard et al. 2016). stellar and stellar populations, but rather because our fitted metallicities are underestimated, which is also 2. In J band, the cloudless Sonora-Bobcat models tend to seen from our Paper I analysis of the late-T bench- over-predict the observed fluxes by an amount which marks HD 3651B and GJ 570D. is larger for later spectral types and cooler Teff. This 2. The inferred ages of our sample, based on their fit- effect is likely caused by missing opacity from clouds (e.g., Morley et al. 2012) or the model assumption that ted {Teff, logg, Z} and evolutionary models, have a median of 50 Myr and a 1σ confidence interval the deep temperature gradient in convective regions of 10 Myr–0.4 Gyr. These values are implausibly lies along an adiabat (e.g., Tremblin et al. 2015). In- younger than the robustly determined ages of nearby cluding clouds or assuming a reduced temperature gra- M8–T5 binaries (Dupuy & Liu 2017) and thus are dient could result in spectra that better match the ob- likely underestimated. servations. 3. The spectroscopically inferred effective temperatures 3. In H band, the cloudless Sonora-Bobcat models tend of our sample show a similar spread (±100 K) at a to under-predict the observed fluxes by an amount given spectral type as compared to empirical effective which is larger for later spectral types, cooler Teff, and temperature scales, but our Teff appear systematically higher Z. This residual might be explained by the dis- hotter (by 50 − 200 K) for >T8 dwarfs. Also, our equilibrium abundance of NH3 (e.g., Saumon et al. derived Teff−SpT relation for late-T dwarfs is weakly 2006; Cushing et al. 2011) which is not included by correlated with fitted logg and Z, as objects with either the Sonora-Bobcat assumption of equilibrium chem- lower logg or higher Z have on average hotter Teff at a istry, although our residuals do not seem to precisely given spectral type. The possible gravity and metallic- coincide with the expected wavelength for NH3 opaci- ity dependence seen in this work might be caused by ties. Also, this H-band residual could merely be a con- the over- and/or under-estimated physical parameters sequence of the spectral-fitting procedure responding from spectral fitting, but it should be further validated to the over-predicted J-band flux. 29

4. In K band, the cloudless Sonora-Bobcat models tend to //github.com/gully/jammer-Gl570D). We thank under-predict the observed fluxes, which likely arises Trent Dupuy for providing age posteriors of brown dwarf bi- from the same reason that cause the J-band residuals. naries with dynamical mass measurements. This work has benefited from The UltracoolSheet at http://bit.ly/ In future work, spectroscopic analysis of late-T dwarfs will UltracoolSheet, maintained by Will Best, Trent Dupuy, benefit from models that include improved opacities, clouds, Michael Liu, Rob Siverd, and Zhoujian Zhang, and devel- reduced vertical temperature gradient, and/or chemical dis- oped from compilations by Dupuy & Liu(2012), Dupuy & equilibrium. Such will also be essential to studying ultra- Kraus(2013), Liu et al.(2016), Best et al.(2018), and Best cool dwarfs with higher or lower effective temperatures than et al.(2021). This work benefited from 2017–2019 Exoplanet late-T dwarfs, for which there are already numerous spec- Summer Program in the Other Worlds Laboratory (OWL) at tra. James Webb Space Telescope can further extend spectro- the University of California, Santa Cruz, a program funded scopic observations down to cooler temperatures and wider by the Heising-Simons Foundation. M.C.L. acknowledges wavelength coverage. In addition, the analysis conducted in the National Science Foundation (NSF) grant AST-1518339. our work can also be extended to other sets of grid models to This work is funded in part by the Gordon and Betty Moore verify whether the physical assumptions made by those mod- Foundation through Grant GBMF8550 to M.C.L. The ad- els can reproduce the observations, and what atmospheric vanced computing resources from the University of Hawaii processes might be included to improve data-model consis- Information Technology Services – Cyberinfrastructure are tency. Since brown dwarfs harbor similar atmospheric pro- greatly acknowledged and Z. Z. thanks the technical sup- cesses as imaged exoplanets, models employed to interpret port received from Curt Dodds. This research was greatly both classes of objects are generated from the same theoret- facilitated by the TOPCAT software written by Mark Tay- ical framework. Dedicated analyses of brown dwarf atmo- lor (http://www.starlink.ac.uk/topcat/). Finally, the authors spheres will therefore help us to robustly characterize direct wish to recognize and acknowledge the very significant cul- spectroscopy of exoplanets and thereby understand their ap- tural role and reverence that the summit of Maunakea has pearance, formation, and evolution. always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. We thank Mark Phillips, Didier Saumon, Caroline Morley, Facilities: IRTF (SpeX) Eugene Magnier, Paul Mollière, Joe Zalesky, Trent Dupuy, Ehsan Gharib-Nezhad for insightful discussions and com- Software: Starfish (Czekala et al. 2015), emcee (Foreman- ments on this work. We thank Ian Czekala, Michael Gully- Mackey et al. 2013), Spextool (Cushing et al. 2004), TOP- Santiago, and Miles Lucas for helpful discussions about CAT (Taylor 2005), Astropy (Astropy Collaboration et al. Starfish. We also thank Michael Gully-Santiago for imple- 2013, 2018), IPython (Pérez & Granger 2007), Numpy menting Starfish for IRTF/SpeX prism data and sharing ini- (Oliphant 2006), Scipy (Jones et al. 2001), Matplotlib tial work on spectroscopic analysis for T dwarfs (https: (Hunter 2007).

APPENDIX

A. BOLOMETRIC LUMINOSITIES: COMPARISON WITH LITERATURE Among our sample, 15 late-T dwarfs have bolometric luminosities in the literature. Filippazzo et al.(2015) constructed SEDs for 14 of our T7−T9 dwarfs by combining optical, near-infrared, and mid-infrared photometry and spectroscopy, and then computed bolometric luminosities by integrating their SEDs, with short-wavelength fluxes linearly extrapolated to zero wavelength and long-wavelength fluxes approximated by a Rayleigh-Jeans tail. Line et al.(2017) performed a retrieval analysis for SpeX prism spectra of 11 of our T7−T8 dwarfs and computed Lbol by integrating the fitted models. Six objects in these two studies now have newer parallaxes with much higher precisions: HD 3651B, PSO J043.5395+02.3995, 2MASS J1217110−031113, Ross 458C, ULAS 1416+1348, and GJ 570D. We thereby scale these objects’ literature Lbol values with the new parallaxes, without modifying the published uncertainties. Figure 19 and Table7 compare our Lbol values for these 15 late-T dwarfs with the literature and suggest a general consistency. Our work produces slightly fainter Lbol, with the differences having a weighted mean and weighted root mean square of 0.057 ± 0.099 dex compared to Filippazzo et al.(2015) and 0 .075 ± 0.044 dex compared to Line et al.(2017). For UGPS J072227.51−054031.2 (UGPS 0722−0540), our estimated Lbol is fainter than the value in Filippazzo et al.(2015) by 8.4σ. As shown in Figure2, our fitted cloudless models do not well match this object’s observed spectrum and our spectroscopically inferred Teff = 680±26 K is much higher than the values of 500−580 K as suggested by Morley et al.(2012) and Tremblin 30

5.0 Filippazzo et al. (2015) Line et al. (2017)

5.2 ) k r 5.4 o W

s i

h 5.6 T (

)

L 5.8 / l o b L ( 6.0 ULAS 1416+1348 g o l

6.2

6.4 UGPS 0722 0540

6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 log(Lbol/L ) (Literature)

Figure 19. Comparison of bolometric luminosities for 15 late-T dwarfs with measurements both by us and the literature (AppendixA). We use dotted lines to connect results for same object. Our computed Lbol’s of these objects are generally consistent with the literature values but are systematically fainter by 0.06 − 0.07 dex. et al.(2015), whose models account for sulfide clouds or diabatic convection (also see Section 6.2) and can better match this object’s spectrum. Our over-estimated Teff for UGPS 0722−0540 might lead to a smaller ratio between the integrated fluxes in mid-infrared wavelengths (approximated by our fitted models) and in near-infrared wavelengths (indicated by our SpeX prism +0.052 data), resulting in an under-estimated Lbol. For ULAS 1416+1348, our derived Lbol = −6.013−0.049 dex is 4.7σ fainter than the value in Filippazzo et al.(2015), but is consistent with that in Line et al.(2017).

B. BOLOMETRIC CORRECTIONS

Table8 presents our bolometric corrections for late-T dwarfs in JMKO, HMKO, and KMKO bands computed using their bolometric ﬂuxes (with Mbol, = +4.74). For this calculation, we exclude six objects that are resolved or candidate binaries, as well as two objects with peculiar spectra (2MASS J0939−2448 and ULAS J1416+1348; see Section4). Comparison with the bolometric corrections of Liu et al.(2010) demonstrates the 1 σ consistency. Filippazzo et al.(2015) provided bolometric corrections using 2MASS bands, so we cannot compare directly to their results. Following Dupuy & Liu(2012), we also compute an estimate of the intrinsic (astrophysical) scatter in the bolometric correction at a given spectral type and band.

C. FORWARD-MODELING ANALYSIS WITH THE TRADITIONAL APPROACH We also conduct the forward-modeling analysis for our late-T dwarfs following the traditional approach described in Paper I. We use the cloudless Sonora-Bobcat model atmospheres over their entire parameter space of [200,2400] K in Teff, [3.25,5.5] dex in logg, and [−0.5,+0.5] dex in Z, and use linear interpolation to synthesize model spectra with an arbitrary set of grid parameters. We determine six physical parameters {Teff, logg, Z, vr, vsini, logΩ} with same priors as our Starfish-based analysis (Section 3.1), and construct our covariance matrix by simply placing squared flux uncertainties of spectra along its diagonal axis. We use emcee to fit our 1.0−2.5 µm spectra with 24 walkers and terminate the fitting process with 3×104 iterations given that such number of iterations exceeds 50 times the autocorrelation length of all the fitted parameters. We incorporate the systematic error of 180 km s−1 into the inferred radial velocity to account for the uncertainty in the wavelength calibration of the SpeX prism data (Section 3.1). We also incorporate the systematic error of 0.4σHMKO into the inferred logΩ to account for the uncertainty in 31

Figure 20. Comparisons of results from Starfish and traditional forward-modeling analyses, with a similar format as Figure2. Left: The upper panel shows the observed spectrum (black) and the median model spectra of those interpolated at parameters drawn from the MCMC chains based on the Starfish (blue) and traditional (purple) methods. The middle and lower panel show the residual of each method (data−model; black). Right: Posteriors of the six physical parameters {Teff, logg, Z, vr, vsini, logΩ} derived from the Starfish-based forward-modeling analysis (blue). We overlay the median values and uncertainties from the traditional method (purple), shown as vertical lines and shadows in the 1-D histograms and as circles and error bars in the 2-D histograms. We use grey vertical and horizontal lines to mark the {Teff, logg, Z} grids pionts of the cloudless Sonora-Bobcat models. Figures of spectral-fitting results for our entire sample (55 late-T dwarfs with 57 spectra) are accessible online.

flux calibration (Section 3.1), where σHMKO is the photometric error in an object’s H-band magnitude. We compute the objects’ radii (R) and masses (M) using their parallaxes and logg and logΩ posteriors and also compute ages by interpolating the Sonora- Bobcat evolutionary models using the fitted {Teff, logg, Z}. We summarize all inferred parameters and their uncertainties in Table9. Figure 20 compares the parameter posteriors and the fitted model spectra (interpolated at parameters drawn from the MCMC samples) derived from the traditional approach to those from the Starfish analysis. While the fitted model spectra by these two methods both match the data, the residuals from the traditional approach significantly exceed measurement uncertainties in some wavelength ranges, indicating that the more sophisticated Starfish covariance matrix better describes the difference between the data and models (also see Figure 5 of Czekala et al. 2015). Also, the fitted models from the two methods are not always identical at several wavelengths, primarily because of their vastly different covariance matrices, which weight residuals and compute the likelihoods differently given the same set of physical parameters. Figure 21 compares eight physical parameters {Teff, logg, Z, vr, vsini, logΩ, R, M} of late-T dwarfs inferred from the two approaches. The weighted mean and weighted root mean square of the Starfish−Traditional parameter differences are −16±23 K −1 −1 for Teff, −0.09 ± 0.23 dex for logg, −0.09 ± 0.10 dex for Z, 45 ± 125 km s for vr, −1 ± 6 km s for vsini, 0.040 ± 0.061 dex for logΩ, 0.03 ± 0.06 dex for R, and −0.17 ± 1.14 dex for M. Therefore, the results from these two spectral-fitting methods are generally consistent. Several objects have significantly different properties based on the two methods: WISE 2209 − 2734, ULAS 1416 + 1348, WISE 0500 − 1223, 2MASS 0939 − 2448, and WISE 0458 + 6434. For WISE 2209−2734, the Starfish analysis derives logg and Z that are smaller than the traditional method by 0.95±0.34 dex and 0.33±0.10 dex, respectively. However, the fitted model spectra from the two methods are similar, given that these parameter offsets follow the logg−Z degeneracy of the cloudless Sonora-Bobcat models that we find based on the stacked posteriors of our late-T dwarf sample (Section 3.2.2). The different results from the two methods are thus likely caused by the different covariance matrices used to compute the likelihood. ULAS 1416+1348 is a wide-orbit companion to a low-metallicity late-L dwarf (see Section 4.2). Its {Teff, logg, logΩ, R, M} are different between the two methods by 1−6σ, but we find the fitted model spectra from Starfish better match the data especially for the blue wing of Y band (Figure 20). Among the remaining three objects, WISE 0458 + 6434 is a resolved 0.5100 T8.5+T9 binary (Section 4.1). Binarity can result in spectral peculiarity and cause the two forward-modeling approaches to derive different parameters. The binarity of WISE 0500−1223 and 2MASS 0939−2448 are uncertain (Section 4.2). In addition, parameter posteriors of WISE 0500−1223 inferred from the traditional method show two peaks, with one peak consistent with the Starfish results. This anomaly might also be related to the low S/N of data (≈ 13 in J band). 32

ULAS 1416+1348 2MASS 0939–2448 Ross 458C

WISE 0458+6434

WISE 0500–1223

WISE 2209–2734

WISE 0500–1223

WISE 0458+6434

ULAS 1416+1348

WISE 0458+6434

WISE 0500–1223

ULAS 1416+1348

2MASS 0939–2448

WISE 2209–2734 ULAS 1416+1348 WISE 0500–1223

WISE 0458+6434 WISE 0500–1223 WISE 0458+6434

Figure 21. Comparison of eight physical parameters of our late-T dwarfs {Teff, logg, Z, vr, vsini, logΩ, R, M} as inferred from Starﬁsh and traditional forward-modeling analysis (masses are compared in the logarithmic scale). We use grey vertical and horizontal lines to mark the {Teff, logg, Z} grids of the cloudless Sonora-Bobcat models. The results from the two methods are generally consistent within the uncertainties, although there are systematic differences in almost all parameters except for vr and vsini. We label the objects that have signiﬁcantly different parameters from the two methods and discuss them in AppendixC.

Table6 compares typical parameter uncertainties between the two methods. The traditional forward-modeling approach produces artificially small parameter errors by factors of 2 − 15 in {Teff, logg, Z, logΩ, R, M} and factors of 1 − 2 in vr and vsini (also see Figure4). We note that the larger error estimates from Starfish are more realistic, given that Starfish accounts for uncertainties from model interpolation and correlated residuals. 33

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Sample of T7 SpT Slit a Table 1 1 T7.5 0.5 50 – – 25 ... 25 38 2 T86 0.5 13 T8 0.5 – 14 – – 25 – ... 25 25 38 ... 25 38 1 T8 0.5 27 – – 25 ... 25 38 9 T7 0.8 60 2017-10-22 HD 1561 25 ... 25 40 9 B T8.5 0.5 24 – – 27 24 25 38 ...... 6 T8 0.5 453 – C (sd) T7.5 – 0.5 48 32 – ... 32 38 – 17,21 17,21 14 38 2 T9 0.5 79 – – 19 ... 23 38 6 T8.5 0.8 40 2017-10-21 HD 31069 22 ... 22 40 08 T7 T8 0.8 0.5 57 27 2017-11-15 HD 18546 – 29 ... – 39 40 34 ... 34 38 92 B?7 T7.5 T7.5 0.54 0.5 29 T7 67 0.5 T7 – 81 0.5 – – 21 – – – 34 – 35 ... – 34 ... 34 38 35 ... 35 38 34 ... 38 35 38 80 T7 T8.5 0.5 0.5 80 782 – – T7 – – 0.5 66 34 34 ...... – 34 34 38 38 – 34 ... 34 38 40 T7.5 0.8 T7 79 0.8 2017-03-13 94 HD 35036 2017-11-02 33 HD 41076 ... 34 33 ... 40 34 40 2 T7 0.5 33 – – 34 ... 34 38 ...... 293258 122343 045624 234017 414203 643452 3322402 T7 0.5 61 – – 5 ... 6 8 39540432448279 T8 pec T8 0.5 0.5 52 50 – – – – 13 ... 5 13 ... 13 6 8 2618235 T7.5 0.5 29 – – 5 ... 6 8 093506 T8 0.5 176 – – 3 ... 6 38 031113 T7.5 0.5 31 – – 1 ... 6 8 171001 T7 0.5 106 – – 3 ... 6 8 − 3995 T8 0.5 107 – – 26,28 ... 26 38 093514 134836 + + − − − 054031 364048 365328 345047 160002 194250 042149 072828 090054 215120 083118 102528 095135 400854 . − − − − − + − + + + − 39 21 89 05 94 66 − − + + + + + − − − + − 02 ...... 52 94 27 51 + 72 29 73 73 62 61 88 73 95 52 90 12 ...... 5395 . WISEPC J022322 WISE J024124 WISE J024512 PSO J043 WISE J004945 2MASS J00501994 WISEPA J012333 WISE J032547 2MASSI J0415195 ObjectHD 3651BWISE J004024 Type C T7.5 0.5 73WISEPA J045853 WISEPA J050003 – – 10 10 12 11 WISE J052126 UGPS J052127 WISE J061437 WISEPA J062309 UGPS J072227 2MASSI J0727182 2MASS J07290002 2MASS J09393548 ULAS J102940 Ross 458CWISEPA J132233 ULAS J141623 C T8 0.5 43 2015-07-07 HD 116960 18 18 23 40 WISE J103907 WISE J105257 2MASS J11145133 2MASSI J1217110 WISE J125448 WISE J125715 WISE J112438 39 Discovery Type SpT Spectra c A0V Star c Obs. Date b IRTF/SpeX Spectra References (continued) -band S/N J ) (UT) Table 1 00 0.8 34 – – 25 31 25 31 ( m) of the object’s spectra. µ 3 SpT Slit . 1 − 2 a . 58 T73 0.82 T8 B?4 T7.5 64 0.5 T7 0.5 2017-07-19 HD 202025 43 T7 0.5 26 25 0.5 99 – – ... 71 25 – – 40 – – – 25 25 – ... 25 ... 25 25 25 ... 38 38 25 ... 38 25 38 2 T7 0.5 39 – –4 2549 T7 ... 25 0.8 T7 38 T8 70 0.5 0.5 2017-07-10 74 HD 210290 69 25 – – ... 25 – – 40 25 25 ...... 25 25 38 38 7 T8 0.5 38 2017-06-13 HD 194982 25 ... 25 40 98 B5 T8 T8 0.5 0.53 T9 33 20 0.8 T7 106 – – 0.5 2017-03-15 HD 165029 115 – – 25,28 2011-04-20 HD 165029 ... 25 25 25 25 40 ... 31 ... 25 25 25 38 38 40 ...... 9 T7.5 0.8 45 2017-07-19 HD 210501 20 ... 15 40 43 T7.51 T8 0.8 0.5 27 T8 46 0.5 – 2017-06-27 HD 174567 67 – 34 2017-06-13 HD 191225 ... 36 34 30 40 ...... 36 34 40 38 1 T7 0.8 53 – – 9 ... 9 38 4 B? T7 0.8 49 2017-05-12 HD 132072 9 ... 9 40 ...... -band peak (1 J 273439 311841 184404 074507 102844 581510 265931 091139 044003 333833 444423 350036 255319 353716 153236 B T7 0.5 141 – – 3 7 6 16 + + − + + − − − − 4057 T7 0.8 40 – – 37135454 ... 37 37 + + + + − 383805 283533 362950 230821 102718 . + + 38 03 10 69 05 74 13 62 10 78 66 05 60 26 + + + + + 47 ...... 79 + 07 40 19 77 63 ...... 3820 . —(1) Burgasser et al. ( 1999 ), (2) Burgasser et al. ( 2000 ), (3) Burgasser et al. ( 2002 ), (4) Burgasser et al. ( 2004 ), (5) Tinney et al. ( 2005 ), (6) Burgasser WISEPA J165311 WISEPA J171104 WISE J180901 WISEPA J195905 SDSS J162838 WISEPA J174124 WISE J181329 WISEPA J185215 WISE J200050 WISEPC J215751 ObjectWISEPC J145715 GJ 570DPSO J224 2MASSI J1553022 Type C T7.5 0.5 37WISEPC J220922 –WISEPC J221354 WISEPC J222623 WISEPC J225540 –WISEPC J231939 ULAS J232123 WISEPC J234026 2WISEPC J234841 2 6 4 SDSS J150411 dwarfs. et al. ( 2006b ), (7) Burgasser etet al. ( 2006c ), al. ( 2007 ), (8) (13) Burgasser etLooper( 2010b ), al. ( 2006a ), (18) et (9) Goldman al. ( 2007 ), etChiu al. ( 2010 ), et (14) (19) (24) al. ( 2006 ),BurgasserLucasGelino (10) et et etMugrauer al. ( 2010 ), al. ( 2011 ), al. ( 2010b ), (20) et (25) Scholz ( 2010b ),(30) al. ( 2006 ),Kirkpatrick (15) (21) Luhman et (11) BurninghamScholz ( 2010a ), et al. ( 2011 ),( 2007 ),Burgasser (22) (26) et al. ( 2012 ), (12) BurninghamLiu (31) al. ( 2010a ), etLuhman (36) etLiu( 2011 ), al. Cushing( 2011b ), al. (16) et (23) et (27) al. ( 2012 ),CushingBurgasser al. ( 2014 ),Mainzer et (32) et (37) et al. ( 2011 ), BurninghamBest al. ( 2011 ), al. ( 2010a ), et (28) et al. ( 2013 ), (17) Scholz al. ( 2015 ), (33) et (38) BurninghamBihain al. ( 2011 ),Burgasser et et (29) &Kirkpatrick al. ( 2013 ), Splat al. et (34) Development( 2017 ), al. ( 2012 ), Team Mace (39) etTinney( 2013a ), al. et (35) al. ( 2018 ),Thompson (40) et This( 2013 ), al. work “B”: Resolved binaries; “B?”: Binarity is likely but not confirmed (Section Signal-to-noise4.2 ); ratio “C”: around the Companions to stars or brown dwarfs. Remaining objects are single field For spectra taken from our own observations, we show the observation date and the A0V telluric standard star. a b c References 40 50 8 50 8 70 3 60 8 1060 9 8 40 5 90 3 30 4 60 2 40 3 30 6,7 09 6,7 6070 8 8 00 8 20 9 0070 9 9 70 9 90 9 20 9 40 9 8015 9 20 6,7 8 2070 1,8 9 70 9 06 6,7 40 3 70 9 40 9 ...... 3 1 2 0 4 1 0 0 2 1 1 5 8 0 2 2 2 2 2 3 2 2 1 1 1 2 3 1 0 4 2 2 5 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 90 10 20 20 50 20 80 50 30 30 20 56 01 60 40 70 40 50 60 80 50 50 20 90 50 10 86 50 70 30 70 79 40 ...... Parallax References ...... 00 122 50 187 13 170 06 89 50 80 80 141 50 56 00 112 80 68 30 45 00 67 10 179 90 41 . . . 50 126 70 78 19 86 90 46 40 139 30 71 ...... ) (mas) 10 94 40 52 34 146 20 175 20 109 80 84 6021 86 242 40 57 57 77 47 107 . . . . 1 2 . . 0 30 39 00 57 84 91 ...... 1 1 2 2 0 1 2 1 2 2 1 5 . . . 1 2 1 0 4 2 1 1 1 1 1 1 1 1 1 0 1 3 1 ± ± ± − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± δ ± ± ± ± ± ± ± ± ± ± ± ± ± ± 00 70 65 81 10 30 30 70 60 30 90 50 80 . . . 40 50 10 02 80 60 ...... 10 40 95 90 20 50 80 03 62 90 07 ...... 58 80 90 ...... 46 62 38 36 65 535 418 156 763 142 120 314 389 133 369 1511 1044 1723 − − − − − − − − − − − − − − − − − − 80 70 75 11 40 70 490 8071 168 352 30 1643 00 20 40 68 50 47 391 90 . . 20 939 15 221 20 535 10 17 80 ...... ) (mas yr 50 148 80 20 291 60 90 70 30 20 60 167 30 88 10 . . . . . 1 1 . 69 129 ...... 0 1 1 1 1 5 2 2 2 0 1 4 1 70 1 . 0 2 0 1 2 δ µ 1 1 1 1 1 5 0 1 2 2 . 1 0 ± ± − 2 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± cos 50 39 ± 06 20 70 30 14 60 40 60 10 15 31 60 . . 50 61 30 50 40 90 ...... α 10 20 00 20 70 50 00 50 40 90 90 ...... 69 ...... 40 . µ . 53 533 906 904 566 419 197 569 632 348 488 462 483 3021 1054 − − − − − − − − − − − − − − − b T9 Dwarfs: Astrometry − Decl. 33:22:28.9 1150 29:32:57.3 780 36:53:28.134:50:47.8 242 – – – ... 09:35:00.4 2214 12:23:43.2 04:56:24.5 05:40:29.8 39:53:45.4 24:48:39.0 573 16:00:03.0 19:42:50.2 317 26:18:27.9 04:21:49.6 03:11:12.2 23:40:17.1 07:28:28.3 1 21:22:08.1 1031 − − − − − − − − − − − − − − − − − − Table 2 continued b . Sample of T7 (hh:mm:ss.ss) (dd:mm:ss.s) (mas yr SpT R.A. Table 2 a 1 T7.5 02:23:22.38 2 T7 14:57:15.02 58:15:10.2 2 T8 05:00:03.04 6 T8 06:23:09.93 1 T8 13:22:33.63 9 T7 01:23:33.25 41:42:03.9 601 9 B T8.5 04:58:53.91 64:34:52.6 207 ...... 6 T8 10:29:40.52 09:35:14.1 3 C (sd) T7.5 14:16:23.97 13:48:36.1 85 2 T9 07:22:27.28 6 T8.5 05:21:27.42 36:40:43.6 569 08 T7 T8 02:41:24.75 02:45:12.62 9 B? T7.5 10:39:07.74 2 T7.5 10:52:57.95 7 T7 11:24:38.11 4 T7 12:54:48.51 4 T7.5 05:21:26.31 10:25:28.4 215 0 T7 06:14:37.75 09:51:35.2 387 2 T7 12:57:15.92 40:08:54.2 294 8 T7 00:40:24.89 09:00:54.5 0 T8.5 00:49:45.65 21:51:19.7 2 T7 03:25:47.73 08:31:18.2 116 ...... 293258 581510 122343 045624 234017 414203 643452 3322402 T7 00:50:21.05 3954043 T8 pec 07:28:59.50 2448279 T8 09:39:35.93 2618235 T7.5 11:14:48.75 093506 T8 04:15:21.27 031113 T7.5 12:17:10.28 171001 T7 07:27:19.16 17:09:51.3 1045 − + 3995 T8 02:54:09.56 02:23:58.6 2562 093514 134836 − + + − − 054031 364048 365328 345047 160002 194250 042149 072828 090054 215120 083118 102528 095135 400854 . − − − − − + − + + + − 39 03 21 89 05 94 66 − + + + − − + + + − − − 02 ...... 94 52 27 51 + 73 62 72 29 73 73 95 12 52 90 88 61 ...... 5395 . ObjectHD 3651B Type C T7.5 00:39:18.91 21:15:16.8 WISE J004024 WISEPA J012333 2MASS J00501994 WISE J004945 WISEPC J022322 WISE J024124 WISE J024512 PSO J043 WISE J032547 2MASSI J0415195 WISEPA J045853 WISEPA J050003 WISE J052126 UGPS J052127 WISE J061437 WISEPA J062309 UGPS J072227 2MASSI J0727182 2MASS J07290002 2MASS J09393548 ULAS J102940 WISE J103907 WISE J105257 2MASS J11145133 WISE J112438 2MASSI J1217110 Ross 458CWISEPA J132233 ULAS J141623 C T8 13:00:41.64 12:21:14.6 WISE J125448 WISE J125715 GJ 570D C T7.5 14:57:15.84 WISEPC J145715 41 80 8 90 9 9040 9 8 30 9 00 9 90 9 20 8 20 9 8010 9 9 90 8 20 8 1040 9 8 1080 8 9 00 9 90 3 50 3 90 3 ...... 2 2 3 0 1 0 1 2 2 2 2 2 2 3 3 5 4 3 2 3 3 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 30 20 10 10 10 70 30 30 60 40 30 80 40 10 40 70 10 60 80 60 50 . . Parallax References ...... 28 214 10 46 70 75 40 75 30 52 90 73 00 66 10 85 60 62 20 72 08 54 20 70 40 83 25 47 . 20 40 80 52 30 49 ...... ) (mas) 90 75 20 131 40 85 90 67 . . 1 . . . . . 2 0 2 2 2 3 1 1 2 5 2 1 1 1 2 1 1 0 4 1 1 ± − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± δ ± ± ± ± 61 20 00 20 70 10 00 20 50 90 55 10 60 12 . 20 40 50 ...... 20 30 20 20 ...... 84 71 35 443 398 506 468 288 193 100 440 463 162 561 250 369 1472 − − − − − − − − − − − − − − − − − 70 166 10 11 60 10 00 50 83 90 ...... ) (mas yr 00 50 95 10 146 00 90 80 00 . 30 4020 131 ...... 0 1 1 2 2 . 2 1 3 1 30 368 . . . 1 δ µ 3 1 0 3 . 1 0 2 1 1 1 1 − 3 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± cos ± 90 30 34 70 90 10 20 17 00 ...... α 30 40 20 80 80 80 60 60 . 80 00 70 ...... 20 . . . µ . 8 76 156 505 590 208 765 116 283 385 − − − − − − − − − − b (continued) Decl. 33:38:33.8 27:34:39.9 31:18:42.0 300 18:44:04.4 74 07:45:08.5 152 10:28:44.1 620 − − − − − − Table 2 b (hh:mm:ss.ss) (dd:mm:ss.s) (mas yr SpT R.A. a 5 T7 22:09:22.09 8 T8 22:55:40.76 3 T7.5 23:19:39.15 2 B? T7 23:40:26.62 4 T7 23:48:41.12 4 T7 21:57:51.35 26:59:31.3 65 49 T7 T8 22:13:54.70 22:26:23.06 09:11:39.4 04:40:03.3 7 T8 19:59:05.65 9 T8 16:53:11.03 44:44:22.7 85 B T8 T9 17:11:04.59 35:00:36.8 17:41:24.23 25:53:19.2 3 T7 18:52:15.82 35:37:16.2 254 ...... 9 T7.5 23:21:23.83 13:54:53.3 78 1 T8 20:00:50.19 36:29:50.1 2 4 T7.5 18:09:01.07 38:38:05.2 3 T8 18:13:29.40 28:35:33.3 4 B?1 T7 15:04:11.81 T7 10:27:15.4 373 16:28:38.99 23:08:17.9 412 ...... 273439 311841 184404 074507 102844 265931 091139 044003 333833 444423 350036 255319 353716 153236 B T7 15:53:01.93 15:32:38.8 + − + + − − − − 4057 T7 14:57:31.68 47:24:20.2 139 135454 + + + + − 383805 283533 362950 102718 230821 . + + 38 10 69 05 74 13 62 10 05 60 26 78 66 + + + + + 47 ...... 79 + 07 40 19 63 77 ...... 3820 . —(1) Tinney et al. ( 2003 ), (2) Burgasser et al. ( 2008b ), (3) Dupuy & Liu ( 2012 ), (4) Faherty et al. ( 2012 ), (5) Leggett et al. ( 2012 ), (6) Gaia Collaboration et al. ObjectPSO J224 Type SDSS J150411 SDSS J162838 2MASSI J1553022 WISEPA J165311 WISEPA J171104 WISEPA J174124 WISE J180901 WISE J181329 WISEPA J185215 WISEPA J195905 WISEPC J215751 WISE J200050 WISEPC J220922 WISEPC J221354 WISEPC J222623 WISEPC J225540 WISEPC J231939 ULAS J232123 WISEPC J234026 WISEPC J234841 ( 2016a ), (7) Gaia Collaboration et al. ( 2018 ), (8) Kirkpatrick et al. ( 2019 ), (9) Best et al. ( 2020a ) “B”: Resolved binaries; “B?”: Binarity is likely but notCoordinates confirmed are (Section provided4.2 ); at “C”: epoch Companions J2000 to with stars equinox or J2000. brown dwarfs. Remaining objects are single field dwarfs. a b References 42 02 24 02 24 02 10 0202 24 02 18 01 17 23 02 17 02 24 02 18 02 24 02 18 02 24 01 23 02 24 02 24 02 10 02 17 02 24 0202 24 02 10 10 02 17 02 10 02 17 02 24 02 10 02 24 02 24 02 17 02 17 02 8 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5] References . ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 44 80 89 49 77 29 82 10 35 22 21 46 62 95 02 22 74 09 58 00 37 98 59 70 59 35 01 84 59 04 76 62 ...... /IRAC Photometry 03 14 02 12 03 13 02 13 04 14 03 14 01 13 02 14 02 12 03 14 03 14 03 14 02 11 01 12 02 13 02 12 03 13 03 14 02 13 02 12 02 12 03 14 02 13 02 12 03 14 03 14 03 14 03 14 02 13 02 13 02 13 04 13 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Spitzer 6] [4 . ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 85 74 67 75 33 92 28 37 19 66 87 08 77 47 46 30 49 48 94 26 08 95 70 69 05 74 81 12 87 01 07 38 ...... 04 19 15 04 19 14 04 19 15 0308 1905 1904 19 14 19 16 15 15 05 19 15 02 19 14 04 19 15 05 19 15 07 19 16 02 19 13 02 19 14 03 19 14 02 19 14 04 19 15 05 19 15 03 19 14 0303 1904 1902 19 14 19 15 15 – – ... 03 19 14 03 19 14 05 19 16 04 19 15 04 19 15 07 19 16 03 19 14 03 19 15 05 19 15 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 References [3 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± W 42 78 96 21 85 43 85 05 27 11 18 38 64 97 96 20 81 24 00 29 02 98 29 53 76 50 35 02 95 58 03 83 ...... 06 14 19 12 09 13 08 14 06 13 04 13 10 14 08 14 04 12 08 14 07 14 12 14 03 11 03 12 04 12 05 12 09 13 11 14 03 13 04 12 15 13 03 12 07 13 04 13 05 12 10 14 08 14 08 14 08 14 04 13 06 13 06 13 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± W 66 99 73 67 04 27 78 45 25 58 48 78 91 29 19 25 84 62 40 45 11 14 44 35 81 03 86 87 84 60 92 99 ...... References 06 25,26 16 09 12,26 15 40 26 16 0606 25,26 12 16 16 0307 1 18,26 16 15 06 25,26 16 05 8 15 05 25,26 16 07 16,26 16 02 14 16 09 5 14 10 26 15 03 2,11 15 08 7 15 09 25,26 16 05 25,26 16 0805 12,2609 25,26 17 13,27 14 14 0310 2,26 9,26 15 16 09 26 15 05 12 15 11 17,26 17 05 16,24,27 16 08 10,26 16 06 25,26 16 10 8 15 17 24,26 15 05 12 15 05 4,26 – – ... 15 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T9 Dwarfs: Photometry ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± − MKO K 21 43 18 90 02 92 40 64 54 00 09 64 83 50 69 07 27 51 07 00 32 83 74 39 73 05 27 59 25 91 80 55 87 ...... Table 3 continued 06 17 03 18 03 15 03 17 06 17 04 16 14 17 05 16 05 16 05 17 04 17 01 17 09 16 10 16 03 15 02 17 07 17 05 16 03 15 11 17 12 18 05 15 04 17 08 16 02 16 10 18 07 17 11 17 05 17 10 15 02 16 02 16 04 16 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± MKO H 14 58 98 63 14 01 65 77 82 08 19 63 96 05 67 90 34 66 45 70 13 27 28 19 29 27 04 30 25 04 72 56 68 ...... Sample of T7 02 17 02 17 03 15 01 17 02 17 02 17 11 16 02 16 05 15 02 17 02 17 01 17 09 15 08 16 03 15 01 16 02 17 02 16 07 17 03 15 02 18 02 15 02 17 07 16 01 16 09 18 01 17 05 17 02 17 10 16 16 16 01 16 03 16 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Table 3 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Near-Infrared MKO Photometry AllWISE Photometry MKO J 79 26 69 75 56 30 89 37 52 83 89 28 61 66 19 52 10 44 13 87 94 32 88 29 92 77 72 10 99 65 44 13 16 ...... 06 16 03 17 03 16 12 16 06 17 06 16 03 15 05 16 05 15 05 16 03 16 02 17 09 15 09 15 06 15 02 16 06 17 05 16 05 14 05 16 09 17 07 17 06 15 09 16 01 15 11 17 05 16 07 17 05 16 10 15 17 16 02 16 06 16 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± MKO Y (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) 81 16 41 72 74 14 58 35 36 72 19 24 47 60 16 37 04 54 02 69 08 71 39 14 00 65 73 08 10 85 29 15 12 ...... T7 17 SpT T8.5 18 a 1 T7.5 18 2 T7 17 1 T8 17 6 T8 18 2 T8 18 9 B T8.5 18 9 T7 18 ...... b b 3 C (sd) T7.5 18 6 T8 18 2 T9 17 6 4 T7 18 7 T7 17 2 T7.5 17 9 B? T7.5 18 8 T8 18 0 2 T7 18 0 T7 17 4 T7.5 15 2 T7 17 0 T8.5 17 8 T7 17 ...... 293258 581510 234017 045624 122343 643452 414203 2618235 T7.5 16 2448279 T8 16 3954043 T8 pec 16 3322402 T7 16 031113 T7.5 16 093506 T8 16 171001 T7 16 + − 134836 093514 3995 T8 17 − − + − + 054031 364048 072828 042149 194250 160002 345047 365328 400854 095135 102528 083118 215120 090054 . − − − − − + − + + − + 03 39 66 94 89 05 21 − + − − − + + + − − + + 02 ...... 94 52 51 27 + 52 90 12 95 73 73 29 72 62 73 61 88 ...... 5395 . WISEPC J145715 ULAS J141623 WISE J125448 WISE J125715 Ross 458CWISEPA J132233 C T8 17 2MASSI J1217110 WISE J112438 2MASS J11145133 WISE J105257 WISE J103907 ULAS J102940 2MASS J09393548 2MASS J07290002 2MASSI J0727182 UGPS J072227 WISEPA J062309 WISE J061437 UGPS J052127 WISEPA J050003 WISE J052126 WISEPA J045853 2MASSI J0415195 WISE J032547 PSO J043 WISE J024512 WISE J024124 WISEPC J022322 WISEPA J012333 2MASS J00501994 WISE J004945 WISE J004024 ObjectHD 3651B Type C T7.5 17 43 02 10 02 10 02 24 02 10 02 10 02 10 02 10 02 10 02 10 02 23 02 17 0202 10 02 10 02 10 02 17 02 17 10 02 24 02 24 02 24 02 24 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5] References . ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 36 62 19 95 21 54 56 90 44 68 79 19 19 87 62 39 90 94 13 06 15 ...... /IRAC Photometry 03 14 02 13 03 14 03 13 03 14 03 14 03 14 03 13 03 14 02 12 02 13 04 14 02 12 04 14 03 14 03 14 03 13 03 13 02 13 03 14 02 12 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Spitzer 6] [4 . ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 87 19 86 92 91 12 79 48 01 22 36 67 46 43 49 82 58 44 51 50 88 ...... 05 19 15 04 19 15 06 19 15 04 19 15 05 19 15 06 19 16 06 19 15 04 19 15 05 19 16 03 19 14 04 19 15 0304 1902 1906 19 15 04 19 16 04 19 14 19 16 15 15 04 19 15 03 19 14 04 19 15 04 19 – – ... 02 19 13 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 References [3 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± W 38 60 11 82 16 51 65 86 49 69 84 82 63 35 98 04 14 92 03 06 65 13 ...... ) using their SpeX prism spectra and the approach MKO Y 09 14 06 13 12 14 09 13 08 14 11 14 08 14 08 13 11 14 06 12 06 13 16 14 04 12 05 14 05 13 13 14 04 14 05 13 04 13 06 14 06 14 04 12 ...... 6 ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± W 65 93 94 67 55 86 58 31 99 08 15 95 80 30 59 58 49 27 29 22 62 93 ...... 364048 + 27 . References 06 25,26 16 06 26 15 01 6 16 08 10,26 16 05 22,26 16 09 12 16 06 25,26 16 06 25,26 16 06 25,26 16 04 13,20,26 15 09 10,26 16 06 25,26 15 2007 15,2606 25,26 15 25,26 17 15 14 26 17 06 25,26 16 03 3 16 03 2 15 08 12 16 06 21,25,26 16 05 8 14 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± (continued) ) and UGPS J052127 MKO K 84 51 16 27 42 24 12 31 35 13 52 93 02 32 93 10 07 72 94 12 08 52 ...... MKO K and Table 3 06 16 04 16 03 17 05 18 03 17 07 17 06 17 06 17 04 17 01 16 04 17 06 17 06 16 06 16 05 16 14 18 05 17 03 16 03 15 05 17 06 17 05 15 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MKO ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± MKO Y H 99 41 15 95 66 45 11 91 49 85 06 31 70 11 73 18 60 63 76 99 43 28 ...... 0 ( . 02 16 03 16 03 17 02 17 01 17 02 17 02 17 02 16 02 17 01 15 13 18 02 16 02 17 02 17 02 16 07 17 02 17 03 16 03 15 01 16 02 17 05 15 365328 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Near-Infrared MKO Photometry AllWISE Photometry MKO 73 J . 62 08 72 56 33 90 76 55 04 44 63 18 39 32 71 91 08 25 34 51 11 82 ...... 05 16 06 16 03 16 06 17 02 17 03 16 05 16 05 16 05 17 05 15 05 16 09 16 14 17 05 16 06 17 05 16 06 17 03 16 06 15 02 16 06 17 10 14 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± MKO Y (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) 69 20 92 56 38 04 81 53 00 31 23 41 67 86 55 10 11 27 37 63 01 78 ...... SpT a —(1) Leggett et al. ( 2002 ), (2) Knapp et al. ( 2004 ), (3) Chiu et al. ( 2006 ), (4) Luhman et al. ( 2007 ), (5) Leggett et ( 2009 ), al. (6) Burningham et al. ( 2010a ), (7) Lucas et al. 4 T7 17 2 B? T7 17 3 T7.5 18 8 T8 18 5 T7 17 9 T8 18 4 T7 17 4 T7 18 7 T8 17 3 T7 17 85 B T8 T9 18 17 9 T8 18 ...... 9 T7.5 17 1 T8 16 4 T7.5 18 3 T8 17 1 T7 17 4 B? T7 17 ...... described in Best et al. ( 2021 ). ( 2010 ), (8) Leggett et al. ( 2010 ),Burningham (9) Gelino et et al. ( 2013 ), al. ( 2011 ), (15) (10) DupuyetKirkpatrick & al. ( 2014 ), et Kraus ( 2013 ), (21) al. ( 2011 ), (16) Best (11) etDupuyMcMahon & al. ( 2015 ), et( 2012 ), Liu (22) al. ( 2013 ), (12) Edge (17) Lawrence etMace et al. ( 2016 ), et al. ( 2012 ), (23) al. ( 2013a ), (13) LeggettUKIDSS (18) et( 2012 ), Consortium Thompson al. ( 2017 ), (14) et (24) ( 2013 ), al. Kirkpatrick (19) etCutri al. ( 2019 ), (25) &Best et et( 2014 ), al. al. ( 2020a ), (20) (26) Cushing Best et al. ( 2021 ), (27) This work “B”: Resolved binaries; “B?”: Binarity is likely but notWe confirmed calculate (Section synthetic4.2 ); photometry “C”: for Companions WISE to J024124 stars or brown dwarfs. Remaining objects are single field dwarfs. 102844 074507 184404 311841 273439 044003 091139 265931 333833 353716 350036 255319 444423 References a b 153236 B T7 16 − − − − + + − + 135454 4057 T7 18 + − + + + 362950 383805 283533 230821 102718 . + + 10 62 13 74 05 69 10 38 60 26 78 66 05 + + + + + 47 ...... 79 + 19 07 40 77 63 ...... 3820 . WISEPC J234841 WISEPC J234026 ULAS J232123 WISEPC J231939 WISEPC J225540 WISEPC J222623 WISEPC J221354 WISEPC J220922 WISEPC J215751 WISE J200050 WISE J181329 WISEPA J185215 WISEPA J195905 WISEPA J171104 WISEPA J174124 WISE J180901 WISEPA J165311 SDSS J162838 2MASSI J1553022 SDSS J150411 PSO J224 ObjectGJ 570D Type C T7.5 15 44 a 029 043 001 025 135 029 015 016 055 003 059 122 040 029 116 075 010 064 033 040 098 067 007 029 020 070 001 194 217 026 073 ...... band 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 systematics + + − + + − + − + + − − − − + + + − + − − + + + + + + + − − − K Ω 004 000 004 059 005 002 004 012 004 067 035 002 004 063 009 017 156 019 018 005 003 056 031 027 017 068 049 038 017 032 049 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 band + − − − − − − − + − − + + − − − − − − − + − + − − − + − − + additive log + J ) (dex) (dex) 1 1365 599 366 1128 381 754 1537 835 264 906 181 1624 1014 747 1015 1020 907 190 288 410 989 207 1677 912 819 797 1008 1273 148 145 169 460 424 1816 549 316 518 209 1626 618 471 90 416 238 598 100 1643 798 512 644 683 819 70 70 79 895 329 447 1048 751 725 1065 − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 8744 3206 7538 3623 8620 6024 8433 2150 8479 1698 6486 8328 8541 8114 8153 6848 5454 1809 5246 8371 5439 6579 7984 8482 8227 8234 8045 5482 5481 5470 7939 ` 14 08 08 11 08 10 13 10 07 11 06 14 11 09 11 10 11 07 07 08 10 08 13 10 11 10 11 13 07 07 07 07 14 08 09 12 09 11 13 10 07 11 07 14 11 09 11 11 12 08 07 08 11 08 14 10 11 09 11 14 07 07 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 06 98 40 19 10 97 30 06 92 80 92 22 62 05 67 64 68 50 66 54 66 23 50 06 24 35 75 93 50 80 11 G ...... a 34 34 33 35 35 34 33 34 34 35 35 34 34 34 34 34 35 35 35 36 34 34 33 34 35 33 35 35 34 34 34 − − − − − − − − − − − − − − log − − − − − − − − − − − − − − − − − 04 04 04 05 04 04 04 04 04 05 04 05 04 04 04 04 04 04 04 05 04 04 04 04 04 04 04 04 04 04 04 05 04 04 04 05 04 05 04 04 04 04 04 05 04 04 04 04 04 04 04 04 04 04 04 04 04 04 05 04 04 04 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 11 00 04 08 07 07 02 09 05 93 05 08 83 04 66 00 02 06 06 02 07 90 75 93 06 99 07 04 06 03 04 ...... N . 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 0 1 0 a 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 048 037 093 138 043 049 052 178 042 028 048 055 057 046 047 048 052 027 032 060 030 038 051 038 051 071 112 034 051 034 145 033 158 056 035 110 109 041 050 060 179 050 029 051 062 054 051 052 045 051 026 032 069 029 036 053 043 051 076 168 031 058 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 063 113 152 066 072 077 182 071 059 071 080 076 070 070 069 072 069 064 091 065 074 072 071 072 096 138 064 075 065 156 072 167 075 063 124 129 065 073 080 189 073 059 072 083 075 072 072 069 073 069 064 096 064 074 074 072 074 101 182 064 077 064 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 902 610 225 326 263 749 860 803 772 986 736 437 220 067 727 543 676 348 390 593 290 739 927 084 380 493 135 651 910 003 783 ...... Ω 19 18 19 19 19 18 19 19 20 20 19 19 19 19 19 20 19 19 20 19 19 19 19 19 19 19 19 19 19 19 19 − − − − − − − − − − − − − − log − − − − − − − − − − − − − − − − − ) ) 103 103 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ( ( ) ) (dex) (dex) (km s 34 50 12 17 16 33 27 20 19 45 19 10 11 20 21 21 19 20 11 19 15 18 10 26 25 40 9 18 18 22 20 46 62 13 21 19 57 30 22 22 63 23 20 26 10 12 21 25 25 20 22 11 22 19 28 10 30 29 69 10 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 106 − 104 i 50 61 19 27 24 49 40 30 28 58 28 15 16 30 32 32 29 30 17 29 23 28 15 39 37 55 14 27 27 33 26 48 63 19 26 25 57 35 28 29 65 30 27 31 15 17 28 32 31 27 29 17 28 25 34 16 35 34 70 15 + − + − − + + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − − + sin 37 60 14 27 50 74 19 27 24 49 40 30 28 67 28 27 v 33 15 16 30 32 32 29 152 30 17 29 23 28 15 39 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 183 186 ) ) ) ) ( ( ) ) ) ) 119 151 143 191 82 127 135 119 110 98 142 110 114 91 76 267 128 181 124 245 131 96 85 137 118 153 144 191 83 127 135 125 88 107 96 140 113 116 93 76 275 128 182 123 243 95 ) ) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 155 168 93 98 165 166 94 96 Table 4 continued ( ( ( ( ( ( ( ( 187 139 96 95 184 144 257 256 ( ( ( ) (km s ( ( ( 222 214 235 226 262 197 221 224 217 199 209 204 226 211 212 201 195 326 220 257 218 301 202 303 225 214 235 228 262 198 220 223 217 202 199 209 204 227 211 213 202 195 324 220 255 217 1 + − 238 245 203 204 204 243 243 202 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − − 203 259 227 203 258 228 + − + − + − + − + − + − + − 197 84 287 359 298 130 566 164 2 397 276 180 91 665 147 387 134 9 60 248 180 1121 242 184 165 743 167 28 0 318 398 r − − − − − − − − − − − ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 07 06 06 13 07 07 07 07 13 14 09 12 12 08 09 09 07 08 12 15 13 11 08 09 11 11 16 10 ) ) ) 15 10 10 11 08 16 14 09 10 09 07 10 13 20 14 15 13 10 13 15 17 07 09 11 11 09 14 14 ) ) ) ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 20 15 11 17 15 ...... − − − − − − − − 0 − − 0 − − − − − − − − − − − − − − 0 − − − − 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (+ (+ (+ (+ (+ (+ (+ (+ ( (+ (+ ( (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ ( (+ (+ (+ 14 10 09 18 12 11 10 12 16 18 18 23 15 17 17 15 15 15 14 14 17 20 18 17 12 15 19 17 16 20 16 18 21 14 14 15 19 15 15 15 14 16 20 19 20 16 15 18 18 15 15 15 14 15 18 22 18 19 17 16 18 17 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − − + − + + − + − + − + − + − + − + − + − + − + − − + − + − + + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 06 22 40 41 24 38 39 40 38 00 15 06 23 22 34 24 23 11 31 33 34 36 11 13 13 32 38 29 23 01 32 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − − − − − − − − − − − − − − − − − − − − − − − − − − − − ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . Starfish-Based Forward-Modeling Analysis: Fitted parameters of T7–T9 Dwarfs 28 20 20 21 22 38 20 18 25 35 28 14 19 18 24 21 18 11 22 10 20 27 34 06 22 20 36 21 16 20 22 24 15 24 27 23 24 23 35 21 17 24 37 23 20 22 19 21 16 25 21 17 16 22 18 20 30 36 11 22 20 44 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 29 42 28 25 34 30 28 29 29 42 28 27 32 40 33 28 29 22 27 26 31 29 26 18 30 17 28 34 41 11 30 33 30 31 30 42 29 26 32 41 31 26 28 26 32 30 26 25 30 25 28 36 43 22 30 28 47 30 25 32 29 30 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − + − + + − + − + − − + − + − + − + − + − + − + − + − + − + + − + − − + − + − + − + − + − + − + − + − + − + − + − + − + − + Table 4 g Z v 61 74 83 52 18 98 98 83 09 58 06 13 15 93 43 15 42 07 91 13 36 50 21 70 60 93 51 74 94 07 47 ...... 4 4 3 3 3 4 4 3 3 4 4 4 3 3 4 4 4 3 3 4 3 4 3 4 3 4 4 4 3 3 3 log ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 17 17 ( ( 38 28 17 22 24 49 21 13 25 29 21 22 16 19 13 25 14 15 20 19 20 27 49 14 21 17 47 17 17 19 16 17 36 35 17 21 23 59 20 14 24 27 22 19 23 16 19 12 23 15 15 19 19 19 25 40 15 22 17 49 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 26 26 42 34 26 30 31 55 29 24 32 35 29 30 26 28 24 32 25 25 28 27 28 33 53 24 29 26 51 26 26 28 26 26 41 41 26 29 30 60 28 24 32 34 30 28 30 26 27 23 31 25 25 28 28 28 32 46 25 29 26 53 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − eff 656 794 847 821 729 879 886 936 933 804 753 756 947 1023 787 909 804 798 873 793 743 727 846 781 956 743 680 896 806 818 T 883 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ) (K) (dex) (dex) (km s 5 5 5 5 5 5 5 5 5 5 5 5 5 8 5 8 5 8 5 5 5 5 5 5 8 8 8 5 5 5 5 ...... 00 ( 0 3954 0 2448 0 2618 0 0311 0 3322 0 0935 0 1710 0 0935 0 0540 0 3640 0 1600 0 1942 0 0421 0 0728 0 2932 0 3653 0 3450 0 1223 0 0456 0 4008 0 0900 0 2151 0 4142 0 0831 0 6434 0 1025 0 0951 0 − − − − − + − 02 0 + + − − − − − + + + + − − − + + − + + − + 2MASS 0727 2MASS 0729 2MASS 0939 ULAS 1029 WISE 1039 WISE 1052 2MASS 1114 WISE 1124 2MASS 1217 WISE 1254 WISE 1257 Ross 458C 0 WISE 0040 WISE 0049 2MASS 0050 WISE 0123 WISE 0223 WISE 0241 WISE 0245 PSO J043 WISE 0325 2MASS 0415 WISE 0458 WISE 0500 WISE 0521 UGPS 0521 WISE 0614 WISE 0623 UGPS 0722 ObjectHD 3651B Slit 0 45 a 067 210 113 041 039 068 100 054 035 085 056 189 031 045 206 161 102 082 058 028 095 068 274 086 014 084 ...... band 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 systematics + + − + + − + + + + − + + + + + + + − − − + + + K + + Ω 002 013 005 003 004 008 072 019 004 005 002 198 005 035 047 052 033 014 030 130 047 048 030 010 019 070 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 band + − − + − − − − − − + + + − − + − + + + + + + + − additive log J − ) (dex) (dex) 1 292 1059 1008 1177 1066 488 881 1226 811 368 76 456 737 957 994 846 170 910 1029 372 354 422 883 236 367 548 1181 480 1211 748 634 80 165 706 1072 788 667 175 264 650 118 486 198 36 224 509 659 437 3 4 − 7 8 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − used in this work (see 8195 7523 8024 8258 5041 8431 7297 824 5306 5526 5272 8526 6118 8097 8254 5468 5351 825 8504 8274 2714 5337 5208 8263 5413 8530 ` MKO 07 11 11 10 13 09 10 11 10 05 09 05 10 09 12 11 10 07 04 10 12 08 09 08 10 08 10 07 08 11 11 12 13 09 10 12 10 08 10 05 10 05 10 09 14 11 04 11 13 08 10 09 H ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 65 07 36 30 26 68 95 73 99 97 22 58 94 41 33 73 94 04 93 91 36 30 79 52 28 25 G ...... a errors of parameters directly obtained 35 35 35 34 34 34 34 35 34 34 35 33 35 35 35 35 35 33 35 35 34 34 35 34 35 33 σ − − − − − − − − − − − − − − − − − − log − − − − − − − − 04 04 04 04 04 04 04 04 04 04 04 05 04 04 04 04 04 04 04 05 04 04 05 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 05 04 04 04 04 04 05 04 04 05 04 04 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 88 06 02 92 06 98 95 93 91 01 05 07 01 04 08 91 01 03 07 18 98 88 14 08 09 97 ...... N . 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 a 1 photometry rather than ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) MKO 047 041 047 090 049 049 043 092 038 045 053 055 039 048 046 037 123 063 097 049 037 035 033 039 036 057 046 034 121 047 044 047 108 051 055 042 042 036 054 114 042 054 055 054 037 049 083 090 055 041 036 032 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( or 070 070 106 071 072 070 084 071 077 079 067 072 068 063 129 073 101 107 075 066 066 063 066 062 078 122 134 073 071 071 117 071 075 070 078 141 085 073 078 079 068 073 068 063 109 105 078 067 066 063 066 062 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − MKO J 630 957 879 868 969 770 711 877 781 950 122 083 150 997 761 761 746 207 994 558 555 918 012 102 972 350 ...... Ω 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 19 19 19 19 20 20 19 19 − − − − − − − − − − − − − − − − − log − − − − − − − − − ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 74 74 ) (dex) (dex) (km s ( ( 12 26 22 44 21 13 23 20 16 13 14 38 25 16 29 17 12 17 27 24 18 16 14 14 20 18 14 35 26 51 24 15 16 14 30 21 18 15 15 54 28 36 18 14 20 28 26 23 26 25 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 78 − 74 i 19 39 33 57 31 21 35 29 24 21 22 52 38 25 44 26 19 27 40 36 27 27 21 21 30 24 20 39 32 54 30 21 22 20 34 28 25 21 21 54 34 40 25 19 25 34 32 29 30 31 + − + − − + + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − − + + − + − + − + − + − + − − + sin (continued) 22 56 38 25 19 39 33 66 31 36 27 27 21 21 21 35 29 24 21 v 30 108 44 26 19 27 40 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 161 164 74 101 234 115 110 89 130 145 103 133 106 115 160 139 59 145 145 386 147 127 225 123 101 138 155 167 75 100 127 232 117 108 88 123 155 123 109 114 164 136 59 136 141 403 146 213 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 105 78 102 79 ( ( ( ( Table 4 ) (km s 206 224 238 244 194 205 219 296 214 210 200 220 234 218 209 212 241 226 189 229 230 435 232 284 235 206 224 237 244 194 205 229 448 231 219 292 214 210 200 219 281 218 209 213 242 225 188 227 1 207 195 195 207 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − − + − + − 921 533 485 199 127 261 107 175 47 367 176 142 89 120 308 431 426 817 183 298 343 499 546 118 132 348 r − − − − − − − − − − − − − ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 11 14 08 08 11 12 12 09 13 09 09 04 08 07 09 08 07 07 18 15 05 08 ) ) ) ) 08 13 09 16 11 09 08 07 11 16 11 09 07 19 25 08 13 15 11 09 09 08 ) ) ) ) ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 12 18 15 17 19 16 14 ...... − − − 0 − − − − − 0 − − − − − − − − − 0 − − 0 − − − 0 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (+ (+ (+ ( (+ (+ (+ (+ (+ ( (+ (+ (+ (+ (+ (+ (+ (+ (+ (+ ( (+ (+ ( (+ (+ 16 19 13 20 14 17 08 17 17 22 15 18 14 15 07 14 11 15 14 19 11 14 17 21 20 12 14 14 16 20 15 20 14 18 22 27 14 18 19 22 15 20 16 15 15 14 15 14 20 15 14 18 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − − − + + + − + − + − + − − + − + + − + − + − + − + − + − + − + − + − − + − + + − + − + − + − + − + − if the objects’ observed spectra are flux-calibrated by their 31 26 21 37 18 29 15 26 42 04 31 05 21 10 36 28 43 38 39 31 33 12 39 30 05 01 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ω − − − − − − − − − − − − − − − − − − − − − − ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 27 23 23 21 22 21 28 19 21 18 19 32 19 21 24 15 22 20 15 19 29 33 18 20 25 15 20 19 27 25 22 22 23 43 19 19 24 27 18 21 19 18 22 28 19 26 16 23 21 15 19 28 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( uncertainties for each fitted parameter with systematic errors incorporated (Section 3.1 ). The formal 1 27 36 28 28 33 30 31 29 30 28 36 38 26 28 29 34 27 29 27 32 25 31 25 29 28 25 34 30 30 30 33 27 29 27 27 32 28 28 32 25 30 29 25 28 34 47 27 27 31 34 26 33 σ ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − + − + + − + − − + − + − + − + − + − + − + + − + − + − + − − + − + − + − + + − + − + − − + − + − + − + g Z v 20 53 09 20 97 92 67 79 54 19 78 66 46 90 62 82 10 21 10 80 61 91 63 98 21 27 ...... 3 4 4 4 4 3 3 4 3 3 3 3 4 4 3 5 4 3 3 3 4 4 4 3 3 3 log ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 18 36 20 24 20 17 24 22 18 19 19 16 42 22 17 24 16 18 17 18 17 22 46 17 32 25 21 18 19 33 21 23 19 41 16 16 24 22 18 19 19 16 46 28 27 23 16 18 17 18 17 22 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 27 41 28 32 28 26 31 30 27 28 27 26 46 30 26 31 26 27 26 27 26 30 52 26 37 32 29 26 28 39 29 31 28 46 26 26 31 30 27 28 27 26 49 34 33 30 25 27 26 27 26 30 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − eff 782 822 918 900 811 899 812 810 860 801 908 829 790 828 930 937 885 910 800 817 821 719 838 822 T 695 882 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ) (K) (dex) (dex) (km s 5 5 8 5 5 8 8 5 5 5 5 8 8 5 5 5 5 8 8 5 8 5 5 8 5 5 ...... 00 —We report the median and 1 0 ( from the spectral fitting process are shown in parentheses. Section 3.1 ). OTE The small shifts to be added to our fitted log N a 1532 0 1354 0 1348 0 0745 0 1028 0 3338 0 2734 0 3118 0 1844 0 2340 0 2553 0 3838 0 2835 0 3537 0 3629 0 2659 0 0911 0 0440 0 5815 0 4444 0 3500 0 1027 0 2308 0 + 47 0 + + + − − − − + + + + − + + − + + + + − + + + WISE 2226 WISE 2255 WISE 2319 ULAS 2321 WISE 2340 WISE 2348 WISE 1741 WISE 1809 WISE 1813 WISE 1852 WISE 1959 WISE 2000 WISE 2157 WISE 2209 WISE 2213 ULAS 1416 WISE 1457 GJ 570D 0 PSO J224 SDSS 1504 2MASS 1553 SDSS 1628 WISE 1653 WISE 1711 ObjectWISE 1322 Slit 46 68 29 19 22 69 18 30 18 22 20 42 21 37 27 39 11 36 10 17 28 19 26 18 24 ± 26 8 19 31 11 13 12 7 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 4 9 3 32 3 ± ± ± ± ± ± ± ± K − − − − − q 7 9 1 11 37 7 4 4 6 24 9 40 8 19 7 37 10 27 9 17 9 14 12 7 5 4 11 37 12 5 17 13 18 4 13 9 2 9 1 7 16 4 3 8 19 4 9 3 6 7 24 9 15 8 27 8 11 7 30 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± H q quantities ) 6 32 4 21 5 25 6 27 6 33 3 16 4 19 3 12 2 16 3 6 3 23 4 25 3 13 3 23 4 13 4 27 4 16 4 33 2 17 5 20 4 26 2 11 2 11 3 10 2 13 3 21 2 11 1 10 3 27 4 23 3 11 4 17 ± ± ± ± ± λλ ( ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± q 11 6 7 2 4 7 9 5 6 10 9 7 10 9 6 10 9 6 7 17 6 7 4 4 5 7 4 2 8 7 8 9 J − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − q 6 6 5 2 7 4 7 4 2 4 4 5 4 5 3 4 6 4 5 3 6 6 2 3 3 2 4 3 4 4 4 4 ± ± ± ± ± 3 3 1 1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Y 3 0 − 0 0 6 7 6 5 6 − 7 7 7 8 10 2 4 7 2 − 9 3 6 4 1 3 − 6 3 6 3 q 4 4 4 4 2 3 3 4 3 4 3 4 5 3 2 3 5 3 5 3 2 4 2 3 2 3 2 1 2 3 3 3 4 5 4 5 2 3 3 4 4 5 3 4 3 6 4 2 2 2 4 5 4 6 4 4 2 3 2 2 3 4 3 4 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + + − − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 2 2 1 5 5 3 5 1 0 7 0 9 7 4 9 2 7 8 8 6 3 1 1 0 1 5 3 9 6 8 6 6 ...... J 6 4 3 2 2 2 3 3 2 3 3 3 2 3 2 3 3 3 2 3 3 2 1 2 2 2 2 1 2 2 2 2 773 2492 461 444 318 85 343 592 2137 1576 749 549 696 2701 + − 30 56 22 23 50 16 7 10 45 32 12 49 62 18 17 63 28 25 7 34 25 135 314 97 67 242 54 26 37 202 137 40 207 311 46 47 318 168 105 23 195 76 + − + − + − + − + − + − 2 1 3 4 7 5 8 13 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + + − − + − + − + − + − + − + − 42 77 709 31 36 74 519 24 12 16 1288 19 402 738 29 68 44 5 106 470 73 92 4 26 91 41 12 40 8 50 38 Age )

064 036 041 060 035 058 049 042 038 046 063 069 033 054 042 046 052 035 040 034 049 029 043 054 051 058 039 038 039 043 029 066 032 043 064 036 063 062 044 040 050 066 045 075 060 031 036 058 047 058 037 042 034 045 057 054 060 041 039 039 045 026 ...... L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − bol L 424 605 444 327 326 227 800 508 682 157 768 422 602 242 755 381 779 002 265 070 139 856 350 287 718 191 918 397 895 347 569 ( ...... Table 5 continued 5 5 5 5 5 5 5 5 5 5 5 5 6 6 5 5 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − log − 97 47 09 26 . . . . 05 44 51 40 . . . . 81 20 33 44 15 28 94 43 61 02 65 14 33 19 29 29 19 07 23 04 29 55 72 27 13 19 86 14 72 32 75 59 67 07 91 58 96 64 30 81 65 58 30 91 06 58 44 43 67 57 04 42 95 27 8 12 9 21 28 31 ...... 1 1 1 1 3 6 0 0 1 0 5 1 1 0 0 1 1 2 1 0 1 1 1 0 1 0 1 1 4 2 2 2 6 21 2 0 3 0 4 12 1 1 1 0 4 2 2 0 2 3 3 0 2 0 3 2 − + + − + − ) (dex) (Myr) (%) (%) (%) (%) (%) + − + − + − + − + − − + + − + − + − + − + − + − + − + − + − + − + − + + − + − − + − + − + − + − + − + − + − + − 37 97 18 Jup . . . 26 30 58 09 27 41 85 91 20 10 16 30 04 76 61 75 41 87 40 09 75 95 32 59 45 51 67 33 ...... M 3 2 13 2 3 6 8 1 0 3 21 0 8 2 2 1 1 1 2 4 0 13 2 2 2 3 0 2 0 3 2 11 06 06 08 06 10 15 07 06 11 16 03 11 13 06 05 06 09 08 04 07 08 05 08 07 07 05 06 05 08 06 14 06 07 09 07 11 19 08 06 12 06 16 08 04 09 14 13 04 07 17 06 08 05 09 09 08 06 06 06 08 07 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )( + − + − + − + − + − + − + − + − + − + − + − + − + + − + − − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − Jup 93 68 70 81 90 02 77 83 74 20 67 97 59 43 78 80 57 45 85 83 96 99 60 90 72 79 62 73 65 88 81 ...... R 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 – – – 6 0 0 0 0 0 0 0 RM0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 )( 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 8 8 5 8 5 8 5 8 8 5 5 5 5 ...... 00 0 0 ( . Starfish-Based Forward-Modeling Analysis: Derived properties of T7–T9 Dwarfs a 0311 0 2618 0 2448 0 3954 0 0935 0 3322 0 1710 0 0935 0 0540 0 3640 0 2340 0 0728 0 0421 0 1942 0 1600 0 0456 0 1223 0 3450 3653 0 2932 0 4008 0 6434 0 0831 0 0951 0 1025 0 4142 0 2151 0 0900 0 Table 5 − − − − + − − 02 0 + − + − + − − − − − + + + − + − − − + + + + WISE 1322 Ross 458C 0 WISE 1257 WISE 1254 WISE 1124 2MASS 1217 2MASS 1114 WISE 1052 ULAS 1029 WISE 1039 2MASS 0939 UGPS 0722 WISE 0614 WISE 0623 2MASS 0727 2MASS 0729 2MASS 0415 WISE 0458 WISE 0325 WISE 0500 UGPS 0521 WISE 0521 PSO J043 WISE 0245 WISE 0241 WISE 0223 WISE 0123 2MASS 0050 WISE 0049 WISE 0040 ObjectHD 3651B Slit 0 47 , as G a 47 12 and 44 19 18 23 20 16 25 15 21 23 13 19 27 26 16 14 21 N ± ± 17 19 20 11 14 95 a ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 10 7 11 ± ± ± ± ± ± K − − − q , and the cloudless Z , 6 18 7 12 9 7 7 10 24 8 19 6 17 6 23 5 10 9 11 9 28 6 22 8 28 11 19 6 8 12 6 0 10 18 5 6 6 16 6 5 10 9 1 7 12 4 5 g . Bolometric luminosity g ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± H , log q quantities eff T ) 4 20 6 28 3 32 3 13 3 15 4 19 3 21 4 24 3 19 3 12 2 15 2 20 4 20 2 14 4 20 5 22 2 15 5 32 2 17 3 12 3 14 2 13 4 22 3 14 2 16 ± ± ± λλ ( ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± q 6 5 6 8 8 6 6 6 10 6 8 14 2 11 8 8 5 8 5 6 6 5 8 6 4 and fitted log J − − − − − − − − − − − − − − − − − − − − − − − − − q , is defined in Equation 2 and computed R ) 4 λλ ( 4 4 4 5 5 4 3 5 3 4 7 3 3 4 3 7 6 3 3 3 3 5 4 3 ± q 1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Y 1 1 3 6 1 − 3 6 6 3 4 5 1 3 5 3 6 2 2 3 6 5 7 4 2 q bands, 6 7 . . ) is derived from 3 3 3 4 4 3 2 2 2 3 3 4 3 2 3 2 4 2 2 3 2 4 3 2 3 4 4 4 5 3 3 3 2 4 4 5 4 2 4 3 5 2 2 3 3 4 4 2 0 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) is derived from the fitted hyper-parameters 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M + − J + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 3 + − + − + − + − + − + + − + − − . YJHK 2 7 3 8 1 6 1 6 1 2 8 1 2 6 1 2 6 2 0 4 4 0 3 3 ...... J 2 2 3 2 3 2 2 2 3 2 2 4 2 2 3 2 10 3 2 2 2 2 2 3 2 . Mass ( Ω 2424 3342 218 77 322 345 79 79 503 311 699 944 336 281 + − 35 10 13 49 8 12 24 21 32 7 6 16 56 56 35 10 20 219 35 39 228 26 34 73 61 91 25 19 122 324 233 158 32 52 + − + − + − + − + − + − 5 18 + − + − + + − − + − + − + − + − + − + − + − + − + − + − + − + + − − + − 52 305 110 401 15 74 20 433 13 18 36 108 33 49 11 10 22 76 84 56 31 8 16 4544 110 Age ) (continued)

056 060 040 051 061 058 093 061 036 042 040 039 048 039 049 064 070 039 028 030 059 039 031 049 105 059 064 040 052 062 061 106 065 038 043 043 041 049 043 051 068 076 042 031 032 063 040 028 052 122 ...... L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / + − + − + − + − + − + + − − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − m SpeX spectra and fitted model spectra to shorter and longer wavelengths spanning bol L µ 477 971 710 012 729 312 434 496 558 609 716 609 343 137 491 395 439 837 453 072 378 544 161 013 229 ( ...... 5 5 4 5 6 5 5 5 5 5 5 5 5 5 6 5 5 5 5 5 5 5 5 5 6 5 . Table 5 2 − − − − − − − − − − − − − − − − − − − − − − − − − log − 0 25 25 26 62 69 ...... 09 37 . . 35 46 95 47 35 03 60 57 75 03 31 45 57 73 29 02 68 42 92 47 29 20 72 79 96 04 87 06 17 07 46 24 44 85 28 51 07 16 11 28 71 00 41 11 8 13 24 18 19 ...... 1 1 1 0 2 1 0 0 0 2 1 0 0 0 1 1 1 3 0 0 4 1 2 2 4 1 4 2 1 1 1 4 2 0 1 1 5 2 3 6 1 1 10 2 − + + − + − ) (dex) (Myr) (%) (%) (%) (%) (%) + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 96 19 63 Jup . . . 70 89 25 95 76 05 26 20 54 78 76 94 19 58 17 84 57 56 03 02 85 47 ...... M 2 21 2 3 0 4 2 12 1 1 1 3 2 0 1 1 2 1 3 7 2 1 7 2 29 07 13 06 06 07 08 11 13 06 05 06 07 07 04 08 10 13 05 05 09 06 07 14 07 08 08 14 06 08 08 10 14 16 07 06 06 08 08 05 09 12 18 06 06 10 06 08 17 08 09 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )( + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − Jup 74 26 67 53 71 87 91 91 82 69 68 77 86 83 54 98 93 60 74 21 79 78 02 97 67 ...... R 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 RM0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 )( 5 5 8 5 5 5 5 8 5 8 5 8 5 5 8 8 5 5 8 5 5 5 8 8 5 ...... 00 0 ( 1532 0 -band peak fluxes (Section 6 ). The measure of spectral-fitting residuals in 1354 0 1348 0 1028 0 0745 0 1844 0 3118 0 2734 0 3338 0 J 0440 0 0911 0 36292659 0 0 28353537 0 0 2553 0 3838 0 3500 0 4444 0 5815 0 2308 0 1027 0 + 47 0 + + − − − − + + − + + − + + + + + + + + + is derived from integrating the objects’ 1 + ) is derived from the objects’ parallaxes and spectroscopically inferred log 3450 lacks a parallax and thus has no radii, mass, or bolometric luminosity results. ) R

− m, with spectra scaled by measured parallaxes (Section 3.2 ). Age is derived from the spectroscopically inferred L µ / bol 50 WISE 2348 WISE 2340 ULAS 2321 WISE 2319 WISE 2255 WISE 2226 WISE 2213 WISE 2209 WISE 2000 WISE 2157 WISE 1959 WISE 1813 WISE 1852 WISE 1741 WISE 1809 WISE 1711 WISE 1653 SDSS 1628 2MASS 1553 GJ 570D 0 SDSS 1504 WISE 1457 PSO J224 ObjectULAS 1416 Slit L ( —Radii ( − 4 . using the objects’ observed spectra and their fitted model spectra (Section 6.2 ). (log 0 Sonora-Bobcat evolutionary models (Section 5.2 ).well as The the measure objects’ of the data-model discrepancy ( OTE WISE 0245 N a 48 ) 9 3 3 2 Jup . . . . ) (M 08 1 04 0 06 1 03 0 Jup RM . . . . Ω 073 0 024 0 049 0 006 0 . . . . log ) (dex) (R 1 i − sin v ) (km s 1 − r 154 220011 183 29 0 20 0 102 127011 29 21 0 20 0 . . . . g Z v 29 0 02 0 22 0 02 0 . . . . log . Typical Parameter Uncertainties for Our Forward-Modeling Anal- —Formal uncertainties are obtained directly from the spectral fitting process and the eff T (K) (dex) (dex) (km s adopted uncertainties arependix C ). We those only show with the formal systematic uncertainties if errors they are incorporated different from (Section the adopted3.1 ones. and Ap- OTE N Table 6 ysis adopted 28adopted 0 3 0 Method Starfish formalTraditional 20formal 0 3 0 49 ) 03 08 07 05 05 08 05 08 05 06 04

...... L 0 0 0 0 0 0 0 0 0 0 0 / ± ± ± ± ± ± ± ± ± ± ± bol L 27 70 30 60 71 77 17 06 48 00 50 ...... ( 5 5 5 5 5 5 5 5 6 5 5 − − − − − − − − − − − 05 06 04 03 08 07 05 05 08 05 08 ...... 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± 49 00 51 27 70 30 60 71 77 16 08 ...... 5 5 5 5 5 5 6 5 5 5 5 Line et al. ( 2017 ) log − − − − − − − − − − − )

L 01 02 1401 – – 0201 – – 06 02 01 02 0201 – – 01 03 – – / ...... a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bol L ± ± ± ± ± ± ± ± ± ± ± ± ± ± ( 55 39 77 74 02 37 57 72 76 30 55 77 54 09 ...... 5 5 5 5 5 5 5 5 5 5 6 5 5 5 − − − − − − − − − − − − − − 01 03 01 02 14 01 02 01 06 02 01 02 02 01 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± 76 29 54 79 55 09 56 39 88 74 02 37 57 72 ...... 5 5 5 5 6 5 5 5 5 5 5 5 5 5 − − Filippazzo et al. ( 2015 ) log − − − − − − − − − − − − 00 7 70 5 4090 8 5 30 6 60 4 40 5 30 5 94 3 . 50 5 32 3 40 5 20 1 53 6 90 5 – – ...... ’s to the more precise parallaxes adopted in this work without modifying the uncertainties 1 2 0 8 4 1 1 0 2 1 1 0 0 2 26 bol ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± L 20 80 50 30 30 20 70 22 80 54 10 10 42 60 00 ...... Parallax Par. Ref. original modiﬁed original modiﬁed ...... 90 94 90 85 46 75 179 109 171 175 242 112 126 187 166 ) 029 038 034 033 042 029 069 063 049 035 036 049 031 039 030 066 062 036 032 052 028 040 032 026 039 034 036 045 031 075

...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L / + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − bol L 768 800 326 605 013 544 161 072 569 397 856 779 422 381 602 ...... ( 5 5 5 5 6 5 5 5 5 5 5 6 5 5 5 − − − − − − − − − − − − − − − 30 09 50 70 40 90 30 60 40 06 40 20 15 50 90 ...... This Work Literature Values ...... 1 1 2 0 8 4 1 0 0 1 0 0 2 2 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 30 20 56 01 10 20 80 50 30 (mas) (dex) (mas) (dex) (dex) (dex) (dex) 79 60 70 86 10 10 ...... Bolometric Luminosities from This Work, Filippazzo et al. ( 2015 ), and Line et al. ( 2017 ) Table 7 3 (sd) T7.5 107 2 T9 242 4 T7 46 . . . 3322402 T7 94 3954043 T8 pec 126 2448279 T8 187 2618235 T7.5 179 031113 T7.5 91 093506 T8 175 171001 T7 112 153236 T7 75 134836 3995 T8 146 054031 102718 . − − − − − + − + + − + —(1) Tinney et al. ( 2003 ), (2) van Leeuwen ( 2007 ), (3) van Leeuwen ( 2007 ), (4) Burgasser et al. ( 2008b ), (5) Dupuy &( 2012 ), Liu (6) Faherty et al. ( 2012 ), (7) Kirkpatrick 02 94 51 + 63 . . . 5395 . provided by the literature. et al. ( 2012 ), (8) Leggett et al. ( 2012 ) We compute the modiﬁed bolometric luminosities by scaling the original literature a References PSO J043 2MASSI J0415195 UGPS J072227 2MASSI J0727182 2MASS J07290002 2MASS J09393548 2MASS J11145133 2MASSI J1217110 Ross 458CULAS J141623 T8 86 GJ 570D T7.5 170 SDSS J150411 2MASSI J1553022 ObjectHD 3651B SpT T7.5 Parallax 89 log 2MASS J00501994 50 rms add σ column gives the MKO K add σ BC for the uncertainty on the correction. rms mean error add σ T9 Dwarfs add σ − MKO H BC rms mean error 1 for the weighted average, as an estimate of the intrinsic (astrophysical) variations. When ≈ 2 χ add . Bolometric Corrections for T7 σ MKO J BC Table 8 mean error (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) obj —For each spectral type bin and photometric band, we tabulate the weighted average, weighted error, and weighted root mean square (rms) of the bolometric corrections, by excluding resolved or candidate binaries and two objects with peculiar spectra (Appendix B ). The additional uncertainty needed to makeusing the these reduced bolometric corrections, we recommend adopting the larger of the weighted error and OTE Spectral Type N [T7, T7.5)[T7.5, T8) 18[T8, T8.5) 10[T8.5, T9) 14[T9, 2.67 T9.5) 2.58 3 0.02 2.57 2 0.03 0.09 0.03 2.59 0.07 0.10 2.29 0.15 0.07 0.10 0.07 0.16 2.32 0.23 2.21 0.02 0.02 0.18 2.20 0.03 0.05 0.00 0.03 2.16 0.00 0.07 2.04 0.00 0.05 0.06 0.08 0.08 2.35 0.20 2.17 0.11 0.02 0.15 2.21 0.03 0.07 0.17 0.03 2.17 0.32 0.17 1.69 0.29 0.09 0.26 0.11 0.29 0.20 0.37 0.18 0.17 N 51 65 43 78 18 4 6 132 32 2 8 + − 4 58 0 10 1 4 0 1 12 2 9 0 5 17 1 17 6 1 2 0 17 0 5 0 0 4 4 105 0 5 4 1 1 13 27 2 7 1 4 0 7 25 5 1 4 0 1 1 + − + − + − + − 0 3 1 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 20 22 16 59 6551 22 72 37 93 428 9 62 434 32 76 64 9 57 79 16 23 19 73 27 222 17 133 23 Age 20 47 b 08 . 38 . 94 51 . . 95 44 165 58 . . 32 10 20 81 22 12 05 06 10 25 51 38 01 13 27 31 11 20 11 25 13 37 04 07 15 28 54 17 10 42 24 46 65 12 66 15 09 15 28 66 62 01 14 42 37 35 36 14 24 10 04 09 16 20 6 76 5 32 ...... + − 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − ) (Myr) + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 86 + − + − + − + − + − + − + − + − . 15 00 Jup . . 62 64 95 49 56 90 43 67 96 69 90 80 92 16 40 20 25 71 81 93 64 27 56 71 86 47 90 ...... M 1 1 3 1 0 2 2 4 8 49 2 3 1 2 165 78 1 2 2 1 0 3 0 0 2 2 1 2 2 2 Derived Parameters 03 01 06 02 05 03 01 05 04 02 01 05 04 03 05 01 01 04 05 04 05 01 04 01 58 29 01 01 02 03 02 04 02 06 07 03 02 06 03 05 06 01 04 17 14 01 04 04 01 01 04 01 01 06 02 04 04 06 01 05 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )( + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − Jup 73 15 59 69 93 85 92 80 67 62 80 89 46 83 59 72 38 80 72 62 61 79 80 65 88 56 85 60 18 79 ...... R 0 0 0 0 0 1 0 0 0 0 0 0 0 0 – – 21 0 0 0 0 0 1 0 0 0 0 0 RM 0 0 0 0 0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 017 009 006 002 013 005 008 003 004 003 012 003 002 025 004 004 002 395 008 593 007 002 030 003 004 007 002 001 005 006 002 004 004 017 010 006 003 013 005 012 002 002 033 011 003 002 006 002 021 004 004 002 097 004 088 006 002 005 007 003 002 006 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 026 018 025 021 021 010 025 013 046 020 041 012 033 012 386 024 587 018 024 041 009 040 009 011 024 009 018 045 028 045 009 013 021 026 021 031 021 021 027 014 039 051 041 009 012 028 045 024 011 036 012 136 019 109 024 040 011 010 024 010 049 021 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + − − + − + + − − + + − − + − + + − − + − + − + + + − − + − − + + − − + − + − + − + − + + − − + − + − + + − − + − + − + − 483 234 104 805 887 836 218 820 923 799 850 358 333 362 941 117 438 227 072 624 604 047 777 920 600 402 401 281 762 572 833 ...... Ω 19 19 19 19 19 19 20 20 19 18 19 19 19 19 19 19 19 19 19 19 20 19 19 20 19 19 19 19 19 19 19 − − − − − − − − − − − − − − − − − − − − − − − − − − − log − − − − ) (dex) ( 1 100 − 103 i 39 21 17 18 19 16 15 21 25 57 22 22 20 13 15 33 28 43 20 19 7 24 13 12 19 14 21 18 20 20 21 26 40 17 17 18 19 20 10 21 92 18 22 7 23 19 24 13 27 17 201 28 243 25 13 12 17 18 19 17 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − sin 25 27 28 25 30 36 58 30 19 38 21 29 82 28 32 10 32 35 19 28 31 29 146 29 32 24 46 55 18 v 23 28 ) ) ) ) ) ) a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 24 85 48 29 23 23 65 15 121 32 15 32 25 64 220 23 123 49 34 19 18 29 24 29 28 23 64 22 85 48 24 20 31 15 116 32 15 18 19 29 35 25 64 282 514 26 20 23 ) ) ) ) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 27 20 25 8 18 26 29 27 8 15 ) ) ( ( ( ( ( ( ( ( ( ( 35 9 31 9 ( ( ) (km s ( ( 189 180 190 199 198 185 183 303 182 187 180 220 183 179 180 184 181 187 181 190 185 309 202 197 185 182 205 180 190 185 199 182 314 437 182 198 184 180 216 180 180 193 181 188 200 188 181 190 1 181 182 181 179 180 181 182 179 180 182 + − + − + − + − + − + − + − + − + − + − + − + − + − + + − + − − + − + − + − + − + − + − + − + − − 182 179 182 179 + − + − + − + − + − + − + − 292 545 77 219 587 168 303 381 212 1 647 80 155 320 194 5 97 476 204 129 66 211 64 769 124 498 615 345 157 291 252 r Table 9 continued − − − − − − − − − − − − Fitted Parameters 01 01 01 00 06 02 01 04 01 02 01 00 04 01 01 00 01 01 01 04 00 35 01 01 37 04 01 01 02 01 02 01 01 03 01 02 01 00 00 04 02 06 01 01 01 06 01 01 00 01 01 01 02 02 01 03 00 01 01 03 01 02 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 45 15 24 26 08 02 50 32 20 04 15 14 24 06 31 04 45 11 35 02 13 02 28 37 17 03 02 14 11 23 14 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − − − − − − − − − − − − − − − − − − − − − − − − − − 25 06 01 02 00 14 00 02 01 00 01 04 01 17 01 47 01 00 74 11 00 01 09 01 02 02 03 02 00 12 00 01 02 08 07 02 02 01 05 02 00 01 12 02 04 01 02 00 04 00 04 02 00 03 01 01 01 01 02 04 00 01 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − g Z v 00 27 67 09 06 83 84 88 70 16 10 43 15 12 94 78 26 46 34 82 46 61 50 01 93 23 67 75 12 87 00 ...... 4 3 3 3 3 4 5 4 4 4 5 3 4 4 4 3 4 4 3 4 3 3 4 3 3 4 4 3 5 log 3 4 1 1 2 2 9 6 3 2 2 6 5 1 0 13 1 5 1 3 1 8 1 24 2 2 17 2 2 1 2 2 1 3 2 1 1 6 3 2 9 5 3 0 110 7 3 1 13 1 5 1 1 2 2 2 1 10 1 1 2 147 1 2 + − . Traditional Forward-Modeling Analysis: Fitted and Derived Properties of T7–T9 Dwarfs + − + − + − + − + − + − + − + + − − + − + − + − + − + + + − − − + − + − + − + − + − + − + − + − + − + − + − + − + − + − eff 938 946 836 809 883 921 860 818 908 667 709 837 911 813 824 815 932 584 640 873 777 966 827 1025 950 760 768 820 926 T 824 914 Table 9 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ) (K) (dex) (dex) (km s 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 8 8 8 5 8 5 8 5 5 5 5 8 ...... 00 0 ( 2618 0 0311 0 2448 0 3954 0 0935 0 3322 0 1710 0 0935 0 0540 0 3640 0 0728 0 1942 0 1600 0 0421 0 0456 0 3653 0 3450 0 1223 0 2932 0 4008 0 1025 0 0951 0 4142 0 0831 0 6434 0 2151 0 0900 0 − − − − + − − 02 0 + + − − − − − + + − + + − + + + − − − + + Ross 458C 0 2MASS 1114 WISE 1052 WISE 1039 WISE 1124 ULAS 1029 2MASS 1217 WISE 1254 WISE 1257 2MASS 0939 2MASS 0729 2MASS 0415 WISE 0458 WISE 0500 WISE 0521 UGPS 0521 WISE 0614 WISE 0623 UGPS 0722 2MASS 0727 WISE 0123 2MASS 0050 WISE 0049 WISE 0040 WISE 0223 WISE 0241 WISE 0245 PSO J043 WISE 0325 ObjectHD 3651B Slit 0 52 23 12 5 21 23 10 9 20 4 7 20 2 4 3 4 1 5 0 1 2 5 6 1 0 4 4 12 7 10 6 0 28 2 2 5 4 2 4 7 1 7 18 1 65 23 5 21 20 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 280 74 64 87 54 39 47 14 23 57 – 19 140 50 23 10 52 82 45 108 62 118 18 34 Age – 49 b 26 52 . . 36 36 . . 76 13 17 4 22 4 . . 17 40 25 07 19 13 73 73 36 17 14 31 16 04 24 52 66 26 19 46 19 10 78 59 41 21 15 07 32 31 07 67 15 36 19 76 38 16 24 04 82 05 12 21 14 11 1 2 ...... + − + − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 3 0 0 1 2 0 + − ) (Myr) + − + − + − + − + − 68 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 62 + − + − + − . . directly from the 97 Jup . 96 68 26 55 36 27 58 37 00 47 67 85 30 92 05 95 26 10 17 23 62 61 39 ...... M Ω 13 2 2 1 3 1 3 1 1 1 1 2 141 5 0 2 3 6 1 131 3 3 3 4 2 3 Derived Parameters 07 05 04 02 02 02 02 03 07 05 10 04 04 02 03 04 01 02 01 09 09 05 02 02 08 11 08 06 02 03 19 04 04 03 02 03 04 04 08 07 01 12 05 02 07 01 02 02 02 09 15 12 and log ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r )( v + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − Jup 16 72 66 49 68 82 64 72 75 79 68 06 82 88 98 14 79 79 73 95 54 49 00 96 02 92 12 Gyr are shown as “–”...... R 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 RM 0 0 1 0 0 > erros of σ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 003 005 005 024 012 008 005 006 003 007 006 004 011 004 002 013 025 006 005 021 004 005 006 009 011 015 003 006 006 004 023 006 038 008 005 008 003 007 006 012 005 011 003 015 027 007 016 008 004 006 013 016 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 012 026 013 031 014 026 031 024 021 025 006 027 018 032 016 060 022 015 029 020 013 013 026 023 059 063 012 031 015 033 026 040 014 031 045 024 023 006 025 027 023 016 073 076 027 016 033 013 020 029 013 064 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + − − + − + − + − + − + − + + − − + + − − + − + − + − + − + − + + − − + − + + − + + − − − + + − − + − 783 895 787 038 017 506 911 016 926 804 793 283 826 033 040 234 062 097 966 406 090 794 010 190 978 567 ...... Ω 19 19 19 20 19 19 19 19 20 19 19 20 19 19 19 20 19 20 19 20 19 20 20 19 19 19 − − − − − − − − − − − − − − − − − − − − − log − − − − − ) (dex) ( 1 69 123 − 119 130 i 24 37 21 19 13 26 19 19 23 16 17 23 14 14 18 28 19 16 25 24 25 21 22 16 19 39 16 19 16 14 13 29 21 16 23 18 14 21 16 16 21 30 19 11 26 23 21 22 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − sin 36 54 28 34 23 31 28 34 104 20 40 30 41 29 26 37 20 32 21 26 26 24 35 v 36 176 33 incorporated (Section 3.1 ). The formal 1 Ω a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 18 17 31 72 20 24 37 15 25 63 44 10 56 72 28 35 59 22 46 38 53 18 29 66 15 17 29 22 37 30 29 99 26 61 21 44 10 46 88 52 38 49 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 26 33 24 32 25 31 32 25 29 25 and log ( ( ( ( ( ( ( ( ( ( (continued) ) (km s r 180 198 211 180 180 190 191 192 182 181 240 198 180 180 184 179 233 202 188 183 187 180 186 195 183 180 180 190 206 185 179 195 181 193 219 181 180 194 184 183 186 223 1 v 181 183 181 180 181 181 180 181 183 181 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − − + − + − + − + − + − 32 99 209 317 462 676 228 359 865 44 36 129 157 66 272 339 193 267 573 358 874 360 118 276 598 104 3450 lacks a parallax and thus has no radii or mass results. Ages r − − − − − − − − − − − − − Table 9 Fitted Parameters 01 01 01 00 01 01 01 01 01 01 02 01 00 01 01 04 03 00 03 03 01 02 01 02 00 02 01 01 01 02 01 00 00 01 01 03 01 01 01 01 05 01 01 05 01 04 02 00 02 04 00 02 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 03 19 19 49 09 00 17 14 19 10 00 10 14 04 20 33 11 18 27 09 00 30 37 30 39 40 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − − − − − − − − − − − − − − − − − − − − 03 02 01 06 00 01 03 03 00 02 00 03 01 11 00 05 09 03 01 01 01 04 01 00 05 07 04 02 02 00 01 07 00 01 03 04 01 02 03 01 10 03 00 12 07 17 01 32 12 02 00 09 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − g Z v 15 41 11 09 16 50 32 73 04 00 11 83 95 71 24 00 90 74 12 92 26 67 07 15 50 02 ...... 4 4 4 4 4 4 3 3 4 3 5 4 3 4 4 3 4 3 4 4 4 3 3 4 log 5 4 3 1 2 3 2 8 11 3 3 4 3 1 2 3 5 5 1 9 5 8 4 3 7 1 1 6 2 1 2 8 3 2 2 3 2 5 4 3 3 2 2 3 2 4 4 1 5 1 9 5 1 6 + − + − + − + − + − + − + − + − + − + − + − + − + + − − + − − + + − + − + − + − + + − + − − + + − − + − eff 907 953 829 805 763 934 807 848 832 923 843 838 867 856 748 807 812 909 934 914 921 612 908 825 T 826 834 errors of each parameter with the systematic errors in σ 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ) (K) (dex) (dex) (km s 5 5 8 5 8 5 5 5 5 5 5 8 5 8 8 5 8 5 8 8 8 5 5 5 5 5 ...... 00 0 ( 1532 0 1354 0 1348 0 1028 0 0745 0 1844 0 2734 0 3118 0 3338 0 2340 0 3537 0 3629 0 2659 0 0911 0 0440 0 2835 0 3838 0 4444 0 2553 0 3500 0 5815 0 1027 0 2308 0 + 47 0 + + − − − − − + + + + + − + + + + + + − + + + spectral ﬁtting process are shown in parentheses. We report the median and 1 Parameters are derived in the same approach as those in5 . Table WISE 0245 a b WISE 2348 WISE 2340 ULAS 2321 WISE 2319 WISE 2255 WISE 1959 WISE 1852 WISE 2000 WISE 1813 WISE 2157 WISE 1809 WISE 2209 WISE 2213 WISE 2226 WISE 1741 2MASS 1553 SDSS 1504 SDSS 1628 PSO J224 ULAS 1416 WISE 1457 GJ 570D 0 WISE 1653 WISE 1711 ObjectWISE 1322 Slit