The Brown Dwarf Kinematics Project (BDKP). VI. Ultracool Dwarf Radial and Rotational Velocities from SDSS/APOGEE High-resolution Spectroscopy

Our UCD sample was curated from APOGEE DR17 (Abdurro'uf et al. 2022) of SDSS-IV (Blanton et al. 2017) based on observations obtained with the 2.5 m Sloan Foundation Telescope at the Apache Point Observatory (Gunn et al. 2006) and the 2.5 m du Pont Telescope at the Las Campanas Observatory (Bowen & Vaughan 1973). The majority of our targets were proposed in two SDSS Ancillary Science Programs, SDSS-III Project 176 (PIs: Suvrath Mahadevan and Cullen Blake; "A Radial Velocity Survey of Bright M Dwarfs with APOGEE: Companions, $v\sin i$, Fe/H") and SDSS-IV Project 288 (PI: Adam Burgasser; "APOGEE-2 and eBOSS Observations of the Lowest-Mass Stars and Brown Dwarfs in the Solar Neighborhood"). ¹⁹ These programs are well aligned with one of the main stellar science goals of the APOGEE survey, which is the identification of close binary systems and substellar companions through RV variables (Blanton et al. 2017). We drew all of our sources from the DR17 allStar catalog, ²⁰ and constructed two UCD samples for our analysis dubbed the "gold" and "full" samples. The analysis in this paper focuses on the gold sample. We present the construction and measurements of the full sample in the Appendix A.

Our "gold" sample was based on sources with classifications of M6 and later as reported in SIMBAD (Wenger et al. 2000), the Late-Type Extension to MoVeRS (Theissen et al. 2016, 2017), Reylé (2018), Best et al. (2021), or reported in this work. We use the most recent spectroscopic classification as our adopted value. Two sources were rejected from the sample based classifications earlier than M6 from additional follow-up spectroscopy discussed below. We validated the remaining sources by visually inspecting optical (SDSS) and near-infrared images (2MASS), and confirmed correct placement on color–magnitude and color–color diagrams combining 2MASS and Gaia EDR3 (Gaia Collaboration et al. 2021a) photometry and astrometry (Figure 1). We further imposed a limit on the median spectral S/N of the APOGEE data to be >10. After removal of spectral type earlier than M6 using our optical spectra (Section 2.2) and color-type relation from Kiman et al. (2019), these criteria resulted in a sample of 931 spectra of 258 sources summarized in Table 1, which is the gold sample.

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Figure 2 displays the observable properties of our gold and full samples. The distribution of targets across the sky is highly dependent on APOGEE's survey pointings. Notably, half of the gold sample includes known members of nearby young moving groups (the Upper Scorpius and Taurus young associations), targeted as part of the programs described in Zasowski et al. (2013, 2017), Beaton et al. (2021), and Santana et al. (2021). The spectral types for our gold sample range from M6 to L2, with 252 late-M dwarfs and six L dwarfs. For the full sample that is not in the gold sample, the Gaia G − G_RP–spectral type relation of Kiman et al. (2019) indicates spectral types of M4 to L4, with 184 late-M dwarfs and 17 L dwarfs. The majority of our gold sample (80%) and full sample (90%) have distances larger than 30 pc, with the M6.5Ve G 51−15 (2MASS J08294949+2646348) being the closest source at a Gaia distance of 3.5810 ± 0.0008 pc. The apparent H-band magnitudes of our targets extend to 14.65, which is slightly fainter than the APOGEE magnitude limit of 13.8 for S/N = 100.

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The majority of both samples (66% and 51%) have APOGEE spectral observations taken over multiple epochs, enabling more precise determinations of RVs and the possibility of measuring RV variations. We defined multiepoch subsamples in our gold and full samples as those with at least four observations satisfying S/N > 10, each separated by at least 0.85 day. A total of 71 sources in the gold sample and 115 sources in the full sample satisfy these criteria, and the RV variables among the subsample are discussed in Section 5.5.

Finally, we note that the requirement for a Gaia detection likely resulted in the rejection of true UCDs in the full sample due to sensitivity limitations (Theissen 2018), and APOGEE targeting was not intended to be uniform across the sky (Zasowski et al. 2013). Therefore, our APOGEE UCD sample is not expected to be magnitude or volume complete, but rather representative of the broader UCD population.

The majority of sources in our full sample lack published spectroscopic classifications. ²¹ To bolster this sample, we obtained additional low-resolution optical spectra with the Kast Double Spectrograph (Miller & Stone 1994) on the Shane 3 m Telescope at the Lick Observatory over multiple nights between 2018 January 21 and 2022 March 11. We used the 600/7500 grating and 2'' slit to obtain 6000–9000 Å spectra at an average resolution of λ/Δλ ≈ 1800. Data acquisition included observations of flat-field and arc lamps for pixel response and wavelength calibration, nightly observations of a spectral flux standard from Hamuy et al. (1992, 1994) for relative flux calibration, and observations of a nearby G2 V or A0 V star at similar air mass for telluric absorption and continuum correction. All data were reduced using the kastredux package ²² using default settings. An example spectrum of the M9 dwarf 2MASS J21272531+5553150 (aka LSPM J2127+5553) is shown in Figure 3. Table 2 summarizes the observations and corresponding measurements. Spectral classifications were determined by the closest match to SDSS dwarf spectral templates from Bochanski et al. (2007b), Schmidt et al. (2014), and Kesseli et al. (2017), and indicate types ranging from M5 to M9. Metallicity index (ζ) measurements (Lépine et al. 2007) are uniformly greater than 0.9 and indicate near-solar metallicities for all sources. We detected Hα emission in all of the sources, and measured relative Hα emission luminosity (${\mathrm{log}}_{10}{L}_{{\rm{H}}\alpha }/{L}_{\mathrm{bol}}$) from equivalent widths and the χ factor relations of Douglas et al. (2014) and Schmidt et al. (2014); these values range over −4.9 < ${\mathrm{log}}_{10}{L}_{{\rm{H}}\alpha }/{L}_{\mathrm{bol}}$ < −3.8.

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Table 2. Shane/Kast Observations of APOGEE Targets

2MASS Source ID	Obs. Date	Air Mass	Exp. Time	S/N a	SpT b	ζc	${\mathrm{log}}_{10}{L}_{{\rm{H}}\alpha }/{L}_{\mathrm{bol}}$ d
	(UT)		(s)
2MASS J14554964+0321420	2021 May 15	1.21	2400	73	M5.0	1.195 ± 0.004	−4.81 ± 0.12
2MASS J15042797+0942464	2021 May 15	1.42	3000	47	M5.0	1.093 ± 0.007	−3.75 ± 0.08
2MASS J07552256+2755318	2021 Nov 27	1.02	3000	130	M6.0	1.118 ± 0.002	−4.01 ± 0.09
2MASS J11210854+2126274	2021 May 15	1.05	2400	97	M6.0	1.048 ± 0.002	−4.11 ± 0.09
2MASS J08080189+3157054	2021 Jan 17	1.11	3000	55	M7.0	1.007 ± 0.003	−4.22 ± 0.16
2MASS J12215013+4632447	2018 Jan 21	1.02	2400	156	M7.0	1.008 ± 0.001	−4.17 ± 0.16
2MASS J13342918+3303043	2020 Feb 5	1.02	3600	84	M7.0	1.031 ± 0.002	−4.91 ± 0.12
2MASS J16572919+2448509	2022 Mar 11	1.06	3000	113	M7.0	0.919 ± 0.002	−4.05 ± 0.13
2MASS J21381698+5257188	2020 Dec 14	1.22	3000	85	M7.0	1.005 ± 0.002	−4.17 ± 0.14
2MASS J12493960+5255340	2021 May 16	1.06	3000	16	M9.0	1.506 ± 0.043	−4.93 ± 0.14
2MASS J19544358+1801581	2020 Aug 14	1.07	3600	98	M9.0	1.207 ± 0.003	−4.70 ± 0.11
2MASS J21272531+5553150	2020 Aug 14	1.06	3600	96	M9.0	1.080 ± 0.003	−4.78 ± 0.10

Notes.

^a Median S/N in the 7200–7400 Å region. ^b Closest match to the SDSS dwarf spectral templates defined in Bochanski et al. (2007b), Schmidt et al. (2014), and Kesseli et al. (2017). ^c Metallicity index defined in Lépine et al. (2007), where ζ > 0.875 indicates a dwarf (solar) metallicity classification. ^d Relative luminosity in Hα emission based on the measured Hα equivalent width and χ correction factors compiled by Douglas et al. (2014) and Schmidt et al. (2014).

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Spectra were drawn from the APOGEE single-epoch, individual visit spectra (apVisit) files (Abdurro'uf et al. 2022), covering chips a (1.657–1.696 μm), b (1.585–1.644 μm), and c (1.514–1.581 μm). APOGEE data reduction is described in detail in Nidever et al. (2015), Holtzman et al. (2018), Jönsson et al. (2020), and Abdurro'uf et al. (2022). Each apVisit spectrum underwent dark, flat-field, cosmic-ray, flux, sky, and telluric corrections, as well as wavelength calibrations. In particular, the wavelength solutions were derived from a combination of sky lines and ThArNe and UNe hollow-cathode lamps (Nidever et al. 2015). Bad pixels were masked out using the bit mask for each chip. ²³ In order to simultaneously calibrate the wavelength solution imprinted in the spectra, the telluric absorption profile has been retained for each reduced apVisit spectrum (HDU7).

To infer the physical properties of our sources—effective temperature (T_eff), surface gravity ($\mathrm{log}g$), RV, and rotational velocity ($v\sin i$)—we employed an MCMC forward-modeling technique that simultaneously models the telluric and stellar absorption present in each APOGEE apVisit spectrum. We used the Spectral Modeling Analysis and RV Tool (SMART; Hsu et al. 2021b), which follows methods previously described in Blake et al. (2010), Burgasser et al. (2016), Theissen et al. (2022), and Hsu et al. (2021a).

Each APOGEE spectrum is forward modeled using the following equation:

$$\begin{eqnarray}\begin{array}{rcl}D[p] & = & \left(C[p]\times \left[\left(M\left[{p}^{* }\left(\lambda \left[1+\displaystyle \frac{{\mathrm{RV}}^{* }}{c}\right]\right),{T}_{\mathrm{eff}},\mathrm{log}g\right]\right.\right.\right.\\ & & \left.\left.\ast \,{\kappa }_{D}({\nu }_{\mathrm{micro}})\ast {\kappa }_{R}(v\sin i)\Space{0ex}{4.0ex}{0ex}\right)\times T\left[{p}^{* }(\lambda ,\mathrm{AM},\mathrm{PWV})\right]\Space{0ex}{4.0ex}{0ex}\right]\\ & & \ast \,\left.{\kappa }_{G}({\rm{\Delta }}{\nu }_{\mathrm{inst}})\Space{0ex}{4.0ex}{0ex}\right)+{C}_{\mathrm{flux}}.\end{array}\end{eqnarray} \tag{ 1 }$$

Here, D[p] is the data model as a function of pixel p; C[p] is a fifth-order polynomial representing a continuum correction; M[p] is the stellar solar-metallicity atmosphere model parameterized by T_eff and $\mathrm{log}g$; p*(λ) is a wavelength-to-pixel conversion function, initially provided by the APOGEE reduction pipeline with an additional constant offset parameter C_Δλ to adjust for chip-to-chip variations; RV^* = RV + v_bary is the RV of the source plus the barycentric motion of the Earth at the observed epoch; c is the speed of light; κ_D is a Gaussian convolution kernel that applies a microturbulence velocity broadening ν_micro, modeled as 2.478 – 0.325 $\times \,\mathrm{log}g$ km s⁻¹ (Zamora et al. 2015); κ_R is a rotational line broadening convolution kernel for projected rotational velocity $v\sin i$, with linear limb-darkening coefficient = 0.6 (Gray 1992); T[p] is the telluric absorption model based on the model grid of Moehler et al. (2014) and parameterized by air mass and precipitable water vapor; κ_G is a Gauss–Hermite convolution kernel used to account for the instrumental line-spread function (LSF), with width ν_inst obtained from the APOGEE pipeline (Nidever et al. 2015; Bovy 2016); and C_flux is a constant additive flux offset.

The log-likelihood function of our model fit was defined as

$$\begin{eqnarray}&&\mathrm{ln}\,{ \mathcal L }=-0.5\times \left[\sum {\chi }^{2}/{C}_{\mathrm{noise}}^{2}+\sum \mathrm{ln}(2\pi {\left({C}_{\mathrm{noise}}\sigma \right)}^{2})\right],\end{eqnarray} \tag{ 2 }$$

where the statistic

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{\left(S[p]-\alpha D[p]\right)}^{2}}{\sigma {\left[p\right]}^{2}},\end{eqnarray} \tag{ 3 }$$

compares the observed spectrum S[p] and uncertainty σ[p] to the scaled forward model D[p], with the scale factor α determined to minimize χ². We include the constant scaling factor C_noise to account for under estimation of observational noise, as well as systematic errors such as missing line features.

Cool dwarfs have abundant molecular absorption lines in their infrared spectra, and hence careful consideration must be made for the choice of stellar model. We explored synthetic model grids from Baraffe et al. (2015, BT-Settl CIFIST models), Husser et al. (2013, ACES models), Gustafsson et al. (2008, MARCS models), and Marley et al. (2018, Sonora models). For the lowest-temperature dwarfs (L dwarfs), we found that the Sonora models outperform other model sets, particular redward of 1.58 μm where FeH absorption is an important source of opacity (Cushing et al. 2003; Souto et al. 2017), although there are still some missing features in chip c.

For each source and model set, we fit all three chips simultaneously, with the nuisance parameters C_flux, C_Δλ , and continuum parameters modeled separately for each chip. The 18 continuum parameters, LSF broadening, ν_micro, and v_bary were fit and applied at the end of the MCMC loop, leaving 13 parameters to be fit by the forward-modeling routine, summarized in Table 3. Priors for these parameters were assumed to be uniformly distributed between bounds based on expected values for late-M and L dwarfs or model parameter limits. We used the emcee code (Foreman-Mackey et al. 2013) to run the MCMC with kernel density estimator (KDE) ²⁴ moves, which allows efficient convergence of best-fit parameters. We deployed 100 chains of 1000 steps each, with the first 800 steps removed for burn-in. At step 600, we used a 3σ-clipping mask of data minus model residuals to remove outlying flux values, which were mostly unmasked bad pixels and discrepant opacities. Chains were visually inspected to ensure convergence, and the typical integrated autocorrelation range was ∼17 steps. We report best-fit parameters as the median of the tails of the chains, with uncertainties derived from the 16% to 84% quantile range.

Table 3. Forward-modeling Parameters

Description	Symbol (Unit)	Priors a	Bounds
Temperature	Teff (K)	(1800, 4000) b	(1200, 7000) b
⋯	⋯	(1500, 2400) c	(200, 2400) c
Surface Gravity	$\mathrm{log}g$ (cm s−2)	(3.5, 5.5)	(3.5, 5.5)
Rotational Velocity	$v\sin i$ (km s−1)	(0, 50)	(0, 100)
Radial Velocity	RV (km s−1)	(−100, +100)	(−200, +200)
Flux Offset d	Cflux	(−0.01, +0.01)	(−104, +104)
Wave Offset d	CΔλ (Å)	(−0.1, +0.1)	(−0.5, +0.5)
Air Mass	AM	(1.0, 3.0)	(1.0, 3.0)
Water Vapor	PWV (mm)	(0.5, 20.0)	(0.5, 20.0)
Noise Factor	Cnoise	(1.0, 5.0)	(1.0, 10.0)

Notes.

^a All priors assume a uniform distribution over the range specified. ^b T_eff range for the BT-Settl models. ^c T_eff range for the Sonora models. ^d The three APOGEE chips were fit individually.

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We note that one-half of our sample are young sources, and hence magnetic emission as traced by the Bracket hydrogen emission series (e.g., Br 16 → 4, λ = 1.556 μm; Br 15 → 4, λ = 1.571 μm; Br 13 → 4, λ = 1.611 μm; Br 11 → 4, λ = 1.681 μm; see Sullivan et al. 2019) could be present in some sources. None of the models used include hydrogen emission; however, any such emission was much weaker compared to other absorption lines, as verified by visual inspection of each best-fit spectrum. ²⁵

To assess the quality of these fits, Figures 4–6 show the best-fit models using Sonora and BT-Settl models for the L2β 2MASS J00452143+1634446, the M9 2MASS J08440350+0434356, and the M7 + M9.5 binary 2MASS J04214955+1929086, respectively. The Sonora models clearly outperform the other models based on the residuals and χ² fit values, even though the best-fit T_eff reaches the T_eff limits of the Sonora model grid. The best-fit models of the BT-Settl, ACES, and MARCS models give similar results, with significant residuals redward of 1.58 μm driving up the χ² values. The Sonora models incorporate the Hargreaves et al. (2010) FeH E⁴Π − A⁴Π transitions (1.58263 μm, 1.59188 μm, and 1.62457 μm bandheads covering from 1.58 to 1.75 μm; Wallace & Hinkle 2001; Cushing et al. 2005), and we thus attribute residuals in the other models to outdated FeH opacities. The other models also yield lower $\mathrm{log}g$ and higher $v\sin i$ values than the Sonora models, which we interpret as compensation for these missing opacities; a similar effect for methane from Yurchenko & Tennyson (2014) has been noted in J-band spectra of T dwarfs (Hsu et al. 2021a).

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For warmer sources (late-M dwarfs) the BT-Settl models provided superior fits to the Sonora and other models, although the telluric absorption strengths are higher for these model fits, again suggesting compensation for missing opacity. Based on these initial fits, we focused our analysis on these two model sets, with an "optimal model" transition, as verified by visual inspection of each fit, around 2700 K ≲ T_eff ≲ 3000 K. At and below these transition temperatures, the Sonora models yield better fits even at its T_eff = 2400 K parameter limit.

Our RV measurements span the range −76.08 to +66.68 km s⁻¹ with a median value of −0.78 km s⁻¹ and a median measurement uncertainty of 0.32 km s⁻¹. To assess sources of systematic uncertainty, we evaluated the scatter in measurements inferred from 13 epochs of observations of the M2 2MASS J16495034+4745402, one of the RV standards in the Deshpande et al. (2013) APOGEE sample. We compared fits based on MARCS, BT-Settl, and ACES models, which had internal per-epoch scatter of 0.18 km s⁻¹, 0.19 km s⁻¹, and 0.16 km s⁻¹, respectively, and an overall scatter between models of 0.18 km s⁻¹. We therefore conservatively assume an overall systematic RV uncertainty of 0.19 km s⁻¹, which has been added in quadrature to the reported RV measurements for all sources, resulting in a final median RV precision of 0.37 km s⁻¹.

We explored whether this precision could be improved by modeling restricted wavelength regions where strong telluric absorption can be used to improve the wavelength calibration. This experiment was motivated by higher fitting scatter in regions that had by slight mismatches between the observed and modeled spectra, possibly due to offsets in the wavelength calibrations derived from arc lamp lines. We fit regions of strong telluric absorption at 16560–16700 Å on chip a, 15980–16160 Å on chip b, and 15100–15500 Å and 15500–16800 Å on chip c (see Figure 4), and included an additional pixel-to-wavelength zero-point offset term in our model. The resulting variance in RV measurements between these regions, 0.23–0.56 km s⁻¹, was worse than fitting all three chips simultaneously, likely due to the lack of strong stellar lines. We conjecture that slight improvements in the APOGEE pipeline wavelength calibration could be realized by combining arc lines, sky emission lines, and telluric absorption lines to compute the overall wavelength calibration.

We quantified the validity of our RV measurements by comparing our measured RVs with those reported in the literature, summarized in Table 5. Excluding published RVs from APOGEE DR16 (Jönsson et al. 2020), a total of 67 sources in our sample have reported RVs in the literature with uncertainties. Figure 7 compares these values. The vast majority (84%) have consistent RVs to within 3σ deviation, while 11 sources are significant outliers.

• 1.  
  Two of these outliers, 2MASS J08294949+2646348 (ΔRV = 5.5 km s⁻¹), 2MASS J16311879+4051516 (ΔRV = 4.4 km s⁻¹), are based on measurements made with lower-resolution data (λ/Δλ ≈ 2000) reported in Terrien et al. (2015b), and may reflect underestimated uncertainties.
• 2.  
  Five other outliers, 2MASS J04201611+2821325, 2MASS J04262939+2624137, 2MASS J04294568+2630468, 2MASS J04330945+2246487, and 2MASS J04363893+2258119 are all members of the Taurus Complex star-forming region and have prior APOGEE measurements reported by Kounkel et al. (2019) that are 2–3 km s⁻¹ lower than our measurements. Cottaar et al. (2014), Cook et al. (2014), and Kounkel et al. (2019) all report a systematic redshift in RV measurements among the lowest-temperature sources in their cluster samples (T_eff ≤ 3400 K), and the last study proposes a systematic correction of ΔRV = 12.84 – 0.0038 × T_eff. Accounting for this offset brings our measurements fully in line with RVs reported in Kounkel et al. (2019), but we did not report these corrected values in this work.
• 3.  
  Another young source, the M6 2MASS J16093019−2059536, a reported member of the Upper Scorpius Association (Slesnick et al. 2006a), also has a significantly different RV from our measurements (RV = −1.3 ± 0.6 km s⁻¹) compared to that reported in Dahm et al. (2012, RV = −5.1 ± 0.6 km s⁻¹). The latter is based on optical high-resolution spectra and cross correlation with the M8 standard VB 10. On the other hand, our measurement is fully consistent with that reported in Jönsson et al. (2020, −0.98 ± 0.09 km s⁻¹), using the cross-correlation method with DR16 APOGEE spectra. The variance between these measurements could again be due to the RV offset found among young low-temperature sources, or variability induced by a binary (only one epoch of APOGEE data was available for this source).
• 4.  
  2MASS J03505737+1818069 (LP 413−53) appears to be the currently known shortest orbital period UCD binary based on the high scatter of individual epoch RV measurements, analyzed in detail in Hsu et al. (2023).
• 5.  
  The remaining outliers, 2MASS J00034394+8606422 and 2MASS J07140394+3702459, appear to be poorly fit by BT-Settl models due to a lack of FeH opacities, whereas the Sonora models provide a much better fit, resulting in a shift of 2–4 km s⁻¹ (depending on the epoch) and bringing our measurement in line with that from the APOGEE pipeline. As the literature measurement from Deshpande et al. (2013) utilized BT-Settl models, we attribute this difference to modeling systematics.

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Table 5. Radial Velocity and $v\sin i$ Measurements with ASPCAP and Literature Comparison

APOGEE ID	〈RV〉 a , b	ASPCAP RV c	Lit. RV	Lit. RV References	$\langle v\sin i\rangle$ a	Lit. $v\sin i$	Lit. $v\sin i$ References
	(km s−1)	(km s−1)	(km s−1)		(km s−1)	(km s−1)
2MASS J00034394+8606422	${7.26}_{-0.33}^{+0.2}$	5.73 ± 2.85	10.8 ± 0.28	(8)	${18.77}_{-0.97}^{+0.95}$	13.2 ± 1.5	(8)
2MASS J00312793+6139333	$-{34.44}_{-0.35}^{+0.2}$	−38.02 ± 0.69	⋯	⋯	<10	⋯	⋯
2MASS J00381273+3850323	${6.62}_{-0.32}^{+0.2}$	3.42 ± 1.58	⋯	⋯	<10	⋯	⋯
2MASS J00452143+1634446	${3.71}_{-0.37}^{+0.22}$	555.81 ± 4.96	3.16 ± 0.83	(14)	${31.64}_{-1.04}^{+0.97}$	32.82 ± 0.17	(4)
2MASS J00514593−1221458	${15.66}_{-0.38}^{+0.24}$	15.25 ± 1.79	15.29 ± 0.11	(20)	${15.85}_{-1.22}^{+1.21}$	⋯	⋯
2MASS J01154176+0059317	${10.78}_{-0.59}^{+0.28}$	11.7 ± 0.41	⋯	⋯	${33.69}_{-1.12}^{+1.57}$	⋯	⋯
2MASS J01215816+0101007	${10.28}_{-0.37}^{+0.22}$	586.83 ± 0.92	⋯	⋯	<10	⋯	⋯
2MASS J01243124−0027556	$-{3.88}_{-0.41}^{+0.23}$	577.03 ± 1.68	0.69 ± 10.14	(19)	${27.7}_{-1.04}^{+1.11}$	⋯	⋯
2MASS J01514363+0046188	${14.44}_{-0.47}^{+0.27}$	14.49 ± 2.0	16.55 ± 10.07	(19)	${18.12}_{-1.0}^{+0.97}$	⋯	⋯
2MASS J02163612−0544175	$-{0.05}_{-0.47}^{+0.23}$	−2.99 ± 3.29	−0.89 ± 0.04	(20)	${11.51}_{-1.03}^{+1.02}$	⋯	⋯

Notes. Measurements from individual spectra over individual or multiple epochs are combined using inverse uncertainty weighting (weight = $1/({\sigma }_{\mathrm{upper}}^{2}+{\sigma }_{\mathrm{lower}}^{2})$); upper and lower uncertainties are also combined using inverse uncertainty-squared weighting. In cases where individual spectra have S/N < 10, spectral data are combined first, then modeled.

^a Our weighted average measurements over all epochs. ^b We did not apply the effective temperature dependent correction to RV from Kounkel et al. (2019) for 2MASS J04201611+2821325, 2MASS J04262939+2624137, 2MASS J04294568+2630468, 2MASS J04330945+2246487, and 2MASS J04363893+2258119. See Section 4.1 for details. ^c RV from DR17 ASPCAP.

References: (1) Lépine et al. (2003); (2) Fűrész et al. (2008); (3) Reiners & Basri (2009); (4) Blake et al. (2010); (5) West et al. (2011); (6) Deshpande et al. (2012); (7) Dahm et al. (2012); (8) Deshpande et al. (2013); (9) Newton et al. (2014); (10) Cottaar et al. (2015); (11) West et al. (2015); (12) Terrien et al. (2015b); (13) Burgasser et al. (2015); (14) Faherty et al. (2016); (15) Reiners et al. (2018); (16) Gilhool et al. (2018); (17) Kesseli et al. (2018); (18) Kounkel et al. (2019); (19) Kiman et al. (2019); (20) Jönsson et al. (2020); and (21) Gaia Collaboration et al. (2021b).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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For completeness, we also compared our measured RVs with those provided by the APOGEE DR17 pipeline Doppler (Nidever 2021), which uses cross correlation with the The Cannon models (Ness et al. 2015) trained from Synspec (Hubeny & Lanz 2017; Hubeny et al. 2021). While performing well for warmer stars, the pipeline is known to have systematic issues with M dwarfs with T_eff ≲ 3500 K (Abdurro'uf et al. 2022). Indeed, roughly half of the sources in our sample with independent literature measurements show a >3σ discrepancy with APOGEE pipeline RVs, and 15 sources in our sample have pipeline RVs > 250 km s⁻¹. We thus consider APOGEE pipeline RVs to be unreliable for these low-temperature objects.

In summary, all of the significant outliers between our APOGEE and literature RV measurements can be explained by methodological or astrophysical causes, and we conclude that our measurements are robust to a median precision of 0.37 km s⁻¹.

Measured $v\sin i$ values for our sample range from 0.4 to 92.8 km s⁻¹, with a median value of 17 km s⁻¹ (Table 4). The distribution of our measurements is shown in Figure 8. To determine our $v\sin i$ detection limit, we compared this distribution for both the Sonora and BT-Settl models and found a sharp transition at 10 km s⁻¹, which we adopt as our vsini detection limit. This limit is more conservative than the 8 km s⁻¹ limit reported in Gilhool et al. (2018) and the 5 km s⁻¹ limit reported in Deshpande et al. 2013). We adopt their final $\langle v\sin i\rangle$ as 10 km s⁻¹ for sources with $\langle v\sin i\rangle$ < 10 km s⁻¹. Our median precision of sources with higher $v\sin i$ (>10 km s⁻¹) values is 0.5 km s⁻¹. In addition to theses statistical uncertainties, we compared multiepoch measurements (N_obs ≥ 3) for all of our non-RV-varying sources, and found that the median standard deviation of $v\sin i$ values per source to be 0.95 km s⁻¹. We adopt this as an estimate of our systematic $v\sin i$ uncertainty and conservatively add it in quadrature to our statistical uncertainties, resulting in a median $v\sin i$ uncertainty of 1.1 km s⁻¹.

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We assessed the reliability of our $v\sin i$ measurements by again comparing to literature values (Figure 7). There are 41 sources with literature measurements, three of which are identified as >3σ outliers:

• 1.  
  2MASS J00034394+8606422 (<10 km s⁻¹; best-fit $v\sin i$ = 18.8 ± 1.0 km s⁻¹ versus 13.2 ± 1.5 km s⁻¹ in Deshpande et al. 2013),
• 2.  
  2MASS J07140394+3702459 (<10 km s⁻¹; best-fit $v\sin i$ = 7.3 ± 1.1 km s⁻¹ versus 12.8 ± 0.5 km s⁻¹ in Deshpande et al. 2013), and
• 3.  
  2MASS J16311879+4051516 (14.8 ± 1.6 km s⁻¹ versus 7.1 ± 1.5 km s⁻¹ in Reiners et al. 2018).

2MASS J00034394+8606422 (LP 2-291) is a known eclipsing binary (P = 13.9182 ± 0.0004 days; Prša et al. 2022), whose $v\sin i$ varies at different epochs possibly due to (unresolved) secondary flux. In these three cases, we find that Sonora models provide a much better fit to the APOGEE data, whereas the literature values are based on comparisons to BT-Settl models. We therefore attribute these deviations to systematics associated with model choice, and adopt our measured $v\sin i$ values for this analysis.

The distribution of $v\sin i$ measurements as a function of spectral type is shown in Figure 9. Median values are approximately constant over the M6–L2 range of ${17}_{-6}^{+25}$ km s⁻¹, with uncertainties computed from the 84th and 16th percentiles. This values is consistent with trends previously reported in the literature (Crossfield 2014; Tannock et al. 2021); e.g., Hsu et al. (2021a) report a median $v\sin i$ of 12.1 km s⁻¹ for M4–M9 dwarfs and 16.2 km s⁻¹ for M9–L2 dwarfs.

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While the majority of our sources have $v\sin i$ < 40 km s⁻¹, there are 14 sources with $v\sin i$ > 60 km s⁻¹.

Thirteen are young brown dwarfs:

• 1.  
  M8 2MASS J04311907+2335047 in the 1–2 Myr Taurus molecular clouds (Slesnick et al. 2006b);
• 2.  
  L0 2MASS J05350162−0521489 (V* V2113 Ori) in the 1–2 Myr Orion Nebula cluster (Meeus & McCaughrean 2005); and
• 3.  
  M7 2MASS J15560497−2106461, M6 2MASS J16003023−2334457, M7.5 2MASS J16025214−2121296, M6.5 2MASS J16044303−2318258, M8 2MASS J16045199−2224108, M6.5 2MASS J16045581−2307438, M6 2MASS J16053077−2246200, M7.5 2MASS J16111711−2217173, M6 2MASS J16124692−2338408, M6.5 2MASS J16131600−2251511, and M6 2MASS J16200757−2359150, in the 10 Myr Upper Scorpius moving group (Pecaut & Mamajek 2016).

These sources are in an age range in which they have largely contracted and spun up, but have not had sufficient time to lose angular momentum through magnetized winds (Kawaler 1988; Barnes 2003; Matt et al. 2015).

The other fast rotator is likely a binary. 2MASS J15010818+2250020 (aka TVLM 513-46546) is a source with known periodic radio variability (Hallinan et al. 2006) and a potential giant planet companion identified by radio astrometry (Curiel et al. 2020), which we also identify as an RV variable in our sample (Gaia renormalized unit weight error (RUWE) = 1.661; ΔRV${}_{\max }\sim 2$ km s⁻¹; Section 5.5). While TVLM 513-46546 exhibits Hα emission and low surface gravity, it does not show Li i absorption (Burgasser et al. 2015) and has been ruled out as a member of the Argus moving group (Section 5.3).

Our fits also provided best estimates of effective temperature and surface gravity based on the particular model used. Of the 258 spectra modeled, 196 were best fit by the Sonora models while 62 were best fit by the BT-Settl models. As noted above, the Sonora grid has a T_eff ceiling of 2400 K, which corresponds to a spectral type of approximately M9 (Filippazzo et al. 2015). As this encompasses the majority of our sample, all of the inferred temperatures from these model fits were close to the model limit, making any inference of T_eff trends impossible. Furthermore, the Sonora model grid is cloudless, with its inferred T_eff typically hotter by ∼300–500 K compared to the BT-Settl model grid (Hsu et al. 2021a), making the inferred T_eff = 2400 K for L2β 2MASS J00452143+1634446 reach (expected T_eff ∼ 2060 K from Filippazzo et al. 2015). On the other hand, the T_eff values inferred from BT-Settl models ranged between 2676 and 3177 K, with a median of 2860 K. Figure 10 shows the T_eff trend as a function of spectral type. Ignoring the Sonora fit values, the best-fit T_eff values from the BT-Settl models show a general decreasing trend toward later spectral types from M6 to M9.5. The outlier in this trend is the young L0 2MASS J05350162−0521489 (T_eff = 3162 ± 13 K; Meeus & McCaughrean 2005). Comparing to the empirical spectral type to T_eff relations from Pecaut & Mamajek (2013) and Filippazzo et al. (2015), our best-fit BT-Settl T_eff values are systematically higher than the empirical T_eff values, which we attribute to the model discrepancies.

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Figure 11 illustrates fit $\mathrm{log}g$ trends as a function of spectral type. Surface gravities from the Sonora model fits scatter across the full model parameter range of 3.5 ≤ $\mathrm{log}g$ ≤ 5.5, with members of young clusters typically (but not consistently) having $\mathrm{log}g$ values close to the minimum. Surface gravities from the BT-Settl model fits have a narrower range of 4.0 ≤ $\mathrm{log}g$ ≤ 5.5, with the young cluster members again having values in the bottom half of this range.

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Given the limited temperature fit range for the Sonora models, and the large scatter in the inferred surface gravities for both models, we do not regard these values as realistic estimates, and do not further investigate their trends. However, we verified that variations in T_eff and $\mathrm{log}g$ about the optimal values had minimal influence on the derived RV and $v\sin i$ values (see also Theissen et al. 2022). Comparing the fits between the BT-Settl and Sonora model sets yields equivalent RVs and $v\sin i$ values on average, with standard deviations of 0.9 km s⁻¹ and 1.5 km s⁻¹, respectively.

We combined our RV measurements with Gaia and ground-based astrometry to compute heliocentric UVW spatial motions following Johnson & Soderblom (1987). We adopt a right-handed coordinate system, with U velocity pointing toward the Galactic center, V velocity pointing in the direction of Galactic rotation, and W velocity pointing toward the Galactic north pole. We also computed velocities in the local standard of rest (LSR) assuming solar UVW LSR velocities of (U_⊙, V_⊙, W_⊙) = (11.1, 12.24, 7.5 km s⁻¹) from Schönrich et al. (2010). These values are visualized in Figure 12 and listed in Table 6. The mean U_LSR = −2.5 ± 1.3 km s⁻¹ and W_LSR = –1.1 ± 0.6 km s⁻¹ velocities of our sample are consistent with 0, while the average V_LSR = −6.0 ± 0.9 km s⁻¹ is slightly negative, as expected for asymmetric drift (Strömberg 1924).

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Table 6. Radial Velocities and Heliocentric Spatial Motions

APOGEE ID	SpT	Adopted RV	U	V	W	P[TD]/P[D] a	Pop a	BANYAN Σ	BANYAN Σ b
		(km s−1)	(km s−1)	(km s−1)	(km s−1)			Prob.	Member
2MASS J00034394+8606422 c	M6.0	${7.26}_{-0.33}^{+0.2}$	−16.2 ± 0.2	7.4 ± 0.3	1.6 ± 0.1	0.01	D	>99%	field
2MASS J00312793+6139333	M7.0	$-{34.44}_{-0.35}^{+0.2}$	−27.4 ± 0.2	−50.0 ± 0.3	29.8 ± 0.1	0.18	D/TD	>99%	field
2MASS J00381273+3850323	M6.0	${6.62}_{-0.32}^{+0.2}$	−31.2 ± 0.2	−4.9 ± 0.3	5.3 ± 0.1	0.01	D	>99%	field
2MASS J00452143+1634446 d	L2.0	${3.71}_{-0.37}^{+0.22}$	−10.8 ± 0.2	−1.5 ± 0.2	1.5 ± 0.3	0.01	D	>99%	ARG
2MASS J00514593−1221458	M6.0	${15.66}_{-0.38}^{+0.24}$	17.5 ± 0.1	13.1 ± 0.1	−9.7 ± 0.4	0.01	D	>99%	field
2MASS J01154176+0059317	M6.0	${10.78}_{-0.59}^{+0.28}$	−35.8 ± 0.4	−7.4 ± 0.3	5.8 ± 0.5	0.01	D	>99%	field
2MASS J01215816+0101007	M6.5	${10.28}_{-0.37}^{+0.22}$	−9.0 ± 0.2	−12.0 ± 0.2	−5.0 ± 0.4	0.01	D	>99%	field
2MASS J01243124−0027556	M7.0	$-{3.88}_{-0.41}^{+0.23}$	17.4 ± 0.2	−16.1 ± 0.3	−0.4 ± 0.4	0.01	D	>99%	field
2MASS J01514363+0046188	M7.0	${14.44}_{-0.47}^{+0.27}$	−15.3 ± 0.2	5.4 ± 0.1	2.8 ± 0.4	0.01	D	98%	field
2MASS J02163612−0544175	M6.0	$-{0.05}_{-0.47}^{+0.23}$	15.2 ± 0.2	0.6 ± 0.0	3.9 ± 0.4	0.01	D	>99%	field

Notes.

^a Galactic thin disk (D), thick disk (TD), intermediate populations (D/TD) are assigned according to probability ratios P(TD)/P(D) < 0.1, P(TD)/P(D) > 10, and 0.1 < P(TD)/P(D) < 10, respectively, following Bensby et al. (2003). ^b BANYAN Σ young moving group name abbreviation follows those defined in Gagné et al. (2018): AB Doradus (ABDMG), Argus (ARG), Carina Near (CARN), Coma Berenices (CBER), Corona Australis (CRA), Hyades (HYA), Taurus (TAU), Upper Centaurus–Lupus (UCL), Upper Corona Australis (UCRA), and Upper Scorpius (USCO). ^c Known or candidate binary. ^d Reported in the literature or a new member of young moving groups. ^e Reported as a member of NGC 1333 in Cantat-Gaudin et al. (2018, 2020), Yao et al. (2018), and Cantat-Gaudin & Anders (2020), but we did not identify because NGC 1333 is not included in BANYAN Σ. ^f Reported as a member of the Castor moving group in Zuckerman et al. (2013), but we did not identify it because the Castor moving group is not included in BANYAN Σ.

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We followed Bensby et al. (2003) to determine the probabilities of kinematic membership for each star to thin disk, thick disk, or halo populations based on the LSR velocities. We specifically separated thin disk, intermediate thin/thick disk, and thick disk membership by using the ratios P[TD]/P[D] < 0.1, 0.1 ≤ P[TD]/P[D] ≤ 10, and P[TD]/P[D] > 10, respectively. As expected, the majority of our sample ²⁷ are thin disk sources (246 sources), with 11 intermediate thin disk/thick disk members and no thick disk members.

We did not detect significant correlations between (U_LSR, W_LSR) or (V_LSR, W_LSR) velocity pairs (p-value = 0.47 and 0.08, respectively, using the Wald test with a t-distribution; Wald 1943; McKinney 2010), but did find a significant positive correlation for (U_LSR, V_LSR) velocities (p-value < 0.001, correlation coefficient R = 0.26), driven largely by our intermediate thin/thick disk members. Excluding these sources significantly reduces the (U_LSR, V_LSR) correlation (p-value = 0.016, R = 0.15). We also found significant positive correlations between total velocity squared, ${v}_{\mathrm{LSR}}^{2}$ = ${U}_{\mathrm{LSR}}^{2}+{V}_{\mathrm{LSR}}^{2}+{W}_{\mathrm{LSR}}^{2}$ and absolute ∣W_LSR∣ velocity for the full sample (p-value < 0.001, R = 0.65) and the thin disk subsample (p-value < 0.001, R = 0.53). ${v}_{\mathrm{LSR}}^{2}$ correlates with the asymmetric drift (Strömberg 1924), while the absolute ∣W_LSR∣ velocity has been used as a proxy of age (Wielen 1977), so this correlation is an indicator of age variation in the sample. While the APOGEE sample selection is not volume complete, these correlations are consistent with our prior analysis of a 20 pc UCD sample (Hsu et al. 2021a).

Galactic orbits can identify sources with different spatial origins, including stars that had drifted radially inward or outward to the solar radius. Starting with the LSR velocities and XYZ Galactic spatial coordinates ²⁸ of each source, Galactic orbits were computed using the galpy package (Bovy 2015), an efficient, ordinary differential equation solver that conserves energy and momentum for Galactic dynamics. We used an axisymmetric potential from Miyamoto & Nagai (1975) and assumed an LSR azimuthal velocity, v_ϕ = 220 km s⁻¹ (Chen et al. 2001; Bovy & Tremaine 2012; Reid et al. 2014). Each orbit was integrated from −5 to +5 Gyr in steps of 10 Myr, and 1000 orbit realizations were computed using Monte Carlo sampling of velocity uncertainties assuming normal distributions. We examined the specific orbital parameters of minimum and maximum Galactic cylindrical radius (${R}_{\min }$, ${R}_{\max }$), maximum Galactic vertical height (∣Z∣), median orbital eccentricity ($e\equiv \langle {R}_{\max }-{R}_{\min }\rangle /\langle {R}_{\max }+{R}_{\min }\rangle$), and median orbital inclination ($\tan i\equiv | Z/\sqrt{{X}^{2}+{Y}^{2}}|$). These parameters are listed in Table 7.

Table 7. Galactic Orbital Parameters

APOGEE ID	Rmin	Rmax	Zmax	e	i
	(kpc)	(kpc)	(kpc)		(deg)
2MASS J00034394+8606422	8.0461 ± 0.0009	10.47 ± 0.03	0.0463 ± 0.0009	0.131 ± 0.0017	0.279 ± 0.005
2MASS J00312793+6139333	5.002 ± 0.008	8.489 ± 0.002	0.3386 ± 0.0008	0.2584 ± 0.0008	2.865 ± 0.014
2MASS J00381273+3850323	7.615 ± 0.006	9.7 ± 0.03	0.0659 ± 0.0013	0.1203 ± 0.0012	0.425 ± 0.011
2MASS J00452143+1634446	7.897 ± 0.013	8.557 ± 0.017	0.023 ± 0.003	0.0401 ± 0.0003	0.16 ± 0.02
2MASS J00514593−1221458	8.0163 ± 0.0011	11.293 ± 0.012	0.113 ± 0.005	0.1697 ± 0.0005	0.65 ± 0.03
2MASS J01154176+0059317	7.047 ± 0.019	9.18 ± 0.03	0.071 ± 0.01	0.1315 ± 0.0016	0.5 ± 0.07
2MASS J01215816+0101007	7.242 ± 0.01	8.3797 ± 0.0009	0.0494 ± 0.0011	0.0728 ± 0.0007	0.359 ± 0.008
2MASS J01243124−0027556	6.883 ± 0.016	8.4725 ± 0.0015	0.0052 ± 0.0015	0.1035 ± 0.0011	0.038 ± 0.011
2MASS J01514363+0046188	8.05 ± 0.005	9.1 ± 0.02	0.031 ± 0.006	0.0613 ± 0.0014	0.2 ± 0.04
2MASS J02163612−0544175	7.982 ± 0.005	9.756 ± 0.003	0.05 ± 0.005	0.1 ± 0.0004	0.31 ± 0.03

Note.

^a The large uncertainties of Galactic orbital parameters of 2MASS J05350162−0521489 are due to its large proper-motion uncertainty (${\mu }_{\alpha }\cos \delta =0\pm 44$ mas yr⁻¹ and μ_δ = 0 ± 40 mas yr⁻¹; Cutri et al. 2021).

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Figure 13 shows the distributions of the derived orbital parameters. The majority of our sample exhibit circular ($\langle e\rangle ={0.07}_{-0.03}^{+0.08}$) and planar orbits ($\langle i\rangle ={0.}^{{\circ }^{}}6{\,}_{-{0.}^{\circ }2}^{+{0.}^{\circ }8}$), residing mostly at the solar Galactic radius ($\langle {R}_{\min }\rangle$= ${7.8}_{-0.7}^{+0.4}$ kpc, $\langle {R}_{\max }\rangle$ = ${8.9}_{-0.6}^{+0.9}$ kpc) and close to the Galactic plane (〈∣Z∣〉 = ${0.08}_{-0.03}^{+0.07}$ kpc). There are 22 sources that have e > 0.2, 10 of which are intermediate thin/thick disk or thick disk members. There are also 13 sources with slightly nonplanar orbits (i > 2°), eight of these being intermediate thin/thick disk members.

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We evaluated young association membership by comparing the 6D heliocentric spatial and velocity coordinates derived above to known nearby systems using the BANYAN Σ web tool (Gagné et al. 2018). Results are summarized in Table 6.

Our APOGEE DR17 sample is highly biased toward young cluster members. Out of 141 confirmed young sources in our sample, 114 are identified as kinematic members of young moving groups, with 138 of these previously reported in the literature, three new young sources, and six ruled out based on BANYAN Σ. The majority of cluster members are associated with Upper Scorpius (10 ± 3 Myr; Pecaut & Mamajek 2016; 75 sources), Taurus (1–2 Myr; Kenyon & Hartmann 1995; 25 sources), and the Hyades moving group (750 ± 100 Myr; Brandt & Huang 2015; six sources).

We identify three moving group members not previously reported in the literature:

• 1.  
  2MASS J05402570+2448090 (G 100-28; 67.2% Argus moving group, 40–50 Myr, Zuckerman 2019; 30.9% Carina Near moving group, ∼200 Myr, Zuckerman et al. 2006),
• 2.  
  2MASS J14093200+4138080 (LP 220-50; 99.6% Argus moving group), and
• 3.  
  2MASS J21272531+5553150 (LSPM J2127+5553; 99.3% Carina Near moving group).

2MASS J05402570+2448090 is a well-studied active M dwarf binary that exhibits flares and Hα emission (Pettersen 1983; Reid et al. 1995; Gizis et al. 2002; Lépine et al. 2013; Gaidos et al. 2014; Terrien et al. 2015a). The secondary is separated at 04720 (Janson et al. 2014; Winters et al. 2021) with a short rotational period of 0.294 day (Newton et al. 2016). It was previously reported as a candidate member of Hyades (Eggen 1993), which is ruled out based on our RV of ${23.1}_{-0.2}^{+0.3}$ km s⁻¹. The uncertainty of its membership between the Argus and Carina Near moving groups is based on the true systematic RV (best RV of 23.3 and 26.4 km s⁻¹, respectively), which requires future RV monitoring to sample its full orbit.

2MASS J14093200+4138080 has been reported as an active M6 dwarf in West et al. (2015; Hα equivalent width = 7.0 ± 1.1 nm), but reported inactive in Newton et al. (2017; Hα equivalent width = −7.2 ± 0.1 nm) and fast rotational period = 0.265 day (Newton et al. 2016). No signatures of youth has been reported for 2MASS J14093200+4138080.

2MASS J21272531+5553150 has been reported as a candidate member of the Carina Near moving group (98.520%) with BANYAN Σ in Seli et al. (2021) using Gaia DR2 astrometry (no RV), so our RV places it as a highly likely member.

On the other hand, we rule out six sources previously associated with clusters which had previously lacked RV data.

• 1.  
  2MASS J00381273+3850323 was reported as a member of the Hyades moving group, which was determined solely from Gaia Data Release 2 (DR2) proper motions and parallax without the RV (Lodieu et al. 2019; Röser et al. 2019). Our measured RV = $+{6.6}_{-0.3}^{+0.2}$ km s⁻¹ is not consistent with the expected RV of Hyades ($+{39}_{-4}^{+3}$ km s⁻¹; Gagné et al. 2018).
• 2.  
  2MASS J04254894+1852479 was reported as a member of the β Pic moving group, based on Gaia DR2 proper motions and parallax only (Gagné & Faherty 2018). Our measured RV = +28.3 ± 0.4 km s⁻¹ is inconsistent with the expected RV of the β Pic moving group (+10 ± 10 km s⁻¹; Gagné et al. 2018).
• 3.  
  2MASS J07140394+3702459 (LSPM J0714+3702) was reported as a member of the Argus moving group and classified as M7β (intermediate gravity class) by Gagné et al. (2015a). With the BANYAN Σ web tool (Gagné et al. 2018), the required RV for 2MASS J07140394+3702459 to be a member of Argus is 20.9 km s⁻¹, which is ruled out by our RV measurement of 35.3 ± 0.2 km s⁻¹.

We also rule out three reported members of the Coma Berenices cluster (${562}_{-84}^{+98}$ Myr; Silaj & Landstreet 2014), 2MASS J12205439+2525568, 2MASS J12263913+2505546, and 2MASS J12265349+2543556, identified in Melnikov & Eislöffel (2012) on the basis of astrometry and photometry alone. Melnikov & Eislöffel (2012) used proper motions from the Lépine Shara Proper Motion catalog (Lépine & Shara 2005), and these sources were selected on the basis of an insignificant proper motion, consistent with the small angular relative motion of the Coma Berenices cluster at large, (${\mu }_{\alpha }\cos \delta$, μ_δ ) = (−12.11 ± 0.05, −9.00 ± 0.12) mas yr⁻¹ (Gaia Collaboration et al. 2018). The measured RV of 2MASS J12205439+2525568 of 14.81 ± 0.29 km s⁻¹ is highly different from the expected RV of the Coma Berenices cluster of −0.1 ± 0.8 km s⁻¹ (Gagné et al. 2018). 2MASS J12263913+2505546 and 2MASS J12265349+2543556 were ruled out largely based on improved astrometry from Gaia.

These cases highlight the importance of RVs for kinematic cluster membership. The 21 known young UCDs not matched by BANYAN Σ are in clusters not included in this web tool, ²⁹ as BANYAN Σ only included young clusters and moving groups within 150 pc. There are five sources in the 1–2 Myr cluster NGC 1333 identified by Gaia DR2 astrometry (Cantat-Gaudin et al. 2018, 2020; Yao et al. 2018; Cantat-Gaudin & Anders 2020); our RV of 15.5–17.5 km s⁻¹ is consistent with the expected cluster motion of ∼11.0–17.6 km s⁻¹ (Tarricq et al. 2021). There are 11 sources in the 3 Myr cluster IC 348 (RV = 15.1–19.5 km s⁻¹; consistent with the expected RVs = 13.0–20.7 km s⁻¹ in Tarricq et al. 2021), and four sources in the ∼2 Myr Orion Nebula cluster (Yao et al. 2018; RV = 25.1–33.4 km s⁻¹; consistent with the expected RVs = 23.4–33.5 km s⁻¹ in Theissen et al. 2022). 2MASS J08294949+2646348 is reported as a member of the Castor moving group (200–700 Myr; López-Santiago et al. 2009; Zuckerman et al. 2013). ³⁰ We include all 141 sources in our "young" sample for our subsequent kinematic analysis.

Returning to our model fit parameters, we note that there is a distinct difference between the $\mathrm{log}g$ values between our young sources ($\langle \mathrm{log}g\rangle ={3.55}_{-0.04}^{+0.42}$ cm s⁻² dex) and our field sources ($\langle \mathrm{log}g\rangle ={5.20}_{-0.56}^{+0.29}$ cm s⁻² dex) based on the Sonora model fits. This is consistent with expectations for the larger radii and lower masses expected for UCDs younger than ∼150 Myr. We also found slightly higher $v\sin i$ values for the young sources ($\langle v\sin i\rangle ={22}_{-8}^{+34}$ km s⁻¹) compared to the field objects ($\langle v\sin i\rangle ={14}_{-5}^{+13}$ km s⁻¹), although these distributions overlap. Again, this is line with expectations for the spin-down timescales of low-mass stars, which can be ≳100 Myr for M ∼ 0.1 M_⊙ (Barnes & Kim 2010; Reiners et al. 2012; van Saders & Pinsonneault 2013; Matt et al. 2015) Similar differences were seen for the BT-Settl model fits, albeit with a much smaller sample size (48 young and 14 field objects).

Ensemble kinematics of a population provides age information, as stellar velocities become increasingly dispersed through dynamical interactions of Galactic structures (Spitzer & Schwarzschild 1953; Wielen 1977; Aumer & Binney 2009; Ting & Rix 2019; Sharma et al. 2021). The increased dispersion over time has been historically captured in empirical age–velocity dispersion relations (AVRs), which can be inverted to derive mean kinematic ages for stellar samples (Hsu et al. 2021a). Here, we considered two functional forms of the AVR in this study: the exponential relation of Wielen (1977) and the power-law relation from Aumer & Binney (2009). We followed the same analysis methodology as described in Hsu et al. (2021a), and the results are summarized in Table 8.

We find an overall velocity dispersion of σ_tot = 27.67 ± 0.15 km s⁻¹, which corresponds to a kinematic age of τ = 1.25 ± 0.10 Gyr using the Aumer & Binney (2009) relation. For the Wielen (1977) relation, the W-weighted velocity dispersion σ_W-tot = 43.4 ± 0.2 km s⁻¹ corresponds to a kinematic age of τ = 3.41 ± 0.04 Gyr. Compared to the late-M age of 4.0 ± 0.3 Gyr in Hsu et al. (2021a) based on the Aumer & Binney (2009) relation, our sample appears to have a younger average age, likely reflecting the sample bias toward young clusters. Removing the 140 young cluster members increases the velocity dispersion to σ_tot = 38.2 ± 0.3 km s⁻¹, corresponding to a kinematic age of 3.30 ± 0.19 Gyr for the Aumer & Binney (2009) relation, in line with prior results (Reiners & Basri 2009; Blake et al. 2010; Burgasser et al. 2015; Hsu et al. 2021a). There is also better agreement in this case with the W-weighted velocity dispersion and Wielen (1977) age of 4.13 ± 0.05 Gyr. We also removed the 11 intermediate thin/thick disk and thick disk sources and 140 young sources, which resulted in an increased velocity dispersion and "thin disk" age of 2.11 ± 0.14 Gyr based on the Aumer & Binney (2009) relation. For segregating these thin disk sources into M dwarfs (102 sources) and L dwarfs (four sources) without young sources, we find similar velocity dispersions and kinematic ages (1.97 ± 0.14 Gyr and 4.3 ± 2.4 Gyr, respectively), albeit with large uncertainties for the latter. Finally, we examined removing the 42 RV variables from the thin disk sample; this had minimal influence on the inferred age (1.36 ± 0.11 Gyr).

We can also discern distinct young and field populations using the velocity probability plot, or probit plot, which ranks the individual velocity components in steps of overall sample standard deviation. A normal distribution would be represented as a straight line in this diagram whose slope equals the sample dispersion (Chambers et al. 1983; Reid et al. 2002; Bochanski et al. 2007a). Figure 14 displays probit plots for each of the UVW velocity components, all of which show two clear linear trends: a shallower "core" sample and a steeper (and hence more dispersed) "wide" sample. A piecewise linear fit to these trends broken at ±1σ components yields total velocity dispersions of σ_tot = 17.3 ± 0.3 km s⁻¹, σ_tot = 39.4 ± 1.2 km s⁻¹, and σ_tot = 45.8 ± 1.7 km s⁻¹ for the shallow, lower wide, and upper wide samples, respectively, corresponding to kinematic ages of 0.20 ± 0.05 Gyr, 3.6 ± 0.4 Gyr, and 5.6 ± 0.6 Gyr, based on the Aumer & Binney (2009) relation. The shallow core sample is fully consistent with the thin disk sources (with young objects); the wide samples have older and similar ages, as expected for field age thin disk sources.

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One of the main stellar science goals of the APOGEE survey is to identify closely separated binary systems, which are crucial for mass measurements and testing binary formation and evolution models. While APOGEE ASPCAP is unable to provide robust RVs in the UCD temperature regime, our RV precisions are sufficient to identify binaries at projected separations ≲ 1.1 au from RV variability, assuming the total mass of 0.2 M_⊙, secondary mass of 5 M_Jup, and RV precision of 0.3 km s⁻¹. Our sample contains 171 sources with at least two epochs of observations, 71 of which have four or more epochs. Of the latter, 26 exhibit evidence of significant RV variations (p < 0.01) based on a χ² test, and we consider these high-probability binary systems. Among the 100 sources with two or three epochs of observations, 11 show significant RV variations, and we consider these promising binary candidates. All of the RV variables are listed in Table 9.

The majority of the RV variables have too few epochs to fully sample a complete orbit, and hence only partial constraints can be made on the orbital parameters. We attempted to make these constraints for each RV variable with at least four epochs of observation using The Joker (Price-Whelan et al. 2017), a Monte Carlo rejection sampler that quantifies single-line RV orbits in terms of period (P), velocity variation semiamplitude (K), eccentricity (e), systemic velocity (v₀), mean anomaly (M₀), and argument of periastron (ω), and identifies a family of orbits consistent with the measurements. We ran The Joker using its default settings. For each system, we initially selected limiting ranges for the minimum and maximum orbital period (${P}_{\min }$ and ${P}_{\max }$, respectively), the maximum RV semiamplitude (K₀), and the number of input samplers (10⁵ ≤ N_samp ≤ 5 × 10⁶). Initial estimates of v₀ and K were determined from the mean and standard deviation of the RV measurements, and both of these quantities were assumed to follow normal distributions with scale factors σ_K , σ_v = 1–4 km s⁻¹ constrained from the ΔRV variations in the observed RV time series. The period distribution was assumed to follow ${ \mathcal P }(P)\propto {P}^{-1}$ following Uehara et al. (2016), Price-Whelan et al. (2017), and Kipping (2018). The eccentricity distribution was assumed to follow a beta distribution:

$$\begin{eqnarray}&&{ \mathcal P }(e)=\displaystyle \frac{{\rm{\Gamma }}(a+b)}{{\rm{\Gamma }}(a)+{\rm{\Gamma }}(a)}{e}^{a-1}{\left[1-e\right]}^{b-1},\end{eqnarray} \tag{ 4 }$$

where Γ is the gamma function and a = 0.867 and b = 3.03 (fixed) following Kipping (2013). The distributions of the mean anomaly and periastron angle were assumed to be uniformly distributed between 0 and 2π. The standard deviation of the RV semiamplitude σ_K prior was assumed to be a normal distribution defined as

$$\begin{eqnarray}&&{\sigma }_{K}^{2}={\sigma }_{K,0}^{2}{\left(\displaystyle \frac{P}{1\,\mathrm{yr}}\right)}^{-2/3}{\left(1-{e}^{2}\right)}^{-1}.\end{eqnarray} \tag{ 5 }$$

This form was chosen because the RV semiamplitude of the primary is K₁ = $\sqrt{\tfrac{G}{(1-{e}^{2})}}{m}_{2}\sin i{\left({m}_{1}+{m}_{2}\right)}^{-1/2}{a}^{-1/2}$ = ${G}^{1/3}{\left(2\pi \right)}^{-1/3}{\left(1-{e}^{2}\right)}^{-1/2}{m}_{2}\sin i{\left({m}_{1}+{m}_{2}\right)}^{-2/3}{P}^{-1/3}$, with G the gravitational constant; m₁ and m₂ the masses of the primary and secondary, respectively; i the orbital inclination; and a the semimajor axis. As such, the adopted prior for σ_K has the advantage that the RV semiamplitude K has a fixed form for a given primary mass independent of period and eccentricity (Price-Whelan et al. 2020).

Results for these fits are provided in Table 10, and individual fits to all RV variables are provided in Appendix B. To assess the robustness of these fits, we computed the relative Bayesian information criterion (ΔBIC) between the best orbital solution and a constant RV, where BIC = ${\chi }^{2}+k\mathrm{ln}n$, χ² is the chi-square quality of fit (see Equation (3)), k is the number of model parameters (six for the full orbit and one for a constant RV), and n the number of RV measurements (Schwarz 1978). ΔBIC ranges of 0–2, 2–6, 6–10, and >10 correspond to insignificant, positive, strong, and very strong evidence against the null hypothesis (constant RV), respectively (Kass & Raftery 1995). Among the 25 sources with significant RV variations we found three to be strong and 22 sources to be very strong; the last set we consider to be highly probable binary candidates. Figure 15 illustrates an example orbit fit for the binary candidate with the largest ΔBIC and K₁, 2MASS J03284407+3120528, which has 18 epochs of observations, from which we were able to constrain a period P_fit = ${4.2}_{-3.0}^{+21.8}$ days, an RV semiamplitude K_fit = 1.3 ± 0.2 km s⁻¹, and an eccentricity e_fit = ${0.20}_{-0.19}^{+0.26}$. There are nine highly probable binaries with estimated orbit periods less than 10 days (Table 10): 2MASS J14005977+3226109 ($P={1.4}_{-0.3}^{+321.9}$ days ³¹ ), 2MASS J13495109+3305136 ($P={2.1}_{-0.7}^{+2.1}$ days), 2MASS J22551142+1442456 ($P={2.8}_{-1.1}^{+4.8}$ days), 2MASS J13232423+5132272 ($P={3.6}_{-2.0}^{+8.0}$ days), 2MASS J03284407+3120528 ($P={4.2}_{-3.0}^{+21.8}$ days), 2MASS J03413641+3216200 ($P={4.3}_{-2.6}^{+3.3}$ days), 2MASS J03440291+3152277 ($P={4.5}_{-2.7}^{+2.2}$ days), 2MASS J03293773+3122024 ($P={7.4}_{-5.8}^{+17.5}$ days), and 2MASS J12481860−0235360 ($P={9.4}_{-1.8}^{+4.5}$ days). We note that the shortest-period confirmed binary in our sample is 2MASS J03505737+1818069 (LP 413−53), which has only one epoch of APOGEE data, but follow-up observations have determined this source to be a 0.71 day period UCD binary (Hsu et al. 2023). The remaining highly probable binaries have estimated orbit periods between 10 and 100 days. It is important to note that the sparse sampling of these binary candidates results in large uncertainties on period (median uncertainty = 29 days) and eccentricity (median uncertainty = 0.21), and more complete sampling of the RV orbit is required to both confirm and robustly constrain orbital parameters.

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Table 10. Binary Candidates and Orbital Parameters

APOGEE ID	Nobs	PPrior	K0, Prior	${\sigma }_{{K}_{0}}$	σv	Nsam, in	P	K1	e	Gaia RUWE	ΔBIC
		(days)	(km s−1)	(km s−1)	(km s−1)		(days)	(km s−1)
ΔBIC > 10: very strong/highly probable multiple systems
2MASS J03282839+3116273	10	1–300	3.0	2.0	2.0	1,000,000	${24.1}_{-21.8}^{+86.9}$	${1.0}_{-0.2}^{+0.3}$	${0.175}_{-0.132}^{+0.245}$	0.995	26
2MASS J03284407+3120528	18	1–1000	6.0	3.0	3.0	10,000,000	${4.2}_{-3.0}^{+21.8}$	${1.3}_{-0.2}^{+0.2}$	${0.195}_{-0.192}^{+0.259}$	1.093	97
2MASS J03290413+3056127	17	1–1000	3.0	2.0	2.0	10,000,000	${73.3}_{-35.5}^{+2.4}$	${0.6}_{-0.1}^{+0.1}$	${0.265}_{-0.181}^{+0.251}$	0.886	43
2MASS J03291130+3117175	15	1–1000	5.0	3.0	3.0	100,000,000	${15.2}_{-0.0}^{+0.0}$	${1.4}_{-0.2}^{+0.4}$	${0.503}_{-0.178}^{+0.093}$	1.170	77
2MASS J03293773+3122024	19	1–500	6.0	3.0	3.0	10,000,000	${7.4}_{-5.8}^{+17.5}$	${0.7}_{-0.1}^{+0.2}$	${0.187}_{-0.137}^{+0.266}$	1.051	30
2MASS J03413332+3157417	17	1–500	5.0	3.0	3.0	10,000,000	${10.5}_{-8.7}^{+69.6}$	${0.6}_{-0.2}^{+0.1}$	${0.144}_{-0.111}^{+0.218}$	4.749	27
2MASS J03413641+3216200	4	1–500	4.0	3.0	3.0	10,000,000	${4.3}_{-2.6}^{+3.3}$	${2.0}_{-0.7}^{+1.3}$	${0.201}_{-0.155}^{+0.281}$	1.081	57
2MASS J03440291+3152277	12	1–500	4.0	3.0	3.0	10,000,000	${4.5}_{-2.7}^{+2.2}$	${0.9}_{-0.2}^{+0.4}$	${0.384}_{-0.259}^{+0.255}$	1.130	24
2MASS J03440599+3215321	7	1–500	4.0	3.0	3.0	10,000,000	${75.0}_{-70.4}^{+182.9}$	${1.0}_{-0.3}^{+0.6}$	${0.168}_{-0.127}^{+0.252}$	0.948	25
2MASS J04161885+2752155	4	1–500	3.0	3.0	3.0	10,000,000	${33.0}_{-30.1}^{+169.9}$	${1.3}_{-0.4}^{+1.0}$	${0.182}_{-0.141}^{+0.259}$	1.157	55
2MASS J05402570+2448090	4	1–800	4.0	3.0	3.0	1,000,000	${27.8}_{-24.8}^{+49.5}$	${2.3}_{-0.9}^{+1.2}$	${0.186}_{-0.143}^{+0.268}$	⋯	52
2MASS J09453388+5458511	10	1–800	4.0	3.0	2.0	5,000,000	${15.1}_{-12.7}^{+119.8}$	${1.0}_{-0.5}^{+0.5}$	${0.315}_{-0.244}^{+0.298}$	0.947	20
2MASS J12261350+5605445	5	1–500	4.0	3.0	3.0	10,000,000	${14.9}_{-10.5}^{+48.3}$	${1.4}_{-0.5}^{+0.9}$	${0.227}_{-0.174}^{+0.299}$	1.253	78
2MASS J12265349+2543556	8	1–100	3.0	3.0	2.0	10,000,000	${12.6}_{-9.1}^{+2.7}$	${0.9}_{-0.2}^{+0.3}$	${0.172}_{-0.132}^{+0.236}$	0.906	50
2MASS J12481860−0235360	14	1–100	3.0	3.0	2.0	10,000,000	${9.4}_{-1.8}^{+4.5}$	${0.8}_{-0.3}^{+0.4}$	${0.346}_{-0.263}^{+0.282}$	0.993	31
2MASS J13122681+7245338	10	1–2000	4.0	3.0	3.0	10,000,000	${7.9}_{-5.9}^{+1046.4}$	${1.3}_{-0.4}^{+0.8}$	${0.188}_{-0.144}^{+0.267}$	1.273	18
2MASS J13202007+7213140	12	20–150	3.0	2.0	3.0	5,000,000	${52.6}_{-16.0}^{+4.1}$	${0.9}_{-0.5}^{+0.6}$	${0.338}_{-0.213}^{+0.268}$	1.332	53
2MASS J13232423+5132272	7	1–50	3.0	2.0	2.0	1,000,000	${3.6}_{-2.0}^{+8.0}$	${0.7}_{-0.3}^{+0.4}$	${0.207}_{-0.154}^{+0.332}$	1.090 and 0.918	23
2MASS J13495109+3305136	13	1–100	5.0	3.0	3.0	10,000,000	${2.1}_{-0.7}^{+2.1}$	${2.0}_{-0.7}^{+1.0}$	${0.25}_{-0.194}^{+0.32}$	0.927	40
2MASS J14005977+3226109	16	1–600	3.0	3.0	3.0	5,000,000	${1.4}_{-0.3}^{+321.0}$	${1.5}_{-0.4}^{+1.0}$	${0.419}_{-0.255}^{+0.258}$	0.986	48
2MASS J15010818+2250020	5	5–50	3.0	3.0	2.0	1,000,000	${23.3}_{-11.0}^{+1.3}$	${1.2}_{-0.3}^{+0.7}$	${0.189}_{-0.142}^{+0.284}$	1.661	92
2MASS J16271825+3538347	4	60–200	5.0	4.0	3.0	5,000,000	${69.7}_{-7.4}^{+17.6}$	${3.4}_{-1.3}^{+2.1}$	${0.524}_{-0.238}^{+0.175}$	0.976	41
2MASS J22551142+1442456	5	1–50	3.0	3.0	2.0	100,000	${2.8}_{-1.1}^{+4.8}$	${1.6}_{-0.5}^{+0.9}$	${0.178}_{-0.135}^{+0.256}$	1.091	18
ΔBIC ≤ 10: strong candidate multiple systems
2MASS J00381273+3850323	6	1–1000	3.0	3.0	3.0	10,000,000	${191.6}_{-174.2}^{+441.8}$	${0.9}_{-0.3}^{+0.7}$	${0.183}_{-0.141}^{+0.267}$	1.053	10
2MASS J04230607+2801194	6	1–500	3.0	3.0	3.0	10,000,000	${8.1}_{-5.7}^{+25.6}$	${0.7}_{-0.3}^{+0.5}$	${0.192}_{-0.148}^{+0.274}$	1.036	7
2MASS J09373349+5534057	20	1–100	4.0	3.0	2.0	5,000,000	${5.7}_{-3.9}^{+13.5}$	${0.4}_{-0.2}^{+0.3}$	${0.271}_{-0.199}^{+0.297}$	1.103 and ⋯	7

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The rotational broadening values inferred from our fits are directly related to rotation period, size, and inclination. While these cannot be disentangled from the spectral measurements alone, some constraints can be made on size and viewing geometry when an independent measure of rotation period, such as photometric variability, is available. The projected radius $R\sin i$ in particular can be inferred directly from $v\sin i$ and period as

$$\begin{eqnarray}&&\displaystyle \frac{R\sin i}{{R}_{\odot }}=0.0198\displaystyle \frac{P}{\mathrm{day}}\displaystyle \frac{v\sin i}{\mathrm{km}\,{{\rm{s}}}^{-1}}.\end{eqnarray} \tag{ 6 }$$

Thanks to the K2 mission (Howell et al. 2014), the Transiting Exoplanet Survey Satellite (Ricker et al. 2015), and ground-based photometric monitoring programs, rotational periods have been measured for several UCDs in our APOGEE sample, in particular for members of the Upper Scorpius cluster. Noting that roughly one-half of our sample is kinematically associated with nearby young clusters with known ages, these measurements can be used to examine radius evolution as a function of time to test evolutionary models (Jackson & Jeffries 2010).

We have compiled period measurements from the literature for 78 APOGEE sources, listed in Table 11. These include 64 young cluster members and 14 field objects. For rotation periods reported without uncertainties, we assumed relative uncertainties of 5%. We note that for these rotation period measurements, the young sources (median period of 0.88 days) have longer rotation periods than the field sources (median period of 0.49 days; Figure 16).

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Table 11. Inferred Projected Radii, Inclination, and Literature Period Measurements

APOGEE ID	Cluster a	Age b	$v\sin i$	Period c	$R\sin i$	Inclination	Period References
		(Myr)	(km s−1)	(days)	(R⊙)	(deg)
2MASS J00452143+1634446	ARG	40–50	31.6 ± 1.0	0.1 ± 0.004	0.06 ± 0.0	23 ± 3	(14)
2MASS J01243124−0027556	field	⋯	27.7 ± 1.2	0.555	0.3 ± 0.03	⋯	(2)
2MASS J03040207+0045512	HYA	750 ± 100	20.0 ± 1.0	1.293	0.51 ± 0.06	⋯	(6)
2MASS J04110642+1247481	HYA	750 ± 100	14.2 ± 1.1	0.897	0.25 ± 0.02	⋯	(5)
2MASS J04214435+2024105	HYA	750 ± 100	32.8 ± 1.4	0.34	0.22 ± 0.02	⋯	(9, 11)
2MASS J04214955+1929086	HYA	750 ± 100	44.4 ± 1.6	0.205	0.18 ± 0.01	⋯	(5)
2MASS J04254894+1852479	field	⋯	31.0 ± 1.0	0.419	0.26 ± 0.02	⋯	(1)
2MASS J04305718+2556394	TAU	1–2	16.0 ± 1.0	1.157	0.36 ± 0.03	60 ± 19	(13)
2MASS J04322329+2403013	TAU	1–2	11.1 ± 1.0	3.364	0.73 ± 0.08	⋯	(13)
2MASS J04330945+2246487	TAU	1–2	16.0 ± 1.0	3.492	1.11 ± 0.1	⋯	(13)
2MASS J04340619+2418508	TAU	1–2	29.5 ± 1.8	0.711	0.41 ± 0.03	60 ± 18	(13)
2MASS J04350850+2311398	TAU	1–2	7.3 ± 1.2	1.498	0.21 ± 0.04	⋯	(13)
2MASS J04351354+2008014	HYA	750 ± 100	23.7 ± 1.1	0.37	0.17 ± 0.01	⋯	(9, 11)
2MASS J04354183+2234115	TAU	1–2	48.2 ± 1.2	0.688	0.65 ± 0.04	⋯	(13)
2MASS J04361038+2259560	TAU	1–2	10.4 ± 1.0	2.933	0.6 ± 0.07	⋯	(13)
2MASS J04363893+2258119	TAU	1–2	16.8 ± 1.0	0.964	0.32 ± 0.03	52 ± 13	(13)
2MASS J04385871+2323595	TAU	1–2	51.2 ± 1.1	0.664	0.67 ± 0.04	⋯	(13)
2MASS J04440164+1621324	TAU	1–2	14.3 ± 1.2	2.172	0.61 ± 0.07	⋯	(13)
2MASS J04464498+2436404	HYA	750 ± 100	17.9 ± 1.0	0.66	0.23 ± 0.01	⋯	(9, 11)
2MASS J05352501−0509095	ORION	≲3	17.9 ± 1.1	1.48	0.52 ± 0.04	⋯	(1)
2MASS J05353193−0531477	ORION	≲3	15.8 ± 1.1	4.36	1.36 ± 0.12	⋯	(1)
2MASS J05402570+2448090	ARG	40–50	30.4 ± 1.0	0.294	0.18 ± 0.01	70 ± 9	(6)
2MASS J07464256+2000321	field	⋯	34.6 ± 1.0	0.086	0.06 ± 0.0	36 ± 5	(3)
2MASS J08072607+3213101	field	⋯	13.7 ± 1.0	0.345	0.09 ± 0.01	61 ± 11	(6)
2MASS J08294949+2646348	CMG	200	11.9 ± 1.0	0.459	0.11 ± 0.01	57 ± 11	(6)
2MASS J10372897+3011117	field	⋯	13.1 ± 1.1	1.012	0.26 ± 0.03	⋯	(6)
2MASS J13022083+3227103	field	⋯	25.7 ± 1.0	0.4	0.2 ± 0.02	⋯	(6)
2MASS J13564148+4342587	field	⋯	16.2 ± 1.1	0.477	0.15 ± 0.01	⋯	(6)
2MASS J14320849+0811313	field	⋯	9.2 ± 1.1	0.757	0.14 ± 0.02	⋯	(6)
2MASS J15010818+2250020	field	⋯	65.3 ± 1.1	0.082	0.11 ± 0.01	69 ± 8	(4)
2MASS J15555600−2045187	USCO	10 ± 3	19.9 ± 1.1	1.7	0.67 ± 0.05	⋯	(7)
2MASS J15560104−2338081	USCO	10 ± 3	13.0 ± 1.3	1.505	0.39 ± 0.03	⋯	(7)
2MASS J15560497−2106461	USCO	10 ± 3	92.8 ± 1.5	0.26	0.48 ± 0.03	⋯	(1)
2MASS J15591135−2338002	USCO	10 ± 3	16.1 ± 1.5	1.216	0.39 ± 0.04	⋯	(1)
2MASS J15592591−2305081	USCO	10 ± 3	23.5 ± 1.0	0.62	0.29 ± 0.02	70 ± 10	(7)
2MASS J16003023−2334457	USCO	10 ± 3	73.6 ± 1.3	0.448	0.65 ± 0.04	⋯	(7)
2MASS J16014955−2351082	USCO	10 ± 3	38.7 ± 1.8	0.527	0.4 ± 0.02	⋯	(7)
2MASS J16020429−2050425	USCO	10 ± 3	57.1 ± 1.2	0.422	0.48 ± 0.03	⋯	(7)
2MASS J16044303−2318258	USCO	10 ± 3	79.3 ± 1.9	0.208	0.33 ± 0.02	⋯	(1)
2MASS 16063110−1904576	USCO	10 ± 3	16.7 ± 1.1	2.301	0.76 ± 0.09	⋯	(7)
2MASS J16090168−2740521	USCO	10 ± 3	59.3 ± 1.4	0.306	0.36 ± 0.02	⋯	(7)
2MASS J16090197−2151225	USCO	10 ± 3	47.8 ± 1.5	0.269	0.25 ± 0.01	62 ± 11	(7)
2MASS J16090451−2224523	USCO	10 ± 3	14.1 ± 1.0	2.181	0.61 ± 0.06	⋯	(7)
2MASS J16093019−2059536	USCO	10 ± 3	17.9 ± 1.0	1.593	0.57 ± 0.03	⋯	(7)
2MASS J16095107−2722418	USCO	10 ± 3	56.1 ± 1.4	0.543	0.6 ± 0.03	⋯	(7)
2MASS J16095217−2136277	USCO	10 ± 3	44.1 ± 1.0	0.702	0.61 ± 0.03	⋯	(7)
2MASS J16095852−2345186	USCO	10 ± 3	34.7 ± 1.7	1.411	0.96 ± 0.07	⋯	(7)
2MASS J16095990−2155424	USCO	10 ± 3	22.0 ± 1.8	0.874	0.38 ± 0.03	⋯	(7)
2MASS J16100541−1919362	USCO	10 ± 3	16.2 ± 1.0	2.552	0.81 ± 0.07	⋯	(1)
2MASS J16105499−2126139	USCO	10 ± 3	57.8 ± 1.2	0.52	0.59 ± 0.03	⋯	(7)
2MASS J16111711−2217173	USCO	10 ± 3	63.9 ± 1.5	0.362	0.46 ± 0.03	⋯	(7)
2MASS J16113837−2307072	USCO	10 ± 3	36.6 ± 1.1	0.72	0.52 ± 0.03	⋯	(7)
2MASS J16114261−2525511	USCO	10 ± 3	55.3 ± 1.1	0.629	0.69 ± 0.04	⋯	(7)
2MASS J16115439−2236491	USCO	10 ± 3	50.3 ± 1.5	0.484	0.48 ± 0.03	⋯	(7)
2MASS J16122703−2013250	USCO	10 ± 3	30.9 ± 1.0	0.888	0.54 ± 0.03	⋯	(7)
2MASS J16124692−2338408	USCO	10 ± 3	60.4 ± 1.4	0.284	0.34 ± 0.02	⋯	(7)
2MASS J16124726−1903531	USCO	10 ± 3	29.0 ± 1.1	1.188	0.68 ± 0.05	⋯	(7)
2MASS J16131211−2305031	USCO	10 ± 3	23.5 ± 1.0	1.154	0.54 ± 0.04	⋯	(7)
2MASS J16132665−2230348	USCO	10 ± 3	19.9 ± 1.9	1.532	0.6 ± 0.06	⋯	(7)
2MASS J16132809−1924524	USCO	10 ± 3	23.1 ± 1.0	1.512	0.69 ± 0.06	⋯	(7)
2MASS J16134027−2233192	USCO	10 ± 3	14.1 ± 1.0	1.716	0.48 ± 0.02	⋯	(1)
2MASS J16134079−2219459	USCO	10 ± 3	19.4 ± 1.0	1.336	0.51 ± 0.03	⋯	(7)
2MASS J16141974−2428404	USCO	10 ± 3	57.4 ± 1.3	0.346	0.39 ± 0.02	⋯	(7)
2MASS J16143287−2242133	USCO	10 ± 3	18.0 ± 1.0	1.823	0.65 ± 0.03	⋯	(7)
2MASS J16152516−2144013	USCO	10 ± 3	16.0 ± 1.0	1.744	0.55 ± 0.04	⋯	(7)
2MASS 16155507−2444365	USCO	10 ± 3	16.0 ± 1.0	2.007	0.63 ± 0.05	⋯	(7)
2MASS J16235470−2438319	USCO	10 ± 3	19.8 ± 1.0	1.765	0.69 ± 0.05	⋯	(7)
2MASS J16262152−2426009	USCO	10 ± 3	23.1 ± 1.1	2.497	1.14 ± 0.08	⋯	(7)
2MASS J16265619−2213519	USCO	10 ± 3	58.7 ± 1.4	0.284	0.33 ± 0.02	⋯	(7)
2MASS J16272658−2425543	ROPH	<2	12.3 ± 1.4	2.884	0.7 ± 0.09	69 ± 10	(7)
2MASS J16281707+1334204	field	⋯	15.8 ± 1.1	0.603	0.19 ± 0.02	⋯	(6)
2MASS J16281808−2428358	ROPH	<2	16.2 ± 1.1	0.796	0.25 ± 0.02	43 ± 22	(7)
2MASS J16311879+4051516	field	⋯	14.8 ± 1.0	0.512	0.15 ± 0.02	⋯	(6)
2MASS J16402068+6736046	field	⋯	16.0 ± 1.0	0.378	0.12 ± 0.01	74 ± 7	(6)
2MASS J17071830+6439331	field	⋯	25.4 ± 1.0	0.151	0.08 ± 0.0	48 ± 7	(15)
2MASS J21272531+5553150	CARN	∼200	17.3 ± 1.0	0.54	0.18 ± 0.02	⋯	(6)
2MASS J21381698+5257188	field	⋯	40.6 ± 1.0	0.183	0.15 ± 0.01	⋯	(6)
2MASS J22021125−1109461	ABDMG	${149}_{-19}^{+51}$	21.3 ± 1.0	0.428	0.18 ± 0.01	⋯	(1)

Notes.

^a Young moving group name abbreviation mostly follows those defined in Gagné et al. (2018): AB Doradus (ABDMG), Argus (ARG), Castor (Castor), Corona Australis (CRA), Carina Near (CARN), Hyades (HYA), ρ Ophiuchi (ROPH), Taurus (TAU), and Upper Scorpius (USCO). The Orion Nebula cluster is labeled as ORION. ^b The age of field dwarfs is assumed between 1 and 10 Gyr. See Section 5.6 for details. ^c We assume a 5% uncertainty for each period without a reported uncertainty. See Section 5.6 for details.

References: (1) Watson et al. (2006); (2) Ivezić et al. (2007); (3) Berger et al. (2009); (4) Crossfield (2014); (5) Douglas et al. (2019); (6) Newton et al. (2016); (7) Rebull et al. (2018); (8) Reiners et al. (2018); (9) Douglas et al. (2019); (10) Vos et al. (2019); (11) Freund et al. (2020); (12) Nardiello (2020); (13) Rebull et al. (2020); (14) Vos et al. (2020); and (15) Rockenfeller et al. (2006).

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Figure 16 illustrates our sample with both $v\sin i$ and rotational periods at different ages. The spin-up trend in rotational periods has been reported in previous studies (Popinchalk et al. 2021; Vos et al. 2022). Our slower $v\sin i$ trend toward older age (10 Myr versus >100 Myr) in our sample is consistent with previous studies (Zapatero Osorio et al. 2006) using their sample of masses between ∼30 and 70 M_Jup. Note that our sample has limited measurements for objects between Upper Scorpius (10 Myr) and Hyades (750 Myr) ages, and the ages are mostly unavailable for late-M dwarfs with $v\sin i$ measurements (Crossfield 2014; Vos et al. 2017; Jeffers et al. 2018; Hsu et al. 2021a).

While opposite to our previously observed $v\sin i$ trend (larger rotations speeds for younger sources; Figure 16), the larger radii of young objects are an important factor. We computed the projected radii of each source using Equation (6), propagating uncertainties by the Monte Carlo method. Since $v\sin i$ values close to our detection limit could potentially yield overestimated projected radii, we conservatively constrained our analysis to sources with $v\sin i$ > 20 km s⁻¹, which encompasses 41 sources in total, including 33 young sources (25 sources in the Upper Scorpius cluster) and seven field objects. The resulting $R\sin i$ values as a function of age are shown in Figure 17.

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Overall, the projected radii are larger for the younger sources (median $R\sin i$ = 0.47 R_⊙) compared to the field objects (median $R\sin i$ = 0.15 R_⊙), and show a consistent decline with age from Taurus at 1–2 Myr, to Upper Scorpius at 10 ± 3 Myr, the Hyades at 750 ± 100 Myr (Brandt & Huang 2015), and field ages (assumed as 5 Gyr). The observed projected radii are qualitatively consistent with the evolutionary models, with predicted values from Baraffe et al. (2003) for the corresponding ages provided in Table 12.

Table 12. Projected and Estimated Radii for Variable Ultracool Dwarfs

Cluster	Age	SpT	RBurrows evol a	RBaraffe evol a	N	$\langle R\sin i\rangle$	References
			(R⊙)	(R⊙)		(R⊙)
Taurus	1–2 Myr	M6–M8	0.265–0.468	0.384–0.665	3	${0.654}_{-0.164}^{+0.012}$	(1)
Upper Scorpius	10 ± 3 Myr	M6–M7.5	0.243–0.344	0.261–0.404	25	${0.48}_{-0.14}^{+0.20}$	(2)
Argus	40–50 Myr	L2	0.135–0.152	0.143–0.153	1	0.062 ± 0.003	(3)
AB Doradus	${149}_{-19}^{+51}$ Myr	M6	0.129–0.136	0.149–0.158	1	0.18 ± 0.01	(4)
Hyades	750 ± 100 Myr	M7–M7.5	0.102–0.105	0.114–0.118	3	${0.180}_{-0.005}^{+0.027}$	(5)
Field	1–10 Gyr	M6–L0	0.094–0.111	0.098–0.125 b	7	${0.15}_{-0.07}^{+0.11}$	(6)

Notes.

^a Based on the Burrows et al. (2001) or Baraffe et al. (2003) evolutionary models for the T_eff range of the sample sources (from the sample spectral types) and individual cluster ages. ^b The highest effective temperature available for Baraffe et al. (2003) is 2786 K at 0.1 M_⊙, corresponding to 0.125 R_⊙.

References: (1) Kenyon & Hartmann (1995); (2) Pecaut & Mamajek (2016); (3) Zuckerman (2019); (4) Bell et al. (2015); (5) Brandt & Huang (2015); and (6) see references in the main text.

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However, we find that over half (26 out of 41 sources) of the observed projected radii over the model radii are larger than 1, with the median radii from theoretical models underestimated by 25%. While our sample is yet small, our inferred $R\sin i$ imply that the radius inflation extends to the very low-mass stars and (young) brown dwarfs (M ∼ 0.05–0.1 M_⊙ using the Baraffe et al. 2003 models).

The radius inflation of higher-mass M dwarfs (M = 0.1–0.6 M_⊙) is widely reported in the literature (ΔR/R ∼ 5%–25%; e.g., Stassun et al. 2012; Jackson & Jeffries 2014; Mann et al. 2015; Venuti et al. 2017; Jackson et al. 2018; Kesseli et al. 2018; Parsons et al. 2018; Khata et al. 2020; Kiman et al. 2024; Swayne et al. 2024). For these higher-mass M dwarfs, the radius inflation is found to be independent of age, metallicity, or multiplicity (Mann et al. 2015). There is correlation between radius inflation and magnetic activity reported in Stassun et al. (2012) and Kiman et al. (2024) for single M dwarfs for M = 0.5–0.6 M_⊙, but no clear trend has been identified in lower mass regime between M = 0.1–0.5 M_⊙. Possible culprits in the theoretical models could be due to opacities and convective modeling (convective overshooting and mixing length; Mann et al. 2015).

For very low-mass stars and brown dwarfs (M ≲ 0.1 M_⊙), the radius inflation issue was also reported but with a much smaller sample size. Although very low-mass individual eclipsing binaries might present consistent (Triaud et al. 2020; Davis et al. 2024) or inflated (Casewell et al. 2020; Buzard et al. 2022) radii compared to theoretical models, the observed radii as a population could appear to inflate by ∼10%–30% (Kesseli et al. 2018; Triaud et al. 2020; Cañas et al. 2022; Carmichael 2023; Davis et al. 2024). Therefore, our empirical projected radii also support that the radius inflation issue extends to masses M ∼ 0.05–0.1 M_⊙, using our sample of 41 very low-mass objects compared to the literature measurements (≲50). While there might be additional systematic uncertainties for $v\sin i$ and rotational periods, the theoretical radii are likely to be underestimated. The difference between the Burrows et al. (2001) and Baraffe et al. (2003) models can be more than 40%. We emphasize that our observed sample of projected radii is small for the majority of the clusters, and a larger sample of projected radii is needed to quantify the underestimated radii across different ages and spectral types.