HIGH-PRECISION DYNAMICAL MASSES OF VERY LOW MASS BINARIES

Characterizing the fundamental properties of brown dwarfs is an important step in unlocking the physics of substellar objects. These very cool objects have internal and atmospheric properties that are quite similar to gas giant planets and that differ fundamentally from those of stars, including partially degenerate interiors, dominant molecular opacities, and atmospheric dust formation (Chabrier & Baraffe 2000). Brown dwarfs also represent a substantial fraction of the galactic stellar content, and are bright and numerous enough to be studied in great detail with current technology (Kirkpatrick 2005). Thus, these substellar objects present an ideal laboratory in which to study the physical processes at work in very low mass (VLM) objects that both approach and overlap the planetary mass regime.

Mass is the most fundamental parameter in determining the properties and evolution of a brown dwarf; unfortunately, it is also one of the most difficult to measure. Masses of brown dwarfs are typically inferred from the comparison of measured luminosities and temperatures with predictions from theoretical models. The most commonly used models are those of Burrows et al. (1997) and Chabrier et al. (2000). However, as shown in Figure 1, masses obtained in this way from different models can be discrepant, especially amongst the lowest mass objects. These discrepancies stem from physical assumptions about the interior and atmospheric properties of these highly complex objects. Examples of uncertainties in the models are atmospheric processes that define the transition regions between spectral types. Specifically, this includes the formation of atmospheric clouds (M to L), the disappearance of clouds (L to T), and the formation of ammonia, which makes the objects, in theory, similar atmospherically to Jupiter (T to Y). Additional sources of uncertainties in the models include, but are not limited to, the equation of state (Chabrier & Baraffe 2000), the initial conditions and accretion history (Baraffe et al. 2009), the treatment of convection and the possible subsequent generation of magnetic fields, which in turn could affect the inferred effective temperatures (Chabrier & Kuker 2006; Browning 2008). An essential step toward properly calibrating these models and constraining their physics is to obtain high precision (≲10%) dynamical mass measurements of brown dwarf binaries.

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The advent of laser guide star (LGS) adaptive optics (AO) on large ground-based telescopes has dramatically increased the number of VLM objects for which high-precision dynamical masses can be obtained. Prior to AO, only one binary brown dwarf had sufficiently precise mass measurements to test the models and this was the case of Two Micron All Sky Survey (2MASS) J053521840546085, an eclipsing binary in Orion, which provided constraints at a very young age (∼Myr; Stassun et al. 2006). Early AO, which used natural guide stars, allowed dynamical mass estimates for two brown dwarfs (Lane et al. 2001; Zapatero Osorio et al. 2004; Simon et al. 2006; Bouy et al. 2004). With LGS AO, much fainter sources can be targeted, allowing ∼80% of known brown dwarf binaries to be observed and a much more systematic look at how the observed properties of brown dwarfs compare with the predictions of models.

To capitalize on the introduction of LGS AO on 10 m class telescopes, we initiated, in 2006, an extensive astrometric and spectroscopic monitoring campaign of 23 VLM binaries (M_tot≲ 0.2 M_☉) in the near-infrared with the Keck/LGS-AO system with the goal of obtaining precision dynamical masses. The astrometric aspect of this project is similar to the work reported on three individual targets by Liu et al. (2008) and Dupuy et al. (2009a, 2009b, 2009c). Our survey includes these targets as well as others to span a wide range of late stellar and substellar spectral types (M7.5 to T5.5) and is the first study to include radial velocity measurements for the LGS AO targets. Our relative astrometric and radial velocity measurements provide estimates of the total system mass for 15 systems. Our absolute radial velocity measurements, add estimates of the mass ratios and hence individual mass estimates for six of these systems. Altogether, this work more than doubles both the current number of system mass measurements and individual mass estimates for VLM objects and, when compared to the models, shows systematic discrepancies.

This paper is organized as follows. Section 2 describes the sample selection and Section 3 provides a description of the astrometric and spectroscopic observations. Section 4 outlines the data analysis procedures and Section 5 describes the derivation of orbital solutions. Section 6 contains the estimates of bolometric luminosities and effective temperatures for the components of the binaries. Section 7 compares the dynamical masses to the predictions from evolutionary models and the implications of our model comparison are discussed in Section 8. We summarize our findings in Section 9.

2.1. Initial Sample

The initial sample for this project was culled from Burgasser et al. (2007), which listed the 68 visual, VLM binaries known as of 2006.¹⁰ Three cuts were applied to this initial list. First, the binaries had to be observable with the Keck telescope LGS AO system, so we imposed a declination >−35° requirement, which reduced the possible number of targets to 61. Second, the operation of the LGS AO system requires a tip-tilt reference source of apparent R magnitude <18 within an arcminute of the source, and therefore VLM binaries without a suitable tip-tilt reference were also cut. This lowered the total number of observable targets to 49, 80% of the northern hemisphere sample.

Third, we required that useful dynamical mass estimates would likely be obtained within three years. To assess the required precision for our dynamical mass estimates, we calculated the predicted masses for the two most commonly used sets of evolutionary models, those of Burrows et al. (1997) and Chabrier et al. (2000), across the entire range of temperatures and luminosities spanned by both models. We calculated the percent difference between the predictions of each model with respect to the prediction of Burrows et al. (1997). The results of this assessment are shown in Figure 1, which displays in color the offset between the models across the Hertzsprung–Russell (H–R) diagram (with the discrepancies averaged in 50 K temperature increments and 0.1 log(L/L_☉ increments). As the figure demonstrates, we found that the difference in the mass predictions of the two models varied anywhere from a few percent to greater than 100%. We therefore chose a precision goal of 10% because at this level the majority of the model predictions could be distinguished and because this level of precision was reasonable to expect given our observing strategy.

To implement our third cut, a series of Monte Carlo simulations were performed. In these simulations, the total system mass for each target was assumed based on the estimated spectral types of the binary components from the original discovery papers, using the Chabrier et al. (2000) models, and held constant for all runs. Additionally, the semimajor axis of the orbit was chosen by sampling from the range of possible values between 1/2 and 2 times the original separation measurement. From these assumptions, a period was calculated, and T_o (time of periapse passage) was randomly selected from the range allowed by this period. All other orbital parameters for an astrometric orbit, which include e (eccentricity), i (inclination), Ω (longitude of the ascending node), and ω (argument of periapsis), were randomly selected from among the complete range of possible values of each parameter.

Using these simulated orbits, it was possible to generate simulated sets of "astrometric data points" corresponding to the likely times of measurement. We planned on two observing campaigns per year, one in June and the other in December. These dates were chosen to coincide with the 2 times per year that NIRSPEC is offered behind the LGS AO system at Keck (see Section 3). The sky coordinates of each binary then determined whether we simulated one or two astrometric and radial velocity data points per year. These simulated measurements were chosen to correspond to appropriate observing dates. All synthetic data points were combined with already existing measurements, the number of which varied from source to source. While the majority of sources initially had only one previous astrometric measurement, a few had up to six. Synthetic astrometric data points were also assigned uncertainties based on the average uncertainty normally obtained for short-exposure measurements of binary stars using the Keck AO system with NIRC2 (σ∼ 1 mas). Although the average uncertainty in relative radial velocity measurements with NIRSPEC+AO (NIRSPAO) was not known at the time, other observations with NIRSPEC suggested using a conservative uncertainty of about 1 km s⁻¹. All data points were then used to run the orbital solution fitter, which uses the Thiele–Innes method (e.g., Hilditch 2001), minimizing the χ² between the model and the measurements (see Ghez et al. 2008). A chi-squared cut of 10 was imposed to account for the fact that in some simulated orbital solutions we could not generate astrometric measurements corresponding to real data points in those systems with multiple measurements. In this way, we were able to utilize more information than simply separation and estimated mass to calculate likelihood of accurate mass measurement in a system. A total of 1000 simulated orbital solutions were created for each system.

From each of these simulations, the predicted uncertainty in dynamical mass could be determined. All those systems for which 66% of the simulations yielded precisions of 10% or better in mass were put in the final sample. This generated a sample of 21 targets that we began monitoring in the spring of 2006. These sources are listed in Table 1. Figure 2 shows the results of our simulations, plotting the percent of solutions with precise mass estimates versus the initial binary separation. The spectral type of the primary component is denoted by symbol shape, and sources included in our initial sample are colored red. The variation in percent of solutions with separation stems from the variation in the estimated masses of the components and the number of previous measurements at the start of our monitoring program.

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Table 1. VLM Binary Sample

Source Name	R.A.	Decl.	Estimated	Discovery	2MASS
	(J2000)	(J2000)	Sp Typesa	Reference	K-band Magnitude
LP 349-25AB	00 27 55.93	+22 19 32.8	M8+M9	11	9.569 ± 0.017
LP 415-20AB	04 21 49.0	+19 29 10	M7+M9.5	7	11.668 ± 0.020
2MASS J05185995−2828372ABb	05 18 59.95	−28 28 37.2	L6+T4	1	14.162 ± 0.072
2MASS J06523073+4710348ABb	06 52 30.7	+47 10 34	L3.5+L6.5	2	11.694 ± 0.020
2MASS J07003664+3157266	07 00 36.64	+31 57 26.60	L3.5+L6	2	11.317 ± 0.023
2MASS J07464256+2000321AB	07 46 42.5	+20 00 32	L0+L1.5	3	10.468 ± 0.022
2MASS J08503593+1057156	08 50 35.9	+10 57 16	L6+L8	3	14.473 ± 0.066
2MASS J09201223+3517429AB	09 20 12.2	+35 17 42	L6.5+T2	3	13.979 ± 0.061
2MASS J10170754+1308398ABc	10 17 07.5	+13 08 39.1	L2+L2	4	12.710 ± 0.023
2MASS J10210969−0304197	10 21 09.69	−03 04 20.10	T1+T5	13	15.126 ± 0.173
2MASS J10471265+4026437AB	10 47 12.65	+40 26 43.7	M8+L0	5	10.399 ± 0.018
GJ 569b AB	14 54 29.0	+16 06 05	M8.5+M9	12	∼9.8
LHS 2397a AB	11 21 49.25	−13 13 08.4	M8+L7.5	10	10.735 ± 0.023
2MASS J14263161+1557012AB	14 26 31.62	+15 57 01.3	M8.5+L1	5	11.731 ± 0.018
HD 130948 BC	14 50 15.81	+23 54 42.6	L4+L4	9	∼11.0
2MASS J15344984−2952274AB	15 34 49.8	−29 52 27	T5.5+T5.5	6	14.843 ± 0.114
2MASS J1600054+170832ABb	16 00 05.4	+17 08 32	L1+L3	4	14.678 ± 0.114
2MASS J17281150+3948593AB	17 28 11.50	+39 48 59.3	L7+L8	4	13.909 ± 0.048
2MASS J17501291+4424043AB	17 50 12.91	+44 24 04.3	M7.5+l0	7	11.768 ± 0.017
2MASS J18470342+5522433AB	18 47 03.42	+55 22 43.3	M7+M7.5	8	10.901 ± 0.020
2MASS J21011544+1756586	21 01 15.4	+17 56 58	L7+L8	4	14.892 ± 0.116
2MASS J21402931+1625183AB	21 40 29.32	+16 25 18.3	M8.5+L2	5	11.826 ± 0.031
2MASS J21522609+0937575	21 52 26	+09 37 57	L6+L6	2	13.343 ± 0.034
2MASS J22062280−2047058AB	22 06 22.80	−20 47 05.9	M8+M8	5	11.315 ± 0.027

Notes. ^aFrom discovery reference. ^bIn all observations of these sources, the binary was never resolved. We report upper limits to the separations of these binaries, but no orbital solutions can be derived. ^cSource cut from sample due to additional astrometry showing that it was not likely to yield a mass to a precision of better than 10% in the required timeframe. References. (1) Cruz et al. 2004; (2) Reid et al. 2006; (3) Reid et al. 2001; (4) Bouy et al. 2003; (5) Close et al. 2003; (6) Burgasser et al. 2003; (7) Siegler et al. 2003; (8) Siegler et al. 2005; (9) Potter et al. 2002; (10) Freed et al. 2003; (11) Forveille et al. 2005; (12) Martín et al. 2000; (13) Burgasser et al. 2006.

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2.2. Sample Refinement

3.1. Astrometric Data

3.2. Radial Velocity Data

4.1. Astrometric Data Analysis

The NIRC2 data were initially processed using standard data reduction techniques for near-infrared images. Frames at differing dither positions were subtracted from each other to remove sky background, followed by the removal of bad pixels, division by a flat field, and correction for optical distortion with a model provided in the pre-ship review document¹¹ using standard IRAF and IDL routines. The binaries were then shifted to a common location in all frames and the images were median combined. Astrometry and flux ratios were obtained using the IDL package StarFinder (Diolaiti et al. 2000). An empirical point-spread function (PSF) is required by the StarFinder fitting algorithm. In the case of two sources, a suitable PSF star falls within the field of view of the NIRC2 observation of the source. However, in the majority of cases, no such PSF source is in the field of view. For these observations, we use either an image of a single star taken near in time to the images of the binary, or if no suitable single star was imaged, we use an idealized Keck PSF degraded to the calculated Strehl ratio of the observation for PSF fitting. For this last case, the PSF is generated by first convolving the idealized PSF with a Gaussian, such that the core is broadened to the appropriate FWHM. Next, a simulated "halo" is generated by adding a Gaussian with FWHM of 05 (average near-infrared seeing halo at Keck), normalized such that the resulting Strehl ratio matches the observations. Internal statistical measurement errors were calculated by fitting the components of the binaries in all individual images that contributed to the combined images and finding the rms of the values derived therein.

Additional systematic uncertainties need to be accounted for when determining the final astrometric and photometric measurements for each binary. First, absolute uncertainties in the plate scale and position of north given above are accounted for in all astrometry. A further, more complicated, source of uncertainty stems from using a PSF that is not imaged simultaneously with each binary, introducing systematic uncertainties in both astrometry and photometry. In particular, the variability of the AO performance over a given night generates time variable PSF structure that can contribute to slight offsets in astrometry and photometry. To estimate the additional uncertainty due to imperfect PSF matching, we performed simulations in which 1000 artificial binaries were generated using images of image single sources with separations and flux ratios spanning the range observed for our sources. These artificial binaries were then fit with StarFinder using either a separate single source from the same night or simulated source as the PSF. This exercise was performed for every night in which observations were taken for both PSF types. Examples of the results of these simulations are shown in Figure 3 (from the night of 2006 May 21). We find in all simulations that median offsets between input and fitted separations are an exponentially decreasing function of the separation, meaning that fits to tighter binaries were more discrepant from the correct values than those to wide binaries. We also find that due to variable structure in AO PSF halos (even after accounting for pupil rotation), the offset in fitted P.A. is a function of the P.A. of the binary. Finally, the median offset in fitted flux ratio with respect to the input flux ratio was essentially constant for all separations and P.A.s. Therefore, for every measurement of each target, we compute the necessary uncertainties from imperfect PSFs based on the relationships detailed above, taking the median values of the measured offsets as the magnitude of the additional uncertainty. The PSF uncertainties have the greatest impact on the tightest systems or on nights when the performance was poor (Strehl ratios ≲20%). In about 25% of our measurements, this PSF uncertainty is larger than our statistical uncertainty. The astrometry and relative photometry for all sources at all epochs is given in Table 4. Those sources that were unresolved in our observations have upper limits on binary separation only. Uncertainties in Table 4 are listed separately for the purpose of illustrating the relative magnitude of each source of uncertainty, but for all further analyses, we add them together in quadrature to give a final uncertainty.

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Table 4. NIRC2 LGS AO Results

Target	Date of	Filter	Separation	Separation	Position Angle	Flux Ratio
Name	Observation (UT)		(pixels)a	(arcseconds)b	(degrees)c	(Ab/Aa)d
2MASS 0518−28	2006 Nov 27	Kp	<6.40	<0.064	...	...
	2007 Dec 2	Kp	<5.55	<0.055	...	...
	2008 Dec 18	Kp	<6.55	<0.065	...	...
2MASS 0652+47	2006 Nov 27	Kp	<3.05	<0.030	...	...
	2007 Dec 2	Kp	<3.73	<0.037	...	...
2MASS 0700+31	2008 Dec 18	Kp	19.999 ± (0.035 ± 0.013)	0.1993 ± 0.0004	279.21 ± (0.14 ± 0.02) [0.14]	3.48 ± (0.10 ± 0.03)
2MASS 0746+20	2006 Nov 27	Kp	29.924 ± (0.058 ± 0.013)	0.2981 ± 0.0006	233.93 ± (0.08 ± 0.03) [0.08]	1.39 ± (0.02 ± 0.04)
	2007 Nov 30	Kp	33.533 ± (0.028 ± 0.057)	0.3341 ± 0.0007	223.54 ± (0.04 ± 0.22) [0.23]	1.39 ± (0.01 ± 0.03)
	2007 Nov 30	J	33.501 ± (0.165 ± 0.072)	0.334 ± 0.002	223.49 ± (0.36 ± 0.26) [0.45]	1.60 ± (0.14 ± 0.04)
	2008 Dec 18	Kp	35.240 ± (0.032 ± 0.002)	0.3511 ± 0.0004	214.31 ± (0.06 ± 0.02) [0.07]	1.39 ± (0.01 ± 0.03)
	2008 Dec 18	H	35.268 ± (0.036 ± 0.039)	0.3514 ± 0.0006	214.38 ± (0.10 ± 0.07) [0.13]	1.50 ± (0.01 ± 0.04)
2MASS 0850+10	2007 Dec 1	Kp	8.927 ± (0.197 ± 0.215)	0.089 ± 0.003	158.71 ± (0.93 ± 0.13) [0.93]	1.81 ± (0.19 ± 0.04)
	2008 Dec 18	Kp	7.611 ± (0.086 ± 0.073)	0.076 ± 0.001	165.87 ± (0.36 ± 0.12) [0.37]	2.12 ± (0.10 ± 0.03)
2MASS 0920+35	2006 Nov 27	Kp	6.583 ± (0.194 ± 0.406)	0.066 ± 0.004	247.14 ± (2.04 ± 0.06) [2.04]	1.36 ± (0.08 ± 0.04)
	2007 Dec 1	Kp	7.561 ± (0.292 ± 0.232)	0.075 ± 0.004	244.91 ± (3.23 ± 0.27) [3.24]	1.19 ± (0.07 ± 0.04)
	2007 Dec 1	J	6.623 ± (1.50 ± 0.66)	0.066 ± 0.016	247.7 ± (1.6 ± 0.2) [1.6]	1.04 ± (0.27 ± 0.05)
	2008 May 30	Kp	6.622 ± (0.079 ± 0.057)	0.066 ± 0.001	249.94 ± (0.53 ± 0.09) [0.54]	1.75 ± (0.09 ± 0.03)
	2008 Oct 20	H	4.714 ± 0.145	0.047 ± 0.001	252.3 ± 3.0	1.20 ± 0.07
	2008 Dec 18	H	3.753 ± 0.335	0.037 ± 0.003	247.6 ± 1.8	1.07 ± 0.05
	2009 Jun 10	H	< 2.63	< 0.0262	...	...
2MASS 1017+13	2006 Nov 27	Kp	8.777 ± (2.403 ± 0.295)	0.087 ± 0.024	83.11 ± (4.98 ± 0.06) [4.98]	1.27 ± (0.63 ± 0.04)
2MASS 1021−03	2008 Dec 18	Kp	14.923 ± (0.032 ± 0.026)	0.1487 ± 0.0004	204.13 ± (0.13 ± 0.02) [0.13]	2.52 ± (0.03 ± 0.03)
2MASS 1047+40	2006 Jun 21	Kp	3.178 ± (0.169 ± 0.153)	0.032 ± 0.002	126.77 ± (4.44 ± 0.05) [4.44]	1.52 ± (0.26 ± 0.02)
	2006 Nov 27	Kp	<4.68	<0.047	...	...
	2007 Dec 2	Kp	<4.68	<0.047	...	...
2MASS 1426+15	2006 Jun 19	Kp	26.565 ± (0.054 ± 0.018)	0.265 ± 0.001	343.07 ± (0.47 ± 0.04) [0.47]	1.81 ± (0.10 ± 0.02)
	2008 May 30	Kp	30.562 ± (0.043 ± 0.015)	0.3045 ± 0.0005	343.55 ± (0.06 ± 0.03) [0.07]	1.82 ± (0.02 ± 0.03)
	2008 May 30	H	30.479 ± (0.107 ± 0.071)	0.304 ± 0.001	343.53 ± (0.28 ± 0.22) [0.36]	2.02 ± (0.05 ± 0.02)
	2009 May 2	Kp	32.389 ± (0.046 ± 0.015)	0.3227 ± 0.0005	343.69 ± (0.06 ± 0.05) [0.08]	1.84 ± (0.02 ± 0.04)
	2009 May 2	H	32.375 ± (0.038 ± 0.017)	0.3226 ± 0.0006	343.84 ± (0.06 ± 0.05) [0.08]	1.91 ± (0.02 ± 0.04)
2MASS 1534−29	2006 Jun 19	J	18.649 ± 0.125	0.186 ± 0.001	15.57 ± 0.29	1.20 ± 0.04
	2008 May 30	Kp	9.571 ± 0.121	0.095 ± 0.001	21.53 ± 0.84	1.23 ± 0.13
	2008 May 30	H	9.549 ± 0.131	0.095 ± 0.001	21.69 ± 0.82	1.38 ± 0.09
	2009 May 4	H	3.919 ± 0.118	0.039 ± 0.001	38.52 ± 3.25	1.28 ± 0.11
2MASS 1600+17	2007 May 20	Kp	<4.02	<0.040	...	...
	2008 May 30	Kp	<3.90	<0.039	...	...
2MASS 1728+39	2007 May 20	Kp	20.496 ± (0.138 ± 0.030)	0.204 ± 0.001	85.08 ± (0.21 ± 0.14) [0.25]	1.83 ± (0.03 ± 0.02)
	2008 May 30	Kp	20.790 ± (0.589 ± 0.026)	0.207 ± 0.006	101.33 ± (0.13 ± 0.03) [0.14]	1.97 ± (0.14 ± 0.03)
	2008 May 30	J	21.467 ± (0.045 ± 0.165)	0.214 ± 0.002	101.85 ± (0.12 ± 0.25) [0.28]	1.34 ± (0.02 ± 0.02)
	2009 May 3	Kp	21.854 ± (0.019 ± 0.110)	0.218 ± 0.001	105.85 ± (0.48 ± 0.15) [0.50]	1.74 ± (0.02 ± 0.02)
	2009 Jun 11	H	21.868 ± 0.034	0.218 ± 0.0004	106.41 ± 0.08	1.52 ± 0.01
2MASS 1750+44	2006 Jun 19	Kp	15.392 ± (0.443 ± 0.050)	0.153 ± 0.004	33.68 ± (2.47 ± 0.07) [2.47]	1.94 ± (0.13 ± 0.02)
	2007 May 17	Kp	17.330 ± (0.161 ± 0.022)	0.173 ± 0.002	42.37 ± (0.28 ± 0.02) [0.28]	1.93 ± (0.02 ± 0.04)
	2008 May 13	Kp	18.556 ± (0.020 ± 0.057)	0.1849 ± 0.0006	52.29 ± (0.05 ± 0.06) [0.08]	1.84 ± (0.02 ± 0.03)
	2008 May 30	J	19.321 ± (0.173 ± 0.201)	0.192 ± 0.003	53.78 ± (0.32 ± 0.16) [0.35]	2.41 ± (0.03 ± 0.02)
	2008 May 30	H	18.575 ± (0.115 ± 0.394)	0.185 ± 0.004	52.64 ± (0.91 ± 1.77) [1.99]	2.04 ± (0.11 ± 0.19)
	2009 May 1	Kp	20.2779 ± (0.035 ± 0.011)	0.2020 ± 0.0004	60.31 ± (0.06 ±0.02) [0.07]	1.83 ± (0.01 ± 0.02)
2MASS 1847+55	2006 May 21	Kp	15.289 ± (0.032 ± 0.076)	0.1523 ± 0.0008	110.90 ± (0.03 ± 0.01) [0.04]	1.30 ± (0.003 ± 0.02)
	2007 May 14	Kp	17.335 ± (0.039 ± 0.059)	0.173 ± 0.007	114.01 ± (0.08 ± 0.04) [0.09]	1.28 ± (0.01 ± 0.02)
	2008 May 20	Kp	19.202 ± (0.147 ± 0.052)	0.191 ± 0.002	116.71 ± (0.44 ± 0.04) [0.44]	1.28 ± (0.04 ± 0.02)
	2008 May 20	J	18.892 ± (0.133 ± 0.209)	0.188 ± 0.003	116.61 ± (0.28 ± 0.15) [0.32]	1.25 ± (0.01 ± 0.10)
	2008 May 20	H	19.245 ± (0.054 ± 0.389)	0.192 ± 0.004	116.64 ± (0.15 ± 2.71) [2.72]	1.30 ± (0.03 ± 0.19)
	2009 May 4	Kp	20.726 ± (0.057 ± 0.008)	0.2065 ± 0.0006	118.74 ± (0.13 ± 0.01) [0.14]	1.28 ± (0.02 ± 0.03)
2MASS 2101+17	2008 May 15	Kp	32.405 ± (0.047 ± 0.014)	0.3229 ± 0.0005	94.47 ± (0.09 ± 0.04) [0.11]	1.31 ± (0.02 ± 0.03)
2MASS 2140+16	2006 May 21	Kp	10.922 ± 0.061	0.1088 ± 0.0006	202.91 ± 0.54 ± 0	1.97 ± 0.04
	2006 Nov 27	Kp	10.803 ± 0.126	0.108 ± 0.001	215.02 ± 1.16	1.94 ± 0.12
	2007 May 14	Kp	10.816 ± 0.044	0.1078 ± 0.0004	223.50 ± 0.25	1.96 ± 0.05
	2007 Dec 1	Kp	10.879 ± 0.209	0.108 ± 0.002	234.02 ± 0.66	1.95 ± 0.12
	2008 May 15	Kp	11.067 ± 0.096	0.111 ± 0.001	243.28 ± 0.56	1.96 ± 0.07
	2008 May 30	J	12.021 ± 0.173	0.120 ± 0.002	241.41 ± 0.45	2.39 ± 0.34
	2008 May 30	H	11.491 ± 0.075	0.115 ± 0.001	242.9 ± 1.6	2.35 ± 0.38
	2008 Dec 19	Kp	11.311 ± 0.390	0.113 ± 0.004	254.68 ± 0.32	1.94 ± 0.19
	2009 Jun 11	Kp	11.478 ± 0.113	0.114 ± 0.001	263.34 ± 0.23	1.93 ± 0.09
2MASS 2152+09	2008 May 30	Kp	32.797 ± (0.413 ± 0.013)	0.327 ± 0.004	117.75 ± (1.04 ± 0.03) [1.04]	1.05 ± (0.20 ± 0.03)
2MASS 2206−20	2006 May 21	Kp	13.068 ± (0.147 ± 0.133)	0.130 ± 0.002	128.99 ± (0.27 ± 0.13) [0.27]	1.04 ± (0.05 ± 0.02)
	2006 Nov 27	Kp	12.747 ± (0.223 ± 0.165)	0.127 ± 0.003	138.65 ± (0.29 ± 0.04) [0.30]	1.06 ± (0.09 ± 0.04)
	2007 May 17	Kp	12.313 ± (0.013 ± 0.035)	0.1227 ± 0.0004	147.68 ± (0.12 ± 0.02) [0.12]	1.03 ± (0.01 ± 0.04)
	2007 Dec 1	Kp	12.199 ± (0.07 ± 0.18)	0.122 ± 0.002	160.40 ± (0.09 ± 0.27) [0.29]	0.97 ± (0.09 ± 0.04)
	2008 May 30	Kp	12.394 ± (0.084 ± 0.042)	0.1235 ± 0.0009	169.58 ± (0.34 ± 0.04) [0.34]	1.11 ± (0.11 ± 0.03)
	2008 May 30	J	11.834 ± (0.104 ± 0.404)	0.118 ± 0.004	170.49 ± (0.35 ± 0.47) [0.59]	1.15 ± (0.03 ± 0.02)
	2008 May 30	H	11.543 ± (0.838 ± 0.453)	0.115 ± 0.009	169.82 ± (0.92 ± 0.35) [0.99]	1.04 ± (0.18 ± 0.19)
	2009 Jun 11	Kp	12.588 ± (0.033 ± 0.123)	0.124 ± 0.001	190.52 ± (0.07 ± 0.03) [0.09]	1.05 ± 0.02
GJ 569B	2009 Jun 11	Kp	9.953 ± (0.047 ± 0.194)	0.099 ± 0.002	79.04 ± (0.20 ± 0.05) [0.20]	1.58 ± (0.02 ± 0.02)
HD 130948 BC	2006 Jun 18e	Hn3	5.401 ± 0.279	0.109 ± 0.006	136.33 ± 3.68	...
	2007 May 11	Kp	10.620 ± (0.058 ± 0.101)	0.1058 ± 0.001	131.63 ± (0.11 ± 0.03) [0.12]	1.21 ± (0.12 ± 0.03)
	2008 Apr 28	Kp	5.068 ± (0.069 ± 0.122)	0.0505 ± 0.001	122.82 ± (4.93 ± 0.05) [4.93]	1.15 ± (0.31 ± 0.04)
	2009 May 9	H	3.775 ± (0.318 ± 0.528)	0.038 ± 0.006	327.1 ± (5.0 ± 1.1) [5.1]	1.20 ± (0.13 ± 0.15)
LHS 2397a	2006 Nov 27	Kp	9.672 ± (4.976 ± 0.259)	0.096 ± 0.050	300.01 ± (9.38 ± 0.38) [9.39]	1.77 ± (0.82 ± 0.04)
	2007 Dec 1	Kp	14.629 ± (0.554 ± 0.155)	0.146 ± 0.006	19.95 ± (2.16 ± 0.13) [2.17]	10.16 ± (0.86 ± 0.04)
	2008 May 30	Kp	15.983 ± (0.758 ± 0.034)	0.159 ± 0.008	37.77 ± (1.68 ± 0.26) [1.70]	12.21 ± (1.49 ± 0.03)
	2008 Dec 18	Kp	19.813 ± (0.101 ± 0.013)	0.197 ± 0.001	50.27 ± (0.11 ± 0.04) [0.12]	13.2 ± (1.0 ± 0.03)
	2008 Dec 18	H	19.908 ± (0.276 ± 0.055)	0.196 ± 0.003	50.94 ± (0.37 ± 0.11) [0.39]	17.4 ± (1.2 ± 0.04)
	2009 Jun 10	Kp	22.026 ± (0.035 ± 0.021)	0.2195 ± 0.0004	59.44 ± (0.08 ± 0.58) [0.59]	12.89 ± (0.39 ± 0.02)
LP 349-25	2006 Nov 27	Kp	12.603 ± (0.049 ± 0.169)	0.126 ± 0.002	234.88 ± (0.17 ± 0.70) [0.72]	1.38 ± (0.02 ± 0.04)
	2006 Nov 27	J	12.439 ± (0.213 ± 0.129)	0.124 ± 0.002	236.67 ± (2.53 ± 0.06) [1.53]	1.64 ± (0.04 ± 0.04)
	2006 Nov 27	H	12.349 ± (0.093 ± 0.150)	0.123 ± 0.002	235.48 ± (0.46 ± 0.06) [0.47]	1.48 ± (0.08 ± 0.04)
	2007 Dec 1	Kp	13.126 ± (0.400 ± 0.169)	0.131 ± 0.004	211.47 ± (1.61 ± 0.23) [1.62]	1.20 ± (0.10 ± 0.04)
	2008 May 30	Kp	12.518 ± (0.120 ± 0.041)	0.125 ± 0.001	197.94 ± (0.40 ± 0.02) [0.40]	1.44 ± (0.04 ± 0.03)
	2008 Dec 19	Kp	8.555 ± (0.080 ± 0.064)	0.085 ± 0.001	172.41 ± (0.81 ± 0.08) [0.82]	1.31 ± (0.08 ± 0.03)
	2008 Dec 19	J	8.944 ± (0.364 ± 0.095)	0.089 ± 0.004	173.17 ± (0.59 ± 0.12) [0.61]	1.63 ± (0.16 ± 0.05)
	2009 Jun 11	Kp	6.653 ± (0.056 ± 0.354)	0.066 ± 0.004	129.62 ± (0.22 ± 0.04) [0.22]	1.34 ± (0.05 ± 0.02)
LP 415-20	2006 Nov 27	Kp	4.616 ± (0.083 ± 0.541)	0.046 ± 0.005	35.11 ± (2.40 ± 0.85) [2.55]	1.77 ± (0.09 ± 0.04)
	2007 Dec 1	Kp	9.617 ± (0.152 ± 0.206)	0.096 ± 0.003	52.45 ± (1.06 ± 0.12) [1.06]	2.53 ± (0.21 ± 0.04)
	2008 Dec 18	Kp	11.215 ± (0.444 ± 0.044)	0.112 ± 0.004	62.56 ± (0.97 ± 0.10) [0.97]	1.42 ± (0.12 ± 0.03)
	2008 Dec 18	H	11.145 ± (0.179 ± 0.066)	0.111 ± 0.002	62.18 ± (1.26 ± 0.04) [1.36]	1.60 ± (0.19 ± 0.04)

Notes. ^aThe first listed uncertainty is that due to the measurement itself, while the second is the systematic uncertainty due to imperfect PSF matching. If only one uncertainty is given, the source had a suitable PSF in the field of view. ^bThe uncertainties given are the empirically estimated statistical uncertainty, the PSF mismatch uncertainty, and the absolute plate scale uncertainty added in quadrature ^cThe first listed uncertainty is that due to the measurement itself, while the second is the systematic uncertainty due to imperfect PSF matching. In the brackets is the combination of these two uncertainties along with the absolute uncertainty of the columns with respect to north. If only one uncertainty is given, the source had a suitable PSF in the field of view. ^dThe first listed uncertainty is systematic uncertainty due to the measurement itself, while the second is that due to imperfect PSF matching. If only one uncertainty is given, the source had a suitable PSF in the field of view. ^eData from the OSIRIS imager, which has a plate scale of 002 pixel⁻¹. This camera has not been fully characterized for distortion. However, the uncertainties on these measurements are such that they should account for distortion on this camera.

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4.2. Spectroscopic Data Analysis

The basic reduction of the NIRSPAO spectra was performed with REDSPEC, a software package designed for NIRSPEC.¹² Object frames are reduced by subtracting opposing nods to remove sky and dark backgrounds, dividing by a flat field, and correcting for bad pixels. As mentioned above, for the purposes of this work we only analyzed order 33, which contains the CO bandhead region. This order is spatially rectified by fitting the trace of each nod of A0 calibrators with third-order polynomials, and then applying the results of those fits across the image. A first-order guess at the wavelength solution for the spectra is obtained using the etalon lamps that are part of the lamp suite of NIRSPAO (this is used as a starting point for our derivation of the true wavelength solution). Order 33 has very few OH sky lines or arc lamp lines to use for this purpose. To obtain the correct values of the wavelengths for the etalon lines, we followed the method described by Figer et al. (2003). The wavelength regime that order 33 encompasses was found to be between ∼2.291 and ∼2.325 μm. The output we used from REDSPEC was therefore a reduced, spatially rectified and preliminarily spectrally rectified fits image of order 33.

As these systems are fairly tight binaries, cross-contamination can be an issue when extracting the spectra. This made the simple square-box extraction provided in REDSPEC unsuitable for these observations. We therefore extracted the spectra by first fitting a Gaussian to the trace of one component of the binary and subtracting the result of this fit from the frame to leave only the other component. The width of the Gaussian is allowed to vary with wavelength, although over the narrow wavelength range covered by order 33, the variation is small. Typically the binaries are separated by more than a FWHM of this Gaussian, making the fit of the bright stars' trace unbiased by the other. In the few cases where the traces were separated by less than about seven pixels, the fitted FWHM would be artificially widened due to the presence of the companion. In these cases, we fixed the FWHM of the Gaussian to that measured for other, more widely separated sources observed on the same night. After the trace of one component was fitted and subtracted from the frame, the trace of the remaining component was then also fit with a Gaussian for extraction. We normalized this Gaussian such that the peak was given a value of one and corresponded to the peak of the trace in the spatial direction. We then weighted the flux of each pixel by the value of the normalized Gaussian at that pixel location, and then added these weighted fluxes together to get our final extracted spectrum. We do not remove telluric absorption from our order 33 spectra because telluric lines are used for radial velocity determination.

Radial velocities are determined from the extracted spectra relative to the telluric absorption features, which provide a stable absolute wavelength reference (e.g., Blake et al. 2007). Our specific prescription is identical to that outlined in detail in J. I. Bailey et al. (2010, in preparation); they demonstrate radial velocity precisions of 50 m s⁻¹ with NIRSPEC spectra for slowly rotating mid-M dwarfs. Here we provide only a brief overview of this method. Each extracted spectrum is modeled as a combination of a KPNO/FTS telluric spectrum (Livingston & Wallace 1991) and a synthetically generated spectrum derived from the PHOENIX atmosphere models (Hauschildt et al. 1999). The model spectrum is parameterized to account for the wavelength solution, continuum normalization, instrumental profile (assumed to be Gaussian), projected rotational velocity (V sin i)¹³ and the radial velocity. The best fit is determined by minimizing the variance-weighted reduced χ² of the difference between the model and the extracted spectrum, once this difference has been Fourier filtered to remove the fringing present in NIRSPEC K-band spectra. This fit is only done using a single order of our NIRSPEC spectra (order 33), since it uniquely contains a rich amount of both telluric and stellar absorption features, sufficient for precise calibration.

To account for any systematic effects from using a template of a given temperature, which can impact the value of v sin i and potentially cause slight shifts in the derived radial velocity, we use multiple templates spanning 300 K in temperature to determine the systematic uncertainty in radial velocity due to our synthetic template. We find that this systematic error amounts to approximately ∼0.2–0.3 km s⁻¹ for most targets.

The measured radial velocities from this method are reported in Table 5. Our uncertainties in radial velocity range from 0.4 to 2.8 km s⁻¹, consistent with the assumptions used in our original Monte Carlo simulations (Section 2). In Figure 4, we show example fits for each of the sources with spectroscopic observations (additional examples are included in online only figures).

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Table 5. Radial Velocity Measurements

Target	Date of	Average SNR	Average SNR	Radial Velocity	Radial Velocity	ΔRV
Name	Observation (UT)	Primary (A)	Secondary (B)	Primary (km s−1)	Secondary (km s−1)	(km s−1)
2MASS 0746+20AB	2006 Dec 16	52	44	55.60 ± 0.68	52.94 ± 0.68	−2.66 ± 0.96
	2007 Dec 4	72	59	55.18 ± 0.60	52.37 ± 1.12	−2.81 ± 1.27
	2008 Dec 19	66	56	56.06 ± 0.85	54.05 ± 2.30	−2.01 ± 2.45
2MASS 1426+15AB	2007 Jun 8	44	33	12.54 ± 0.43	14.41 ± 1.27	1.87 ± 1.34
	2008 Jun 1	50	36	12.67 ± 0.36	15.39 ± 1.40	2.72 ± 1.45
	2009 Jun 12	41	29	12.78 ± 0.49	15.00 ± 0.68	2.22 ± 0.84
2MASS 1750+44AB	2008 May 31	48	36	−17.52 ± 0.39	−15.89 ± 0.54	1.63 ± 0.67
	2009 Jun 12	41	31	−17.09 ± 0.53	−15.25 ± 1.31	1.84 ± 1.41
2MASS 1847+55AB	2007 Jun 8	69	60	−23.88 ± 0.32	-20.46 ± 0.29	3.42 ± 0.43
	2008 Jun 1	69	60	−24.15 ± 0.21	−20.09 ± 0.46	4.06 ± 0.51
	2009 Jun 13	39	36	−24.68 ± 0.60	−19.63 ± 1.00	5.05 ± 1.17
2MASS 2140+16AB	2007 Jun 9	43	28	13.90 ± 0.30	11.07 ± 1.21	−2.83 ± 1.25
	2008 May 31	58	40	13.62 ± 0.27	12.26 ± 1.62	−1.36 ± 1.68
	2009 Jun 13	38	26	13.47 ± 0.28	10.97 ± 2.00	2.50 ± 2.01
2MASS 2206−20AB	2007 Jun 9	47	39	13.66 ± 0.36	13.28 ± 0.48	−0.38 ± 0.66
	2008 Jun 1	54	48	13.14 ± 0.39	13.46 ± 0.51	0.32 ± 0.64
	2009 Jun 12	47	44	13.37 ± 0.24	12.75 ± 0.37	−0.62 ± 0.44
GJ 569b AB	2007 Jun 9	89	82	−10.49 ± 0.20	−4.90 ± 0.50	5.59 ± 0.54
	2009 Jun 13	86	67	−8.97 ± 0.36	−6.83 ± 0.27	2.14 ± 0.45
HD 130948Bc	2007 Jun 9	44	33	4.57 ± 2.61	−0.72 ± 1.05	−5.29 ± 2.81
LHS 2397aAB	2007 Dec 4	68	27	34.43 ± 0.86	34.84 ± 2.24	0.41 ± 2.40
	2008 May 31	114	44	33.85 ± 0.27	36.30 ± 0.86	2.45 ± 0.90
	2008 Dec 19	85	31	33.79 ± 0.37	35.30 ± 2.49	1.51 ± 2.52
	2009 Jun 12	103	33	33.51 ± 0.66	34.27 ± 2.02	0.76 ± 1.22
LP 349-25AB	2006 Dec 16	58	45	−11.91 ± 1.33	−6.57 ± 2.50	5.34 ± 2.83
	2007 Dec 4	63	58	−11.11 ± 3.00	−5.50 ± 3.02	5.67 ± 2.12
	2008 Dec 19	105	84	-9.89 ± 1.51	−6.78 ± 1.84	3.11 ± 0.98
	2009 Jun 12	114	98	−8.16 ± 0.49	−7.27 ± 1.35	0.89 ± 1.44
LP 415-20A	2008 Dec 19	42	32	41.13 ± 0.91	40.41 ± 1.06	−0.72 ± 1.40

Note. Velocities are in the heliocentric reference frame.

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The orbital analysis of the 24 stars monitored for this study fall into the following three categories.

• 1.  
  There are nine stars that do not have sufficient kinematic information to do any orbital analysis. Of the stars that fall into this category, three were unresolved in all epochs of our NIRC2 observations. These sources may have orbits that take them below the Keck diffraction limit during this study. We note that the initial separation measurements, made with HST, were 017 (2MASS 0652+47), 0057 (2MASS 1600+17), and 0051 (2MASS 0518−28). Another binary in this category (2MASS 1047+40) was resolved in the first epoch of NIRC2 observations, but unresolved in the subsequent epochs. The remaining five sources in this category have been resolved in all NIRC2 observations, but have not yet shown significant astrometric curvature due to either their late addition to the sample (2MASS 0700+31, 2MASS 1021−03, 2MASS 2101+17, 2MASS 2152+09) or, in the case of 2MASS 1017+13, large projection effects. We are continuing to monitor the first four but have stopped observing the latter target, as updated Monte Carlo simulations with the new epoch of data showed that this source was no longer "likely" to yield a mass with the necessary precision on a few years timescale. The unresolved and resolved astrometric measurements of these nine systems are reported for completeness in Table 4.
• 2.  
  There are 15 stars that have enough kinematic information measured to solve for the system's relative orbit, from which the total mass and eccentricity of the system can be inferred. All of them have at least four independent measurements; four of them have only astrometric data and 11 of them have multiple epochs of astrometry and at least one epoch of radial velocity measurements. The relative orbit analysis is described in Section 5.1.
• 3.  
  There are six systems, which are a subset of those in category (2), that have three or more radial velocity measurements for the individual components, allowing for estimates of the system's absolute orbital parameters from which individual masses can be derived. We have shown that radial velocity measurements are possible for additional five stars. Two other sources, which have been unresolved in NIRC2 observations, have K magnitudes that are comparably bright (K≲ 12.0). Further measurements of these systems are likely to also yield individual masses. The absolute orbital analysis is described in Section 5.2.

5.1. Relative Orbit Model Fits

To derive total mass estimates from relative orbital solutions for our sources, we combine our relative astrometric measurements from Section 4.1, previous astrometry reported in the literature, and the relative radial velocity between the components as determined in Section 4.2. As described in Ghez et al. (2008), our model for the relative orbit always contains six free parameters: period (P), semimajor axis (a), eccentricity (e), time of periapse passage (T_o), inclination (i), P.A. of the ascending node (Ω), and longitude of periapse passage (ω). We can remove the degeneracy in the values of Ω and ω which exists without information in this third dimension for the 11 sources that have radial velocity information. The radial velocity data also allow distance to be a free parameter in the fit for those sources without a previously measured parallax (five systems). For those systems with a parallax measurement (nine systems), we do not allow distance to be a free parameter, but rather we constrain it to be consistent with the parallax distance and its uncertainties. The uncertainties on parallax measurements are smaller than those from fitting for distance as a free parameter, and the values are consistent in all cases. The distances, either used or derived in our fits, are given in Table 6. In the case of one system, 2MASS 0920+35, we had neither radial velocities nor a parallax measurement, so we use instead the photometric distance as determined from the relationship in Cruz et al. (2003), which is based on J-band photometry and spectral type (here assumed to have an uncertainty of ±2 spectral subclasses). The best-fit orbital parameter values are found by minimizing the total χ², which is found by summing the χ² of each data type (χ²_tot = χ²_ast + $\chi ^2_{\rm{rv}}$; see Ghez et al. 2008 for more details on this fitting procedure).

Table 6. Astrometric Orbital Parameters

Target	Fixed Distance	Fit Distance	Total System	Period	Semimajor	Eccentricity	To	Inc.	Ω	ω	Best-fit
Name	(pc)	(pc)	Mass (M☉)a	(years)	Axis (mas)		(yr)	(degrees)	(degrees)	(degrees)	Reduced χ2
2MASS 0746+20AB	12.21 ± 0.05b	...	0.151 ± 0.003	12.71 ± 0.07	237.3+1.5−0.4	0.487 ± 0.003	2002.83 ± 0.01	138.2 ± 0.5	28.4 ± 0.5	354.4 ± 0.9	0.88
2MASS 0850+10AB	38.1 ± 7.3c	...	0.2 ± 0.2	24+69−6	126100−32	0.64 ± 0.26	2016+9−24	65 ± 12	96 ± 27	236+117−171	2.85
2MASS 0920+35AB	24.3 ± 5.0d	...	0.11 ± 0.11	6.7+3.33.4	69 ± 24	0.21+0.65−0.21	2003.43 ± 1.15	88.6 ± 2.4	69.0 a± 1.5	317+43−300	0.92
2MASS 1426+15AB	...	34 ± 13	0.11+0.08−0.11	1985+2141−1945	2273 ± 1560	0.85+0.10−0.41	1998 ± 24	88.3 ± 0.8	344.8 ± 0.4	282+78−210	1.89
2MASS 1534−29AB	13.59 ± 0.22e	...	0.060 ± 0.004	23.1 ± 4.0	234 ± 30	0.10 ± 0.09	2006.4 ± 3.0	85.6 ± 0.4	13.4 ± 0.3	25+154−25	1.57
2MASS 1728+39AB	24.1 ± 2.1c	...	0.15+0.25−0.04	31.3 ± 12.7	220 ± 26	0.28+0.35−0.28	2017 +4−22	62 ± 7	118+11−9	94 ± 15	2.60
2MASS 1750+44AB	...	37.6 ± 12.3	0.20 ± 0.12	317 ± 240	728 ± 375	0.71 ± 0.18	2004.3 ± 1.8	44 ± 10	99 ± 6	267 ± 26	1.63
2MASS 1847+55AB	...	29.8 ± 7.1	0.18+0.35−0.13	44.2 ± 18.7	237 ± 36	0.1+0.5−0.1	2020+6−28	79+4−2	125 ± 3	68 ± 30	0.63
2MASS 2140+16AB	...	25 ± 10	0.10 ± 0.08	20.1+5.3−1.6	1419−6	0.26 ± 0.06	2012.0+0.5−2.0	46.22.5−8.7	104 ± 7	223+10−47	0.50
2MASS 2206−20AB	26.67 ± 2.63f	...	0.16 ± 0.05	23.78 ± 0.19	168.0 ± 1.5	0.000+0.002−0.000	2000.0+1.9−3.2	44.3 ± 0.7	74.8 ± 1.0	326+28−52	2.32
GJ569B ab	9.81 ± 0.16g	...	0.126 ± 0.007	2.370 ± 0.002	90.8 ± 0.8	0.310 ± 0.006	2003.150 ± 0.005	33.6 ± 1.3	144.8 ± 1.9	77.4 ± 1.7	1.43
HD 130948BC	18.18 ± 0.08g	...	0.109 ± 0.002	9.83 ± 0.16	120.4 ± 1.4	0.16 ± 0.01	2008.6 ± 0.2	95.7 ± 0.2	313.3 ± 0.2	253.3 ± 3.9	2.12
LHS 2397a AB	14.3 ± 0.4h	...	0.144 ± 0.013	14.26 ± 0.10	215.8 ± 1.5	0.348 ± 0.006	2006.29 ± 0.04	40.9 ± 1.2	78.0 ± 1.5	217.7 ± 2.6	1.47
LP 349-25AB	13.19 ± 0.28i	...	0.121 ± 0.009	7.31 ± 0.37	141 ± 7	0.08 ± 0.02	2002.5 ± 0.8	118.7 ± 1.5	213.8 ± 1.1	109+37−22	2.15
LP 415-20AB	...	21 ± 5	0.09 ± 0.06	11.5 ± 1.2	108 ± 24	0.9 ± 0.1	2006.5 ± 0.2	55 ± 12	200 ± 40	73 ± 50	1.47

Notes. ^aDerived from period and semimajor axis. ^bDistance from parallax measurement by Dahn et al. (2002). ^cDistance from parallax measurement by Vrba et al. (2004). ^dNo parallax measurement or radial velocity data exists—spectrophotometric distance used here. ^eDistance from parallax measurement by Tinney et al. (2003). ^fDistance from parallax measurement by Costa et al. (2006). ^gDistance from Hipparcos parallax for high mass tertiary companion. ^hDistance from parallax measurement by Monet et al. (1992). ⁱDistance from parallax measurement by Gatewood & Coban (2009).

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After the best fit is determined, the uncertainties in the orbital parameters are found via a Monte Carlo simulation. First, 10,000 artificial data sets are generated to match the observed data set in number of points, where the value of each point (including the distance when determined from parallax measurements) is assigned by randomly drawing from a Gaussian distribution centered on the best-fit model value with a width corresponding to the uncertainty on that value. Each of these artificial data sets is then fit with an orbit model as described above, and the best-fit model is saved. The resulting distribution of orbital parameters represents the joint probability distribution function of those parameters. We obtain the uncertainties on each parameter as in Ghez et al. (2008), where the distribution of each parameter is marginalized against all others and confidence limits are determined by integrating the resulting one-dimensional distribution out to a probability of 34% (1σ) on each side of the best-fitting value. On occasion, when one or more parameters are not well constrained, the best-fit value does not correspond to the peak of the probability distribution. However, in almost all cases the best-fit value for a parameter is within 1σ of the peak. The few fit parameters in which this is not the case are normally represented by bifurcated or poorly constrained flat distributions (see for example the distributions of e and ω for 2MASS 1847+55 AB, online version of Figure 6).

The resulting best-fit orbital parameters and their uncertainties are given in Table 6. We find the orbital solutions for 15 of the systems in our sample. The orbital solutions are shown, along with both the astrometric and relative radial velocity data points, in Figures 5(a)–(e). The dotted blue lines represent the 1σ range of separations and relative radial velocities allowed at a given time based on the orbital solutions from the Monte Carlo. The distributions of orbital parameters for three sources are shown in Figures 6(a)–(c), chosen to be representative of the full sample. The rest of the distributions are shown online. The shaded regions on the histograms show the 1σ ranges of each parameter. If the distances were sampled from previous parallax measurements, they are denoted with a red histogram. These figures are alphabetically ordered based on the sources' names.

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5.2. Absolute Orbit Model Fits

For six systems in our sample, sufficient absolute radial velocity measurements (at least three) have been made, in conjunction with their relative orbits, to derive the first estimates of their absolute orbits, and hence the individual masses of the binary components. Common parameters between absolute and relative orbits, namely the P, T_o, e, and ω make it possible to only have to fit two free parameters: the semi-amplitudes of the velocity curve for the primary (K_Primary) and the systemic velocity (γ). K_Secondary is derived from the constraint that K_Primary + K_Secondary = 2π a sini/P (1 − e²)^1/2.

To first obtain the best-fit solution for these parameters, we use our radial velocities from Table 5 and fix the values of P, a, T_o, e, i, and ω to the values obtained in the relative orbit fitting to perform a least-squares minimization between the equations for the spectroscopic orbit of each component and our data. We fully map χ² space (where in this case χ²_tot = χ²_Primary + χ²_Secondary) by first sampling randomly 100,000 times from a uniform distribution of K_Primary and γ that are wide enough to allow mass ratios between 1 and 5 (where M_primary/M_Secondary = K_Secondary/K_Primary) for all sources except LHS 2397a AB, for which we allow for mass ratios between 1 and 10. To determine the uncertainties on our fit parameters, we again perform a Monte Carlo simulation. We use the distributions of P, a, T_o, e, i, and ω derived from our astrometric orbit Monte Carlo as inputs into the fits to account for the uncertainty in these parameters. We also then resample our radial velocity measurements to generate 10,000 artificial data sets such that the value of each point is assigned by randomly drawing from a Gaussian distribution centered on the true value with a width corresponding to the uncertainty on that value (as was done with the astrometric data). We then find the best-fit solution for each of these data sets (coupled with the sampled parameters from the astrometric fits). As with the astrometric orbit, we find the uncertainties by marginalizing the resulting distribution of each parameter against all others and integrating the resulting one-dimensional distribution out to a probability of 34% on each side of the best-fitting value.

The resulting best-fit orbital parameters for the absolute motion and their uncertainties are given in Table 7. The absolute orbital solutions are shown with the absolute radial velocity data points in Figure 7 and the distributions of orbital parameters for LHS 2397a AB, as a representative example, are shown in Figure 8. All other distributions are shown in the online version of the figure. By combining our mass ratio distribution derived with these data with the total system mass derived in Section 5.1, we have computed the first direct measurements of the individual masses of the components for five of these six systems. These individual masses are given in Table 7.

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Table 7. Absolute Orbital Parameters

Target	Fit Parameters			Derived Properties
Name	KPrimary	Center-of-mass	Best-fit	KSecondary	Mass Ratio	MPrimary	MSecondary
	(km s−1)	Velocity (km s−1)	Reduced χ2	(km s−1)	(MPrimary/MSecondary)	(M☉)	(M☉)
2MASS 0746+20AB	1.0+3.0−0.1	54.7 ± 0.8	0.44	4.1+0.1−3.1	4.0+0.1−3.8	0.12+0.01−0.09	0.03+0.09−0.01
2MASS 2140+16AB	0.8 ± 0.3	13.0 ± 0.2	0.9	3.1 ± 1.1	4.0+0−0.1	0.08 ± 0.06	0.02+0.08−0.02a
2MASS 2206−20AB	0.8 ± 0.2	13.3 ± 0.2	2.2	3.1 ± 0.4	4.0+0.0−0.2	0.13 ± 0.05	0.03+0.07−0.02a
GJ 569b AB	2.7 ± 0.3	−8.0 ± 0.2b	0.56	3.8 ± 0.4	1.4 ± 0.3	0.073 ± 0.008	0.053 ± 0.006
LHS 2397a AB	1.7 ± 1.2	34.6 ± 1.4	0.41	2.6 ± 1.4	1.5+7.1−1.4	0.09 ± 0.05	0.06 ± 0.05
LP 349-25 AB	4.5 ± 0.9	−8.0 ± 0.5	0.8	2.2 ± 0.9	0.5 ± 0.3	0.04 ± 0.02	0.08 ± 0.02

Notes. Using our absolute radial velocities in conjunction with the parameters from our relative orbital solutions, we fit for K_Primary and γ. We then use those values to find K_Secondary and the mass ratio. We combine the mass ratio and the total system mass from the relative orbits to find component masses. ^aUpper uncertainty set using the uncertainty in M_Primary and M$_{\rm{Tot}}$. ^bSet to our value.

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5.3. Eccentricity Distribution

Using the distributions from our Monte Carlo simulations in Section 5.1, we can begin to examine the bulk orbital properties of our sample of VLM objects. In particular, we can determine the distribution of eccentricities for our sample (nine of which are constrained to better than 30%), which may shed light on the formation of VLM binaries. To determine our eccentricity distribution, we performed a Monte Carlo analysis in which we randomly sampled one value of eccentricity per source from the distributions in Section 5.1. In each trial, the total number of sources per bin was calculated in bins of width 0.1 over the range of values from 0 to 1. We performed 10,000 of these trials, which gave a distribution for each bin of the number of expected sources. This distribution provided a predicted number of sources in each bin along with an uncertainty. We then combined distributions such that the final bin width was 0.2. The resulting distribution for our 15 sources is shown in the left panel of Figure 9.

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Though we have a small sample, the eccentricities of the binaries in our sample appear to follow a rough trend with orbital period. In the right panel of Figure 9, we plot the eccentricity of our sources as a function of period. In addition, we have overplotted in Figure 9 the periods and eccentricities from Duquennoy & Mayor (1991) for solar-like field stars with periods >1000 days (all sources in our sample meet this criteria except GJ 569B, which has a period of 865 days and an eccentricity of 0.31). Duquennoy & Mayor (1991) also found that eccentricity appeared to be a function of period, with longer period systems tending toward higher eccentricities (albeit with fairly large scatter past the tidal circularization period of ∼10 days). A two-dimensional Kolmogorov–Smirnov (K–S) test between our distribution and the distribution from Duquennoy & Mayor (1991) shows that the two samples have an 11% chance of being drawn from the same distribution (therefore being consistent to within 1.6σ), suggesting the distributions are statistically consistent (again with the caveat that we have a much smaller sample than those authors). Thus, although our sample appears to have a slight overabundance of moderate to low eccentricities compared to the expected distribution of eccentricities if a population is dynamically relaxed of f(e) ∼ 2e (shown as a red line in the left panel of Figure 9), the similar trend in our sources with period to Duquennoy & Mayor (1991) suggests that the VLM objects may ultimately have a similar eccentricity distribution.

5.4. Individual System Remarks

5.4.1. 2MASS 0746+20AB

2MASS 0746+20 AB originally had its total system mass derived by Bouy et al. (2004). Those authors found a total mass of 0.146^+0.016_−0.006 M_☉. Our new astrometric and radial velocity data have allowed us to improve this total mass estimate by a factor of 4 to 0.151 ± 0.003 M_☉, or to a precision of 2%. This measurement represents the most precise mass estimate for a VLM binary yet determined. Our individual mass estimates are the first for this system and the first for a binary L dwarf.

Though we have achieved superb precision in total mass for this system, the uncertainties in our radial velocity measurements are large compared to the current difference between in the velocities of the components (as can be seen in Figure 7). This has limited the precision we can currently achieve for the individual component masses, meaning cannot yet resolve the debate on whether the secondary is a brown dwarf or a low mass star (Gizis & Reid 2006). However, our measurements can shed some light on the result by Berger et al. (2009), in which a radius for the primary component of the system was estimated using periodic radio emission from the system. Based on the assumptions that the emission was coming from the primary, that the previously reported, spatially unresolved V sin i measurements reflected the V sin i (∼25 km s⁻¹) of the primary, and that the rotation axis of the primary was aligned with the binary inclination, Berger et al. (2009) derive a radius of 0.76 ± 0.10 R_Jup for this system. These authors note that this radius is about 30% smaller than expected based upon the models. Though we do not report definitive V sin i measurements for each component from spatially resolved spectra, providing an assumption-free value for this type of analysis, preliminary work comparing all K-band spectra across multiple orders for this system to high-resolution atmosphere models suggests that the rotational velocity of the secondary component may be ∼35 km s⁻¹. If it were the case that the radio emission was coming from this component of the binary instead of the primary (which does seem to have V sin i ∼ 25 km s⁻¹), this would increase the derived radius to something more in line with predictions. We leave the full analysis of our spectroscopy to derive quantities such as spatially resolved V sin i for a future publication.

5.4.2. 2MASS 0850+10AB

2MASS 0850+10 AB has two independent measurements of its distance via parallax. The first was from Dahn et al. (2002; 25.6 ± 2.5 pc) and the second was from Vrba et al. (2004; 38.1 ± 7.3 pc). These values are about 1.5σ discrepant from each other, a fact noted by Vrba et al. (2004). We performed a full Monte Carlo analysis as described in Section 5.1 using both estimates for distance. Since the current uncertainties in the period and semimajor axis for this system are large, the impact on the total mass estimate of choosing one distance over the other is negligible. We choose to present the values of mass as derived from the Vrba et al. (2004) distance estimate here because the larger uncertainties on this value make it the marginally more conservative choice.

5.4.3. 2MASS 0920+35AB

2MASS 0920+35 AB was discovered to be binary by Reid et al. (2001) using HST. A follow-up monitoring campaign of the system was performed by Bouy et al. (2008) using both HST and the VLT in conjunction with their facility AO system. In each of the five observations performed by Bouy et al. (2008), the system was unresolved (2002 October, 2003 March, 2005 October, 2006 April). These authors postulated that the binary was therefore perhaps on a highly inclined orbit with a period of roughly 7.2 years. When our monitoring of the system began in 2006, the system was again resolved, and remained resolved for all of our measurements until our most recent in 2009 June. We therefore utilize both the resolved and unresolved measurements to perform our orbit fits. First, we fit the resolved astrometric measurements for relative orbital parameter solutions as described in Section 5.1. We then took the output orbital solutions for those trials and calculated the predicted separation of the binary at each of the epochs in which it was unresolved—if the predicted separation was above the detection limits given by Bouy et al. (2008) or our 2009 June 10 measurement, it was thrown out. These unresolved measurements therefore provided tighter constraints on the orbital parameters for this system.

The results of the Monte Carlo simulation for this system are shown in Figure 6(b). As shown in this figure, the resulting distribution of periods has a strong bifurcation, whereby ∼45% of the solutions favor an orbital period of ∼3.3 years and a very high eccentricity, and 55% favor the best-fit solution of ∼6.7 years and more modest eccentricities. Since these solutions are nearly equally preferred but quite distinct, we display the best fit of both solution families in Figure 5(a). The two solution sets cause the current mass uncertainty to be fairly high. The mass distributions for the two sets overlap, creating the continuous distribution seen in Figure 5(a). The long tail out to masses greater than 1 M_☉ is generated by the short period solution set. An additional astrometric measurement before mid-2010 should distinguish between the two sets, as it will not be resolved for periods of ∼6.7 years but it will be resolved for periods of ∼3.3 years. Further, we have found the inclination of this system to be nearly edge on, meaning it has a non-negligible chance of being an eclipsing system (see Section 8).

5.4.4. 2MASS 1534−29AB

The first derivation of the orbit of 2MASS 1534−29AB was performed by Liu et al. (2008), where they calculated a total system mass of 0.056 ± 0.003 M_☉. By combining our astrometry with that reported by Liu et al. (2008), we find a slightly higher, but consistent, total system mass of 0.060 ± 0.004 M_☉. We note that if we perform our analysis on only the astrometry given in Liu et al. (2008), we obtain a mass of 0.056 ± 0.004 M_☉, consistent with their values.

5.4.5. 2MASS 2140+16AB and 2MASS 2206−20AB

We have acquired sufficient radial velocity data to make the first calculations of the absolute orbits of these systems. However, the uncertainty in the radial velocities is comparable to the difference between the values. Because of this, the best fit is typically the one that minimizes K_Primary, which in turn maximizes K_Secondary. This leads to relatively high predicted mass ratios. Though there is some spread in the value of mass ratio, as shown in the online version of Figure 8, the mass ratio is quite peaked at this high value. This leads not only to mass values for the secondary that are likely too low given their approximate spectral types, but also uncertainties that are too small for the secondary given the uncertainty in the mass of the primary. For these two systems, we therefore extend the uncertainty in the secondary mass by combining in quadrature the uncertainty in the total system mass and the uncertainty in the mass of the primary component. We have noted that we have taken this approach in Table 7 and have shaded the histograms in Figure 8 to reflect our chosen uncertainties. Though these first estimates of individual mass are fairly uncertain, they will improve with continued monitoring of these systems. We note that the total mass of 2MASS 2206–20AB was also derived by Dupuy et al. (2009a), who derived a value of 0.15^+0.05_−0.03 M_☉, consistent with the value derived here.

5.4.6. GJ 569Bab

The first derivation of the relative orbit of GJ 569Bab was performed by Lane et al. (2001), and was followed with improvements by Zapatero Osorio et al. (2004) and Simon et al. (2006). The work of Zapatero Osorio et al. (2004) and Simon et al. (2006) also contained spatially resolved, high-resolution spectroscopic measurements for this system, which is one of two targets in our sample that is an NGS AO target. Zapatero Osorio et al. (2004) derived the first estimate of the individual component masses of this system using their J-band spectroscopic measurements. Simon et al. (2006) made their radial velocity measurements in the H-band and noted that their derived center of mass velocity (−8.50 ± 0.30 km s⁻¹) is discrepant from that of Zapatero Osorio et al. (2004; −11.52 ± 0.45 km s⁻¹) by ∼3 km s⁻¹. Simon et al. (2006) postulate that this stems from the choice of lines used to make their measurements. Zapatero Osorio et al. (2004) use the K I doublet location referenced to laboratory wavelengths, while Simon et al. (2006) perform cross-correlation of their full order 48 (λ = 1.58–1.60 μm) and 49 (λ = 1.55–1.57 μm) spectra with spectral templates. Simon et al. (2006) also note that the relative radial velocities are consistent with what is predicted based on astrometry.

We now note that our spectra, measured in the K band and fit for radial velocity as described in Section 4.2, appear to be systematically offset from both the measurements of Zapatero Osorio et al. (2004), but consistent to within 1.25σ of Simon et al. (2006). We find a center-of-mass velocity using just our data points of −8.05 ± 0.20 km s⁻¹, which is the most consistent of the three sets of measurements with that of the M2V primary of this tertiary system (−7.2 ± 0.2 km s⁻¹). We find, as shown in Figure 5(d), that the relative velocities are consistent with what is expected for the relative orbit. Thus, the velocity differences truly seem to be driven by an offset in their absolute value. It is possible that these velocity offsets between Zapatero Osorio et al. (2004) and all other measurements may be related to the orbit of this binary around GJ 569A. To examine whether this is the case, we fit a rough relative orbital solution to all astrometric data in the literature for GJ569AB (Forrest et al. 1988; Martín et al. 2000; Lane et al. 2001; Simon et al. 2006) and the relative velocities between the components as determined by the difference between the velocity of GJ 569A and each of the three measurements for GJ569B. We assume for this exercise that the total system mass is ∼0.5 M_☉. We find that the best-fit orbit that can be obtained for all data has a reduced χ² of 5.1, driven entirely by the velocity point from Zapatero Osorio et al. (2004). This is demonstrated in Figure 10, where we plot the predicted velocity from the best-fit orbit with each systemic radial velocity measurement. The Zapatero Osorio et al. (2004) lies far from the best fit, while at the same time driving it to require a very high eccentricity (0.9) and a time of periapse passage close to the time of the measurements. Thus, we conclude that the differences in systemic velocity are likely due to systematics in absolute radial velocity calibration as described by Simon et al. (2006) and not orbital motion.

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In order to use all the radial velocity measurements to calculate individual masses, we opt to shift all data points from Zapatero Osorio et al. (2004) and Simon et al. (2006) such that their center-of-mass velocity is consistent with ours. We also increase the uncertainties in these values such that they incorporate the uncertainties in our value of systemic velocity and in the systemic velocity derived in each work, which we combine in quadrature. We then use these shifted velocities in conjunction with our measurements to derive the absolute orbit, which is shown in Figure 7. The application of this offset results in a very nice fit with a reduced χ² of 0.56. We find a mass ratio of 1.4 ± 0.3, which is lower than the value of 5.25 found by Simon et al. (2006). Those authors postulated that since the mass of the primary appeared to be so much higher than that of the secondary, that the primary may be a binary itself (something potentially suggested by the wider lines seen in GJ 569Ba). Our values of primary and secondary masses suggest that the sources actually have fairly similar masses of 0.073 ± 0.008 M_☉ and 0.053 ± 0.006 M_☉. We cannot, however, definitively rule out that GJ 569Ba is comprised of two components, as suggested by Simon et al. (2006) and Kenworthy et al. (2001), although this possibility is more unlikely given the mass we find for GJ 569Ba.

5.4.7. HD 130948BC

The first derivation of the relative orbit of HD 130948BC was performed by Dupuy et al. (2009b), where they calculated a total system mass of 0.109 ± 0.003 M_☉. By combining our astrometry with that reported by Dupuy et al. (2009b), we find an identical, but slightly more precise, total system mass of 0.109 ± 0.002 M_☉. Although we only have one radial velocity measurement for this system, which is insufficient to calculate individual masses for the components, our radial velocity measurement allows us to resolve the degeneracy in the values of ω and Ω.

5.4.8. LHS 2397a AB

The first derivation of the relative orbit of LHS 2397a AB was performed by Dupuy et al. (2009c), where they calculated a total system mass of 0.146^+0.015_−0.013 M_☉. Combining our astrometry with that reported by Dupuy et al. (2009c), we also find a consistent total mass of 0.144^+0.013_−0.012 M_☉. Performing our analysis on just the astrometry given in Dupuy et al. (2009c), we find a slightly different, but consistent, mass of 0.150^+0.014_−0.013. Dupuy et al. (2009c) also use their results in conjunction with a bolometric luminosity and the evolutionary models (both Burrows et al. 1997 and Chabrier et al. 2000) to derive the individual component masses. Here, we derive the first individual mass estimates free of assumptions, which allows for a direct comparison to the models. We find component masses of 0.09 ± 0.06 M_☉ for the primary and 0.06 ± 0.05 M_☉ for the secondary. The well-mapped velocity curve of the primary allows for this absolute orbit to be relatively well defined with a comparable number of radial velocity measurements to other sources that do not yet have well defined absolute orbits.

In order to compare the predictions of theoretical evolutionary models to our dynamical mass measurements, estimates of both the effective temperature and bolometric luminosity are required. With input of these parameters, the evolutionary models can be used to derive a mass and an age for a source. Thus, we must derive these parameters for all binary components.

Our method for deriving both of these quantities relies on the spatially resolved photometry we have obtained with our imaging data. In our NIRC2 data, we have measured the flux ratio of the binary components in the J, H, and K' bands, given in Table 4. We convert these flux ratios into individual apparent magnitudes using the unresolved photometry for these sources from 2MASS (Cutri et al. 2003). The apparent magnitudes can then be converted into absolute magnitudes using the distances from Table 6. We also find the absolute magnitudes for each system in all other photometric bands for which spatially resolved measurements exist. The majority of these measurements were made in the optical with HST. The absolute photometry for all sources is given in Table 8.

The determination of effective temperature for these sources is complex. Generally speaking, spectral type is not as accurate a proxy for the temperature of brown dwarfs as it is among hydrogen burning stars, with derived temperatures spanning several hundred Kelvins for different sources of the same spectral type (Leggett et al. 2002; Golimowski et al. 2004; Cushing et al. 2008). We therefore opt to perform spectral synthesis modeling using atmospheric models on each source individually, which given sufficient wavelength coverage allows for lower temperature uncertainties for most objects than would be achieved by using a temperature versus spectral-type relationship. Though this introduces a model assumption into our comparison of these sources to evolutionary models, we can use our mass estimates to determine the consistency of the atmospheric and evolutionary models with each other. For our sources of late-M to -L spectral types, we derive effective temperature using the DUSTY form of the PHOENIX atmosphere models (Allard et al. 2001). These models, in which all refractory elements are assumed to form dust grains and create thick dust clouds, have been shown to reproduce well the colors and spectra of these types of objects.

Updated opacities and grain size distributions, which are used in our analysis, have improved the correspondence of these models to observations (T. Barman et al. 2010, in preparation; Rice et al. 2010). Among the 30 individual components in our dynamical mass sample, 21 have previously determined late-M to early-L spectral types for which the DUSTY models are appropriate. For the two sources in our sample of mid-T spectral type, we use the COND version of the PHOENIX atmosphere models, which have been shown to reproduce the colors and spectra of T dwarfs well. In these models, all refractory elements have been removed from the atmosphere through an unspecified "rain out process," resulting in dust-free atmospheres and blue near-infrared colors. The treatment of L/T transition sources is discussed below. Due to the close proximity of these sources, we assume that extinction is negligible.

Since temperature can be most effectively constrained by comparing synthetic atmosphere data over a broad range in wavelengths, we elect to use our spatially resolved photometry to perform the spectral synthesis modeling. The wavelength and bandpass information for each of our photometric measurements in Table 8 were used in conjunction with the PHOENIX models to generate a grid of synthetic photometry for objects with T_Eff = 1400–4500 K for DUSTY and T_Eff = 300–3000 K for COND, with log g = 4.0–5.5. This range of surface gravity should be appropriate for all sources in our sample (McGovern et al. 2004; Rice et al. 2010). We then use this grid to fit the measured photometry for each source, allowing for interpolation between finite grid points. Since the grid contains surface flux densities, the model values must be scaled by (radius/distance)². Since the distance is known, the radius becomes a simple scaling parameter to fit simultaneously with gravity and effective temperature. Uncertainties in the derived temperature and radius are then calculated via Monte Carlo simulation, in which 20,000 new photometric data points are generated by sampling from a Gaussian distribution centered on each apparent magnitude with a width given by the uncertainty in each magnitude. The apparent magnitudes are then converted to absolute magnitudes using a distance sampled from a Gaussian distribution centered on the values given in Table 6. These data points are then fit in the same manner, and confidence limits on effective temperature and radius are then calculated by integrating our resulting one-dimensional distribution out to a probability of 34% on each side of the best-fitting value. The best-fit spectral energy distributions (SEDs) from the atmosphere models are shown overplotted on the photometry for each source in Figure 11. The one-dimensional probability density functions (PDFs) for temperature and radius are shown in Figure 12. Although surface gravity is also allowed to vary, we do not have sufficient photometric precision to distinguish between values of surface gravity for these field binaries, and the distributions of surface gravity are essentially flat.

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View figure set (23 panels)

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View figure set (23 panels)

In addition to the temperature uncertainties resulting from our photometric uncertainties, there are several systematic uncertainties in temperature that must be accounted for. First, the intrinsic uncertainty in the models themselves is estimated to be on the order of 50 K. We therefore combine this uncertainty in quadrature with the uncertainties from our Monte Carlo simulations for all sources. In addition, the lack of optical photometry for sources with spectral types earlier than L2 tends to bias the derived temperatures toward cooler values than is calculated for sources with optical photometry. The systematic offset is on average ∼200 K. Therefore, for those systems in this spectral type range with no optical photometry, we add in quadrature an additional uncertainty of 200 K. In Figure 13, we plot our derived effective temperatures as a function of spectral type. We also combine the results of Golimowski et al. (2004), Cushing et al. (2008), and Luhman et al. (2003) to illustrate previous measures of temperature versus spectral type. This relationship is plotted in red in Figure 13, along with error bars representing the range of allowed values by these works. This comparison demonstrates that in the cases where our photometry is well constrained and spans a broad range of wavelengths, the temperatures we derive using atmospheric modeling have lower uncertainties than we would be able to obtain using spectral type.

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Because we have a derived temperature and radius, we can also calculate the PHOENIX model predicted bolometric luminosity. However, this would also generate a model dependence in our value of luminosity. Instead, we elect to determine bolometric luminosity using the K-band bolometric corrections provided by Golimowski et al. (2004). These corrections are a function of spectral type and were derived using sources with photometric measurements over a broad range of wavelengths, integrating under their SEDs. The only assumption required to use these corrections is that spectral type is a good proxy for K-band bolometric corrections. In contrast to predicted effective temperature, the change in the K-band bolometric correction with spectral type is quite gradual with lower scatter. In addition, Liu et al. (2008) and Dupuy et al. (2009c) showed that by deriving bolometric luminosities from the SEDs of four sources, they obtain values fully consistent with those they would have obtained using the bolometric corrections of Golimowski et al. (2004). To be conservative, we assume an uncertainty in the spectral type of each source of ±2 spectral subclasses to determine our uncertainty in bolometric correction. Even with this assumption, the bolometric correction uncertainty is never the limiting factor in our bolometric luminosity uncertainty. Generally, the uncertainty is dominated by the distance uncertainty. Our estimates of bolometric luminosity from using these bolometric corrections are given in Table 8. To demonstrate the correspondence between the luminosities calculated in this way and the luminosities predicted by the atmosphere models, we plot the percent difference between the luminosities derived in each method in Figure 14. The measured scatter around perfect correspondence of these values is smaller than or consistent with our uncertainties. We therefore feel confident that our model-independent estimates of bolometric luminosity are appropriate for these sources and consistent with our other methodology.

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In principle, our high-resolution spectroscopy can also be used to calculate effective temperature. However, the narrow wavelength coverage in the near-infrared provides relatively loose constraints on temperature, with many temperatures being allowed by our K-band spectra. We have, however, performed a few comparisons of our K-band spectra to the same models we use for the photometric fitting and find that the results are consistent, with the photometry providing lower uncertainties than the spectroscopy alone. Ultimately, the best temperatures would be derived by fitting a combination of the photometry and the spectroscopy. However, such fitting has known challenges associated with how data are weighted (Cushing et al. 2008; Rice et al. 2010). In the future, we hope to perform fitting of this kind, combining all spectral data.

For the seven sources in our sample in the L/T transitions region, we must take a different approach to obtaining effective temperatures. The DUSTY and the COND models can be thought of as boundary conditions to the processes occurring in brown dwarf atmospheres, meaning each atmosphere is either fully dusty or completely dust free. There is no transitional dust phases represented in the current versions of these models. Though we attempted to fit sources in this region via the method described above with both models, we obtained very high temperatures (≳1900 K) and unphysically small radii (≲0.5 R_Jup). Therefore averaging the predictions of the two models does not work. For these sources, we elect to use the bolometric luminosity of the source and assume a radius with a large uncertainty, chosen to conservatively span the values derived in our atmospheric model fitting (1.0 ± 0.3 R_Jup). A radius in this range is also what is expected for these objects theoretically. In Figure 15, we plot the radii from our fits as a function of spectral type. Although there is a lot of scatter in this relationship due to the mixed ages in our sample, the large uncertainty we have assumed for radii at the L/T transition region should account for this variation. The result of assuming a radius is higher temperature uncertainties for these objects. All derived temperatures and radii are given in Table 8.

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The derived temperatures and bolometric luminosities are used to determine the model-predicted mass for each source in our sample. We consider both the Chabrier et al. (2000) evolutionary models, called DUSTY and COND, and the Burrows et al. (1997) evolutionary models (TUCSON). The DUSTY and COND evolutionary models are named as such because they use the boundary condition between the interior and the atmosphere provided by the DUSTY and COND atmosphere models, respectively. Thus, the evolutionary and atmosphere models are not strictly independent. By comparing to the Chabrier et al. (2000) evolutionary models, we are consistently testing model predictions because we have used the same atmospheric models in our analysis. Comparisons to the Burrows et al. (1997) models require a caveat, as we do not have access to the atmospheric models employed by those authors. However, we still perform the comparison to test the correspondence of these models to our measurements, as the effect of the atmospheric model boundary condition should only have a minor impact on the evolutionary predictions (Chabrier & Baraffe 2000).

To do this comparison, we first interpolate over the surface defined by the grids of temperature, luminosity, mass, and age provided by the evolutionary models using spline interpolation. Then, the temperature and luminosity point on the interpolated surface closest to our input value of temperature and luminosity are determined. For the sources from late-M to mid-L, we calculate the predictions of the DUSTY version of the Chabrier et al. (2000) models, while for the T dwarfs we use the COND version on these models. For the L/T transition objects, we calculate the predictions of both sets of Chabrier et al. (2000) models. The Burrows et al. (1997; TUCSON) models do not assume a different atmospheric treatment for different spectral types and assume that dust species have condensed out of the atmosphere across the entire substellar regime. We therefore compare the predictions of these models to all objects in our sample. These comparisons provide the predicted mass and age for each source.

To determine the uncertainties in each model prediction, we sample from temperatures and luminosities defined by the uncertainties in these quantities for each sources, accounting for the correlations between bolometric luminosity and temperature (Konopacky et al. 2007).¹⁴ The range of masses and ages predicted from this sampling, marginalized against the other parameters, provides the uncertainties. The values of mass predicted by each model are provided in Table 9.

Table 9. Evolutionary Model Predictions

Target	MPrimary	MSecondary	M$_{\rm{Total}}$	MPrimary	MSecondary	M$_{\rm{Total}}$	MPrimary	MSecondary	MTotal
Name	Tucson (M☉)	Tucscon (M☉)	Tucscon (M☉)	DUSTY (M☉)	DUSTY (M☉)	DUSTY (M☉)	COND (M☉)	COND (M☉)	COND (M☉)
2MASS 0746+20AB	0.050 ± 0.01	0.050 ± 0.01	0.10 ± 0.01	0.07 ± 0.01	0.06 ± 0.01	0.13 ± 0.01	...	...	...
2MASS 0850+10AB	0.04 ± 0.04	0.04 ± 0.04	0.08 ± 0.06	0.04+0.04−0.03	0.04+0.04−0.03	0.08+0.06−0.04	0.03+0.05−0.03	0.03+0.05−0.03	0.06+0.07−0.04
2MASS 0920+35AB	0.04 ± 0.04	0.03+0.05−0.03	0.07+0.06−0.05	0.04+0.04−0.03	0.03+0.05−0.03	0.07+0.06−0.04	0.03+0.05−0.03	0.03+0.05−0.03	0.06+0.07−0.04
2MASS 1426+15AB	0.03+0.04−0.02	0.04+0.04−0.03	0.07+0.06−0.04	0.04+0.04−0.02	0.05 ± 0.03	0.09+0.05−0.04	...	...	...
2MASS 1534−29AB	0.06 ± 0.02	0.06+0.03−0.02	0.12+0.04−0.03	...	...	...	0.04+0.04−0.01	0.04+0.04−0.01	0.08+0.06−0.01
2MASS 1728+39AB	0.04 ± 0.04	0.03+0.05−0.03	0.07+0.06−0.05	0.04 ± 0.04	0.04 ± 0.04	0.08 ± 0.06	0.03+0.05−0.03	0.03+0.04−0.03	0.06+0.06−0.04
2MASS 1750+44AB	0.01+0.02−0.01	0.01+0.02−0.01	0.02+0.03−0.01	0.02 ± 0.02	0.02 ± 0.02	0.04± 0.03	...	...	...
2MASS 1847+55AB	0.02+0.03−0.02	0.01 ± 0.01	0.03+0.03−0.02	0.03+0.05−0.02	0.01+0.02−0.01	0.04+0.05−0.02	...	...	...
2MASS 2140+16AB	0.04+0.05−0.03	0.07+0.03−0.05	0.11 ± 0.06	0.06 ± 0.04	0.07+0.03−0.04	0.13+0.05−0.06	...	...	...
2MASS 2206−20AB	0.032 ± 0.010	0.026+0.007−0.010	0.058+0.012−0.014	0.047+0.016−0.012	0.037+0.011−0.009	0.084+0.019−0.015	...	...	...
GJ569B ab	0.01 ± 0.01	0.02 ± 0.02	0.03± 0.02	0.02+0.01−0.02	0.03+0.03−0.02	0.05± 0.03	...	...	...
HD 130948BC	0.030 ± 0.010	0.032 ± 0.010	0.062 ± 0.014	0.035 ± 0.010	0.037+0.013−0.010	0.072+0.016−0.014	...	...	...
LHS2397a AB	0.02 ± 0.01	0.04 ± 0.04	0.06 ± 0.04	0.03 ± 0.01	0.04 ± 0.04	0.07 ± 0.04	...	0.03+0.04−0.03	0.06+0.04−0.03a
LP 349-25AB	0.01+0.02−0.01	0.01 ± 0.01	0.02+0.02−0.01	0.02 ± 0.02	0.02 ± 0.02	0.04 ± 0.03	...	...	...
LP 415-20AB	0.06 ± 0.04	0.05+0.04−0.03	0.11+0.06−0.05	0.08+0.02−0.05	0.06+0.03−0.04	0.14+0.04−0.06	...	...	...

Note. ^aTotal mass found by adding DUSTY prediction for primary to COND prediction for secondary.

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The majority of the sources in the sample have little to no age information—hence, we look for whether the models predict that the components are coeval as opposed to correct age prediction by the models. For the two sources with age information, HD 130948 BC (∼500 Myr; Gaidos 1998) and GJ 569Bab (∼100 Myr; Simon et al. 2006), the uncertainties on these ages are such that both models predict ages for these systems that are consistent with these values. For all sources in the sample, all binary components are consistent with being coeval within the uncertainties by both models. Figure 16 shows the predicted ages of the binary components in the DUSTY and the TUCSON models plotted versus each other. A line of 1:1 correspondence is overplotted. The relatively large uncertainty in age estimates stems from the fact that the model isochrones become more closely packed with increasing age. Because of this fact, empirical age estimates for field objects provide relatively weak constraints on the models. Thus, it is not surprising that both models predict all binary components are coeval. Stronger constraints on the ages predicted by the evolutionary models are likely to be made using younger sources, for which isochrones are less dense.

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Since the highest precision measurements are currently in total system mass, the model predictions can be compared most effectively to these measurements. To do this, we add the model masses derived for each component together and add their uncertainties in quadrature. The combined mass predictions are also given in Table 9. For seven systems, all models underpredict the total system mass by greater than 1σ. These seven systems have the smallest uncertainties in dynamical mass and primary component spectral types earlier than L4. For seven other systems, all models considered predict masses that are consistent with the dynamical mass within 1σ. These systems all have mass uncertainties over 60%, and also generally higher temperature uncertainties. Finally, the mass of one system is overpredicted by both models by greater than 1σ and is the only system with a mid-T spectral type. All systems in our sample with mass precisions better than 60% are therefore discrepant with the models by more than 1σ. To illustrate the apparent dependence in the direction of the mass discrepancy with spectral type, we have plotted the percent difference between the model prediction and the total dynamical mass for each model. These plots are shown in Figure 17. While the significance of the discrepancy is only on the order of 2σ (at most) for each individual target, the fact that there is a systematic trend (as a function of spectral type) suggests that the problem is more profound. Indeed, the hypothesis that the evolutionary models predict the correct masses for all systems can be tested by computing the associated reduced-χ² ($\frac{1}{(N-1)} \sum ^N\frac{(M_{\rm dyna}-M_{\rm model})^2}{\sigma _{\rm dyna}^2+\sigma _{\rm model}^2}$). The probability of obtaining the observed value of χ² = 4.81 with 15 measurements is ∼5 × 10⁻⁹. In other words, the significance level of the discrepancy between empirical and modeled systems masses is very high if one considers the entire sample as a whole.

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We can test the predictions of the models a bit further by considering our handful of individual mass measurements. Although our individual mass measurements do not yet have the high precision we have achieved in total mass, we can already see for the most precise cases that the discrepancy holds. That is, for the primary components of 2MASS 2206-20AB, GJ 569Bab, and LHS 2397a AB (which have the highest precision in component mass), the models underpredict the mass. These three systems all have approximate spectral types of M8. We also see that the secondary component LP 349-25 AB has its mass underpredicted by the models, and the TUCSON models underpredict the mass of GJ569Bb. These systems are both of approximate spectral type M9. For further illustration of these points, we again plot the percent difference between the masses predicted by the models and the dynamical masses, this time plotting the individual component mass. These plots are shown in Figures 18.¹⁵ Although the uncertainties are larger, the trends we saw among total system mass holds. These figures also demonstrate the power of using individual component masses to perform model comparisons, allowing for the investigation of where discrepancies lie without assumptions (this is particularly apparent in the case of LHS 2397a AB, which has an M8 primary and an L7.5 secondary). In addition, individual component masses effectively double the sample of sources that can be used for comparison (here, we have compared 12 sources, already approaching the 15 we can do with total system masses). Emphasis in the future will be placed on obtaining more precise individual mass estimates for these systems to see if these trends persist.

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We note that discrepancies between the evolutionary and atmosphere models have been noted before for three of our systems. The systems HD 130948BC, LHS 2397a AB, and 2MASS 1534−29 AB had their relative orbits derived by Dupuy et al. (2009b, 2009c), and Liu et al. (2008). These authors use spectral synthesis by Cushing et al. (2008) as a proxy for performing atmospheric fitting on the sources, and see that there is an offset between the evolutionary and atmosphere models, though they make the comparison only in terms of temperature (using their luminosities and total system masses with a mass ratio assumption to derive an evolutionary model predicted temperature). Thus, though their approach is different, they arrive at similar conclusions.

We have found systematic discrepancies between our measured dynamical masses and the predicted masses from theoretical evolutionary models, where overall the M and L dwarfs have higher dynamical masses than predicted and one T dwarf has a lower dynamical mass than predicted by evolutionary models. We determined the mass predicted by each evolutionary model using our measured parameters of luminosity and temperature, which are related to each other through the canonical equation L = 4πR²σT⁴. Our observed bolometric luminosity is the most constrained of these parameters and does not rely on evolutionary or atmospheric models; therefore, it is the least likely parameter to contribute to disagreement. Instead, the radius and temperature are the most likely cause of the discrepancy between the predicted evolutionary model masses and our dynamical masses, either those predicted by the evolutionary model or those from our atmospheric model fits. In this section, we explore temperature and radius and discuss other assumptions used with both atmospheric and evolutionary models which may give rise to differences between our measured masses and predicted modeled mass. For reference, we show in Figure 19 the location of GJ 569Ba of the H–R diagram, whose individual mass measurement was underpredicted by the evolutionary models. We show the location of the line of constant mass for a 0.07 M_☉ source as given in both the LYON and TUCSON models, which should align with the position GJ 569Ba if there was no discrepancy. The direction of the offset between these lines and the position of GJ 56Ba is representative of the direction of the offset for all discrepant systems of M and L spectral types. Though we cannot make a corresponding plot for the case of our overpredicted T dwarf system, for which we do not have individual component masses, the direction of the offset is opposite to that of GJ 569Ba.

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We first consider the case in which the driver for the discrepancy is primarily the evolutionary models, which begin with mass and age as input parameters and then predict quantities of radius, temperature, and, in turn, the luminosity. For the sources in the late-M through mid-L spectral types that are discrepant, the mass tracks that agree with our dynamical masses lie at higher temperatures and/or lower luminosities than our input values. To bring these sources into agreement would require either a decrease in the evolutionary model-predicted temperature for these sources of ∼100–300 K or an increase in the radii by a factor of ∼1.3–2.0. Meanwhile, for the discrepant T dwarfs, the correct mass lines lie at lower temperatures and/or higher luminosities than our input values. To bring them into agreement would require an increase in the evolutionary model predicted temperatures by ∼100 K, or a decrease in the radii by a factor of ∼1.5.

There are a number of implications for the physics of the evolutionary models. If the radii are off, this suggests that the mass–radius relationship in these models might be off. The predicted radius for a source is driven almost entirely by the assumed equation of state, with a very minor dependence on the assumed atmosphere (Chabrier & Baraffe 1997). An update of the equation of state from that given by Saumon et al. (1995), which is the equation of state used by both Burrows et al. (1997) and Chabrier et al. (2000), could potentially modify the mass–radius relationship, although improvements with using new experiments are unlikely to have a major impact in the brown dwarf mass regime.

Meanwhile, if the required change is in the predicted effective temperature, this implies that adjustments need to be made to the efficiency of interior energy transport, or an offset is needed in the interior/atmospheric boundary condition. Magnetic activity, which is not included in the models, could possibly inhibit the efficiency of convection, lowering the effective temperatures of these objects (Chabrier & Baraffe 2000). This may be important for the discrepant sources of spectral-type M or L, which have lower temperatures than predicted by the evolutionary models. Several studies have shown that cooler temperatures are measured for low mass eclipsing binaries than predicted by evolutionary models (e.g., Stassun et al. 2007; Morales et al. 2008), and the lack of accounting for activity is thought to be a likely culprit for this discrepancy.

The other case we consider is that in which the discrepancy is caused by the temperatures and radii predicted by the PHOENIX atmosphere models. In our spectral synthesis modeling, these two parameters are linked through the bolometric luminosity. Because our luminosity is a fixed, model-independent quantity, the parameter that matters in this case is temperature, because the radius is effectively set by the measurement of L_Bol and enters only as a scaling factor for the SEDs which are shaped by temperature. Therefore, if the discrepancy is caused by the atmosphere models, it is through the temperature prediction. In this case, the temperatures predicted would be too low in the case of the M and L dwarfs by ∼100–300 K, and too high in the case of the T dwarf by about ∼100 K. A change in temperature would cause the atmosphere model-predicted radius to change as well, but in a way that again maintains correct L_Bol.

The PHOENIX atmosphere models we have used are thought to represent the limiting cases in terms of atmospheric dust treatment. If the temperatures predicted for the discrepant M and L dwarfs are too low, it implies that the dust clouds are too opaque, trapping too much radiation. For the T dwarfs, removal of all refractory elements from the atmosphere may have resulted in a drop in opacity that allows too much radiation to escape, causing a higher than predicted temperature. Recent work by Helling et al. (2008) has shown that the treatment of dust clouds in atmospheric models has a dramatic effect on the output photometry. Though Helling et al. (2008) only compared two test cases, one at 1800 K and the other at 1000 K, a rough comparison between the colors of our discrepant sources and those test cases show that models with thinner dust clouds and uniform grain sizes may bring the temperatures into alignment with what is predicted by the evolutionary models.

Therefore, there are a number of scenarios in which a slight change in the input physics to either the evolutionary models or the atmosphere models (possibly both) could generate agreement between our dynamical masses and the model predicted masses. We also note that if the discrepancies between the models and the dynamical masses continue to follow the same trend, the implication for pushing into the planetary mass regime is that, like the T dwarfs, the masses of planets would be overestimated by the evolutionary models. For instance, in the case of the directly imaged extrasolar system, HR 8799, relatively high masses of 7, 10, and 10 M_Jup have been derived using evolutionary models (Marois et al. 2008). These higher masses have generated some difficulty in terms of allowing for systemic stability over long timescales (Goździewski & Migaszewski 2009; Fabrycky & Murray-Clay 2010). The decrease of these masses by only a few tens of percent would imply the system was much more stable over long timescales. Thus, this work may have important implications for the masses derived for directly imaged extrasolar planets using evolutionary models.

In order to place further constraints on these models, more measurements of brown dwarf radii are required in addition to mass. Thus far, only one eclipsing binary brown dwarf has been reported (Stassun et al. 2006), providing the only empirical measurement of a brown dwarf radius (for a very young system in Orion). In our sample, one source in our sample, 2MASS 0920+35 AB (which has components in the L/T transition region), is on a highly inclined orbit with an inclination of 886 ± 12. Assuming that the components have a radius of 1 R_Jup, the system will be an eclipsing system if it has an inclination between 8989 and 9015. Based on our full relative orbital solution distribution, we find that the system has a 6.8% probability of eclipsing. If we consider only those solutions with a ∼6.5 year period, which is the best-fit period, we find that the system has only a 3.1% chance of eclipsing. If we instead only consider those solutions with a ∼3.5 year period, the system has an 11.3% chance of eclipsing. In Figure 20, we plot the total probability distribution of eclipse dates, considering both periods. The highest probability of eclipse occurred in 2009 April. The next most likely date of eclipse is in mid-2012. The duration of the eclipse would most likely be between 2 and 4 hr. If this system does eclipse, it will provide for a direct measurement of its radius, allowing for a very powerful test of models at the L/T transition region.

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We have calculated relative orbital solutions for 15 VLM binary systems, using a combination of astrometric and radial velocity data obtained with the Keck Observatory LGS AO system. For ten of these systems, this is the first derivation of the relative orbits, one of which gives the most precise mass yet measured for a brown dwarf binary. We have also calculated the absolute orbital solutions for six systems, five of which are the first for those systems, representing the first individual component masses for several L dwarfs.

The masses we have calculated based on these orbital solutions and our derived temperatures have allowed us to perform the first comprehensive comparison of a sample of VLM objects to theoretical evolutionary models. All systems with mass precision better than 60% show discrepancies with the predictions of evolutionary models. We find that for six systems, their total system masses are underpredicted by both evolutionary models considered. In these systems, 11 of the 12 components have spectral types earlier than L4. We find that one binary T dwarf has its total system mass overpredicted by the evolutionary models. We postulate that for those systems in which we see a discrepancy, the possible cause is either an incorrect radius prediction by the evolutionary models, an incorrect temperature prediction by the evolutionary models, or an incorrect temperature prediction by the atmospheric models.

Future work that would illuminate the apparent mass discrepancies include (1) improving the precision of the dynamical masses, with a particular emphasis on obtaining more individual masses across a broader range of spectral types and (2) obtaining radius measurements for our sources, which we can potentially pursue through the calculation of surface gravity (which, with individual mass, provides a means for calculating radius), or measure directly if 2MASS 0920+35AB is eclipsing. Such measurements will allow us to test the evolutionary and atmospheric models independently.

The authors thank observing assistants Joel Aycock, Heather Hershley, Carolyn Parker, Gary Puniwai, Julie Rivera, Chuck Sorenson, Terry Stickel, and Cynthia Wilburn, and support astronomers Randy Campbell, Al Conrad, Jim Lyke, and Hien Tran for their help in obtaining the observations. We thank Jessica Lu, Tuan Do, Sylvana Yelda, Marshall Perrin, and Will Clarkson for helpful discussions of this work. We also thank an anonymous referee for helpful suggestions for the improvement of this document. Support for this work was provided by the NASA Astrobiology Institute, the Packard Foundation, and the NSF Science & Technology Center for AO, managed by UCSC (AST-9876783). Portions of this work were performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Q.M.K. acknowledges support from the NASA Graduate Student Research Program (NNG05-GM05H) through JPL and the UCLA Dissertation Year Fellowship Program. This publication makes use of data products from the 2MASS, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The W. M. Keck Observatory is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors also recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.