DYNAMICAL MASS OF THE M8+M8 BINARY 2MASS J22062280 − 2047058AB^*,†,‡,§

We monitored 2MASS J2206 − 2047AB using the LGS AO system at the Keck II Telescope on Mauna Kea, Hawaii (Wizinowich et al. 2006; van Dam et al. 2006), using the facility near-infrared camera NIRC2 in its narrow field-of-view mode. At each epoch, we obtained data in one or more filters covering the standard atmospheric windows from the Mauna Kea Observatories (MKO) filter consortium (Simons & Tokunaga 2002; Tokunaga et al. 2002). The LGS provided the wavefront reference source for AO correction, with the exception of tip-tilt motion. The LGS brightness, as measured by the flux incident on the AO wave front sensor, was equivalent to a V ≈ 10.1–10.7 mag star. The tip-tilt correction and quasi-static changes in the image of the LGS as seen by the wave front sensor were measured contemporaneously by a second, lower bandwidth wave front sensor monitoring 2MASS J2206 − 2047, which saw the equivalent of an R ≈ 15.8–16.2 mag star.

Our procedure for obtaining, reducing, and analyzing our images is described in detail by Dupuy et al. (2009b). Table 1 summarizes our observations of 2MASS J2206 − 2047AB, and typical images from each data set are shown in Figure 1. The binary separation, position angle (P.A.), and flux ratio were determined using the same three-component Gaussian representation of the point-spread function (PSF) as described in Dupuy et al. (2009b). We used the astrometric calibration from Ghez et al. (2008), with a pixel scale of 9.963 ± 0.005 mas pixel⁻¹ and an orientation for the detector's +y-axis of +013 ± 002 east of north, and applied the distortion correction developed by B. Cameron (2007, private communication), which changed the results below the 1σ level.

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To assess systematic errors in our PSF-fitting procedure, we also applied it to simulated Keck images of 2MASS J2206 − 2047AB that were created using images of PSF reference sources with similar FWHM and Strehl, summarized in Table 2. After being reduced in an identical fashion to the science images, the individual dithered images were shifted with interpolation and added to themselves to match the observed binary configuration at each epoch. These simulated images were then fitted in an identical manner to the science images, and the rms and mean of the truth-minus-fitted parameters determined the uncertainty and systematic offset.

For the 2008 May and December K_S-band data, the simulations predicted insignificant systematic offsets (≲ 0.3σ) and rms errors that were somewhat smaller than the rms of individual science dithers. For these data sets, we adopted the science rms values for the errors in all binary parameters and did not apply the Monte Carlo offsets. For the 2008 September data (J, H, K_S, and L' bands), our simulations yielded significant systematic offsets (as large as 2.4σ in L' band) and errors that were consistent with or somewhat smaller than the rms of individual science dithers. Such offsets are expected, particularly in high-Strehl images with prominent Airy rings, as our multi-Gaussian PSF model is known to be an imperfect representation of the data. Because we had data in four bandpasses at a single epoch, we were able to check that the Monte Carlo offsets brought the astrometry into better agreement—if they did not, our simulations would not have correctly assessed our systematic errors. We used the Monte Carlo-derived errors to compute χ² for the separation and P.A. measurements and found that after applying the systematic offsets χ² improved from 17.3 to 2.4 for the separation and from 7.8 to 1.3 for the P.A. (n.b., since there are 3 degrees of freedom, the median value of χ² is 2.4). Thus, we applied the systematic offsets from our Monte Carlo simulations as they brought the multi-band astrometry into agreement, and we adopted the Monte Carlo errors if they were larger than the rms of measurements from individual images.

A summary of the astrometry and flux ratios derived from the Keck data is given in Table 3. We used the data set with the smallest astrometric errors at each epoch in the orbit fit. As discussed in Section 3.1, the resulting orbit fit changes insignificantly if we vary the input astrometry by: (1) using the errors and offsets derived from Monte Carlo simulations rather than simply using the rms of individual dithers for the errors; and (2) using a different data set for a given epoch.

Bouy et al. (2003) reported binary parameters for 2MASS J2206 − 2047AB based on their HST discovery images; however, we have chosen to re-analyze this data because in our previous work we have found that our PSF-fitting technique can yield somewhat more precise astrometry (Liu et al. 2008). Also, Bouy et al. (2003) did not derive individual measurement errors for each binary in their sample, and accurate uncertainties are critical for orbit fitting. We retrieved the HST archival images of 2MASS J2206 − 2047AB obtained with the WFPC2 Planetary Camera (PC1) on UT 2000 August 13 (GO-8581, PI: Reid). These comprise two 30 s exposures in the F814W bandpass and one 500 s F1042M exposure. We only used the F814W images for deriving astrometry because (1) the smaller PSF enables better deblending of this tight binary and (2) a pair of images offers better cosmic ray rejection than a single image. We used TinyTim (Krist 1995) to generate PSF models which were fit to the data in a similar fashion to our previous work (Liu et al. 2008; Dupuy et al. 2009a). From our PSF fitting of the two images, we determined the binary separation, P.A., and flux ratio.²

To determine uncertainties and potential systematic offsets for our measurements, we simulated images of 2MASS J2206 − 2047AB using images of single ultracool objects from other HST/WFPC2 programs (GO-8563, PI: Kirkpatrick; GO-8581, PI: Reid; and GO-8146, PI: Reid). We only used objects with an equivalent or higher signal-to-noise ratio (S/N) compared to the science data so that we could degrade the S/N of the single images to match the science data. We also restricted ourselves to observations consisting of two or more images to allow robust rejection of cosmic rays. We only shifted images by an integer number of pixels to preserve the somewhat undersampled nature of the WFPC2 data. However, we were able to reproduce the binary configuration of 2MASS J2206 − 2047AB to within 0.2 pixels of the actual (Δx, Δy) separation of (+3.2, −1.6) pixels by carefully selecting appropriate pairs of input PSFs whose sub-pixel positions were determined in advance by single TinyTim PSF fitting.

We fitted the simulated binary images with TinyTim PSFs in the same way as the science data. The resulting rms scatter of the truth-minus-fitted parameters gave their errors, and the mean gave their systematic offsets. For both the separation and P.A., the offsets were small compared to the errors (−0.5 ± 1.8 mas and −01 ± 11), but we found an offset in the flux ratio that was large compared to its rms error (−0.10 ± 0.02 mag). Since these errors in deblending the binary are due to small imperfections in the PSF model, it is natural that they would have the largest impact on the flux ratio, not positional measurements, since the cores of the PSFs are well separated for this binary. In fact, because this binary has a flux ratio near unity, the systematic offset we found "flips" the binary, changing the component that is identified as the primary, and this flip brings the astrometry into agreement with the astrometry from other epochs.

We applied the systematic offsets from our simulations to the binary parameters, resulting in a separation of 161.1 ± 1.8 mas, a P.A. of 575 ± 11, and a flux ratio of 0.06 ± 0.02 mag (Table 3). We can compare these parameters to those derived by Bouy et al. (2003), who found a separation (163.0 ± 2.8 mas) and P.A. (575 ± 03) consistent with our measurements. Our separation uncertainty is slightly smaller, and our P.A. uncertainty is larger. From their Figure 2, it is clear that our separation and P.A. errors are actually consistent with their analysis, and the apparent discrepancy between our errors only arises because they condense their detailed study of binary parameter uncertainties to a single number for the separation error and two numbers for the P.A. error (03 above separations of 150 mas; 12 below 150 mas). For example, our P.A. error of 11 is intermediate between their two values, which is reasonable for a binary with a separation very close to their cutoff between the two regimes. However, the flux ratio of 0.36 ± 0.07 mag derived by Bouy et al. (2003) is inconsistent with ours at 4σ. Because the measurement of the flux ratio is most sensitive to imperfections in the PSF model, it is natural that our different PSF-fitting methods would disagree most on this parameter. We note that they apply a large 0.17 mag systematic offset to their flux ratio, which again is a single number condensed from more detailed analysis (see their Figure 3). This offset could account for 2.3σ of the discrepancy, which would bring our two flux ratios into reasonable agreement. The F814W flux ratio does not enter substantially into the following analysis, and so the discrepancy between our value and that of Bouy et al. (2003) has no impact on our results.

2MASS J2206 − 2047AB was imaged on UT 2001 September 22 by the Hokupa'a curvature AO system at the Gemini-North Telescope on Mauna Kea, Hawaii. Analysis of these data has previously been presented by Close et al. (2002); however, we have conducted our own analysis in an attempt to reduce the astrometric errors. We retrieved these raw data from the Gemini science archive and registered, sky-subtracted, and performed cosmic ray rejection on the images. Figure 1 shows a typical image of one of the 15 K' band 10 s exposures, which were used to derive the astrometry for 2MASS J2206 − 2047AB.

We used the same analytic PSF-fitting routine as for the Keck data to fit the Gemini images of 2MASS J2206 − 2047AB. Adopting an instrument pixel scale of 19.98 ± 0.08 mas pixel⁻¹ (F. Rigaut 2001, private communication), we found a separation of 167.7 ± 1.0 mas, where the uncertainty is the standard deviation of measurements from individual dithered images. This is in good agreement with the 168 ± 7 mas separation reported by Close et al. (2002). The improvement in the separation error may be attributed to the fact that Close et al. (2002) used the scatter among J-, H-, and K'-band images, whereas we have restricted our measurement to the bandpass with highest quality images (K' band). We found a K'-band flux ratio of 0.04 ± 0.02 mag, which is consistent (at 1.1σ) with the flux ratio of 0.08 ± 0.03 mag derived by Close et al. (2002).

Unfortunately, we are not able to derive the correct value for the binary P.A. from the archive images. As reported by Close et al. (2002), the image rotator was turned off for these observations so that the pupil is aligned with the detector. Thus, there is an arbitrary rotation in the images, which is not recorded in the FITS headers, and this rotation changes during each data set. We were able to remove this changing rotation by subtracting the parallactic angle from the binary P.A. measured in the individual dithers, and the rms scatter among the resulting P.A. measurements was 04. Adding in quadrature the 03 error in the absolute orientation of Hokupa'a/QUIRC that was adopted by Close et al. (2002) results in a P.A. error of 05, which is identical to their P.A. uncertainty.

Given the good agreement between our derived parameters and those derived by Close et al. (2002), we adopt our own separation measurement because of the smaller uncertainty but the Close et al. (2002) P.A. measurement because of our inability to reconstruct the orientation of the archival images. As discussed in Section 3.1, this choice is validated by the resulting orbit fit having a reduced χ² near unity.

We retrieved archival images of 2MASS J2206 − 2047AB obtained with the VLT at Paranal Observatory on UT 2003 July 11 and 2006 June 28. These data were taken with the NACO adaptive optics system (Lenzen et al. 2003; Rousset et al. 2003) using the N90C10 dichroic and S13 camera (13.221 ± 0.017 mas pixel⁻¹)³ at both epochs. We registered, sky-subtracted, and performed cosmic ray rejection on the raw archival images. The 2003 data comprise four H band 60 s exposures, and the 2006 data comprise 14 K_S band 60 s exposures. Typical images from each data set are shown in Figure 1.

We used the same analytic PSF-fitting routine as was used for the Keck data to fit the VLT images, and the results are summarized in Table 3. The uncertainties were determined from the standard deviation of measurements from individual dithers. Since no PSF star was contemporaneously observed with 2MASS J2206 − 2047AB, we were unable to assess any additional systematic errors in the VLT astrometry by fitting simulated binary images. However, in our previous work we have found that the rms scatter among dithered VLT images has been a good representation of errors derived from such simulations (Dupuy et al. 2009b).

Our observations together with archival data span 8.3 yr of the orbit of 2MASS J2206 − 2047AB. We used a Markov Chain Monte Carlo (MCMC) approach, described in detail by Liu et al. (2008), to determine the probability distributions of all orbital parameters. All the chains had lengths of 2 × 10⁸ steps, and the correlation length of our most correlated chain, as defined by Tegmark et al. (2004), was 5.4 × 10⁴ for the orbital period. This gives an effective length of the chain of 3.7 × 10³, which in turn gives statistical uncertainties in the parameter errors of about $1/\sqrt{3.7\times 10^3}$ = 1.6%, i.e., negligible. Figure 2 shows the resulting MCMC probability distributions, and the best-fit parameters and their confidence limits are given in Table 4. The single best-fit orbit has a reduced χ² of 1.07 (7 degrees of freedom) and is shown in Figures 3 and 4.

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Applying Kepler's Third Law (M_tot = a³/P²) to the period and semimajor axis distributions gives the posterior probability distribution for the total mass of 2MASS J2206 − 2047AB, which has a median of 0.152 M_☉, a standard deviation of 0.003 M_☉, and 68.3(95.4)% confidence limits of ^+0.003_−0.003(^+0.007_−0.006) M_☉ (Figure 5). In Figure 6, it is evident that the tight correlation between the two parameters P and a is responsible for the very precise total mass (2%), despite the fact that the parameters are not independently determined as precisely (16% and 12%, respectively). The MCMC probability distribution of the total mass does not include the uncertainty in the parallax (9.1%), which by simple propagation of errors would contribute an additional ⁺³²₋₂₂% uncertainty in the mass. We account for this additional error by randomly drawing a normally distributed parallax value for each step in the chain. The resulting mass distribution is asymmetric, and our final determination of the total mass is 0.15^+0.05_−0.03(^+0.13_−0.06) M_☉ at 68.3(95.4)% confidence.

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As an independent verification of our MCMC results, we also fit the orbit of 2MASS J2206 − 2047AB using the linearized least-squares routine ORBIT (described in Forveille et al. 1999). All of the orbital parameters are consistent between the ORBIT and MCMC results, and the resulting total mass and χ² were identical. Using ORBIT, we tested whether varying the input astrometry and corresponding uncertainties affected the orbital solution. We tried a variety of permutations: using the published HST and/or Gemini astrometry (rather than our own); using Keck astrometry with and without Monte Carlo offsets and with rms or Monte Carlo-derived errors; and excluding one of the two VLT epochs. The resulting orbits had masses of 0.145–0.157 M_☉, with errors of 0.003–0.020 M_☉, and χ² of 1.07–2.32. Thus, the dynamical mass and corresponding error are not significantly impacted by the input astrometry, as all variation is much smaller than the parallax error, and our default solution has a lower χ² and mass uncertainty than any other plausible trial permutation.

Using the integrated-light optical spectrum of 2MASS J2206 − 2047, both Gizis et al. (2000) and Crifo et al. (2005) found a spectral type of M8.0 ± 0.5. Without resolved spectroscopy of the binary, we cannot directly determine the spectral types of the two components; however, our resolved photometry shows that they are nearly identical. To quantify the potential difference in spectral types, we compiled 2MASS photometry for the single M8.0, M8.5, and M9.0 objects with parallaxes from Monet et al. (1992) and Dahn et al. (2002). Between the spectral types M8.0 and M8.5, we found a difference in absolute magnitude of ΔM_J = 0.67 ± 0.19 mag, ΔM_H = 0.68 ± 0.20 mag, and $\Delta {M_{K_S}}$ = 0.64 ± 0.22 mag, where the uncertainty is the rms of objects in each bin added in quadrature. Between spectral types M8.0 and M9.0, we found only slightly larger magnitude differences. Our best measurement of the flux ratio of 2MASS J2206 − 2047AB in each of these bands is ΔJ = 0.06 ± 0.02 mag, ΔH = 0.065 ± 0.017 mag, ΔK_S = 0.067 ± 0.010 mag.⁴ Thus, we find that the J-, H-, and K_s-band photometry is inconsistent with the two components of 2MASS J2206 − 2047AB having different spectral types at 3.0σ, 3.1σ, and 2.6σ, respectively. Therefore, we adopt spectral types of M8.0 ± 0.5 for both components.

Because of the identical spectral types and colors of the components of 2MASS J2206 − 2047AB, we computed their individual bolometric luminosities simply from the flux ratio and integrated-light bolometric luminosity. For the former, we used the most precise flux ratio available, which was 0.067 ± 0.010 mag from our 2008 September Keck K_S-band data. There is no published value for the bolometric luminosity of 2MASS J2206 − 2047, so we computed it from its near-infrared spectrum and estimated L'-band photometry. We obtained the spectrum on UT 2008 July 6 using IRTF/SpeX (Rayner et al. 2003) in SXD mode, which has five orders spanning 0.81–2.42 μm (R = 1200). We calibrated, extracted, and telluric-corrected the data using the SpeXtool software package (Vacca et al. 2003; Cushing et al. 2004). In Figure 7, we show our spectrum alongside spectra from Cushing et al. (2005) and Rayner et al. (2009) of the M dwarf spectral standards defined by Kirkpatrick et al. (1991). As expected from the M8.0 ± 0.5 optical spectral type of 2MASS J2206 − 2047, the best matching standard is the M8 dwarf VB 10. To estimate the L'-band photometry, we compiled the K − L' colors of M dwarfs and early L dwarfs from Golimowski et al. (2004), which essentially follow a linear relation with spectral type. Fitting all single M1–L1 objects (i.e., excluding the binaries LHS 333AB, LHS 2397aAB, and 2MASS J0746+2000AB) and weighting by their photometric errors, we found

$$\begin{eqnarray} K_{\rm MKO}-\mbox{${L^\prime }$}& = & 0.111 + 0.0526\times {\rm SpT}, \end{eqnarray} \tag{ 1 }$$

where K − L' is in mag, spectral type (SpT) is defined such that M0 = 0 and L0 = 10, and the rms about the fit was 0.07 mag. From this relation, we estimated a K − L' color for 2MASS J2206 − 2047 of 0.53 ± 0.07 mag, resulting in an L'-band magnitude of 10.73 ± 0.08 mag.⁵

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To derive the integrated-light bolometric luminosity, we numerically integrated our SpeX spectrum and the L'-band photometric point at 3.8 μm, interpolating between the gaps in the data, neglecting flux at shorter wavelengths, and extrapolating beyond L' band assuming a blackbody. We determined the luminosity error in a Monte Carlo fashion by adding randomly drawn noise to our data over many trials and computing the rms of the resulting luminosities. We accounted both for the noise in the spectrum and the errors in the 2MASS photometry used to flux-calibrate it. Before accounting for the error in the distance, we found a total bolometric luminosity of log(L_bol/L_☉) = −2.982 ± 0.006 dex. After accounting for this error, the symmetric parallax uncertainty results in slightly asymmetric luminosity errors, giving log (L_bol/L_☉) = −2.98^+0.08_−0.07 dex. Using the flux ratio to apportion this luminosity to the two binary components results in individual luminosities of −3.27^+0.08_−0.07 dex and −3.30^+0.08_−0.07 dex. In the following analysis, we correctly account for the covariance between these quantities (via the flux ratio), enabling more precise determinations of relative quantities (such as ΔT_eff) due to the more precise luminosity ratio (Δlog(L_bol) = 0.027 ± 0.004 dex).

Because the two components of 2MASS J2206 − 2047AB have essentially identical fluxes and colors, we can determine the effective temperatures and surface gravities of both by fitting atmospheric models to its integrated-light spectrum.⁶ We used the PHOENIX-Gaia (Brott & Hauschildt 2005) and the Ames-Dusty (Allard et al. 2001) solar-metallicity atmospheric models to fit our IRTF/SpeX SXD spectrum of 2MASS J2206 − 2047. The PHOENIX-Gaia models include updated line lists compared to the Ames-Dusty models, but they do not include the effects of dust. The treatment of dust in the Ames-Dusty models is an extreme limiting case (no dust settling), but models with a more sophisticated treatment of dust are not yet publicly available. For the PHOENIX-Gaia models, we used grids of synthetic spectra ranging in T_eff from 2000 to 3500 K (ΔT_eff = 100 K) and log (g) from 3.5 to 5.5 (Δlog (g) = 0.5). For the Ames-Dusty models, we used grids of synthetic spectra ranging in T_eff from 1500 to 3400 K (ΔT_eff = 100 K) and log (g) from 4.0 to 6.0 (Δlog (g) = 0.5).

Our fitting procedure utilized a Monte Carlo approach based on that of Cushing et al. (2008) and Bowler et al. (2009). To account for the heterogeneous resolution of our SXD spectrum, we Gaussian-smoothed synthetic spectra in separate spectral ranges corresponding to the different SXD orders. When fitting our near-infrared spectrum (0.81–2.42 μm), we excluded a small region from 1.82–1.88 μm not covered by the instrument. We flux-calibrated our observed spectrum using 2MASS J, H, and K_S photometry. For each Monte Carlo trial, we applied small flux shifts to the observed spectrum corresponding to the spectroscopic (SpeX) and photometric (2MASS) measurement errors and then found the best-fitting model by minimizing the χ² statistic. This process was repeated 10³ times, after which we tallied the fraction of times each model yielded the best fit (f_MC). Fractions near 1.0 indicate that only a single model fit the data well. The results of our fitting procedure are given in Table 5, and the best-fit spectra for each set of models are shown in Figures 8 and 9.

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Both sets of models gave best-fit effective temperatures of 2800 K, but the best-fit PHOENIX-Gaia model had a lower surface gravity (log (g)= 4.5) than the Ames-Dusty model (log (g)= 6.0; the grid maximum). Although the f_MC value for the best-fit PHOENIX-Gaia model was only 0.46, the next best-fit model (f_MC = 0.27) had a temperature and gravity different by only one grid step (T_eff= 2700 K, log (g)= 4.0). We also fitted the observed spectrum separately over individual bandpasses (Y, J, H, and K), and these fits yielded similar results to the entire spectrum, typically within 100 K and 0.5 dex (i.e., one model grid step). Thus, we adopt errors of ±100 K and ±0.5 dex on the best-fit parameters in order to account for the impact of the measurement errors on the fit as well as uncertainties in modeling a limited spectral range.

In addition to the effective temperature and surface gravity, the radius can be derived from atmospheric model fitting when the distance is known. This is because the scaling factor used to shift the synthetic spectrum to the observed flux-calibrated spectrum is a free parameter equal to R²/d². Accounting for the flux ratio between the two components, the error in the distance, and the rms scatter in this scaling factor over the 10³ trials, we found identical radii of 0.096 ± 0.009 R_☉ from the PHOENIX-Gaia models and 0.095 ± 0.009 R_☉ from the Ames-Dusty models.⁷

In this section, we consider whether the space motion or activity of 2MASS J2206 − 2047 can provide useful constraints on the age of the system. There are three values of its radial velocity in the literature: (1) 16.3 ± 2.7 km s⁻¹ derived by Reid et al. (2002) from cross-correlation of the optical spectrum with radial velocity standards; (2) 8.0 ± 2.0 km s⁻¹ also from Reid et al. (2002) but derived from the central wavelength of the Hα emission line; and (3) 10.8 ± 1.3 km s⁻¹ derived by Guenther & Wuchterl (2003) using the central wavelengths of unspecified spectral lines. Reid et al. (2002) attributed the discrepancy between their two radial velocities to the fact that 2MASS J2206 − 2047 is a fast rotator (vsin i = 22 km s⁻¹) with an asymmetric Hα profile, confusing their estimate of the Hα centroid. 2MASS J2206 − 2047's binarity was unknown to Reid et al. (2002), and we note that this could be partially responsible for the discrepancy. For example, if one component dominated the Hα emission, the Hα centroid would be offset from the cross-correlation velocity (which likely represents the average velocity of the two components) by 1.8 km s⁻¹, assuming a mass ratio of unity. However, this alone is insufficient to account for the 8.3 km s⁻¹ discrepancy observed.

We used the cross-correlation radial velocity (16.3 ± 2.7 km s⁻¹) from Reid et al. (2002) and the parallax and proper motion from Costa et al. (2006) to derive the heliocentric velocity of 2MASS J2206 − 2047: (U, V, W) = (+7.8 ± 1.6, +1.7 ± 1.1, −15.0 ± 2.1) km s⁻¹. We adopted the sign convention for U that is positive toward the Galactic center and accounted for the errors in the parallax, proper motion, and radial velocity in a Monte Carlo fashion. For comparison, we compiled all objects of spectral type M7 or later that have the radial velocities, parallaxes, and proper motions necessary for computing space motions (described in detail in Section 3.4 of Dupuy et al. 2009b). 2MASS J2206 − 2047 is only 1.3σ away from the mean of this population's space motion ellipsoid (Figure 10). Thus, its space motion is not significantly different from other ultracool dwarfs, implying an age consistent with the population of ultracool dwarfs as a whole. Several authors have attempted to estimate the age of this population, typically comparing the distribution of tangential velocities (V_tan, which requires only a proper motion and distance determination) to the well-studied nearby populations of FGKM stars. The resulting age for the population of ultracool dwarfs estimated in this way has been found to be 2–4 Gyr (Dahn et al. 2002; Faherty et al. 2009).⁸

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We have also assessed 2MASS J2206 − 2047's membership in the Galactic populations of the thin disk (1–10 Gyr; e.g., Wood & Oswalt 1998) and thick disk (∼10 Gyr; e.g., Ibukiyama & Arimoto 2002) using the Besançon model of the Galaxy (Robin et al. 2003). Our method is described in Section 3.4 of our study of LHS 2397a (Dupuy et al. 2009b); and for 2MASS J2206 − 2047, we found a membership probability of > 99.9% for the thin disk and < 0.1% for the thick disk.

Finally, the fact that 2MASS J2206 − 2047 is chromospherically active (log (L_Hα/L_bol) = −4.59, −4.54; Gizis et al. 2000 and Reid et al. 2002, respectively) could also potentially provide an age constraint, as the activity of M dwarfs changes with age. West et al. (2008) showed that the fraction of active M dwarfs as a function the vertical distance above the Galactic plane (z) provides a constraint on the activity lifetime of M dwarfs, given a model of how thick disk heating pumps up z over time. West et al. (2008) found that the activity lifetime increases monotonically with M dwarf spectral type, and the latest type for which they were able to determine a robust lifetime was M7 (8.0^+0.5_−1.0 Gyr). This provides a weak constraint on the age of 2MASS J2206 − 2047, as its activity is therefore expected to last for at least ≳8 Gyr.

As described in detail by Liu et al. (2008) and Dupuy et al. (2009a), the total mass of a binary along with its individual component luminosities can be used to estimate the age of the system from evolutionary models. This age estimate can be surprisingly precise when both components are likely to be substellar since their luminosities depend very sensitively on age. However, with spectral types of M8.0 ± 0.5, both components of 2MASS J2206 − 2047AB are likely to be stars unless the system is quite young.

We derived an age of 0.4^+9.6_−0.2 Gyr from both the Tucson and Lyon models (Figure 11). Because the median total mass is 0.15 M_☉, which is roughly the limit at which both components would be brown dwarfs, the median age derived from models is correspondingly young. However, the 1σ uncertainty in the total mass reaches 0.20 M_☉, in which case both components would be main-sequence stars. In this case, since stars do not dim over time as brown dwarfs do, the luminosities of both components do not strongly constrain the age of the system. Thus, while the lower bound of our uncertainty on the model-derived age corresponds to the age a pair of brown dwarfs would need to match the observed luminosities and total mass, the upper limit is essentially unconstrained. Our limit of 10 Gyr comes from the fact that the evolutionary models are computed only up to this age. In our analysis, which uses a Monte Carlo approach to compute model-derived properties, we found that about 30% of the time the randomly drawn observed luminosities were too low to match the randomly drawn total mass. In other words, models would never predict that such massive objects (≳0.09 M_☉ stars) could be as faint as the components of 2MASS J2206 − 2047AB, and in such cases we assigned an age of 10 Gyr.

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Given the near unity flux ratio of 2MASS J2206 − 2047AB, we expect the mass ratio to also be very close to unity. We used evolutionary models to estimate the mass ratio of 2MASS J2206 − 2047AB (q ≡ M_B/M_A) by constraining the model-derived individual masses of 2MASS J2206 − 2047A and 2MASS J2206 − 2047B to add up to the observed total mass, while still matching their observed luminosities. The Tucson models gave q = 0.981^+0.012_−0.007, while the Lyon models gave a consistent value of 0.982^+0.008_−0.006. The resulting individual masses (Table 7) are essentially identical to those resulting from an assumed mass ratio of unity, with the exception of the upper confidence limits. This is due to the effect described in Section 4.1 where about 30% of the randomly drawn total masses and individual luminosities were inconsistent with any models. In these cases, we assigned the highest individual masses for which the luminosities were consistent with the models. In Section 5.1, we consider the issue of plausible individual masses in more detail.

Without radius measurements for 2MASS J2206 − 2047A and 2MASS J2206 − 2047B, we cannot directly determine their effective temperatures or surface gravities.⁹ In Section 3.4, we derived these properties by fitting atmospheric model spectra to the integrated-light spectrum, and we have also used evolutionary models to estimate these properties in the same fashion as our model-derived age and individual masses. The Tucson models give effective temperatures for 2MASS J2206 − 2047A and 2MASS J2206 − 2047B of 2660⁺⁹⁰₋₁₀₀ K and 2640⁺⁹⁰₋₁₀₀ K, and surface gravities of 5.26^+0.09_−0.17 and 5.26^+0.10_−0.17 (cgs). The Lyon models give systematically lower but formally consistent temperatures of 2550⁺⁹⁰₋₁₀₀ K and 2530⁺⁹⁰₋₁₀₀ K and surface gravities of 5.17^+0.12_−0.18 and 5.17^+0.12_−0.18 (cgs). (Note that the upper confidence limits are likely affected by the same truncation within our Monte Carlo method as discussed in Section 4.2 for the individual masses.) The differences between the two sets of models are due to the fact that the Tucson models predict radii that are 9% smaller than predicted by Lyon models (Table 7).

Compared to the effective temperature of 2800 ± 100 K derived from spectral synthesis fitting, the Tucson models are consistent (at 1.0σ–1.2σ), but the Lyon model temperatures are 1.9σ–2.0σ lower. This is illustrated in Figure 12, which shows the atmospheric model-derived temperatures in comparison to the evolutionary tracks on the Hertzsprung–Russell (H–R) diagram. As a result of this temperature discrepancy, the Tucson and Lyon model-predicted radii are larger than derived from the atmospheric model scaling factors by 14% and 24%, respectively. These could be brought into better agreement if the system was older than the median model-derived age of 0.4 Gyr, as evolutionary models would predict smaller radii and thus higher effective temperatures (corresponding to the 2σ lower/upper limits in Table 7 for radii/temperatures). Finally, atmospheric model fitting did not yield consistent surface gravity estimates: the dust-free PHOENIX-Gaia models gave log (g) = 4.5, and the Ames-Dusty models gave log (g) = 6.0 (the maximum allowed by the model grid). These are, respectively, lower and higher than the evolutionary model-derived surface gravities of log (g) = 5.0–5.4.

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4.3.1. Comparison to Field Dwarfs

The Lyon evolutionary models provide predictions of the fluxes of ultracool dwarfs in standard filter bandpasses as a function of model mass and age.¹¹ We derived the model-predicted near-infrared colors of both components of 2MASS J2206 − 2047AB in the same fashion as the individual masses, effective temperatures, and age (i.e., using the combined observational constraints of the total mass and individual luminosities). Figure 13 shows the observed colors of 2MASS J2206 − 2047AB on color–magnitude diagrams in comparison to model tracks and other field dwarfs. Compared to the observed colors of the components of 2MASS J2206 − 2047AB, only H–K is consistent with the Dusty models, and both J–K and J–H are about 0.2–0.3 mag redder than predicted by Dusty. Since field M8 dwarfs have very similar colors to the components of 2MASS J2206 − 2047AB, the Lyon Dusty models will generally provide inaccurate estimates of the fundamental properties of late-M dwarfs from their near-infrared colors.

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We have directly measured the total mass of 2MASS J2206 − 2047AB to be 0.15^+0.05_−0.03 M_☉. However, the entire range of formally allowed masses is not consistent with some of its other properties. For example, at the 1σ upper limit in M_tot, both components of 2MASS J2206 − 2047AB would be 0.10 M_☉ stars, but they would then be 0.7 mag fainter at K band than the faintest object of comparable mass (Gl 234B: 0.1034 ± 0.0035 M_☉, M_K = 9.26 ± 0.04; Delfosse et al. 2000; Ségransan et al. 2000). In contrast, the objects closest in K-band brightness to 2MASS J2206 − 2047AB are GJ 1245C (0.074 ± 0.013 M_☉, M_K = 9.99 ± 0.04; Henry et al. 1999) and LHS 2397aA (0.0848 ± 0.0011 M_☉,¹² M_K = 10.06 ± 0.07; Dupuy et al. 2009b). Thus, it is more likely that the components of 2MASS J2206 − 2047AB have individual masses in this range; in which case, the total mass would be well below the formally allowed value of 0.20 M_☉.

The 1σ lower limit of M_tot = 0.12 M_☉ corresponds to a pair of 0.06 M_☉ brown dwarfs, and masses at or below this value are also disfavored. Reid et al. (2002) found an upper limit of 0.05 Å for lithium absorption at 6807 Å in the integrated-light spectrum of 2MASS J2206 − 2047, indicating that both components have depleted their initial lithium. As discussed by Chabrier et al. (1996), lithium can only be depleted in objects more massive than ≈0.06 M_☉. Moreover, even more massive objects require a finite amount of time to become lithium depleted: according to Chabrier et al. (1996), a 0.070 M_☉ object takes 0.2 Gyr to destroy 99% of its initial lithium, with lower mass objects taking longer. Given the constraint of the individual luminosities of 2MASS J2206 − 2047AB, the low-mass tail of the M_tot distribution corresponds to young ages. At the median mass of 0.15 M_☉, the model-derived age is 0.4 Gyr; and at the 1σ lower bound of 0.12 M_☉, the age is 0.2 Gyr. Below this 1σ limit, the components of 2MASS J2206 − 2047AB would be inconsistent with the lithium non-detection: (1) they would have had insufficient time to destroy their initial lithium, and/or (2) they should be low enough mass that they would never destroy any lithium. Thus, regardless of the precise location of the lithium-fusing boundary, the formally allowed low-mass tail of the M_tot distribution is not physically plausible.

Since the components of 2MASS J2206 − 2047AB are nearly identical, testing models using our measured total mass is straightforward. However, future measurements may constrain the binary's mass ratio directly. Given that the flux ratio is so near unity, the mass ratio would not be feasible to measure from astrometric monitoring of the photocenter since the center of light would be imperceptibly different from the center of mass. Thus, the mass ratio must be determined through radial velocity monitoring. This is also challenging as the binary is currently approaching ΔV = 0 km s⁻¹ and will not reach the next peak in the radial velocity curve for P/4≈ 9 yr, in 2018. Until then, the radial velocities of the two components will remain below 1.4 km s⁻¹. Since the velocity of each component must be measured to 7% in order to determine the mass ratio to 10%, radial velocity measurements with a precision better than 0.1 km s⁻¹ are needed. This is just at the limit of state-of-the-art techniques using current instrumentation (Blake et al. 2007) but should be well within reach of future near-infrared spectrographs with precision goals of 1 m s⁻¹ (Jones et al. 2008).