RADIO EMISSION AND ORBITAL MOTION FROM THE CLOSE-ENCOUNTER STAR–BROWN DWARF BINARY WISE J072003.20–084651.2^*

WISE J0720–0846 was observed with the VLA (project code 14B–313, PI Burgasser) in the compact C-configuration (baselines of 0.035–3.4 km) on 2014 November 11 from UT 08:48:50 to 14:47:48. The WIDAR correlator was set up for C-band continuum observations with two basebands, each having eight 128 MHz sub-bands centered at 5.0 and 6.0 GHz, for a frequency range of 4.488–6.512 GHz and a total bandwidth of ≈2 GHz. Each sub-band had four polarization products (RR, LL, RL, LR) and sixty-four 2 MHz channels; a 5 s dump rate was used. After observation of our primary calibration source 3C48 to set the absolute flux scale and measure the complex bandpass, we conducted a sequence of 4 minute cycles, with 3 minutes on WISE J0720–0846 and 1 minute on the gain calibrator QSO J0730–1141. This observational strategy optimizes image quality by frequent monitoring (and hence correction) of phase fluctuations.

All data were reduced with the Astronomical Image Processing Software package (AIPS; Greisen 2003, p. 109) following best practices for wide-band radio data reduction. Radio-frequency interference (RFI) was present throughout the observation, being especially persistent and strong in the sub-bands covering 5.782–6.512 GHz, and also present at varying levels in several other sub-bands. All data affected by RFI were removed. In addition, sub-band 0 of each baseband failed to produce robust data, and these measurements were also removed. After data flagging, our average frequency was 5.27 GHz and our total bandwidth was reduced to ≈1.2 GHz.

The final cleaned and calibrated data were analyzed by imaging and performing time-series analysis on the uv-data. Figure 1 shows imaging of the entire data set (all unflagged times and frequencies) at the coordinates of WISE J0720–0846, which reveals a weak, elongated source. This is interpretted to be a blend between weak quiescent emission from WISE J0720–0846 and an unrelated faint background source, each having similar flux densities of ≈15 μJy.⁵ Using the AIPS task DFTPL and the procedures described in Osten & Wolk (2009), we analyzed the time-series data at the peak flux position, shown in Figure 2. This series reveals weak but nonzero emission over the entire course of the observation, with a short burst of emission at UT 13:21:37. Masking out the burst, we measure a mean quiescent flux density of 15 ± 3 μJy with no evidence of statistically significant variability.⁶ This is the weakest quiescent emission detected for a very low mass dwarf to date, comparable to the quasi-quiescent emission reported for the T6.5 dwarf 2MASS J10475385 + 2124234 (Williams et al. 2013, 16 ± 5 μJy). The apparent flux density translates into a specific radio luminosity of log₁₀L_ν = (7 ± 2) × 10¹¹ erg s⁻¹ Hz⁻¹ at the 6.0 ± 1.0 pc distance of WISE J0720–0846, and log₁₀ν_pkL_ν/L_bol = −8.49 ± 0.19 assuming log₁₀L_bol/L_⊙ = −3.60 ± 0.05 (B15; I15) andν_pk = 5.28 GHz. The specific luminosity is 1–2 orders of magnitude below measurements of previously detected late M and L dwarfs (Antonova et al. 2013) and the relative luminosity falls below the activity-rotation relation of McLean et al. (2012) given this source's v sin i = 8.6 ± 0.8 km s⁻¹ (Section 3.4). Indeed, in terms of rotation and radio power, WISE J0720–0846 is similar to the M7 dwarf VB 8 (Krishnamurthi et al. 1999) and the M8 dwarf DENIS J1048–3956 (Burgasser & Putman 2005), both nearby late M dwarfs with weak but variable magnetic emission and modest rotation rates. The weakness of the quiescent emission prevents us from determining either its spectral behavior across the 4.4–6.5 GHz band or its polarization.

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A close-up view of the radio burst around UT 13:21 is shown in Figure 2. The burst appears to have begun during observations of the secondary calibrator, so our data sample it only at or after its peak emission. Reimaging of the radio data in the period around this burst shows it to be associated with a bright point source at the coordinates expected for the brown dwarf (Figure 3).⁷ The burst is highly polarized, with emission in Stokes I of 175 ± 28 μJy and Stokes V of –150 ± 28 μJy, implying left-handed circular polarization of 84${}_{-20}^{+16}$% (1σ equivalent uncertainties); i.e., consistent with full polarization. There is no evidence of spectral variation in the burst to within our measurement uncertainties.

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We modeled the time series emission as both a rotating spot ("pulse") feature with a Gaussian profile, and as an exponentially declining flare, ${f}_{\nu }\propto {e}^{-t/\tau }$ after peak emission. The former yields a peak flux density of 310 ± 40 μJy, ≈20 times greater than the quiescent emission, and consistent with a total emitted energy of (1.3 ± 0.4) × 10²⁴ erg over the 4.5–6.5 GHz band (assuming a flat spectrum). The flare model yields a decay time constant of τ = 3.5 ± 0.9 minutes and, depending on whether the burst initiated at the start or end of the calibration period, a total emitted energy of (1–3) × 10²⁴ erg.

The burst emission occurs for at most 3 minutes during our 5 hr of observation of WISE J0720–0846, or 1% of the total time on-source. This is consistent with the flaring duty cycle inferred from the aperiodic white light bursts reported in B15. Thus, we favor an infrequent flare mechanism for this emission as well. We nevertheless note that the pulse/spot model implies an emission region ≈1°–2° in longitudinal extent (assuming a rotation period of ≈14 hr; see Section 4), which is consistent with previous periodic pulse detections (Hallinan et al. 2007; Williams & Berger 2015). Longer-term monitoring would be required to distinguish between the flare and pulse hypotheses.

WISE J0720–0846 was monitored over four nights on 2014 February 18, 20, 22 and 24 (UT) using the facility CCD camera on the 1 m Nickel telescope at Lick Observatory (Table 1). The CCD was configured for 2 × 2 binning for a pixel scale of 037 pixel⁻¹. After acquisition and centering at I-band, WISE J0720–0846 was monitored without dithering through the narrow-band Hα filter (λ_c = 6557 Å, Δλ = 15 Å), with integration times of 900 s on February 18 and 20 and 300 s on February 22 and 24. Total monitoring periods per night spanned 2.63–3.97 hr, but due to overheads and pauses during occasional clouds the on-source time totalled 7.25 hr over the entire observing run. Bias frames and quartz flat field lamps were also acquired each night for detector calibration.

Data were reduced using standard image reduction techniques. Aperture photometry of WISE J0720–0846 and nearby non-saturated stars was measured using a variable aperture scaled to encapsulate 80% of each source's peak brightness, and an annulus of 63–100 pixels (23''–36'') was used to subtract foreground emission. We did not observe a photometric calibrator during these observations, so no attempt was made to measure absolute Hα fluxes. Instead, we used the mean flux of non-variable stars in the field of view to compute a reference light curve, and used this to normalize the photometry for WISE J0720–0846 over the course of each night. Uncertainties were dominated by source photometry, of order 20%–50%.

We found no significant flux variations over the course of the four nights of observation. At the measured noise level, we can rule out bursts 2–3 times above quiescent emission. B15 reported a single Hα flare with a line flux five times greater than quiescent emission, and an order of magnitude brighter than the local continuum. Such a flare would have been easily detected in our observations had it occured. Assuming a flare period of ≈5 minutes, we infer a flare indicidence rate of <1%, similar to the white light flare rates reported in B15 and the radio flare reported here. All of these observations reinforce the conclusion that WISE J0720–0846 is a weakly active and infrequently bursting source.

WISE J0720–0846 was re-observed with the Near-infraRed Camera 2 (NIRC2) and LGSAO system (van Dam et al. 2006; Wizinowich et al. 2006) on the Keck II 10 m Telescope on 2015 January 11 (UT) in mostly clear and windy conditions with 08 seeing. The narrow field of view (FOV) camera was used to obtain dithered observations in broad-band MKO⁸  J, H and K_s filters, and the medium-band CH₄s filter sampling 1.54–1.65 μm. The R = 16.8 mag field star USNO 0812–0137390 was used to correct for tip-tilt aberrations. We achieved better Strehl ratios than observations reported in B15, 10%–20% depending on wavelength. This allowed us to easily resolve the candidate companion reported in that study in all four filters, as shown in Figure 4.

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To extract relative photometry and astrometry, we performed point source function (PSF) fitting of each individual image using a two-dimensional asymmetric Moffat profile optimized to the PSF of the primary component (i.e., with the secondary masked). Measurements are reported in Table 2. We find a separation of 197 ± 3 mas at position angle 2567 ± 06, wider than, and at a marginally distinct position angle as, the candidate source previously reported (139 ± 14 mas at 262° ± 2°). As the source detected in these data would have been easily resolved in prior observations, we conclude that statistically significant relative motion has been observed between the two epochs.

Table 2.  Resolved Photometry and Astrometry of WISE J0720–0846

Parameter	Value
ΔJ	2.92 ± 0.07
${\rm{\Delta }}{\mathrm{CH}}_{4}{\rm{s}}$	3.05 ± 0.08
ΔH	3.85 ± 0.11
ΔKs	4.07 ± 0.20
ρ (mas)	197 ± 3
ρ (AU)	1.19 ± 0.21
PA (°)	256.7 ± 0.6
Primary SpTa	M9.5 ± 0.5
Secondary SpTa	T5.5 ± 0.5

Note.

^aBased on spectral template fitting with templates constrained to have the same relative flux scaling as measured in the JHK_s NIRC2 bands.

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Relative motion can be due to differential motion between two physically unrelated sources or orbital motion in a gravitationally bound binary. Figure 5 displays the estimated center of mass motion and component positions between our 2014 and 2015 imaging data, assuming systemic astrometry from B15 and q = 0.4 (see below). It is clear that the predominant motion of both sources is co-aligned with the proper motion of the system, particularly in declination; the change in position angle is inconsistent with the secondary being an unmoving background source at the 16σ confidence level. We therefore determine that WISE J0720–0846 is a co-moving binary system, and identify significant astrometric orbital motion over a one year period.

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Our new observations provide greatly improved relative photometry of the two components in the four bands measured, and refine the coarse ΔH estimate from B15 (Table 2). There is a significant difference in relative brightnesses in the CH₄s and H-band filters consistent with strong CH₄ absorption in the secondary. We used the relative photometry to better constrain the spectral types of the components through spectral template fitting. Following the procedures described in Burgasser et al. (2011), we combined 512 M7-L1 and 125 T0-T7 spectral templates from the SpeX Prism Library (SPL; Burgasser 2014), scaled so that that relative spectrophotometry agreed to within 1σ of the NIRC2 JHK_s measurements.⁹ We compared the 16394 binary templates that satisfied these constraints to the combined light SpeX spectra of WISE J0720–0846 from both Kirkpatrick et al. (2014) and B15, following the methods described in the latter paper. Figure 6 shows the best fitting template to the B15 spectrum. Both analyses yield identical results, with component types M9.5 ± 0.5 and T5.5 ± 0.5. The secondary is a half subtype later than, but formally consistent with, the classification reported in B15.

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New high resolution optical and near-infrared spectroscopy of WISE J0720–0846 were obtained with the Near InfraRed Spectrometer (NIRSPEC; McLean et al. 2000) on the Keck II telescope on 2014 December 8 (UT), and with the Hamilton echelle spectrograph (Vogt 1987) on the Lick Observatory Shane 3 m telescope on 2015 March 10 (UT). Data were acquired and reduced as described in B15. The NIRSPEC spectrum (Figure 7) is of high quality (median S/N = 82) and similar to data reported in B15. The Hamilton data had a somewhat lower signal-to-noise ratio (S/N) than prior observations (S/N ≈ 5 at 7500 Å) due to a shorter total integration of 3000 s.

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Both spectra were analyzed as described in B15 for radial motion and, for the Hamilton data, Hα emission equivalent width (EW). Despite the low S/N, cross-correlation of the Hamilton spectrum with contemporaneous observations of the RV standard GJ 251 yielded an RV = +86.4 ± 0.5 km s⁻¹, significantly different than the mean motion reported in B15 (+82.5 ± 0.4 km s⁻¹). Marginal Hα emission was observed in these data, with EW = −2 ± 1 Å, consistent with the lowest emission states observed in B15.

For the NIRSPEC data, we re-analyzed the observations reported here and the 2014 observations using an updated Markov Chain Monte Carlo (MCMC) adaptation of the forward-modeling method described in Blake et al. (2010) and B15. We used the BT-Settl atmosphere models (Allard et al. 2012) with updated solar abundance values from Caffau et al. (2011) over an effective temperature (T_eff) range of 1600–2900 K and surface gravity (log g) range of 4.5–5.5 (cgs). The Solar atlas of Livingston & Wallace (1991) was used to model telluric absorption features. Table 3 summarizes the RV and rotational velocities (v sin i) inferred from these analyses, while Figure 7 shows the best-fit model, with T_eff = 2700 K and log g = 5.5, to the most recent NIRSPEC observations. The rotational velocity measurements are mutually consistent, with a mean value of v sin i = 8.6 ± 0.4 km s⁻¹. However, like the Hamilton observations, the most recent NIRSPEC RV measurement of +85.2 ± 0.2 km s⁻¹ is significantly distinct from the mean RV reported in B15 (+83.7 ± 0.4 km s⁻¹) and follows the trend of increasing recessional motion. With these independent measurements, we conclude that WISE J0720–0846A is an RV variable gravitationally perturbed by its brown dwarf secondary.

We note that the best-fitting models for the forward-modeling analyses were consistently in the ranges T_eff = 2500–2800 K and log g = 5.0–5.5. The temperatures are somewhat higher than those expected for an M9.5 dwarf (T_eff ≈ 2300 K; Stephens et al. 2009), but the surface gravity is consistent with the lack of Li i absorption and low-surface gravity spectral features, indicating an age ≳1 Gyr. We verified that lower temperature and lower surface gravity models yielded identical RVs, so this measurement appears to be insensitive to the specific model over the parameter range examined here. A robust investigation of the atmospheric parameters of this source is deferred to a later study.

With WISE J0720–0846AB verified as a gravitationally bound binary with orbital motion detected in all three spatial dimensions, we can begin to constrain the orbital properties of the system and the physical properties of its components. We adapted the MCMC analysis described in Burgasser et al. (2012) to include both RV and relative astrometry measurements. We employed an orbit model with nine parameters,

$$\begin{eqnarray}&&{\boldsymbol{\theta }}=\left(P,a,e,i,\omega ,{\rm{\Omega }},{M}_{0},q,{V}_{\mathrm{COM}},d\right)\end{eqnarray} \tag{ 1 }$$

where P is the period of the orbit in years, a is the semimajor axis in AU, e is the eccentricity, i is the inclination, ω is the argument of periastron, Ω is the longitude of nodes, M₀ is the mean anomaly at epoch τ₀ = 2014.0896 (MJD¹⁰ = 56675.982), q ≡ M₂/M₁ is the system mass ratio, V_COM is the center of mass (systemic) RV in km s⁻¹, and d is the distance in pc. The primary RV as a function of time t, V₁(t), is

$$\begin{eqnarray}&&{V}_{1}(t)={K}_{1}\left[e\mathrm{cos}\omega +\mathrm{cos}(T(t)+\omega )\right]+{V}_{\mathrm{COM}}\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{K}_{1}=\displaystyle \frac{2\pi a\mathrm{sin}i}{P\sqrt{1-{e}^{2}}}\displaystyle \frac{q}{1+q}\end{eqnarray} \tag{ 3 }$$

and the true anomaly T(t) is related to the eccentric anomaly E(t) through

$$\begin{eqnarray}&&\mathrm{tan}\displaystyle \frac{T(t)}{2}=\sqrt{\displaystyle \frac{1+e}{1-e}}\mathrm{tan}\displaystyle \frac{E(t)}{2}\end{eqnarray} \tag{ 4 }$$

which is solved by Kepler's equation:

$$\begin{eqnarray}&&M(t)-{M}_{0}=2\pi \displaystyle \frac{t-{\tau }_{0}}{P}=E(t)-e\mathrm{sin}E(t).\end{eqnarray} \tag{ 5 }$$

The angular separation vector from primary component to secondary component, ${\boldsymbol{\rho }}$ = (Δα(t), Δδ(t)) is determined from

$$\begin{eqnarray}&&{\rm{\Delta }}\alpha (t)=\displaystyle \frac{a}{d}\left[A(\mathrm{cos}E(t)-e)+F\sqrt{1-{e}^{2}}\mathrm{sin}E(t)\right]\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\delta (t)=\displaystyle \frac{a}{d}\left[B(\mathrm{cos}E(t)-e)+G\sqrt{1-{e}^{2}}\mathrm{sin}E(t)\right]\end{eqnarray} \tag{ 7 }$$

where Δα and Δδ are the angular separations on sky measured in arcseconds, and A, B, F and G are the Thiele-Innes constants (Innes 1907; van den Bos 1927):

$$\begin{eqnarray}&&A=\mathrm{cos}\omega \mathrm{cos}{\rm{\Omega }}-\mathrm{sin}\omega \mathrm{sin}{\rm{\Omega }}\mathrm{cos}i\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&B=\mathrm{cos}\omega \mathrm{sin}{\rm{\Omega }}+\mathrm{sin}\omega \mathrm{cos}{\rm{\Omega }}\mathrm{cos}i\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&F=-\mathrm{sin}\omega \mathrm{cos}{\rm{\Omega }}-\mathrm{cos}\omega \mathrm{sin}{\rm{\Omega }}\mathrm{cos}i\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&G=-\mathrm{sin}\omega \mathrm{sin}{\rm{\Omega }}+\mathrm{cos}\omega \mathrm{cos}{\rm{\Omega }}\mathrm{cos}i.\end{eqnarray} \tag{ 11 }$$

Note that the total system mass (M_tot = a³/P² in solar masses) and component masses (M₁ = M_tot/[1−q], M₂ = qM₁) could in principle be uniquely inferred from these parameters if sufficiently constrained.

We selected an initial parameter set that visually coincided with the observations through manual experimentation. We then computed a chain of 10⁷ parameter sets using the Metropolis–Hastings algorithm (Metropolis et al. 1953; Hastings 1970), at each step varying parameter ${\theta }_{j}\to {\theta }_{j}^{\prime }$ by drawing a random offset from a normal distribution

$$\begin{eqnarray}&&P({\theta }_{j}^{\prime }| {\theta }_{j})\propto {e}^{-\frac{{({\theta }_{j}^{\prime }-{\theta }_{j})}^{2}}{2\beta {j}^{2}}}\end{eqnarray} \tag{ 12 }$$

where ${\boldsymbol{\beta }}$ is the set of jump steps.¹¹ We applied additional parameter constraints of 0.5 year < P < 30 year, e < 0.8 and 4 pc < d < 8 pc to eliminate improbable regions of parameter space; note that the eccentricity cutoff is beyond the e ≈ 0.6 limit suggested in empirical data by Dupuy & Liu (2011). We also limited the component masses to 0.055 M_⊙ < M₁ < 0.15 M_⊙ given the spectral classification of the primary and lack of Li i absorption in its optical spectrum (B15; I15), and M₂ < 0.075M_⊙ given the substellar nature of the secondary. Orbit models were compared to the data using a χ² statistic that combined both RV and relative astrometric measurements:

$$\begin{eqnarray}{\chi }^{2} & = & \displaystyle \sum _{i=1}^{{N}_{\mathrm{RV}}}\displaystyle \frac{{({\mathrm{RV}}_{i}^{(\mathrm{obs})}-{\mathrm{RV}}_{i}^{(\mathrm{model})})}^{2}}{{\sigma }_{\mathrm{RV},i}^{2}}\\ & & +\displaystyle \sum _{j=1}^{{N}_{\mathrm{ast}}}\displaystyle \frac{{({\rm{\Delta }}{\alpha }_{j}^{(\mathrm{obs})}-{\rm{\Delta }}{\alpha }_{j}^{(\mathrm{model})})}^{2}}{{\sigma }_{{\rm{\Delta }}\alpha ,j}^{2}}\\ & & +\displaystyle \sum _{j=1}^{{N}_{\mathrm{ast}}}\displaystyle \frac{{({\rm{\Delta }}{\delta }_{j}^{(\mathrm{obs})}-{\rm{\Delta }}{\delta }_{j}^{(\mathrm{model})})}^{2}}{{\sigma }_{{\rm{\Delta }}\delta ,j}^{2}}\end{eqnarray} \tag{ 13 }$$

where N_RV and N_ast are the number of RV and astrometric measurements, respectively, and σ the measurement errors. Model values were calculated at the same epochs as the observations. The criterion to adopt successive parameter sets was $U(0,1)\leqslant {e}^{-0.5({\chi }_{({j}^{\prime })}^{2}-{\chi }_{(j)}^{2})}$, where U(0, 1) is a random number drawn from a uniform distribution between 0 and 1. We compared models to data after each individual parameter change rather than changing all parameters, a procedure we found greatly improved the acceptance rate, which declined from 10% to 1% through the chain. Separate fits were made to the NIRSPEC + NIRC2 and Hamilton + NIRC2 measurements, and to all of the data. Note that in the last case we did not take into account a possible velocity offset between the RV datasets (Ford 2005).

To test for convergence, we monitored parameter autocorrelation and the evolution of subchain variance through the chain. We also generated M = 10 chains of length 2n = 10⁶ steps on the Hamilton + NIRC2 data, varying the initial parameter set ${{\boldsymbol{\theta }}}_{\mathrm{init}}$ as

$$\begin{eqnarray}&&{{\boldsymbol{\theta }}}_{\mathrm{init}}={{\boldsymbol{\theta }}}_{p}+N(0,1){{\boldsymbol{\sigma }}}_{p}\end{eqnarray} \tag{ 14 }$$

where ${{\boldsymbol{\theta }}}_{p}$ and ${{\boldsymbol{\sigma }}}_{p}$ are the median and half quantile ranges of the posterior parameter distributions (see below) and N(0, 1) is a normal distribution centered on zero with unit variance. We quantified the within chain and between chain variance of all parameters using the Gelman & Rubin (1992) scale reduction factor

$$\begin{eqnarray}&&{R}_{k}^{2}=1+\displaystyle \frac{1}{n}\left(\displaystyle \frac{{B}_{k}}{{W}_{k}}-1\right)\end{eqnarray} \tag{ 15 }$$

where ${W}_{k}={\bar{\xi }}_{k}$ is the within chain variance for parameter θ_k, equal to the average of parameter variances ${\xi }_{{jk}}=\displaystyle \frac{1}{n-1}{\sum }_{i=1}^{n}{({\theta }_{{ijk}}-{\bar{\theta }}_{{jk}})}^{2}$ for each of the ten chains using the second half of the chains as the sample; and ${B}_{k}=\displaystyle \frac{n}{M-1}{\sum }_{j=1}^{M}{({\bar{\theta }}_{{jk}}-{\bar{\bar{\theta }}}_{k})}^{2}$ is the between chain variance, equal to the variance in parameter averages ${\bar{\theta }}_{{jk}}$. Scale reduction factors were within 5% of unity for all modeled parameters with the exception of P (1.35), a (1.37) and ω (1.51). As described below, these three parameters were weakly constrained, and this analysis suggests that the primary MCMC chains may not converge for our limited datasets, re-emphasizing that the orbital results presented here should be considered preliminary.

Figure 8 shows the best-fit orbits from our separate analyses of the RV and imaging datasets. Table 4 lists the best-fit orbital and component parameters, as well as the median and 16% and 84% quantiles of the parameter distributions, after eliminating the first 10% of each chain. Figures 9 through 11 display the distributions and correlations of parameters P, a, e, i, q and M_tot for all three datasets.

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Table 4.  Parameters from Orbital Analysis

	NIRSPEC+NIRC2		Hamilton+NIRC2		All Data
Parameter	Best-fit	Median	Best-fit	Median	Best-fit	Median
Modeled Parameters
Best χ2	5.83	⋯	2.84	⋯	27.6	⋯
Pa (year)	10.4	6.1${}_{-2.6}^{+5.1}$	2.9	4.1${}_{-1.3}^{+2.7}$	3.1	4.5${}_{-1.4}^{+2.7}$
a (AU)	2.4	1.7${}_{-0.5}^{+0.9}$	1.1	1.3${}_{-0.3}^{+0.5}$	1.1	1.4${}_{-0.3}^{+0.5}$
ea	0.79	0.72${}_{-0.09}^{+0.06}$	0.79	0.77${}_{-0.04}^{+0.02}$	0.80	0.76${}_{-0.04}^{+0.03}$
i (°)	92.1	94.1${}_{-1.7}^{+2.2}$	92.8	93.6${}_{-1.4}^{+1.6}$	94.9	94.4${}_{-1.6}^{+1.8}$
ω (°)	109	83${}_{-24}^{+20}$	70	76${}_{-17}^{+17}$	54	74${}_{-16}^{+16}$
Ω (°)	81.7	82.2${}_{-2.1}^{+2.3}$	82.4	82.7${}_{-2.1}^{+2.1}$	82.2	83.3${}_{-2.2}^{+2.2}$
M0 (°)	5	20${}_{-11}^{+15}$	10	13${}_{-6}^{+8}$	24	17${}_{-7}^{+8}$
q	0.36	0.61${}_{-0.18}^{+0.21}$	0.61	0.77${}_{-0.17}^{+0.15}$	0.91	0.75${}_{-0.17}^{+0.16}$
VCOM (km s−1)	87.2	87.3${}_{-0.9}^{+0.9}$	87.3	87.4${}_{-0.8}^{+0.8}$	87.4	87.5${}_{-0.7}^{+0.7}$
da (pc)	4.5	6.7${}_{-1.0}^{+0.8}$	4.5	5.4${}_{-0.7}^{+0.8}$	6.8	6.2${}_{-0.7}^{+0.7}$
Inferred Parameters
Mtot (M⊙)	0.13	0.13${}_{-0.03}^{+0.04}$	0.17	0.15${}_{-0.03}^{+0.03}$	0.16	0.15${}_{-0.03}^{+0.03}$
M1(M⊙)	0.098	0.080${}_{-0.018}^{+0.032}$	0.105	0.082${}_{-0.015}^{+0.026}$	0.081	0.081${}_{-0.016}^{+0.028}$
M2 (M⊙)	0.035	0.051${}_{-0.014}^{+0.013}$	0.064	0.064${}_{-0.010}^{+0.008}$	0.074	0.062${}_{-0.011}^{+0.009}$
K1 (km s−1)	3.0	4.4${}_{-1.0}^{+1.2}$	7.2	6.3${}_{-1.0}^{+1.1}$	8.7	6.0${}_{-0.8}^{+1.1}$
K2 (km s−1)	8.4	7.3${}_{-1.6}^{+2.2}$	11.9	8.5${}_{-1.6}^{+2.1}$	9.5	8.2${}_{-1.6}^{+2.1}$

Note.

^aParameter was constrained to a limited value range in MCMC analysis.

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The best-fit orbits are exceptionally good fits for the individual RV datasets, with χ² = 5.83 for nine data points (zero degrees of freedom) for NIRSPEC + NIRC2 and χ² = 2.84 for twelve data points (three degrees of freedom) for Hamilton + NIRC2. Note that these low χ² values largely reflect the underconstained nature of the solution in orbital phase; i.e., a relatively large family of solutions is formally consistent with these data. The combined dataset yields a somewhat poorer best fit, with modest disagreement between contemporary NIRSPEC and Hamilton RVs driving the χ² values. Nevertheless, this solution is nearly identical to that of the Hamilton + NIRC2 dataset. While the best-fit solutions between the NIRSPEC + NIRC2 and the Hamilton + NIRC2 datasets are distinct, the median parameters are consistent within the uncertainties for all datasets. Since parameters inferred from the Hamilton + NIRC2 data are best constrained, we refer to these in the following discussion.

Several of the parameters are very well constrained, most notably the orbital inclination (i = 936⁺¹⁶₋₁₄), the longitude of ascending node (Ω = 827 ± 21) and the center of mass RV (V_COM = +87.4 ± 0.8 km s⁻¹). The tight constraints on i and Ω stem from the near-radial motion of the secondary between the two NIRC2 epochs. The position angle of the secondary changed by only 5° ± 2° between these two observations but the secondary has moved 40% further away; this is possible only when the orbit is viewed very close to edge-on or e ≈ 1. As discussed below, even with an eccentricity at the proscribed limits, the inclination remains close to 90°. Similarly, Ω is constrained by the orientation of the orbit on the sky. The tight constraint on V_COM arises from the small variance in observed RVs and consistency in the values measured in the earlier epochs. The NIRSPEC + NIRC2 and Hamilton + NIRC2 datasets yield nearly identical values for all three of these parameters.

Parameter distributions for P, e and q are more weakly constrained, and the latter two abut the imposed limits, emphasizing that we do not yet have sufficient coverage of the orbit of WISE J0720–0846AB to robustly determine them. To gain some insight on how the quality of the fits vary as e changes, we performed additional MCMC chains on Hamilton + NIRC2 data using models with fixed e = 0.2, 0.4, 0.6 and 0.8. Figure 12 and Table 5 display the results of this experiment. We found that all parameters remain consistent with the unconstrained MCMC analysis, although q and M₂ values are modestly smaller for the e = 0.2. Parameter uncertainties are considerably larger for the e = 0.8 case, in part reflecting the lower χ² values of viable solutions. It is clear that the e = 0.2 and e = 0.4 models are poor representations of the RV data, with best-fit χ² values that are significantly worse and can be eliminated with >95% confidence as compared to the unconstrained model.¹² This analysis supports an eccentic orbit for WISE J0720–0846AB, but a firm constraint on its value should emerge with further monitoring.

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Table 5.  Parameters from Orbital Analysis of Hamilton + NIRC2 Data with Eccentricity Fixed

Parameter	e = 0.2	e = 0.4	e = 0.6	e = 0.8
Best χ2	39.3	25.1	10.4	2.8
Pa (year)	4.4${}_{-0.7}^{+0.8}$	4.3${}_{-0.8}^{+1.0}$	4.2${}_{-0.9}^{+1.2}$	3.8${}_{-1.3}^{+3.0}$
a (AU)	1.40${}_{-0.16}^{+0.18}$	1.38${}_{-0.18}^{+0.23}$	1.4${}_{-0.2}^{+0.3}$	1.3${}_{-0.3}^{+0.6}$
i (°)	94.1${}_{-1.3}^{+1.5}$	94.4${}_{-1.3}^{+1.4}$	94.2${}_{-1.3}^{+1.5}$	93.5${}_{-1.6}^{+2.0}$
qa	0.69${}_{-0.20}^{+0.19}$	0.80${}_{-0.18}^{+0.14}$	0.82${}_{-0.17}^{+0.13}$	0.73${}_{-0.18}^{+0.16}$
VCOM (km s−1)	85.9${}_{-0.6}^{+0.5}$	86.5${}_{-0.5}^{+0.5}$	87.0${}_{-0.5}^{+0.5}$	87.3${}_{-0.9}^{+0.9}$
Mtot (M⊙)	0.14${}_{-0.03}^{+0.03}$	0.14${}_{-0.02}^{+0.03}$	0.15${}_{-0.02}^{+0.02}$	0.15${}_{-0.03}^{+0.03}$
M1 (M⊙)	0.083${}_{-0.018}^{+0.032}$	0.079${}_{-0.014}^{+0.024}$	0.079${}_{-0.013}^{+0.022}$	0.083${}_{-0.017}^{+0.028}$
M2 (M⊙)	0.059${}_{-0.014}^{+0.011}$	0.064${}_{-0.011}^{+0.008}$	0.066${}_{-0.010}^{+0.007}$	0.062${}_{-0.011}^{+0.009}$

Note.

^aParameter was constrained to a limited value range in MCMC analysis.

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