The Young Substellar Companion ROXs 12 B: Near-infrared Spectrum, System Architecture, and Spin–Orbit Misalignment^*

We acquired moderate-resolution (R ≡ λ/δλ ≈ 6000) J-band spectroscopy of ROXs 12 B spanning 1.15–1.36 μm with the Near-infrared Integral Field Spectrometer (NIFS; McGregor et al. 2003) at the Gemini-North 8.1 m telescope on UT 2016 April 18 and 22 (Gemini Program ID: GN-2016A-Q-37). The facility adaptive optics (AO) system ALTAIR (Christou et al. 2010) provided diffraction-limited correction using the Gemini laser guide star system with ROXs 12 as a bright (R = 13.5 mag) tip-tilt reference. NIFS has a 3'' × 3'' field of view with $0\buildrel{\prime\prime}\over{.} 1\times 0\buildrel{\prime\prime}\over{.} 04$ rectangular spaxels. To better sample the point-spread function (PSF) wing of the host star, we rotated the instrument so that the short side of the spaxels was oriented in the direction of the ROXs 12 A and B P.A. with the host star located immediately off the detector (see Figure 2). Our observations of ROXs 12 B were then carried out in an ABBA pattern by nodding $1\buildrel{\prime\prime}\over{.} 5$ orthogonal to the binary P.A.

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We acquired a total of 40 minutes (eight exposures of 300 s each) with the J-band grating centered at 1.25 μm and ZJ filter on UT 2016 April 18 in queue mode. On UT 2016 April 22, an additional 70 minutes (14 exposures of 300 s each) were taken with the same configuration. On both nights, the A0V standard HD 145127 was targeted at a similar airmass for telluric correction. Our observations are summarized in Table 1.

Table 1.  Spectroscopic Observations

Object	Date	Telescope/	Filter	Slit Width	Plate Scale	Tot. Exp.	Resolution	Standarda
	(UT)	Instrument		(arcsec)	(mas pixel−1)	(minutes)	(=λ/$\delta \lambda$)
ROXs 12 B	2016 Apr 18+22	Gemini-North/NIFS	J	⋯	100 × 40	110	6000	HD 145127
ROXs 12 B	2016 May 22	Keck/OSIRIS	Kbb	⋯	50	50	3800	HIP 69021
ROXs 12 B	2016 May 22	Keck/OSIRIS	Hbb	⋯	50	60	3800	HIP 93691
ROXs 12 B	2016 May 22	Keck/OSIRIS	Jbb	⋯	50	90	3800	HIP 93691
ROXs 12 A	2011 Jun 28	Keck/HIRES	KV418	0.861	⋯	5	48000	⋯
ROXs 12 A	2016 Jul 26	McDonald 2.7 m/IGRINS	⋯	0.98	⋯	30	45000	HD 155379
2M1626–2527	2011 Apr 29	IRTF/SpeX	⋯	0.3	⋯	8	2000	HD 144925
2M1626–2527	2014 May 21	Mayall/RC-Spec	GG495	1.5	⋯	20	2600	HZ 44
2M1626–2527	2015 May 03	Keck/HIRES	GG475	0.861	⋯	10	48000	⋯
2M1626–2527	2016 Jul 26	McDonald 2.7 m/IGRINS	⋯	0.98	⋯	30	45000	HD 155379

Note.

^aTelluric or radial velocity standard.

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Basic data reduction was carried out with NIFS reduction packages in IRAF that include flat-fielding, bad-pixel interpolation, sky subtraction, and image rectification to data cubes. ROXs 12 B is clearly visible in each cube but overlaps with the PSF wing from the host star. To remove this low-level contaminating flux, we performed PSF subtraction in the same fashion as in Bowler et al. (2014) for NIFS data of ROXs 42B b. For each column of each spectral channel, we masked out the companion and then fit and subtracted various parameterized PSF models to the 2D wing of ROXs 12 A using the Levenberg–Marquardt least-squares curve-fitting package MPFIT (Markwardt 2009). Gaussian, Lorentzian, and Moffat PSF models were tested to examine their influence on the extracted spectra. All three profiles resulted in similar spectra; ultimately, we adopted the Gaussian model, as this produced the lowest systematic oversubtraction or undersubtraction in the region surrounding ROXs 12 B (Figure 2). After PSF subtraction, we extracted the spectra from each cube using aperture photometry and median-combined them after scaling individual spectra to their median values. Telluric correction was performed using the xtellcor_general routine in the Spextool data reduction package for the InfraRed Telescope Facility (IRTF) SpeX instrument (Vacca et al. 2003; Cushing et al. 2004). These steps were carried out separately for each night, and the resulting spectra were subsequently combined by calculating the weighted mean and uncertainty of both spectra. Note that standards were taken immediately before and after the science observations, and the signal-to-noise ratio (S/N) levels were low for individual spectra, so we performed telluric correction on the coadded spectra from each night instead of carrying this out for each separate exposure. The final NIFS spectrum has an effective on-source integration time of 110 minutes and is shown in Figure 3 after having been dereddened by A_V = 1.8 mag—the extinction to ROXs 12 A measured by Rizzuto et al. (2015) based on moderate-resolution optical spectroscopy—following the extinction curve from Fitzpatrick (1985).

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We targeted ROXs 12 B with the OH-Suppressing InfraRed Imaging Spectrograph (OSIRIS; Larkin et al. 2006; Mieda et al. 2014) using natural guide star adaptive optics (NSG AO; Wizinowich 2013) at the Keck I telescope on 2016 May 22 UT. Broadband filters were used with the 50 mas spaxel^–1 plate scale, producing a 16 × 64 spaxel (08 × 32) rectangular field of view with a spectral resolution of R ≈ 3800. The sky was clear with seeing between 05 and 08 throughout the night. To minimize contamination from the host star, we oriented the long axis of the array to be orthogonal to the binary P.A. Our observations consisted of nodded ABBA sequences with about $1^{\prime\prime}$ offsets for pairwise sky subtraction. A total of 50 minutes of integration time was acquired in the Kbb band (10 300 s exposures), 60 minutes in the Hbb band (12 300 s exposures), and 90 minutes in the J band (18 300 s exposures). Multiple A0V standards were targeted between filter changes at an airmass similar to that of the science observations (see Table 1).

Basic data reduction was carried out using the OSIRIS data reduction pipeline. Images were flat-fielded, corrected for bad pixels, and assembled into data cubes using the latest rectification matrices provided by Keck Observatory. We then extracted the spectra from the data cubes using aperture photometry at each wavelength at the centroided location of the target in median-collapsed cubes. Spectra were median-combined after scaling them to their individual median levels. Telluric correction was then carried out with the xtellcor_general routine in Spextool (Vacca et al. 2003; Cushing et al. 2004). Each spectral band was then flux-calibrated using the photometry of ROXs 12 B (Figure 4), which was derived by transforming the 2MASS photometry of ROXs 12 A to the MKO system using relations from Leggett et al. (2006) and then using the relative photometry of ROXs 12 AB from Kraus et al. (2014). The resulting photometry is reported in Table 2.

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Table 2.  Physical, Kinematic, and Photometric Properties of the ROXs 12 Triple System

Property	ROXs 12 A	ROXs 12 B	2M1626–2527	Reference
2MASS ID	J16262803–2526477	⋯	J16262774–2527247	(1)
B (mag)	16.172 ± 0.141	⋯	17.1 ± 0.7	(2)
V (mag)	14.346 ± 0.066	⋯	15.8 ± 0.3	(2)
R (mag)	13.5	⋯	15.3	(3)
$r^{\prime}$ (mag)	13.5	⋯	15.8	(4)
I (mag)	11.870 ± 0.04	⋯	12.91 ± 03	(5)
${J}_{2{\rm{MASS}}}$(mag)	10.282 ± 0.024	15.82 ± 0.03a	11.021 ± 0.024	(1), (6)
${H}_{2{\rm{MASS}}}$ (mag)	9.386 ± 0.026	14.83 ± 0.03a	9.930 ± 0.026	(1), (6)
${K}_{s,2{\rm{MASS}}}$ (mag)	9.099 ± 0.025	14.14 ± 0.03a	9.211 ± 0.025	(1), (6)
$L^{\prime}$ (mag)	[9.1 ± 0.1]b	13.2 ± 0.1	[9.2 ± 0.1]b	(6)
W1 (mag)	8.805 ± 0.023	⋯	8.360 ± 0.023	(7)
W2 (mag)	8.712 ± 0.021	⋯	7.831 ± 0.021	(7)
W3 (mag)	8.393 ± 0.042	⋯	5.983 ± 0.017	(7)
W4 (mag)	6.558 ± 0.089	⋯	3.784 ± 0.025	(7)
${\mu }_{\alpha }$cosδ (mas yr−1)	−12.7 ± 2.3	⋯	−11.9 ± 2.3	(8)
${\mu }_{\delta }$ (mas yr−1)	−29.0 ± 2.3	⋯	−30.5 ± 2.3	(8)
vrad (km s−1)c	−6.09 ± 0.16	⋯	−6.03 ± 0.16	(9)
vrad (km s−1)d	−5.67 ± 0.12	⋯	−6.97 ± 0.15	(9)
$v\sin i$ (km s−1)d	8.2 ± 0.7	⋯	4.5 ± 1.0	(9)
log(Lbol/L⊙)e	−0.57 ± 0.06	−2.87 ± 0.06	−0.81 ± 0.06	(9)
Mass (M⊙)	${0.65}_{-0.09}^{+0.05}$	0.0167 ± 0.0014	${0.5}_{-0.1}^{+0.1}$	(9)
Teff (K)	3900 ± 100	${3100}_{-500}^{+400}$	3700 ± 150	(9)
Prot (d)	9.1 ± 0.4	⋯	3.30 ± 0.05	(9)
Age (Myr)	${6}_{-2}^{+4}$	${6}_{-2}^{+4}$	${8}_{-4}^{+7}$ Myr	(9)
Spectral type	M0 ± 0.5	L0 ± 2	M1 ± 1	(9), (10)
i* (deg)f	${77}_{-9}^{+7}$	⋯	${17}_{-4}^{+5}$	(9)
R* (R⊙)g	1.14 ± 0.07	⋯	0.96 ± 0.08	(9)

Notes.

^aThe J- and H-band photometry for ROXs 12 B is on the MKO filter system. The K_s-band photometry assumes K_s ≈ K' to convert the contrast measurement from Kraus et al. (2014) to an apparent magnitude. ^bEstimated $L^{\prime}$-band magnitudes assuming K–$L^{\prime} =0.0\,\pm \,0.1$ mag for M0 and M1 spectral types (Golimowski et al. 2004). ^cFrom our IGRINS high-resolution near-infrared spectra. ^dFrom our HIRES high-resolution optical spectra. ^eAssumes a distance of 137 ± 10 pc. ^fLine-of-sight stellar inclination. ^gRadius from bolometric luminosity and effective temperature.

References. (1) 2MASS (Cutri et al. 2003); (2) APASS DR9 (Henden et al. 2016); (3) USNO-B1.0 (Monet et al. 2003); (4) CMC15 (Muiños & Evans 2014); (5) DENIS; (6) Kraus et al. (2014); (7) AllWISE Data Release (Cutri et al. 2014); (8) HSOY (Altmann et al. 2017), which includes data from Gaia DR1 (Gaia Collaboration et al. 2016); (9) this work; (10) Rizzuto et al. (2015).

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We obtained high-resolution (R = 45000) 1.45–2.45 μm spectra of ROXs 12 A and 2M1626–2527 on UT 2016 July 26 with the Immersion GRating INfrared Spectrometer (IGRINS; Park et al. 2014; Mace et al. 2016) at the McDonald Observatory's 2.7 m Harlan J. Smith Telescope to measure radial velocities for both components of this common proper-motion pair in order to test whether they share common space velocities. Six exposures of 300 s each were acquired while nodding in an ABBA pattern along the slit, totaling 30 minutes of integration for each target. The A0V star HD 155379 was observed on the same night. Spectra were extracted and reduced using version 2.1 of the IGRINS reduction pipeline¹³ following the description in Bowler et al. (2017). The S/N per pixel of both spectra ranges from 80 to 100. In summary, after bias subtraction and flat-fielding, the spectra were optimally extracted and cross-correlated with over 100 other IGRINS spectra of comparable spectral type taken over the past 2 yr to measure radial velocities. The final radial velocities for ROXs 12 A and 2M1626–2527 are −6.09 ± 0.16 and −6.03 ± 0.16 km s⁻¹, respectively, where the uncertainties are dominated by the transformation to the external reference frame (as opposed to relative uncertainties, which are 0.04 km s⁻¹).

We observed 2M1626–2527 with the Ritchey–Chretien Spectrograph (RC-Spec) using the T2KA CCD at the Kitt Peak National Observatory's 4 m Mayall telescope on UT 2014 May 21. The 15 × 98'' slit was used with the BL420 grating, resulting in a resolving power of R ≈ 2600 spanning 6300–9200 Å. The slit was oriented in a fixed north–south direction throughout the night. We targeted 2M1626–2527 near transit at an airmass of 1.86 to minimize effects from differential atmospheric refraction. However, because the slit was not oriented exactly at the parallactic angle, some slit losses may have occurred that may have affected the slope of our RC-Spec spectrum. These data are therefore useful for spectral classification but not for detailed reddening measurements. A single exposure was acquired with the GG495 filter and a total integration time of 1200 s.

The raw data was first bias-subtracted and flat-fielded to remove pixel-to-pixel sensitivity variations. Night sky lines were removed using median sky values in the spatial direction on either side of the spectrum. We then extracted the spectrum of the science target by summing flux in the spatial direction. The overall throughput response was then corrected using the spectrophotometric standard HZ 44. Finally, wavelength calibration was achieved using HeNeAr lamp observations taken throughout the night. The optical spectrum of 2M1626–2527 is discussed in more detail in Section 3.4.

2M1626â–2527 was targeted with the IRTF's SpeX instrument (Rayner et al. 2003) in short cross-dispersed mode on UT 2011 April 29. We used the $0\buildrel{\prime\prime}\over{.} 3$ slit rotated to the parallactic angle, which yielded a mean resolving power of ≈2000 across the 0.8–2.4 μm spectrum. Four nodded pairs were obtained with 60 s per exposure in an ABBA pattern. The A0V standard HD 144925 was observed prior to our science target. Standard data reduction, including spectral extraction, wavelength calibration, and telluric correction, was carried out using the Spextool package (Vacca et al. 2003; Cushing et al. 2004). The near-infrared spectrum of 2M1626–2527 is discussed in more detail in Section 3.4.

We obtained high-resolution (R ≈ 48,000) optical spectra of ROXs 12 A and 2M1626–2527 with the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) at Keck Observatory on UT 2011 June 28 and UT 2015 May 3, respectively. Both spectra were optimized for long wavelengths with the red cross-disperser. A single 300 s exposure of ROXs 12 A was taken with the C1 decker using the KV418 order-blocking filter, resulting in a slit size of 70 × 0861 on the sky and a wavelength range of 4310–8770 Å. For 2M1626–2527, we obtained one 600 s exposure spanning a wavelength range of 4800–9220 Å using the C1 decker, the GG475 order-blocking filter, and a slit size of 70 × 0861.

Basic data reduction and spectral extraction were carried out following the description in Kraus et al. (2011). The extraction pipeline MAKEE was used for bias subtraction, flat-fielding, cosmic-ray rejection, spectral extraction, and wavelength calibration. Zero-point corrections to the wavelength solution were derived by cross-correlating the telluric band of the O7V star S Mon. Radial velocities and projected rotational velocities (vsini_*) were then measured for both targets using early–M dwarf radial velocity (RV) standards from Chubak et al. (2012) after removing orders with strong telluric features or low S/N. A broadening function (Rucinski 1999) was measured relative to these standards to find the absolute RV and line broadening, which was then translated into projected rotational velocities using rotationally broadened template spectra to empirically correlate line broadening and projected rotational velocities following Kraus et al. (2017). Our final RV values from HIRES are −5.67 ± 0.12 km⁻¹ for ROXs 12 and −6.97 ± 0.15 km s⁻¹ for 2M1626–2527, comparable to the radial velocities we found from our IGIRNS spectra. The slight differences (<1 km s⁻¹) between the HIRES and IGRINS RVs are likely caused by jitter in these young and active stars. Our vsini_* measurements from this analysis are 8.2 ± 0.7 and 4.5 ± 1.0 km s⁻¹ for ROXs 12 and 2M1626–2527, respectively.

ROXs 12 (EPIC 203640875) and 2MASS J16262774–2527247 (EPIC 203637940) were both observed with K2 (Howell et al. 2014), the extended mission of the Kepler spacecraft, during Campaign 2 between 2014 August 23 and November 10 (GO 2052, PI: K. Covey; GO 2063, PI: A. Kraus). The raw long-cadence (30 minutes) photometry for ROXs 12 was corrected for systematic features caused by pixel-to-pixel drift and spacecraft thruster firings using the reduction pipeline described in Vanderburg & Johnson (2014). For 2MASS J16262774–2527247, we used the raw light curve instead of applying corrections, because this star exhibits very large photometric variations—up to ∼50% in amplitude—that are far larger than the systematics present in the data. Light curves for both targets are presented in Section 3.5.

ROXs 12 was observed on UT 2011 April 24 using NIRC2's nine-hole aperture mask together with NSG AO at Keck Observatory as part of the multiplicity survey of Ophiuchus members by Cheetham et al. (2015). The observations consisted of four 20 s images (5 s × 4 coadds) in the Kp filter using NIRC2's narrow camera mode. After basic reduction (bias subtraction, flat-fielding, and bad-pixel correction), we reexamined the images and found that the interferogram from ROXs 12 B is clearly visible in all four frames. This serendipitous detection opens the opportunity to search for a potential binary companion to ROXS 12 B at the Keck diffraction limit—analogous to brown dwarf–brown dwarf binary companions to stars like HD 130948 BC (Potter et al. 2002) and Indi Bab (Scholz et al. 2003; McCaughrean et al. 2004)—albeit at modest flux ratios because of flux loss inherent to this technique and ROXs 12 B's intrinsic faintness.

Figure 5 displays the coadded image of the ROXs 12 AB system after applying the distortion solution and north orientation from Yelda et al. (2010). Using the aperture-masking pipeline described in Kraus et al. (2008), we did not find evidence for an additional companion to ROXs 12 B down to angular scales of about 30 mas. The corresponding 99%-level contrast curve in the Kp band over which we can exclude companions is {0.08, 1.77, 2.02, 2.03, 1.97, 1.98} mag at {15, 30, 60, 120, 200, 280} mas. This corresponds to mass limits of ≈10 M_Jup at about $0\buildrel{\prime\prime}\over{.} 1$ based on the evolutionary models of Baraffe et al. (2015) for an age of $\approx 6\,\mathrm{Myr}$ (see Section 3.6). It therefore appears that ROXs 12 B is single down to physical separations of ≈4 au, which is well within its Hill radius of about 50 au. Contrast limits for the host star from these same data are about 3 mag deeper; these are shown in Figure 5 and presented in Cheetham et al. (2015).

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Our moderate-resolution spectra of ROXs 12 B from Gemini-North/NIFS (J band) and Keck/OSIRIS (J, H, and K bands) are shown in Figures 3 and 4. The NIFS spectrum shows prominent atomic and molecular absorption features from K i, Fe i, Mn i, Al i, H₂O, and FeH, as well as hints of broad VO absorption at ≈1.2 μm. Notably absent is Paβ emission at 1.282 μm, a signature of active accretion from a circumsubstellar disk; such emission has been detected in several other young brown dwarf companions over the past few years (e.g., Seifahrt et al. 2007; Bowler et al. 2011).

We compare our NIFS spectrum to brown dwarfs from the Brown Dwarf Spectroscopic Survey (BDSS; McLean et al. 2003) in Figure 6. Relative to field objects, ROXs 12 B exhibits shallower K i doublets at 1.169/1.178 μm and 1.244/1.253 μm, as well as diminished FeH molecular absorption features at ≈1.195–1.205 and 1.239 μm—all signs of low surface gravity (e.g., Gorlova et al. 2003; McGovern et al. 2004; Slesnick et al. 2004; Allers & Liu 2013). The pseudocontinuum is a good match to the L2 template spectrum. Equivalent widths of absorption lines are reported in Table 3.

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Table 3.  ROXs 12 B NIFS J-Band Equivalent Widths

${\lambda }_{0}$ a	Line	EW
(μm)	ID	(Å)
1.1693	K i	1.10 ± 0.09
1.1773	K i	2.60 ± 0.10
1.1887	Fe i	0.89 ± 0.04
1.1976	Fe i	0.73 ± 0.05
1.2436	K i	0.88 ± 0.04
1.2526	K i	1.26 ± 0.05
1.2903	Mn i	0.42 ± 0.12
1.3127	Al i	0.54 ± 0.06
1.3154	Al i	0.47 ± 0.03

Note.

^aNominal air wavelength.

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Our OSIRIS spectrum broadly resembles a late-M and early-L dwarf with deep H₂O bands at $\approx 1.4$ and $\approx 1.9$ μm, FeH, ¹²CO, Na i, and weak K i. The triangular H-band shape is typical of low-gravity brown dwarfs and giant planets (e.g., Lucas et al. 2001; Kirkpatrick et al. 2006; Allers et al. 2007), offering independent evidence that the system is young; although, note that the triangular H-band shape does not strictly track low surface gravity (Allers & Liu 2013).

In Figure 7, we compare our OSIRIS and NIFS spectra to very-low-gravity (VL-G) templates from Allers & Liu (2013). The full 1.2–2.4 μm spectrum is most similar to the young L0 template, but each bandpass individually shows inconsistent matches: the J band resembles L1–L2 objects, the H band is closest to the M8 template, and the K band appears similar to M8–M9 objects. Altogether, we adopt a near-infrared spectral type of L0 ± 2 for ROXs 12 B. Note that the OSIRIS broadband filter bandpasses do not cover the spectral regions used in the Allers & Liu (2013) classification system, so we adopt visual-based classifications for ROXs 12 B.

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Equivalent widths of the J-band potassium doublets from our NIFS spectrum are shown relative to ultracool objects in Figure 8. High-gravity field objects possess deep lines that trace out an upper envelope of equivalent widths, whereas young objects exhibit weaker alkali features as a result of their low-pressure atmospheres. Compared to low-mass stars and brown dwarfs spanning a range of ages and surface gravities, ROXs 12 B has shallow potassium lines similar to those of VL-G objects from Allers & Liu (2013) and Martin et al. (2017), as well as young (≲10 Myr) companions near and below the deuterium-burning limit, as measured from published spectra (Table 4).

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We implement two methods to assess the effective temperature and surface gravity of ROXs 12 B using atmospheric models. The first approach is based on maximum-likelihood fitting of the BT-Settl models (CIFIST2011bc version; Allard et al. 2011) following the general description in Bowler et al. (2009). In summary, the grid of models are first Gaussian-smoothed to the resolving power of the OSIRIS spectrograph (R ≈ 3800) and resampled onto the same wavelength grid as the data. A reduced χ² value (${\chi }_{\nu }^{2}$) is calculated for each synthetic spectrum in the grid spanning T_eff = 1100–4000 K (ΔT_eff = 100 K) and log g = 2.5–5.5 dex [cgs units throughout this work] (Δlog g = 0.5 dex). Similarly, a corresponding radius is derived for each model by making use of the factor that scales the emergent model spectrum to the observed flux-calibrated spectrum, as this quantity is also equal to the ratio of the object's radius to its distance squared. Here we adopt a distance of 137 ± 10 pc, where the mean value reflects the distance measured to the dark cloud Lynds 1688 by Ortiz-León et al. (2017) and the uncertainty reflects the uncertainty of the membership of this system in the Ophiuchus versus Upper Sco star-forming regions (see Section 4.1).

In addition to fitting the entire 1.2–2.4 μm spectrum, we also fit the individual J, H, and K bands using this method. The results are shown in Figure 9. The models qualitatively match the data quite well but produce different quantitative results depending on the bandpass, with effective temperatures ranging from 1600 to 2800 K. The bottom panel of the figure shows the entire 1.2–2.4 μm spectrum; the best model does a poor job of reproducing the data by underestimating the flux at shorter wavelengths and overpredicting it in the K band. Despite the generally good agreement of the models with the individual bands, the inferred gravities and effective temperatures disagree depending on the spectral region being considered. Two regions of local minima are clearly visible for the effective temperature, one at ≈1500–1700 K and one at ≈2600–2900 K. For the entire spectrum, the best-fitting model is visually a poor match to the data despite formally producing the lowest ${\chi }_{\nu }^{2}$ value. Similarly, the inferred radii span 1.7–5.1 R_Jup depending on the spectral bandpass used in the fits. Only the H and K bands are consistent with expectations of ≈2 R_Jup from hot-start evolutionary models. The implied masses based on surface gravity and radii are also unphysical, ranging from 3.8 M_Jup for the H band to 3300 M_Jup for the entire spectral fit. These discrepancies make it difficult to interpret the results; systematic errors in the models mean that the data are not standard deviates about the model, and therefore the ${\chi }_{\nu }^{2}$ statistic does not correspond to a maximum likelihood. Instead, these fits are more sensitive to the overall shape of the pseudocontinuum rather than individual absorption features that may be more useful for constraining surface gravity and effective temperature.

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For this reason, we explored another approach based on maximizing the cross-correlation between models and data. This method better incorporates information from atomic and molecular lines, which may be more sensitive to temperature and surface gravity, compared with the ${\chi }^{2}$ treatment. Cross-correlation encodes line positions and line ratios but is mostly insensitive to uncertainties in the calibration of the continuum, as well as broadband spectral variations (e.g., Brogi et al. 2012).

The same set of models described above was convolved with a Gaussian kernel to a resolution of R = 10,000. This is high enough to provide a high-enough resolution template for both the OSIRIS and the NIFS spectra but low enough to avoid interpolation errors. Prior to cross-correlation, a high-pass filter was applied to both the data and the models to remove broadband variations. The models were then Doppler-shifted to radial velocities between −900 and +900 km s⁻¹ (in steps of 30 km s⁻¹), linearly interpolated to the spectral range of the data, and cross-correlated with our observed spectra. Each spectral channel in the data was weighted by the inverse of its relative error to prevent noisy sections of the spectra from dominating the analysis.

The cross-correlation functions (CCFs) obtained with the full set of model spectra are shown in Figure 10. Not only does the S/N vary between different bands, but the peak of the cross-correlation seems to occur at different positions in the spectral sequence, confirming the previous observation that different bands could lead to different best-fitting parameters for the atmosphere.

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To formally compute confidence intervals (CIs) on the model parameters, we incorporated the values of the CCFs into a likelihood function L following the approach of Zucker (2003). Since in this case we are combining four different bands, we adopt the following effective CCF:

$$\begin{eqnarray}&&{\bar{C}}^{2}=1-{\left\{\displaystyle \prod _{i}[1-{C}_{i}^{2}]\right\}}^{1/4},\end{eqnarray} \tag{ 1 }$$

where the product is performed over the CCF of the four available spectral bands. The effective CCF and the number of spectral channels N are used to compute a log-likelihood function

$$\begin{eqnarray}&&\mathrm{log}(L)=-\displaystyle \frac{N}{2}\mathrm{log}[1-{\bar{C}}^{2}],\end{eqnarray} \tag{ 2 }$$

and the corresponding likelihood values L are used to derive CIs.

The bottom left panel in Figure 11 shows the two-dimensional map of $\mathrm{log}(L)$ (blue), and the corresponding 1σ, 2σ, and 3σ CIs are labeled. The top left and bottom right panels show the projected probability distributions for $\mathrm{log}(g)$ and T_eff, respectively. The resulting effective temperature of ROXs 12 B is ${T}_{\mathrm{eff}}={3100}_{-500}^{+400}$ K, and the bimodality observed with the previous model comparison is no longer present. However, only an upper limit of $\mathrm{log}(g)\lt 4.0$ is obtained for the surface gravity. Although this result points to a low-gravity object, the likelihood also rises at high values of $\mathrm{log}(g)$.

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The inferred effective temperature from our cross-correlation analysis is warmer than expected for a young L0 object, although the distribution is broad and encompasses the range of anticipated values. For example, using the effective temperature–spectral type relation for young objects from Filippazzo et al. (2015) yields 2260 ± 60 K, assuming no error on the spectral type. If we instead adopt an uncertainty of two subtypes, the median effective temperature of the resulting distribution is 2160 K with a 1σ highest posterior density range¹⁴ of 1810–2770 K.

We determine the luminosity of ROXs 12 B using our flux-calibrated OSIRIS spectrum together with bolometric corrections using BT-Settl model synthetic spectra at shorter (λ < 1.2 μm) and longer (λ > 2.4 μm) wavelengths. The synthetic spectrum is scaled to the short- and long-wavelength endpoints of our spectrum and then integrated to derive a bolometric flux. Because the results from atmospheric model fits are either inconclusive (for the ${\chi }^{2}$ analysis) or have large uncertainties (from the CCF analysis), we utilize the T_eff = 2300 K model with $\mathrm{log}g=4.0$ dex based on relations from Filippazzo et al. (2015) for young L0 objects—although the resulting luminosities are only weakly sensitive to the input model. The bolometric luminosity of ROXs 12 B is $\mathrm{log}{L}_{\mathrm{bol}}/{L}_{\odot }=-2.87\pm 0.06$ dex, which takes into account uncertainties in individual spectral measurements, the flux-calibration scale factor, and the estimated distance to the system. In comparison, using a 3100 K model produces a similar bolometric luminosity of −2.82 ± 0.06 dex.

Based on the bolometric luminosity of the companion and the system age (${6}_{-2}^{+3}$ Myr; Section 3.6), the inferred mass of ROXs 12 B is 17.5 ± 1.5 M_Jup from Burrows et al. (1997) hot-start evolutionary models. This mass is consistent with the value of 16 ± 4 M_Jup found by Kraus et al. (2014).

2M1626–2527 appears to constitute a wide tertiary to ROXs 12 AB separated by $37^{\prime\prime}$, or 5100 au, at a distance of about 140 pc. The proper motions of this pair from Hot Stuff for One Year (HSOY; Altmann et al. 2017) are consistent with one another: ${\mu }_{\alpha }\cos \delta =-12.7\pm 2.3$ and ${\mu }_{\delta }=-29.0\pm 2.3$ mas yr⁻¹ for ROXs 12 and ${\mu }_{\alpha }\cos \delta =-11.9\pm 2.3$ and ${\mu }_{\delta }=-30.5\pm 2.3$ mas yr⁻¹ for 2M1626–2527. Similarly, we measure consistent radial velocities for both components from our IGRINS spectra: −6.09 ± 0.16 km s⁻¹ for ROXs 12 and −6.03 ± 0.16 km s⁻¹ for 2M1626–2527. This is far less than the velocity dispersion of ≈1 km s⁻¹ for ρ Ophiuchus cluster members (Wilking et al. 2015), offering further evidence that these two stars are not simply members of the same cluster but are likely gravitationally bound. The identical radial velocities also suggest that both stars are probably single. Indeed, deep adaptive-optics imaging by Bryan et al. (2016) did not reveal any additional members of this system. The kinematic, photometric, and physical properties of this triple system comprising ROXs 12 A, ROXs 12 B, and 2M1626–2527 are listed in Table 2.

Figure 12 shows the spectral energy distributions of ROXs 12 A, ROXs 12 B, and 2M1626–2527 based on optical through infrared photometry in Table 2. Flux-density measurements for all three targets are dereddened by A_V = 1.8 mag.

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Solar metallicity BT-Settl model atmospheres from Allard et al. (2012) are overplotted in Figure 12. Spectral types are converted to effective temperatures using empirical calibrations for pre-main-sequence stars. ROXs 12 A has a spectral type of M0.0 ± 0.5 (Bouvier & Appenzeller 1992; Rizzuto et al. 2015), which corresponds to an effective temperature of ${3900}_{-90}^{+60}$ K on the Herczeg & Hillenbrand (2014) scale and ${3770}_{-70}^{+100}$ K on the Pecaut & Mamajek (2013) scale; we adopt an effective temperature of 3900 ± 100 K for ROXs 12 A here. Without a large sample of radius measurements for young stars, it is difficult to assess the size of potential systematic errors in these spectral type–effective temperature scales. However, we note that even systematic errors as large as 300 K for ultracool dwarfs (e.g., Dupuy et al. 2010) do not substantially affect the broad results or conclusions regarding the luminosities, radii, or stellar inclinations throughout this paper.

The agreement of the model atmosphere with the dereddened photometry of ROXs 12 A is generally quite good, spanning the optical to mid-infrared wavelengths. But the SED clearly shows a large excess at 22 μm, presumably from disk emission originating from the host star ROXs 12 A or, less likely, a warm disk surrounding the companion ROXs 12 B that is only prominent at 22 μm. If this excess emission originates from ROXs 12 A and peaks at ∼22 μm, then the dust temperature is ∼230 K and is located at ≈0.7 au following the heuristic relations from Wyatt (2008). If it originates from ROXs 12 B, then it must be located much closer in, at ∼0.05 au.

For ROXs 12 B, the dereddened J, H, Kp, and $L^{\prime}$ photometry from Kraus et al. (2014) is in good agreement with the 2300 K BT-Settl model, which was chosen based on spectral type–effective temperature relations for young brown dwarfs (see Section 3.2). There is no evidence of excess emission from ROXs 12 B that would indicate that it hosts a circumsubstellar disk. This contrasts with many other young brown dwarfs and PMCs that harbor accretion subdisks, as evidenced by optical/near-infrared emission lines and thermal infrared excess (e.g., Bowler et al. 2011; Bailey et al. 2013; Zhou et al. 2014).

No spectral type has been published for 2M1626–2527. Our own low-resolution optical spectrum shown in Figure 13 reveals a late-type spectrum with clear signs of active accretion, including strong Hα emission (EW = −46 Å), as well as various transitions from He i, [O i], and Ca ii, all in emission (Table 5). Aside from Hα emission, the optical spectrum of 2M1626–2527 is relatively featureless shortward of ≈7000 Å and appears to be heavily veiled. This likely originates from the boundary layer of an accretion disk (e.g., Basri & Batalha 1990; Hartigan et al. 1991). The near-infrared spectrum of 2M1626–2527 shown in Figure 13 also points to a late (M-type) spectrum with strong emission lines from Ca ii, He i, and H i (the Paschen series and Br γ; see Table 5).

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Table 5.  Emission-line Equivalent Widths

${\lambda }_{0}$ a	Line	EW
(Å)	ID	(Å)
ROXs 12 A: Keck/HIRES
6562.8	Hα	−1.41 ± 0.02
8498.0	Ca ii	−0.40 ± 0.01
8542.1	Ca ii	−0.60 ± 0.01
8662.1	Ca ii	−0.42 ± 0.02
2M1626–2527: Keck/HIRES
4861.3	Hβ	−26.2 ± 0.2
4921.9	He i	−0.53 ± 0.03
5015.7	He i	−0.51 ± 0.03
5875.6	He i	−2.54 ± 0.03
6562.8	Hα	−53.17 ± 0.13
6678.2	He i	−1.12 ± 0.02
8446b	O i	−0.47 ± 0.04
8498.0	Ca ii	−1.1 ± 0.01
8542.1	Ca ii	−1.29 ± 0.02
8662.1	Ca ii	−0.82 ± 0.01
2M1626–2527: Mayall/RC-Spec
6300.3	[O i]	−0.9 ± 0.3
6562.8	Hα	−46.5 ± 0.5
6678.2	He i	−1.33 ± 0.12
7002b	O i	−0.4 ± 0.2
7065.2	He i	−0.9 ± 0.3
8446b	O i	−1.09 ± 0.05
8498.0	Ca ii	−2.00 ± 0.05
8542.1	Ca ii	−1.91 ± 0.04
8662.1	Ca ii	−1.46 ± 0.04
2M1626–2527: IRTF/SpeX
8446b	O i	−0.7 ± 0.3
8498.0	Ca ii	−1.9 ± 0.3
8542.1	Ca ii	−1.6 ± 0.2
8662.1	Ca ii	−0.8 ± 0.2
9546.2	H i Pa8 ()	−0.9 ± 0.3
10049.8	H i Pa7 (δ)	−1.5 ± 0.2
10830.3	He i	−1.1 ± 0.2
10938.2	H i Pa6 (γ)	−1.7 ± 0.2
128218.1	Paβ	−1.85 ± 0.13
21661.2	Brγ	−1.5 ± 0.2

Notes.

^aNominal air wavelength. ^bLines are blended in our spectrum.

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The relative depths of the TiO bandheads spanning 7050–7150 Å are highly sensitive to temperature and can be used as an indicator of spectral type for 2M1626–2527 despite the heavy veiling at shorter wavelengths. Figure 14 shows a detailed view of this region compared to K4–M4 templates from Bochanski et al. (2007) for the M dwarfs and Mann et al. (2013) for the K dwarfs. The relatively pronounced TiO bands in 2M1626–2527 are stronger than the K4–M0 templates and weaker than the M2–M4 templates, with M1 being the best match. We therefore assign 2M1626–2527 a spectral type of M1 with an uncertainty of one subclass to reflect the imperfect agreement with this field template. The corresponding effective temperature is ${3720}_{-160}^{+180}$ and 3630 ± 140 K using the Herczeg & Hillenbrand (2014) and Pecaut & Mamajek (2013) conversions, respectively. We adopt T_eff = 3700 ± 150 K for 2M1626–2527. A model effective temperature of 3700 K and reddening of A_V = 1.8 mag is a good fit to the photometry of 2M1626–2527 in the optical and near-infrared, but beyond about 3 μm there is clearly very strong excess emission from a protoplanetary disk (Figure 12).

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A few selected regions of our HIRES spectra of ROXs 12 A and 2M1626–2527 are shown in Figure 15. ROXs 12 A shows relatively weak Hα emission (EW = −1.41 ± 0.02 Å) considering its young age. This is similar to the value of 1.2 Å found by Bouvier & Appenzeller (1992) and −2.00 ± 0.02 Å from Rizzuto et al. (2015). These values are consistent with the lower envelope of Hα emission for pre-main-sequence M0 stars spanning the youngest star-forming regions, like Taurus (Kraus et al. 2017), to somewhat older populations, like Upper Sco (Rizzuto et al. 2015). In our HIRES spectrum, 2M1626–2527 shows strong, broadened Hα emission (EW = −53.17 ± 0.13 Å) comparable to our measurement with RC-Spec (EW = −46.5 ± 0.5 Å). Both stars show Li i 6708 Å absorption; we measure an equivalent width of 0.54 ± 0.01 Å for ROXs 12 A—in good agreement with the value of 0.52 ± 0.02 Å from Rizzuto et al. (2015)—and EW(Li) = 0.32 ± 0.02 Å for 2M1626–2527. The Ca ii infrared triplet is also seen in emission for both sources, with ROXs 12 A exhibiting broad absorption with superimposed central cores in emission. The Ca ii line profiles from 2M1626–2527 show a broad, blueshifted component in emission, offset from the central narrow-line emission. This suggests that we are viewing a face-on accretion disk producing a broadened component along our line of sight together with a narrow zero-velocity peak where material hits the star. This geometry is consistent with the mostly pole-on stellar inclination we find for 2M1626–2527 in Section 4.2.

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The K2 light curves for ROXs 12 A and 2M1626–2527 are shown in Figure 16. ROXs 12 A exhibits remarkably strong periodic changes with peak-to-peak semi-amplitude variations of ≈14%. The most straightforward interpretation of these modulations is that they are caused by large, long-lived regions of inhomogeneous spot coverage coming in and out of view as the star rotates. A Lomb–Scargle (LS) periodogram (Lomb 1976; Scargle 1982; Horne & Baliunas 1986) reveals a strong peak period of 9.0998 days, which we adopt as the rotation period of ROXs 12 A. This is near the upper envelope of (but consistent with) rotation period measurements of low-mass pre-main-sequence stars during the first ≈1–10 Myr of their evolution (e.g., Rebull et al. 2004; Irwin et al. 2011; Bouvier et al. 2014). Several flaring events are clearly visible over the course of these observations. The K2 light curve of ROXs 12 A phased to 9.100 days (Figure 17) exhibits only slight changes in the shape of the light curve from period to period, indicating minimal evolution of starspots on timescales less than one period.

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The light curve of 2M1626–2527 shows both low-amplitude (few percent) and high-amplitude (≈40%) variations on timescales ranging from hours to days (Figure 16). This stochastic behavior is characteristic of young accreting stars and is thought to be caused by time-dependent mass accretion events that produce transient hot spots on the stellar photosphere (e.g., Cody et al. 2014; Stauffer et al. 2016). Cody et al. (2017) found that this phenomenon is fairly common, occurring among ≈9% of ρ Oph and Upper Sco members with strong infrared excesses. Although there are no obvious signs of periodicity in the light curve, the LS periodogram shows a significant peak power at 3.297 days. To explore whether this represents a real underlying modulation superimposed on stochastic variations, we applied the same periodogram analysis to smaller portions of the light curve, which should return the same approximate peak period near 3.3 days if the modulations are authentic and consistent over time. The inset plot in the lower panel of Figure 17 shows the peak period, second-highest peak, and third-highest peak starting with 20% of the light curve and progressively including more data in 10% bins. Beginning with 30% of the data, the peak periods range between 3.28 and 3.37 days. This suggests that the underlying modulations are real and are probably caused by rotationally modulated starspots. Interestingly, this means that 2M1626–2527 is rotating about three times faster than ROXs 12 A despite having an accreting disk, which has been shown, on average, to slow the rotation rates of young stars via star-disk interactions (e.g., Herbst et al. 2002; Cieza & Baliber 2007). This is unusual but not entirely unexpected given the broad spread in periods for diskless and disk-bearing young stars.

We estimate the uncertainties for our light-curve periods following Kovács (1981) and Horne & Baliunas (1986). For a single signal with Gaussian noise, they found that the error in the frequency measurement ν is δν = 3σ/($4\sqrt{N}{TA}$), where σ is the standard deviation of the noise about the signal, N is the number of data points, T is the period, and A is the signal semi-amplitude. From this, the period (P) uncertainty is δP = P²δν. We fit a sine curve to the phased data to find A from the K2 light curves. The rms noise was approximated by computing the standard deviation of the residuals between the time-series photometry and a Gaussian-smoothed version of the same data. From this analysis, we find formal uncertainties of 0.009 day for ROXS 12 and 0.002 day for 2M1626–2527. The high precision of these errors is primarily a result of the large number of periods sampled during the observing window (9–24 full cycles), the large number of data points in these light curves (≈3400), and the relatively high semi-amplitudes for both systems (10%–14%).

Stars exhibit differential rotation between their poles and equators, which can make it difficult to infer the equatorial rotation period from light curves alone if the location of their starspots is unknown. For example, the Sun has a longer rotation period of about 34 days at the poles compared to 25 days at its equator. Our period measurements for ROXs 12 A and 2M1626–2527 may therefore be overestimating the equatorial rotation periods of these objects, depending on in which latitudinal regions the starspots were located. Reinhold & Gizon (2015) showed that the pole-equatorial shear (ΔΩ = 2π(1/P_min—1/P_max)) of Kepler stars with effective temperatures comparable to those of ROXs 12 A and 2M1626–2527 ranges from about 0.01 to 0.10 rad day⁻¹, with a typical value of about 0.03 rad day⁻¹. This typical shear would imply a range of maximum and minimum periods of 8.7–9.5 days for ROXs 12 A and 3.25–3.35 days for 2M1626–2527. To take into account possible effects of differential rotation, we adopt larger rotation period uncertainties of 9.1 ± 0.4 days for ROXs 12 and 3.30 ± 0.05 days for 2M1626–2527 when using these values in the context of equatorial rotation period estimates.

Bolometric luminosities for ROXs 12 A and 2M1626–2527 are derived using the 3900 and 3700 K BT-Settl models, respectively, shown in Figure 12. The models are first scaled to the H-band dereddened flux and then integrated from 0.2 to 200 μm to find the bolometric flux. A distance of 137 pc is assumed based on Very Large Baseline Array (VBLA) parallax measurements of Ophiuchus members in the Lynds 1688 dark cloud (Ortiz-León et al. 2017), and we adopt a ±10 pc uncertainty to encompass ambiguity in Ophiuchus versus Upper Sco membership. Uncertainties are derived by integrating models 100 K warmer and 100 K cooler than the nominal stellar effective temperatures and calculating the mean difference between these and the stars' bolometric fluxes. Surface gravities of $\mathrm{log}g=4.0$ dex are chosen based on expectations from evolutionary models (e.g., Baraffe et al. 2015), as substantial deviations from this are unphysical. This yields luminosities of $\mathrm{log}{L}_{\mathrm{bol}}/{L}_{\odot }=-0.57\pm 0.06$ dex for ROXs 12 A and −0.81 ± 0.06 dex for 2M1626–2527.

Stellar masses and ages are estimated using Baraffe et al. (2015) evolutionary models. For a given effective temperature and luminosity, we identify the corresponding mass and age by finely interpolating the grid of evolutionary models (Figure 18). This process is repeated in a Monte Carlo fashion to build a distribution of masses and ages assuming normally distributed errors in T_eff and log(L_bol/L_⊙). We infer a mass of ${0.65}_{-0.09}^{+0.05}$ M_⊙ and an age of ${6}_{-2}^{+4}$ Myr for ROXs 12 A. For 2M1626–2527, we find a mass of ${0.50}_{-0.10}^{+0.10}$ M_⊙ and an age of ${8}_{-4}^{+7}$ Myr.

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The question of whether this system belongs to the younger Ophiuchus association or the older, more expansive Upper Sco region has implications for the inferred mass of the substellar companion ROXs 12 B. ROXs 12 has historically been regarded as a member of the ≈0.5–2 Myr Ophiuchus star-forming region since its identification by Bouvier & Appenzeller (1992). This region is centered on the dark cloud Lynds 1688, and its >300 members comprise a range of evolutionary stages spanning embedded protostars, accreting T Tauri stars, transition disks, and an older surface population that merges with the ≈5–10 Myr Upper Sco subgroup (Wilking et al. 2008). ROXs 12 and 2M1626–2527 are located about 04 from the L1688 cloud core in (or behind) an extended region of modest extinction (A_V = 1–2 mag; Cambrésy 1999).

Unfortunately, because the internal velocity dispersions of cluster members are $\approx 1.0$ km s⁻¹ ($\approx 1.5$ mas yr⁻¹; Kraus & Hillenbrand 2008; Wilking et al. 2015), the kinematics of Ophiuchus and Upper Sco are nearly indistinguishable using current proper-motion surveys. Mamajek (2008) found a mean proper motion of ${\mu }_{\alpha }\cos \delta =-10\pm 1.5$ and ${\mu }_{\delta }=-27\pm 1.5$ mas yr⁻¹ (1 mas yr⁻¹ systematic error) for Ophiuchus cloud members. Based on the UVW space velocities of Upper Sco from Rizzuto et al. (2011), the expected proper motion of an Upper Sco member at the sky position of ROXs 12 is ${\mu }_{\alpha }\cos \delta =-11.9\pm 1.5$ and ${\mu }_{\delta }=-24.4\pm 1.5$ mas yr⁻¹. (Note that because of the large angular extent of Upper Sco on the sky, the proper motions of its members vary slightly across this region, so we use the association's three-dimensional space velocity at the sky position of ROXs 12 to calculate the expected proper motion.) The measured proper motions of ROXs 12 A and 2M1626–2527 (Table 2) agree with both of these regions to within about 2σ. Unsurprisingly, we find Sco-Oph membership probabilities of 99% for both stars following the Bayesian membership analysis described in Rizzuto et al. (2011, 2015).

Regardless of kinematics, the ages inferred for ROXs 12 A and 2M1626–2527 from their positions on the H-R diagram are most consistent with Upper Sco. Pecaut & Mamajek (2016) adopted a median age of 10 ± 3 Myr for Upper Sco with a large intrinsic age spread of ±7 Myr, which is consistent with recent findings by Fang et al. (2017). Pecaut et al. also found evidence of an age gradient in this subgroup, with ROXs 12 positioned near the transition point between ≈5 and 9 Myr ages in the northwest and ≈11 and 15 Myr ages in the southeast. Our independently inferred ages for ROXs 12 A (${6}_{-2}^{+4}$ Myr) and 2M1626–2527 (${8}_{-4}^{+7}$ Myr) from Section 3.6 (also see Figure 18) appear to agree with typical ages of Upper Sco members in the vicinity of the Ophiuchus cloud. In the near future, this membership analysis and age will be refined with parallactic distances and precise proper motions from Gaia.

The radii of ROXs 12 A and 2M1626–2527 can be determined with the Stefan–Boltzmann law using their effective temperatures and bolometric luminosities. This yields values of 1.14 ± 0.09 R_⊙ for ROXs 12 A and 0.96 ± 0.10 R_⊙ for 2M1626–2527, where uncertainties are propagated analytically.

A lower limit on the radius of ROXs 12 A can also be inferred from the measured rotation period (P_rot) and projected rotational velocity (v_p = v sin i_*). By equating v and 2πR_*/P_rot, the stellar radius R_* and associated uncertainty ${\sigma }_{{R}_{* }}$ are

$$\begin{eqnarray}&&{R}_{* }({i}_{* })=\displaystyle \frac{{P}_{\mathrm{rot}}{v}_{p}}{2\pi \sin {i}_{* }},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\sigma }_{{R}_{* }}\approx {R}_{* }{\left({\left(\displaystyle \frac{{\sigma }_{{P}_{\mathrm{rot}}}}{{P}_{\mathrm{rot}}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{v}_{p}}}{{v}_{p}}\right)}^{2}+{\left({\sigma }_{{i}_{* }}\displaystyle \frac{\cos {i}_{* }}{\sin {i}_{* }}\right)}^{2}\right)}^{1/2}.\end{eqnarray} \tag{ 4 }$$

In cases where the inclination is unknown, Equation (3) becomes a lower limit on R_* if an edge-on orbit is assumed (i_* = 90°). Adopting P_rot = 9.1 ± 0.4 days and v_p = 8.2 ± 0.7 km s⁻¹ for ROXs 12 A implies R_* > 1.47 ± 0.14 R_⊙. There is tension between this radius and the radius implied from the Stefan–Boltzmann law at the 2.0σ level. The most likely origin of this discrepancy is that (1) the inferred effective temperature for ROXs 12 A is too high, (2) our v sin i_* measurement is too high, and/or (3) the rotation period samples high-latitude rather than equatorial starspots.

The line-of-sight inclination of ROXs 12 A can be determined more precisely using joint constraints from the "spectroscopic" (Stefan–Boltzmann) radius and the "rotational" (R_*(i_*)) radius. Equating these two radii and solving for the stellar inclination gives

$$\begin{eqnarray}&&{i}_{* }={\sin }^{-1}\left(\sqrt{\displaystyle \frac{{\sigma }_{\mathrm{SB}}}{\pi {L}_{\mathrm{bol}}}}{v}_{p}{P}_{\mathrm{rot}}{T}_{\mathrm{eff}}^{2}\right),\end{eqnarray} \tag{ 5 }$$

where ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant. Uncertainties are incorporated in a Monte Carlo fashion to build a line-of-sight inclination distribution, which peaks at 77° and is truncated at 90° (Figure 19). Monte Carlo realizations that result in $\sin {i}_{* }$ values >1 are excluded. The mode and 68.3% (1σ equivalent) highest posterior density interval is ${{77}_{-9}^{+7}}^\circ$. The 95.4% (2σ equivalent) range spans 60°–88°. Note that these line-of-sight stellar inclinations are symmetric about 90° and produce mirrored observational signatures at higher inclinations.

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Similarly, using P_rot = 3.30 ± 0.05 days and v_p = 4.5 ± 1.0 km s⁻¹ for 2M1626–2527 implies R_* > 0.29 ± 0.07 R_⊙. This agrees with the spectroscopic radius of 0.96 ± 0.10 R_⊙. The joint constraint on the inclination distribution for 2M1626–2527 is shown in Figure 19. The mode and 68.3% highest posterior density interval is ${{17}_{-4}^{+5}}^\circ$, and the 95.4% range spans 9°–27°.

The difference between these two inclination distributions, ${\rm{\Delta }}{i}_{* }$, is the degree to which these two stars are misaligned. For ROXs 12 A and 2M1626–2527, we find ${\rm{\Delta }}{i}_{* }={{60}_{-11}^{+7}}^\circ$, indicating that these stars are strongly misaligned. This is not surprising for two young stars separated by over 5000 au, and it adds to mounting evidence that misalignments between the components of wide binaries is a common phenomenon (e.g., Hale 1994; Jensen & Akeson 2014; Williams et al. 2014). This indicates that either the initial fragmenting cloud core that produced this system did not act as a uniformly corotating collapsing body, that this wide companion was captured, or that the rotational inclinations of these systems caused them to evolve with time.

ROXs 12 B has a projected separation of 240 au, which corresponds to an orbital period of 4600 yr and angular orbital motion of 008 yr⁻¹ assuming a circular face-on orbit. Because of the long baseline since the initial discovery epoch of ROXs 12 B by Ratzka et al. (2005) in 2001, enough time elapsed for Kraus et al. (2014) and Bryan et al. (2016) to detect small but significant orbital motion from this companion. Despite the small amount of orbital coverage, Bryan et al. (2016) were able to constrain the orbital elements of ROXs 12 B using the efficient rejection sampling algorithm for Keplerian orbits described in Blunt et al. (2017).

Of particular interest for this study is the posterior distribution of the orbital inclination, i_p, which peaks at about 135° with a broad tail extending from about 90° to 180°, as shown in Figure 20. The inclination of the stellar rotation axis of ROXs 12 A, i_*, can be compared with i_p to offer clues about the stellar obliquity—the relative orientation of the stellar spin axis and orbital angular momentum vector. The true (deprojected) angle between the stellar spin and planetary orbital axes, ψ, is related to the sky-projected spin–orbit angle, λ, as follows:

$$\begin{eqnarray}&&\psi ={\cos }^{-1}(\cos {i}_{* }\cos {i}_{p}+\sin {i}_{* }\sin {i}_{p}\cos \lambda ).\end{eqnarray} \tag{ 6 }$$

Diagrams showing the geometric configuration can be found in, e.g., Ohta et al. (2005) and Fabrycky & Winn (2009). The three angles λ, i_*, and i_p must be known to determine ψ. The projected obliquity (λ) is regularly measured for large transiting planets using the Rossiter–McLaughlin effect (e.g., Queloz et al. 2000; Winn et al. 2005; Johnson et al. 2009). In those cases, i_p ≈ 90°, so $\cos \psi \approx \sin {i}_{* }\cos \lambda$, and therefore the projected obliquity (λ) is a measurement of the lower bound on the true obliquity (ψ), assuming i_* is unknown. Similarly, if λ is unknown, as is the case for ROXs 12 AB, the lower limit on ψ can be determined from the absolute difference between i_p and i_*: ψ $\geqslant | {i}_{* }-{i}_{p}| \equiv {\rm{\Delta }}i$ (see Appendix). Therefore, a system can have spin–orbit misalignment if ${\rm{\Delta }}i$ is zero, but a nonzero value of ${\rm{\Delta }}i$ means that the system must be misaligned by at least that amount (Glebocki & Stawikowski 1997). In this sense, knowing both i_p and i_* offers comparable information about ψ, as do measurements of the Rossiter–McLaughlin effect.

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The probability distribution functions for i_*, i_p, and ${\rm{\Delta }}i$ for ROXs 12 A and B are shown in Figure 20. Note that since i_* is symmetric about 90°, it mirrors Figure 19 and is therefore bimodal. The distribution of ${\rm{\Delta }}i$ values peaks at 49° with a 1σ minimum CI of 17°–69°. From this, we calculate that the probability that ${\rm{\Delta }}i$ is greater than 10° is 94%. This distribution represents the minimum values of true obliquity angles in this system and indicates that ROXs 12 A and B are likely to have spin–orbit misalignment, although alignment (${\rm{\Delta }}i=0^\circ$) cannot be ruled out from the observations.

The presence of disks around widely separated brown dwarfs and PMCs offers a convenient way to directly study the formation and late stages of growth of these objects. These subdisks produce a variety of observational signatures: ultraviolet excess emission from accretion of hot gas (Zhou et al. 2014), hydrogen-line emission like Hα and Paβ (e.g., Bowler et al. 2011; Wu et al. 2015a), and thermal excess emission in the mid-infrared (Bailey et al. 2013). Submillimeter emission from warm dust has only been detected for the ambiguous companion to FW Tau (Caceres et al. 2015; Kraus et al. 2015) despite ongoing searches targeting several other disk-bearing substellar companions (e.g., Bowler et al. 2015; MacGregor et al. 2017; Wu et al. 2017).

Although we find no convincing evidence that ROXs 12 B harbors a subdisk, the lack of Paβ emission in this individual object can nevertheless be used to more broadly assess the global frequency of low-mass companions with accretion rates large enough to result in Paβ line emission. The sample of young companions near and below the deuterium-burning limit that also have moderate-resolution J-band spectroscopy has increased over the past few years to the point where we can begin to quantify the statistical properties of these objects as a population. Note that the strength of the Paβ emission line is expected to vary strongly with effective temperature, which alters the pseudocontinnuum level, as well as mass accretion rate. In addition, both the S/N and resolving power of a spectrum can influence the detection of an emission line. For this analysis, we ignore these effects and simply count reported Paβ detections and nondetections as discrete Bernoulli trials.

If we isolate a sample of young companions with ages ≲15 Myr, when such subdisks might still be expected to be actively accreting, and companion masses ≲20 M_Jup, there are 11 low-mass brown dwarfs and planetary-mass objects with published moderate-resolution spectroscopy. Five of these show Paβ in emission: GSC 6214-210 B (Bowler et al. 2011; Lachapelle et al. 2015), CT Cha B (Schmidt et al. 2008; Bonnefoy et al. 2014), GQ Lup B (Seifahrt et al. 2007), DH Tau B (Bonnefoy et al. 2014; Wolff et al. 2017), and FW Tau b (Bowler et al. 2014). Six companions do not appear to have Paβ in emission: 1RXS J1609–2105 B (Lafrenière et al. 2010), ROXs 42 Bb (Bowler et al. 2014), 2M0441+2301 Bb (Bowler & Hillenbrand 2015), SR 12 C (Bowler et al. 2014), USco CTIO 108 B (Bonnefoy et al. 2014), and ROXs 12 B (this work). Binomial statistics implies a frequency of 46% ± 14% for five detections out of 11 trials.

There are several caveats about these accreting companions that are worth noting. Paβ emission was not observed in the moderate-resolution spectroscopy of GQ Lup B presented by McElwain et al. (2007) and Lavigne et al. (2009), which suggests that the emission observed by Seifahrt et al. (2007) was variable. Similarly, Wolff et al. (2017) also found variable emission for DH Tau B. This suggests that other companions without Paβ emission could be observed during a quiescent state of low accretion. Published mass estimates for GQ Lup B range from ∼10 to 40 M_Jup, so this companion is consistent with (but may fall above) the cutoff of 20 M_Jup we use in this analysis. Finally, the mass of FW Tau b is uncertain, as its spectrum shows substantial veiling and indications of an edge-on disk (Bowler et al. 2014).

Accretion subdisks are apparently very common among companions near the deuterium-burning limit. The incidence of both accreting and nonaccreting circumsubstellar disks in general is likely much higher—and perhaps universal—for these objects.

The three-dimensional orbital and rotational architecture of planetary systems provides fundamental insight into their formation and subsequent dynamical evolution. Massive giant planets (≳1 M_Jup) formed in circumstellar disks should inherit the orientation of the disks' angular momentum vectors, which are expected to initially be aligned with those of the host stars. Similarly, the orbital planes of planets should also align with the equatorial planes of host stars in the absence of internal (other planets) or external (wide binary companions or passing stars) perturbers. There is also now substantial evidence that the tail end of the star formation process can produce companions in the planetary-mass regime near the opacity limit for fragmentation (e.g., Todorov et al. 2010; Bowler & Hillenbrand 2015). Just as binary companions that formed from the turbulent fragmentation of molecular cloud cores tend to have misaligned orientations (e.g., Brinch et al. 2016; Lee et al. 2016; Offner et al. 2016), the same random spin axes could also be imprinted on PMCs that form in this fashion. Together these two pathways—formation in a disk versus turbulent cloud core—have direct observational implications for wide substellar companions: in the absence of outside influences, massive isolated objects formed in a disk can be expected to exhibit spin–orbit alignment, whereas misalignment implies formation from turbulent fragmentation. Our result that ROXs 12 B is likely misaligned with the spin axis of its host star suggests that it was formed from the cloud fragmentation process, as opposed to disk instability, but this interpretation is somewhat complicated by the wide stellar tertiary, as discussed below.

It is also possible that ROXs 12 B could have formed from disk instability if it later underwent dynamical interactions with another body, for instance, through gravitational scattering or Kozai–Lidov librations with the wide stellar companion 2M1626–2527 (Kozai 1962; Lidov 1962; Naoz 2016). The timescale for Kozai–Lidov secular oscillations for a planet with an outer tertiary component is governed by the period of the planet, the masses of the host star and tertiary, the semimajor axes of the planet and tertiary, and the eccentricity of the host–tertiary orbit (e.g., Holman et al. 1997). For the ROXs 12 triple system, this characteristic timescale reduces to ≈6 × 10⁷ ${(1-{e}_{\mathrm{ter}}^{2})}^{3/2}$ yr, where e_ter is the eccentricity of the ROXs 12 A and 2M1626–2527 pair. As long as this eccentricity is below about 0.9, oscillation timescales are likely too long to have substantially influenced the orbital inclination of ROXs 12 B, given the young age of the system.

Similarly, prior to the planet formation process, wide stellar companions can gravitationally torque protoplanetary disks and result in spin–orbit misalignments with planets when they eventually form (e.g., Batygin 2012; Lai 2014). It is therefore conceivable that at a very early stage, a massive disk around ROXs 12 A could have been torqued by 2M1626–2527 to produce the misalignment we now observe in this system. For a protoplanetary disk around ROXs 12 A, the disk precession period would be roughly 80 Myr following Batygin (2012) or about 20 Myr for maximal misalignment, assuming that the outer disk radius ends at the location of ROXs 12 B. A more extended disk or a nonzero eccentricity for 2M1626–2527 can reduce this timescale even further, well within the age of ≈6 Myr of this system, implying that the spin–orbit misalignment we observe for ROXs 12 B could plausibly be a result of an initial torque on a disk around ROXs 12 A caused by 2M1626–2527.

In addition to ROXs 12 B, only a few other systems with widely separated substellar companions have information available about their stellar obliquity angles. There is some evidence that the orbital axis of the brown dwarf companion GQ Lup B may be misaligned with both the stellar rotation axis and the stellar disk inclination (Ginski et al. 2014; Schwarz et al. 2016; MacGregor et al. 2017; Wu et al. 2017), although it is unclear how significant this potential spin–orbit misalignment is due to discrepancies in the rotation period and radius of the host star. HR 8799 is an excellent example of a system with increasingly robust constraints on the spin axis of the host star, orbital inclination of its four imaged planets, and inclination measurements for its multibelt debris disk. Interestingly, each of these components appears to be mutually consistent within uncertainties (e.g., Reidemeister et al. 2009; Matthews et al. 2013; Booth et al. 2016; Konopacky et al. 2016), suggesting aligned angular momentum vectors and planet formation in a disk. As the orbits of directly imaged planets become better constrained in the future, stellar obliquity measurements like the one we have carried out here for ROXs 12 AB will help identify the dominant formation pathway(s) for these objects as a population analogous to Rossiter–McLaughlin measurements for transiting planets.