The Mass-Luminosity Relation in the L/T Transition: Individual Dynamical Masses for the New J-Band Flux Reversal Binary SDSSJ105213.51+442255.7AB
Trent J. Dupuy, Michael C. Liu, S. K. Leggett, Michael J. Ireland, Kuenley Chiu, David A. Golimowski
aa r X i v : . [ a s t r o - ph . S R ] M a r The Mass–Luminosity Relation in the L/T Transition:Individual Dynamical Masses for the New J -Band Flux Reversal BinarySDSS J105213.51+442255.7AB ∗ , † Trent J. Dupuy, Michael C. Liu, S. K. Leggett, Michael J. Ireland, Kuenley Chiu, and David A. Golimowski ABSTRACT
We have discovered that SDSS J105213.51+442255.7 (T0 . ± .
0) is a binary in Kecklaser guide star adaptive optics imaging, displaying a large J -to- K -band flux reversal(∆ J = − . ± .
09 mag, ∆ K = 0 . ± .
05 mag). We determine a total dynamicalmass from Keck orbital monitoring (88 ± M Jup ) and a mass ratio by measuring thephotocenter orbit from CFHT/WIRCam absolute astrometry ( M B /M A = 0 . ± . ± M Jup for the L6 . ± . ± M Jup for the T1 . ± . L bol / ∆ log M = 0 . +0 . − . ). Thisprovides the first observational support that cloud dispersal plays a significant role inthe luminosity evolution of substellar objects. Fully cloudy models fail our coevalitytest for this binary, giving ages for the two components that disagree by 0.2 dex (2.0 σ ).In contrast, our observed masses and luminosities can be reproduced at a single ageby ”hybrid” evolutionary tracks where a smooth change from a cloudy to cloudlessphotosphere around 1300 K causes slowing of luminosity evolution. Remarkably, suchmodels also match our observed J HK flux ratios and colors well. Overall, it seems thatthe distinguishing features SDSS J1052+4422AB, like a J -band flux reversal and high-amplitude variability, are normal for a field L/T binary caught during the process of * Data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnershipamong the California Institute of Technology, the University of California, and the National Aeronautics and SpaceAdministration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. † Based on data obtained with WIRCam, a joint project of CFHT, Taiwan, Korea, Canada, France, at the Canada-France-Hawaii Telescope, which is operated by the National Research Council of Canada, the Institute National desSciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The University of Texas at Austin, Department of Astronomy, 2515 Speedway C1400, Austin, TX 78712, USA Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA Gemini Observatory, Northern Operations Center, 670 N. A’ohoku Place, Hilo, HI, 96720 USA Research School of Astronomy & Astrophysics, Australian National University, Canberra ACT 2611, Australia C3 Energy, 1300 Seaport Boulevard Suite 500, Redwood City, CA 94063, USA Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA . +0 . − . Gyr) and surface gravity (log g = 5 . Subject headings: astrometry — binaries: close — brown dwarfs — parallaxes — stars:fundamental parameters — stars: individual (SDSS J105213.51+442255.7)
1. Introduction
Perhaps the most important but uncertain process to model for substellar objects is the for-mation, growth, and dispersal of condensate clouds. When present in the photosphere, clouds area dominant opacity source and thereby control basic observable properties like broadband colors,magnitudes, and spectra. In brown dwarfs, clouds appear to grow in influence going from early tolate-type L dwarfs then begin dispersing with early-type T dwarfs, resulting in drastic changes innear-infrared spectra. The most prominent features of this transition are that
J HK colors becomesignificantly bluer (e.g., Leggett et al. 2002) and J -band fluxes become brighter (e.g., Dahn et al.2002) going from late L to early T dwarfs. Early analysis of such observations indicated that theL/T transition occupies a narrow range of effective temperature ( T eff ), although the underlyingphysical process explaining the brightening at J band was debated. For example, rapid changesin cloud sedimentation efficiency (Knapp et al. 2004) or breakup caused by convection (Burgasseret al. 2002) could reproduce the J -band brightening at a single T eff , while Tsuji & Nakajima (2003)suggested that mass/age spreads in the population of field brown dwarfs were responsible, notchanges in the cloud themselves.The discovery of binaries at the L/T transition in which one component was directly observedto be brighter at 1.0–1.3 µ m ( Y or J ) but fainter at other wavelengths ( I , H , or K ; Gizis et al. 2003;Liu et al. 2006) provided the first unambiguous evidence that the J -band brightening must occuralong a single isochrone. Such flux reversals require a significant flux redistribution as brown dwarfscool, most likely brought on by changes in cloud opacity. Recent discoveries of large-amplitudevariables at near-infrared wavelengths, so far only reliably detected in the L/T transition (e.g.,Artigau et al. 2009; Radigan et al. 2012, 2014), support the idea that cloud clearing is spatiallyheterogeneous in the photosphere. However, without well determined masses and/or ages for anysystems that display a J -band flux reversal or weather, the alternative possibility of unusual cloudproperties, e.g., due to surface gravity, exists. In fact, there still remains only one object in theL/T transition with a precise dynamical mass measurement (LHS 2397aB, photometric spectraltype estimate of L7; Dupuy et al. 2009c).For nearly all substellar objects found to date, evolutionary models are the sole means forestimating their physical properties, typically by using the observed luminosity and an adoptedage to yield a model-dependent mass (and temperature, radius, and surface gravity). Such modelsrequire an assumption for the photosphere opacity as a boundary condition, and one of the keychallenges is the treatment of condensate clouds. The formation, growth, and settling of dust con- 3 –densates likely occurs at many different levels in the atmosphere and would thus also be influencedboth by the local physical conditions and the bulk motions of the gas via convection (Freytag et al.2010). There are numerous approaches to modeling these complex processes and parameterizingthem so that they can be incorporated into one-dimensional atmospheric models (e.g., see recentreviews by Helling & Casewell 2014; Marley & Robinson 2014). However, the currently availableevolutionary models assume one of two limiting cases for treatment of the dust. Either dust existsin chemical equilibrium with the gas, resulting in thicker clouds at cooler temperatures (Chabrieret al. 2000), or the grains rapidly fall out of the photosphere as soon as they form, leaving behinddust-free gas (Burrows et al. 1997; Baraffe et al. 2003). Attempting to match the observationsdescribed above that dust clouds disperse over a narrow range of T eff , Saumon & Marley (2008)computed evolutionary models where the atmosphere is interpolated between the fully cloudy andcloud free limiting cases as objects cool from 1400 K to 1200 K. Despite the limitations of thesevarious simplifying assumptions, current models are at least broadly in accord with the observedsubstellar sequence in open clusters (e.g., Lodieu et al. 2014; Bouy et al. 2015) and in the field (e.g.,Tinney et al. 2003; Saumon & Marley 2008), although discrepancies are obvious in regimes wherephotospheric condensates play a more significant role, especially the L/T transition.More stringent test of the theoretical models are now within reach. The past decade sawa growing number of substellar visual binaries with dynamical masses measured via astrometricmonitoring (e.g., Lane et al. 2001; Bouy et al. 2004; Liu et al. 2008; Dupuy et al. 2009a,b,c, 2010;Konopacky et al. 2010). The most powerful tests to date come from brown dwarf binaries in a hier-archical triple with a main-sequence star, where the subtellar binary orbit gives its dynamical totalmass and the primary star gives the system age from gyrochronology. For the two known systemswhere this is possible, the models seem to predict luminosities that are systematically 0.2–0.4 dexlower than observed (Dupuy et al. 2009b, 2014). However, without individual masses the mass–luminosity relation is unconstrained, and thus a complementary test would be to obtain massesand luminosities for a coeval binary system, even in the absence of an age determination. Previouswork has resulted in individual masses for late-M dwarfs, showing broad agreement with the mass–luminosity relation as models approach the substellar boundary. Further tests of evolutionarymodels are sorely needed, especially in the L/T transition where they are routinely employed tocharacterize planetary-mass discoveries, e.g., 2MASSW J1207334 − − J -band flux reversal binary, SDSS J105213.51+442255.7AB(hereinafter SDSS J1052+4422AB), along with high-precision dynamical masses of the individualcomponents based on resolved orbital monitoring from Keck and absolute astrometry from the Individual masses have been determined for two field late-M dwarf systems: one from resolved radial velocitiesand relative astrometry (Gl 569Bab; Zapatero Osorio et al. 2004; Simon et al. 2006; Konopacky et al. 2010), and onefrom absolute astrometry (LHS 1070BC; Seifahrt et al. 2008; K¨ohler et al. 2012). The young brown dwarf eclipsingbinary 2MASS J05352184 − . ± . . ± . J band, although their integrated-light observations couldnot determine which component was responsible for the variability. Our mass determination forSDSS J1052+4422AB is therefore the first for a J -band flux reversal binary and the first for abrown dwarf displaying significant weather. More generally, our results are also the first individualmass measurements for any field L or T dwarfs. This is distinct from the aforementioned resultson dynamical total masses, as the only individual masses in this spectral type range are for twosubstellar companions to stars measured from absolute astrometry (Gl 802B; Ireland et al. 2008), orrelative astrometry combined with radial velocities (HR 7672B; Crepp et al. 2012). There are alsoa number of stellar model-dependent mass determinations for brown dwarfs in eclipsing systems(Deleuil et al. 2008; Anderson et al. 2011; Bouchy et al. 2011a,b; Johnson et al. 2011; Siverd et al.2012; D´ıaz et al. 2013; Montet et al. 2014). However, all of these companions lack the spectralinformation available for field L and T dwarfs that enables the strongest tests of substellar models.
2. Discovery and Astrometric Monitoring of SDSS J1052+4422AB2.1. Keck/NIRC2 LGS AO
We observed SDSS J1052+4422 on 2005 May 1 UT with the then recently commissionedlaser guide star adaptive optics (LGS AO) system at the Keck II telescope (Bouchez et al. 2004;Wizinowich et al. 2006; van Dam et al. 2006). We used the facility near-infrared camera NIRC2,obtaining five dithered images in K ′ band. SDSS J1052+4422 appeared to be marginally resolved(peanut shaped) in these images, indicating that it was likely a binary. In follow-up imaging on2006 May 5 UT, SDSS J1052+4422 was more obviously resolved because it had moved to a widerprojected separation of 70 mas, as compared to 42 mas in 2005. We obtained data in the Mauna KeaObservatories (MKO) J , H , and K S photometric bandpasses (Simons & Tokunaga 2002; Tokunagaet al. 2002) and discovered that while the western component was brighter in K S and perhaps H band, the eastern component was in fact brighter in J band. In keeping with the convention withprevious J -band flux reversal binaries (e.g., Liu et al. 2006; Looper et al. 2008), we will refer to thecomponent brighter in K S -band as the primary (SDSS J1052+4422A).We continued Keck AO monitoring of SDSS J1052+4422AB in order to determine its orbitalparameters and thereby a total dynamical mass. Our observations are a combination of normalimaging and data taken with the 9-hole non-redundant aperture mask installed in the filter wheelof NIRC2 (Tuthill et al. 2006). On some nights we obtained data using the natural guide star(NGS) AO system, because the tip-tilt star is bright enough ( R ≈ . ′′ away) that it can sometimes be used as an NGS. The analysis of our data was the 5 –same regardless of whether we observed in NGS or LGS mode.Our procedure for reducing and analyzing NIRC2 imaging data is described in detail in ourprevious work (Dupuy et al. 2009a,b,c, 2010). To summarize briefly, we measure binary parametersusing a three-component Gaussian representation of the point-spread function. We derive uncer-tainties by applying our fitting method to artificial binary images constructed from images of singlestars with similar full-width half-maxima (FWHM) and Strehl ratios, as well as by checking thescatter between individual dithered images. We use the NIRC2 astrometric calibration from Yeldaet al. (2010), which includes a correction for the nonlinear distortion of the camera and has a pixelscale of 9 . ± .
002 mas pixel − and an orientation for the detector’s + y -axis of +0 . ◦ ± . ◦ J and H bands are consistent with variability at the ≈ J = − . ± .
09 mag, ∆ H = 0 . ± .
07 mag,and ∆ K = 0 . ± .
05 mag. The only other ultracool binary known to have such a large J -bandflux reversal is 2MASS J14044948 − J = − . ± .
08 mag; Looper et al. 2008; Dupuy& Liu 2012).
We have been monitoring SDSS J1052+4422AB as part of the Hawaii Infrared Parallax Pro-gram at the CFHT in order to measure the precise distance needed for a dynamical mass determina-tion. Our methods for obtaining high-precision astrometry from the facility near-infrared wide-fieldimager WIRCam (Puget et al. 2004) are described in detail in Dupuy & Liu (2012). We have ob-tained a total of 427 J -band images centered on SDSS J1052+4422AB over 21 epochs spanning6.79 yr. At each epoch, we measured the position of SDSS J1052+4422AB in integrated light alongwith 30 other stars in the field having signal-to-noise ratios (S/N) >
23. The subset of 26 stars thatappear in the SDSS-DR9 catalog (Ahn et al. 2012) were used for the absolute astrometric calibra-tion of the linear terms (pixel scales in x and y , rotation, and shear). We simultaneously fit forthe proper motion and parallax of all stars in the field and found no other sources co-moving with 6 –SDSS J1052+4422AB down to J = 18 . . ′′ ± . ′′ χ = 428 .
3. Measured Properties of SDSS J1052+4422AB3.1. Orbital Parameters & Parallax
We performed a joint analysis of our two astrometric data sets for SDSS J1052+4422AB:resolved measurements from Keck AO and integrated-light positions from CFHT/WIRCam. Allbut one of the seven visual binary orbit parameters are shared in common between the Keck andCFHT data. Since our Keck data only gives us the position of one binary component relativeto the other, we fit for the total semimajor axis ( a = a + a ) that is the sum of the individualcomponent’s semimajor axes about the center of mass. In our CFHT data, we only see the motionof the photocenter, the amplitude of which depends on the flux ratio and mass ratio of the binary.We therefore fit for a photocenter semimajor axis ( α ) that we will later use to derive the systemmass ratio. We also fit for the five usual parameters needed for our CFHT parallax data: R.A. zeropoint and proper motion, Dec. zero point and proper motion, and parallax. Therefore, there are atotal of 13 parameters in the joint fit of our Keck and CFHT data.To determine probability distributions for the orbit and parallax parameters, we performeda Markov Chain Monte Carlo (MCMC) analysis. Unlike our previous work, we used the Pythonimplementation of the affine-invariant ensemble sampler emcee v2.1.0 (Foreman-Mackey et al.2013). This allows for more efficient exploration of our 13-dimensional parameter space than ourown custom MCMC tools that used a Metropolis-Hastings jump acceptance criterion with Gibbssampling. We adopted uniform priors in the logarithms of period and semimajor axis (log P , log a ),eccentricity ( e ), argument of periastron ( ω ), PA of the ascending node (Ω), mean longitude at areference time ( λ ref ), and the ratio of the photocenter semimajor axis to the total semimajor axis( α/a ). The reference time is t ref = 2455197 . T = t ref − P × ( λ ref − ω ) / ◦ . We assume randomly distributed viewingangles by adopting an inclination prior uniform in cos i . We adopt uniform priors in the propermotion and R.A. and Dec. zero points and a uniform spatial volume prior in the parallax. Thelatter is justifiable as SDSS J1052+4422 was discovered well above the magnitude limits of the 7 –SDSS survey ( m lim − M ≈ . ≤ walkers of 10 steps each, saving only every hundredth step and discarding the first 10% of steps as the burn-intime for each walker.The best-fit parameters and credible intervals derived from our MCMC posterior distributionsare given in Table 3. We found an orbital period of 8 . ± .
025 yr (0.29% error) and totalsemimajor axis of 70 . ± .
24 mas (0.34% error), and accounting for the slight covariance betweenthese parameters results in an uncertainty in the dynamical total mass of 1.1% from our orbitdetermination alone. As a check on our new MCMC methods, we performed a separate MCMCanalysis on just the Keck data using our own Metropolis-Hastings code (Dupuy et al. 2014). Theresulting 1 σ credible intervals for the seven visual binary parameters were consistent to within afraction of 1 σ . The resolved orbit of SDSS J1052+4422AB is shown in Figure 3 along with ourKeck astrometry.The additional parameters we fitted to our integrated-light CFHT data provide the propermotion and parallax relative to our grid of astrometric reference stars, as well as the size of thephotocenter’s orbit ( α = − . ± . π abs − π rel = 1 . ± . J -bandmagnitude range of our images, according to the much larger modeled Besan¸con population. Addingthis to the relative parallax results in an absolute parallax of 38 . ± . . ± . µ R . A . = − ± − and ∆ µ Dec . = − ± − . As a check, we input our absolute propermotion and parallax to the BANYAN II v1.3 web tool (Malo et al. 2013; Gagn´e et al. 2014) butfound no linkage to the seven kinematic associations in their solar neighborhood model. We quote the photocenter semimajor axis as a negative value because the photocenter motion is the opposite ofwhat is seen in typical pairings of stars, brown dwarfs, or planets. Normally, the less massive component is fainter andthus the center-of-light follows the brighter, more massive component’s motion. In the case of SDSS J1052+4422AB,the center of J -band light follows the secondary component. This can be seen when comparing Figures 3 and 4 where,e.g., in 2007 the secondary is seen in Keck data to be southeast of the primary and in CFHT data the photocentershift is also to the southeast. Combining our measured parallactic distance with the total semimajor axis and orbital periodgives a precise total system mass for SDSS J1052+4422AB of 88 ± M Jup (6% error). We can alsocompute the mass ratio and thereby individual component masses by considering the photocentermotion seen in our integrated-light CFHT data. We found the ratio of the photocenter semimajoraxis to the total semimajor axis was α/a = − . ± . α/a = f − β . The first parameter is the ratio of thesecondary’s mass to the total mass, f = M B / ( M A + M B ), and the second parameter is the ratioof secondary’s flux to the total flux, β = L B / ( L A + L B ). Our J -band flux ratio measured fromKeck is ∆ J = − . ± .
09 mag, which corresponds to β = 0 . ± . f gives0 . ± .
022 and thus a mass ratio of q ≡ M B /M A = 0 . ± .
07. This in turn gives individualmasses of 49 ± M Jup for SDSS J1052+4422A and 39 ± M Jup for SDSS J1052+4422B. Therefore, wevalidate for the first time that assumed primary component in a J -band flip system is indeed moremassive, and the mass ratio is surprisingly low. We also directly determine that both componentsare unambiguously substellar ( < M Jup ; Chabrier & Baraffe 1997).
In order to fully characterize the SDSS J1052+4422AB system and aid in computing bolo-metric corrections for the components, we have determined the component spectral types throughdecomposition of its integrated-light spectrum. Burgasser et al. (2008) published a SpeX prismspectrum of SDSS J1052+4422 in integrated light ( R = 120) which we obtained from the SpeXPrism Libraries. We performed spectral decomposition analysis using the method described in Sec-tion 5.2 of Dupuy & Liu (2012). Briefly, we started with all possible pairs of the 178 IRTF/SpeXprism spectra from the library of Burgasser et al. (2010). For each template pairing we determinedthe scale factors needed to minimize the χ compared to our observed spectrum. This resultedin a set of J -, H -, and K -band flux ratios for each pairing, which we compared to the flux ratioswe measured from our Keck AO images (∆ J = − . ± .
09 mag, ∆ H = 0 . ± .
07 mag, and∆ K = 0 . ± .
05 mag). We excluded template pairs that disagreed significantly with our mea-sured flux ratios, p ( χ ) < .
05, and then examined the ensemble of template pairs that providedthe best spectral matches.The best match to our spectrum was provided by the templates SDSSp J010752.33+004156.1(L6) and SDSS J175024.01+422237.8 (T1.5), where we use the infrared types reported by Burgasseret al. (2010). This best-fit spectral template match is shown in Figure 5. The next best matchesuse primary templates with types ranging from L4.5:: (2MASSW J0820299+450031, typed in theoptical as L5 by Kirkpatrick et al. 2000) to L8.5 and secondary templates with types ranging from http://pono.ucsd.edu/~adam/browndwarfs/spexprism . ± . . ± . By combining our Keck flux ratios with published MKO system photometry for SDSS J1052+4422AB(Chiu et al. 2006) and our CFHT parallax, we are able to estimate the component luminosities.Given the fact that the flux ratio flips between J and K bands, we first consider the bolometricluminosity ( L bol ) implied by each bandpass separately. We used the polynomial relations betweenspectral type and bolometric correction (BC) from Liu et al. (2010). To determine the uncertaintyin the bolometric correction we allow for spectral type uncertainties in a Monte Carlo fashion, com-pute the rms, and then add the published rms scatter about the polynomial relation in quadrature.In J band we find bolometric corrections of 1 . ± .
16 mag and 1 . ± .
24 mag for the primary andsecondary, respectively. This BC difference exactly compensates for the fact that the secondary isbrighter in J band, resulting in nearly identical luminosities of log( L bol /L ⊙ ) = − . ± .
07 dex and − . ± .
10 dex, respectively. Similarly, in H band where our photometry is consistent with thetwo components having equal flux, the BC compensates and gives log( L bol /L ⊙ ) = − . ± .
04 dexand − . ± .
04 dex. We find comparable results using K band of log( L bol /L ⊙ ) = − . ± .
05 dexand − . ± .
06 dex.We chose to use the luminosities derived from our K band photometry because it is the leastlikely to be affected by the variability observed by Girardin et al. (2013) in J band, and we havemany more K -band flux ratio measurements than at J or H bands. Our K -band flux ratio hasthe smallest uncertainty, and the scatter in the BC K relation (0.08 mag) is almost as small as forBC H (0.07 mag). We note however that the L bol estimates in all bands are consistent within theuncertainties.Table 4 provides a summary of all of the directly measured properties of the SDSS J1052+4422ABsystem. Figure 6 shows the components of SDSS J1052+4422AB on a color–magnitude diagram incomparison to other field L and T dwarfs with measured distances.
4. Model-Derived Properties for SDSS J1052+4422AB
With a precisely determined total dynamical mass (6%), component masses (7%), and com-ponent luminosities (15%–20%), we can derive all other physical properties ( T eff , log g , age, etc.)by invoking evolutionary models. Only one set of models currently incorporates cloud dispersal atthe L/T transition, which is particularly relevant for SDSS J1052+4422AB. SM08 “hybrid” modelsassume the photosphere smoothly transitions from cloudy to cloudless as objects cool from effectivetemperatures of 1400 K to 1200 K. Because SDSS J1052+4422A is expected to be cloudy based onits late-L spectral type, and SDSS J1052+4422B likely still possesses some cloud opacity at the 10 –photosphere, we also consider the SM08 fully cloudy ( f sed = 2) and Lyon Dusty (Chabrier et al.2000) models.To derive model properties from the individual masses and luminosities only requires a straight-forward bilinear interpolation of model tracks. But this could result in very different ages if modelsdo not accurately predict the mass–luminosity relation for our objects. Because we are also in-terested in deriving properties under the assumption of coevality, we also use our (more precise)total mass and individual luminosities, ignoring our measured mass ratio, to derive properties fromevolutionary models in the same fashion as in our previous work (Liu et al. 2008; Dupuy et al.2009b). In this coeval analysis, at each point on a log(age) grid we use the luminosity of each com-ponent to calculate their model-predicted mass, T eff , surface gravity, radius, lithium abundance,and near-infrared colors. This is done in a Monte Carlo fashion such that we use 10 values for acomponent’s L bol , resulting in 10 mass estimates at each age. We then step through each of these10 L bol pairs, considering the full range of ages for that pair, sum the component masses as afunction of age, and determine the age that matches the measured total mass by interpolating thecurve. This is also done in a Monte Carlo fashion by repeating this step 10 times using randomlydrawn values for the measured M tot from our MCMC posterior. This results in 10 model-derivedvalues for every parameter and accounts for the errors in both L bol and M tot while appropriatelytracking their covariances via the common uncertainty in the distance.We report the median, 1 σ , and 2 σ credible intervals of the model-derived parameter distribu-tions in the case where we used the individual masses and in the case where we used the total massassuming coevality (Table 5). One of the fundamental predictions of substellar evolutionary models is how luminosity changeswith age for a given mass (or changes with mass at a given age). Thus, by measuring the componentmasses and luminosities of SDSS J1052+4422AB we can test whether models successfully give thesame age for the two components. (By a typical field age of ∼ ∼ . +0 . − . Gyr and 0 . +0 . − . Gyr forthe primary and secondary of SDSS J1052+4422AB, respectively. Accounting for the covariance indistance and mass ratio, the age difference is ∆ log t = 0 . ± .
10 dex, 2.0 σ discrepant with beingcoeval. The SM08 cloudy models give similar ages to Lyon Dusty but somewhat more coeval with∆ log t = 0 . ± .
10 dex (1.6 σ different from coeval). In contrast to both of these cases, the SM08hybrid models give ages consistent with coevality at 0.9 σ , ∆ log t = 0 . ± .
12 dex. 11 –The more realistic assumption of SM08 hybrid models that clouds disappear as temperaturescool from 1400 K to 1200 K results in higher luminosities at a given mass and age during thetransition. This higher luminosity is not simply due to less cloud opacity. The difference inentropy between a cloudy 1400 K brown dwarf and a cloudless 1200 K brown dwarf is greater thanthe entropy difference of two brown dwarfs at those temperatures that are both cloudy (Saumon& Marley 2008). Therefore, luminosity evolution should appear to slow down as brown dwarfscool through the L/T transition because it takes longer to shed this excess entropy, causing aphase of increased luminosity compared to either cloudy or cloudless models. This means that themass–luminosity relation at a given age becomes shallower in the L/T transition, so that a givenluminosity ratio could correspond to a mass ratio further from unity, like the one we measureddirectly (0 . ± .
07, Section 3.2). Therefore, it is not surprising that the SM08 hybrid models giveages in better agreement with coevality for SDSS J1052+4422AB.If we force coevality by ignoring our measured mass ratio, then we find single best matchingmodel-derived ages of 1 . +0 . − . Gyr (SM08 hybrid) and 0 . +0 . − . Gyr (Lyon Dusty). Figure 7 showsthe mass–luminosity relation predicted by models at these respective coeval ages, illustrating thefundamental difference in the predicted luminosity evolution between these two models. Over themass range 40–50 M Jup , the Lyon Dusty isochrone has a power-law slope of ∆ log L bol / ∆ log M =3 .
1. In contrast, for the SM08 hybrid models this slope is only 1.3. Our directly measured massesfor SDSS J1052+4422AB imply a power-law slope ∆ log L bol / ∆ log M = 0 . +0 . − . over the same ≈ M Jup mass range. Thus, we find a mass–luminosity relation in the L/T transition that isin much better agreement with SM08 hybrid models than fully cloudy models. In fact, our slopeseems to be even shallower than the hybrid models and is even nominally consistent with a invertedrelation (∆ log L bol / ∆ log M <
0) within the 1 σ uncertainty.Finally, we note that another way of framing the coevality test is to compare the model-derivedmass ratios with our observed value of 0 . ± .
07. When using just our total dynamical mass andindividual luminosities, both cloudy models give similar mass ratios of 0 . +0 . − . (SM08) and 0 . ± .
05 (Lyon). These are much closer to unity than we observe because the steeper mass–luminosityrelation predicted by cloudy models gives a very small difference in mass for a correspondinglysmall difference in observed luminosity (∆ log L bol = 0 . ± .
07 dex). In comparison, SM08 hybridmodels predict a mass ratio of 0 . +0 . − . that is somewhat larger than but consistent with ourmeasured value at 0.9 σ . Combining evolutionary model radii with a measured luminosity and mass readily producesestimates of effective temperature ( T eff ∝ L bol − / R − / ) and surface gravity ( g ∝ M R − ), re-spectively. There are only small differences between the radii predicted at a given age by themodels considered here ( . .
1% in T eff and . g . Moreimportant to the model-derived radii is whether we force coevality, in which case the secondary 12 –is predicted to be only slightly larger ( ≤ σ range), so its predicted radius is 3%–6% larger. Therefore, we adopt the coevalmodel-derived temperatures and surface gravities of SDSS J1052+4422AB, using the SM08 hybridmodels that are most consistent with coevality.The model-derived temperature of the L6 . ± . ±
30 K, while theT1 . ± . +40 − K. Their predicted surface gravities are log g = 5 . +0 . − . dex and5 . +0 . − . dex, respectively. Interestingly, the mean evolutionary model-derived temperature of thetwo components ( ≈ µ m spectrum of SDSS J1052+4422ABfrom Stephens et al. (2009) who found T eff = 1300 K (acceptable range of 1200–1400 K) andlog g = 5 . T evoleff = 1430 ±
40 K and T atmeff = 1400 K; Dupuy et al. 2009c). Finally,we note that the model-derived temperatures for SDSS J1052+4422AB align very well with theassumption made in SM08 hybrid models that the L/T transition occurs over the temperaturerange 1200–1400 K. We have independently measured the
J HK colors of the components of SDSS J1052+4422ABby combining our Keck flux ratios with the photometry from Chiu et al. (2006). All colors agreewithin 1 σ of the predictions of the SM08 hybrid models whether we enforce coevality or not,although there is somewhat better agreement when deriving colors directly from the individualmasses and luminosities (non-coeval). This agreement is remarkable as all other ultracool dwarfswith dynamical mass determinations to date have typically shown & ≈ J − K colors for the components of SDSS J1052+4422AB, whichis not surprising given their assumption of maximal dust clouds. The reason that the SM08 hybridmodels agree with our observed J HK colors is because these evolutionary models also predict a J -band flux reversal for a system like SDSS J1052+4422AB. The model-derived flux ratios from theindividual masses and luminosities are ∆ J = − . +0 . − . mag and ∆ K = 0 . +0 . − . mag, which arequite similar to our measured values (∆ J = − . ± .
09 mag, ∆ K = 0 . ± .
05 mag). Figure 8shows our observed colors and magnitudes for SDSS J1052+4422AB compared to SM08 hybridevolutionary model tracks. 13 –
According to Chabrier et al. (1996), most of the initial supply of a ≥ M ⊙ brown dwarf’slithium is destroyed via fusion by an age of ≤ ≤ . ± . M ⊙ and 0 . +0 . − . , so the Lyonmodels predict that they should have retained almost all of their lithium (Li / Li ≥ .
55 at 2 σ for the primary). However, even if both components of SDSS J1052+4422AB are lithium bearing,they may not possess a significant amount of atomic lithium that would be readily detectable viathe Li I doublet at 6708 ˚A. At temperatures . ≈ σ upper limit on ourmodel-derived temperature for SDSS J1052+4422A ( T eff < . ± . − ǫ Indi Ba (T1), although this may be due to the fact that it is massive enough to have depletedits lithium (Cardoso et al. 2009). Thus, it is unclear whether SDSS J1052+4422AB would showevidence for atomic lithium in its integrated-light spectrum. High-resolution optical spectroscopy ofSDSS J1052+4422AB would provide a unique, joint test of the theoretical lithium-fusing mass-limitand atmospheric model predictions of the chemical depletion of lithium.
5. Conclusions
We have discovered that SDSS J1052+4422AB (L6.5+T1.5) is a J -band flux reversal binary.We present precise individual dynamical masses by combining resolved Keck AO orbital monitoringspanning 9.0 yr with integrated-light CFHT/WIRCam astrometric monitoring spanning 6.8 yr, thefirst such masses for any field L or T dwarfs. Despite spectral types that are similar and luminositiesthat are indistinguishable within the errors, we find a surprisingly low mass ratio of q = 0 . ± . . ± .
01 K¨ohleret al. 2012), also measured from astrometry, which highlights the greater potential of astrometry formeasuring precise individual masses as compared to radial velocities. For example, our mass ratiois based on a total of only 2.4 hr of integration time on a 4-m-class telescope, yet it is more precisethan the q = 0 . +0 . − . measured for the ≈ ± M Jup for SDSS J1052+4422A (L6 . ± .
5) and 39 ± M Jup for SDSS J1052+4422B (T1 . ± . J -band flux reversal binary or high-amplitude variable with a dynamical mass 14 –measurement, providing a precise benchmark for the cloud dispersal phase of substellar evolution.We validate that the component fainter in J band is in fact more massive and that both componentsare unambiguously substellar ( < M Jup ). Perhaps the most striking result is the shallow mass–luminosity relation in the L/T transition implied by our data (∆ log L bol / ∆ log M = 0 . +0 . − . over ≈ M Jup ). This disagrees with the mass–luminosity relation predicted by fully cloudy models,providing the first direct observational support that cloud dispersal plays an important role inluminosity evolution. We quantify this as a coevality test using our measured individual massesand luminosities to derive an age from evolutionary models for each component and test if themodels successfully give the same age for both components. Lyon Dusty models give ages thatare different by 0 . ± .
10 dex, a 2.0 σ discrepancy. In comparison, hybrid models from Saumon& Marley (2008), in which the dispersal of clouds causes a slowing of luminosity evolution, givescomponent ages different by 0 . ± .
12 dex and thus consistent at (0.9 σ ).In fact, these SM08 hybrid evolutionary models paint a remarkably self-consistent picture forthe properties of SDSS J1052+4422AB. The models assume that clouds disperse as temperaturescool from 1400 K to 1200 K. From our measured luminosities and SM08 model-derived radii wefind T eff = 1330 ±
30 K for the L6 . ± . +40 − K for the T1 . ± . J HK colors of the components, including thereversal in flux ratio observed between J and K bands. In addition, the T eff of 1300 K found forSDSS J1052+4422AB by Stephens et al. (2009), who used the same atmospheres in their spectralsynthesis modeling as are used by SM08 evolutionary models, is in excellent agreement with ourtemperatures derived from luminosities and model radii. We note that without an independentmeasurement of the age of SDSS J1052+4422AB, we cannot rule out a constant systematic offsetin the SM08 hybrid model luminosities, as our coevality test only constrains slope of the mass–luminosity relation. For example, mid-L dwarfs appear to be 0.2–0.4 dex more luminous thanpredicted by models at a given mass and age (Dupuy et al. 2009b, 2014). If this holds true for L/Ttransition objects, then the age we derive from SM08 models would be underestimated by a factorof ≈ J -band flux re-versal and high-amplitude variability, are normal for a field L/T binary caught during the processof cloud dispersal. SDSS J1052+4422AB’s model-derived age of 1 . +0 . − . Gyr is typical of fieldbrown dwarfs (e.g., Zapatero Osorio et al. 2007), and the component surface gravities are corre-spondingly unexceptional, log g = 5 . J -bandflux reversal systems, will require more individual mass measurements for late-L to early-T typebrown dwarfs. Fortunately, such masses will likely be available in the near future as our CFHTastrometric monitoring continues. Orbit determinations typically require ≈
30% coverage of theorbital period, and we have been obtaining CFHT data on our Keck dynamical mass sample for ≈ .
20 yr should soon have photocenter semimajoraxis measurements that will enable precise individual dynamical masses to further map out the 15 –substellar mass–luminosity relation.Our results lend further support to the growing evidence that clouds have a significant impacton the luminosity evolution of substellar objects. A shallow mass–luminosity relation in the L/Ttransition suggests that even when the age and luminosity of an object are constrained its massmay be difficult to estimate precisely. This adds another obstacle to estimating masses for directlyimaged extrasolar planets in this spectral type range (e.g., HR 8799b; Bowler et al. 2010; Barmanet al. 2011). The L/T transition corresponds to the breakup of mostly silicate and iron clouds.At cooler temperatures, clouds composed of sulfides emerge ( T eff .
900 K; Morley et al. 2012) andwater ice clouds possibly at .
350 K (Morley et al. 2014). Even though sulfide clouds are expectedto be thinner, in principle they could impact luminosity evolution in a comparable way as we havenow observed for silicate clouds, implying similar alterations to the mass–luminosity relation formuch colder brown dwarfs. Directly measured individual masses for late-T and Y dwarf binariesshould be able to test this idea.We thank the referee for a timely and very helpful review. It is a pleasure to thank AntoninBouchez, David LeMignant, Marcos van Dam, Randy Campbell, Gary Punawai, Peter Wizinowich,and the Keck Observatory staff for their efforts assisting on our first LGS AO night that resulted inthe discovery of SDSS J1052+4422AB. We also thank Joel Aycock, Al Conrad, Greg Doppmann,Heather Hershley, Jim Lyke, Jason McIlroy, Julie Riviera, Hien Tran, and Cynthia Wilburn forassistance with subsequent Keck LGS AO observing. We greatly appreciate the CFHT staff fortheir constant observing support and dedication to delivering the highest quality data products.This work was supported by a NASA Keck PI Data Award, administered by the NASA ExoplanetScience Institute. M.C.L. acknowledges support from NSF grant AST09-09222. Our research hasemployed the 2MASS data products; NASA’s Astrophysical Data System; the SIMBAD databaseoperated at CDS, Strasbourg, France; and the SpeX Prism Spectral Libraries, maintained by AdamBurgasser at . Finally, the authors wish to recognizeand acknowledge the very significant cultural role and reverence that the summit of Mauna Keahas always had within the indigenous Hawaiian community. We are most fortunate to have theopportunity to conduct observations from this mountain.
Facilities:
Keck:II (LGS AO, NGS AO, NIRC2), CFHT (WIRCam) IRTF (SpeX) 16 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
20 – K’ J H K S K K’ J K H Fig. 1.— Contour plots of our Keck AO images from which we derive astrometry and flux ratios(Table 1). Contours are in logarithmic intervals from unity to 10% of the peak flux in each band.The image cutouts are all the same size and have the same native pixel scale, and we have rotatedthem such that north is up for display purposes. 21 –
K’ K H K K K Fig. 2.— Keck/NIRC2 images of the interferograms produced when observing SDSS J1052+4422ABwith the 9-hole aperture mask. The binary can be seen by eye as an elongation or double peak inthe center of the point-spread function in all but one epoch. In data from 2010 May 22 UT thebinary is very tight (39 . ± . ◦ )caused by atmospheric dispersion given the modest airmass (1.32) of the observation. These imagecutouts are all the same size, have the same native pixel scale, have been rotated such that northis up, and are shown with a square-root stretch. 22 – ∆α cos δ (arcsec)−0.10−0.05 ∆ δ ( a r cs e c ) SDSS J1052+4422AB M tot = 88 ± M Jup −1200120 P o s i t i on ang l e ( o ) O − C S epa r a t i on ( m a s ) −303 O − C Fig. 3.—
Left:
Keck AO relative astrometry for SDSS J1052+4422AB along with 100 randomlydrawn orbits from our MCMC analysis individually plotted as thin lines. Error bars for the datapoints are smaller than the plotting symbols. The short dotted line indicates the time of periastronpassage, the long dashed line shows the line of nodes, and small empty circles show predicted futurelocations.
Right:
Measurements of the projected separation and PA of SDSS J1052+4422AB. Thebest-fit orbit is shown as a solid line. The bottom panels show the observed minus computed( O − C ) measurements with observational error bars. 23 – ∆α cos δ (arcseconds)−1.0−0.8−0.6−0.4−0.20.0 ∆ δ ( a r cs e c ond s ) SDSS J1052+44AB −50−2502550 ∆ α c o s δ ( m a s ) orbit & proper motion subtracted ∆ α c o s δ ( m a s ) parallax & proper motion subtracted −50−2502550 ∆ δ ( m a s ) orbit & proper motion subtracted ∆ δ ( m a s ) parallax & proper motion subtracted Fig. 4.—
Left:
CFHT/WIRCam integrated-light astrometry for SDSS J1052+4422AB (blue circles)along with the best-fit model incorporating proper motion, parallax, and photocenter orbital motion(dotted line).
Middle, Right:
The same astrometry except with the best-fit proper motion andorbital motion removed, leaving just the parallax (top), and with the best-fit proper motion andparallax removed, leaving just the orbital motion of the photocenter (bottom). Error bars areplotted on all panels, but they are typically only visible in the plots displaying orbital motion. 24 – N o r m a li z ed f l u x SDSS J1052+4422AB
SDSSp J010752.33+004156.1 (L6.0)SDSS J175024.01+422237.8 (T1.5) µ m)−101 ∆ m ag Fig. 5.— Integrated-light spectrum of SDSS J1052+4422AB (black) and best matching componenttemplates (colored lines). The bottom subpanel shows the observed J -, H -, and K -band broadbandflux ratios used to constrain the decomposition (filled black circles with errors) and the resultingflux ratios computed from the best matching template pair (open colored squares). 25 – -0.5 0.0 0.5 1.0 1.5J-H (mag)16151413 M J ( m ag ) L4-L9.5T0-T3.5 ≥ T4 Fig. 6.— Color–magnitude diagram showing the components of SDSS J1052+4422AB (yellow stars)along with field L and T dwarfs with measured distances (open circles). Both components havetypical colors and magnitudes for their spectral types (L6 . ± . . ± . ; Dupuy & Liu 2012), and we only plot objects withuncertainties <
10% in parallax and <
70 60 50 40 30 20Mass (M
Jup )10 −5 −4 Lu m i no s i t y ( L S un ) SM08 hybrid
70 60 50 40 30 20Mass (M
Jup )Lyon Dusty
Fig. 7.— Our directly measured individual masses and luminosities for the components ofSDSS J1052+4422AB compared to predictions from SM08 hybrid (left) and Lyon Dusty (right)evolutionary models. Model tracks are shown for the single coeval system age that best matchesthe total mass and individual luminosities. The unexpectedly shallow mass–luminosity relationimplied by our data are better described by the SM08 hybrid models that show a slowing of lumi-nosity evolution for objects in the L/T transition, while Lyon Dusty models are inconsistent withcoevality at 2.0 σ . (Note that we do not plot a confidence range for models as that would effectivelybe double-plotting our errors, since the age of the plotted isochrone is derived from our observedtotal mass and component luminosities.) 27 – −1 0 1 2J−K (mag)15.515.014.514.013.5 M J ( m ag ) SM08 hybrid +0.21 −0.17
Gyr M H ( m ag ) M K ( m ag ) Fig. 8.— Measured colors and absolute magnitudes of the components of SDSS J1052+4422ABcompared to predictions from SM08 hybrid evolutionary models. Model tracks are shown for thecoeval system age that best matches the total mass and individual luminosities (solid) and agesat ± σ of this value (dotted). Unlike previous generations of evolutionary models, the predictedcolors and magnitudes of SM08 hybrid match our observations remarkably well. Table 1. Relative astrometry and photometry for SDSS J1052+4422AB from Keck/NIRC2 AODate Airmass Filter FWHM Strehl ratio ρ PA ∆ m (UT) (mas) (mas) ( ◦ ) (mag)2005 May 1 1.342 K ′ ± . ± .
02 42 . ± . . ± . . ± . J ± . ± .
008 69 . ± . . ± . − . ± . H ± . ± .
02 70 . ± . . ± . . ± . K S ± . ± .
06 70 . ± . . ± . . ± . K ± . ± .
05 79 . ± . . ± . . ± . K ′ ± . ± .
04 80 . ± . . ± . . ± . J ± . ± .
016 80 . ± . . ± . − . ± . K ± . ± .
05 72 . ± . . ± . . ± . H ± . ± .
02 55 . ± . . ± . − . ± . K ′ · · · · · · . ± . . ± . . ± . H · · · · · · . ± . . ± . . ± . K · · · · · · . ± . . ± . . ± . K · · · · · · . ± . . ± . . ± . K · · · · · · . ± . . ± .
29 0 . ± . K · · · · · · . ± . . ± . . ± . NIRC2STREHL . Masking observations have no FWHM and Strehl listed. 29 –Table 2. Integrated-light astrometry for SDSS J1052+4422AB from CFHT/WIRCamDate R.A. Dec. σ R . A . cos δ σ Dec . Airmass Seeing(UT) (deg) (deg) (mas) (mas)2008 Feb 17 163.05659646 +44.38217395 1.8 2.3 1.100 0 . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′ . ′′
55 30 –Table 3. Derived orbital and parallax parameters for SDSS J1052+4422AB
Parameter Best fit Median 68.3% c.i. 95.4% c.i.Visual binary orbital parametersOrbital period P (yr) 8.614 8.608 8.583, 8.632 8.560, 8.658Semimajor axis a (mas) 70.59 70.67 70.43, 70.91 70.20, 71.16Eccentricity e i ( ◦ ) 62.0 62.1 61.7, 62.4 61.4, 62.7PA of the ascending node Ω ( ◦ ) 126.7 126.8 126.5, 127.2 126.2, 127.5Argument of periastron ω ( ◦ ) 186.5 187.3 185.6, 188.9 184.0, 190.5Mean longitude at 2455197.5 JD λ ref ( ◦ ) 113.4 113.4 112.9, 113.8 112.5, 114.2Additional integrated-light astrometric parametersR . A . − . − − − . − +44 . − − µ R . A ., rel (mas yr − ) 24.51 24.56 24.36, 24.77 24.16, 24.97Relative proper motion in Dec. µ Dec ., rel (mas yr − ) − − − − − − π rel (mas) 36.87 36.67 36.06, 37.29 35.42, 37.90Photocenter semimajor axis α (mas) − − − − − − χ in theMCMC chain, χ = 50 .
7, 59 degrees of freedom) along with the median of the posterior distribution and theshortest intervals containing 68.3% and 95.4% of the chain steps (i.e., 1 σ and 2 σ credible intervals). The time ofperiastron passage corresponding to these λ ref and ω posteriors is T = 55842 ±
13 MJD (2011 Oct 7 UT). Forclarity, the R.A. and Dec. zero points are reported relative to their best-fit values. R.A. and Dec. zero pointsare reported at equinox J2000.0 and epoch 2010.0. Without resolved radial velocities there is a 180 ◦ ambiguityin Ω, ω , and λ ref .
31 –Table 4. Measured Properties of SDSS J1052+4422ABProperty SDSS J1052+4422A SDSS J1052+4422B Ref. d (pc) 26 . ± . . +0 . − . M tot ( M Jup ) 88 ± q ≡ M B /M A . ± .
07 1Mass ( M Jup ) 49 ± ± . ± . . ± . J (mag) 16 . ± .
06 16 . ± .
05 1,2 H (mag) 15 . ± .
05 15 . ± .
05 1,2 K (mag) 14 . ± .
04 15 . ± .
04 1,2 J − H (mag) 1 . ± .
08 0 . ± .
07 1,2 H − K (mag) 0 . ± .
06 0 . ± .
06 1,2 J − K (mag) 1 . ± .
07 0 . ± .
06 1,2 M J (mag) 14 . ± .
07 14 . ± .
06 1,2 M H (mag) 13 . ± .
06 13 . ± .
06 1,2 M K (mag) 12 . ± .
05 13 . ± .
06 1,2BC K (mag) 3 . ± .
10 2 . ± .
13 1,3log( L bol / L ⊙ ) − . ± . − . ± .
06 1∆ log( L bol ) 0 . ± .
07 1Parallax (mas) 38 . ± . µ R . A . (mas yr − ) +19 ± µ Dec . (mas yr − ) − ± π = 1 . ± . µ R . A . = − ± − , ∆ µ Dec . = − ± − .References. — (1) This work; (2) Chiu et al. (2006); (3) Liu et al. (2010). Table 5. Evolutionary model-derived properties for SDSS J1052+4422AB
Saumon & Marley (2008) hybrid SM08 cloudy ( f sed = 2) Lyon Dusty (Chabrier et al. 2000)Property Median 68.3% c.i. 95.4% c.i. Median 68.3% c.i. 95.4% c.i. Median 68.3% c.i. 95.4% c.i.Using individual masses and luminosities t A (Gyr) 1.22 0.99, 1.43 0.82, 1.69 0.95 0.79, 1.07 0.69, 1.30 1.01 0.84, 1.16 0.72, 1.36 t B (Gyr) 0.99 0.79, 1.17 0.63, 1.39 0.66 0.55, 0.77 0.45, 0.90 0.66 0.54, 0.76 0.45, 0.90log( t A / yr) 9.09 9.01, 9.17 8.93, 9.24 8.98 8.91, 9.04 8.86, 9.12 9.01 8.94, 9.08 8.87, 9.14log( t B / yr) 9.00 8.92, 9.09 8.82, 9.16 8.82 8.75, 8.89 8.67, 8.96 8.82 8.74, 8.89 8.67, 8.97∆ log t (dex) 0.09 − − − − T eff , A (K) 1340 1310, 1370 1280, 1400 1320 1280, 1360 1250, 1400 1360 1320, 1390 1280, 1430 T eff , B (K) 1270 1230, 1300 1200, 1330 1240 1190, 1270 1160, 1320 1260 1220, 1300 1180, 1340∆ T eff (K) 70 30, 110 −
10, 150 90 40, 140 −
10, 180 100 50, 150 0, 190log( g A ) (cgs) 5.14 5.10, 5.18 5.05, 5.22 5.12 5.07, 5.16 5.03, 5.21 5.15 5.10, 5.20 5.05, 5.24log( g B ) (cgs) 5.00 4.96, 5.05 4.92, 5.09 4.96 4.91, 5.01 4.86, 5.05 4.99 4.94, 5.03 4.89, 5.08 R A ( R Jup ) 0.947 0.929, 0.965 0.912, 0.983 0.970 0.953, 0.987 0.934, 1.004 0.939 0.923, 0.957 0.901, 0.973 R B ( R Jup ) 0.972 0.950, 0.991 0.934, 1.017 1.023 1.003, 1.041 0.985, 1.063 0.991 0.972, 1.010 0.955, 1.030(Li/Li ) A · · · · · · · · · · · · · · · · · · ) B · · · · · · · · · · · · · · · · · · Y − J ) A (mag) 1.205 1.200, 1.212 1.187, 1.214 1.201 1.194, 1.212 1.176, 1.215 · · · · · · · · · ( Y − J ) B (mag) 1.182 1.165, 1.206 1.134, 1.215 1.16 1.13, 1.18 1.12, 1.21 · · · · · · · · · ( J − H ) A (mag) 1.04 0.94, 1.14 0.86, 1.21 0.98 0.90, 1.13 0.71, 1.16 2.51 2.41, 2.61 2.31, 2.71( J − H ) B (mag) 0.73 0.57, 0.91 0.37, 1.02 0.55 0.32, 0.69 0.26, 0.92 2.78 2.68, 2.90 2.56, 3.00( H − K ) A (mag) 0.71 0.56, 0.87 0.44, 0.99 0.62 0.48, 0.87 0.26, 0.94 2.05 1.96, 2.15 1.85, 2.25( H − K ) B (mag) 0.29 0.09, 0.50 − − − J − K ) A (mag) 1.8 1.5, 2.0 1.3, 2.2 1.6 1.4, 2.0 1.0, 2.1 4.56 4.36, 4.75 4.16, 4.96( J − K ) B (mag) 1.0 0.7, 1.4 0.2, 1.7 0.6 0.1, 0.9 0.1, 1.5 5.11 4.93, 5.32 4.69, 5.48( K − L ′ ) A (mag) 1.25 1.21, 1.29 1.18, 1.34 1.28 1.22, 1.32 1.18, 1.39 1.92 1.82, 2.03 1.72, 2.13( K − L ′ ) B (mag) 1.36 1.29, 1.42 1.25, 1.50 1.42 1.35, 1.50 1.28, 1.57 2.23 2.12, 2.35 1.98, 2.44 Table 5—Continued
Saumon & Marley (2008) hybrid SM08 cloudy ( f sed = 2) Lyon Dusty (Chabrier et al. 2000)Property Median 68.3% c.i. 95.4% c.i. Median 68.3% c.i. 95.4% c.i. Median 68.3% c.i. 95.4% c.i.Using total mass, individual luminosities, and assuming coevalityAge ( t , Gyr) 1.11 0.91, 1.28 0.76, 1.49 0.81 0.69, 0.91 0.60, 1.04 0.84 0.69, 0.94 0.61, 1.09log( t/ yr) 9.04 8.97, 9.12 8.89, 9.18 8.91 8.85, 8.97 8.79, 9.02 8.92 8.86, 8.99 8.79, 9.05 M A ( M Jup ) 47 43, 51 40, 55 45.8 42.8, 48.6 40.3, 51.7 45.5 42.6, 48.2 40.0, 51.5 M B ( M Jup ) 41 38, 44 35, 48 42.6 39.7, 45.2 37.5, 48.4 43.1 40.4, 45.9 37.6, 48.7 q ≡ M B /M A T eff , A (K) 1330 1300, 1360 1270, 1400 1310 1270, 1340 1240, 1380 1340 1300, 1370 1270, 1410 T eff , B (K) 1270 1240, 1310 1200, 1350 1250 1210, 1290 1170, 1340 1280 1230, 1320 1200, 1370∆ T eff (K) 60 0, 100 −
50, 160 60 20, 120 −
50, 160 60 10, 120 −
50, 170log( g A ) (cgs) 5.10 5.06, 5.15 5.01, 5.19 5.06 5.02, 5.10 4.98, 5.14 5.09 5.04, 5.13 5.00, 5.18log( g B ) (cgs) 5.04 5.00, 5.09 4.95, 5.13 5.03 4.98, 5.06 4.95, 5.11 5.06 5.02, 5.10 4.97, 5.14 R A ( R Jup ) 0.958 0.940, 0.974 0.924, 0.993 0.991 0.975, 1.007 0.959, 1.022 0.958 0.939, 0.969 0.930, 0.993 R B ( R Jup ) 0.960 0.942, 0.977 0.927, 0.997 0.998 0.983, 1.013 0.967, 1.027 0.963 0.944, 0.975 0.934, 0.997(Li/Li ) A · · · · · · · · · · · · · · · · · · ) B · · · · · · · · · · · · · · · · · · Y − J ) A (mag) 1.205 1.200, 1.213 1.185, 1.215 1.200 1.191, 1.214 1.171, 1.216 · · · · · · · · · ( Y − J ) B (mag) 1.184 1.167, 1.206 1.138, 1.214 1.17 1.14, 1.19 1.12, 1.21 · · · · · · · · · ( J − H ) A (mag) 1.01 0.90, 1.13 0.80, 1.20 0.92 0.81, 1.08 0.65, 1.14 2.57 2.47, 2.67 2.37, 2.76( J − H ) B (mag) 0.77 0.60, 0.95 0.38, 1.07 0.64 0.40, 0.81 0.29, 1.01 2.73 2.62, 2.85 2.48, 2.96( H − K ) A (mag) 0.67 0.48, 0.83 0.38, 0.98 0.54 0.36, 0.73 0.18, 0.89 2.10 2.01, 2.20 1.91, 2.29( H − K ) B (mag) 0.3 0.1, 0.6 − − − J − K ) A (mag) 1.7 1.4, 2.0 1.2, 2.2 1.5 1.2, 1.8 0.8, 2.0 4.67 4.48, 4.87 4.28, 5.03( J − K ) B (mag) 1.1 0.7, 1.5 0.3, 1.8 0.8 0.3, 1.2 0.1, 1.7 5.0 4.8, 5.2 4.5, 5.4( K − L ′ ) A (mag) 1.26 1.22, 1.30 1.18, 1.35 1.29 1.24, 1.34 1.20, 1.40 1.99 1.89, 2.09 1.78, 2.18( K − L ′ ) B (mag) 1.35 1.28, 1.41 1.23, 1.50 1.40 1.33, 1.49 1.25, 1.55 2.16 2.04, 2.27 1.91, 2.37(mag) 1.35 1.28, 1.41 1.23, 1.50 1.40 1.33, 1.49 1.25, 1.55 2.16 2.04, 2.27 1.91, 2.37