LSPM J1112+7626: detection of a 41-day M-dwarf eclipsing binary from the MEarth transit survey
Jonathan M. Irwin, Samuel N. Quinn, Zachory K. Berta, David W. Latham, Guillermo Torres, Christopher J. Burke, David Charbonneau, Jason Dittmann, Gilbert A. Esquerdo, Robert P. Stefanik, Arto Oksanen, Lars A. Buchhave, Philip Nutzman, Perry Berlind, Michael L. Calkins, Emilio E. Falco
aa r X i v : . [ a s t r o - ph . S R ] S e p Draft version November 15, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
LSPM J1112+7626: DETECTION OF A 41-DAY M-DWARF ECLIPSING BINARY FROM THE MEARTHTRANSIT SURVEY
Jonathan M. Irwin, Samuel N. Quinn, Zachory K. Berta, David W. Latham, Guillermo Torres,Christopher J. Burke, David Charbonneau, Jason Dittmann, Gilbert A. Esquerdo, and Robert P. Stefanik
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA
Arto Oksanen
Hankasalmi Observatory, Jyv¨askyl¨an Sirius ry, Vertaalantie 419, FI-40270 Palokka, Finland
Lars A. Buchhave
Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark andCentre for Star and Planet Formation, Natural History Museum of Denmark, University of Copenhagen, DK-1350 Copenhagen,Denmark
Philip Nutzman
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
Perry Berlind, Michael L. Calkins and Emilio E. Falco
Fred Lawrence Whipple Observatory, Smithsonian Astrophysical Observatory, 670 Mount Hopkins Road, Amado, AZ 85645, USA
Draft version November 15, 2018
ABSTRACTWe report the detection of eclipses in LSPM J1112+7626, which we find to be a moderately bright( I C = 12 . ± .
05) very low-mass binary system with an orbital period of 41 . ± . M = 0 . ± .
002 M ⊙ and M = 0 . ± .
001 M ⊙ in an eccentric ( e =0 . ± . I -band, which is probably due to rotational modulation of photospheric spots on one of thebinary components. This paper presents the discovery and characterization of the object, includingradial velocities sufficient to determine both component masses to better than 1% precision, and aphotometric solution. We find that the sum of the component radii, which is much better-determinedthan the individual radii, is inflated by 3 . +0 . − . % compared to the theoretical model predictions,depending on the age and metallicity assumed. These results demonstrate that the difficulties inreproducing observed M-dwarf eclipsing binary radii with theoretical models are not confined tosystems with very short orbital periods. This object promises to be a fruitful testing ground for thehypothesized link between inflated radii in M-dwarfs and activity. Subject headings: binaries: eclipsing – stars: low-mass, brown dwarfs INTRODUCTION
Detached, double-lined eclipsing binaries (EBs) pro-vide a largely model-independent means to precisely andaccurately measure fundamental stellar properties, par-ticularly masses and radii. In the best-observed systemsthe precision of these can be at the < .
35 M ⊙ ,the boundary at which field stars are thought to becomefully convective (e.g. Chabrier & Baraffe 1997). Fur-thermore, observations of many of the best-characterizedM-dwarf EBs indicate significant discrepancies with thestellar models.The components of CM Dra, the best characterizeddouble-lined system in the fully-convective mass range jirwin -at- cfa -dot- harvard -dot- edu (Eggen & Sandage 1967; Lacy 1977; Metcalfe et al. 1996;Morales et al. 2009), have radii approximately 5 − −
10% level, and effective tempera-tures 3 −
5% lower than the models predict, is nearlyubiquitous among the best-characterized objects (e.g.Torres et al. 2010, and references therein; Morales et al.2010; Kraus et al. 2011), although it is interesting to notethat Carter et al. (2011) find a smaller inflation for KOI-126B and C, members of a fascinating system containinga short-period M-dwarf binary orbiting a K-star, wherethe system undergoes mutual eclipses allowing a precisedetermination of the radii of both M-dwarfs.Two main hypotheses have been advanced to ex-plain the inflated radii in low-mass eclipsing bina-ries: metallicity (Berger et al. 2006), and the ef-fect of magnetic activity (the “activity hypothe-sis”, e.g. L´opez-Morales & Ribas 2005; Ribas 2006;Chabrier et al. 2007). Irwin et al.As discussed by L´opez-Morales (2007), one of the maindifficulties with metallicity as an explanation is that cur-rent models of M-dwarfs yield radii differing only by avery small amount (an increase of approximately 3%)between [M / H] = − . Observationally, the difficulty of determining metal-licities for M-dwarfs (e.g. Bonfils et al. 2005;Woolf & Wallerstein 2006; Bean et al. 2006) renders themetallicity hypothesis difficult to test in practice, andthis is further exacerbated by the double lines andrapid rotation in most M-dwarf eclipsing binary spec-tra. Nonetheless, it is interesting to note that CM Dra isthought to be metal poor (Viti et al. 1997, 2002), whichwould increase the size of the radius discrepancy for thisobject, rather than decreasing it, and more generally,this is expected to also be true of other eclipsing bina-ries in the sample, assuming they follow the metallicitydistribution of field stars (e.g. Carney et al. 1989).Given these lines of reasoning, it seems unlikely thatmetallicity is the only explanation for the inflated radii(although it probably plays a role, particularly in cre-ating scatter in the effective temperatures). This hasled to the “activity hypothesis” becoming the preferredexplanation in the literature for the inflation. It is mo-tivated by noting that nearly all EB systems have shortorbital periods. For example, below 0 .
35 M ⊙ , the longestperiod system is 1RXS J154727.5+450803 at 3 .
55 days(Hartman et al. 2011). At such short periods, the com-ponents are expected to have been tidally synchronizedand the orbits circularized by field ages. The effect ofthe companion and tidal locking is likely to significantlyincrease the activity levels. The available observationalevidence supports this, with the eclipsing binaries be-low 0 .
35 M ⊙ all showing signs of high activity and tidaleffects including synchronous, rapid rotation, large am-plitude out of eclipse modulations, H α emission, X-rayemission, and in most cases, circular orbits.Magnetic activity could have an effect eitherby inhibiting convection (Mullan & MacDonald 2001;Chabrier et al. 2007), or due to reduced heat fluxresulting from cool photospheric spots. Of these,Chabrier et al. (2007) find the structure of objects be-low 0 .
35 M ⊙ is almost independent of variations in themixing length parameter, which they use to implementinhibition of convection, so the latter possibility (inhibi-tion of radiation due to the effect of spots) seems morelikely the explanation in the mass domain of particularinterest here. It is important to note that spots also in-fluence the solution of eclipsing binary light curves. Thepossibility of such systematic errors in the observationsis discussed by Morales et al. (2010), and we shall returnto it later in the analysis.An obvious way to test the activity hypothesis is tomeasure radii for components of eclipsing binaries withlong orbital periods, where tides are unimportant, and As discussed by L´opez-Morales (2007), it is possible the effecton the radius is suppressed by a missing source of opacity in themodels. Such effects are known to exist as the same models failto reproduce the observed optical spectra and colors of M-dwarfs(Baraffe et al. 1998). We thank the referee for bringing this point to our attention. the stars should rotate close to the rates expected forsingle stars in the absence of a binary companion. Unfor-tunately, such an endeavor is difficult due to the reducedgeometric alignment probability and limited availabilityof long-duration photometric monitoring of M-dwarfs atrapid cadences, where the majority of wide-field variabil-ity surveys are carried out in visible bandpasses, oftenyielding poor signal to noise ratios for M-dwarfs, whichare faint at these wavelengths.The longest-period M-dwarf EB with a published orbitis T-Lyr1-17236 (Devor et al. 2008), which has an 8 . .
68 and 0 .
52 M ⊙ . Theradii are indeed consistent with the theoretical models,but the observational uncertainties are still quite largein the discovery paper. This object is currently beingobserved by the NASA Kepler mission, so a more precisesolution should be forthcoming.In light of the difficulty of identifying longer-periodEBs, several authors have attempted to test the activityhypothesis using the existing EB sample to search for cor-relations between e.g. orbital period or activity measuresand radius inflation (L´opez-Morales 2007; Kraus et al.2011). While there does appear to be evidence for sucha correlation above 0 .
35 M ⊙ , below this mass the presentsample is limited by small number statistics and thenarrow range of orbital periods spanned. The Keplermission appears to have identified a large number ofvery long-period EB systems (Prˇsa et al. 2011; see alsoCoughlin et al. 2011 where more detailed solutions us-ing the Kepler photometry are presented for low-massobjects from this sample), but we are not aware of pub-lished dynamical masses and model-independent radii forthese at the present time, and many of the M-dwarfs (es-pecially below 0 .
35 M ⊙ ) are secondaries in systems withmass ratios much less than unity, where detecting thesecondary spectrum to measure radial velocities may bechallenging.An alternative method to determine M-dwarf radii isby making interferometric measurements of angular di-ameters for (usually isolated) stars. Combined withtrigonometric parallaxes and minor assumptions aboutthe stellar limb darkening, the physical radius of thestar can be inferred in a model-independent fashion.The difficulty with this method comes in estimatingthe stellar mass. Dynamical measurements are usuallynot available, so one has to resort to inferring massfrom other measurable stellar properties, necessarily withlarger uncertainties. Typically, absolute magnitude innear-infrared passbands is used, which is found to corre-late well with stellar mass and has a relatively small scat-ter. The empirical polynomials of Delfosse et al. (2000)are the most widely used, although there is some argu-ment in the literature as to how accurately these predictthe mass, especially over a range of metallicity and spec-tral type (the uncertainties are probably ≥ OBSERVATIONS AND DATA REDUCTION
Initial detection
LSPM J1112+7626 was targeted as part ofroutine operation of the MEarth transit survey(Nutzman & Charbonneau 2008; Irwin et al. 2009a)during the 2009-2010 season, with the first observationstaken on UT 2009 December 2. Exposure times forthe LSPM J1112+7626 field were 27 s, taking threeexposures per visit (the total exposure times are tailoredfor each target to achieve sensitivity to a particularplanet size for the assumed stellar radius), with visits ata cadence of 20 minutes.Eclipses were first detected in LSPM J1112+7626 onUT 2010 April 3 (this was a primary eclipse), and subse-quently confirmed on UT 2010 April 28th by observingthe majority of a secondary eclipse egress. Both eclipseswere detected by an experimental automated real-timeeclipse detection and followup system we began to trialduring the 2009-2010 season (this will be described in de-tail in a future publication when it has matured). Con-fident that the object was an eclipsing binary, but stillwith an ambiguous orbital period, we immediately com-menced radial velocity monitoring. The radial velocitiesultimately determined which of the possible orbital pe-riods was correct, in conjunction with a further eclipse(which was also a secondary) observed on UT 2010 June8 with MEarth and the Clay telescope (see § Secondary eclipses
By the time the orbital ephemeris was well-known, theobserving season on LSPM J1112+7626 from Arizonawas ending, and the weather had begun to deterioratedue to the arrival of the monsoon. We therefore tookadvantage of the high declination of our target and com-menced an observing program to target the secondaryeclipses using the Harvard University 0 .
4m Landon T.Clay telescope, which is located on the roof of the ScienceCenter on the Harvard campus near Harvard Square,Cambridge, Massachusetts. This instrument is normallyused for undergraduate teaching. It is equipped with ane2v CCD47-10 1024 × × ≈ . ′′ I -band filter to mitigate the effects ofsky background and atmospheric extinction. The filteravailable was built to the prescription of Bessell (1990)using Schott RG9 glass to define the bandpass, and shall hereafter be referred to as I Bessell . While this filter wasintended to approximate the Cousins I passband (here-after I C ), the combination of the glass filter and CCDquantum efficiency yields a system response with signif-icant sensitivity at very red wavelengths not present inthe original Cousins passband, and thus is a poor ap-proximation to the Cousins filter for very red stars suchas M-dwarfs. We shall return to this issue in § −
60 s. The typical cadence was 50 s and the FWHMof the stellar images was 2 − −
20C operating tem-perature of the CCD, and the data were flat-fielded us-ing twilight flat-fields in the usual way. No defringing orabsolute calibration of the photometry were attempted.Fringing was clearly visible in the images, but at a lowamplitude (3% peak-to-peak) and should not have a sig-nificant effect on the photometry due to the stabilizedtelescope pointing.We present the full light curve data-set used in thispaper in Table 1, and times of minimum light for theeclipses where a clear minimum was seen in Table 2.
Primary eclipse
Using the timing of the single visit within the pri-mary eclipse of UT 2010 April 3 from MEarth (see § § §
3) it was possible to predict theprimary eclipse times with reasonable accuracy. Dur-ing 2010-2011 full eclipses were observable at reasonableairmass ( < .
0) from eastern Europe, Scandinavia, andwestern Asia, with the high declination of the targetfavoring northerly latitudes during winter to maximizedark hours at reasonable airmass.The primary eclipse was observed using the 40 cm tele-scope at Hankasalmi Observatory, Finland on UT 2011January 15, starting approximately 10 minutes beforefirst contact, and continuing until well after last con-tact. Data were taken using a similar I Bessell filter to thesecondary eclipse observations (see § § TABLE 1Light curve data. ID a Orbit b HJD UTC
I σ ( I ) ∆ mag c FWHM d Airmass MeridFlag e x y CM f (days) (mag) (mag) (mag) (pix) (pix) (pix) (mag)C 1 2455355 . − . . .
000 2 .
10 1 . .
427 378 .
595 0 . . − . . − .
002 2 .
19 1 . .
715 378 .
567 0 . . − . . − .
000 2 .
18 1 . .
936 378 .
442 0 . . − . . − .
033 2 .
26 1 . .
904 378 .
255 0 . . − . . − .
010 2 .
20 1 . .
900 378 .
023 0 . Note . — Table 1 is published in its entirety in the electronic edition of the Astrophysical Journal. A portion isshown here for guidance regarding its form and content. a Data-set identifier. C: Clay telescope secondary eclipse (magnitudes for each eclipse / night were separately nor-malized to zero median) H: Hankasalmi primary eclipse (normalized to zero median as above) O: MEarth out ofeclipse (magnitudes in the instrumental system) P: MEarth primary eclipse (715nm long-pass, normalized as above)S: MEarth secondary eclipse (normalized as above) b Orbit number (from zero at T ). Given only for the data-sets containing eclipses. These are the numbers used inthe figures showing the light curves. c Photometric zero-point correction applied by the differential photometry procedure. Note: this has already beenincluded in the I column and is for reference purposes only (e.g. detecting frames with large light losses due to clouds). d FWHM of the stellar images on the frame. Zero in cases where the FWHM could not be reliably estimated. e Flag for detecting “meridian flip” on German equatorial mountings. Used only for the MEarth data, where the flagis 0 when the telescope was observing in the usual orientation for the east side and 1 for the west side of the meridian.Note that there is not a one-to-one relation with the sign of the hour angle because the MEarth mounts can trackapproximately 5 degrees across the meridian before needing to change sides of the pier, so this flag can sometimes be0 even for (small) positive hour angle. f For the MEarth data only. Gives the “common mode” interpolated to the Julian date of the exposure. Derived fromthe average differential magnitude of all the M-dwarfs observed by all 8 MEarth telescopes in a given time interval.This should be scaled and subtracted from the I column to correct for the suspected variations in the MEarth bandpasswith humidity and temperature (see text). TABLE 2Times of minimum light.
HJD UTC Uncertainty Cycle Eclipse Instrument(days) (s)2455577 . . . . . . . Note . — Estimated using the method ofKwee & van Woerden (1956), over an interval of ± − in normalized orbital phase. These times of minimum arenot used in our analysis (we fit for the ephemeris directlyfrom the data) and are reported only for completeness.Exposure times of 60 s were used, yielding a cadence ofapproximately 2 minutes, and the target was re-centeredevery 10 exposures to keep the drift below ≈
50 pixels.Data were reduced using 2-D bias frames, and masterdark and flat-field frames. Light curves were then derivedusing the same methods and software as for the otherdata-sets.
Out of eclipse and additional secondary eclipses
Following the discovery of atmospheric water vaporinduced systematic effects in the MEarth photometry(see Irwin et al. 2011), a customized I -band filter wasprocured for the start of the 2010-2011 season, with abandpass designed to eliminate the strong telluric wa- ter vapor absorption at wavelengths >
900 nm, by usingan 895 nm (50% transmittance point) interference cutoffon the same RG715 glass substrate as the original filter(where the long-pass glass filter defines the blue end ofthe bandpass). This was predicted to reduce the effectto largely insignificant levels, and the resulting systemresponse approximates the Cousins I bandpass (as tab-ulated by Bessell 1990; see later in this section).Observations of LSPM J1112+7626 using this new fil-ter commenced on UT 2010 October 29, and continueduntil UT 2011 May 18. Exposures of 2 ×
97 s were used at20 minute cadence for times out of eclipse. 5283 usefuldata points were obtained at airmasses <
2, after dis-carding 400 bad frames (pointing errors, severe light lossdue to clouds, and large scatter in the zeropoint solutionfor the differential photometry).Partial secondary eclipses were obtained with the sameMEarth telescope on UT 2010 November 19, UT 2011February 9, and UT 2011 May 2. Due to known system-atic effects caused by persistence in the MEarth CCDs,leading to offsets in magnitude depending on the cadence(see Berta et al. 2011), and to remove any residual fromthe out-of-eclipse variations, we allow these high-cadenceobservations to have different photometric zero-pointsthan the out of eclipse data (and each other) in the finallight curve analysis.The out-of-eclipse data (excluding the high-cadenceeclipse observations) were searched for periodic varia-tions using the method outlined in Irwin et al. (2011).Correction for the “common mode” effect was still per-formed, as we discovered during the season that replacingthe filters had in fact increased the size (and changed theSPM J1112+7626 5
Fig. 1.— “F-test periodogram” for the MEarth 2010-2011 sea-son out of eclipse observations, plotting the square root of the F-test statistic comparing the null hypothesis of no variation to thealternate hypothesis of a sinusoidal modulation, as a function offrequency (top) and the corresponding window function (bottom).The vertical line indicates the best-fitting period of 64 . sign) of the systematic effect rather than eliminating it.This problem is still under investigation, but we suspectit is due to humidity and/or temperature dependence ofthe filter bandpass, an effect also seen in the originalSloan Digital Sky Survey (SDSS) Photometric Telescopefilters (Doi et al. 2010).After correcting for the “common mode” effect, theMEarth data show a strong near-sinusoidal modulationwith a 65 day period (see Figure 1; the data are shownin §
4) and a peak-to-peak amplitude of approximately0 .
02 mag. Marginal evidence for a similar periodicity isfound in the 2009-2010 data, although the phase is notconsistent and the amplitude appears to be much smaller.It is unclear if the amplitude change results from thediffering filter systems, but the change in phase (if real)may be indicative of evolution of the photospheric spotpatterns giving rise to the modulations.An improvement resulting from the new MEarth fil-ter is a greatly reduced color term when standardizingthe photometry onto the Cousins system. The scatter inthis relation also appears to be significantly lower, whichmay also result in part from improvements made to ourflat-fielding strategy during the 2010-2011 season. TheMEarth observing software automatically obtains mea-surements of equatorial photometric standard star fieldsfrom the catalog of Landolt (1992) every night, and theseare used to derive photometric zero-points and calibratedphotometry. The typical scatter in these zero-point so-lutions on a photometric night is 0 .
02 mag. We find atypical atmospheric extinction of 0 .
05 mag / airmass fromMEarth data, which has been assumed henceforth.A significant issue with the equatorial standard fieldsis they contain very few red stars suitable for definingthe red end of the photometric response, and so colorequations derived only from these equatorial fields may TABLE 3Summary of the photometric and astrometricproperties of the LSPM J1112+7626 system.
Parameter Value α h m s . δ +76 ◦ ′ . ′′ µ α cos δ b . ′′
131 yr − µ δ b − . ′′
108 yr − g c . ± . r c . ± . z c . ± . I Cd . ± . J . ± . H . ± . K . ± . a Equinox J2000.0, epoch 2000.0. b From L´epine & Shara (2005). c From SDSS Data Release 7. Note that theseuncertainties do not account for variability, andare therefore underestimates. d From § e We quote the combined uncertainties from the2MASS catalog, noting that the intrinsic variabil-ity of our target means in practice that these areunderestimates.be significantly in error when attempting to standard-ize M-dwarf photometry. Thankfully, MEarth serendip-itously targeted a number of M-dwarfs with photome-try available on the Cousins system from Bessel (1990).We selected all such objects in the 2010-2011 MEarthdatabase with data available on photometric nights forcalibration. Two objects with clearly unreliable MEarthdata were discarded, leaving 8 objects. The followingcolor equation was derived from these stars, which span I C − K colors of 2 . − . I C = I MEarth − (0 . ± . . ± . I MEarth − K )In the case of LSPM J1112+7626, we estimate I C =12 . ± .
05 using this relation, where we have inflatedthe uncertainty to account for variability and any errorsintroduced by the varying MEarth bandpass as discussedabove. The observed I C − J , I C − H and I C − K colors arelater used to infer effective temperatures and luminositiesfor the members of the binary, and thus the distance tothe system. Literature data: proper motion and photometry
In Table 3, we summarize the photometric and astro-metric system properties gathered from literature dataand our photometry. This star lies within the area cov-ered by the Sloan Extension for Galactic Understand-ing and Evolution (SEGUE) stripe 1260 photometry, sowe retrieved the PSF magnitudes from Data Release 7(Abazajian et al. 2009). This object is flagged as sat-urated in i -band (the image also shows a very unusualPSF and may be corrupted), and the u -band photometryof M-dwarfs in SDSS is corrupted by a well-documentedred leak problem, so we omit these filters.The moderately high proper motion of our target al- Irwin et al.lows the contribution of any background objects to thesystem light to be constrained using previous epochs ofimaging. In Figure 2, we show imaging data with theposition and size of the MEarth photometric aperture(which is representative of the aperture sizes used for allthe photometry) overlaid. The DSS1 image indicates theaperture is unlikely to contain any significant flux frombackground objects, and the SDSS image (which has thehighest angular resolution) does not appear elongated. Intermediate-resolution optical spectroscopy:spectral typing and metallicity constraints
A single epoch intermediate-resolution ( R ≃ . / mm grating and 3 ′′ slit was usedto obtain two exposures of 300 s. The spectra were re-duced using standard procedures for long slit spectra inIRAF (Tody 1993), using a HeNeAr arc exposure takeninbetween the two target exposures to ensure accuratewavelength calibration. The 2-D spectra were dividedby a flat field before extraction to remove the majorityof the fringing seen at the reddest wavelengths. Relativeflux calibration was performed using the spectrophoto-metric standard BD+26 2606. We show the resultingspectra in Figure 3.Using the TiO5 band index defined by Reid et al.(1995), we find a composite spectral type of M3.7 (notethat the uncertainty in this value is at least 0 . The Hammer (Covey et al. 2007), we findM3-M4 (visual inspection indicates the observed spec-trum is closer to M4). The spectral type and the mea-sured colors are in good agreement using the empiricalcolors of Galactic disk M-dwarfs from Leggett (1992) forthe Johnson/Cousins/CIT system and West et al. (2005)for the SDSS/2MASS.It is also clear from Figure 3 that no H α line is seen inthe spectrum. Such behavior is typical of inactive fielddwarfs at these spectral types when observed at low reso-lution, where the equivalent width of the H α absorptionline becomes close to zero at M4-M5 (e.g. Gizis et al.2002; see their Figure 5, noting that the TiO5 indexwas 0 .
42 for LSPM J1112+7626). The presence of H α emission in M-dwarf spectra is usually indicative of highactivity levels.The relative strengths of the TiO and CaH bands inoptical spectra have been used to identify and classifyM-subdwarfs (Gizis 1997), and more generally proposedas a method for estimating metallicities of M-dwarfs (e.g.Woolf & Wallerstein 2006). While detailed calibrations(e.g. using band indices) do not yet appear to be avail-able, we performed a visual comparison between Figure3 and the spectra shown in Gizis (1997). The strongTiO bands seen in our target indicate it is not extremelymetal poor. We therefore estimate it has metallicity[Fe / H] > −
1, and is probably closer to solar (or greater).
High-resolution optical spectroscopy: radialvelocities IRAF is distributed by the National Optical Astronomy Ob-servatories, which are operated by the Association of Universitiesfor Research in Astronomy, Inc., under cooperative agreement withthe National Science Foundation. S D SS r . M E a r t h . Fig. 2.—
Images of LSPM J1112+7626 centered on the positionas measured from the MEarth data. The circle shows the approx-imate position and size of the 8 ′′ (radius) photometric apertureused to derive the MEarth light curves. Data are from the firstand second epoch Palomar sky surveys as provided by the Dig-itized Sky Survey (DSS; top and center panels), SDSS, and theMEarth master image (bottom panel). The approximate epochsare shown on the images, and all four panels have the same center,scale and alignment on-sky. High-resolution spectroscopic observations were ob-tained using the TRES fiber-fed ´echelle spectrograph onthe FLWO 1 . . ′′ R ≃
44 000.Observations commenced on UT 2010 May 1, and con-SPM J1112+7626 7
Fig. 3.—
Intermediate-resolution FAST spectra ofLSPM J1112+7626 showing the TiO and CaH molecularbands typically used for M-dwarf spectral classification. Twoexposures were obtained and are shown offset in the verticaldirection for clarity. Overlaid in gray are the locations of theH α line, and the CaH2, CaH3 and TiO5 band indices defined byReid et al. (1995). tinued until UT 2011 April 13, with 43 epochs acquired.Total exposure times ranged from 45 to 80 minutes, bro-ken into sequences of 3 or 4 individual exposures. Thespectra were extracted using a custom built pipeline de-signed to provide precise radial velocities for ´echelle spec-trographs. The procedures are described in more detailin Buchhave et al. (2010), and will be summarized brieflybelow.In order to effectively remove cosmic rays, the sets ofraw images were median combined. We used a flat fieldto trace the ´echelle orders and to correct the pixel to pixelvariations in CCD response and fringing in the red-mostorders, then extracted one-dimensional spectra using the“optimal extraction” algorithm of Hewett et al. (1985)(see also Horne 1986). The scattered light in the two-dimensional raw image was determined and removed bymasking out the signal in the ´echelle orders and fittingthe inter-order light with a two-dimensional polynomial.Thorium-Argon (ThAr) calibration images were ob-tained through the science fiber before and after eachstellar observation. The two calibration images werecombined to form the basis for the fiducial wavelengthcalibration. TRES is not a vacuum spectrograph and isonly temperature controlled to 0 . ◦ C. Consequently, theradial velocity errors are dominated by shifts due to pres-sure, humidity and temperature variations. In order tosuccessfully remove these large variations ( > . − ),it is critical that the ThAr light travels along the sameoptical path as the stellar light and thus acts as an effec-tive proxy to remove these variations. We have thereforechosen to sandwich the stellar exposure with two ThArframes instead of using the simultaneous ThAr fiber,which may not exactly describe the induced shifts in thescience fiber and can also lead to bleeding of ThAr lightinto the science spectrum from the strong argon lines, es-pecially at redder wavelengths. The pairs of ThAr expo-sures were co-added to improve the signal to noise ratio,and centers of the ThAr lines found by fitting a Gaussianfunction to the line profiles. A two-dimensional fifth or-der Legendre polynomial was used to describe the wave- length solution. SPECTROSCOPIC ANALYSIS
Radial velocities were obtained using the two-dimensional cross-correlation algorithm todcor (Zucker & Mazeh 1994), which uses templates matchedto each component of a spectroscopic binary to simul-taneously derive the velocities of both stars, and anestimate of their light ratio in the spectral bandpass.We used a single epoch observation of Barnard’sstar (Barnard 1916; also known as Gl 699) taken onUT 2008 October 20 as the template for the todcor analysis. Barnard’s star has a spectral type of M4(Kirkpatrick et al. 1991), which should be a reasonablematch to both components. Correlations were performedusing a wavelength range of 7063 to 7201˚A in order 41of the spectrum to derive the velocities, since this regioncontains strong molecular features (mostly from TiO),which are rich in radial velocity information. We as-sume a barycentric radial velocity of − . ± . − for Barnard’s star (where the stated uncertainty reflectsour estimate of the systematic errors), derived frompresently unpublished CfA Digital Speedometer mea-surements spanning 17 years.To derive the light ratio, we ran todcor using a rangeof fixed values of this quantity, seeking the maximum cor-relation over all epochs. In this way, consistency betweenthe different epochs is enforced, in accordance with theexpectation that the true value of this quantity shouldbe (approximately) constant. This procedure improvesthe stability of the solution and minimizes systematic er-rors in the velocities. We find a best-fitting light ratioof L /L = 0 . ± .
05, where we adopt a conservativeestimate of the uncertainty.The radial velocities derived from our analysis are re-ported in Table 4. We omit four velocities with peakcorrelation values below C peak = 0 . § .
26 km s − for the primary, and 0 .
58 km s − for thesecondary, with an additional systematic uncertainty, es-timated to be 0 . − (see above), in the velocityzero-point system.We find no evidence for “peak pulling” effects closeto the systemic velocity, which might indicate the needfor additional rotational broadening of the template tomatch the target. Barnard’s star is a very slow rotator(e.g. Benedict et al. 1998), so this and the high corre-lation values obtained without need for additional rota-tional broadening indicate both components of our targethave low v sin i , as expected. MODEL
For EB systems showing moderate or high eccentrici-ties, the commonly exploited property of being able toseparate the photometric and spectroscopic solutions nolonger holds as the eccentricity affects both. Typically,the orbital period P , epoch of primary eclipse T , and e cos ω (where e is eccentricity and ω is the argument ofperiastron) are better determined by the photometry (byeclipse timing; the latter is related to the departure of thesecondary eclipses in phase from being exactly halfway Irwin et al. TABLE 4Radial velocity data.
HJD UTC a Phase b v v C peakd (days) (km s − ) (km s − )2455317 . . − . − .
49 0 . . . − . − .
08 0 . . . − . − .
77 0 . . . − . − .
62 0 . . . − . − .
71 0 . . . − . − .
09 0 . . . − . − .
76 0 . . . − . − .
66 0 . . . − . − .
34 0 . . . − . − .
12 0 . . . − . − .
19 0 . . . − . − .
31 0 . . . − . − .
02 0 . . . − . − .
98 0 . . . − . − .
52 0 . . . − . − .
84 0 . . . − . − .
82 0 . . . − . − .
53 0 . . . − . − .
72 0 . . . − . − .
42 0 . . . − . − .
40 0 . . . − . − .
08 0 . . . − . − .
75 0 . . . − . − .
98 0 . . . − . − .
16 0 . . . − . − .
07 0 . . . − . − .
00 0 . . . − . − .
22 0 . . . − . − .
11 0 . . . − . − .
05 0 . . . − . − .
66 0 . . . − . − .
85 0 . . . − . − .
76 0 . . . − . − .
63 0 . . . − . − .
88 0 . . . − . − .
32 0 . . . − . − .
15 0 . . . − . − .
78 0 . . . − . − .
03 0 . a Heliocentric Julian Date of mid-exposure, in theUTC time-system. b Normalized orbital phase. c Barycentric radial velocity of stars 1 and 2. d Peak normalized cross-correlation from tod-cor . These are later used for weighting the radialvelocity points in the solution.between the primary eclipses, although note that thecommonly-used one-to-one relation between these quan-tities is approximate), and the e sin ω component is usu-ally best constrained by the velocities (but also affectsthe relative durations of the two eclipses, and in gen-eral eccentricity also influences their shape). Therefore,in order to leverage the best possible constraints on thesystem orbit, we adopt a joint solution of all the lightcurves and the radial velocities simultaneously. This alsogreatly simplifies the error analysis.We proceed by first discussing the basic model and as-sumptions in § § § § § § § Basic model and assumptions
To model the system, we used the light curve genera-tor from the popular jktebop code (Southworth et al.2004a,b), which is in turn based on the eclipsing bi-nary orbit program ( ebop ; Popper & Etzel 1981). Thismodel approximates the stars using biaxial ellipsoids(Nelson & Davis 1972), which is appropriate for well-detached systems such as the present one. We modifiedthe model to perform the computations in double preci-sion, to account for light travel time, which is necessarydue to the large semimajor axis, and to incorporate asimple prescription for the effect of photospheric spots(discussed in § P as measured in the helio-centric frame (i.e. without the correction) in the tables.We fit all of the Clay, Hankasalmi, and 2010-2011 sea-son MEarth data, the radial velocities, and the spectro-scopic light ratio simultaneously. We also include thesingle primary eclipse of UT 2010 April 3 to improvethe constraint on the ephemeris, assuming the bandpasswas approximately the same as the I Bessell filter (thismakes little difference in practice due to the sparse cov-erage). The remainder of the MEarth 2009-2010 seasondata were omitted as these are of quite poor quality andwere taken in a different bandpass from the remainder,so do not usefully constrain the model after accountingfor the extra free parameters which must be added inorder to fit them.Table 5 summarizes the assumptions made in our lightcurve model. The light curve parameter set generally fol-lows jktebop , and has been chosen to minimize correla-tions between parameters where possible. We add a term k l ( X −
1) to the model magnitudes, where X is airmass,and l is an index for the different light curves, to ac-count for differential atmospheric extinction between thetarget and comparison stars, presumably due to color-dependence. A different k l value was allowed for eachlight curve l . This has only been used for fitting the Claysecondary eclipse curves and 2010-2011 MEarth data (see § TABLE 5Parameters and priors used in the basic model.
Parameter Value Prior a Description J MEarth varied uniform Central surface brightness ratio (secondary/primary) in I MEarth J Bessell varied uniform Central surface brightness ratio (secondary/primary) in I Bessell ( R + R ) /a varied uniform Total radius divided by semimajor axis R /R varied uniform Radius ratiocos i varied uniform (isotropic) Cosine of orbital inclination e cos ω varied uniform in e , ω Eccentricity times cosine of argument of periastron e sin ω varied uniform in e , ω Eccentricity times sine of argument of periastron L /L . ± .
05 Gaussian I MEarth -band light ratio u − . b u ′ . b u . b u ′ . b β .
32 . . . Primary gravity darkening exponent c β .
32 . . . Secondary gravity darkening exponent c κ computed . . . Primary reflection effect coefficient κ computed . . . Secondary reflection effect coefficient L b Φ 0 . . . Tidal lead/lag angle δ ◦ . . . Integration ring size d F varied e uniform Primary rotation parameter a varied e uniform Primary out-of-eclipse sine coefficient b varied e uniform Primary out-of-eclipse cosine coefficient F varied e uniform Secondary rotation parameter a varied e uniform Secondary out-of-eclipse sine coefficient b varied e uniform Secondary out-of-eclipse cosine coefficient q varied uniform Mass ratio K + K varied modified Jeffreys Sum of radial velocity semiamplitudes K a = 0 .
02 km s − Prior P ( K + K ) ∝ / ( K + K + K a ) γ varied uniform Systemic radial velocity P varied uniform Orbital period (heliocentric) T varied uniform Epoch of primary conjunction (HJD UTC) T sec computed . . . Epoch of secondary conjunction (HJD UTC) z l varied uniform Baseline magnitude for light curve l f s l varied Jeffreys Error scaling factor for light curve l f k l varied uniform Airmass coefficient for light curve l (where used) f C l varied uniform Common mode coefficient for light curve l (where used) f s r varied Jeffreys Error (if C peak = 1) for radial velocity curve r fa See § b Varied in later sections. c Values appropriate for convective atmospheres. d Our tests indicate this value results in a maximum error of 3 × − mag in the eclipse depth(compared to a model with 0 . ◦ ), which should be sufficient for our photometry. e Parameters varied only for one of the stars in each simulation. f l and r are indices for the different light curves and radial velocities for each component of thesystem.(Diaz-Cordoves & Gimenez 1992), of the form: I λ ( µ ) I λ (1) = 1 − u (1 − µ ) − u ′ (1 − √ µ ) (1)where I λ ( µ ) is the specific intensity, and µ = cos θ , where θ is the angle between the surface normal and the lineof sight. As discussed by van Hamme (1993), the squareroot law is a better approximation to the specific inten-sity distribution given by model atmospheres of late-typestars in the NIR than the other common two-parameterlaws (quadratic and logarithmic) implemented in jkte-bop . We also verified this by comparing the Claret(2000) four-parameter law (which we assumed to be the best representation of the original phoenix model) withthe two-parameter laws for typical M-dwarf stellar pa-rameters.For our baseline model, we fixed the limb darkening co-efficients to values for the I C filter derived from phoenix model atmospheres by Claret (2000), using T eff = 3130 Kfor the primary, and T eff = 3010 K for the secondary,log g = 5 . I filters (the errorintroduced by this assumption is examined in § § L = 0), but see also § Method of solution
To derive parameters and error estimates, we adopta variant of the popular Markov Chain Monte Carlo(MCMC) analysis frequently applied in cosmology andfor analysis of exoplanet radial velocity and light curvedata (e.g. Tegmark et al. 2004; Ford 2005). We useAdaptive Metropolis (Haario et al. 2001) rather than thestandard Metropolis-Hastings method with the Gibbssampler. The adaptive method allows adjustment ofthe proposal distributions while the chain runs, and isthus simpler and faster to operate, obviating the needfor manual tuning of the proposal distributions typicalof the more conventional methods. This method uses amultivariate Gaussian proposal distribution to perturball parameters simultaneously.Compared to the simpler Monte Carlo and bootstrap-ping methods (e.g. as implemented in jktebop ) thesemethods have an important advantage for the presentcase where a heterogeneous set of light curves and ra-dial velocities must be analyzed: it is possible to esti-mate the appropriate inflation of the observational un-certainties self-consistently and simultaneously with theother model parameters (e.g. Gregory 2005). This allowsthe correct relative weighting of the various observationaldata-sets to be decided essentially by their residuals fromthe best-fitting model, and the uncertainty in this weight-ing to be propagated to the final parameters. Of course,this assumes the correct model has been chosen for thedata, so these methods must be used carefully.In the Monte Carlo simulations, we used the standardMetropolis-Hastings acceptance criterion, accepting thenew point with probability: P acc = (cid:26) , P ( M i +1 | D ) ≥ P ( M i | D ) P ( M i +1 | D ) P ( M i | D ) , P ( M i +1 | D ) < P ( M i | D )where P ( M i | D ) and P ( M i +1 | D ) are the posterior prob-abilities of the previous and current points, respectively, M i represents the i th model, and D the data. The previ-ous point in the chain was repeated if the new point wasnot accepted (this is required to correctly implement themethod). The appropriate ratio of posterior probabilities(“Bayes factor”) was: P ( M i +1 | D ) P ( M i | D ) = P ( M i +1 ) P ( M i ) exp( − ∆ χ /
2) (2)where the factors in the right-hand side are the ratio ofprior probabilities, and the usual (least squares) likeli-hood ratio, respectively.Priors assumed for each parameter are detailed in Ta-ble 5, and are chosen to be uninformative. In most cases,this choice is a uniform improper prior (labeled “uni-form” in the table). For the orbital inclination, we adopta uniform prior on cos i , which produces an isotropic dis-tribution of orbit normals, and for the eccentricity, weuse a uniform prior on e and ω , rather than on the jumpparameters e cos ω and e sin ω (the latter would producea prior in e, ω proportional to e ; e.g. Ford 2006).As discussed by Gregory (2005), the appropriate choiceof uninformative prior for “scale parameters” such as ra- dial velocity amplitude or orbital period is a Jeffreys prior(or a modified Jeffreys prior for parameters which can bezero). This enforces scale invariance (equal probabilityper decade). In the present case, some of these parame-ters (particularly the period) are so well-determined thechoice of prior is unimportant, so for simplicity we haveonly used non-uniform priors on the less well-determinedparameters where the prior is important.As discussed earlier, and in Gregory (2005), it is possi-ble to fit for the appropriate inflation of the observationaluncertainties simultaneously with the model parameters.This has been done via the s k parameters shown in Table5. It is conventional to apply this inflation by multipli-cation for light curves, and for RV by adding an extra“jitter” contribution in quadrature. We follow these con-ventions here. Note that additional factors appear in thelikelihood ratio of Eq. (2) when doing this, which havebeen subsumed into the ∆ χ in the equation.For the RV, it is difficult to estimate reliable uncer-tainties from cross correlation functions, so instead wederive them during the simulation. We weight eachpoint by C (the square of the peak normalized cross-correlation value; see Table 4), where we find C peak isapproximately proportional to the signal to noise ratio intypical observations at similar signal to noise and peakcorrelation values to those seen in this work. This proce-dure is essentially equivalent to photon-weighting, exceptthe cross-correlation accounts for all sources of uncer-tainty rather than only photon noise. The uncertaintycorresponding to a peak correlation of unity is derivedduring the simulation, and we allow separate values ofthis quantity for each star, labeled s for the primary,and s for the secondary, in the tables. The resultingper-data-point uncertainties used in the fitting are givenby σ ij = s i /C peak , j for the i th star and j th radial velocitypoint.All Monte Carlo simulation results reproduced in thispaper are derived from chains of 2 × points in length,where the first 10% of the points were used to initial-ize the parameter covariance matrix for the AdaptiveMetropolis method (starting from the initial parame-ters and covariance matrix derived using a Levenberg-Marquardt minimization ), and the next 40% were usedto “burn in” the chain. This was found to be sufficientto ensure the chains were very well converged, and all ofthese first 50% of the points were discarded, leaving theremainder (10 points) for parameter estimation. Corre-lation lengths in all parameters were <
200 points.We report the median and 68 .
3% central confidenceintervals as the central value and uncertainty for all pa-rameters. Reduced χ values for all the model fits wereunity, by construction. Spots
In LSPM J1112+7626, the presence of spots is clearlyindicated by the observed out of eclipse modulation de-scribed in § SPM J1112+7626 11ric parameters J , ( R + R ) /a and i (and by exten-sion, R /R ). Spots occulted during eclipse temporar-ily reduce the depth as they are crossed, and spots notocculted during eclipse increase it, because the surfacebrightness of the photosphere under the eclipse chord isgreater than would be inferred from the out-of-eclipsebaseline level.The difficulty in solving for physical system param-eters arises because the true spot distribution is notusually known. Out-of-eclipse modulations, and in non-synchronized systems, modulation of the eclipses them-selves, are sensitive only to the longitudinal inhomogene-ity in the spot distribution, except in special cases wherespots are crossed during eclipse and cause detectable de-viations from the usual light curve morphology (this canbe difficult to detect in systems with radius ratios close tounity, because the deviations then last a large fraction ofthe eclipse duration). Any longitudinally homogeneouscomponent, such as a polar spot viewed equator-on, ora homogeneous surface coverage of small spots, cannotusually be detected from light curves.This can cause undetected systematic errors in theradii and effective temperatures derived from the lightcurves. Morales et al. (2010) discuss this issue in somedetail for the best-measured literature EBs, finding thiseffect could produce up to 6% systematic errors in theradii for stars with spots of 30% filling factor, when thespots are concentrated at the pole.The influence of the spectroscopic light ratio is notcommonly discussed with regard to spots, but this is im-portant because it provides complementary informationto the photometry, on the difference in spot coveragebetween the two stars. For example, consider the caseof a large coverage of longitudinally homogeneous, po-lar spots that are not crossed during eclipse. If thesespots are distributed on both stars, the light ratio is un-affected but both eclipses will appear deeper. However,if the spots are on only one star, the effect on the pho-tometry is the same, but the light ratio is altered be-cause the spotted star appears darker. This will producea discrepancy between the light ratio derived only fromthe light curve parameters, and the spectroscopic value.More generally, it is possible to place constraints on thedifference in the overall spot coverage between the binarycomponents, using the spectroscopic information.One unusual feature of the present system among low-mass eclipsing binaries showing signs of spots is the spinand orbit appear to not be synchronized. Because of this,it is possible to measure eclipses at different rotationalphases, and thus different spottedness of the visible stel-lar hemispheres. While this does not necessarily resolvethe difficulty of determining the longitudinally homoge-neous component of the spot distribution, it does pro-vide additional information on the longitudinally inho-mogeneous component, specifically which star the spotsare located on. This information is not usually avail-able if only out of eclipse modulations are seen. It iscommon to assume the spots are on both binary com-ponents, which may be reasonable for near equal masssystems at short periods where tidal synchronization isexpected to have occurred, but this is not a reasonableassumption for LSPM J1112+7626, where only one out-of-eclipse modulation is seen. The observed rotationalevolution of M-dwarfs (e.g. Irwin et al. 2011) indicates it is extremely unlikely the two binary components couldhave the same rotation period (and phase!) by chance inthe absence of tidal effects, which are not expected.Unfortunately, at the time of writing, only a singleprimary eclipse is available, so we are unable to takefull advantage of the non-synchronized spin and orbitat present. Also, while multiple secondary eclipses wereobserved, and residuals from our best-fitting model (as-suming no changes in eclipse shape or depth) are seen,it is not clear if many of these are simply the result ofsystematic errors in the photometry, given that equallylarge deviations are seen out of eclipse. Constraining thespot properties by this method will be an important areafor future work, and observing in multiple bandpasseswould be advantageous as it may provide information onthe spot temperatures. Spot model
Conventionally, a Roche model such as the one im-plemented in the popular Wilson-Devinney program(Wilson & Devinney 1971) would be used to model asystem with spots as these usually include a circularspot model and perform the necessary surface integralsover the stars; however the treatment of proximity ef-fects is completely unnecessary in the present system,and these models are extremely computationally inten-sive, especially for systems with eccentric orbits, whichwould make a detailed Monte Carlo based error analysisprohibitive.It is also not clear that the circular spot modelwith a small number of spots is realistic for late-type dwarfs. Observed light curves very rarely showthe characteristic “eclipse-like” features with flat base-line at maximum light, as would be predicted for asmall number of near-equatorial spots. This indi-cates either that these objects have very large, po-lar spots (e.g. Rodono et al. 1986), such that someof the spot is always in view to the observer to pro-duce the continuous photometric modulations seen inlight curves, or that the surfaces of these objects havemany small spots (e.g. Barnes & Collier Cameron 2001;Barnes et al. 2004) with the photometric modulationsarising from a longitudinal inhomogeneity of the spotdistribution. Modeling an ensemble of spots, as in thelatter case, in detail would be prohibitive, as even singlespot models are usually degenerate given limited lightcurve data. In our case, the degeneracies in fitting spotmodels would be further exacerbated by the availabilityof only single-band light curve information, meaning spottemperature and size would be essentially degenerate.Given these difficulties, we take an alternative ap-proach for the solution presented in this paper. Ratherthan trying to model the unknown spot distribution indetail, we simply attempt to incorporate the effect ofspots into the uncertainties on the final parameters. Weconsider two models: (a) spots on the non-eclipsed partof the photosphere on both stars; and (b), a case in-tended to approximate the effect of spots covering thewhole photosphere, again for both stars. In both cases,the models are required to reproduce the observed out ofeclipse modulation. It is necessary to run two models ineach spotted case because it is not known which star theout of eclipse modulation originates from.We assume a simple sinusoidal form for the modula-2 Irwin et al.tions, which appears to be an adequate description of theavailable light curve data. The functional form adoptedwas:∆ L i L i = a i sin (cid:18) πF i tP (cid:19) + b i cos (cid:18) πF i tP (cid:19) − q a i + b i (3)where t is time, L i is the light from star i , a i and b i are constants expressing the amplitude and phase of themodulation, and F i is the “rotation parameter”, the ratioof the rotation frequency to the orbital frequency. Thisexpression is constructed to yield ∆ L i = 0 at maximumlight, corresponding to the conventional assumption thatno spots are visible when the star is brightest, which givesthe minimum spot coverage necessary to reproduce theobserved modulations. jktebop computes the final system light (and thus thechange in magnitude) by summing the out of eclipse lightand then subtracting the eclipsed light. By modulatingonly the out of eclipse light in this summation using Eq.(3), hypothesis (a) can be implemented, and hypothesis(b) can be implemented by also modulating the eclipsedlight (changing the surface brightness under the eclipsechord) when the eclipsed star is star i . Results
The effect of applying our spot model on the eclipsesis shown in Figures 4 and 5, which were calculated us-ing the parameters of LSPM J1112+7626, comparing amodel with spots to a model with identical physical pa-rameters, but without spots. This figure demonstratesthat observing multiple eclipses at high precision wouldallow the star hosting the spots to be identified.We fit all four spotted models to the full data-setfor LSPM J1112+7626 using the procedures already de-scribed. The results are given in Table 6, and Figures 6,7 and 8 show the data with a representative model, cor-responding to hypothesis (a), the “non-eclipsed spots”model, on the primary, overplotted.The 2010 November 19 partial secondary eclipse fromMEarth (numbered 5 in the figure) provides the mostdiscriminating power between the various models, par-ticularly taken in conjunction with the 2011 May 2 event(numbered 9), which dominates the fit. As shown in Fig-ure 7, these were taken at opposite extremes of the outof eclipse variation, with the 2010 November 19 eventat maximum light and the 2011 May 2 event close tominimum light. Examining the sizes of the s parame-ter in Table 6, the “eclipsed spots” model on the sec-ondary, which is the only model presenting a significantdetectable deviation in the secondary eclipses (as shownin Figure 5) is disfavored by the data (see Figure 9), hav-ing an s MEarth − value larger by almost 3 σ thanthe “non-eclipsed spots” model on the secondary. In-deed, the observations indicate that this eclipse needs tobe made as shallow as possible in the model in order toproduce a good fit, possibly even slightly shallower thanthe “non-eclipsed spots” model is able to produce (thediscrepancy may arise from the imperfect modeling of theout-of-eclipse variation by a sinusoid). The depth of thiseclipse therefore argues that if the spots causing the outof eclipse modulation are on the secondary, they shouldbe at latitudes not crossed by the eclipse chord.It is interesting to note that, because the radius ratio isquite close to unity, the range of allowed combinations of Fig. 4.—
Predicted deviation in the primary (top) and secondary(bottom) eclipses resulting from spots on the primary star. Thesolid curves correspond to hypothesis (a) where the spots are noteclipsed, and the dashed curves to (b) where they are eclipsed.The two sets of curves are identical in the lower panel. The cyclenumbers (integer part of the normalized orbital phase) are shownon the right of each panel. spot latitude and inclination angle for the secondary thatwould explain the observed out of eclipse modulationsis therefore quite narrow, compared to the much widerrange of possibilities on the primary that are still allowed.This mildly argues in favor of spots on the primary asbeing the more likely explanation, purely from geometry.Nevertheless, with the presently-available data, theother models do not show any clear disagreement withthe observations, so we consider (from a purely observa-tional point of view) that all three are equally likely atpresent.
Effect of additional spots
The model we have considered so far uses the minimumspot coverage necessary to reproduce the observed out-of-eclipse modulations. As noted, e.g. by Barnes et al.(2011) and other authors, studies of cool stars sensitiveto the total spot coverage typically find much higher fill-ing factors than photometric spot models. Therefore itis likely these stars have a significant longitudinally ho-mogeneous spot component.In this section, we consider the effect of adding ad-ditional spots in a longitudinally homogeneous fashionwhere they would not be detectable through modula-tions in the photometry. For simplicity, we have placedthese spots on the same star and at the same latitudesas the longitudinally inhomogeneous component, notingthat we are predominantly interested in the effect on thecomponent radii, where it does not matter which of theSPM J1112+7626 13
TABLE 6Derived parameters and uncertainties for the four spot configurations with no additional spots.
Non-eclipsed spots Eclipsed spotsParameter a Primary Secondary Primary Secondary J MEarth . ± . . ± . . ± . . ± . J Bessell . ± . . ± . . ± . . ± . R + R ) /a . +0 . − . . +0 . − . . +0 . − . . +0 . − . R /R . ± .
017 0 . ± .
018 0 . ± .
017 0 . ± . i . +0 . − . . +0 . − . . +0 . − . . +0 . − . P (days) 41 . ± . . ± . . ± . . ± . T (HJD UTC) 2455290 . ± . . ± . . ± . . ± . F . ± . . ± . a (mag) 0 . ± . . ± . b (mag) 0 . ± . . ± . F . . . 0 . ± . . ± . a (mag) . . . 0 . ± . . ± . b (mag) . . . 0 . ± . . ± . s OOEb . ± .
014 1 . ± .
014 1 . ± .
014 1 . ± . s MEarth − . ± .
058 1 . ± .
057 1 . ± .
058 1 . ± . s MEarth − . ± .
087 1 . ± .
087 1 . ± .
087 1 . ± . s MEarth − . ± .
055 1 . ± .
056 1 . ± .
055 1 . ± . s MEarth − Primary . ± .
36 1 . ± .
35 1 . ± .
36 1 . ± . s Hankasalmi . ± .
049 1 . ± .
049 1 . ± .
049 1 . ± . s Clay − . ± .
046 1 . ± .
046 1 . ± .
047 1 . ± . s Clay − . ± .
048 1 . ± .
048 1 . ± .
050 1 . ± . s Clay − . ± .
038 1 . ± .
038 1 . ± .
038 1 . ± . s Clay − . ± .
034 1 . ± .
034 1 . ± .
034 1 . ± . k MEarth − . ± . − . ± . − . ± . − . ± . C MEarth . ± . . ± . . ± . . ± . k Clay − − . ± . − . ± . − . ± . − . ± . k Clay − . ± . . ± . . ± . . ± . k Clay − . ± . . ± . . ± . . ± . k Clay − − . ± . − . ± . − . ± . − . ± . e cos ω . ± . . ± . . ± . . ± . e sin ω . ± . . ± . . ± . . ± . q . ± . . ± . . ± . . ± . K + K ) (km s − ) 55 . ± .
087 55 . ± .
087 55 . ± .
087 55 . ± . γ (km s − ) c − . ± . ± . − . ± . ± . − . ± . ± . − . ± . ± . s (km s − ) 0 . ± .
015 0 . ± .
015 0 . ± .
015 0 . ± . s (km s − ) 0 . ± .
033 0 . ± .
032 0 . ± .
032 0 . ± . i ( ◦ ) 89 . +0 . − . . +0 . − . . +0 . − . . +0 . − . e . ± . . ± . . ± . . ± . ω ( ◦ ) 50 . ± .
18 50 . ± .
19 50 . ± .
19 50 . ± . a (R ⊙ ) 43 . ± .
069 43 . ± .
070 43 . ± .
070 43 . ± . L /L . ± .
025 0 . ± .
025 0 . ± .
025 0 . ± . T sec (HJD UTC) 2455314 . ± . . ± . . ± . . ± . M (M ⊙ ) 0 . ± . . ± . . ± . . ± . M (M ⊙ ) 0 . ± . . ± . . ± . . ± . R + R ) (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Parameter names as defined in Table 5 and in the text. b The acronym OOE is used as shorthand for “out of eclipse”. c Errors quoted for the γ velocity are the random error (scatter), followed by our estimated systematic error inthe velocity zero-point, from the assumed barycentric velocity of Barnard’s star.4 Irwin et al. Fig. 5.—
Predicted deviation in the primary (top) and sec-ondary (bottom) eclipses resulting from spots on the secondarystar. Curves as Figure 4. The two sets of curves are identical inthe upper panel. possible locations is used for the longitudinally inhomo-geneous spot component.As discussed earlier, the spectroscopic light ratio is sen-sitive to differences in the overall spot coverage betweenthe two stars in some cases. We have explored this limit,and the effect of adding varying quantities of spots onthe component radii, by subtracting an extra term c i inEq. (3) representing the fraction of stellar light removedby the longitudinally homogeneous spot component.For single late-type active dwarfs, it is typical to findspot filling factors of 20 −
50% from observations sensi-tive to the entire surface spot coverage (e.g. O’Neal et al.2004), although we note the available information for M-dwarfs is very limited. Photometric observations of M-dwarfs indicate spot temperature contrast of up to 10%(e.g. Rockenfeller et al. 2006, who find 4 − −
20% in the I -band stellar light, assuming parameters appropriate tothe present system, with the lower end of the range pre-sumably more typical for less active stars. We there-fore consider values of c i = 0 . .
2. The latter cor-responds approximately to the spot levels discussed byMorales et al. (2010) for the existing short-period eclips-ing binaries. The results are given in Tables 7 and 8.Firstly, it is clear from Table 7 that adding additional“eclipsed spots” has no effect on the radii or light ratioand merely alters the surface brightness ratio (this meansit also changes the effective temperature ratio). This isalso true when placing these spots on the secondary, butwe do not show results for this model since it is unable toreproduce the observed eclipse light curves, as discussedin § R , and decreasing R . The light ra-tio behaves as expected, becoming closer to unity asthe primary is darkened by the addition of spots. Both c = 0 . c = 0 . § c i = 0 . c i = 0 and 0 . c i ). These parametersand estimated uncertainties are reported in § Bandpass mismatch
A concern when combining light curves taken with dif-ferent instruments, is the effect of any difference in thephotometric bandpasses. In the present case, where theprimary and secondary eclipses were (by necessity) mea-sured using different instruments, this predominantly af-fects the ratio of the eclipse depths, and thus the param-eter J , the ratio of central surface brightnesses.In order to determine the approximate size of the band-pass mismatch, we measured the difference in magnitudebetween LSPM J1112+7626 and a nearby, much bluercomparison star, at 11 h m s .
15 +76 ◦ ′ . ′′ J − K ) = 0 .
27 and is thus probably in the Fspectral class. It is non-variable at the precision of theMEarth data (rms scatter 0 .
003 mag).Our measured magnitude differences were∆ I (Hankasalmi) = 0 .
36, ∆ I (Clay) = 0 .
35, and∆ I (MEarth) = 0 .
55. The excellent agreement betweenthe two I Bessell filters (especially in light of the observedout of eclipse variations of our target) indicates they aremost likely an extremely good match.Assuming a simple linear scaling, we estimate thechange in J corresponding to the mismatch between thetwo I Bessell filters is approximately 1 /
10 of that betweenthe I Bessell and I MEarth filters, or δJ ≈ . Limb darkening
Differences between I Bessell and I C We first examine the effect of assuming the limb dark-ening law is the same in the two I filters. To do this,we compare the fitting results using our baseline modelto using the coefficients for the z ′ SDSS passband fromClaret (2004), where the I Bessell band should lie inbe-tween these two extremes.Using the observed colors of LSPM J1112+7626, weestimate that approximately 1 / I C and z ′ results is appropriate for the error intro-duced by our assumption of the I C limb darkening lawSPM J1112+7626 15 Fig. 6.—
Eclipse light curves used in the model, with the best fit overlaid (the model is the first of the assumptions in Table 6, but theother model light curves are almost identical in appearance). In the top panels, the model is shown with red dots at the epochs of theobservations, and in the other panels it is plotted as a continuous curve. The figure has been divided horizontally into one column for eachcombination of eclipse and passband, with the primary eclipses (all in I Bessell ) plotted on the left, the secondary eclipses in I MEarth inthe center, and in I Bessell on the right. Each column contains three panels showing the raw differential photometry (top), the same aftercorrection to flatten the out of eclipse baseline (center; these corrections are the airmass term for the Clay and MEarth secondary eclipses,and the “common-mode” term for the MEarth secondary eclipses) and the residual in the bottom panel. We have offset the differenteclipses vertically for clarity, and the cycle number (integer part of the normalized orbital phase) is shown on the right. The vertical barsindicate the approximate locations of the first and last contact points.
TABLE 7Derived parameters and uncertainties showing the effect of additional spots on the primary.
Non-eclipsed spots Eclipsed spotsParameter c = 0 c = 0 . c = 0 . c = 0 c = 0 . c = 0 . J MEarth . ± . . ± . . ± . . ± . . ± . . ± . J Bessell . ± . . ± . . ± . . ± . . ± . . ± . R + R ) /a . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R /R . ± .
017 0 . ± .
016 0 . ± .
017 0 . ± .
017 0 . ± .
018 0 . ± . i . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . i ( ◦ ) 89 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . L /L . ± .
025 0 . ± .
025 0 . ± .
032 0 . ± .
025 0 . ± .
026 0 . ± . R + R ) (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Fig. 7.—
MEarth 2010-2011 season out of eclipse light curve. Vertical panels as Fig. 6, with the best-fitting sinusoidal model overplotted.Arrows indicate the phases corresponding to the observed eclipses.
TABLE 8Derived parameters and uncertainties showing the effect ofadditional spots on the secondary.
Non-eclipsed spotsParameter c = 0 c = 0 . c = 0 . J MEarth . ± . . ± . . ± . J Bessell . ± . . ± . . ± . R + R ) /a . +0 . − . . +0 . − . . +0 . − . R /R . ± .
018 0 . ± .
016 0 . ± . i . +0 . − . . +0 . − . . +0 . − . i ( ◦ ) 89 . +0 . − . . +0 . − . . +0 . − . L /L . ± .
025 0 . ± .
022 0 . ± . R + R ) (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . . +0 . − . SPM J1112+7626 17
Fig. 8.—
Radial velocity curves used in the model, with the best fit overlaid (top) and residual (bottom). The primary velocities areshown with square symbols, and the secondary velocities with round symbols. The residuals for the two stars have been offset for clarity.
Fig. 9.—
As Figure 6, except for the “eclipsed spots” model onthe secondary, showing the deviations in the MEarth 2010 Novem-ber 19 event.
TABLE 9Comparison of derived parameters and uncertaintiesfor I C and z ′ limb darkening laws using the“non-eclipsed spots” model on the primary. Parameter I C z ′ J MEarth . ± . . ± . J Bessell . ± . . ± . R + R ) /a . +0 . − . . +0 . − . R /R . ± .
017 0 . ± . i . +0 . − . . +0 . − . i ( ◦ ) 89 . +0 . − . . +0 . − . L /L . ± .
025 0 . ± . R + R ) (R ⊙ ) 0 . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . R (R ⊙ ) 0 . +0 . − . . +0 . − . in fitting the I Bessell data. Comparing the results forthe two bands in Table 9, we find this source of error inthe radii is negligible (although it does affect J and i atclose to 1 σ ). While the radius sum error is comparable tothe individual estimated uncertainties in the table, theuncertainty in this parameter for our adopted solutioncombining six spot configurations is larger, and the limbdarkening contribution is then less important. Errors in the atmosphere models
We have assumed limb darkening coefficients fixed tothe theoretical values from phoenix model atmospheresthroughout the analysis. We note that the same atmo-sphere models have issues reproducing the observed spec-tra of M-dwarfs in the optical, which raises the possibil-ity of systematic errors in the limb darkening law. In theliterature, this is conventionally addressed by analyzingphotometry in multiple passbands, which is sensitive todifferences in the limb darkening as a function of wave-8 Irwin et al.length. We do not have multi-band photometry, but notethis could be an important area for future work.It has long been recognized (e.g. Nelson & Davis 1972and references therein) that changes in the radius of theeclipsed star mimic changes in the limb darkening. Ina system with near-equal stars such as the present, itis reasonable to expect any systematic error in the limbdarkening from the models should affect both stars sim-ilarly, and thus to first order, the dominant effect willbe on the sum of the radii (and the inclination, but anyuncertainties here have negligible effect on the final pa-rameters).We verified this using simulations where the squareroot limb darkening coefficients were varied, finding astrong correlation between ( R + R ) /a and the integralof the specific intensity over the stellar disc, which is1 − u/ − u ′ / J and cos i ). This quantityis the normalization term in the eclipse depths in thephotometric model, so it is not surprising that it shoulddirectly influence the quantities derived from the abso-lute eclipse depth. This is predominantly a concern forthe interpretation of the sum of the radii: we find evenquite large changes in the limb darkening law do notsignificantly alter the individual radii compared to theiruncertainties in our adopted model. Third light
We now examine our assumption of no third light.While the proper motion evidence (see Figure 2) arguesit is unlikely there are any background stars contribut-ing at a significant level in the photometric aperture, therelatively low proper motion of our target does not allowthis possibility to be completely eliminated given the lim-ited angular resolution of the first epoch imaging data.Common proper motion companions are also permittedat intermediate separations below the angular resolutionof the SDSS data, but still in wide enough orbits or withmass ratios much less than unity, where they would notbe detected by the presence of additional lines in thespectra or radial velocity drift over the approximately 1year baseline available.Therefore, to examine any constraints on third lightwhich can be placed directly from the light curves, andthe effect on the derived parameters, we ran an addi-tional set of Monte Carlo simulations using the basic“non-eclipsed spots” model on the primary discussed in § L to vary. For simplicity, auniform prior was assumed. We find L < .
029 at 95%confidence. As discussed by Nelson & Davis (1972), thedominant parameter mimic is between third light andinclination, and our results confirm this, finding the un-certainties in cos i and the central surface brightness ratioparameters were slightly inflated. The other light curveparameters and radii derived from these chains are in-distinguishable within the uncertainties from the resultswhere L was not varied. Summary and adopted system parameters
Of the sources of uncertainty we have considered, spotsdominate. Since the stars hosting the spots and the spotconfiguration are mostly unknown, we adopt the union ofthe six spotted solutions discussed in § TABLE 10Adopted physical parameters and uncertainties,combining six spot configurations.
Parameter ValueC08 a L96 a M (M ⊙ ) 0 . ± . M (M ⊙ ) 0 . ± . R + R ) (R ⊙ ) 0 . +0 . − . R (R ⊙ ) 0 . +0 . − . R (R ⊙ ) 0 . +0 . − . log g . ± . g . ± . T eff , (K) 3061 ±
162 3191 ± T eff , (K) 2952 ±
163 3079 ± T eff , /T eff , . +0 . − . . +0 . − . L bol , (L ⊙ ) 0 . ± . . ± . L bol , (L ⊙ ) 0 . ± . . ± . m − M ) 3 . ± .
22 3 . ± . d (pc) 48 . ± . . ± . U b (km/s) 56 . ± . . ± . V (km/s) − . ± . − . ± . W (km/s) − . ± . − . ± . a Effective temperatures and bolometric cor-rections used. C08: Casagrande et al. (2008);L96: Leggett et al. (1996). A 150 K system-atic uncertainty was assumed for the effectivetemperatures, and the color equations given inthe 2MASS explanatory supplement were usedto convert the observed
JHK photometry tothe CIT system when using the L96 tabula-tion. b Adopting the definition of positive U to-ward the Galactic center. Calculated using themethod of Johnson & Soderblom (1987).equal weight. The final parameter estimates and theiruncertainties are reported in Table 10, derived from theposterior samples produced by the Monte Carlo simula-tions, adopting the median and 68 .
3% central confidenceintervals as the central value and uncertainty (see § γ velocity from Table6, and position and proper motions from Table 3. Wefind this object is in the old Galactic disk population,following the method and definitions in Leggett (1992).For very precise work, even matters as seemingly trivialas physical constants can be important, so we briefly dis-cuss our assumptions in this regard. We adopt the 2009IAU values of G M ⊙ and the astronomical unit (see theAstronomical Almanac 2011), and follow Cox (2000) inadopting the value of the solar photospheric radius fromBrown & Christensen-Dalsgaard (1998) . For the solareffective temperature, we use the value of 5781 K fromBessell et al. (1998), which is based on measurements of A useful compilation of solar data may be foundin “Basic Astronomical Data for the Sun”, by Eric Ma-majek (University of Rochester, NY, USA), available at . SPM J1112+7626 19the solar constant by Duncan et al. (1982).The final constant, the solar bolometric absolutemagnitude, is a more thorny issue, as discussed byBessell et al. (1998), and extensively by Torres (2010).When using tables of bolometric corrections, it is vitalto adopt the consistent value of this quantity in accor-dance with the table. We therefore use the appropriatevalues in each case in the following section.
Effective temperatures, bolometric luminosities anddistance
In order to estimate effective temperatures (and thus,bolometric luminosities, and the distance), an externalestimate of the effective temperature of one of the bi-nary components is required, in addition to bolomet-ric corrections. For M-dwarfs, there are large sys-tematic uncertainties in the effective temperature scale(e.g. Allard et al. 1997; Luhman & Rieke 1998), withdisagreement at the few hundred degrees Kelvin levelamong different authors. Many of the early difficultieswere due to the lack of model atmospheres (which areusually needed to integrate the full SED from the avail-able measurements to obtain L bol ), and significant im-provement on this front was made in the 1990s. Theavailability of a much larger sample of angular diame-ter measurements from interferometry should further im-prove the situation as this provides a much more directmethod to estimate T eff for single stars, but there arerelatively few temperature scales available at the presenttime using this information.We show results from two different inversions to illus-trate the typical range of parameters. Both scales weuse were derived with model atmospheres rather thanblackbodies, and in both cases we use the measured I C − J , I C − H and I C − K colorsfrom Table 3 in conjunction with the component radiiand I C -band light ratio from the adopted eclipsing bi-nary solution. The results from these different pairingsof bandpasses were found to be consistent within the un-certainties, so we report the union of the results.The two sets of effective temperature and bolometriccorrections adopted were from Casagrande et al. (2008),which is a recent determination using interferometric an-gular diameter measurements to derive T eff , and fromLeggett et al. (1996), which is the basis for several recentworks and tabulations of the effective temperature scale,including Luhman & Rieke (1998), Luhman (1999), andKraus & Hillenbrand (2007).For the Leggett et al. (1996) scale, we fit polynomi-als to their tabulated data, omitting three objects theseauthors found to be metal poor from our fits: Gl 129,LHS 343, and LHS 377. We follow Leggett et al. (1996)in adopting a systematic uncertainty of 150 K, whichappears to be consistent with the differences we find be-tween the two determinations.Our results for LSPM J1112+7626 are reported in Ta-ble 10. DISCUSSION
We now compare our results for LSPM J1112+7626with theoretical models from Baraffe et al. (1998) andother literature objects with shorter periods summarizedin Table 11. We first show the conventional mass-radius
Fig. 10.—
Mass-radius relation for components of detached,double-lined, double-eclipsing binary systems below 0 . ⊙ .LSPM J1112+7626 is shown as bold, black points, and the lit-erature systems from Table 11 are shown in gray, excludingCCDM J04404+3127B,C, which does not have a full solution avail-able at present, but including KOI 126. The lines show stellar evo-lution models from Baraffe et al. (1998) for 10 Gyr age, [M / H] = 0(solid line), [M / H] = − . / H] = 0 modelwith the radius inflated by 5%, corresponding to ρ = 1 .
05 (dottedline).
Fig. 11.—
Mass-effective temperature relation for componentsof detached, double-lined, double-eclipsing binary systems below0 . ⊙ . Lines and points as Figure 10. KOI 126B,C and1RXS J154727.5+450803 do not have reported effective temper-atures, so they do not appear on this diagram. The dotted lineshows the prediction from assuming ρ = 1 .
05 and that the bolo-metric luminosity is preserved. diagram in Figure 10, and then the corresponding mass-effective temperature ( T eff ) diagram in Figure 11.We also perform a quantitative comparison with thesame models. We follow Torres (2007) in defining a pa-rameter equal to the ratio of the observed radius to thatpredicted by the models given the observed mass, al-though we denote this parameter by ρ instead, to avoidany potential confusion with the β parameter defined by0 Irwin et al. TABLE 11Detached, double-lined, double-eclipsing main sequence EB components below . ⊙ a Name Period
M R T eff [M / H] Source b (days) (M ⊙ ) (R ⊙ ) (K)SDSS-MEB-1 A 0 .
407 0 . ± .
020 0 . ± . ±
130 . . . 1SDSS-MEB-1 B 0 . ± .
022 0 . ± . ± c .
771 0 . ± .
016 0 . ± . ±
110 . . . 2GJ 3236 B c . ± .
015 0 . ± .
015 3242 ± .
27 0 . ± . . ± . ±
70 [ − , − .
6] 3CM Dra B 0 . ± . . ± . ± .
63 0 . ± .
014 0 . ± .
02 3058 ±
195 . . . 4LP 133-373 B same same 3144 ±
206 . . .MG1-2056316 B 1 .
72 0 . ± .
001 0 . ± . ± .
002 3320 ±
180 . . . 5KOI 126 B d .
77 0 . ± . . ± . . ± .
08 6KOI 126 C d . ± . . ± . e .
05 . . . . . . . . . . . . 7CU Cnc B 2 .
77 0 . ± . . ± . ±
150 . . . 81RXS J154727.5+450803 A 3 .
55 0 . ± . . ± . . ± . a This mass criterion has been applied to keep the number of objects in the table and plots manageable,and is not intended to necessarily be physically meaningful beyond being appropriate for comparisonto the present system. b (1) Blake et al. (2008), (2) Irwin et al. (2009b), (3) Morales et al. (2009), (4) Vaccaro et al.(2007), (5) Kraus et al. (2011), (6) Carter et al. (2011), (7) Shkolnik et al. (2010), (8) Ribas (2003);Delfosse et al. (1999), (9) Hartman et al. (2011). c Parameters determined giving equal weight to all three models, following Hartman et al. (2011). d While not double-lined, this object is a special case as it is still possible to solve for the masses andradii of both M-dwarfs independent of M-dwarf models. The period given is that for the inner M-dwarfbinary as this is presumably the appropriate one for estimation of activity levels. e Full solution not available.Chabrier et al. (2007) and used by Morales et al. (2010),which is not the same.As argued in Torres (2007), there is evidence that thebolometric luminosities from the models are not seri-ously in error. If this is the case, a corresponding fac-tor of ρ − / must be applied to the effective tempera-tures from the models when inflating the radii, in orderto preserve the bolometric luminosity. We show resultsfor this scenario in the tables and plots. Note, how-ever, that the bolometric luminosity is not preservedin several recent theoretical works (Chabrier et al. 2007;Morales et al. 2010; MacDonald & Mullan 2011), so thetemperatures derived from this procedure should not beconsidered definitive.As is often the case in eclipsing binaries, while the indi-vidual masses of the objects are well-determined from theradial velocities, the sum of the radii is much better de-termined than the individual radii, modulo limb darken-ing uncertainties as discussed in § ρ values for the individual stars, a“composite” value comparing the model-predicted radiussum given the individual observed masses, to the mea-sured radius sum. This quantity, which we call ρ , hasmuch smaller formal uncertainties than the individual ρ and ρ values, and measures essentially a “weighted av-erage inflation” for the whole system. These quantities,and the corresponding effective temperatures predictedfrom preserving the bolometric luminosity, are given fora range of assumptions for the age and metallicity in Ta-ble 12. While we caution over-interpretation of the ρ results due to limb darkening uncertainty, the individualvalues should be much less prone to this problem. Mass-radius comparison
We first limit our comparisons to the mass-radiusplane. If the metallicity is solar, the sum of radii is in-flated at the 7 σ level compared to the models, even for10 Gyr age, which produces the largest model radii. Thediscrepancies are not as significant in the individual radii,with the primary 2 σ and the secondary 3 σ larger thanthe model predictions. We also note that the “eclipsedspots” model on the secondary, which we discarded in § α emission (e.g. Terndrup et al. 2000;Hartman et al. 2010, and references therein), neither ofwhich are seen.Metallicity may provide a viable explanation for theobserved radii given the present uncertainties (both inthe observations and the models, where for the latter itis possible the effect of metallicity on radius is under-predicted; see § / H] ≈ − . §
1) and the kinematicevidence, we proceed to examine other possible causes ofSPM J1112+7626 21
TABLE 12Quantitative mass-radius comparison of LSPM J1112+7626 with models.
Age [M / H] ρ ρ T eff , T eff , T eff , /T eff , (Gyr) ρ (K) (K)1 0 . . ± .
016 1 . ± .
018 3426 ±
27 3255 ±
26 0 . ± . − . . ± .
016 1 . ± .
018 3621 ±
28 3387 ±
27 0 . ± . . . ± .
015 1 . ± .
017 3453 ±
27 3293 ±
27 0 . ± . − . . ± .
016 1 . ± .
018 3648 ±
29 3436 ±
28 0 . ± . . . ± .
015 1 . ± .
017 3468 ±
27 3324 ±
27 0 . ± . − . . ± .
016 1 . ± .
017 3666 ±
29 3475 ±
28 0 . ± . . . +0 . − . ±
13 3291 ±
12 0 . ± . − . . +0 . − . ±
14 3439 ±
13 0 . ± . . . +0 . − . ±
13 3323 ±
12 0 . ± . − . . +0 . − . ±
14 3477 ±
13 0 . ± . . . +0 . − . ±
13 3344 ±
12 0 . ± . − . . +0 . − . ±
15 3504 ±
13 0 . ± . the inflated radii.It is tempting to suggest that the inflation inLSPM J1112+7626 results from elevated activity levelsin one or both components. While the lack of H α emis-sion, unless this is rendered undetectable by the limitedsignal-to-noise ratio of our spectra, may be a difficultyfor this scenario to explain, it should be noted that moremodest levels of activity can result in weakened H α emis-sion, or even absorption, but may still produce sufficientspot coverage to explain the inflation.This system straddles (in mass) the full convectionlimit, where a substantial increase in the observed activ-ity lifetimes for M-dwarfs occurs (e.g. West et al. 2008).This would argue that the secondary is more likely tobe active and thus the source of the inflation. The sec-ondary lines in the spectrum are also more difficult toobserve due to its lower luminosity, which would makean H α line easier to hide in the noise. Identification ofthe star hosting the out of eclipse modulations may shedadditional light on which of the components might beresponsible for the inflation.As discussed by Morales et al. (2010), the radius in-flation inferred from eclipsing binary analyses often im-plies extremely high spot filling factors. We estimate that ρ = 1 .
05 corresponds to β = 0 .
1. Assuming a spot tem-perature contrast of T eff , spot /T eff , phot = 0 .
9, this wouldrequire a filling factor of approximately 30%. This ismuch larger than needed to produce the observed outof eclipse modulation, and would mean there is a sub-stantial longitudinally symmetric spot component, if theinflation is indeed caused by spots. We showed in § R andincreasing R . In practice, this may slightly reduce therequired filling factor depending on where the spots arelocated, but unless the spot coverage is much larger thanwe assumed, the systematic errors in the radii resultingfrom such spots are still insufficient to explain the obser-vations without invoking inflation.Finally, on suggestion of the referee, we note that thisobject may indicate the need to revisit the equation ofstate for very low mass stars, which was discussed as apossibility to explain the radius inflation before the ac-tivity hypothesis gained prevalence (e.g. Lopez-Morales 2004; Torres & Ribas 2002). The equation of state wasa significant source of difficulty in early work attempt-ing to model very low mass stars (Chabrier & Baraffe1995), and although substantial progress on this fronthas been made (see the reviews by Chabrier & Baraffe2000; Chabrier et al. 2005), it is still an open question. Mass- T eff comparison Comparisons in only the mass-radius plane ignore im-portant information contained in the effective tempera-tures. The correct physical model must explain all of theobservations simultaneously, so we now proceed to exam-ine the temperatures. This comparison is more problem-atic than in mass-radius for a number of reasons. Themain observational issues are the difficulty of constrain-ing effective temperatures (see § §
1, unlike the radius, the effective temper-ature predicted from models does depend quite stronglyon metallicity, because the bolometric luminosity is afunction of metallicity. As the metallicity is decreased,the model bolometric luminosity and effective temper-ature increase, and the radius decreases by a smallamount. While metallicity does complicate the inter-pretation of the mass- T eff diagram, this a key argumentagainst metallicity as the explanation for all of the in-flated radii in the eclipsing binary sample, as discussedalready in §
1. In this regard, CM Dra is puzzling asseveral authors have claimed this object is metal poor,which would further exacerbate the effective temperaturediscrepancy with the models.Given the present uncertainties, it is not clearif the effective temperature difference betweenLSPM J1112+7626 and the models is significant.However, it does lie at the low-temperature end of theeclipsing binary results, and this lends support to thehypothesis that this object is more metal rich than theother eclipsing binaries.
Future work
It is clear that LSPM J1112+7626 offers one of the bestprospects to observationally test the causes of inflatedradii in eclipsing binaries, but further observations are2 Irwin et al.needed. The most fruitful avenues would be to pursuea determination of the metallicity, improvement of thesystem parameters with a particular focus to investigat-ing potential systematics in the radii (e.g. due to spots),and constraining the activity levels in the components byindependent means (for example, X-ray emission, or theCa ii H and K lines).While we have mentioned the non-synchronized spinand orbit, we were unable to take full advantage of thisproperty with the available observational data. As shownin § . The SDSS is managed bythe Astrophysical Research Consortium for the Partic-ipating Institutions. The Participating Institutions arethe American Museum of Natural History, Astrophysi-cal Institute Potsdam, University of Basel, University ofCambridge, Case Western Reserve University, Universityof Chicago, Drexel University, Fermilab, the Institute forAdvanced Study, the Japan Participation Group, JohnsHopkins University, the Joint Institute for Nuclear As-trophysics, the Kavli Institute for Particle Astrophysicsand Cosmology, the Korean Scientist Group, the Chi-nese Academy of Sciences (LAMOST), Los Alamos Na-tional Laboratory, the Max-Planck-Institute for Astron-omy (MPIA), the Max-Planck-Institute for Astrophysics(MPA), New Mexico State University, Ohio State Univer-sity, University of Pittsburgh, University of Portsmouth,Princeton University, the United States Naval Observa-tory, and the University of Washington. Facilities:
FLWO:1.5m (FAST, TRES)
REFERENCESAbazajian, K. N., et al. 2009, ApJS, 182, 543Allard, F., Hauschildt, P. H., Alexander, D. R., & Starrfield, S.1997, ARA&A, 35, 137Andersen, J. 1991, A&A Rev., 3, 91Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998,A&A, 337, 403Barnard, E. E. 1916, AJ, 29, 181Barnes, J. R., & Collier Cameron, A. 2001, MNRAS, 326, 950Barnes, J. R., James, D. J., & Collier Cameron, A. 2004,MNRAS, 352, 589Barnes, J. R., Jeffers, S. V., & Jones, H. R. A. 2011, MNRAS,412, 1599Bean, J. L., Sneden, C., Hauschildt, P. H., Johns-Krull, C. M., &Benedict, G. F. 2006, ApJ, 652, 1604Benedict, G. F., et al. 1998, AJ, 116, 429Berger, D. H., et al. 2006, ApJ, 644, 475Berta, Z. K., Charbonneau, D., Bean, J., Irwin, J., Burke, C. J.,D´esert, J.-M., Nutzman, P., & Falco, E. E. 2011, ApJ, 736, 12 Bessel, M. S. 1990, A&AS, 83, 357Bessell, M. S. 1990, PASP, 102, 1181Bessell, M. S., Castelli, F., & Plez, B. 1998, A&A, 333, 231Blake, C. H., Torres, G., Bloom, J. S., & Gaudi, B. S. 2008, ApJ,684, 635Bonfils, X., Delfosse, X., Udry, S., Santos, N. C., Forveille, T., &S´egransan, D. 2005, A&A, 442, 635Boyajian, T. S., et al. 2011, in Cool Stars, Stellar Systems andthe Sun XVI, ed. C. Johns-Krull, A. West, & M. Browning,Astronomical Society of the Pacific Conference Series, in press(arXiv:1012.0542)Brown, T. M., & Christensen-Dalsgaard, J. 1998, ApJ, 500,L195+Buchhave, L. A., et al. 2010, ApJ, 720, 1118Carney, B. W., Latham, D. W., & Laird, J. B. 1989, AJ, 97, 423Carter, J. A., et al. 2011, Science, 331, 562Casagrande, L., Flynn, C., & Bessell, M. 2008, MNRAS, 389, 585Chabrier, G., & Baraffe, I. 1995, ApJ, 451, L29+