A refined analysis of the low-mass eclipsing binary system T-Cyg1-12664

0.88 MB·

by  R. Ramón Iglesias-Marzoa

Astronomy&Astrophysicsmanuscriptno.TCyg1paper_def cESO2017
(cid:13)
January25,2017
A refined analysis of the low-mass eclipsing binary system
⋆
T-Cyg1-12664
RamónIglesias-Marzoa1,2,MercedesLópez-Morales3,MaríaJ.Arévalo1,4,JeffreyL.Coughlin5,andCarlosLázaro1,4
1 AstrophysicsDepartment,UniversidaddeLaLaguna,E-38205LaLaguna,Tenerife,Spain
2 CentrodeEstudiosdeFísicadelCosmosdeAragón,PlazaSanJuan1,E-44001,Teruel,Spain
e-mail:[email protected]
3 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
7 4 InstitutodeAstrofísicadeCanarias,E-38200,LaLaguna,Tenerife,Spain
1 5 SETIInstitute,189BernardoAve,MountainView,CA94043,USA
0
2 ReceivedSeptember15,1996;acceptedMarch16,1997

n
a ABSTRACT
J
4 Context.Theobservational mass-radiusrelationofmainsequence starswithmassesbetween 0.3and1.0 M revealsdeviations
2 betweenthestellarradiipredictedbymodelsandtheobservedradiiofstarsindetachedbinaries∼. ⊙
Aims.Wegenerateanaccuratephysicalmodelofthelow-masseclipsingbinaryT-Cyg1-12664intheKeplermissionfieldtomeasure
] thephysicalparametersofitscomponentsandtocomparethemwiththepredictionoftheoreticalstellarevolutionmodels.
R Methods.WeanalyzetheKeplermissionlightcurveofT-Cyg1-12664toaccuratelymeasurethetimesandphasesoftheprimaryand
S secondaryeclipse.Inaddition,wemeasuretherotationalperiodoftheprimarycomponentbyanalyzingtheout-of-eclipseoscillations
thatareduetospots.Weaccuratelyconstraintheeffectivetemperatureofthesystemusingground-basedabsolutephotometryinB,
.
h V,R ,andI .WealsoobtainandanalyzeVR I differentiallightcurvestomeasuretheeccentricityandtheorbitalinclinationofthe
C C C C
p system,andapreciseT ratio.Fromthejointanalysisofnewradialvelocitiesandthoseintheliteraturewemeasuretheindividual
eff
- massesofthestars.Finally,weusethePHOEBEcodetogenerateaphysicalmodelofthesystem.
o
Results.T-Cyg1-12664isaloweccentricitysystem,locatedd=360 22pcawayfromus,withanorbitalperiodofP=4.1287955(4)
tr days,andanorbitalinclinationi=86.969 0.056degrees.Itiscomp±osedoftwoverydifferentstarswithanactiveG6primarywith
s T =5560 160K,M =0.680 0.045M ,±R =0.799 0.017R ,andaM3VsecondarystarwithT =3460 210K,M =0.376 0.017
a Meff1,andR±=0.3475 01.0081R±. ⊙ 1 ± ⊙ eff2 ± 2 ±
[ Co⊙nclusion2s.Thep±rimarystar⊙isanoversizedandspottedactivestar,hotterthanthestarsinitsmassrange.Thesecondaryisacool

1 starnearthemassboundaryforfullyconvectivestars(M 0.35 M ),whoseparametersappeartobeinagreementwithlow-mass
v stellarmodel. ∼ ⊙
5
Keywords. Stars:fundamentalparameters–Stars:low-mass–Binaries:eclipsing–Binaries:spectroscopic
3
8
6
01. Introduction estimatemassesofabout0.62 M and0.32 M fortheprimary
. and secondary. Çakırlıetal. (201⊙3) reanalyzed⊙ this system us-
1T-Cyg1-12664(KIC 10935310, 2MASS J19513982+4819553)
ing new RV measurements, the high precision Kepler mission
0
is a low-mass detached eclipsing binary (LMDEB) at
7 lightcurvesandR-bandground-basedphotometryandobtained
1α=19:51:39.824,δ=+48:19:55.38(J2000.0).Itwasfirstdiscov- thefollowingvaluesforthemassesandradiiofthestars: M1 =
v:(eTrerdESin;AthleonTs-Coyegt1alfi.e2l0d0o4f),thaendTritaniss-aAlstloaninticthEexKopelpalneertmSiusrsvioeny 0M.68,0R±0=.002.189M7⊙,0R.011=2R0.6.1I3n±t0h.e0ir07stRud⊙y;,awndhiMle2th=e0ra.3d4iu1s±o0f.0t1h2e
Xifieldofview.Becauseofitsintermediateorbitalperiod( 4.129 pri⊙mar2yisconsi±stentwith⊙mainsequencelow-massstellarmod-
∼
d) and low mass ratio (q 0.55), this system can serve as a els, the radiusof the secondarydefiesinterpretationas a main-
r ≃
abenchmarkforlow-mass(M<1M )stellarevolutionmodelsthat sequencestar.
haveproblemswiththeradiipredi⊙ctedforthestars.
The discovery and first analysis of T-Cyg1-12664 was re- AfteritspublicationinDevoretal.(2008a),weincludedT-
ported by Devoretal. (2008a) and Devor (2008) using TrES Cyg1-12664inourobservingprogramtorefinethephysicalpa-
photometric data. They obtained a small number of radial ve- rametersofseveralLMDEBs.Herewepresentathoroughanal-
locity (RV) observations, including only one RV measurement ysisofthesystembasedontheKeplerphotometry,newground-
of the faint secondary star. Their RV measurements show that basedmultibandphotometryandRV measurements,andaspe-
theperiodinitiallyidentifiedintheTrESphotometry(8.26days) cificphotometriccalibrationcarriedouttoobtainunbiasedcol-
was in fact of 4.129 days, and a secondary eclipse would be ors and precise, photometry-based effective temperatures. The
buried in the TrES data. From that first analysis Devor (2008) outlineofthisworkisasfollows.InSects.2and3wedescribe
the Kepler and the ground-basedRV and photometricobserva-
⋆ Tables 1, 2, 3 and 10 are only available in electronic form at the tions. Section 4 describes a careful analysis of the system, in-
CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via cludingspecificsectionsforaconsistentphotometriccalibration,
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ effectivetemperature,a carefultreatmentofthe thirdlight,and
Articlenumber,page1of18
A&Aproofs:manuscriptno.TCyg1paper_def
Fig.1.Fitofanorder300Legendrepolynomial totheKeplerdatain Fig.2.EffectofthedetrendingoverthephasedlightcurveofQ6quarter.
quarter6.Theupperpanelshowstheout-of-eclipselightcurveandthe Upperpanelshowstherawlightcurveandlowerpanelshowsthelight
fit(redline).Thelowerpanelshowstheresidualsofthefit. curveafterapplyingtheLegendrepolynomialfittedinFig.1.
2.2.Ground-basedobservations
theorbitaleccentricity.Finally,theabsoluteparametersandthe
distanceofthesystemarecomputedinSect.6,withSect.6.3de- WeobservedT-Cyg1-12664withtheCAMELOTcameraatthe
votedtocomparingourresultstothestellarmodelsandprevious IAC80 telescope in Tenerife, Spain, in VR I bands, over 14
C C
results. nights,between2009-04-04and2010-10-01UT(JD2454936.6
– JD 2455471.5), when eclipses occurred. We also requested
a routinary observation program at the same telescope to sam-
pleout-of-eclipsephases.CAMELOTisequippedwithaback-
illuminatedE2V2048x2048sensorwithsquared13.5µmpixels,
2. Light-curveobservations
whichresultsinaplatescaleof0.”304/pixanda10.’4x10.’4field
ofview.
2.1.Keplerobservations
AlthoughCAMELOThastwo readoutchannels,allthe ob-
servations were made using one channel to avoid systematic
T-Cyg1-12664 was observed by the Kepler spacecraft in Long
effects between the object and the comparison stars. Exposure
Cadence mode from Q0 through Q17 (JD 2454953.54 –
times ranged from 100 s to 180 s in V band, 65 s to 95 s in
JD 2456424.00). The observations were reduced using the
R band, and 60 s to 90 s in I band, dependingon the seeing
Kepler mission pre-search data conditioning (PDC) pipeline C C
andtheatmospherictransparency.Eachnight,wemonitoredT-
(Jenkinsetal. 2010), which produces high quality light curves
Cyg1-12664 continuously over an altitude limit of 30 degrees
liketheonesillustratedinFigs.1and2forQ6.Theinstrumen-
(airmass 2) abovethe horizon.We also acquired bias frames
tal systematics and spot-related activity in this star are readily
≤
anddomeflatsinallthefiltersatthebeginningofeachnight.
visible in the Kepler light curveswith out-of-eclipsevariations
We used standardreductiontechniques,includingbias sub-
deeperthanthesecondaryeclipse.
straction, flat-field correction, and alignment of all images to
We detrended the light curves from each quarter following
a common reference system, to process the frames. We ana-
aproceduresimilartoSlawsonetal.(2011),byfittingLegendre
lyzedthe correctedframesusingthe photaperturephotometry
polynomialstotheout-of-eclipsedata.Theresultforonequarter
packageinIRAFandacustomdifferentialphotometrypipeline
(Q6) is shown in Fig. 2. For T-Cyg1-12664 we had ephemeris
designed to deal with large sets of frames. The pipeline first
information from preliminary works, so we could exclude the
performs aperture photometry for a set of preselected compar-
time intervals affected by eclipses, and it was not necessary to
ison stars and the target, for a range of aperture sizes. Us-
applytheiterativeprocessesdescribedinSlawsonetal.(2011).
ing the flux within each aperture, the pipeline computes the
Because of the effects of the spacecraft safe-modes and trig-
signal-to-noiseratio(S/N)ofthetargetusingtheCCDequation
gers, some quarters were divided in smaller intervals, but this
(Merline&Howell1995,eq.25),andselectstheapertureradius
detrending process was applied to all the Kepler mission pho-
that produces the maximum S/N. This aperture is then used to
tometryencompassinganintervalof1470days.TypicalLegen-
extractthefluxofallthestarsinthatframe.Thisprocedureac-
drepolynomialsorderswerebetween30and300,dependingon
counts for seeing and transparency variations along the night.
thecomplexityofthevariationsanddurationofthelight-curves
The CCD equationwas also used to computethe formalerrors
intervals.Theresultofthisdetrendingisillustratedinthebottom
in the photometry.We imposeda limit of 9 pixelsto the maxi-
panelofFig.2.Thisdetrendedlightcurve,providedinTable1,
mumapertureradiusto avoidcontaminationbynearbystarsas
wassubsequentlyanalyzedtogetherwiththeground-basedpho-
isthecaseforT-Cyg1-12664(seeSect.4.5).Theselectedcom-
tometrydescribedinSect.2.2.
parison stars provedto be stable during all the observing runs.
Wenotethatthisdetrendingprocesshasthedisadvantageof We generatedV, R , and I differentiallight curvesof the tar-
C C
potentially removing informationfrom the light curve, such as getdividingitsfluxbythecombinedfluxofallthecomparison
heating effects and proximity effects that are due to the shape stars. Owing to the small field of view of the telescope no dif-
of the stars. However, as we demonstrate in subsequent sec- ferentialextinctioneffectsweretakenintoaccount.OurVR I
C C
tions, these effects are negligible in T-Cyg1-12664,because of photometry(givenin Table 2) includessevenprimaryand four
thelargeseparationbetweenthestars. secondaryeclipseswhich,whenaddedtothosefromtheKepler,
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RamónIglesias-Marzoaetal.:Arefinedanalysisofthelow-masseclipsingbinarysystemT-Cyg1-12664
Çakırlıetal.(2013)andTrESphotometry,enableustoimprove determineF (λ)andF (λ)bycubicsplinesinterpolationofthe
1 2
theperiodofthesystemandsearchforperiodvariations. originalreferencespectra.
For reference spectra, we used the normalized (flattened)
synthetic spectra from Munarietal. (2005). The Munarietal.
3. Radialvelocityobservations (2005) grid we used covers models with 3500 T 10,000
eff
≥ ≥
in steps of 250 K, 0.0 logg 5.0 in steps of 0.5 dex,-2.5
We observedT-Cyg1-12664aspartof a largerprogramto pro- ≥ ≥ ≥
[M/H] 0.5instepsof0.5dex,and0 V 100km/sinsteps
rot
duce accurate radial velocity curves of low-mass binaries (see ≥ ≥ ≥
of10km/s.Toensurethatwefoundtheglobalminimuminboth
Coughlin2012). We collectedradialvelocityobservationsover
selected reference spectra and their velocities, we performed a
two seasons (June – December 2010 and May – September
globalgridsearch loopingoverallpossible combinationsof α,
2011),withtheDualImagingSpectrograph(DIS)onthe3.5-m
V ,V ,T ,T ,usingacommon[M/H]forbothreferencespec-
1,j 2,j 1 2
at Apache Point Observatoryand the R-C Spectrographon the
tra,andloggandV foreachreferencestar.Wecomparedthe
rot
4-m at Kitt Peak National Observatory (KPNO). With DIS we
radial velocities found by this method with those obtained us-
observed using its red channel with the R1200 grating, which
ingthewell-testedTODCORcode(Zucker&Mazeh1994),and
givesaresolutionof0.58Åperpixel,orR 10,000.Thewave- findconsistentresultsforallthebinariesinourprogram.
≈
length range was set to 5900 - 7100 Å for the first observing The errors in the derived radial velocities were computed
∼
season,and 5700-6800Å forthesecond.Forthe R-CSpec- using a bootstrapping re-sampling method iterated over 10000
∼
trograph,weusedtheKPC-24gratinginsecondorder,resulting times. The adopted errors correspond to the 1σ confidence in-
inaresolutionof0.53Åperpixel,orR 10,000,withthewave- tervalofthedistributionofbootstrappingsolutions.Toestimate
length rangeset to 5700- 6750Å. W≈e collected a total of 31 theerrorsinthederivedtemperatures,weusedthespectroscopic
individual spectra f∼or T-Cyg1-12664, in addition to bias, dark, quality-of-fitparameter(López-Morales&Bonanos2008;Behr
andflatfieldcalibrationframes.WealsocollectedHeNeArlamp 2003),definedas
frames before or after each target observation to use as wave-
rms2
lengthcalibrations. z= √N 1, (2)
Allrawframeswerebias,dark,andflat-fieldcorrected.For rms2min − 
the DISobservations,column1023is a dead column,and thus
we replaced those values by a linear interpolation between the where N is the numberof data points, rms2 is the standard de-
two neighboring columns. We also removed cosmic rays from viation of the fit under consideration,and rms2 is the best fit
min
all frames using a Laplacian edge algorithm implemented in found.Thezparameterissimilartoareducedχ2 intheabsence
the lacos_spec IRAF package (see vanDokkum 2001). Fi- ofknownerrorsontheindividualpoints.Bydefinitionz= 0at
nally, we extracted one-dimensional spectra from each image, thebest-fit,andthe1σconfidenceintervalcorrespondstoz=1.
includingskybackgroundsubstraction,usingtheIRAFpackage IntheparticularcaseofT-Cyg1-12664,thespectrumofthe
apall. After wavelength calibration, all science spectra were secondaryistooweak,andwewereunabletodetectanysignif-
flattened and normalized by fitting a 20-piece cubic spline fit icant radial velocity measurements for the secondary star over
overteniterations,duringwhichpoints3σabovethefitor1.5σ anyreasonableparameterranges.Thus,wetreatedthesystemas
below the fit were rejected. For each spectrum, we calculated asingle-linedbinarybyfixingthevalueofαto9999andonly
thebarycentricJuliandateinterrestrialtime,BJD(TT),andcor- fitting for the parametersof the primary star V , T , [M/H],
1,j eff1
rectedthemtothereferenceframeofthesolarsystembarycenter logg and V . The resultant radial velocities are listed in Ta-
rot
usingtheIRAFtaskbcvcorr. ble3.Inaddition,thebestfitoftheprimary’sspectratothegrid
Forallbinariesweattemptedtoextracttheradialvelocityof of synthetic spectra gives a Teff=5750 250 K, logg=4.5 0.5,
eachstarineachframe,V1,j andV2,j,where1and2denotethe [M/H] =-0.5±0.5 and vrsini=40±10±km s−1. This Teff±is in
primaryandsecondarystars,and jistheframenumber,usinga agreementwiththeoneobtainedinSect.4.2usingphotometric
newexploratorymethod:insteadofusingcross-correlations,we colors. The metallicity appearsto be lower than the typical for
directlyfitreferencespectratoeachobservedspectrumviaatra- starsneartheGalacticplane(seeSection6.2),whichwouldhave
ditionalstandarddeviationminimization,whichisequivalentto near-solarmetallicity.Weattributethisresulttotherathercoarse
minimizingχ2,assumingallobservedpointshaveequalerrors. samplinginmetallicityofourmodels,andthereforeadoptsolar
Specifically,foreachobservedspectrum,wefitforthevelocity metallicityforthissystemintheremaininganalysis.
ofeachreferencespectra,aswellasfortheirluminosityratio,α Wenotethat,althoughthisχ2minimizationtechniqueyields
= L /L ,foratotalofthreefreeparameters.Duringthefitting, valuesof the radialvelocitiesand other stellar parameterscon-
1 2
theoriginalobservedspectrumisneverchanged,andthusitcan sistent with those obtained via other methods in the case of T-
have any arbitrary number and distribution of wavelength and Cyg1-12664, the technique can be prone to systematics intro-
fluxvaluepairs.Givenatotalof M observedspectra,eachwith duced by the presence of correlated noise or not well modeled
N points, taken at times t , we want to find the values of V , spectralinstrumentalprofiles,especiallyinthecase oflowS/N
j j 1,j
V ,andαateacht ,thatminimizethefunction spectra.Itisthereforeadvisabletohaveresultscheckedagainst
2,j j
othermethodslikeTODCOR.
XjM=1XiN=1j F0,i− α·F1(cid:16)λ0,i·(1− Vc1α,j)+(cid:17)+1F2(cid:16)λ0,i·(1− Vc2,j)(cid:17)2, 44..1A.Pnhaolytosmisetorifctchaelibsryastiotenmandcolors
(1)
InTable4,wecompilealltheavailablephotometryforT-Cyg1-
where F isthefluxofanobservedspectrumata givenwave- 12664 in public catalogs. We found inconsistencies between
0,i
length, λ , F (λ), and F (λ) are the fluxes of reference stars someofthepassbandmagnitudesreportedbydifferentcatalogs,
0,i 1 2
1 and 2 at a given wavelength, and c is the speed of light. We e.g.theBandVmagnitudesreportedbytheGSCandNOMAD
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Table4. PhotometryinseveralbandsforT-Cyg1-12664obtainedfrom Table5.ObservedLandoltfields.Thenumberofmeasurementsisthe
photometriccatalogs. globalcountfordifferentairmasses.
Bandpass Magnitude Source Night Field N N Airmassrange
stars meas
V 13.520 0.190 GSCa 2012-07-26 SA110SF3 8 19 1.130-2.519
B 14.040±0.416 GSC2.3.2a PG1633+099 8 9a 1.055-2.590
V 13.108±0.297 GSC2.3.2a 2012-08-18 SA110SF3 8 4b 1.158-2.435
± PG0231+051 6 9c 1.086-1.690
B 13.530 NOMADb,c
PG1657+078 6 6 1.076-2.542
V 13.120 NOMADb,c
(a)8measurementsfortheBandVfilters.(b)6measurementsfortheI
R 13.500 NOMADb C
filter.(c)10measurementsfortheV filter.
B1 14.460 USNO-B1d
B2 14.020 USNO-B1d Table 6. Standard transformation coeficients from Eq. 3 for the two
R1 13.130 USNO-B1d nights.Theuncertaintiesareshowninparenthesesaffectingthelasttwo
R2 13.500 USNO-B1d digits.
I2 13.210 USNO-B1d
r 12.400 USNO-A2e Parameter 2012-07-26 2012-08-18
b 13.100 USNO-A2e b0 2.7185(30) 2.7947(65)
B 13.530 UCAC3e,f b1 0.2262(18) 0.2250(38)
R2 12.512 UCAC3e,f b2 0.0101(39) 0.0430(90)
b -0.0204(22) -0.0342(49)
I 11.858 UCAC3e,f 3
b -0.0038(13) -0.0186(27)
U 13.905 0.021 Everettetal.(2012) 4
± v 2.7752(25) 2.8607(44)
B 13.864 0.024 Everettetal.(2012) 0
± v 0.1184(13) 0.1170(24)
V 13.280 0.018 Everettetal.(2012) 1
± v 0.0436(30) 0.0575(56)
g 13.541 SloanDigitalSkySurvey 2
′ v 0.0025(13) -0.0082(27)
r 13.052 SloanDigitalSkySurvey 3
′ v 0.01621(82) 0.0163(15)
i 12.911 SloanDigitalSkySurvey 4
′ r 2.8779(29) 2.9391(46)
z 12.843 SloanDigitalSkySurvey 0
′ r 0.0877(13) 0.0905(24)
D51 13.327 Keplermissiong 1
r -0.0662(61) 0.029(11)
Kepler 13.100 Keplermission 2
r -0.0096(21) -0.0208(40)
J 11.911 0.024 2MASSh 3
H 11.582±0.015 2MASSh r4 0.0940(29) 0.0501(56)
K 11.529±0.021 2MASSh i0 3.2383(37) 3.3249(52)
s ± i1 0.0426(17) 0.0482(29)
i -0.2608(78) -0.209(12)
Notes. 2
(a)Identifier:GSC0356101711. i3 0.0024(26) -0.0326(45)
(b)Identifier:1383-0349082. i4 0.0717(39) 0.0865(73)
(c)MagfromYB2.
(d)Identifier:1383-0345184.
(e)Identifier:U135011189502. packageapphot,andadoptinganapertureof14”(a13pixelra-
(f)MagfromSuperCosmos. diusintheTCPimages),tomatchtheLandoltaperturewidths.
(g)Bandcenteredin510nm,seeBrownetal.(2011). Theextinctioncorrectionandtransformationtothestandard
(h)Identifier:2MASSJ19513982+4819553. photometric system was performed simultaneously solving the
setofequations
catalogsdisagreeby0.4–0.5mags.Thedifferencesaretoolarge m =B +b +b X +b (B V) +b X (B V) +b (B V)2
to be explained by stellar activity, or even accounting for the B 0 1 B 2 − 3 B − 4 −
m =V +v +v X +v (B V) +v X (B V) +v (B V)2
depthoftheeclipses( 0.20intheKeplerband).Wedonothave V 0 1 V 2 − 3 V − 4 −
an explanationforthe∼se discrepanciesbut,concernedaboutin- m =R +r +r X +r (V R )+r X (V R )+r (V R )2
RC C 0 1 RC 2 − C 3 RC − C 4 − C
troducinglargeerrorsinthedeterminationofparametersforthis m =I +i +i X +i (V R ) +i X (V R ) +i (V R )2
system,weproceededtoderivenew,consistentcolors. IC C 0 1 IC 2 − C 3 IC − C 4 − C
(3)
We performed photometric calibrations of several objects,
including T-Cyg1-12664, over two photometric nights in July wheretheinstrumentalmagnitudesappearontheleft-handside
and August 2012, during out-of-eclipse phases. We used the of the equations, the absolute magnitudesare denoted as B, V,
Trömso CCD Photometer in full frame mode (TCP, Östensen R , and I , and the airmass values as X. The equations were
C C
2000; Östensen&Solheim 2000), at the IAC80 telescope solved using least-square techniques, and the resulting coeffi-
(CAMELOTwasnotavailableatthetimeoftheseobservations). cients for each night are summarized in Table 6. The absolute
The TCP is built arounda Texas Instruments1024x1024CCD magnitudesobtainedforT-Cyg1-12664astheaverageofthere-
sensor and produces a 9.’6 field of view, with a plate scale of sultsforeachnightaresummarizedinTable7.Inthatsameta-
0.”537pixel−1,attheIAC80’sfocalplane. ble,wealsolistcolorindexes,includingthenear-infraredJHKS
WeobservedseveralLandoltstandardfields(Landolt2009), bandsfrom2MASS(Skrutskieetal.2006).
coveringobjectswithawiderangeofcolorsandairmasses,and Our BandV calibratedmagnitudesareslightlyfainterthan
using Johnson-Cousins B, V, R , and I filters. A log of the the ones from Everettetal. (2012). We attribute the small dif-
C C
observationsis shown in Table 5. We measured the flux of the ferences (∆B=0.073, ∆V=0.019), to stellar activity variations,
standardstarsviastandardaperturephotometry,usingtheIRAF as seen in Fig. 2. In the Kepler light curve these variationsare
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RamónIglesias-Marzoaetal.:Arefinedanalysisofthelow-masseclipsingbinarysystemT-Cyg1-12664
Table7.CalibratedmagnitudesandcolorindicesforT-Cyg1obtained Table 8. Mean effective temperature estimations resulting from our
from our photometric calibration. The infrared magnitudes are taken photometriccalibration,2MASSandSDSSphotometryandourspec-
from2MASS(seeTable4). troscopyforT-Cyg1-12664. Theinterestellarreddeningwasnottaken
intoaccountforthiscomputation.
Band Adoptedvalue(mag)
B 13.937 0.002 Calibrations Colorindices Teff(K)
± Empirical
V 13.299 0.005
RICC 1122..591215±±±00..000045 CCABooarrlxxilbe((sa22ts00er00&o00s))Ma(2a0rt1in2e)zRoger(1988) ((((VVBB−−−VVKK))SS,))(,R(J−−ICH)) 5555462891620000±±±3261500900
VVB−−−VRICC 000...376783488±±±000...000000765 MFIvueakzsuiac´gniaetateaetlta.la(.2l.(02(020800)161)) Mode(((lVggd′′−e−−−pKerrn′′S))d)ent 555455662047±±±±124584b
VRIJJCC−−−−HKKKISSCS 10000.....749337182804292±±±±00000.....000002022326182 BHLeeojsuesdueanllseh(1eet9lta7el9.t)(a1l9.9(280)00) ((((((VVBBBJ−−−−−−HVVVKK))))SS,,,,))((((,,VVVJ((RJ−−−−C−KRII−CCKSC)))IS),,C,,)(()(,RH,V(CH−−−−IKICCKS)),)S) 555456278000±±±322093000
− ± Thiswork Spectroscopy 5750 250
H−KS 0.053±0.026 Meanvalue(adopted) 5560±±160
Notes.
of ∆m 0.022, which is very close to the variation we see (a)InterpolationfromTable7.6.(b)Forlogg=4.5,[M/H]=-0.5).
Kp
∼
inV band.The B V indexobtainedbyEverettetal.(2012)is
−
0.584 0.030,whichisslightlybluerthanours,mainlyasacon-
±
sequenceofthedifferenceintheBmeasurements.
4.2.Effectivetemperature
We derived the effective temperature of the system, T , using
eff
theempiricalandmodeldependent,temperature–colorrelations
listed in Table 8, our BVR I colors derived in Sect. 4.1, and
C C
the colors of the system published in 2MASS (Skrutskieetal.
2006), and the SDSS (Eisensteinetal. 2011). All those colors
weremeasuredoutsideofeclipses,whichguaranteesthatnosys-
tematicswere introducedby the dimmingof the system during
eclipseevents.Westillhavetheeffectofstellarvariability,which
resultsinsmallvariationsofthecolorindexes.Thosevariations
areaccountedforintheT errorsreportedinTable8.
eff
Table8summarizesthevaluesoftheT derivedusingeach
eff
temperature–colorrelation.Weassumednegligiblereddeningin
the directionof T-Cyg1-12664when computingthe colorsbut,
in any case, the effect of the reddening over the distances in-
Fig.3.Comparisonbetweentheabsoluteopticaland2MASSphotom-
volved will be smaller than the uncertainties in colors among
etrywith thesynthetic photometry obtained for a Kurucz model with
theopticalandIRbands,asaconsequenceofthosecolorsbeing
T =5500K,logg=4.5,and[M/H]=-0.5.Theuncertaintiesintheup-
measuredindifferentepochsandtheintrinsicphotometricvari- eff
perpanelaresmallerthanthepointsizes.
ability of the system. The mean effective temperature adopted
forthesystem,computedastheaverageofalltheresultsinTa-
ble8,is5560 160K.Thisvalueiswithin1σofthevalueofTeff paredtheopticalandnear-infraredcolorsmeasuredforT-Cyg1-
±
obtainedfromfitting our spectra in Sect. 3, andcorrespondsto 12664 to a Kurucz (1993) model template with T =5500 K,
eff
aspectraltypeG6V,basedonthetabulatedvaluesofMamajek logg = 4.5, and [M/H]=-0.5. The result is shown in Fig. 3,
(2015). We adopted this result as the spectral type for the pri- wherethe fitbetweenthe modelandthe observationsproduces
mary. an rms of 0.037 mags. We also repeated the same comparison
Our Teff andspectraltype of the primarydo notagreewith withthecalibratedphotometryinTable7andatemplatemodel
thosederivedbyDevor(2008)andÇakırlıetal.(2013).Inthose withthesamemetallicityandlogg,butwithT =4250Kfrom
eff
studiestheauthorsclaimaK5Vspectraltypeforthesystemand Çakırlıetal.(2013).Thebestfitinthiscaseproducesanrmsof
a mean Teff=4320 100 K. The main difference between their 1.029mags,withaclearsystematictrend.Basedontheseresults
±
calculations and ours is their use of B and V magnitudesfrom weconcludethatthevalueof5560 160Kisagoodestimateof
±
theUSNOcatalogandGSC2.3.2,whichwenoticedinSect.4.1, theactualT forT-Cyg1-12664.
eff
areunreliable.
4.4.Rotationandsynchronicityparameter
4.3.Spectralenergydistribution
The Kepler raw light curves of the system (see, for example,
Given the difference between the spectral type obtained in this Fig. 2) reveals periodic photometric variations owing to spots,
work and those in the literature, and as an additional check to and with a period clearly different from that of the binary. In
the absolute photometry values derived in Sect. 4.1, we com- addition,and giventhe largedifferencesin luminositybetween
Articlenumber,page5of18
A&Aproofs:manuscriptno.TCyg1paper_def
the primary and secondary stars, we can attribute those photo- Table9.FluxesandcolorsforT-Cyg1-C,thecontaminantintheKepler
passband.
metricvariationstospotsontheprimarystar.Therefore,wecan
measure the rotation period of the primary, assuming that the
spotsrotatejointlywiththephotosphere.Afternormalizingeach Parameter Value
quarteroftheKeplerrawlight-curve,andremovingtheeclipses B 18.600+/-0.039
using the ephemeris equation of the system (see Sect. 4.6), we V 17.822+/-0.029
calculatedLomb-Scargleperiodograms(Scargle1982)foreach R 16.975+/-0.020
normalized light curve. A prominent peak is seen in the peri- I 16.009+/-0.017
odogram from which we obtained rotation periods in a range B V 0.778+-0.049
−
between 3.840778 d and 4.250872 d. We attribute those vari- V RC 0.847+-0.035
−
ations to changes in the different spot configurations over the VC IC 1.813+-0.034
−
time span ofthe observations,andadopteda meanrotationpe- RC IC 0.966+-0.026
−
riodof P = 3.968 0.091d.Thatrotationperiodresultsina
rot
± Notes.ThecolorindexB-VisnotusedinthecomputationoftheT of
synchronicityparameter F = ω /ω = 1.041 0.024,using eff
rot orb ± thecontaminatingstar(seetext).
theperiodofthebinarycomputedin Sect.4.6. Thisvalueof F
is too small to producea significant deviation in the computed
stellar radius and lets us assume that the primary star is rota- (R /D)2 =(5.65 0.14) 10 22overtheintegratedVR I bands
− C C
tionallysynchronizedwiththeorbitalperiod.This Prot mustbe (se∗eMoro&Mu±nari200×0).Theuncertaintywascomputedper-
takenasanestimate,giventhatthestarisnotarigidbodywith formingMonte-Carlofitsover100iterations.Weusedthesame
awelldefinedrotationperiodand,thus,thisperioddependson KuruczspectrumusedinthesyntheticSED,scaledbythecom-
thelatitudeofthespots.ForeachofthetwomainspotsintheQ6 puted dilution factor and integrated over the Kepler response
interval,we measuredthe timestmin of eachassociatedminima function1 to estimate the contaminant flux due to T-Cyg1-C.
andplottedthedifferenceT0 tminversusthesequentialnumber The uncertainty was also computed with a Monte-Carlo anal-
−
ofeachminima,withT0definedasthetimeofaprimaryeclipse. ysis, as in the dilution factor case, and the obtained value is
Wefitastraightlinetoeachspot’sdatasetandobtainedslopesof FKP = (5.88 0.15) 10−9 erg cm−2 s−1. For T-Cyg1-12664
3.9659 0.0092d and 3.988 0.010d, which confirmsthe pres- we followed t±he same×procedure but using a dilution factor of
± ±
enceofdifferentialrotation,asalreadyrevealedbytheobserved (R/D)2 = (2.7291 0.0082) 10 21, whichresultsina Kepler
−
rangeofrotationalperiods.In addition,thespotwith thelower fluxofF =(2.500±6 0.0071×) 10 7ergcm 2s 1.
K − − −
rotationperiodistheonewithlowerlatitudeintheQ6-1epoch Using these fluxe±s for the b×inary and the contaminant, the
(see Sect. 5.5), which suggests that the primary has solar-type thirdlightratiointheKeplerbandresultsin
differential rotation (see, for example, Reinhold&Arlt 2015;
Lanzaetal.2014, andreferencestherein). lKp =0.02297 0.00058.
3 ±
Asacheckofthefeasibilityofthisresult,wecomparedthis
4.5.Thirdlight
valuewiththeestimatedamountofthirdlightintheV,R ,and
C
T-Cyg1-12664 has an optical companion at a distance of 4.”6 I bands using images taken the night of 2010-08-28at an or-
C
as pointedby Çakırlıetal. (2013), which is clearly seen in our bitalphase of 0.285.The valuesobtainedfor the differencesin
images. We call this contaminant star T-Cyg1-C in the rest of magnitude between T-Cyg1-12664 and T-Cyg1-C were ∆V =
the text. The aperture of our ground-based photometry avoids 4.619 0.028,∆R = 4.186 0.018,and∆I = 3.463 0.013,
C C
contaminationbylightfromthisstar.Thus,ourVR I differen- which±translateintothirdligh±tratiosateachpassband ±
C C
tialphotometryisfreeofthisthirdlightcontribution.However,
Keplerusesaphotometricapertureof 10 11pixforT-Cyg1- lV =0.0140 0.0004;lRC =0.0207 0.0003;
12664(roughly3x3pix),dependingo∼nthe−season,withaplate 3 lI±C =0.03963 0.0005. ±
scaleof3”98/pix.Giventhatthiscontaminantisnotlistedinthe 3 ±
KeplerInputCatalog(KIC)itwouldn’thavebeenaccountedfor These results are of the same order of magnitude as those
duringKepler’sthirdlightcalibration,andtheKeplerlightcurve obtained for the K band. No other nearby stars are observed
P
ofT-Cyg1-12664isaffectedbythefluxofT-Cyg1-C. in the IAC80 images and there are no signs of apsidal motion
ToestimatetheamountofthirdlightintheKeplerpassband, in the eclipse timings (primary or secondary) in the time span
we hadtorelyonthefluxofthe contaminantinV, RC, and IC. oftheobservations(seeSect.4.6).Thus,weassumedthatthere
We applied the photometriccalibration performedon the night is no other significant source of third light in the system and
of2012-07-26toT-Cyg1-C,toobtainitsmagnitudesandcolors we adopted the value for the Kepler band as the only source.
listedinTable9.FromthosecolorsweadoptedTeff =3761 56 Nonetheless,duringthemodellingofthissysteminSect.5.5we
±
KforT-Cyg1-C,basedonthecalibrationsofCox(2000),Bessell ranteststo ensurethatnoothersourceofthirdlightaffectsthe
(1991,1979)andLejeuneetal.(1998).Wedidn’tusetheB V BVR I bands, for example, if T-Cyg1-12664were a multiple
− C C
color index since their Teff is not consistent with the results of systemwithathirdobjectthatisnotresolvedinourimages.All
the other color indices and the B-band magnitude is too faint. thesetestsyieldednegativeresults.
Thus,werelyonlyonthecolorsthatmakeuseofV,R ,andI
C C
photometry.From the obtainedT , and assumingit is a main-
eff
sequence star, we adopteda spectral type of M0V for this star, 4.6.Orbitalperiodanalysisandephemeris
usingthetablesinMamajek(2015).
WecalculatedthecentraltimesofeclipseinboththeKeplerde-
To estimate the Kepler band flux, K , for T-Cyg1-C we fit
p trendedlightcurvesandtheIAC80lightcurves.Tofindthecen-
an spectral energy distribution (SED) using the observed col-
traltimesofeclipseintheKeplerlightcurve,wefiteitherthird
ors and a Kurucz model spectrum with T =3750 K and so-
eff
lar metallicity and obtain a geometric dilution parameter of 1 Avaliableathttp://keplergo.arc.nasa.gov/CalibrationResponse.sh
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RamónIglesias-Marzoaetal.:Arefinedanalysisofthelow-masseclipsingbinarysystemT-Cyg1-12664
Table11.RVfittingresults.
Parameter Value
AdjustedQuantities
Pa(d) 4.1287955 4 10 7
−
± ·
T (HJD) 2454936.705 0.011
p
ea 0.0365 ±0.0014
ωa(deg) 92±.8 2.2
±
γ(km/s) -6.28 0.53
±
K (km/s) 48.07 0.74
1
±
K (km/s) 87.0 2.5
2
±
DerivedQuantities
M sin3i(M ) 0.677 0.046
1
M sin3i(M⊙) 0.374±0.017
2
q= M /M ⊙ 0.553±0.018
2 1
±
a sini(106km) 2.728 0.042
1
±
Fig.4.Histogramofthesecondaryeclipsedistributioninphase(lower a sini(106km) 4.93 0.14
2
axis)andintime(upperaxis).Allthesecondaryeclipsesaresistemat- asini(106km) 7.66±0.15
icallybelowphase0.5.Theverticaldashed linesshow thepositionof ±
OtherQuantities
thefourIAC80I -bandsecondaryeclipses.
C χ2 58.657
χ2 1.128
red
or fourth order polynomials to the points in each eclipse. The Nobs (primary) 47
timingsoftheKeplereclipseshaverelativelylargeuncertainties Nobs (secondary) 9
( 3min)becausetheeventsaresparselysampled,giventhe29.4 Timespan(days) 1507.86
m∼inutes sampling rate of the long cadence Kepler data. In the rms1 (km/s) 5.079
caseoftheIAC80lightcurves,whicharebettersampled,wefit- rms2 (km/s) 3.492
tedsixthorderpolynomialsforboththeprimaryandsecondary
eclipses. Timesfor thesecondaryeclipseswere determinedus- Notes.(a)Parameterfixedbeforehand.
ing the I -band light curve since it shows the deepest eclipses
C
andallowsabetterdeterminationoftheminima.Alltheeclipse
timesarelistedinTable10.
the complete set of keplerian parameters [P,T ,e,ω,γ,K ,K ]
p 1 2
To compute a new ephemerides equation for the system,
for the primaryand secondaryRV curvesby χ2 function mini-
we also added the eclipse times published by Devoretal.
mization.The uncertaintiesin the parameterscan be computed
(2008a)andÇakırlıetal.(2013).Whennecessary,weconverted
from the Fisher matrix or using a Markov-Chain Monte Carlo
the times of minima from HJD to BJD using the method of
method (MCMC). We chose this last method since it can deal
Eastmanetal.(2010).Wealsoconservativelyassignedanuncer-
withcorrelationsbetweenparameters.MCMCwasrunover106
taintyof0.01daystotheeclipsesofDevoretal.(2008a),given
samplesforeachsolutionfoundwithrvfit.
thedispersionoftheirphotometry.Alldatacombinedprovideda
totalof228primaryeclipsesoveratimebaselineofeightyears. We ran a first fit to obtain initial parameter values and to
Wefitallthemeasurementsbyastraightline,eliminatingtwoof check the significance of an elliptical orbit. In this first fit, we
theKeplerobservations,whichpresentedlongdeviationscaused fixedonlytheorbitalperiodtothevalueobtainedinSect.4.6and
bybadsamplingoftheeclipses.Theresultofthatfitisthefol- leftalltheotherparametersfree.Thisfirstsolutionwascompat-
lowingupdatedephemeridesequation: ible with a low eccentric, e = 0.069 0.016,orbit and a mass
±
ratioofq = 0.558 0.018.Thisellipticalorbitisconfirmedby
±
thesmalldisplacementinphaseofthesecondaryeclipsesshown
MinI(BJD)=2455415.61831(5)+4.1287955(4)n (4) inSect.4.6.
Theresidualstothisfitshownosignsofdeviationfromastraight The dispersion in the RV measurements and the sinusoidal
line,whichsuggestnothirdbodyispresentinthesystem,atleast appearanceoftheRVdatadistributioncanhidethesubtleeffect
with orbital periods of the order of the obsevations timescale. ofasmalleccentricity.Tocircumventthislimitation,wedecided
Finally,weshowahistogramwiththephasedistributionsofthe to constrainthe eccentricityand the argumentof the periastron
secondaryeclipsesinFig.4. Alltheeclipsesaresystematically usingtheKeplerlightcurves(seeSect.5.2)andfixthemtothe
belowphase0.5,withameanphaseof∆φ=0.49894 0.00025, valuese=0.0365 0.0014andω=92.8 2.2deg.Thisnewfit
whichsuggeststhatthebinaryisslightlyeccentric. ± yieldedthevalues±inTable11.Thefitted±RV curveisshownin
Fig.5.
5. Radialvelocityandlightcurvefits
5.2.LightcurvespreliminaryanalysiswithJKTEBOP
5.1.RVfittingprocedure
We fit our RVs jointly with the primary and secondary WeusedthedetrendedKeplerdatatoobtainaninitialmodelof
measurements from Çakırlıetal. (2013) and Devor (2008). thelight-curvewithoutspotsandtofixsomeinitialparameters,
We fit a keplerian orbit to the data using the rvfit code namelyecosωandesinω.Thisallowsustostartourcombined
(Iglesias-Marzoaetal.2015).Thiscodeisbasedonanadaptive analysis of the multiband light curves with initial values near
simulatedannealing(ASA)algorithmandcansimultaneouslyfit thefinalsolutiononcethespotsareincludedintheanalysis.For
Articlenumber,page7of18
A&Aproofs:manuscriptno.TCyg1paper_def
Fig.6.JKTEBOPfittotheKeplerdetrendedlight-curve.Adetailedplot
ofthefitsintheeclipsesisshowninFig.7
Fig. 5. RV curve fit and residuals. Thecontinuous and dashed curves
arethefitsfortheprimaryandsecondarystars,respectively.Thefilled
and empty circles are the RV measurements for the primary and sec-
ondary star respectively. The crosses are the RV measurements from
Çakırlıetal.(2013).Thedottedlineintheupperpanelisthevaluefor
γ.ThephasewascomputedwithEq.4.
this goal we used JKTEBOP2 (Southworthetal. 2004), which
is based on the EBOP code (Popper&Etzel1981; Etzel1981)
and models the stars as triaxial ellipsoids. We fixed q, P and
Fig. 7. Detail of the Fig. 6 showing the fit in the eclipses. Note the
T0 fromtheephemerisEq.4,andthesolutionoftheRVcurves changeintheverticalscaleinthetwoplotsandthesmalldisplacement
(Sect.5.1).Wefitthefollowingparameters:thesumandratioof ofthesecondaryeclipsefromphase0.5,indicatedwithaverticaldashed
fractionalradii,i.e.r1+r2andr1/r2,theorbitalinclinationi,the line.
quantities esinω, ecosω, the surface bright of the secondary,
andthelightscalefactor.
Inthisfirststage,theeffectofheatingbetweenthetwocom-
ponents of the binary was not taken into account, since it was Toavoidproblemscausedbythelongintegrationtimeofthe
eliminated from the light curves when we detrended the out- Keplerdata.we split upeachobservationintosix pointsto un-
of-eclipse variations. Therefore, the model coefficients associ- dergoanumericalintegration.TheeffectofthisKeplerlonginte-
ated with this effect were fixed to zero. In any case, we ran grationtimeinlightcurvesofeclipsingbinaries(Coughlinetal.
tests to confirm this effect was not noticeable. The third light 2011) and exoplanetsystems(Kipping2010) is a knownissue.
contribution, l3, was fixed to the value calculated in Sect. 4.5. Theseauthorsfindthatundersamplingaffectsthemorphological
Testsfittingl3yieldedsolutionswithhigherthirdlightcontribu- shapeoftheeclipses(transits),insuchawaythattheyappearto
tions, butpoorerfits. We adopteda square-rootlimb darkening beshallowerandhavealongerduration.Thiseffectismostpro-
(LD)law(Diaz-Cordoves&Gimenez1992;vanHamme1993), nounced in eclipsing binaries with short periods and low sum
withtheLDcoefficientsinterpolatedfromthetablesprovidedby of the fractional radii (r + r ) values. T-Cyg1-12664, with a
1 2
Claret&Bloemen (2011) and PHOENIX stellar models using r +r 0.1isintherangewherethiseffectcanbenoticed.
1 2
thenominalT of5560 160Kand3350 50Kfortheprimary ≃
eff
and secondary compone±nts. We adopted±those values running TheheavydetrendingappliedtotheKeplerlightcurvepro-
preliminary PHOEBE tests with the optical IAC80 curves and duces points deviating largely from the model in phases with
theRVsolutionforacircularorbit.ThePHOENIXmodelswere stronglightcurvecurvaturenearthe centerof the eclipses (see
selected based in their lower T limit, althoughthe LD coeffi- Fig.7).Thus,aconservativesigma-clippingof10σwasapplied
eff
cientwere computedonlyforZ=0.0.Thelogg valueswereset in the fitting processto discard these points. With this prelimi-
to4.5and5.0foreachcomponent,respectively.Forthegravity- naryfit,wearrivedatthesolutionshowninFigs.6and7,where
brighteningcoefficientsweusedvaluesfromClaret&Bloemen i = 86.87 0.14, e = 0.0365 0.0014, and ω = 92.8 2.2.
± ± ±
(2011)forthesameTeff. The resulting fractional radii are r1 = 0.07282±0.00098 and
r =0.03300 0.00050.Theuncertaintiesinthefittedparameter
2
±
2 JKTEBOPiswritteninFORTRAN77andthesourcecodeisavail- werecomputedusinga bootstrapanalysiswith 5 000synthetic
ableathttp://www.astro.keele.ac.uk/ jktcodes/jktebop.html datasets.
∼
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RamónIglesias-Marzoaetal.:Arefinedanalysisofthelow-masseclipsingbinarysystemT-Cyg1-12664
Fig.8.LightcurvesoftheT-Cyg1-12664intheepochQ6-1andfitted Fig.9.Samefigureas8,butfortheQ6-2epoch.Notethejumpinthe
modelwiththeparametersofthefirstcolumninTable12.Fromtopto Keplerphotometryatphase0.55(seetext).
bottom,filtersI ,R ,V,andKeplerband.Theground-basedobserva-
C C
tionsaredisplacedverticallytoshrinktheplot.Thelowerpanelsshow
theresidualsofthesefitsinthesameorder. Toavoidpotentialproblemsintroducedbythe 28minlow-
≃
cadence duty cycle of the Kepler light-curve, we performed a
firstfitusingonlytheIAC80lightcurvesandtheinitialparame-
tersderivedfromthedetrendedKeplerlightcurveinSect.5.2to
5.3.FinallightcurveanalysiswithPHOEBE
getafirstsetofparametersforthebinary(radius,effectivetem-
peratures,etc).Withtheseresults,werananewfit,thistimein-
To properly model the multiband light-curves of T-Cyg1-
cludingtheKeplerlightcurvetoproperlymodelthespots.This
12664, we used PHysics Of Eclipsing BinariEs (PHOEBE,
two-stepapproachhelpsthefittingofthesystembecause1)the
Prša&Zwitter 2005; Prša 2011). Given the low value of q for
IAC80VR I lightcurvesarebestsuitedtoconstrainingtheef-
thissystem,theRocheloberelativeradiiforeachstar(Eggleton C C
fectivetemperatures,radii,andinclinationofthesystembecause
1983)arer 0.43andr 0.33,wellabovethevaluesob-
RL1 ≃ RL2 ≃ of their high cadence during eclipses and finer spectral resolu-
tained for the relative radii from the preliminary fit with JK-
tion and 2) the Kepler lightcurve have a better coverage of the
TEBOP, so we selected a detached eclipsing binary model in
out-of-eclipsephases,whichallowsustomodelthespots.Inthe
PHOEBE.
secondfit,thephysicalandgeometricalparametersofthebinary
We fit the IAC80 VR I light curves simultaneously with arefixedandweonlyfitthespots.Weiteratethistwo-steppro-
C C
theKeplerlightcurve,butonlyusingepochsoftheKeplerlight cess untila satisfactorymodelis obtainedand variationsin the
curvecoevalwiththeground-basedobservations.Thisisanec- resultantparametersremainwithintheuncertaintiesforatleast
essarysteptoensurethatwearemodelingthesamespotconfig- threeconsecutivefits.Thisapproachhasthedisadvantageofbe-
uration.AlmostallIAC80observationsarecoevalwithKepler’s ing slower than the simultaneous fit of all the light curves but
Q6. We identified three epochs in which a primary and a con- ourpreliminarytestshowedthatthefittingresultswerenotably
tiguoussecondaryeclipseswereobservedsimultaneouslyinall better from the residuals point of view. The main advantage is
fourbands.Oneepochwasdiscardedowingtounidentifiedsys- thatweretainboththephysicalinformationthatarecontainedin
tematicsintheIAC80data,whichaffecttheleveloftheprimary theground-basedlightcurvesandtheexcellenttimecoverageof
eclipse.TheothertwowereselectedforanalysiswithPHOEBE. thespotconfigurationcontainedintheKeplerlightcurves.
Articlenumber,page9of18
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Table12.T-Cyg1-12664parameterscomputedfortwodifferentepochs.
Parameter Q6-1epoch Q6-2epoch
Timespan 2455376.51579- 2455380.70480-
2455380.64350 2455384.83250
Geometricandorbitalparameters
P(d)(fixed) 4.1287955(7)
T0(BJD)(fixed) 2455415.61831(5)
∆φ -0.000800 0.000024 -0.000700 0.000023
± ±
q(fixed) 0.553 0.018
±
γ(kms−1)(fixed) -6.28±0.53
i(deg) 87.003 0.012 86.931 0.010
± ±
e(fixed) 0.0365 0.0014
±
a(R )(fixed) 11.03 0.21
⊙ ±
ω(deg)(fixed) 92.8 2.2
±
Ω1 14.391±0.057 14.257±0.038
Ω2 19.461±0.072 18.937±0.061
F1(fixed) 1.041±0.024
F2(fixed) 1.000
Fractionalvolumetricradii
r1vol 0.07240±0.00060 0.07311±0.00040
r2vol 0.03039±0.00024 0.03129±0.00022
Radiativeparameters
Teff1(K)(fixed) 5560±160
Teff2(K) 3540±160 3380±160
Albedo(fixed) 0.5
β1,β2 0.4353,0.3710 0.4353,0.3837
lKp(fixed) 0.02297 0.00058
3 ±
Limb-darkeningcoeficients(square-rootlaw)
x1,y1(bol) 0.3811,0.3636 0.3811,0.3636
x2,y2(bol) -0.0427,0.8284 -0.0427,0.8284
x1,y1(KPband) 0.3598,0.3869 0.3598,0.3869
x2,y2(KPband) 0.0928,0.8096 0.06280.8721
x1,y1(Vband) 0.4786,0.3242 0.4786,0.3242
x2,y2(Vband) 0.1759,0.7896 0.1544,0.8349
x1,y1(RCband) 0.3177,0.4299 0.3177,0.4299
x2,y2(RCband) 0.1592,0.7278 0.1061,0.8169
x1,y1(ICband) 0.1795,0.4965 0.1795,0.4965 Fig.10.RepresentationofthespotconfigurationsofT-Cyg1-12664in
x2,y2(ICband) -0.1726,1.0385 -0.1848,1.0833 the(v,w)plane for thetwoepochs analyzed: thelower plot isfor the
Spot1parameters(primarystar) Q6-1epochandtheupperplotisfortheQ6-2epoch,whichisdisplaced
Colatitude(deg) 81.3±7.4 19.51±0.74 byanoffsetof0.2inthewaxis.Bothaxeshaveunitsofrelativeradius.
Longitude(deg) 18.2 1.5 20.9 1.8
± ± Thestarsarerepresentedatorbitalphase0.020,justaftertheprimary
Radius(deg) 19.97 0.40 29.39 0.45
± ± eclipse,andthesecondarystarisorbitingtowardstherightside.
Tspot/Tsurf 0.98141±0.00087 0.9686±0.0014
Spot2parameters(primarystar)
Colatitude(deg) 34.9 4.1 93 72
± ± chronicityparameterfortheprimaryF wassetto1.041 0.024,
Longitude(deg) 286.31 7.3 291.8 9.0 1
Radius(deg) 14.5 0±.66 8.1 ±1.5 asobtainedinSect.4.4.Althoughthisisaverysmallva±lueand
± ±
Tspot/Tsurf 0.9801±0.0047 0.9862±0.0037 it would not have practical effects on the shape of the star, we
ParameterscomputedfromMCMC included it for completeness. For the secondary star, the syn-
i∆(φdeg) -08.070.000297++−000...0002005002284 -08.60.09022961+−+000...00000000932107 chronicityparameterF2wasassumedtobe1.0.
ΩΩ12 1148..4987+−+−00000.....46230808830 1148..3604+−+−00000.....744300252068 agreTeemff1enwtawsisthetthtoe5sp5e6c0t±ro1s6c0opKic, avsalcuoemapnudtethdeincoSloercst.d4e.r2iv,eind
Teff2(K) 3548−[phone]−+4552 from the absolute photometry. The albedos of the two com-
Fractionalvolumetr−icradiifromMCMC − ponents were set to A = A = 0.5 as given by Rucin´ski
1 2
r1vol 0.0719+00..00002342 0.0729+00..00002317 (1969), since the temperatures and spectral types are compat-
r2vol 0.0312+−00..00001140 0.0318+−00..00001116 ible with those of stars with convective envelopes. The values
Derivedphysical−radiifromMCMC −
of the gravity-brightening exponents, β and β , and those of
RR21((RR⊙⊙)) 00..374943+−+−0000....000011337308 00..385014+−+−0000....000011244984 (th2e01L1D)focroeeffiacchiepnatsssbwaenrde.iAnsteirnpothlaetecdasfe1roomfthCela2JrKetT&EBBOloPempreen-
liminary light-curve analysis we used a square-root LD law
(Diaz-Cordoves&Gimenez 1992). For the primary star these
Giventhe largenumberof parametersin a completebinary coefficient values were kept fixed since T does not change
eff1
systemmodel,wetriedtofixasmanyofthemaspossiblebefore- throughout the fitting process. For the secondary, we updated
hand,usingallavailableinformation.WefixedtheperiodandT the coefficientsvalues whenevernecessary.As in Sect. 5.2, we
0
tothevaluesintheephemerisequationinSect.4.6.Themassra- used the same set of PHOENIX-based LD coefficients. This is
tioqandγwerefixedfromtheRVsolutioninTable11.Theval- because, during the fitting process, the T evolves below the
eff2
uesforeandωwerefixedfromthepreliminaryJKTEBOPanal- 3500KlimitoftheKuruczLDtables.Thisoccurredforallfit-
ysis of the Kepler detrended light-curve in Sect. 2.1. The syn- tingattemptsusingthePhoebe(2010)orvanHamme(1993)LD
Articlenumber,page10of18