EPIC 220204960: A Quadruple Star System Containing Two Strongly Interacting Eclipsing Binaries

6.8 MB·

by  S. Rappaport

MonthlyNoticesoftheRoyalAstronomicalSociety,ACCEPTED2016JANUARY17
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
EPIC220204960: AQUADRUPLESTARSYSTEMCONTAININGTWOSTRONGLYINTERACTINGECLIPSING
BINARIES
S.RAPPAPORT1,A.VANDERBURG2,T.BORKOVITS3,B.KALOMENI1,4,J.P.HALPERN5,H.NGO6,G.N.MACE7,B.J.FULTON8,
A.W.HOWARD9,H.ISAACSON10,E.A.PETIGURA11,D.MAWET9,M.H.KRISTIANSEN12,13,T.L.JACOBS14,D.LACOURSE15,
A.BIERYLA2,E.FORGÁCS-DAJKA16,L.NELSON17
MonthlyNoticesoftheRoyalAstronomicalSociety,Accepted2016January17
ABSTRACT
WepresentastronglyinteractingquadruplesystemassociatedwiththeK2targetEPIC220204960. TheK2
7 targetitselfisaK =12.7magnitudestaratT (cid:39)6100Kwhichwedesignateas“B-N”(bluenortherlyimage).
p eff
1
The host of the quadruple system, however, is a K (cid:39)17 magnitude star with a composite M-star spectrum,
0 which we designate as “R-S” (red southerly image)p. With a 3.2(cid:48)(cid:48) separation and similar radial velocities and
2
photometric distances, ‘B-N’ is likely physically associated with ‘R-S’, making this a quintuple system, but

n that is incidental to our main claim of a strongly interacting quadruple system in ‘R-S’. The two binaries in
a ‘R-S’ have orbital periods of 13.27 d and 14.41 d, respectively, and each has an inclination angle of (cid:38)89◦.
J From our analysis of radial velocity measurements, and of the photometric lightcurve, we conclude that all

9 four stars are very similar with masses close to 0.4M . Both of the binaries exhibit significant ETVs where
(cid:12)
1 thoseoftheprimaryandsecondaryeclipses‘diverge’by0.05daysoverthecourseofthe80-dayobservations.
Viaasystematicsetofnumericalsimulationsofquadruplesystemsconsistingoftwointeractingbinaries,we

] concludethattheouterorbitalperiodisverylikelytobebetween300and500days.Ifsufficienttimeisdevoted
R
toRVstudiesofthisfainttarget,theouterorbitshouldbemeasurablewithinayear.
S
Subject headings: stars: binaries (including multiple): close—stars: binaries: eclipsing—stars: binaries:
.
h general—stars: binaries: visual
p
-
o 1. INTRODUCTION term (i.e., (cid:46) few years) perturbative dynamical interactions
r amongtheconstituentstars;and(iii)enableustolearnmore
t Higher-ordermultiplestarsystemsareinterestingtostudy
s about longer-term dynamical interactions that can actually
for several reasons. Such systems (i) provide insights into
a
alter the configuration of the system (e.g., via Kozai-Lidov
[ star-formation processes; (ii) allow for a study of short-
cycles; Kozai 1962; Lidov 1962). These multi-component

1 stellar systems can be discovered, studied, and tracked via a
1DepartmentofPhysics,andKavliInstituteforAstrophysicsandSpace
v wide variety of techniques including historical photographic
Research,MassachusettsInstituteofTechnology,Cambridge,MA02139,
1 USA,[email protected] plates (e.g., Frieboes-Conde & Herczeg 1973; Borkovits &
8 2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Hegedüs 1996), searches for common proper motion stellar
2 Cambridge,MA02138USA;[email protected] systems (e.g., Raghavan et al. 2012); ground-based photo-
5 3BajaAstronomicalObservatoryofSzegedUniversity, H-6500Baja, metric monitoring programs searching for gravitational mi-
Szegediút,Kt.766,Hungary;[email protected]
0 4Department of Astronomy and Space Sciences, Ege University, crolensingevents(MACHO;e.g.,Alcocketal.2000;OGLE;
1. 35100,˙Izmir,Turkey;[email protected] e.g., Pietrukowiczetal.2013)orplanettransits(e.g., Super-
0 5Department of Astronomy, Columbia University, New York, NY; WASP,Lohretal.2015a; HATNet,Bakosetal.2002; KELT,
7 [email protected] Pepperetal.2007),high-resolutionimagingorinterferomet-
1 6CaliforniaInstituteofTechnology,DivisionofGeologicalandPlane- ric studies (e.g., Tokovinin 2014a, 2014b), and spectroscopy
tarySciences,1200ECaliforniaBlvdMC150-21,Pasadena,CA91125,
: aimedatmeasuringradialvelocities(Tokovinin2014a).
v USA;[email protected]
i 7McDonald Observatory and the Department of Astronomy, Perhapsthequickestpathwaytodiscoveringclosemultiple
X The University of Texas at Austin, Austin, TX 78712, USA; interacting star systems is via the study of eclipsing binaries
[email protected] whose eclipse timing variations (‘ETVs’) indicate the pres-
r
a 8Institute for Astronomy, University of Hawai’i, 2680 Woodlawn ence of a relatively nearby third body or perhaps even an-
Drive,Honolulu,HI96822,USA;[email protected]
other binary. In a series of papers based on precision Ke-
9Astronomy Department, California Institute of Technology, MC
249-17,1200E.CaliforniaBlvd.,Pasadena,CA91125,USA pler photometry (see, e.g., Borucki et al. 2010; Batalha et
10Department of Astronomy, University of California at Berkeley, al.2011),some220triple-starcandidateswerefoundviatheir
Berkeley,CA,94720-3411,USA ETVs(Rappaportetal.2013;Conroyetal.2014;Borkovitset
11Hubble Fellow, Astronomy Department, California Institute of al.2015;Borkovitsetal.2016). SeveraloftheKeplerbinary
Technology,Pasadena,California,USA
systemsturnedouttobemembersofquadruplesystemscon-
12DTU Space, National Space Institute, Technical University of
Denmark,Elektrovej327,DK-2800Lyngby,Denmark sisting of two gravitationally bound binaries (KIC 4247791:
13Brorfelde Observatory, Observator Gyldenkernes Vej 7, DK-4340 Lehmann et al. 2012; KIC 7177553: Lehmann et al. 2016:
Tølløse,Denmark andquintupleEPIC212651213: Rappaportetal.2016). One
1412812SE69thPlaceBellevue,WA98006 oftheKeplersystems,KIC4150611/HD181469,isarranged
15750752ndPlaceNEMarysville,WA98270
16Astronomical Department, Eötvös University, H-1118 Budapest, asatriplesystemboundtotwootherbinaries(Shibahashi&
PázmányPéterstny.1/A,Hungary Kurtz2012,andreferencestherein;Prsaetal.2016).
17Department of Physics and Astronomy, Bishop’s University, 2600 Otherinterestingquadruplestarsystemsinclude: 1SWASP
CollegeSt.,Sherbrooke,QCJ1M1Z7
2 Rappaportetal.2016
J093010.78+533859.5 (Lohr et al. 2015b); the young B-star features. In addition, some of us (MHK, DL, and TLJ) vi-
quintuple HD 27638 (Torres 2006); HD 155448 (Schütz et suallyinspectedalltheK2lightcurvesforunusualstellaror
al.2011); 14Aurigae(Barstowetal.2001); σ2 CoronaeBo- planetarysystems.
realis(Raghavanetal.2009);GGTau(DiFolcoetal.2014); Within a few days after the release of the Field 8 data set,
andHIP28790/28764andHIP64478(Tokovinin2016). EPIC220204960wasidentifiedasapotentialquadruplestar
Perhapsthetwoquadruplesinabinary-binaryconfiguration system by both visual inspection and via the BLS algorith-
(i.e., ‘2+2’) with the shortest known outer periods are V994 mic search. After identifying four sets of eclipses in the K2
Her (1062 days; Zasche & Uhlaˇr 2016) and VW LMi (355 lightcurve,were-processedthelightcurvebysimultaneously
days; Pribulla et al. 2008). ξ-Tau (145 days; Nemravova et fitting for long-term variability, K2 roll-dependent systemat-
al.2016)isaquadrupleina‘2+1+1’configurationwhichputs ics, and the four eclipse shapes in the light curves using the
it in a somewhat different category. The scale of dynamical methoddescribedinVanderburgetal. (2016). Fortherestof
perturbationsofonebinarybytheothercanbecharacterized the analysis, we use this re-processed light curve and divide
bytheparameter: P2 /P ,whereP andP arethebinary awaythebest-fitlong-termvariability,sinceitwasdominated
bin out bin out
andouterperiod,respectively. Thevaluesofthisquantityare byaninstrumentaltrend.
0.004dand0.18dforV994HerandVWLMi,respectively. The basic lightcurve is shown in Fig. 1, where three fea-
The value of this parameter for ξ-Tau, where the binary is tures are obvious by inspection. (1) All four eclipses of the
largelyperturbedbyasinglestar,is0.35d. two binaries have very similar depths, though the secondary
InthisworkwereportthediscoverywithK2ofastrongly eclipseintheAbinaryhasabout3/4thedepthoftheprimary.
interacting quadruple system consisting of two eclipsing bi- (2)Theperiodsofthetwobinariesarequitecomparablewith
naries, with orbital periods of 13.27 d and 14.41 d and all PA=13.27dandPB=14.41d. (3)Theeclipsedepthsarere-
fourMstarshavingverysimilarproperties. Bothbinariesex- markablyshallowat∼0.4%. Weratherquicklyinferredthat
hibitstrongETVsfromwhichweinferanouterperiodof∼a thecoincidenceofthesimilarsetsofextraordinarilyshallow
yearthat,inturn,impliesP2 /P ≈0.54(P /yr)−1 d. Such eclipses indicates a dilution effect from a neighboring star,
bin out out rather than two precisely inclined orbits that happen to pro-
a substantial value of this parameter could turn out to be the
duce such tiny eclipse depths. Quantitatively, we note that
largestamongtheknownsampleofquadruples.
foreclipsingbinarieswithtwosimilarstarstheaprioriprob-
This work is organized as follows. In Sect. 2 we describe
ability of an undiluted eclipse of 0.4% is only ∼0.02. The
the 80-day K2 observation of EPIC 220204960 with its two
probabilityofthisoccurringbychanceintworelatedbinaries
physically associated eclipsing binaries. Our ground-based
observations of the two stellar images associated with this isonly5×10−4.
target are presented in Sect. 3. These include classification The primary and secondary eclipses in both binaries are
spectra and Keck AO imaging. In Sect. 4 we discuss the six close to being equally spaced, but are measurably different
radial-velocityspectrathatwewereabletoobtain,andthere- from being equal. We define the fractional separations be-
sultant binary orbital solutions. The discovery of significant tweeneclipsesas,∆ts,p/Porb=(tsec−tpri)/Porb,wheretsec and
andsubstantialETVsintheeclipsesofbothbinariesarepre- tpri are times of sequential secondary and primary eclipses,
sentedinSect.5.Weuseaphysically-basedmodeltoevaluate and tsec >tpri. The fitted fractional separations between the
theeclipsingbinarylightcurvesinSect.6,andtherebydeter- two eclipses are: 0.4633±0.0001 and 0.4797±0.0001, for
minemanyofthesystemparametersofthebinariesnotavail- the A and B binaries, respectively. We can then utilize the
able from the radial velocities, as well as independent mass approximateexpression(goodto2ndorderineccentricitye):
determinations. In Sect. 7 we re-introduce a method for si-
(cid:20) (cid:21)
π ∆t 1
multaneouslymodelingthetwoeclipsingbinarylightcurves, ecosω(cid:39) s,p − (1)
and the results are compared with those derived in Sect. 6. 2 P 2
orb
InSect.8wesimulatevianumericalintegrationsthedynami-
calinteractionsofthefourstarsinthequadruplesystem,and whereω istheargumentofperiastronoftheprimarycompo-
set substantial constraints on the outer period of the two bi- nent(derivedfromaTaylorseriesexpansionofEqn.14;from
naries orbiting each other. The dynamical perturbations of Sterne 1939), to say that ecosωA (cid:39)−0.0577 and ecosωB (cid:39)
each binary on the other are assessed analytically in Sect. 9. −0.0319,fortheAandBbinaries,respectively.
Wesummarizeourresultsanddrawsomefinalconclusionsin Wecanalsoutilizeinformationfromtherelativewidthsof
Sect.10. thetwoeclipses,w1 andw2,tofindameasureofesinω. For
smalleandarbitraryω:
(1−w /w )
esinω(cid:39) pri sec (2)
2. K2OBSERVATIONS (1+w /w )
pri sec
As part of our ongoing search for eclipsing binaries, we
downloadedallavailableK2ExtractedLightcurvescommon (see, e.g., Kopal 1959, Chapt.VI). From the K2 photom-
toCampaign8fromtheMAST18.WeutilizedboththeAmes etry, we determine that wA,pri/wA,sec = 1.13±0.05, and
pipelineddatasetandthatofVanderburg&Johnson(2014). wB,pri/wB,sec =1.09±0.04. Therefore, eAsinωA =−0.061±
Thefluxdatafromall24,000targetsweresearchedforperiod- 0.023 and eBsinωB = −0.042±0.020. Thus, based on the
icitiesviaFouriertransformsandtheBLSalgorithm(Kovács limits obtained from Eqns. (1) and (2) we can constrain the
etal.2002). Thefoldedlightcurvesoftargetswithsignificant orbitaleccentricitiesandargumentsofperiastronoftheAand
peaks in their FFTs or BLS transforms were then examined Bbinariestobe
byeyetolookforunusualobjectsamongthosewithperiodic
0.058(cid:46)e (cid:46)0.10 and 0.032(cid:46)e (cid:46)0.07
A B
18 http://archive.stsci.edu/k2/data_search/search.
php ω (cid:39)230+10deg and ω (cid:39)240+10deg
A −30 B −40
InteractingQuadrupleStarSystem 3
FIG.1.—K2fluxdataforEPIC220204960.Theeclipsesofthe13.27-day
‘A’binaryarecoloredinblue,whilethoseofthe14.41-day‘B’binaryare
inred. Allfoureclipsesareofcomparablyshallowdepth. Wenotethatthis
lightcurvecontainsthelightofthebrightnortherlybluestellarimagedesig-
nated‘B-N’(seeFig.2). At∼3(cid:48)(cid:48)separationfromthe‘A’and‘B’binaries,
thefluxesarenotseparablewithK2.
TABLE1
PROPERTIESOFTHEEPIC220204960SYSTEM FIG.2.—SDSSimageshowingtheregionnearEPIC220204960.Wehave
designatedthebrighterbluishcoloredimagetothenorthas‘B-N’whilethe
Parameter 220204960‘B-N’ 220204960‘R-S’ fainterreddishimagesome3(cid:48)(cid:48)tothesouthisdesignatedas‘R-S’.The‘R-S’
RA(J2000) 00:48:32.65 00:48:32.67 imagehostsbothbinariesinaboundquadruplesystem.
Dec(J2000) 00:10:18.59 00:10:15.20
Kp 12.66 ...
ua 15.08 24.64 The SDSS image of EPIC 220204960 is shown in Fig. 2.
Bb 13.31 ... Thebrighterbluishimagetothenorth(hereafter‘B-N’)dom-
ga 13.02 18.01 inatesthelight,butnotethefainterreddishimagesome3(cid:48)(cid:48) to
Gb 12.58 16.82 thesouth(hereafter‘R-S’).Wesummarizetheavailableprop-
Vb 12.76 ... ertiesofthesetwostarsinTable1.
Rb 12.63 ... ThroughtheKeplerbandpass,the‘R-S’imagerangesfrom
ra 12.71 16.44
between 2.8 and 5 magnitudes fainter than the ‘B-N’ image.
za 13.37 15.51
Whenwecarefullyintegratethesemagnitudes,aswellasour
ib 12.54 ...
detailed spectra (see Sect. 3.2), more quantitatively over the
Jc 11.75 14.2
Keplerbandpass, wefindafluxratioof45±10(90%confi-
Hc 11.54 ...
Kc 11.44 13.4 dence)betweenthe‘B-N’and‘R-S’images.Aswewillshow,
W1d 11.28 ... thisdifferenceissufficienttoexplaintheextremedilutionof
W2d 11.30 ... theeclipsesprovidedthatbothbinariesarehostedwithinthe
W3d 11.40 ... ‘R-S’image.
W4d ... ...
Distance(pc)e 560±150 600±150 3.2. MDMSpectra
µα(mas yr−1)f −0.1±1.3 ... On 2016 August 31 UT, two 1500-s spectra of EPIC
µδ(mas yr−1)f −8.5±1.4 ... 220204960 were obtained with the Ohio State Multi-Object
Spectrograph(OSMOS)onthe2.4mHiltnertelescopeofthe
Notes.(a)TakenfromtheSDSSimage(Ahnetal.2012).(b)FromVizieR
http://vizier.u-strasbg.fr/;UCAC4(Zachariasetal.2013). MDMObservatoryonKittPeak,Arizona. Inlong-slitmode,
(c)2MASScatalog(Skrutskieetal.2006).(d)WISEpointsourcecatalog a1.(cid:48)(cid:48)2slitwasalignedwiththetwostellarimagesforthefirst
(Cutrietal.2013).(e)Basedonphotometricparallaxonly.Thisutilized exposure.Thesecondexposurehadtheslitorientedeast-west
adaptedVmagnitudesof12.76and17.1forthetwostellarimages,the
throughimage‘R-S’.Avolumephaseholographicgrismpro-
bolometricluminositiesforthefourMstarsgiveninTable5,thebolometric
magnitudeofthe‘B-N’imageinferredfromTable2,andappropriate vided a dispersion of 0.72 Å pixel−1 and a resolution of 2.9
bolometriccorrectionsfortheMstarsinquestion.(f)FromUCAC4
Å on a Silicon Technology Associates STA-0500 CCD with
(Zachariasetal.2013);Smart&Nicastro(2014);Huberetal.(2015).
4064×406415µpixels. Thewavelengthcoverageis3967–
Thus,notonlyarethebinariesverysimilarinotherrespects,
6876 Å. The dispersion solution was derived from 28 com-
theybothhavesmall,butdistinctlynon-zeroeccentricities.
We return to a more detailed quantitative analysis of the parisonlinesofHgandNe,yieldingrmsresidualsof0.02Å,
lightcurvesofthetwobinariesinSections5,6,and7. although a systematic error of up to 0.4 Å could be present
duetoinstrumentflexure.
Thespectraforboththe‘B-N’imageand‘R-S’imageare
3. GROUNDBASEDOBSERVATIONS shown in Fig. 3. The east-west slit was used here to extract
the spectrum of ‘R-S,’ as it had less contamination from ‘B-
3.1. SDSSImage
N’. There is no detectable leakage of the spectrum of ‘B-N’
4 Rappaportetal.2016
into‘R-S’,astheprominentBalmerabsorptionlinesin‘B-N’
are absent in ‘R-S.’ Although the narrow slit and sky condi-
tions were not conducive to absolute spectrophotometry, the
standard star HD 19445 was used for flux calibration. The
equivalentslitmagnitudeof‘B-N’isV ≈12.6,inreasonable
agreementwiththevalueinTable1(V =12.76).
It is clear that the spectrum of ‘R-S’ is that of an early M
star. ExaminingthePickles(1998)atlasofstellarspectra,we
findabestmatchwithanM2.5Vtype. Although,itisworth
notingthatthisisactuallyacompositespectrumoffour,very
likelysimilar,stars. Bycontrast,the‘B-N’imageisthatofa
G2Vstar.
3.3. SpectralClassificationofthe‘B-N’ImagefromTRES
Spectrum
We observed the blue northern component of EPIC
220204960 with the Tillinghast Reflector Echelle Spectro-
graph (TRES) on the 1.5 meter telescope on Mt. Hopkins,
AZ.1500-sand2000-sexposuresweretakenon2016July13
UT and 2016 Oct. 24 UT, respectively. These yielded spec-
tra with signal-to-noise ratio of ∼30 per resolution element
at 520 nm, and a spectral resolving power of R = 44,000.
We reduced the spectra following Buchhave et al. (2010).
A portion of one spectrum is shown in Fig. 4. We mea-
suredanabsoluteradialvelocityforthe‘B-N’imageofEPIC
220204960bycross-correlatingtheobservedTRESspectrum
against a suite of synthetic model spectra based on Kurucz
(1992) model atmospheres. The velocities for the two mea-
surements were −4.505 and −4.516 km s−1, consistent with
no change at 11±50 m s−1. These have been corrected for
thegravitationalblueshifttothebarycenter. Theyalsohavea
residual,systematic,error(incommon)of100ms−1. FIG.3.—MDM2.4-mspectraofthe‘R-S’image(toppanel)and‘B-N’star
(bottompanel).Thespectrahavebeencorrectedforthethroughputefficiency
We measured the stellar parameters of the ‘B-N’ image asafunctionofwavelength.Thereferencefluxdensity,F0,is10−14ergscm−2
using the Stellar Parameter Classification code (SPC, Buch- s−1Å−1.Theratioofdetectedfluxinthetwospectrais∼100.Thisimpliesa
have et al. 2010, 2012). SPC cross correlates an observed ratioof∼60intheKeplerbandpassaftercorrectingfortheredfluxbetween
spectrum against a grid of synthetic spectra based on Ku- 6800Å and8500Å thatisnotincludedinthespectrum.
ruczatmosphericmodels(Kurucz1992).Theanalysisyielded
T =6085±72K,logg=4.23±0.02,[m/H]=0.16±0.13,
eff
andvsini=7.6±0.2kms−1(seeTable2). ∼45intheKeplerband,isanindicationofhowredthe‘R-S’
imageis.
3.4. AdaptiveOpticsImaging Theevidencepresentedinthenextsectionshowsthatboth
binaries are actually hosted by the ‘R-S’ image. In the bot-
Weobtainednaturalguidestarobservationsofboththe‘B-
tompanelofFig.5,weshowazoomed-inimageofthe‘R-S’
N’ and ‘R-S’ components of EPIC 220204960 on 2016 July
component. Thisblown-upimagelooksdistinctlysingle,and
19UTtobettercharacterizethisquadruplesystem. Weused
showsnosignofthecoreevenbeingelongated. Wehavecar-
thenarrowcamerasetting(10milliarcseconds,mas,perpixel)
ried out simulations of close pairs of comparably bright im-
of the NIRC2 camera (PI: Keith Matthews) on Keck II. We
ages, at a range of spacings, and we conclude from this that
useddomeflatfieldsanddarkframestocalibratetheimages
separationsbetweenthetwobinariesof(cid:38)0.05(cid:48)(cid:48) canbecon-
andremoveartifacts.
servatively ruled out. At a source distance of some 600 pc,
Weacquired12framesofEPIC220204960ineachofthe
thissetsanupperlimitontheprojectedphysicalseparationof
J and K bands (central wavelengths of 1.250 µm and 2.145
s ∼30AU.
µm, respectively) for a total on-sky integration time of 240
seconds in each band. Figure 5 shows a stacked K band
s 4. AFEWRADIALVELOCITYMEASUREMENTS
image of both components of this target (cf. the SDSS im-
age in Fig. 2). The northerly ‘B-N’ image is separated by Because the ‘R-S’ image, which hosts all four M stars, is
3.(cid:48)(cid:48)359±0.(cid:48)(cid:48)002fromthesoutherlyredimage‘R-S’ataposi- relatively faint, we have been able to obtain only six spectra
tionangleof174.60±0.03degreeseastofnorth. Photometry at five independent epochs of the quality required for radial
andK bandastrometrywerecomputedviaPSFfittingusing velocity measurements. Two were taken with the IRGINS
s
a combined Moffat and Gaussian PSF model following the spectrograph mounted on the Discovery Channel Telescope,
techniquesdescribedinNgoetal.(2015)andtheNIRC2dis- while four others were acquired with the HIRES spectrome-
tortionsolutionpresentedinServiceetal.(2016). The‘B-N’ teronKeck. Bycoincidence, thesecondofthetwoIGRINS
component is 2.43±0.03 magnitudes brighter than ‘R-S’ in spectrawastakenwithinthreehoursofthefirstoftheHIRES
the K band (2.50±0.01 magnitudes in J). The fact that the spectra,andthereforethesenearlysimultaneousspectraserve
s
‘B-N’/‘R-S’ flux ratio is only ∼10 in the NIR, compared to asaconsistencycheckbetweenthetwosetsofdata.
InteractingQuadrupleStarSystem 5
FIG.4.—200Å segmentoftheoverallTRESspectrumofEPIC220204960usedtocharacterizethe‘B-N’image. Dataareplottedinbluewhilethefitted
modelcurveisshowninred.TheresultsofthemodelfitaresummarizedinTable2.
We observed the red southern component of EPIC
TABLE2
220204960 with the High Resolution Echelle Spectrometer
PROPERTIESOFSTELLARIMAGE‘B-N’
(HIRES,Vogt1994)ontheKeckItelescopeonMaunaKea.
Parameter Value We used the standard California Planet Search observing
Teff[K]a 6085±72 setup with the red cross disperser and the C2 0.(cid:48)(cid:48)86 decker
logg[cgs]a 4.23±0.02 (Howard et al. 2010). We obtained 20-minute exposures on
M[M(cid:12)]b 1.20±0.07 2016Oct.10,Nov.21,andNov.26anda15-minuteexposure
R[R(cid:12)]b 1.35±0.18 on Nov. 5, yielding signal-to-noise ratios that were typically
L[L(cid:12)]b 2.3±0.7 between5and20perpixelbetween500and800nm.
γ[km/s]a −4.510±0.062
ThecrosscorrelationbetweenthefirstoftheHIRESspectra
vsini[km/s]a 7.6±0.2
[m/H]a 0.16±0.13 andthetemplatefromareferenceMstarisshowninFig.6.
FBN/FRSc 45±10 WeclearlydetectfoursignificantpeaksintheCCFwhichwe
Notes.(a)TakenfromtheanalysisoftwoTRESspectraacquiredon2016 identifyasbelongingtothefourMstarsinthequadruplestar
July13and2016Oct.24(seeSect.3.3).(b)DerivedfromTeffandlogg system.
usingtheYonsei–Yaletracks(Yietal.2001)foranassumedsolar
composition.(c)BasedontheMDMspectra(seeSect.3.2),andthe
4.3. RadialVelocities
magnitudesgiveninTable1.
4.1. IGRINSSpectra Wecross-correlatedthefourHIRESandtwoIGRINSspec-
tra of the red southern image of EPIC 220204960 with high
The Immersion Grating Infrared Spectrometer (IGRINS)
signal-to-noise template spectra of bright, nearby M-dwarfs.
employs a silicon immersion grating for broad spectral cov-
For HIRES, we used a spectrum of GL 694, while for
erageathigh-resolutioninthenear-infrared. Thedesignpro-
IGRINS, we used a spectrum of LHS 533. We placed the
videshighthroughputandanunprecedentedR≈45,000spec-
crosscorrelationfunctionsonanabsolutevelocityframeus-
trumofboththeHandKbands(1.45-2.5µm). IGRINSwas
ing the measured absolute RVs of these two template stars
initially commissioned on the 2.7m Harlan J. Smith Tele-
fromNideveretal. (2002).
scope at McDonald Observatory (Park et al. 2014; Mace et
WesummarizeinTable3allsixsetsofRVmeasurements
al. 2016) before being deployed to the Discovery Channel
taken at five independent epochs. In first discussing these
Telescope (DCT) in September 2016. The ‘R-S’ image was
measurements we refer to only five sets of measurements
observedonceduringIGRINScommissioningattheDCTon
since the first of the HIRES spectra is nearly simultaneous
UT2016Sept.19andagainduringregularscienceoperation
in time with the second of the IGRINS spectra. Thus, in all
onUT2016Oct.10.TheseobservationsweretakeninABBA
thereare5spectra×4CCFpeaksthateachmustbeidentified
nod sequences with 900s and 1200s exposure times. The
withaparticularstarinoneofthetwobinaries.Toaccomplish
spectra were optimally extracted using the IGRINS Pipeline
this,wechosetwopeaksfromeachCCFtorepresentthestars
Package (Lee & Gullikson 2016). Dome-flats were taken at
inbinaryA,withitsknownorbitalperiod,temporarilyignor-
the start of the night and wavelengths were determined us-
ing the other two peaks in the first pass. We then fit simple
ing sky lines. Telluric correction by A0V stars at similar
circularorbitsto(4×3)5/2=124,416distinctcombinations
air masses to EPIC 220204960 provide a flattened spectrum
ofchoicesofstarswithCCFpeaks19. OncetheCCF-peakto
with a signal-to-noise of 30-40 per resolution element. The
starassignmentshavebeenmadethatworkbestforbinaryA,
longerexposuretimesrequiredforthisfaintertargetresulted
thereareonly16independentcombinationsremainingtotry
inhigherOHresidualsinthespectrumfrom2016Oct.10.
19ThenamingconventioninthefirstCCFisamatterofdefinition,hence
4.2. HIRESSpectrum
the1/2factor.
6 Rappaportetal.2016
FIG.6.—Crosscorrelation(redcurve)betweentheHIRESspectrumand
the template from a reference M star. After subtracting off a pedestal of
broaderfeatures,thegreencurveshowsthefourpeaksmoreclearlythatcor-
respondtothefourMstarsinthequadruplestarsystem.Theinferredradial
velocities,whichrangefrom−50kms−1to+50kms−1,areaboutasexpected
nearquadratureforthetwobinaries.
nary’saccelerationinitsouterorbit. Inpractice,however,we
have found that the RV points are neither numerous enough
nor sufficiently accurate to derive values for ω or e that are
nearly as good as we are able to derive from the lightcurve
analysis(seeSects.2,6,and7). Toalesserextent,thesame
isalsotrueofτ.
WethereforerestrictedourfitsoftheRVdatapointstothe
four parameters: K , K , γ, and γ˙ while fixing ω, τ, and e
1 2
at the values given in Tables 5 and 6. The fits were carried
outwithanMCMCroutinethatisdescribedinmoredetailin
Sect.6. TheresultsofthefitsareshowninFig.7andTable
3. The plotted error bars in Fig. 7 are just the empirical rms
scatter of the data points about the model curve because we
have no other independent way of assessing them. Note the
linear trend (γ˙) for both binaries, but of opposite signs, in
Fig.7.
FIG.5.—Toppanel: Keck-AOimageinKs-bandofEPIC220204960,in- InadditiontotheKvelocitiesanduncertaintiesgiveninTa-
cludingthebrighterblueimagetothenorth,‘B-N’,andthefainterredim- ble 3, we also list the four constituent stellar masses that we
age3.(cid:48)(cid:48)4tothesouth,‘R-S’.Azoom-inonthe‘R-S’imagewhichhoststhe inferfromtheK-velocities. Allfourstarsseemquiteconsis-
quadruplesystem,isshowninthebottompanel.Ifthetwobinariesweresep-
aratedby0.(cid:48)(cid:48)1ormore,thecoreoftheimagewouldbecleanlysplitintotwo tentwith∼0.4M(cid:12)late-Korearly-Mstars. Welatercompare
objects. Aseparationofeven0.(cid:48)(cid:48)05wouldproduceanoticeablyelongated thesestellarmasseswiththosefoundfromouranalysisofthe
centralcore,whichisnotseen. photometric lightcurves. The results are in reasonably good
agreementandhavecomparableuncertainties.
Theγ-velocitiesofthetwobinariesarefoundtobe: γ (cid:39)
forbinaryB. A
+6kms−1andγ (cid:39)−14kms−1. Wecanusethesetwovalues
Eachbinaryfitutilizedfourfreeparameters: thetwostellar B
tocomputethe‘effective’γ oftheCMofthequadruplesys-
K-velocities, K andK , thebinary’sγ-velocity, andalinear
trend, γ˙, to rep1resent p2ossibly detectable acceleration of the temfromγquad(cid:39)(γA+γB)/2(cid:39)−4kms−1. Sincethisagrees
very well with the γ velocity of star ‘B-N’ (see Table 2) we
binary in its outer orbit. Only a few such combinations of
stellarIDandCCFpeakyieldeddecentχ2 valuesandphysi- take that as an indication that the two stellar images are part
ofaphysicallyboundgroupoffivestars. Finally,withregard
callysensibleresultsforthebinarybeingfitted,butwherethe
to the γ-velocities, we can also use them to estimate the or-
remaining(i.e.,unused)CCFpeakscouldbereasonablyfitto
bitalspeedofthetwobinariesaroundtheircommoncenterof
thestarsintheotherbinary. Weselectedonechoiceofstellar
mass. A rough estimate of the instantaneous projected (i.e.,
IDswithCCFpeaksthatyieldedthebestfitforbothbinaries.
radial) speed of each binary in its orbit can be found from
ThatparticularsetofRVsmatchedwithstellarcomponentsis
K (cid:39)(γ −γ )/2(cid:39)10kms−1
summarizedinTable3. quad A B
OncetheidentificationofCCFpeakswithindividualstars
hasbeenuniquelymade,therearethen10RVpointsthatare 5. ECLIPSETIMINGVARIATIONS
associatedwitheachbinary(seeTable3andFig.7). Inprin- Inordertoanalyzethelightcurves,wefirstfoldedthedata
ciple,weshouldthenfitthesecurveswith7freeparameters: foreachbinaryaboutthebest-determinedorbitalperiod. We
K , K , γ, γ˙, ω, τ, and e, where τ is the time of periastron quicklydiscovered,however,thatregardlessofwhatfoldpe-
1 2
passage and, again, γ˙ (assumed constant) represents the bi- riodweused,oneeclipseortheotherwasmisshapenorpar-
InteractingQuadrupleStarSystem 7
TABLE3
RESULTSFROMRADIALVELOCITYSTUDY
StarA-1 StarA-2 StarB-1 StarB-2
RadialVelocityMeasurementsa:
BJD-2450000 Spectr.
7650.7427 +46.7 −32.2 +18.4 −54.7 IGRINS
7671.8570 −29.3 +47.8 −52.6 +18.1 IGRINS
7671.9812 −27.1 +49.1 −50.6 +20.0 HIRES
7697.9627 −36.3 +49.6 +0.6 −23.3 HIRES
7713.8823 +6.4 −4.2 −25.7 +6.4 HIRES
7718.8820 +21.0 −27.0 −27.0 +16.4 HIRES
OrbitFitsb:
K[kms−1] 39.5±2.0 46.5±2.0 41.6±2.5 42.8±2.5
γc[kms−1] +6.0±0.8 −13.7±1.0
γ˙d[cms−2] −0.16±0.03 +0.15±0.04
γquade[kms−1] −3.8±1.3
Kquadf [kms−1] 9.9±1.3
ConstituentStellarMasses:
mass[M(cid:12)] 0.47±0.05 0.40±0.05 0.45±0.06 0.44±0.06
Notes.(a)Uncertaintiesaredifficulttoestimate.Empirically,wefoundthaterrorbarsontheindividualRVvaluesof∼3kms−1yieldedgoodχ2values.Fora
descriptionofhowweassignedspecificpeaksinthecross-correlationtospecificstarsseetext.(b)Theorbitfitsforeachbinaryinvolvedfourfreeparameters:
K1,K2,γ,andγ˙.Theorbitalperiod,eccentricity,andargumentandtimeofperiastronweretakenfromthelight-curveanalysis(seeTable5).(c)Centerofmass
velocityofeachbinary.(d)Accelerationofthecenterofmassofeachbinary.(e)γquadistheradialvelocityoftheCMoftheentirequadruplesystem.This
assumesthatmassesofthetwobinariesareapproximatelyequal.(f)Kquadistheprojectedradialspeedofeitherbinaryinitsorbitaroundthequadruplesystem.
Thisalsoassumesthatthemassesofthetwobinariesarethesame.
FIG.8.—Eclipsetimingvariationsinthearrivaltimesofallfoureclipses
inthetwobinariesthatcompriseEPIC220204960. Ineachcasethemean
orbital period for each binary has been used to produce the ETV curves.
NotethestrongdivergenceoftheETVcurvesfortheprimaryandsecondary
eclipsesofboththeAandBbinaries.SeeTable4forasummaryofperiods
andETVs.
tially filled in. This was true for both binaries. In order to
understand the cause, we then fit each of the 20 observed
eclipses(approximately5eachfortheprimaryandsecondary
eclipsesofbothbinaries),tofindaccuratearrivaltimes.
To find the arrival times we fit each eclipse with the fol-
lowingnon-physical,butsymmetricfunction(i.e.,hyperbolic
secant; Rappaport et al. 2014), that has a shape sufficiently
close to the eclipse profile, f(t), to allow for a precise mea-
surementoftheeclipsecenter:
FIG.7.—RadialvelocitymeasurementsfromtwoIGRINSandfourHIRES
spectra.ThesecondoftheIGRINSspectrahasnearlythesameepochasthe (cid:2) (cid:3)−1
f(t)(cid:39)B+2D exp[(t−t )/w]+exp[−(t−t )/w] (3)
HIRESspectrum. Toppanelisforthe13-dAbinaryandbottompanelfor 0 0
the14-dBbinary.ThesolidcurvesarethebestfitswithonlytheKvelocity
The four free parameters are: B, the out-of-eclipse back-
ofeachstar,theγvelocity(blackhorizontalline),andγ˙ asfreeparameters
foreachbinary,whileω,τ,andearetakenfromTable6. ground, D, the eclipse depth, t0 the time of the center of the
eclipse,andwacharacteristicwidthoftheeclipse.
Aftersubtractingofftheexpectedtimesofeclipseusingthe
8 Rappaportetal.2016
FIG.9.— K2eclipseprofilesfortheprimaryandsecondaryeclipsesinboththeAbinary(toppanels; Porb=13.27d)andtheBbinary(bottompanels;
Porb=14.41d).Eachprofilecontainsdatafrom∼5eclipses.Orbitalphasezeroisarbitrary,butcorrectlygivestherelativephasesoftheprimaryandsecondary.
Theabsolutetimesoftheprimaryeclipse(definedasthoseintheleftpanels)aregiveninTable4. Theredcurveisabestfittingmodelwhichincludes6
independentparametersforeachbinarysystem(seeSect.6).
meanorbitalperiodsofP =13.2735dandP =14.4158d,we
A B TABLE4
findtheeclipsetimingvariations(hereafter‘ETVs’)shownin ETVDIVERGENCESANDAPPARENTORBITALPERIODSa
Fig.8. WeweresurprisedtofindthattheETVcurvesforthe
primary and secondary eclipses, for both binaries, ‘diverge’ Parameter StarA-1 StarA-2 StarB-1 StarB-2
soclearlyandbysuchalargeamountoverthecourseofonly ETV[d] +0.024 −0.024 −0.024 +0.024
80 days. For both binaries, the divergence in the ETV times ETVslope[d/d] 0.00033(4) −0.00033(3) −0.00031(4) 0.00031(9)
amounts to plus and minus 0.025 days for the primary and ∆Porb[d] 0.0044 −0.0044 −0.0045 0.0045
secondary eclipses, respectively. In terms of slopes to the ApparentPorb[d] 13.26913 13.27789 14.41130 14.42024
Epochs[BJD]b 7401.864 7394.718 7403.021 7395.497
ETV curves, these correspond to plus and minus ∼0.00032
days/day for both binaries, where the plus and minus signs Notes.(a)Derivedfromthe20totaleclipsesofbinariesAandB.‘ETV’
referstothetotaleclipsetimingvariationsoverthe80-dayK2observations.
are for the primary and secondary eclipses. Finally, we can
∆Porbreferstothedifferencebetweenthemeanapparentorbitalperiodand
determineanapparent‘local’(intime)periodforeacheclipse. thatderivedindependentlyfortheprimaryandsecondaryeclipses.(b)The
Theseare:13.26913d,13.27789d,14.41130d,and14.42024 epochtimesareactuallyBJD–2450000.
d. Thesedelays,slopes,andapparentperiodsaresummarized
significantlyaffectourabilitytodeterminequantitiessuchas
inTable4.
eclipse spacing (related to ecosω) or the eclipse profiles. In
Finally, weusethesefourperiodstofoldthedata, onefor
fact, the meaning of the divergence in the ETVs is precisely
each eclipse, in order to produce the eclipse profiles that we
thefactthatωischangingbyasmall,butmeasurableamount
usetofitfortheorbitalparameters. Forthesefoldsweusean
overthecourseofthe80-dayobservationinterval.
epochnearthemidpointofthe80-dayK2observations. Be-
Theresultsoffoldingthedataaboutfourdifferentperiods
causetheprimaryeclipseprofileisproducedusingaslightly
leadstothefourprofilesshowninFig.9.Notehowsimilarall
different period from that of the secondary eclipse, the rela-
foureclipseslookintermsofwidth, shape, anddepth. Only
tivephasingbetweenthetwoeclipsesisonlywelldefinedat
theeclipsedepthforthesecondarystarinbinaryAispercep-
the center time of the K2 observations. However the phase
tibly more shallow than the other three. In spite of the fact
drifts of one eclipse with respect to the other over this time
that only a small portion of the lightcurve is shown around
period amount to only ∼0.0015 cycles, and thus they do not
each eclipse, the orbital phases of one eclipse with respect
InteractingQuadrupleStarSystem 9
TABLE5
PROPERTIESOFTHEQUADRUPLESTARS
Parameter BinaryA BinaryB
Porba[days] 13.2735±0.0044 14.4158±0.0045
semimajoraxisb[R(cid:12)] 22.8±1.3 23.7±0.8
inclinationb[deg] 89.5+0.4 89.7±0.3
−0.4
ecosωa 0.0577±0.0001 0.0318±0.0001
eb 0.061+0.017 0.033+0.007
−0.003 −0.002
ωb[deg] 208+20 192+32
−36 −26
tprimeclipsea[BJD] 2457401.864±0.003 2457403.021±0.003
3rd-lightfactorc 90+23 97+11
−25 −19
individualstars A1 A2 B1 B2
massb[M(cid:12)] 0.49−+00..0076 0.38+−00..0079 0.45+−00..0056 0.42+−00..0056
radiusb[R(cid:12)] 0.45−+00..0065 0.35+−00..0067 0.41+−00..0045 0.39−+00..0054
Teffb[K] 3600+−18100 3460+−7600 3540+−7555 3500+−6405
luminosityb[L(cid:12)] 0.031+−00..001130 0.016+−00..000087 0.023−+00..000078 0.020−+00..000066
loggb[cgs] 4.82+0.06 4.92+0.08 4.86+0.05 4.89+0.05
−0.05 −0.06 −0.04 −0.04
Notes.(a)BasedontheK2photometry.(b)DerivedfromananalysisoftheK2photometriclightcurve(seeSect.6)andtheuncertaintiesare90%confidence
limits.ThisanalysisutilizedtheanalyticfittingformulaeforR(m)andTeff(m)giveninequations(4)and(5).WhenwemodifytheR(m)relationslightlyto
accountforthesomewhatlargerradiimeasuredforanumberofstarsinclosebinaries(seeApp.AandFig.16),thecitedmasseswoulddecreaseby
∼0.03−0.04M(cid:12).(c)Fromphotometricmeasurementsofthe‘B-N’and‘R-S’fluxratio.
to the other, shown on the x axes are correct, at least for the inmass(i.e.,(cid:46)0.5M )thattheyaresubstantiallyunevolved
(cid:12)
mid-timeofthe80-dayobservation. atthecurrentepoch.Thisthenallowsustodetermineboththe
stellarradiusandluminosityfromthemass(andanassump-
6. PHYSICALLYBASEDFITSTOLIGHTCURVES
tion about the metallicity). These 6 parameters are adjusted
InthissectionwefitthelightcurvesshowninFig.9toex-
via a Markov Chain Monte Carlo (‘MCMC’) routine, which
tractasmanyofthesystemparametersascanbeconstrained
usestheMetropolis-Hastingsalgorithm(see,e.g.,Ford2005;
bytheeclipsedepths,shapes,andrelativephasing. Wedonot
Madhusudhan & Winn 2009, and references therein; Rappa-
attempttofittheout-of-eclipseregionsofthelightcurvesfor
portetal.2016)inordertofindthebestfittingvaluesandtheir
effects such as ellipsoidal light variations (‘ELVs’), Doppler
uncertainties.
boosting, or illumination effects (see, e.g., van Kerkwijk et
In somewhat more detail, each step in the MCMC proce-
al.2010;Carteretal.2011). Thereasonsforthisaretwofold.
duregoesasfollows.Wefirstchooseaprimaryandsecondary
First,withorbitalperiodsaslongas13-14days,sucheffects
massfromwithinauniformpriorrangingfrom0.2−0.7M .
are quite small, i.e., at the ∼ten parts per million level (by (cid:12)
Theinclinationangle,i,ischosenfromwithinauniformprior
comparisonwiththeeclipseswhicharetypically4000ppm), rangingfrom87◦to90◦,whiletheargumentofperiastron,ω,
andthesearefurtherseriouslydilutedbythelightfromthe‘B-
canrangeover0to2π. Thedilutionfactorforeitherbinaryis
N’image. Second,thefidelityoftheK2photometryatthese
chosenfromwithintherange60–120(equivalenttoa3rdlight
lowfrequencies,i.e.,ontimescalesof(cid:38)10daysisnottobe
of 0.987–0.992). Note that because there are two binaries
trusted at these low levels, and in any case they are largely
withinthe‘R-S’image,thisdilutionfactorisabouttwicethe
filteredoutintheprocessingofthedata.
ratiooffluxeswefindfor‘B-N’/‘R-S’.Finally,thetimeofpe-
Because of the very large dilution factor in these eclipses
riastronpassage,τ,ischosenoverasmallrangebasedonthe
(due to the presence of the ‘B-N’ image in the photometric
fact that for nearly circular orbits τ (cid:39)t +P (ω/2π−1/4)
aperture), theso-called“3rdlight”(‘L3’)parameterisinthe ecl orb
wheret istheeclipsetime.
range 0.985-0.992, as we detail below. In principle, binary ecl
Once the masses have been chosen, we compute the or-
lightcurveemulatorssuchasPhoebe(Prša&Zwitter2005)
bital separation from Kepler’s 3rd law using the known or-
canfitforthe3rdlightasafreeparameter. Inpractice,how-
bital period. The stellar radii and effective temperatures are
ever, we have found that when L3 is so large, and two bi-
calculatedfromanalyticfittingformulaeforlow-massmain-
nary lightcurves are combined photometrically, Phoebe is
sequence stars. Initially, we utilized the expressions of Tout
not able to find accurate values for either L3 or the remain-
etal.(1996)whichcovertheentiremainsequence(0.1-100
der of the binary parameters. Thus, we adopt a more physi-
M ), but later switched to our own relations derived more
(cid:12)
callymotivatedapproachtofittingthelightcurveswhichuses
explicitlyforstarsonthelowermainsequence. Welaterveri-
supplemental information to ensure that the L3 parameter is
fiedthatthetwosetsoffittingformulaeactuallyproducefairly
meaningful.
similarresults. OurfittingformulaeforR(m)andT (m),dis-
Theapproachweutilizetofittheeclipsesiscloselyrelated eff
cussedinAppendixA,areoftheform:
to the one presented by Rappaport et al. (2016) in the study
of the quadruple system in EPIC 212651213. However, it is 5
(cid:88)
sufficientlydifferentthatweoutlineourprocedurehere. log[R(m)]= c logn(m) (4)
n
Inbrief,thegoalistousetheinformationinthetwoeclipses
n=1
foreachbinary,includingtheirorbitalphaseseparation,tofit
for 6 free parameters: the two masses, the argument of peri- b m4.5+b m6+b m7+b m7.5
astron, the inclination angle, time of periastron passage, and Teff(m)= 1 1+b2m4.5+3b m6.54 K (5)
thirdlight.(Theeccentricityisfoundfromthechoiceofωand 5 6
thealreadydeterminedvalueofecosω –seeSect.2.) Wedo wheremisthemassinM ,RisinunitsofR ,andthecon-
(cid:12) (cid:12)
thisundertheassumptionthatallthestarsaresufficientlylow stantcoefficientsc andb aregivenintheAppendix.
n n
10 Rappaportetal.2016
Afterthisprocesshasbeenrepeatedmanytimes,theprob-
abilitydistributionsfortheparametersofbothstarsinthebi-
naryunderconsideration, aswellasthosefori, ω, andeare
evaluated. The best fitting values and their 1σ uncertainties
arelistedinTable5. Thebestfitstothelightcurvesofthetwo
binariesareshownindetailinFig.9.
Mostoftheparameteruncertainties,asdeterminedfromthe
MCMCanalysis,areunremarkable,andaregiveninTable5.
However,inthecaseofBinaryA,themassesofthetwostars
aresignificantlydifferent. InFig.10weshowthecorrelation
between M and M in both binaries. Note that the region
1 2
ofuncertaintyintheM −M planeforBinaryAliesentirely
1 2
below the M =M line. This is related to the fact that for
1 2
low-massstarsinthemassrange0.3−0.5M theT (M)re-
(cid:12) eff
lationisremarkablyflat(Baraffe&Chabrier1996;Baraffeet
al. 1998). Since the ratio of eclipse depths depends only on
the values of T for the two stars (assuming circular orbits
eff
or where ω (cid:39)0), in order to explain the 25% more shallow
eclipsedepthforstar2inBinaryA,thefittingcodeneedsto
considerablyreducethevaluesofM comparedtoM .
2 1
AratherclearpictureemergesoffourquitesimilarM-stars
intwoimpressivelyalikebinaries. Weweregratifiedthatso
much information could be extracted from the measurement
of only ∼10 eclipses for each binary. In particular, we find
impressiveagreementwiththemassesderivedindependently
fromtheRVmeasurements(Table3),andnotethattheuncer-
taintiesofbothdeterminationsareactuallyquitecomparable.
Finally, inregardtotheMCMCfits, wehaverunthecode
withthedilutionfactorasafreeparameterwithalargeprior
range of values (i.e., (cid:38)60) as well as with a narrow enough
range so as to force a match with the observed ratio of the
‘B-N’/‘R-S’ fluxes (i.e., dilution = 90±20). The extraction
ofthebasicphysicalresultsforthebinariesisaffectedonlyin
anincidentalway.
7. SIMULTANEOUSLIGHTCURVESOLUTION
In this section we present an approach to simultaneously
modelingthelightcurvesoftwointeractingbinarieswithina
single photometric aperture. As we shall see, this approach
isquitecomplementarytothephysically-basedlightcurveso-
lutions discussed in Sect. 6. For this purpose we modified
ourWilson-Devinney-andPhoebe-basedlightcurveemula-
tor (see e.g., Wilson & Devinney 1971; Wilson 1979; Wil-
froFmIGt.h1e0o.u—tpuCtoorfrethlaetiMonCpMloCtfibtettwoetheeneMcl1ipasnesd.M2 forbothbinariestaken son 2008; Prša & Zwitter 2005), Lightcurvefactory
(Borkovits et al. 2013), to solve both binary lightcurves si-
multaneously. Thepracticaldifficultyofsuchasimultaneous
The binary lightcurve is generated for two spherical stars analysis is that it requires at least twice the number of pa-
that are limb darkened with a quadratic limb-darkening law rameters to be adjusted (or even more) than in a traditional
usingcoefficientsappropriateforearlyMstarsandtakenfrom analysis of a single EB lightcurve (in this regard, see the
Claret&Bloemen(2011). Asdiscussedabove, noELVs, il- discussion of Cagaš & Pejcha 2012, which to our knowl-
lumination, or Doppler boosting effects were computed be- edge is the only prior paper which reports a simultaneous
causethewideorbitandthelowfrequencybehaviorofthese lightcurveanalysisoftwoblendedEBs). However, whenei-
features would not reveal such effects. The lightcurve was theroverlappingeclipsesarepresent,ortherearelargeout-of-
computedin2-minutesteps,andthenconvolvedwiththeKe- eclipsevariationsinthelightcurve(s)whichmakethesimple,
plerlongcadencetimeof29.4minutes. phase-folding-baseddisentanglement(see,e.g.,Rappaportet
AftertheMCMCparametershavebeenchosenforagiven al. 2016) impossible, a simultaneous analysis becomes in-
binaryrealization,weusethevalueofthedilutionfactorinthe evitablyimportant.Inourcurrentsituationthisisnotthecase.
currentMCMCsteptoscalethemodellightcurveaccordingly. Aswasillustratedintheprevioussections,thelightcurvesof
The model lightcurves are then compared to the observed the two EBs can, by chance, be nicely separated. On the
lightcurves with χ2 as the quantitative measure of agree- other hand, an important coupling remains between the two
ment. The Metropolis-Hastings jump conditions (see, e.g., lightcurveseveninthiscase,namelythefluxratioofthetwo
Ford2005)arethenusedtodecidewhetheragivenstepwill EBs. If the two EB-lightcurves are solved separately for the
be accepted or not. If the step is accepted, then that set of two systems itwould mean that thevalue of the ‘third-light’
parametersisstoredaspartoftheparameterdistributions. parameterineachsolutionwoulddependontheresultsofthe