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Mon.Not.R.Astron.Soc.000,1–??(2014) Printed10June2015 (MNLATEXstylefilev2.2) Circumbinary planets - why they are so likely to transit David. V. Martin1(cid:63), Amaury. H.M.J. Triaud2,3,4 1ObservatoiredeGene`ve,Universite´deGene`ve,51chemindesMaillettes,Sauverny1290,Switzerland 2DepartmentofPhysics,andKavliInstituteforAstrophysicsandSpaceResearch,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA 5 3CentreforPlanetarySciences,UniversityofToronto,DepartmentofEnvironmentalandPhysicalSciences,1265MilitaryTrail,Toronto,OntarioM1C1A4, 1 Canada 0 4FellowoftheSwissNationalScienceFoundation 2 n u Accepted.Received J 9 ABSTRACT ] P Transitsonsinglestarsarerare.Theprobabilityrarelyexceedsafewpercent.Further- E more, thisprobability rapidly approaches zeroat increasing orbital period.Therefore transit . surveyshavebeenpredominantlylimitedtotheinnerpartsofexoplanetarysystems.Herewe h demonstratehowcircumbinaryplanetsallowustobeattheseunfavourableodds.Byincorpo- p rating the geometry and the three-body dynamics of circumbinary systems, we analytically - o derive the probability of transitability, a configuration where the binary and planet orbits r overlap on the sky. We later show that this is equivalent to the transit probability, but at an t s unspecified point in time. This probability, at its minimum, is always higher than for single a star cases. In addition, it is an increasing function with mutual inclination. By applying our [ analyticaldevelopmenttoeclipsingbinaries,wededucethattransitsarehighlyprobable,and 2 in some case guaranteed. For example, a circumbinary planet revolving at 1 AU around a v 0.3 AU eclipsing binary is certain to eventually transit - a 100% probability - if its mutual 1 inclinationisgreaterthan0.6◦.Weshowthatthetransitprobabilityisgenerallyonlyaweak 3 function of the planet’s orbital period; circumbinary planets may be used as practical tools 6 forprobingtheouterregionsofexoplanetarysystemstosearchforanddetectwarmtocold 3 transitingplanets. 0 1. Key words: binaries: close, eclipsing, spectroscopic – astrometry and celestial mechanics: 0 celestialmechanics,eclipses–planetsandsatellites:detection,dynamicalevolutionandsta- 5 bility,fundamentalparameters–methods:analytical,numerical,statistical 1 : v i X 1 INTRODUCTION al.2011withtransitphotometry).Therearepresentlytentransit- r a ingcircumbinaryplanetsknown,allfoundbytheKeplertelescope Intheburgeoningsearchforextra-solarplanets,circumbinaryplan- (Welshetal.2014). etsrepresentsomeofthemostexoticsystemsfoundtodate.They Theadvantageoffindingcircumbinaryplanetsintransitisthat pose astronomers with interesting questions regarding their de- theycanyieldanunambiguousdetection,thankstoauniquesig- tectability (Schneider 1994), abundance (Armstrong et al. 2014; naturethatishardtomimicwithfalsepositives.Thephotometric Martin&Triaud2014),formation(Pierens&Nelson2013;Kley measurementoftheradiuscanbecomplementedwithtransittim- &Haghighipour2014),habitability(Haghighipour&Kaltenegger ing variations (TTVs), eclipse timing variations (ETVs) or spec- 2013; Mason et al. 2014), orbital dynamics (Leung & Hoi Lee troscopytoobtainthemassandbulkdensity,whichareimportant 2013)andstability(Dvorak1986;Dvoraketal.1989;Holman& fromaformationperspective.Transitsalsoopenthedoortoatmo- Wiegert1999). sphericcharacterisation(Seager&Deming2010),themeasureof Answerstothesequestionsarereliantonplanetdetections.So the Rossiter-McLaughlin effect (Queloz et al. 2000; Fabrycky & far there have been reported discoveries from several techniques, Winn2014),andthedetectionofexomoons(Kippingetal.2012). includingPSRB1620-26(Thorsettetal.1999withpulsartiming), Itwillbeshowninthispaperthatcircumbinaryplanets,be- HD202206(Correiaetal.2005withradialvelocimetry),DPLeo- yond their exoticity, are useful astronomical tools. Their particu- nis(Qianetal.2010witheclipsetimingvariations),Ross458(Bur- largeometryandorbitaldynamicsleadtopotentiallymuchhigher gasser et al. 2010 with direct imaging), and Kepler-16 (Doyle et transitprobabilitiesincomparisonwithsinglestars.Thereisalsoa weakerdependenceonorbitalperiod,allowingustoextendtransit studiestotheouterregionsofstellarsystems. (cid:63) E-mail:[email protected] Thepaperisstructuredasfollows.InSect.2weintroducethe (cid:13)c 2014RAS 2 MartinandTriaud geometryofcircumbinaryplanets.NextinSect.3weanalysethe y orbitaldynamicsofcircumbinarysystemsandtheeffectsontheir observability.Wethendefinetheconceptoftransitabilityandana- lyticallyderiveacriterionforitsoccurrenceinSect.4.Following this,weconvertthiscriterionintotheprobabilityofacircumbinary f systemexhibitingtransitabilityinSect.5,similartotheworkdone 2 periapse fwoersainnagllyessetathrse(sBpoecruiaclkcia&seSoufmemcleiprssi1n9g8b4i;nBarairense.s2007).InSect.6 a ( 1 - e ) Asanobserver,theobservablequantityisatransit,nottran- ω sitability.ThisiswhyinSect.7weconnectthetwoconcepts,ver- x ifyingthatasystemexhibitingtransitabilityiseffectivelyguaran- a ( 1 + e ) 1 teed to transit, albeit at an unspecified point in time. Some illus- trativetransitwaittimesarecalculated,revealingthattheymaybe withinafewyearsformanysystems.InSect.8wediscusssomeap- apoapse plicationsandlimitationsofourwork,beforeconcludinginSect.9. 2 GEOMETRY WewilltreatacircumbinarysystemasapairofKeplerianorbits Figure1.Planarorbitalelementsofatwo-bodysystem. inJacobicoordinates,withtheadditionoffirst-orderdynamicalef- fects(Sect.3).Theinnerorbitisthestellarbinary(subscript“bin”). z Theouterorbitistheplanetaroundthebinarycentreofmass(subscript“p”).EachKeplerianorbitisanellipsecharacterisedbyfour orbital elements: the semi-major axis a, eccentricity e, argument towards observer ofperiapsisωandtrueanomaly f.Thesequantitiesaredefinedin Fig.1.Thissetoffourisnotunique,andoftenwewillusethepe- ΔI riodT insteadofthesemi-majoraxis.Theorientationofeachorbit inthreedimensionsisdefinedusingtwoextraangles:theinclina- tionIandlongitudeoftheascendingnodeΩ.InFig.2wedepictthe I 3Dorientationofabinaryandplanetorbit.Wetaketheobserverto bin belookingdownthez-axis.Aneclipsingbinary,forexample,cor- y respondstoI ≈π/2.Throughoutthispaperweuseradiansunless bin otherwisespecifiedwitha◦symbol. Ω bin Theorientationoftheplanetaryorbitwithrespecttothebinary ΔΩ I p ischaracterisedbytwoquantities:themutualinclination Ω p cos∆I=sinIbinsinIpcos∆Ω+cosIbincosIp, (1) xx sky plane andthemutuallongitudeoftheascendingnode Figure2.Acircumbinaryplanetinamisalignedorbit(blue,outer)around abinarystarsystem(pink,inner).Themisalignmentischaracterisedbythe ∆Ω=Ωbin−Ωp, (2) mutualinclination,∆I,andthemutuallongitudeoftheascendingnode,∆Ω. Theobserverislookingdownthez-axisfromabove,andhencethegreyx-y whicharealsoshowninFig.2.Whenusingtransitphotometryor planedenotestheplaneofthesky. radial velocimetry, the observer is sensitive to ∆Ω but not to the individual quantities Ω and Ω . Throughout this paper can we bin p thereforetakeΩ =0andallowΩ tovary. bin p the binary plane (∆Ω = 0 to 2π), whilst maintaining ∆I = const (Schneider1994;Farago&Laskar2010;Doolin&Blundell2011). TheprecessionperiodT accordingtoSchneider(1994)is 3 DYNAMICORBITS prec A static Keplerian orbit is insufficient for accurately describing a 16(cid:32) a (cid:33)2 1 circumbinary planet. Owing to perturbations from the binary, the Tprec=Tp 3 ap cos∆I, (3) orbital elements defined in Sect. 2 vary on observationally rele- bin vanttimescales.Weincludeinourderivationthemostprominent where the stars are assumed to be of equal mass. An alternative, oftheseeffects:aprecessionintheplanet’sorbitalplane.Thisbe- morecomplexderivationcanbefoundinFarago&Laskar(2010). haviourisdescribedbyatime-variationin∆I,and∆Ω.Werestrict Theplanetorbitisstableaslongasitisnottooclosetothebinary. ourselvestocircularbinariesandplanets1.Inthiscase,theorbital AnapproximatecriterionfromtheworkofDvorak(1986);Dvorak planeoftheplanetrotatesataconstantratearoundthenormalto etal.(1989);Holman&Wiegert(1999)is 1 Itistechnicallyamisnomertospeakofcircularcircumbinaryorbits,since ever,thisonlyhasaverysmalleffectonthetransitgeometry.SeeSect.8.5.2 perturbationsfromthebinarycauseeptovaryevenifinitiallyzero.How- forfurtherdetail. (cid:13)c 2014RAS,MNRAS000,1–?? Analyticcircumbinarytransitprobability 3 130 1 125 0.8 120 ) 0.6 g 115 e π 0.4 (d 110 / ) 0.2 p105 Ω I ∆ ( 0 100 n si −0.2 95 I ∆ −0.4 90 −0.6 0 20 40 60 t (ydeaayrss) −0.8 Figure4.VariationofIpovertimefortwocircumbinarysystems(ap=0.3 −1 AUinblackandwhitedashes,ap = 0.6AUinblue).Thehorizontalred −1 −0.5 0 0.5 1 linedenotesthebinary’sorbitalplaneinclinationonthesky,Ibin.Thegrey ∆Icos(∆Ω)/π regioncorrespondstowhentheplanetisintransitability. Figure3.Surfacesofsectionofthemutualinclination,∆I,andmutuallon- gitudeoftheascendingnode,∆Ω,betweenthebinaryandplanetorbital planes. a (cid:38)3a . (4) p bin InFig.3wedemonstratehowprecessioneffects∆I and∆Ω usingnumericalN-bodyintegrations2.Weranasetofsimulations with∆Ωstartingat90◦and∆Ivariedbetween0◦and180◦insteps of10◦ and,eachcorrespondingtoadifferentcurveinFig.3.The starsareofmass1M and0.5M witha =0.07AU.Theplanet (cid:12) (cid:12) bin Figure5.Anexamplecircumbinarysystemexhibitingtransitability. isamasslesstestparticlewitha =0.3AU.Thegreen,innercurves p areforprogradeorbitswithclockwiseprecession.Theblue,outer curvesareforretrogradeorbitswithanti-clockwiseprecession.The gapbetweenthegreenandbluecurvescorrespondsto∆I = 90◦, expressionsinEq.3producesprecessionperiodsof6.0yrand67.1 yr,showingittobereasonablyaccurate. i.e.forapolarorbitwheretheprecessionperiodbecomesinfinitely long(Eq.3). Theseorbitaldynamicshaveobservationalconsequences.The inclinationplanetonthesky,I varieswithtimeaccordingto 4 CRITERIONFORTRANSITABILITY p (cid:32) (cid:33) 2π Transitability is an orbital configuration where the planet and bi- I =∆Icos t +I , (5) p T bin naryorbitsintersectonthesky,liketheexampleshowninFig.5. prec In this scenario transits are possible but not guaranteed on every wheretistime.AnexampleisshowninFig.4fortwocircumbi- passageoftheplanetpastthebinary,becauseoftherelativemotion narysystems.Thebinaryinbothsystemshasequalmassstarswith ofthethreebodies.ThisterminologywasfirstintroducedinMartin M = M = 1M , a = 0.1 AU, I = 110◦, and Ω = 0◦. A B (cid:12) bin bin bin &Triaud(2014),wewhereformallydefinedandelaboratedupon Theplanetisamasslessbodywithstartingvalues I = 130◦ and p a concept that had been already used in several studies (Schnei- Ω = 0◦.Themutualinclinationis20◦.Thetwoplanetsshownin p der 1994; Welsh et al. 2012; Kratter & Shannon 2013). In Fig. 4 thefigurehavedifferentvaluesfora :0.3AUfortheblack,dashed p thegreyregiondenotesthetimespentintransitability.Duringeach sinusoidand0.6AUfortheblue,solidsinusoid. precessionperiod,therewillbezero,oneortwointervalsoftran- ThemaximumandminimumvaluesofI areindependentof p sitability,orpermanenttransitabilityinonlytwoscenarios:1)I a . The planet semi-major axis does, however, strongly influence bin p andI arebothveryclosetoπ/2and2)polarorbitswhereI =0 theprecessionperiod:T = 5.6yrfora = 0.3AUand65.9yr p bin prec p andI =π/2. fora =0.6AU,accordingtotheN-bodysimulation.Theanalytic p p We work to derive a criterion that predicts whether or not a circumbinaryplanetwillentertransitabilityatanypointduringthe 2 All N-body simulations in this paper are done using a fourth-order precessionperiod.Asafirstapproximation,weknowthattheplanet Runge-Kuttaalgorithm,whereenergylossduetoitsnon-symplecticnature isintransitabilitywhentheplanetorbitisperpendiculartotheplane waskepttonegligiblelevels. ofthesky: (cid:13)c 2014RAS,MNRAS000,1–?? 4 MartinandTriaud π sky! I = . (6) p 2 plane Thisisthemostconservativecasepossible,sinceitignoresthefi- towards! nite extent of the binary. According to Eq. 5, I is guaranteed to p observer reachπ/2if |90°-I | p X ∆I>(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)π2 −Ibin(cid:12)(cid:12)(cid:12)(cid:12)(cid:12). (7) XB A ΔI |90°-Ibin| Xp I This is the first-order criterion for transitability. Three things are bin apparent:1)mutualinclinationsaidtransitability,whichiscontrary toconventionalviewsontransitgeometries,2)thiscriterionisin- dependentoftheplanetperiodand3)thiscriterioniseasiesttoful- Figure6.Aside-onviewofacircumbinarysysteminthelimitingcaseof filatI ≈ π/2,i.e.foreclipsingbinaries.Thiscriterionwasalso transitability,where∆Ω = 0.Thetwoextremeverticalpositionsofeach bin derived by Schneider (1994), who was the first author to analyse starontheskyaredrawnindifferentcolours.Inthisexample,theplanetis barelyintransitabilityonthesecondarystar,butnotontheprimary. circumbinarytransitprobabilitiesinthepresenceofprecession. Thesecondlevelofcomplexityistoincludethefullextentof the stellar orbits, meaning that a value of I offset from π/2 may p stillexhibittransitability.Considerthelimitingcaseoftransitabil- aretheindividualsemi-majoraxesforthetwostarsand ity.Thisiswhentheplanetandbinaryorbitsbarelyoverlapwhen M |Ip−π/2|isataminimum(dIp/dt=0).Wecalculatetheorientation µB,A= MA+B,AMB (14) ofthebinaryandplanetorbitsinthisconfiguration.TakeEq.1and rearrangeittoisolatethetermcontaining∆Ω: arethereducedmasses.Similarlyfortheplanet, (cid:12) (cid:12) (cid:12) π(cid:12) cos∆I−cosI cosI Xp=apsin(cid:12)(cid:12)(cid:12)Ip− 2(cid:12)(cid:12)(cid:12). (15) cos∆Ω= sinI sinbinI p. (8) ThereistransitabilityonstarsAand/orBwhen bin p InEq.8onlyI and∆Ωaretime-dependentquantities.Differenti- p atingbothsidesofEq.8withrespecttotimeleadsto X <X , (16) p A,B Accordingtotheorbitaldynamics(Eq.5),theplanetwillentertran- −sin∆Ωd∆Ω =(cid:34)cosI sinI dIp sinI sinI sitabilityatsomepoint(fulfillingEq.16)ifthefollowingcriterion dt bin p dt bin p ismet: (cid:35) /−(si(nc2osI∆Isi−n2coIs)I.bincosIp)sinIbincosIpddItp (9) ∆I>(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)π2 −Ibin(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−βA,B(Ibin), (17) bin p wheretheangleβ isafunctionofthebinaryextentonthesky,as BysubstitutingdI /dt=0intoEq.9weget A,B p seenbytheplanet.Thisisthesecond-ordercriterionfortransitabil- ity.CombineEqs.11and17toget d∆Ω sin∆Ω =0. (10) dt (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)π (cid:12) From Sect. 3 it is known that d∆Ω/dt = const (cid:44) 0, and hence (cid:12)(cid:12)Ibin−Ip(cid:12)(cid:12)>(cid:12)(cid:12)(cid:12)2 −Ibin(cid:12)(cid:12)(cid:12)−βA,B(Ibin). (18) sin∆Ω = 0, implying that ∆Ω = 0. This means that the limiting ByinsertingEqs.12and15intoEq.16andrearrangingtomatch caseoftransitabilityoccurswhentheascendingnodesofthebinary theformofEq.18,weobtain andplanetorbitsarealigned.Thissimplifiesthegeometryandcal- culations.AccordingtoSect.3,∆Ωisguaranteedtoequalzeroat (cid:34) (cid:12) (cid:12) (cid:35) sinocmlienaptoioinntcdaulrciunlgattihoenpinreEceqs.s1ioinspsiemripoldifi.Wedhteon∆Ω=0themutual βA,B(Ibin)=sin−1 aaAp,B sin(cid:12)(cid:12)(cid:12)(cid:12)π2 −Ibin(cid:12)(cid:12)(cid:12)(cid:12)+ RaAp,B , (19) wherethequantitiesinsidethesquarebracketsaresufficientlysmall ∆I=|I −I |. (11) thatwecanusethesmallangleapproximation. bin p Thesizeofβ determineshoweasyitisforasystemtoful- A,B InFig.6weshowacircumbinarysysteminthelimitingcase filthetransitabilitycriterioninEq.17.Dependingontheseparate oftransitability.Thisisa“side-on”viewoftheorbit,withtheob- valuesofβ andβ ,itispossibletofulfilthecriterionforjustone A B serverlocatedtotherightofthepage.Usingthisdiagram,wedefine ofthestars.Generallyβ >β exceptforeclipsingbinaries. B A halftheprojectedheightsontheskyofthetwostellarorbitstobe Totestthevalidityofthesecond-ordertransitabilitycriterion werannumericalN-bodysimulationsonthousandsofhypothetical (cid:12) (cid:12) (cid:12) π(cid:12) circumbinarysystems.ThedetailsareshowninAppendixA.The XA,B=aA,Bsin(cid:12)(cid:12)(cid:12)Ibin− 2(cid:12)(cid:12)(cid:12)+RA,B, (12) analyticcriterionisshowntobeveryaccurate,withanerrorless than0.1%.Allerrorcaseswerenearthelimitoftheinequalityin where Eq.17.Errorsariseduetosmallvariationsinthesemi-majoraxis a =a µ (13) and eccentricity, which in the limiting case of transitability may A,B bin B,A (cid:13)c 2014RAS,MNRAS000,1–?? Analyticcircumbinarytransitprobability 5 leadtoacontraryresulttothepredictionofEq.17.Thisiselabo- (a) rateduponinSect.8.5.2. celestial! sphere 5 PROBABILITYOFTRANSITABILITY planet at! Foragivensetoforbitalparameters,wecancalculatetheproba- inclination! extremum bilitythatagivenobserverwillobservetransitability,atsometime ϑ A duringtheprecessionperiod.Asaninitialapproximation,weuse ζ A the first-order transitability criterion (Eq. 7). The orientation of a A circumbinarysystemontheskyisuniformlyrandom.Ittherefore followsthatcosI hasauniformdistribution,andhencetheprob- bin abilitydensityfunctionis ϑAζAA 2ΔI p(Ibin)=sinIbin. (20) 2ΔI+2ζAA ByintegratingthisbetweentheboundsspecifiedbyEq.7,weob- tainanapproximateprobabilityoftransitability: planet at! inclination! (cid:90) π/2 extremum P = sinI dI A,B bin bin π/2−∆I =sin(∆I). (21) The probability is period-independent and non-zero, except for strictlycoplanarsystems. (b) Thenextstepistoincludethefiniteextentofthebinaryor- R bit.Theprobabilityoftransitabilityisafunctionofthesizeofthe A solid angle subtended on the celestial sphere such that the planet a a p ϑ and stellar orbits are seen overlapping, including the full orbital A A evolution.WedemonstratethisinFig.7a.Tosimplifythediagram, ΔI ζAA a weonlydrawtheanglesfortransitabilityontheprimarystar,but A R thecalculationproceedsidenticallyforthesecondary.Theplanet A orbitisshownintwodifferentpositions,correspondingtothetwo extrema of I separated by 2∆I (Eq. 5). The angles ζ and θ p A,B A,B Figure7.In(a)weshowaside-onviewofacircumbinarysystemwiththe aretheanglessubtendedbytheplanetintransitabilityoneachstar. They are functions of how big the stellar orbit is, as seen by the planetshownatitstwoextremevaluesofIp.Bothstarsareplottedtwice indifferentcoloursatclosestandfarthestseparationfromtheplanet.The planet,atclosest(ζ )andfarthest(θ )separation.InFig.7bwe A,B A,B anglessubtendedbytheorbitonthecelestialsphereareonlyshownforthe zoomintoseehowtheanglesaredefined3 primarystar,toavoidclutter.Thehatchedregioncorrespondstoobservers whowouldseetheplanetintransitabilityatsomepointintime.In(b)we (cid:32)a sin∆I+R (cid:33) zoomintoseehowθandζaredefined. ζ =tan−1 A,B A,B (22) A,B a −a cos∆I p A,B and bandinFig.7ais (cid:90) π/2 (cid:32)a sin∆I+R (cid:33) PA,B= sinIbindIbin θ =tan−1 A,B A,B . (23) π/2−∆I−ζ A,B ap+aA,Bcos∆I =sin(∆I+ζA,B) Thelargerangleζiswhatcorrespondstothelimitoftransitability =sin(cid:32)∆I+ aA,Bsin∆I+RA,B(cid:33), (24) andhenceθdoesnotappearinanyfurtherequations. a −a cos∆I p A,B Allobserverswithinthehatchedbandwilleventuallyseetran- where because ζ is generally small we can apply the small angle sitability.Theprobabilityofanobserverbeingwithinthehatched approximation to remove the tan−1 function4. Equation 24 is the probabilityoftransitabilityontheprimaryand/orsecondarystars, 3 Thecurrentequationsarecorrectedfromtyposthatappearedinthepub- 4 Relatedtotheerrormentionedinfootnote3,Eq.24hasbeencorrected lishedversion:inEq.22thereusedtobea+signinthedenominatorand fromthepublishedversionwhichincorrectlycontainedan+signinthe inEq.23thereusedtobea−signinthedenominator.Wearesorryforthe denominator.Thiserrorwaspurelytypographicalandallresultsshownin error. thepaperwerecalculatedusingthecorrectequation. (cid:13)c 2014RAS,MNRAS000,1–?? 6 MartinandTriaud forabinaryofanyorientation.Theinclusionofζ addsaperiod- 1 Eq. 234 (A) dependencythatisabsentinEq.21. 0.9 Eq. 234 (B) InFig.8awedemonstrateEq.24onanexamplecircumbinary Eq. 245 (A) system,comprisedofabinarywith MA = 1M(cid:12),RB = 1R(cid:12), MB = 0.8 Eq. 245 (B) 0.5M(cid:12), RB = 0.5R(cid:12) and abin = 0.082 AU (Tbin = 7 days). The 0.7 Eq. 201 planetsemi-majoraxisisvariedfrom0.24AUto2AU.Thethree mutualinclinationsare0◦,5◦and10◦. 0.6 As∆Iisincreasedtheprobabilityoftransitabilityisincreased ,B A0.5 significantly. For the misaligned cases, PB > PA. In Fig. 8b we P 0.1 zoominonthecoplanarcase.Asacomparison,weshowthetransit 0.4 0.08 probabilityonasinglestarofradiusR ,calculatedusing A,B 0.3 0.06 0.04 R 0.2 P = A,B. (25) 0.02 A,B ap 0.1 0 0 2 4 Forcoplanarsystemstheprobabilityoftransitabilityreduces 0 0 10 20 30 40 50 60 to (cid:54) I (deg) R P = A,B , (26) Figure9.TheprobabilityoftransitabilityonstarsAandBcalculatedusing A,B ap−aA,B Eq. 24, and the approximation using Eq. 21. The bottom right image is zoomedintosmallmutualinclinations.Inthisplotthehorizontallinesare whichcomesfromsetting∆I=0inEq.24andusingasmallangle theequivalentsinglestartransitprobabilities(Eq.25). approximationtoremovethesinfunction.Thisequationmatches Welshetal.(2012),whoderivedananalyticestimateforthetran- sitprobability5 undertheassumptionof∆Ω = 0andstaticorbits, althoughitwasdulynotedthatcircumbinaryorbitsprecess.Equa- (cid:12)(cid:12)π (cid:12)(cid:12) R +R −2αR tpiaorntic2u6lairslyclaotssehtoortthpeersioindgsl.eTshtiasripsrboebcaabuilsietythbeutstsalrisghartleybhroiguhgehrt, sin(cid:12)(cid:12)(cid:12)2 −Ibin(cid:12)(cid:12)(cid:12)(cid:54) A aBbin B, (28) closertotheplanetsbytheirorbitalmotion. where α determines whether the criterion is for grazing eclipses ThecrucialdifferencetothesinglestarcaseisthatEq.25de- (α=0),fulleclipses(α=1)oranythinginbetween.Sinceeclipses creasestowardszeroforlargesemi-majoraxes,butforcircumbi- occurwhenI ≈π/2,wecanapplythesmallangleapproximation bin narysystemsthelimitofEq.24is inEq.28todeducethatthedistributionofI foreclipsingbinaries bin isuniformbetween lim P =sin∆I, (27) A,B ap→∞ π δ ± , (29) whichisequaltothefirst-orderderivationinEq.21.Thisapprox- 2 abin imateprobabilityisalsoapplicabletosystemswithalarge∆I be- wheretosimplifytheequationwehavedefined causetheangleζ,whichencompassestheperiod-dependence,be- δ=R +R −2αR . (30) comesrelativelysmall. A B B InFig.9wedemonstratehowtheprobabilityoftransitability Knowing that the binary eclipses with this uniform random varieswith∆I,usingthesamecircumbinarysystemasinFig.8a, distribution of Ibin, the probability of transitability is the fraction butfixingap=0.26AU.Inthebottomrightofthisfigurewezoom ofeclipsingbinarieswithIbin suchthattheinequalityinEq.17is innear∆I=0◦.Thecurveforthesecondarystarisseentoovertake satisfied. First, multiply Eq. 28 by PA,B and insert it into Eq. 17, thatoftheprimaryataround∆I=3◦. usingβA,BfromEq.19,toobtain (cid:32) (cid:33) δ a δ R ∆I=P −sin−1 P µ bin − A,B . (31) A,Ba A,B B,A a a a 6 CONSEQUENCESFORECLIPSINGBINARIES bin p bin p TheinequalityfromEq.17hasdisappearedsincewearecalculat- Eclipsingbinariesareonlyasmallfractionofthetotalbinarypopu- ing P for a given ∆I. Use the small angle approximation and lationbuttheeasiestbinariestodetectphotometrically.Itwassug- A,B rearrangetoobtain gestedbyBorucki&Summers(1984)thattheyarefavourabletar- gets for transit surveys because they positively bias the planetary (cid:32) (cid:33) 1 1 R orbit towards being aligned with the line of sight. In this section ∆I=P δ −µ − A,B. (32) wederivetheprobabilityoftransitability, PA,B,underassumption A,B abin B,Aap ap thatthebinaryisknowntoeclipse.Indoingsowequantifywhat BysolvingforP weget A,B wsc5nirotaiItatnseubfirsfiiialronicstgntyt,tnfthiohosaerttieearndanadsmieienerecis.Svltiaepttcisoote.nf4iuwsluafipslofinonrdttehhreeivpcirnaosgbeaEboqilf.it7eyc:oltifhpetsriacnnrgistietbarbiinoilanitryfi,eosdr.etsrTpahintee- PA,B= 1δ(cid:32)a∆1biIn+−RµaABp,,ABa1p(cid:33) iiff∆∆II(cid:62)<∆∆IIlim , (33) lim (cid:13)c 2014RAS,MNRAS000,1–?? Analyticcircumbinarytransitprobability 7 (a) (b) 0.25 0.025 Eq. 234 (A) Eq. 234 (A) Eq. 234 (B) Eq. 234 (B) 0.2 ΔI = 10° Eq. 21 0.02 Eq. 245 (A) Eq. 245 (B) 0.15 0.015 B B , , A A ΔI = 5° P P 0.1 0.01 ΔI = 0° 0.05 0.005 ΔI = 0° 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a (AU) a (AU) p p Figure8.In(a)weshowtheprobabilityoftransitabilityonstarsAandB(Eq.24)asafunctionofap,forthreedifferentmutualinclinations.Thehorizontal dashedlinesarecalculatedusingthefirst-orderapproximateprobability(Eq.21).In(b)wezoominonthe∆I = 0◦ case.Asacomparison,weshowthe equivalentsinglestarprobabilityindashedlines(Eq.25). wherewedefine ismorelikelytofindplanetstransitingasinglestarwhereanother (cid:32) (cid:33) transiting planet has already been found, compared to around a 1 1 R ∆I =δ −µ − A,B, (34) randomstar.Thisisbecausethemutualinclinationdistributionof lim a B,Aa a bin p p multi-planetsystemsisnotisotropicbutweightedtowardscopla- narity6. This is analogous to a circumbinary system, if one con- in order to truncate P at 1. As an example, for a binary with A,B solar and half-solar masses and radii and a = 0.3 AU orbited sidersthesecondarystarasthe“innerplanet”.Thereis,however, byaplanetata = 1.0AU,α = 0.5andcobipnlanarorbits,Eq.33 afundamentaldifferencebetweensingleandbinarystars:planets yields P = 0.3p3 and P = 0.19. Coplanar orbits correspond to orbitingsinglestarsdosooneffectivelystaticorbits. A B a minimum value of P . A slight increase in ∆I to 0.5◦ raises A,B theseprobabilitiesto0.96and0.89,respectively.For∆I = 1◦ the probability on both stars is 1. The circumbinary geometry is evi- 7 CONNECTINGTRANSITABILITYTOTRANSITS dentlyveryfavourablefortransitabilityoneclipsingbinaries.We noteforreferencethatthemeanmutualinclinationinthetransiting 7.1 Doestransitabilityguaranteetransits? circumbinary planets found so far is 1.73◦ (see Table 1) and that theSolarSystemmutualinclinationdistributionroughlyfollowsa Transitabilityaloneisnotdetectableviaphotometry,onerequires 1◦Rayleighprofilerelativetotheinvariantplane(e.g.Clemence& anactualtransit.Afundamentalelementofthedefinitionoftran- Brouwer1955;Lissaueretal.2011) sitability is that transits are possible but not guaranteed on any For∆I >∆I twothingsoccur:1)transitsareguaranteedon given passing of the binary orbit. By having a transit probability lim eclipsingbinariesofanyorientationand2)transitsbecomepossible between0and1foreachpassing,itisintuitivetothinkthatatran- onnon-eclipsingbinaries.InFig.10weplot∆I asafunctionof sitwilleventuallyhappenifobservedcontinuouslyforasufficiently lim a anda ,forabinarywithsolarandhalf-solarmassandradius longtime.ThisconclusionwassharedbySchneider(1994);Welsh bin p and a between 0.007 and 0.2 AU, where the lower limit corre- etal.(2012);Kratter&Shannon(2013);Martin&Triaud(2014). bin spondstoacontactbinary:a =R +R .Fortheeclipsecriterion Wetestedthishypothesisbynumericallysimulatingcircumbi- bin A B weusedα=0.5.Thewhiteemptyspaceontheleftistheunstable nary systems over 50 precession periods and looking for transits regionaccordingtoEq.4. incaseswheretransitabilityoccurred.Thiswasfirstdoneforcom- Thebinarysemi-majoraxisisthebiggestfactorinthecalcu- pletelyrandomsystemstakenfromthetestsinAppendixA.Thede- lation.ForthesystemsinFig.10wherecircumbinaryplanetshave tailsareprovidedinAppendixB.Lessthan0.3%ofsystemsman- beenfoundsofar(a >0.08AU),∆I islessthan3◦. agedtoevadetransit.Alloftheseexceptionalcasescorresponded bin lim For closer binaries ∆I rises sharply, reaching a maximum to the limit of transitability, where the planet only spends a very lim of38◦foracontactbinary.Transitsonveryshort-periodeclipsing short time in transitability, and hence the chance of transiting on binariesareofcoursepossiblebutEq.34showsthatnotallsuch binaries can be transited unless there is significant misalignment. InSect.8.4weapplythisworktotheKeplerdiscoveriessofar. 6 Ragozzine&Holman(2010)founditresemblesaRayleighdistribution. There are similarities between transits on eclipsing binaries Thedistributionof∆Iincircumbinarysystemsispresentlyunknown,be- and studies of multi-transiting systems orbiting single stars (e.g. cause the detections so far have been highly biased towards coplanarity Ragozzine&Holman2010;Gillonetal.2011).Geometrically,one (Martin&Triaud2014). (cid:13)c 2014RAS,MNRAS000,1–?? 8 MartinandTriaud 0.2 g n25 1 i t i s t n i a20 m r i t ) l (AU 0.1 lity 2 tems15 bin abi sys 25 a t 4 10 s f 20 o 15 e 8 g Numerical (A) 10 a 5 16 t Numerical (B) 5 32 n 00 0.5 1 1.5 2 rce EEqq.. 223344 ((AB)) 00 2 4 6 8 a (AU) e 0 p P 0 50 100 150 200 250 300 Observing time (yr) Figure10.3Dhistogramoftheminimummutualinclinationneededinde- greestoguaranteetransitsonEBsofanyorientation,atdifferentbinaryand Figure11.Thepercentageofsystemsseentransitingasafunctionoftime planetsemi-majoraxes(Eq.34). fromnumericalsimulationsof10,000circumbinarysystems,andtheana- lyticpredictions(Eq.24).Systemsarecountedastransitingafterthedetec- tionofasingletransit.Thezoomedfigureinthebottomrightcornershows thepercentageoftransitingsystemsoverKepler-likeobservingtimes.The anygivenorbitissmall.Transitsareexpectedtooccureventually, blackverticallinedenotestheprecessionperiod. butafteratimelongerthanwhatwassimulated. Second,weconstructedsystemswith4:1and5:1periodcom- mensurabilities,specificallydesignedtomakeplanetspermanently to know how long an observer must wait to see a transit. It is a evade transit. The evasion percentage increased slightly but re- strongfunctionoftheprecessionperiod,sincethatdetermineshow mainedlessthan1%.Itisprobablethatthisvaluewouldeventually spacedaparttheregionsoftransitabilityare. droptozero,butlongersimulationswouldberequired.Theplanets An analytic calculation of the time-dependent transit proba- inevitablytransitbecauseexactperiodcommensurabilitiesarenot bility is outside the scope of this paper, and has been previously sustainable,owingtoperturbationsfromthebinaryontheplane- labelledimpossible(Schneider&Chevreton1990).Weinsteaduse taryorbit(Sect.8.5.2).AsideforHD202206whichhasaperiod numericalN-bodysimulations. rationear5:1(Correiaetal.2005),periodcommensurabilitieshave InFig.11wedemonstratethepercentageofsystemsseentran- notbeenobserved.Thissystemmaynotberepresentative,sinceit sitingstarsAand/orBasafunctionoftime,using10,000simulated straddlestheborderbetweenacircumbinaryandatwo-planetsys- circumbinary systems. The primary and secondary stars are solar tem; the secondary “star” has a minimum mass of 15MJup. It has andhalf-solarinmassandradius,T =7days,T =40daysand alsobeentheorisedbyKley&Haghighipour(2014)thatcircumbi- ∆I = 10.Overtime,thepercentagebinoftransitingspystemsreaches naryplanetsshouldformbetweenintegerperiodratios,notinthem. thevaluepredictedbyEq.24,inagreementwiththeconclusions Whilst not an exhaustive proof, our tests indicate that in the ofSect.7.1.Mostofthetransitingsystemshavedonesowithina vastmajorityofcases,transitabilityindeedleadstotransit,albeitat singleprecessionperiod(here∼sevenyears). anunspecifiedpointintime. Asanextendedtest,wetookthesystemsfoundtransitinginin Sect.7.1andcalculatedthetimetakenforprimaryandsecondary transits to occur. The results are provided in Table C1. Whilst a 7.2 Transitsovertime largermutualinclinationleadstomoreplanetstransiting,theme- Theprobabilityoftransitabilityisequivalenttotheprobabilityof dianwaittimeisincreased.Generally,asignificantnumberofsys- transit, granted the observer has infinite time. Unfortunately, due temsarefoundtransitingwithinKepler-likemissiontimes. to limitations in technology, funding and human life-expectancy, onemuststrivetocapturetransitswithinafinitetime.Wecalcu- latesomeexampleobservationtimesneededinordertoobservea 8 DISCUSSIONANDAPPLICATIONS transit. Inthecaseofasinglestar,aftercontinuousobservationsofa 8.1 Thecircumbinaryplanetsdiscoveredsofar timeequaltoT ,theplaneteitherwillorwillnothavetransited. p Our first application is to calculate the transitability probabilities Thisisnotthecaseforcircumbinaryplanets,fortworeasons: for the Kepler discoveries so far, assuming of course that we do • Theplanetmaycurrentlybeoutsideoftransitability,butwill not have a priori knowledge of transits and eclipses7. In Table. 1 precessintotransitabilityatalatertime. wecalculatetheprobabilityoftransitabilityonbinariesofanyori- • Theplanetmaycurrentlybeinsidetransitability,butthecon- entation(Eq.24)andoneclipsingbinaries(Eq.33),whereforthe junctionrequiredforatransithasnotyetoccurred. latterweusedα=0.5todefineeclipses.Theequivalentsinglestar The fraction of circumbinary planets transiting therefore in- creaseswithtime,uptoavaluespecifiedbyEq.24.Itisimportant 7 Otherwiseyouwouldhaveaboringtablefullof100%’s. (cid:13)c 2014RAS,MNRAS000,1–?? Analyticcircumbinarytransitprobability 9 Table1.ProbabilitiesoftransitforthecircumbinaryplanetsdetectedsofarbyKepler. Name MA MB RA RB abin ap ∆I PA,B%(all) PA,B%(EBs) PA,B%(single) (M(cid:12)) (M(cid:12)) (R(cid:12)) (R(cid:12)) (AU) (AU) (deg) A B A B A B Kepler-16 0.69 0.20 0.65 0.23 0.22 0.71 0.31 1.04 0.91 75.5 66.1 0.42 0.15 Kepler-34 1.05 1.02 1.16 0.19 0.23 1.09 1.86 4.18 4.16 100 100 0.50 0.47 Kepler-35 0.89 0.81 1.03 0.79 0.18 0.60 1.07 3.11 2.94 100 100 0.78 0.61 Kepler-38 0.95 0.27 1.78 0.27 0.15 0.46 0.18 2.28 0.79 41.2 14.4 1.80 0.27 Kepler-47b 1.04 0.46 0.84 0.36 0.08 0.30 0.27 1.93 1.26 39.5 25.8 1.30 0.56 Kepler-47c 1.04 0.46 0.84 0.36 0.08 0.99 1.16 2.48 2.32 50.8 47.6 0.39 0.17 Kepler-64 1.50 0.40 1.75 0.42 0.18 0.65 2.81 6.54 6.66 100 100 1.25 0.30 Kepler-413 0.82 0.52 0.78 0.48 0.10 0.36 4.02 8.98 9.18 100 100 1.01 0.62 KIC9632895 0.93 0.19 0.83 0.21 0.18 0.93 2.30 4.68 5.09 100 100 0.49 0.12 Refs:Doyleetal.(2011);Welshetal.(2012);Oroszetal.(2012a,b);Schwambetal.(2013);Kostovetal.(2013,2014) Welshetal.(2014) Note:Kepler-47disexcludedbecauseithasnotyetbeenpublishedandlacksavaluefor∆I. probabilitywascalculatedusingEq.25.Inthetableweincludeall EBs on which there would be transitability by a putative planet, necessary variables for the calculations. In more than half of the with∆Ibetween0◦and10◦anda /a =3,10,20.Theresultsare p bin cases,transitsareguaranteedoneclipsingbinariesofanyorienta- only shown for the primary star, since the plot for the secondary tion. starisindistinguishable. Consistentwithearliersections,thebiggestfactoristhemu- tual inclination, with a higher ∆I leading to a greater chance of 8.2 Multi-planetcircumbinarysystems transitability.Transitabilityisfavouredforsmallervaluesofa ,but p thisdependencydiminishesatlargermutualinclinations. Onlyonemulti-planetcircumbinarysystemhasbeendiscoveredso The Kepler EB catalog would benefit from extended photo- far(Kepler-47,Oroszetal.2012b).Kratter&Shannon(2013)con- metric observations by the future PLATO telescope (Rauer et al. sideredaneclipsingbinarywithaknowntransitingplanet,andcal- 2014),inordertofindnewcircumbinaryplanetsthathavemoved culatedthelikelihoodofasecondplanetbeingseentransiting.They intotransitabilityduringthe∼eightyearsbetweenmissions.This derived an analytic probability for whether or not the binary and mayincludeadditionalplanetsinknowncircumbinarysystems. planetorbitswouldoverlaponthesky,undertheassumptionthat thebinaryisperfectlyedge-on(I =π/2).Infact,whattheycal- bin culatedwastheprobabilityoftransitability.Theirderivationdoes notincludeprecession,andconsequentlyunderestimatestheprob- 8.4 Onthedearthofplanetsaroundshort-periodbinaries ability. Anobservedtrendhasbeenthelackofcircumbinaryplanetsaround BasedontheworkinSect.6,anyadditionalplanetswith∆I theclosestbinaries;theshortest-periodbinaryhostingaplanetis greaterthanthefirsttransitingplanetareguaranteedtoentertran- Kepler-47 with T = 7.4 days (Orosz et al. 2012b). This is de- sitabilityatsomepoint. bin spite the median10 period of the EB catalog being 2.8 days. This raisesvariousquestionsabouttheabilitytoformplanetsinsuchan 8.3 Kepler’seclipsingbinarycatalog environment,particularlyinthepresenceoftertiarystellarcompan- ion,asisoftenthecaseforverytightbinariesaccordingtotheory The Kepler telescope, with its four years of continuous observa- (Mazeh&Shaham1979;Fabrycky&Tremaine2007)andobser- tions and exquisite precision, has provided the most comprehen- vations(Tokovininetal.2006). sive catalog of EBs to date (Slawson et al. 2011). We used the The reason why EBs are preferentially found at short peri- online beta version of this catalog8 to test our transitability crite- ods despite a smaller natural occurrence (Tokovinin et al. 2006) riononhypotheticalorbitingplanets.Fromthecatalogweobtained isbecausethereisagreaterrangeof I thatallowforaneclipse bin M1,M2,R1,R2 and abin, which were derived from stellar temper- (Eq.29).WhentheEBishighlyinclined,however,theplanetitself atures calculated in Armstrong et al. (2013) using a method ex- needs a greater misalignment in order for transitability to occur. plainedinArmstrongetal.(2014).Onlysystemswithamorphol- WedemonstratethisinFig.13,wherewecalculatetheminimum ogyparameterlessthan0.5wereused,correspondingtodetached ∆IneededtoseetransitabilityontheprimarystarofeachEBinthe EBs(seeMatijevicetal.2012fordetails).Thebinaryinclination Keplercatalog,takinga /a = 3.5.Asareference,weshowthe p bin wasrandomisedbetweentheboundsdefinedinEq.299. meanandmaximummutualinclinationsfromtheKeplerdiscover- The remaining quantities needed for Eq. 17 are ∆I and ap. iessofar,althoughthesearebiasedtowardsbeingsmall(Martin& Giventhedistributionofcircumbinaryplanetsispresentlypoorly Triaud2014). knownandsubjecttostrongbiases,weconsideredawiderangeof ForT <7.4days,amisalignmentof1.73◦resultsin60%of bin potentialvalues.InFig.12wecalculatedthepercentageofKepler planetsmissingtransitability.Forthosesystemsmisalignedenough for transitability on the shortest period binaries, there should be 8 http://keplerebs.villanova.edu/maintainedbyAndresPrsaetal. 9 ItispossibletoobtainatruevalueofIbin,howevertheonlypublished versionisinthenow-outdatedcatalogofSlawsonetal.(2011),andcontains 10 Inthepublishedversionofthepaperweaccidentallywrotethatthiswas errors. the“mean”period. (cid:13)c 2014RAS,MNRAS000,1–?? 10 MartinandTriaud Percentage with transitability on A11234567890000000000000 5 ∆ I 1(0deg) aaa ppp1 5=== 312 00a aab ibbniinn2 0 ∆Minimum I for transitability (deg)110505 1 P P∆1 0bI i=(n4d=.a7∆0y. 2s4I°) = d1(a.Ky7es3p1 °l 0(e(0Krme−ep4al1ne3)r)−47) bin Figure12.ThepercentageofEBsfoundbyKepleronwhichtherewould Figure13.Theminimummutualinclinationneededfortransitabilityon betransitabilitybyaputativeplanetwithdifferentvaluesof∆Iandap/abin. eachoftheEBsfoundbyKepler. transitswithintheKeplertimeseries,fortworeasons:1)thepre- cessionperiodisonlyacoupleofyearslong11,sotheplanetand 1 binary orbits would have intersected at least once during the Ke- plermissionand2)theseareverytightsystems,sowearelikelyto 0.8 haveobservedoneorprobablymoretransitswhilstintransitability. Thedearthofplanetsmayalsobeexplainedbystellarnoise 0.6 inthelightcurves.Binarieswithperiodsthisshortareexpectedto 0.4 betidallylocked,whichleadstofasterrotationandincreasedstar π / spots,whichmayinhibitdetections. ) 0.2 Ω Thecurrentnulldetectionlikelyremainssignificantforplan- ∆ 0 etsthataremisalignedbyatleastafewdegrees,buttheremayre- ( n mainsomecoplanaronesthatareundetectablebyKepler.Transits i −0.2 s oncontactbinariesrequireanevenhigherlevelofmutualinclina- I tion.Discoveriesaroundcontactbinariesmayalsobehinderedby ∆ −0.4 Kepler’s30-minutecadence,whichispotentiallytoolongtoade- −0.6 quatelysampleitsorbitinthesearchfortransits. −0.8 8.5 Limitations −1 −1 −0.5 0 0.5 1 8.5.1 EccentricSystems ∆Icos(∆Ω)/π Theadditionofeccentricity,toboththebinaryandplanetorbits,in- troducestwocomplexities.First,thegeometryiscomplicatedsince Figure14.ThesamecircumbinarysystemsasinFig.3butwithebin=0.5. welosecircularsymmetry,andtherearetwoadditionalanglesto consider:ω andω (theargumentsofperiapse).Furthermore,the bin p orbitaldynamicscauseω tobetime-dependent,furthercomplicat- p The simulations show that the error is only of order ∼ 2% (Ap- ingthesituation. pendixD).Furthermore,theresultssuggestthateccentricityactu- Second,theprecessioncycleismorecomplexwhenthebinary allymakestransitabilitymorelikely,althoughamoredetailedstudy iseccentric.InFig.14wedemonstratetheprecessionofthesame systemasinFig.3,butwithe = 0.5.Themutualinclinationis isneededtoconfirmthis. bin nolongerconstant.Therearetwoislandsoflibration,centredon ∆Ω = 0 and ∆I = π/2 (red) and ∆I = −π/2 (magenta), within 8.5.2 Additionaldynamicaleffects which ∆Ω does not circulate through 0 to 2π. We therefore lose twooftheassumptionsmadeinSect.4. Acircumbinarysystemisathree-bodyproblemandhencenotsolv- Werannumericalsimulationstotesttheabilityofourcriterion ableanalytically.OuranalytictreatmentofitasapairofKeplerians inEq.17topredicttransitability,inthecaseofeccentricsystems. plusorbitalprecessionencompassesthemajorityofthephysics,but neglectssomesmalleramplitudeeffects. Thesemi-majoraxisandperiod,whichweassumedtobecon- 11 IfweassumetheobservedoverdensityofplanetsatTp∼5Tbinextends stant, experience slight variations over time because of perturba- toveryclosebinaries. tions from the binary. There are also small variations in the ec- (cid:13)c 2014RAS,MNRAS000,1–??