Transit probability of precessing circumstellar planets in binaries and exomoons

1.2 MB·

by  David V. Martin

Mon.Not.R.Astron.Soc.000,1–??(2016) Printed17January2017 (MNLATEXstylefilev2.2)
Transit probability of precessing circumstellar planets in binaries
and exomoons
David. V. Martin
ObservatoiredeGene`ve,Universite´deGene`ve,51chemindesMaillettes,Sauverny1290,Switzerland,E-mail:[email protected]
7
1 Accepted.Received
0
2
ABSTRACT
n
a
J Over two decades of exoplanetology have yielded thousands of discoveries, yet some
types of systems are yet to be observed. Circumstellar planets around one star in a binary
6
1 have been found, but not for tight binaries ((cid:46) 5 AU). Additionally, extra-solar moons are
yettobefound.Thispapermotivatesfindingbothtypesofthree-bodysystembycalculating

] analyticandnumericalprobabilitiesforalltransitconfigurations,accountingforanymutual
P
inclinationandorbitalprecession.Theprecessionandrelativethree-bodymotioncanincrease
E
the transit probability to as high as tens of per cent, and make it inherently time-dependent
h. overaprecessionperiodasshortas5-10yr.Circumstellarplanetsinsuchtightbinariespresent
p a tempting observational challenge: enhanced transit probabilities but with a quasi-periodic
- signaturethatmaybedifficulttoidentify.Thismayhelpexplaintheirpresentnon-detection,
o
ormaybetheysimplydonotexist.Whilstthispaperconsidersbinariesofallorientations,itis
r
t demonstratedhoweclipsingbinariesfavourablybiasthetransitprobabilities,sometimestothe
s pointofbeingguaranteed.Transitsofexomoonsexhibitasimilarbehaviourunderprecession,
a
[ butunfortunatelyonlyhaveonestartotransitratherthantwo.

1 Keywords: binaries:close,eclipsing–astrometryandcelestialmechanics:celestialmechan-
v ics,eclipses–planetsandsatellites:detection,dynamicalevolutionandstability,fundamental
9 parameters–methods:analytical
2
1
4
0
. 1 INTRODUCTION does have theoretical merit (Kraus et al. 2016). Each star is ex-
1
pected to have its own circumstellar disc, but if the two stars are
0 Weknowroughlyeverystarhostsatleastoneplanet(Petiguraet
close enough this will be tidally truncated to roughly 15-35% of
7 al.2013),andthat∼50%ofstarsexistinbinariesorhigher-order
1 thebinaryseparation(Artymowicz&Lubow1994).Binariescloser
multiples (Duquennoy & Mayor 1991). It is with natural inquisi-
: than∼10−30AUmayeventruncatethediscinteriortoitssnow
v tionthatwepondertheexistenceofplanetsinmulti-starsystems,
line, suggesting that giant planet formation is inhibited (Kraus et
i aconceptdatingbackasfarasFlammarion(1874).Forveryclose
X binaries((cid:46)0.5AU)planetsareknownonwiderorbitsaroundboth al. 2012),althoughonemustrecallthatthepreviouslymentioned
KOI-12575.3AUbinaryhostsa1.45M planet.
r stars-acircumbinaryorp-typeplanet(e.g.Kepler-16,Doyleetal. Jup
a However,despitethepaucityofcircumstellarplanetsintighter
2011).Inthispaperweconsiderthealternative:acircumstellaror
binaries,onemustrememberthatthehistoryofexoplanetsisoneof
s-typeplanetthatorbitsjustoneofthetwostarsinawiderbinary.
surprises.Thediscoveryofevenasinglecircumstellarplanetorbit-
Observationshaveuncoveredaninfluenceofthebinaryseparation;
stellar companions closer than (cid:46) 100 AU reduce the occurrence ingwithinaverytightbinarywouldalreadyposeintriguingtheo-
reticalquestions,whilstasampleofmanywouldrevolutioniseour
rate of planets whilst planets in wider binaries seemingly have a
understanding of the robustness of planet formation. One should
distributionsimilartothataroundsinglestars(Eggenbergeretal.
therefore be motivated, rather than deterred, to find such plan-
2007).Thetightestbinaryknowntohostacircumstellarplanetis
ets. The recent paper Oshagh et al. (2016) followed this philos-
the5.3AUKOI-1257(Santerneetal.2014)1,andnocircumstellar
ophy, assessing the detectability of circumstellar planets orbiting
planetsareknownineclipsingbinaries.
close eclipsing binaries using a combination of radial velocities
Theobservedpaucityofcircumstellarplanetsintightbinaries
andeclipsetimingvariations.Thisalleviatesdegeneraciesbetween
theplanetmassandsemi-majoraxisthatexistwhenusingsolely
1 AlthoughonemayalsoconsidertheborderlinecaseofWASP-81.Itcon- eclipsetimingvariations.
tainsahot-JupiterorbitingaSolarmassstar,withanouter2.4AUBrown TheworkofOshaghetal.(2016)hasrootsinaconceptually
Dwarfcompanionwithaminimummassof56.6MJup(Triaudetal.2016). similar three-body system: extra-solar moons. Methods to detect
(cid:13)c 2016RAS
2 Martin
0.5
0.25
host (cid:49)
(cid:54)
n
i
a aout s 0
in
(cid:47)
/
I
satellite (cid:54)
−0.25
companion
−0.5
Figure1.Geometryoftheinnerrestrictedthree-bodyproblem.Forcircum- −0.5 −0.25 0 0.25 0.5
(cid:54)I/(cid:47) cos(cid:54)(cid:49)
stellarplanetsinbinariestheredmasslesssatelliteistheplanetandthetwo
bluebodiesarestars.Inthisimagetheplanetisarbitrarilyorbitingthesec-
Figure2.N-bodysimulationsover5,000yearsofamasslesscircumstellar
obnludearhyossttaisr.tFhoerpelaxnoemtoaonndsththeelirgehdtmblauseslceosmspsaatneilolinteisisthtehestmaro.on,thedark pinlaitniaelt∆aIn.dStywstoemSosldaervsotiadrsofwliatrhgeaiKno=za0i-.L1idAoUv,cayocultes=ar2etAhUecaenndtradlicffiercrelenst
drawnwithsoldlines,where∆I0rangesfrom10◦(red)to40◦(purple)in
stepsof10◦.SystemswithKozai-Lidovcycleshaveamorecomplicated
themhavefocusedonthetransittiminganddurationvariationsthey evolutionandaredrawnwithdashedlinesfor∆I0rangingfrom50◦(dark
induceonatransitingexoplanet(Sartoretti&Schneider1999;Kip- green)to80◦(orange).Inallsimulations∆Ω0=0◦.
pingetal.2009)andtransitsofthemoonsthemselves(Sartoretti&
Schneider1999).However,despitesearchattemptsusingthehigh
precision and lengthy continuous photometry of Kepler (Kipping theouterorbitcontains100%oftheangularmomentum.Thetran-
etal.2015),detectionshavenotyetbeenforthcoming.IntheSolar sitprobabilitiesderivedinthispaperwouldalsoworkforamore
Systemnomoonsareknownaroundplanetslessthan1AUfrom massivesatelliteaslongastheinnerorbit’sangularmomentumre-
theSun,butanotherlessonfromtwodecadesofexoplanetsurveys mainedmuchlessthantheouter.Theorientationofeachorbitwith
hasbeentoavoidneighbourhood-inducedpreconceptions. respecttotheobserverischaracterisedbytheinclinationIandthe
This paper is an analytic and numerical study of the transit longitudeoftheascendingnodeΩ.Theorbitaldynamicsbetween
probabilityofbothcircumstellarplanetsinbinariesandexomoons thetwoorbitsaredictatedbytherelativeorientation,whichwede-
in planetary systems. The mathematics and geometry derived are fineusingthemutualinclination,
doneinfullgeneralitytobeequallyapplicabletobothcases.Ad-
vancesaremadeonexistingstudiesbyincludingNewtonianthree-
bodyorbitalprecessionoftheplanet/moonandconsideringallmu- cos∆I=cos∆ΩsinIinsinIout+cosIincosIout, (1)
tual inclinations. This is shown to both increase the transit prob- where∆Ω = Ω −Ω .Transitobservationsareonlysensitiveto
in out
abilityandmakeittime-dependent,inawaysimilartocircumbi- ∆ΩandnottheindividualvaluesofΩsowearbitrarilysetΩ =0.
out
naryplanets(Schneider1994;Martin&Triaud2014,2015;Martin
Bothorbitsareassumedtobecircular.
2017).Theworkisapplicabletoallbinaries,notjustthosewhich
Theinnerorbitfeelsgravitationalperturbationsfromtheouter
eclipse.
orbitandconsequentlyevolveswithtime,whilsttheouterorbitre-
ThegeometryandorbitaldynamicsareintroducedinSect.2.
mains constant. In Fig. 2 we illustrate the evolution of the orien-
Next,inSect.3wederivetheprobabilityoftransitingthehostbody
tation of the inner orbit with respect to the outer, using N-body
thatthesatellite(planetormoon)orbits,beforeinSect.4deriving
simulationsofa0.1AUplanetorbitingonestarina2AUbinary
theprobabilityoftransitingtheother,companionbody.Finallyin oftwoSolarstars,withdifferentmutualinclinations.Itisseenthat
Sect.5N-bodysimulationsareruntotestavarietyofexamplesand for∆I(cid:46)40◦thereisaprecessionof∆Ωoverafull360◦,whilst∆I
illustratetheobservationalsignature,beforeconcluding. remainsconstant(theevolutiontracesoutacircleinthe∆Icos∆Ω
vs∆Isin∆Ωdomain).If∆Iisinitiallygreaterthan(cid:38)40◦thenthe
precessionbehaviourisdifferentundertheinfluenceoftheKozai-
2 PROBLEMSETUP Lidoveffect(Lidov1961;Kozai1962),inwhichcase∆Ibeginsto
varyalongwithe ,evenforinitiallycircularorbits.Constancyof
Weconsidertheinnerrestrictedthree-bodyproblem,illustratedin in
∆I isfundamentalinthispapersowearerestrictedto∆I (cid:46) 40◦.
Fig.1.Namely,thereisamasslesssatellite(thecircumstellarplanet
Assumingthis,theobservationalconsequenceofthisprecessionis
inabinaryortheexomoon)onacloseorbitofperiodT andsemi-
in
thatI oscillatesaroundtheconstantI ,
majoraxisa aroundahostbody(fortheexomoonthehostisthe in out
in
planet)ofmass M andradiusR .Onawiderorbit(T ,a )
host host out out (cid:32) (cid:33)
isanoutercompanion(fortheexomoonthecompanionisthestar) 2πt
I (t)=∆Icos +I , (2)
with Mcomp andRcomp.Itisan“innerrestricted”problembecause in Tprec out
(cid:13)c 2016RAS,MNRAS000,1–??
Circumstellarplanetsinbinariesandexomoons 3
(a)FixedMcomp=1M(cid:12),variableMhost (a)
10000
Tout = 2000 d celestial!
sphere
) 1000 Tout = 1000 d
r
(y Tout = 500 d

c 100
e
pr Tout = 200 d
T
10 Tout = 100 d 2θA
1 2ΔI
0 0.2 0.4 0.6 0.8 1
M (M )
host Sun
M (M )
host Sun 2ζ
(b)FixedMhost=1M(cid:12),variableMcomp 2θ
A
10000 Tout = 2000 d
T = 1000 d
out
) 1000
r Tout = 500 d
y
(
ec 100 Tout = 200 d (b)
r
p
T T = 100 d
10 out
1
0 0.2 0.4 0.6 0.8 1
M (M )
comp Sun
M (M )
comp Sun
(c)
Figure3.PrecessionperiodcalculatedbyEq.3whereinallsimulations a
in
Tin = 10dand∆I = 30◦,whilstTout = 100d(black),200d(red),500 ζ ΔI
d (green), 1000 d (magenta) and 2000 d (blue). In (a) Mcomp is fixed at
1M(cid:12)andMhostvariesbetween0(likeforanexomoon)and1M(cid:12)(likefor Rcomp
acircumstellarplanetinabinary).In(b)insteadMhostisfixedat1M(cid:12)and
Mcompvariesbetween0(likeforasinglestarplanetwithanouterplanetary aout
companion)and1M(cid:12)(likeforacircumstellarplanetinabinary).
Figure4.(a)Side-onviewofasatellite(red)orbitingitshost(darkblue)
whichisinturninorbitaroundthecompanionbody(lightblue).Thesatel-
liteorbitvariesduringprecession,andisillustratedinthetwoextremeori-
entations.Darkandlightbluehatchedregionsprojectedontothecelestial
spherecorrespondtoobserverswhowillseetransitsonthehostandcom-
panionbody,respectively.(b)Front-onviewofanexampletransitonthe
where t is time, Tprec is the precession period and the semi- companion.(c)Zoomedside-onviewusedinderivingEq.7.
amplitudeofoscillation∆Iisconstant.Equation2assumesΩ =0
in
att=0.
TherateofprecessionwascalculatedbyMardling(2010)to 3 PROBABILITYTOTRANSITTHEHOST
be
Foracircumprimary/circumsecondaryplanetthehostbodybeing
4M +M T2 1 orbited is a star. For an exomoon, however, the host body is the
T = host comp out , (3) planetandthephotometricsignalofamoon-planettransitislikely
prec 3 M T cos∆I
comp in beyond present detection capabilities, but Cabrera & Schneider
(2007) remain optimistic. Regardless, the geometry, mathematics
wherebothorbitsareassumedtobecircular.InFig.3weevaluate andnotationusedareapplicabletobothtypesofsystem.
Eq.3forTin = 10d,Tout between100and2000d,∆I = 30◦ and Thechanceofthesatelliteontheinnerorbittransitingitshost
variousMhostandMcompbetween0and1M(cid:12).InFig.3aweseethat issolelydictatedbytheirrelativeorientation;themovementofthe
thereisonlyaweakdependenceonMhost,andthatprecessionofan hostintheouterorbitisirrelevant.Thetransitcriterionis
exomoonisslightlyfasterthanofacircumstellarplanetinabinary.
ThiscontrastswithFig.3bwhereTprecbecomesverylargeasMcomp (cid:12)(cid:12) π(cid:12)(cid:12) (cid:32)R (cid:33)
asitgoestozero.IntheexamplesshownTprecmaybeasshortas5- (cid:12)(cid:12)(cid:12)Iin− 2(cid:12)(cid:12)(cid:12)<θ=sin−1 ahost . (4)
10yr,whichisroughlycomparabletotheoriginalKeplermission’s in
lifetime.ThenodalprecessionoftheMoonaroundtheEarthtakes Ifthisorientationwereconstantthentheprobabilityoftransiting
18yr,where∆I=5.14◦. the host body would be simply akin to that of a planet around a
(cid:13)c 2016RAS,MNRAS000,1–??
4 Martin
singlestar: P = R /a .However,inclosebinariesonemust smallerthanthatforthehostbody.Theangleζwhichcharacterises
host host in
accountforvariationofthesatellite’sorbitalorientation.InFig.4a thisregionisdefinedmoreclearlyinFig.4candis
we illustrate a host body (dark blue) and companion body (light
blue)movingbackandforthalongthesolidlineconnectingthem, (cid:32)a sin∆I+R (cid:33)
with dotted circles indicating their positions at the opposite edge ζ =tan−1 ain −a cosc∆omIp . (6)
of their orbit. The red satellite orbits its host body with two ori- out in
entations drawn, corresponding to the extremes of its inclination, Like in Sect. 3 one may integrate over this angle to calculate the
I = I ±∆I,whichitreachesduringitsprecessionperiod.The probabilityoftransitability:
in out
satelliteorbitsubtendsanangleof2∆Ionthecelestialsphere.This
∆I,inadditiontothesizeofthehostbodyrelativetothesatellite, (cid:90) π/2
θ=sin−1(Rhost/ain),definestheregionofobserversonthecelestial Pcomp= sinIoutdIout,
sphere(darkbluehatchedregion)thatwillseeatransitonthehost π/2(cid:34)−ζ (cid:32)a sin∆I+R (cid:33)(cid:35)
atsomepointduringtheprecessionperiod. =sin tan−1 in comp , (7)
Tocalculatetheprobabilitythatthesatellitewilltransititshost aout−aincos∆I
atsomeunspecifiedpointintimeistocalculatetheprobabilityof ≈ ainsin∆I+Rcomp.
anobserverbeingwithinthedarkbluehatchedregioninFig.4a. a −a cos∆I
out in
Weassumethatthethreebodysystemisrandomlyorientedonthe
Aswasseenfortransitsonthehoststar,amutualinclination
celestialsphere,whichischaracterisedbyauniformdistributionof
increases the transit probability. There is also the expected result
cosI ,andhenceaprobabilitydensityfunctionofsinI .Wethen
out out thatcompanionbodiesthatarelargerandcloseraremorelikelyto
integratesinI overtheangle∆I+θ:
out betransited,althoughthecompanioncannotbetoocloseotherwise
(cid:90) π/2 the system will be unstable (e.g. Holman & Wiegert 1999). The
Phost= sinIoutdIout, efficiency of transitability, i.e. the chance that overlapping orbits
π/2(cid:34)−∆I−θ (cid:32) (cid:33)(cid:35) (5) lead to transits within a given time, is a function of the various
R
=sin ∆I+sin−1 host . orbital parameters and is not trivial to calculate, as was seen in
a
in the case of circumbinary planets (Martin & Triaud 2015; Martin
The integral is just done over a quarter of the projected celestial 2017).InSartoretti&Schneider(1999)thisiscalledthe“orbital
sphere(π/2)butsymmetrypermitsthis.Alarge∆IincreasesP probability”andequationsarederivedinthecaseofexomoonsbut
host
asthesatellitesubtendsagreaterrangeofanglesonthesky.Likein withouttheinclusionoforbitalprecession.Inthispresentpaperwe
thesinglestarcase,orbitingclosetoalargehostbodyincreasesthe usenumericalmethodsforthis.
likelihood.For∆Ibeyondafewdegreestransitswillnotbecontin-
ualbutratherwillcomeandgoovertimeasI changes.However,
in
ifthesatelliteisevertotransitthenitisguaranteedtodosowithin 5 ANALYSISANDAPPLICATIONS
asingleprecessionperiod.
5.1 Comparisonwithnumericalsimulations
Asimpletestisruntoshowthevalidityoftheanalyticwork.The
percentageofsystemstransitingover35yearsiscalculatednumer-
4 PROBABILITYTOTRANSITTHECOMPANION
icallyasaMonteCarloexperimentof2000N-bodysimulations3
Ifbothorbitswereperfectlyalignedwiththeobserver(I =I = where the three-body orientation with respect to the observer is
out in
90◦)thenthesatellitewouldtriviallytransitboththehostandcom- randomised isotropically. A system is recorded to have a transit
panion.However,transitsonthecompanionarestillpossibleforin- if the projected sky position of the satellite intersects the host or
clinedcases.InFig.4bisanexample,showingafront-onobserver companion disc at least once. The detectability of the transit sig-
perspective of a red satellite orbiting the dark blue host, transit- nal,i.e.itsdepth,durationandfrequency,isnotaccountedforin
ingthelightbluecompanion.However,unlikeinSect.3wherethe thetransitprobability.However,takingthesatelliteasapointbody
host body is stationary with respect to the satellite, the compan- is equivalent to demanding at least half of the satellite’s disc in-
ion is moving. This means that overlapping orbits of the satellite tersectsthehost/companiondisc,andhencehighlygrazingtransits
and companion, accounting for its radius, make transits possible areexcluded.Theexpectedtimingoftransitsisdiscussedbrieflyin
butnotguaranteedoneverypassing.Wecallsuchaconfiguration Sect.5.4.
“transitability”aswasdoneinMartin&Triaud(2014)todescribe ThenumericalresultsarethencomparedwithEqs.5and 7.
theconceptuallysimilarcaseofinclinedcircumbinaryplanets.Sar- Twoexamplesystemsaretested:forbothT = 11d,T = 100
in out
toretti&Schneider(1999)callthisthe“geometricprobability.” d,∆I = 30◦ andthecompanionbodyisaSun.Inonesystemthe
Toworkoutiftransitsarepossiblewecalculatethelimitsof hostbodyisalsoaSun,andintheotheritisaJupiter.Owingto
transitability,correspondingtowhentheprojectedseparationofthe Kepler’s laws, a = 0.0968 AU for the circumstellar planet and
in
starsisminimisedlikeinFig.4b.Thesatellite,whoseorbitistiedto a = 0.0095 AU for the exomoon4. The angle between the two
in
itshost,must“reach”overtothecompaniontotransit.Thisreach
ismaximisedwhenΩ =02.DuringtheprecessionperiodΩ =0
in in
(andhence∆Ω=0too)attheextremitiesI =I ±∆I. 3 A4th-orderRunge-Kuttawitha30minutetimestep,whichisthestan-
in out dardKeplerobservingcadenceandconservesenergytowithin∼10−12%.
InFig.4atheregionoftransitabilityonthecompanionbody
All Monte Carlo simulations run in this paper are done with 2000 ran-
isdenotedasalightbluehatchedregion.Generallythisregionis
domisedsystems.
4 Anexomoonat0.0968AUwithallotherparameterskeptthesamewould
beunstablebecausethehillsphereofaJupiter-massplanetismuchsmaller
2 Ωinrotatestheprojectedsatelliteorbitcounter-clockwise. thanaSolar-massstar.
(cid:13)c 2016RAS,MNRAS000,1–??
Circumstellarplanetsinbinariesandexomoons 5
60 Circumprimaryplanetsimulations
g
n g30 Tout (days)
ti50 transits on host in 100
si circumstellar planet it25 200
an40 exomoon ns 500
r a
t r20 1000
s 30 t
m transits on companion s
e m15
syst20 ceixrocmuomosntellar planet yste10
% of 10 of s 5

0 %
0 5 10 15 20 25 30 35
0
time (yr) 0 2 4 6 8 10
time (yr)
Figure5.ExampleN-bodysimulatedtransitpercentagescomparedwith
analytictheory,foracircumstellarplanetinabinary(Tin=11d,Tout=100 Figure6.N-bodysimulationsoftransitsonthehoststar(solidlines)and
d,∆I = 30◦,twoSolarstars)andanexomoon(sameparametersbutthe companionstar(dashedlines)foracircumprimaryplanetinabinarywith
hostisaJupiter).Horizontaldashedlinesaretheanalyticequations(Eq.5, Solarandhalf-Solarstars.InallsimulationsTin=11d(planetperiod)and
upper and Eq. 7, lower) and solid lines are N-body simulations. For the ∆I = 15◦.DifferentcoloursindicatedifferentbinaryperiodsTout:100d
binarycasedarkandlightredaretransitsonthehostandcompanionbodies, (red),200d(green),500d(magenta)and1000d(blue).
respectively,andforexomoonsdarkandlightbluearehostandcompanion
transits.Thetwoblackverticaldashedlinesindicatetheprecessionperiod
forthebinarycase(7.67yr)andexomoon(3.83yr).Theanalyticequation
fortransitsonthehostcoincidentallygivesalmostanidenticalvalueinboth planetisarbitrarilychosentoorbittheprimarystarandhasafixed
cases,andhencethedarkblueanddarkredhorizontaldashedlinesoverlap. T =11dandmutualinclination∆I=15◦.
in
Transits on the host star that the planet orbits initially have
a probability of R /a , which then increases over time due to
host in
orbits,∆Ω,israndomisedbetween0and360◦,meaningthateach precession.ForTout =100dacompleteprecessionperiodiscom-
pletedwithinthe10yearobservingwindow.Attheotherextreme,
phase of the precession period is uniformly sampled. The results
forT = 1000dtheprecessionperiodissolongthatthetransit
areshowninFig.5. out
probabilitybarelyincreasesover10years.Ultimately,accordingto
In all four cases (i.e. the binary and exomoon cases and for
Eq.5thetransitprobabilitywillreachthesamevalueregardlessof
hostandcompaniontransits)theanalytictheoryaccuratelymatches
T , but practically, accounting for orbital precession is only im-
theamountoftransitsafterobservingformanyyears.Aspredicted, out
portantforbinariesofacoupleofhundreddaysperiod.Transitson
thepercentageofsystemstransitingthehostbodyplateausinless
the companion star are systematically less likely than transits on
thanaprecessionperiod.Thepercentageofsystemstransitingthe
host star almost instantaneously (within one T = 11 d) rises to thehoststar,andsimilarlytendtobefavouredintighterbinaries.
in
R /a ,sincethisistheprobabilityforastaticorbitsanspreces-
host in
sion.Coincidentallythisratioisalmostthesameinthetwosystems
tested,despiteR anda beingsignificantlydifferentinthetwo 5.2.2 Effectoftheinnerperiod
host in
cases, and hence the dark red and dark blue curves both start at
WhilstincreasingT wasshowntoalwaysdecreasetransitproba-
nearlythesamepercentageatt = 0andalsoplateauatnearlythe out
bilitiesonbothstars,forT itislessstraightforward.Equation5
samepercentageaccordingtoEq.5.Whatisdifferentbetweenthe in
predictsP todecreaseasT increases,whilstconverselyEq.7
host in
two cases is the precession period, with the exomoon precessing
predictsahigherP forlongerT .Thesepredictionsaretruein
comp in
faster. The percentage of systems with transits on the companion
thelongterm,butasweknow,therateofprecessioniskeyintran-
body rises slower and continues to do so after the precession pe-
sitsoccurringwithinanobservabletimeframe,andT ∝ 1/T .
prec in
riod, as expected, and is significantly higher for the binary case
N-bodysimulationsareshowninFig.7foracircumprimaryplanet
(lightred)thantheexomooncase(lightblue).Thislatterresultis withdifferentT andafixedT =100d.Fortransitsonthehost
in out
becausea ismuchlargerinthebinarycase(0.0968AU)compared
in star we see there is a balance: short-period planets are geometri-
totheexomoon(0.0095AU).
callyclosertothestar(that’sgood)butprecessslower(that’sbad).
A short observing time (< 1 year) favours short T but after 10
in
years the 10 day planet transits the most. Only the 20 day planet
5.2 Circumstellarplanetsinbinaries
completesafullprecessionperiod.Fortransitsonthecompanion
5.2.1 Effectoftheouterperiod staritissimpler:longerperiodplanetsaremorelikelytotransit,as
theyarenotonlygeometricallyfavourablebutprecessfaster.
Severaltestsareruntocalculatethetime-dependenttransitprob-
abilityofcircumstellarplanetsinbinaries.Numericalsimulations
are run over 10 years to cover the existing Kepler mission (four 5.2.3 Effectofthemutualinclination
years)andwhatmaybeconsideredreasonableforfuturemissions.
First,inFig.6wetesttheeffectofthebinaryperiod,T ,between Akeyresultofthisworkistheimportanceof∆I.InFig.8therela-
out
100 and 1000 d. The binary is constructed with a Solar primary tionshipbetweenthetransitprobabilityand∆I isshownforan11
andhalf-Solarsecondary(intermsofbothmassandradius).The daycircumprimaryplanetina100daybinarywith∆I between0
(cid:13)c 2016RAS,MNRAS000,1–??
6 Martin
Circumprimaryplanetsimulations Exomoonsimulations
35
g g30
n n
i30 i
t t
nsi25 nsi25 Tout (days)
a a 100
r r20
t20 t 200
ms Tin (day1s) 10 ms 15 500
e15 e 1000
t 3 20 t
s s10
y10 5 y
s s

of 5 of 5

% %
0 0
0 2 4 6 8 10 0 2 4 6 8 10
time (yr) time (yr)
Figure7.N-bodysimulationsoftransitsonthehoststar(solidlines)and Figure9.N-bodysimulationsoftransitsonthehostplanet(solidlines)and
companionstar(dashedlines)foracircumprimaryplanetinabinarywith companionstar(dashedlines)foranexomoonaroundaJupiterplanetthat
Solarandhalf-Solarstars.InallsimulationsTout = 100d(binaryperiod) isinturnorbitingaSolarstar.InallsimulationsTin =11d(moonperiod)
and∆I = 15◦.DifferentcoloursindicatedifferentplanetperiodsTin:1d and∆I =15◦.DifferentcoloursindicatedifferentplanetperiodsTout:100
(black),3d(red),5d(green),10d(magenta)and20d(blue). d(red),200d(green),500d(magenta)and1000d(blue).
Circumprimaryplanetsimulations
star)aresignificantlyreducedintheexomooncase.Thisisbecause
g70 ΔI (deg) for the same Tin = 11 d period ain is much smaller for an exo-
n moon (0.0095 AU) than for the circumsecondary planet (0.0968
i60 40
t AU).Thisreducedprobabilityonthestarisunfortunatesincethose
i 20
ns50 10 aretheeventslikelytobeactuallyobservable.
a
r 5
t40
s 0 5.4 Observationalsignature
m
e30
t TwoexamplesfromthecircumprimarysimulationsinFig.6with
ys20 ∆I=15◦,Tin=11dandTout=100dareusedtoillustratethetim-
s
ingoftransitsonthehostandcompanionstar,theresultsofwhich
f
o10 areshowninFig.10a.Bothofthesebinarieseclipse.Thevariation

% of Iin is shown to follow Eq. 2, except for tiny additional short-
0 periodfluctuations.Consecutivesequencesof∼ 18transitsonthe
0 2 4 6 8 10
time (yr) hostoccurwhenI isnear90◦.Thistransientsignalmaybemissed
in
ifsearchalgorithmsarenottunedtoanalysethelightcurveinseg-
Figure8.N-bodysimulationsoftransitsonthehoststar(solidlines)and ments.Transitsonthecompanionarerarer,sparselydistributedand
companionstar(dashedlines)foracircumprimaryplanetinabinarywith canoccuratatanyI .Inbothexamplesthereareonlythreetransits
in
Solarandhalf-Solarstars.InallsimulationsTin=11d(planetperiod)and onthecompanion,whichwastypicalofthe2000simulationsrun;
Tout = 100d(binaryperiod).Differentcoloursindicatedifferentmutual planetsthattransitedthecompaniondidsoanaverageof3.8times.
inclinations∆I:40◦ (black),20◦ (red),10◦ (green),5◦ (magenta)and0◦
Thesewillbehardertodetectandcharacterise,inasimilarwayto
(blue).
inclinedcircumbinaryplanets(Martin&Triaud2015;Kostovetal.
2014;Martin2017).
WenextshowinFig.10bthecomparableresultsfortwoofthe
and40◦.Fortransitsonthehoststartheinitialvalueisthesameir- exomoonsimulationsfromFig.9,whereagainweuse∆I = 15◦,
respectivelyof∆I,butthehigherthe∆Ithehigherthetransitprob- T = 11dandT = 100d.TheamplitudeofthevariationofI
abilityisovertime.Evenamere5◦ mutualinclinationcandouble in out in
is the same as for the circumprimary planets in Fig. 10a, but the
the transit probability on the host star within just four years. For
precessionperiodisslightlyshorterasexpectedfromEq.3.Tran-
coplanarplanetsthetransitprobabilityisstaticandminimiseddue
sitsonthehostplanetagainoccurinconsecutiveblockswhenthe
tothelackofprecession.Fortransitsonthecompanionstarwesee moon’sinclinationreaches90◦,butashasbeenstressedthroughout
asimilartrendof∆Iaidingtransitprobabilities.
thispaperthisislikelynotanobservablephenomenon.Transitson
thecompanionstar,however,occuratanyI andinthesetwoex-
in
amplesinFig.10bareverynumerous:14inthetopsimulationand
5.3 Exomoons
37inthebottom,thelattercorrespondingtoonemoontransitevery
In Fig. 9 we run similar simulations to Fig. 6 but with a Jupiter orbitofitsplanet.ThesetwoexamplesinFig.10barerepresenta-
hostand Solarcompanion.Transitson thehostbody(the planet) tiveofthe2000simulationsrun,asmoonsthatwereobservedto
occurataslightlyfasterratethanthecircumprimaryplanetcase, transitthecompanionstaronaveragedidso21times,muchmore
owing to a shorter T , but transits on the companion body (the thantheaverageof3.8inthecircumprimaryexample.Thediffer-
prec
(cid:13)c 2016RAS,MNRAS000,1–??
Circumstellarplanetsinbinariesandexomoons 7
110
30
100
25
90
20
t
g) 80 un15
de 70 Co10
( 110

n 5
Ii100
0
90 70 80 90 100 110
I (deg)
80 out
700 2 4 6 8 10 (a)HistogramofIoutforalltransitingcircumprimaryplanetsfromthesimu-
time ((yyera)rs) lationsinFig.6with∆I =15◦,Tin =11dandTout =100d.Thedarkblue
histogramisforplanetstransitingthehoststarandthelightbluehistogram
(a) Time variation of Iin for two example circumprimary planets from the isfortransitsonthecompanionstar.Thehistogrambinwidthsare1◦.When
simulationsinFig.6with∆I = 15◦,Tin = 11dandTout = 100d,where Iout≈90◦theplanetisorbitinginaneclipsingbinary.
thebinaryisknowntoeclipse.Transitsonthehostandcompanionstarare
denotedbydarkblueandlightbluediamonds,respectively.Thedarkblue 30
diamondslookmergedtogetherbecauseonthehostthereareconcentrated
25
batchesof∼18transitsseparatedbyonlyTinwheneverIin≈90◦.
110 20
t
n
100 u15
o
C
10
90
5
) 80
g 0
e 70 80 90 100 110
d 70
( 110 I (deg)
n out
i
I 100
(b)Thesameas(a)butforexomoonsimulationsfromFig.9with∆I=15◦,
90 Tin = 11dandTout = 100d.WhenIout ≈ 90◦ theexomoonisorbitinga
transitingplanet.
80
Figure11.
70
0 2 4 6 8 10
time (yr)
(b)Thesameas(a)butfortwoexomoonsimulationstakenfromFig.9with
∆I =15◦,Tin =11dandTout =100d.Inbothexamplestheplanethosting
the moon is known to transit the star. Like in (a) transits on the host (the
planet)comeinlargeconsecutivebuncheswhenIin ≈90◦buttransitsonthe
companion(thestar)aremuchmorenumerousthanin(a),owingtoasmaller almostguaranteedtobewithinbothofthehatchedregionsdesig-
ainforthesameTin. nating transits on the host and companion, and hence we have a
favourablebias.
Figure10. Tofurtherdemonstratethis,wetaketransitingcircumprimary
planetsfromthesimulationsinFig.6,with∆I = 15◦,T = 100
out
dandT = 11d,andplotahistogramof I ,showninFig.11a
in out
enceisthatfortheexomoona =0.0095is∼10timessmallerthan forbothhost(darkblue)andcompaniontransits(lightblue).For
in
forthecircumprimaryplanetwherea =0.0968AU,simplydueto transitsonbothbodiesthereisseentobearangeofI forwhich
in out
Kepler’slaws.Wethereforehaveaninterestingtrade-off:widening transits are permissible, centred around 90◦. This range is much
thesatellite’sorbitincreasesincreasesitschanceoftransitabilityon largerfortransitsonthehostthanthecompanion.Withinthisrange
thecompanion,andhencethelong-termtransitprobabilityaccord- thehistogramappearsroughlyflat,butmorethoroughsimulations
ingtoEq.7,butdoingsodecreasestheefficiencyoftransitability, wouldbeneededtoprovethis.
makingthesatellitehardertodetect. Inthe2000simulationstherewere618planetstransitingthe
hoststarand95%ofthemdidsoonnon-eclipsingbinaries.Thisis
duetothewiderangeofI permittingtransitsandthesimplefact
5.5 Advantageofeclipsingbinariesandexomoonsaround out
thatmostbinariesdonoteclipse.Thisnumberisreducedto67%
transitingplanets
fortransitsonthecompanion.Whilsteclipsingbinariestherefore
Theresultsthroughoutthispaperarecalculatedassumingistrorop- arenotanecessityfortransits,theyarehighlybeneficialasinthis
icallydistributedsystems(auniformdistributionofcosI ).What smallsampletransitsoccurredonthehostandcompanionatarate
out
happens if we instead know that I ≈ 90◦, i.e. a circumstellar of100and97%,respectively,withinthesimulated10yr.
out
planetwithinaneclipsingbinaryoranexomoonaroundatransit- Eventhougheclipsingbinariesarebiasedtowardsshortperi-
ingplanet?FromFig.4itisevidentthatobserversonthecelestial ods,andonemaynotexpectcircumstellarplanetsinsaya<100d
spherewhoseethecompanionbodyinfrontofthehostbodyare binary,theKeplermissiondiscovered167eclipsingbinarieswith
(cid:13)c 2016RAS,MNRAS000,1–??
8 Martin
periodsof100dandlonger5.Iftheydohostplanetsthereisavery 7 ACKNOWLEDGEMENTS
highchanceofthemtransiting,albeitwithacomplicatedsignature.
ThankyoutomyPhDsupervisorStephaneUdryforconstantsup-
InFig.11bweproducethesamehistogrambutforexomoon
port and discussions, Amaury Triaud for our circumbinary work
simulations from Fig. 9, again with ∆I = 15◦, T = 100 d and
out thatinspiredthispaper,RosemaryMardlingfortheorbitalpreces-
T = 11 d. Compared with the circumprimary case the range of
in siontimescale,andJeanSchneiderfortheFlammarion(1874)ref-
I permittingtransitsonthehostisroughlythesame.However,
out erence. This work was aided by the thoughtful comments of an
transitsonthecompanionstaronlyoccurinamuchnarrowerrange
anonymous referee. Finally, I thank the financial support of the
ofI around90◦.Consequently,only38%ofexomoonstransited
out SwissNationalScienceFoundationoverthepastfouryears.
the star did so when the planet did not transit. This suggests that
forsearchesfortransitingexomoonsitisrecommendedbutnotre-
quiredthattheplanettransits.
REFERENCES
Finally,anotheradvantageofasatellitethatorbitsaneclips-
ing binary or a transiting planet is that those eclipses/transits Artymowicz,P.&Lubow,S.H., 1994,ApJ,421,651
would also be identified by a continuous transit survey like Ke- Cabrera,J.&Schneider,J., 2007,A&A,464,1133
pler,TESSorPLATOandcanprovidecomplementaryinformation Cochran,W.D.,Hatzes,A.P.,Butler,P.R.&Marcy,G.W. 1997,
likeeclipse/transittimingvariations(Sartoretti&Schneider1999; ApJ,483,1
Kippingetal.2009,2015;Oshaghetal.2016). Doyle,L.R.,Carter,J.A.,Fabrycky,D.C.,etal. 2011,Science,
333,1602
Duquennoy,A.&Mayor,M., 1991,A&A,248,485
Eggenberger,A.,Udry,S.,Chauvin,G.,etal., 2007,A&A,474,
6 CONCLUSION
273
TheequationsandN-bodysimulationsinthispaperhavedemon- Faigler,S.&Mazeh,T., 2011,MNRAS,415,3921
stratedthatforcircumstellarplanetsinclosebinariesandexomoons Flammarion,C, 1874,LaNature-Revuedessciences,161
inplanetarysystems,orbitalprecessionenhancestransitprobabili- Holman,M.J.,&Wiegert,P.A.1999,AJ,117,621
tiesandmakestheminherentlytime-dependent.Thetransitproba- Kipping,D.M., 2009,MNRAS,392,181
bilityofthesatellite(exomoonorcircumstellarplanet)onthehost Kipping, D. M., Schmitt, A. R., Huang, X., Torres, G., et al.,
body(fortheexomoonthatisitsplanet)risesroughlylinearlyover 2015,ApJ,813,14
aprecessionperiodbeforeplateauingatthevaluecalculatedinthis Kostov,V.B.,McCullough,P.R.,Carter,J.A.,etal. 2014,ApJ,
paper.Thisprobabilitymaybeashighastensofpercent,largely 784,14
as a function of the mutual inclination, and may be reached over Kozai,Y., 1962ApJ,67,591
aprecessionperiodasshortas5-10yrforouterorbitsof100-200 Kraus,A.L.,Ireland,M.J.,Hillenbrand,L.A.&Martinache,F.,
d.Transitscomeandgo,exhibitingconsecutivesequenceswhen- 2012,ApJ,745,19
everthesatellite’sinclinationisnear90◦.Transitsonthecompan- Kraus, A. L., Ireland, M. J., Huber, D., Mann, A. W. & Dupuy,
ionbody(fortheexomoonthatisitsstar)aresystematicallyless T.J., 2016,ApJ,152,8
likely but still have a time-dependence, which rises even slightly Lidov,M.L., 1961,Iskusst.SputnikiZemli,8,5
beyondtheprecessionperiod.Theymayoccuratanypointduring Mardling,R.A., 2010,MNRAS,407,1048
the satellite’s precession period but depending on the orbital pa- Martin,D.V.&Triaud,A.H.M.J., 2014,A&A,570,A91
rametersmaybeinasparsesequenceanddifficulttocharacterise. Martin,D.V.&Triaud,A.H.M.J., 2015,MNRAS,449,781
This odd transit signature draws parallels with inclined cir- Martin,D.V., 2017,MNRAS,465,3235
cumbinaryplanets(Martin&Triaud2015;Martin2017),suchas Oshagh,M.,Heller,R.,Dreizler,S., 2016,arXiv:1610.04047
themarginallyinclined(4◦)Kepler-413(Kostovetal.2014).Even Petigura,E.A.,Howard,A.W.&March,G.W., 2013,PNAS,
if large circumstellar planets in close binaries have individually 110,19273
significant transits, they may be missed by automated detection Prsa,A.,Batalha,N.,Slawson,R.W.,etal., 2011,AJ,141,3
pipelinesthatrelyontransitregularity.Forexomoonsthereisthe Santerne, A., Hebard, G., Deleuil, M., et al., 2014, A&A, 571,
added difficulty of the transits themselves generally being unde- A37
tectablysmall,andacleverdynamicalphase-foldingwouldbenec- Sartoretti,P.&Schneider,J., 1999,A&AS,134,553
essarytorecoverasignal. Schneider,J., 1994,Planet.SpaceSci.,42,539
Searches for circumstellar planets in close binaries are en- Triaud,A.H.M.J.,Neveu-VanMalle,M.,Lendl,M.,Anderson,
hanced (and sometimes guaranteed) if the binary is known to D.R.,etal., 2016,arXiv:1612.04166
eclipse, particularly if transits are desired on both stars, but this
is not a necessity. The sample of almost 3,000 eclipsing binaries
found throughout the Kepler mission, with more to come partic-
ularlyfromPLATO,provideshopeoffindingsuchplanetsifthey
exist.Tofindexomoonstransittheirstaritishighlyrecommended
thattheplanettransitstoo.
Finally, we note that this work may be applied to
eclipses/transitsofclosetriplestarsystemsandsimilar-massbinary
planets.
5 OriginalsampleinPrsaetal.(2011)anduptodatecatalogavailableat
http://keplerebs.villanova.edu/.
(cid:13)c 2016RAS,MNRAS000,1–??