On the formation of hot and warm Jupiters via secular high-eccentricity migration in stellar triples

0.75 MB·

by  Adrian S. Hamers

MNRAS000,1–14(2017) PreprintJanuary10,2017 CompiledusingMNRASLATEXstylefilev3.0
On the formation of hot and warm Jupiters via secular
high-eccentricity migration in stellar triples
Adrian S. Hamers1⋆
1InstituteforAdvancedStudy,SchoolofNaturalSciences,EinsteinDrive,Princeton,NJ08540,USA
Accepted2017January5.Received2016December5;inoriginalform2016October21
7
ABSTRACT
1
HotJupiters(HJs)areJupiter-likeplanetsorbitingtheirhoststarintightorbitsofafewdays.
0
Theyarecommonlybelievednottohaveformedinsitu,requiringinwardsmigrationtowards
2
the hoststar. One ofthe proposedmigrationscenariosis secular high-eccentricityor high-e

n migration, in which the orbit of the planet is perturbed to high eccentricity by secular pro-
a cesses, triggering strong tidal evolution and orbital migration. Previous theoretical studies
J
have considered secular excitation in stellar binaries. Recently, a number of HJs have been

6 observedinstellartriplesystems.Inthelatter,theseculardynamicsaremuchrichercompared

tostellarbinaries,andHJscouldpotentiallybeformedmoreefficiently.Here,weinvestigate
]
P this possibility by modeling the secular dynamicaland tidal evolution of planets in two hi-
E erarchicalconfigurationsinstellartriplesystems.WefindthattheHJformationefficiencyis
higher compared to stellar binaries, but only by at most a few tens of per cent. The orbital
.
h propertiesof theHJs formedinthe simulationsareverysimilar to HJs formedin stellar bi-
p
naries,andsimilarly tostudiesofthe latterwe findnosignificantnumberofwarmJupiters.
-
o HJsareonlyformedinoursimulationsfortripleswithspecificorbitalconfigurations,andour
r constraints are approximatelyconsistent with current observations. In future, this allows to
t
s ruleouthigh-emigrationinstellartriplesifaHJisdetectedinatriplegrosslyviolatingthese
a constraints.
[
Keywords: planetsandsatellites:dynamicalevolutionandstability–planet-starinteractions

1 –gravitation
v
3
3
7
1 1 INTRODUCTION Chatterjeeetal.2008;Ford&Rasio2008;Juric´&Tremaine2008;
0 Nagasawaetal.2008;Beaugé&Nesvorný2012);
1. Hot Jupiters (HJs) are Jupiter-like planets orbiting their host star (ii) secular Lidov-Kozai (LK) oscillations (Lidov 1962; Kozai
ontightorbits,downwardof10d.Despitethefirstdetectionofa
0 1962) induced by a distant companion star or an additional
HJovertwodecadesago(Mayor&Queloz1995),itisstillunclear
7 (massive) planet on an inclined orbit (Wu&Murray 2003;
howtheseplanetscouldhaveformed.Itiscommonlybelievedthat
1 Fabrycky&Tremaine 2007; Naozetal. 2012; Petrovich 2015a;
planetformationintheprotoplanetarydiskphaseisnotefficientso
: Andersonetal.2016;Petrovich&Tremaine2016);
v closetothestar(e.g.Linetal.1996),althoughsomerecentstud-
i (iii) secularexcitationinducedbyacloseandcoplanar,butec-
X ies have considered the possibility of in situ formation through centricplanetarycompanion(Petrovich2015b;Xue&Suto2016),
core accretion (Leeetal. 2014; Boleyetal. 2016; Batyginetal.
r and
a 2016). Discounting the latter possibility, then the proto-HJ must
(iv) secularchaosinmultiplanetsystemswithatleastthreeplan-
haveformedinregionsfurtherawayfromthestar,i.e.beyondthe
etsin mildlyinclined and eccentric orbits(Wu&Lithwick2011;
icelineofonetoseveralAU,andsubsequentlymigratedinwards.
Lithwick&Wu2011,2014;Hamersetal.2016a).
Two main migration scenarios have been considered: (1)
diskmigratione.g.Goldreich&Tremaine1980;Lin&Papaloizou TheorbitalperiodsofHJspeakaround∼3−5d.Inthecase
1986;Bodenheimeretal.2000;Tanakaetal.2002),and(2)migra- ofdiskmigration,migrationinprincipleproceedsuntiltheplanet
tion induced by tidal dissipation inthe HJ, requiring high orbital isengulfed by thestar,and truncationof thediskneeds tobein-
eccentricity,alsoknownas‘high-e’migration.Severalmechanisms vokedinordertoexplaintheobservedorbitalperioddistribution.
havebeenproposedtodrivethehigheccentricityrequiredforhigh- Incontrast,high-emigrationinducedbysecularprocessesnaturally
emigration,including predictsa‘stalling’orbitalperiodof∼ 3−5d.Also,fordiskmi-
grationtheobliquity(theanglebetween thestellar spinandorbit
(i) close encounters between planets (Rasio&Ford 1996;
of the planet) is expected to be (close to) zero, unless the stellar
spin axis was initiallymisaligned with respect to the protoplane-
⋆ E-mail:[email protected] tarydisk(e.g.Bateetal.2010;Foucart&Lai2011;Batygin2012;
(cid:13)c 2017TheAuthors
2 Hamers
Figure1.Schematicrepresentationsofhierarchicalorbitsofplanetsinstellartriplesystems,usingmobilediagrams(Evans1968).Notethatthesediagrams
onlydepictthehierarchyofthesystem,andnottherelativesizesandorientationsoftheorbits.Forhierarchicalfour-bodysystems,therearegenerallytwo
distinct hierarchical configurations (‘2+2’and‘3+1’). However, taking intoaccount that thestarsaredistinct fromtheplanet, there areinthiscase four
physicallydistinctconfigurations shownineachofthepanels.Inthiswork,onlytheconfigurations inpanels1and2areconsideredinthecontextofHJ
formationthroughsecularhigh-emigration.
Lai2014).Incontrast,largeobliquities,evenretrograde,havebeen otherstars.Thebinarityofthedistantpaircanaffecttheorbitalevo-
observed(e.g.Winnetal.2011;Mazehetal.2015),andtheseare lutionoftheplanetarounditshoststar.Inparticular,Pejchaetal.
morenaturallyexplainedinthecaseofsecularhigh-emigration,in (2013) showed that in these systems the parameter space for ex-
whichtheorbitoftheplanetiscontinuouslychangingitsorienta- citinghigheccentricitiestriggeredbyflipsoftheorbitalplanesis
tionwithrespect tothestar before theevolution isdominated by larger compared to the equivalent triple systems, potentially en-
tidaldissipation. hancingtheformationrateofHJscomparedtothecaseofasingle
stellarcompanion.
However, secular high-e migration generally faces the prob-
The observations of HJs in stellar triples mentioned above
lemthatthetheoreticaloccurrenceratesareaboutanorderofmag-
constituteonlyoneexampleoftheconfigurationsinwhichplanets
nitudetoolowcomparedtoobservations.Also,ahighefficiencyof
canorbitstarsinstellartriplesystems.Forhierarchicalfour-body
tidaldissipationintheHJisrequired,andthepredictedorbitalperi-
systems,thereexisttwoclassesoflong-termstableconfigurations:
odsareontheshortendoftheobservationsunlesstidaldissipation
the‘2+2’ and the ‘3+1’ configurations1. In the‘2+2’ or ‘binary-
isveryefficient(e.g.Petrovich2015a;Andersonetal.2016).
binary’ configuration (cf. panel 1 of Fig.1), two binary pairs are
Recently, HJs have been found orbiting stars in stellar boundinawiderorbit.Inthe‘3+1’or‘triple-single’configuration
triple systems. The three HJs are WASP-12b (Hebbetal. 2009; (cf.panels2-4ofFig.1),ahierarchicaltriple,composedofaninner
Bergforsetal.2013;Bechteretal.2014),HAT-P-8b(Lathametal. andouterorbit,isorbitedbyamoredistantfourthbody.
2009; Bergforsetal. 2013; Bechteretal. 2014), and KELT-4Ab Inthecaseof‘2+2’systems,theplanetorbitsoneofthestars,
(Eastmanetal. 2016). In these systems, the HJs orbit the tertiary andonlyonephysicallyuniqueconfigurationispossible(cf.panel
star in a stellar triple system, i.e. the HJhost star is orbited by a
moredistantpairofstars.Fromadynamicalpointofview,sucha
systemisahierarchicalfour-bodysysteminthe‘2+2’or‘binary-
binary’ configuration, where one of the binaries is composed of 1 Here,wedonotconsiderthepossibilityofnon-hierarchical orbitssuch
asinglestar+planet,andtheotherbinaryiscomposedofthetwo astrojanorbits.
MNRAS000,1–14(2017)
HJs intriples 3
1ofFig.1).For‘3+1’systems,threephysicallydistinctconfigura- Symbol Description (Rangeof)value(s)
tionsexist.Theplanetcanorbit
m0 Stellarmass 1M⊙
(i) one of the starsinthe innermost binary (i.e.an ‘S-type’ or m1 Planetarymass 1MJ
‘circumprimary’orbit;cf.panel2ofFig.1); m2 Stellarmass q2m0a
(ii) the(centerofmassofthe)innermostbinary(i.e.a‘P-type’ m3 Stellarmass q3m0a
or‘circumbinary’orbit;cf.panel3ofFig.1); R0 Stellarradius 1R⊙
(iii) the(centerofmassofthe)triplesystem(i.e.a‘circumtriple’ R1 Planetaryradius 1RJ
R2 Stellarradius (m2/M⊙)0.8R⊙b
orbit;cf.panel4ofFig.1). R3 Stellarradius (m3/M⊙)0.8R⊙b
η Tidaldisruptionfactor 2.7c
Note that the nomenclature ‘S-type’ and ‘P-type’, introduced by
Dvorak(1982),strictlyonlyappliestobinarystarsystems. tV,⋆ Stellarviscoustime-scale 5yr
Incase(i),theplanetisorbitingasinglestar;itcanapproach tV,1 Planetaryviscoustime-scale 0.0137,0.137,1.37yr
kAM,⋆ Stellarapsidalmotionconstant 0.014
thisstaratashortdistance(i.e.afractionofanAU)whilemain-
kAM,1 Planetaryapsidalmotionconstant 0.25
tainingdynamicalstability,i.e.notdestroyingthehierarchyofthe rg,⋆ Stellargyrationradius 0.08
system. However, in cases (ii) and (iii), the orbit of the planet is rg,1 Planetarygyrationradius 0.25
verylikelyunstableiftheplanetweretoapproachoneofthestars Ps,⋆ Stellarspinperiod 10d
atsuch ashort distancebecause of theperturbations bytheother Ps,1 Planetaryspinperiod 10hr
stars.Therefore,for‘3+1’systems,likelyonlycase(i)isrelevant θ0 Stellar obliquity (stellar spin- 0◦
forHJformationthroughsecularhigh-emigrationbecausethelat- planetaryorbitangle)
terrequiresrepeatedclosepassagestoastar.Nevertheless,itmight a1 Planetaryorbitalsemimajoraxis 1-4AU
bepossibletoformHJsthroughtidalcaptureofdynamicallyunsta- P2 Periodoforbit2 1-1010dd
bleplanetsbystars,triggerede.g.bysecularevolution.Wedonot P3 Periodoforbit3 1-1010dd
e1 Planetaryorbitaleccentricity 0.01
consider the latter possibility here. In case (i), similarlyto ‘2+2’ ei Orbitieccentricity(i,1) 0.01-0.9e
four-bodysystems,coupledLKoscillationsbetweentheorbitscan i1 Planetaryorbitalinclination 0◦
result inextremelyhigh eccentricitiesandsecular chaotic motion ii Orbitiinclination(i,1) 0-180◦
(Hamersetal.2015). ωi OrbitiArgumentofpericenter 0-360◦
The above considerations suggest a higher formation effi- Ωi Orbit i Longitude of ascending 0-360◦
ciencyofHJsinstellartriplescomparedtostellarbinaries2,andthis node
motivatesstudyoftheefficiencyofHJformationthrough secular aMassratioqisampledfromaflatdistributionbetween0.1and1.0.
high-emigrationinstellartriples.Inthispaper,weinvestigatethis bKippenhahn&Weigert(1994).
through numerical simulations,focussing onthe‘2+2’and‘3+1’ cGuillochonetal.(2011);cf.equation2).
(i)configurations(i.e.panels1and2ofFig.1).Wepayparticular dLognormaldistribution(Raghavanetal.2010).
emphasisoncomparingtheefficiencyofHJproductiontothecase eRayleighdistributionwithanrmswidthof0.33(Raghavanetal.2010).
ofastellarbinary(i.e.panels1or2ofFig.1withm removed).
3 Table1.Description ofquantities relevant forSection4(seealsoFig.1)
Furthermore,weconsidertheconditionsnecessaryforHJfor- andtheirassumed(rangesof)value(s).Thestellarquantitieswiththe‘⋆’
mation through high-e migration in triples, and show that these subscriptapplytoallthreestars,i.e.i∈{0,2,3}.Theorbitalelementsare
conditionsexcludetheformationofHJsthroughhigh-emigration definedwithrespecttoanarbitraryreferenceframe.Notethattheranges
in triples with certain configurations. Although not presently the oforbital parameters are the ranges before applying dynamical stability
case,afuturedetectionofaHJviolatingtheseconditionswouldbe constraints.
strongindicationforanotherformationmechanismatworksuchas
diskmigration.
aretwotothreeorbitsinthesystem,labeled1to3.Forexample,
Thestructureofthispaperisasfollows.InSection2,wede-
for ‘2+2’ systems, the semimajor axis of the orbit of the planet
scribethe secular method and other assumptions. Wegive anex-
arounditsstarisdenotedwitha ,thesemimajoraxisoftheorbitof
ample of the rich secular dynamics of planets in stellar triplesin 1
starsm andm isdenotedwitha ,andthesemimajoraxisofthe
Section3.InSection4,wepresenttheresultsfromthepopulation 2 3 2
‘superorbit’ofthetwoboundpairsisdenotedwitha .
synthesisstudy.WediscussourresultsinSection5,andconclude 3
inSection6.
2.2 Seculardynamics
2 METHODSANDASSUMPTIONS InHamersetal.(2015),theequations describingthesecular evo-
lution of hierarchical four-body systems were given for both
2.1 Configurationsandnotation the ‘2+2’ and ‘3+1’ configurations. Here, we used the updated
code SecularMultiple (Hamers&PortegiesZwart 2016) inter-
Inpanels1and2ofFig.1,thehierarchicalconfigurationsofplan-
faced in the AMUSE framework (PortegiesZwartetal. 2013;
etsintriplesconsideredinthisworkaredepictedschematically.In
Pelupessyetal. 2013). The SecularMultiple code applies to ar-
additiontothesetwoconfigurations,weincludedthecaseofastel-
bitrary hierarchies and numbers of bodies; evidently, its underly-
larbinary,i.e.panels1or2ofFig.1withm removed.Invariably,
3
ingalgorithmreduces tothesameequations for four bodies. The
weusem todenotethemassofthestarthattheplanetisorbiting,
0
method is based on an expansion of the Hamiltonian in terms of
andm todenotetheplanetarymass.Theothertwostellarmasses
1
ratiosoftheorbitalseparations(ratiosofallcombinationsofsmall
aredenotedwithm andm .Dependingontheconfiguration,there
2 3
tolargeseparations),whichareassumedtobesmall.Subsequently,
the Hamiltonian is averaged assuming Keplerian motion, and the
2 Nottakingintoaccounttherelativefrequenciesofbinariesandtriples. resultingequationsofmotionfortheorbitalvectorsekandhkforall
MNRAS000,1–14(2017)
4 Hamers
orbitskareintegratednumerically.NotethatunlikeinHamersetal.
(2015),SecularMultipleincludesthe‘cross’termsattheoctupole
order that depend on all three orbits simultaneously, and which
werenotincludedintheintegrationsofHamersetal.(2015).Here, 2+2 e1 2+2 i13 70
inadditiontoincludingalltermsattheoctupoleorder,wealsoin- 0.8 e2 i23
e3 60
cludedtermscorrespondingtopairwisebinaryinteractionsupand
0.6 50
includingthefifthorderintheseparationratios.Relativisticcorrec- g
e
tionswereincludedtothefirstpost-Newtonian(PN)order,neglect- ei 40/d
ingtermsinthePNpotentialassociatedwithinteractionsbetween 0.4 30irel
binaries. 20
0.2
10
2.3 Tidalevolution binary e1 binary i12 70
0.8 e2
Weincludedtidalevolutionfortheplanetandstarsdirectlyorbiting 60
otherbodies.Forthe‘2+2’configuration,tidesweretakenintoac- 0.6 50
countinbodies0and1inorbit1,andbodies2and3inorbit2(cf. eg
Fig.1).Forthe‘3+1’configuration,tidesweretakenintoaccount ei0.4 3400i/drel
onlyforbodies0and1inorbit1.Fortidalevolution,weonlycon-
sidered two-body interactions, i.e. we neglected perturbations of 0.2 20
otherbodiesonthedirecttidalevolution.Evidently,indirectcou- 10
plingistakenintoaccountbythechangingoftheeccentricitiesdue
0 2 4 6 8 0 2 4 6 8 10
tosecularevolution.
t/Gyr t/Gyr
We adopted the equilibrium tide model of Eggletonetal.
(1998),alsotakingintoaccountthespinevolution(magnitudeand
direction)ofallbodiesinvolvedintidalevolution.Fortheplanet-
hosting star,weassumed zeroinitialobliquities. Weincluded the
2+2 2+2
effectofprecessionoftheorbitsduetotidalbulgesandrotation.
The equilibrium tidemodel is described in termsof thevis- 0.8 102
cous time-scale t (or equivalent related time-scales), the apsidal U
motion constant VkAM, the gyration radius rg and the initial spin 0.6 101 A/p,i
period P. Our assumed values for the planet and stars, if ap- ei r
s 0.4 100 U;
plicable, are given in Table 1. Most of the values are adopted A
/
from Fabrycky&Tremaine (2007). For all tidal interactions, we 0.2 10−1ai
assumedaconstant tidalviscoustime-scalet duringthesimula-
V
tions. For high-e migration, the viscous time-scale of the planet,
tV,1,isthemostimportantquantity.Apartfromitssimplicity,atem- binary binary
porallyconstanttVduringhigh-emigrationfollowsfromtheequa- 0.8 102
tionsofmotionwithanumberofphysically-motivatedassumptions U
(Socrates&Katz2012). 0.6 101 A/
p,i
ei r
0.4 100 U;
A
/
3 EXAMPLEOFDIFFERENCESINSECULAR 0.2 10−1ai
EVOLUTIONINSTELLARTRIPLESVERSUS
STELLARBINARIES
2.35 2.45 2.55 2.65 2.35 2.45 2.55 2.65 2.75
BeforeproceedingwiththepopulationsynthesisstudyinSection4, t/Gyr t/Gyr
weherebrieflydemonstratesomeofthedifferencesinthesecular
evolutionofplanetsinstellartriplesystemscomparedtostellarbi-
naries. In particular, we consider a planet in atriplein the‘2+2’ Figure2.ExampleevolutionofasysteminwhichaHJisformedthrough
configuration in which the orbit of the planet is excited to high high-e migration in a stellar triple system (the ‘2+2’ configuration, cf.
eccentricity, and subsequently a HJ is formed. In contrast, in the Fig.1),whereasintheequivalentbinarysystem,noHJisformed(cf.Sec-
equivalent case of a stellar binary, i.e. with orbit 2 replaced by a tion3).Inthetopfourpanels,theeccentricities ei (leftcolumn)andincli-
nations relative totheouterorbit(rightcolumn)areshownasafunction
pointmass,theeccentricityoftheorbitoftheplanetislessexcited,
of time (refer to the legends). The first (second) row corresponds to the
andnoHJisformed(within10Gyr).
triple(binary)configuration.Inthebottomfourpanels,azoom-inisshown
For theexample system, weselected asystemforming aHJ
aroundthetimeatwhichtheHJisformedinthetripleconfiguration.Ec-
inthe‘2+2’tripleconfigurationinthepopulationsynthesisofSec- centricities areshownintheleftcolumn;intherightcolumn,solidlines
tion4.Theparametersarem0 = 1M⊙,m1 = 1MJ,m2 ≈ 0.80M⊙, showthesemimajoraxesanddashedlinesshowthepericenter distances.
m ≈0.42M ;a =1AU,a ≈1.28AU,a ≈107AU;e =0.01, Thecolorsinthebottomfourpanelshavethesamemeaningasintheleft
3 ⊙ 1 2 3 1
e = 0.31,e = 0.32;i = 0◦,i ≈ 23.4◦,i ≈ 57.4◦;ω ≈ 174◦, columninthetopfourpanels.
2 3 1 2 3 1
ω ≈ 344◦,ω ≈ 246◦;Ω ≈ 9.70◦,Ω ≈ 83.1◦,Ω ≈ 92.8◦ and
2 3 1 2 3
t ≈0.014yr.TheotherparametersaregiveninTable1.Notethat
V,1
MNRAS000,1–14(2017)
HJs intriples 5
fortheseinitialparameters,theinitialinclinationbetweenorbits1 Petrovich&Tremaine 2016). With some approximations, high-e
and3isi ≈57.4◦,andi ≈34.5◦fororbits2and3. migration fractions in stellar binaries can also be calculated ana-
13 23
In theequivalent binary configuration, all orbital parameters lytically(Muñozetal.2016).
associatedwithorbit2areremoved, andtheyarereplacedbythe Consistentwithformationbeyondtheiceline,theplanetwas
parametersoforbit3inthetripleconfiguration. Thebinarycom- initiallyplacedonanearlycircularorbit(e = 0.01)withasemi-
1
panionmass,m˜ (herethetildeindicatesthebinarycase),isnow majoraxisa rangingbetween1and4AUaroundeitherthetertiary
2 1
givenbythesumofthemasseswithinorbit2inthetripleconfigu- star(inthecaseofthe‘2+2’configuration)ortheprimarystar(in
ration,i.e.m˜ =m +m ≈1.22M .Effectively,orbit2inthetriple thecaseofthe‘3+1’configuration).Inthecaseofastellarbinary,
2 2 3 ⊙
configurationisthenreducedtoapointmass. theplanetwasplacedonanorbitaroundtheprimarystar(S-type
In Fig.2, we show the time evolution of the eccentricities, orbit).Subsequently,wefollowedtheseculardynamicalevolution
relative inclinations and semimajor axes for the triple and binary ofthesystem,alsotakingintoaccounttidalevolution.
cases. In the binary case, the initial relative inclination of the
orbit of the planet with respect to its ‘outer’ orbit 2, is ˜i ≈
12
57.4◦. The maximum eccentricity induced in the former orbit,
4.1 Initialconditions
e˜ ∼ 0.7, is consistent with the simplest, lowest-order expec-
1,max
tation,[1−(5/3)cos(˜i12)2]1/2 ≈ 0.72.Thismaximumeccentricity For each of the three configurations, we selected three differ-
isnothighenoughtoinducestrongtidaldissipationinorbit1(the ent values of the viscous time-scale t of the planet and for
V,1
semimajoraxisa˜1remainsconstant),nortriggeratidaldisruption. each combination, we generated NMC = 4000 systems through
Asexpected,therelativeinclination˜i13 oscillatesbetweentheini- Monte Carlo sampling. The viscous time-scales considered were
tialvalueand≈40◦. t ≈ 0.014,0.14 and 1.4 yr, respectively. For gas giant planets
V,1
Ifthestellarcompaniontom0isreplacedbyabinary,thenthe andhigh-emigration,Socratesetal.(2012)providedtheconstraint
evolutionoforbit1isverydifferent.InsteadofregularLKoscilla- t . 1.2 ×104hr ≈ 1.4yr,byrequiringthataHJat5discircu-
V,1
tions,e1 oscillatesmuchmoreirregularly.Whereasthemaximum larizedinlessthan10Gyr.OurlargestvalueoftV,1correspondsto
inclination between orbit 1 and its outer orbit was limited to the thisvalue;theothervaluescorrespondto10and100moreefficient
initial value in the binary case, here this maximum inclination is tidaldissipation.Notethatt ≈1.4yrcorrespondstoatidalqual-
V,1
muchhigher,reachingvaluesofupto≈80◦.Themaximumeccen- ityfactor of Q ≈ 1.1×105 (cf.equation 37fromSocratesetal.
1
tricitiesreached inorbit 1 aremuch higher, exceeding 0.9. At an 2012),andQ ∝t .
1 V,1
ageof≈ 2.7Gyr,e1 becomeshighenoughfortidaldissipationto WemadethefollowingassumptionsintheMonteCarlosam-
becomeimportant,andaHJisrapidlyformed. pling. The mass and radius of the planet were set to m = 1M
1 J
These differences between the binary and triple cases can and R = R, respectively. The mass and radius of the star host-
1 J
be ascribed to the nodal precession of orbit 3 induced by LK ingtheplanetweresettom = 1M andR = 1R ,respectively.
0 ⊙ 0 ⊙
oscillations acting between orbits 2 and 3. This nodal preces- The masses of the companion stars, m and m , were computed
2 3
sion can enhance the LK oscillations between orbits 1 and 3 if fromm = q m andm = q m ,whereq andq weresampled
2 2 0 3 3 0 2 3
the ratio of the LK time-scales of associated with the orbital fromflatdistributionsbetween0.1and1.0.Theradiiofthecom-
pairs (1,3) and (2,3) are commensurate. The latter condition can panion stars, R and R (of importance for stellar tidesand colli-
2 3
be quantified by considering the ratio of LK time-scales as in sions),werecomputedusingtheapproximatemass-radiusrelation
Hamers&PortegiesZwart(2016),i.e. R =(m/M )0.8R fori∈{2,3}(Kippenhahn&Weigert1994).
i i ⊙ ⊙
Theinitialorbitoftheplanetarounditshoststarwasassumed
P a 3/2 m +m 3/2
R ≡ LK,13 ≈ 2 0 1 , (1) tohaveaneccentricitye =0.01andasemimajoraxisa between
2+2 P a ! m +m ! 1 1
LK,23 1 2 3 1and4AU,sampledfromaflatdistribution.TheorbitalperiodsP
2
where we neglected factors of order unity in estimating the LK andP3weresampledfromanormaldistributioninlog10(Pi/d)with
time-scales. If R ≪ 1, orbit 2 can be considered effectively a ameanof5.03andwidthof2.28(Raghavanetal.2010),andlower
2+2
pointmass,i.e.thebinarynatureofthisorbitcanbeneglected(see andupper limitsof 1and 1010 days, respectively. Thesemimajor
alsoSection2.4ofHamers&PortegiesZwart2016).Intheinter- axesa2 and a3 werecomputed fromtheseorbitalperiods andthe
mediateregimeofR2+2∼1,theevolutionisgenerallycomplexand sampled masses using Kepler’s law. The eccentricities e2 and e3
chaoticasdemonstratedinFig.2,andpotentiallyveryhigheccen- werecomputedfromaRayleighdistributionwithanrmswidthof
tricitiescanbeattainedinorbit1.ForthesysteminFig.2,R is 0.33between0.01and0.9,approximatingthedistributionsfound
2+2
indeedclosetounity,i.e.R ≈1.1. byRaghavanetal.(2010).
2+2
TherichdynamicsoftheregimeR ∼1willbeexploredin In the case of ‘2+2’ systems, sampled orbital combinations
2+2
moredetailinalaterpaper. were rejected if a /a was smaller than the critical value for dy-
3 2
namicalstabilityaccordingtothecriterionofMardling&Aarseth
(2001).Here,wetreatedorbits2and3asanisolatedtriplesystem
withthe mass of the ‘tertiary’ given by m +m , which is likely
4 POPULATIONSYNTHESIS 0 1
agoodapproximationgiventhesmallrelativemassoftheplanet.
Using the methods described in Section2, we carried out a Also,werequiredthata /a belargerthanthecriticalvaluefordy-
3 1
population synthesis of planets in hierarchical stellar triple sys- namical stability according to the criterion of Holman&Wiegert
tems,consideringtwodistincthierarchicalconfigurations(cf.pan- (1999),computedtreatingthestarsorbitingoutsidetheplanetasa
els 1 and 2 in Fig.1). For reference, and in order to investi- singlebodywithmassm +m .
2 3
gate the differences, we also carried out simulations of a planet Similarly,inthecaseof‘3+1’systems,stabilityoforbits2and
in a stellar binary mimicking previous studies of high-e migra- 3wasassessedusing thecriterionofMardling&Aarseth(2001),
tioninstellarbinaries(Wu&Murray2003;Fabrycky&Tremaine treatingorbit1asapointmasswithmassm +m .Fortheplanetary
0 1
2007; Naozetal. 2012; Petrovich 2015a; Andersonetal. 2016; orbit,weagainappliedthecriterionofHolman&Wiegert(1999),
MNRAS000,1–14(2017)
6 Hamers
thistimetreatingorbits1and2asanisolatedsystem(i.e.neglecting
themassm ).
3
For the reference case of a stellar binary, only the aa22;;ssiimm aa22;;oobbss aa33;;ssiimm aa33;;oobbss
Holman&Wiegert(1999)criterionwasappliedtoorbits1and2.
Without loss of generality, the inclination of orbit 1 was set
toi = 0◦;theotherinclinationsi andi weresampledfromflat 2+2
1 2 3 0.8
distributionsincos(i).Theargumentsofpericenterω andthelon-
i i
gitudesoftheascendingnodesΩ weresampledfromflatdistribu-
i 0.6
tionsbetween0◦and360◦.Thesechoicescorrespondtocompletely F
D
randomorbitalorientations. C
0.4
For the other fixed initial parameters, we refer to Table1.
Regarding the stars, we adopted a constant viscous time-scale of
0.2
t = 5yr, following Fabrycky&Tremaine (2007). Assuming a
V,⋆
tidalfrequencyof4d,thiscorrespondstoatidalqualityfactorof
0.0
Q ∼6×108orQ′ ≡3Q/(4k )∼3×106,whichistypicalfor
⋆ ⋆ AM,⋆
3+1
Solar-typestars(Ogilvie&Lin2007).
0.8
In Fig.3, we show the initial distributions of the semimajor
axesforthethreeconfigurations.Thickblue(thingreen)solidlines
0.6
indicatethesampleddistributionsofa (a );eachrowcorresponds F
2 3 D
toadifferentconfiguration(notethatforastellarbinary,or3bod- C
0.4
ies,onlya applies).Dashedlinesshowdistributionsfromobserved
2
F-andG-typestellartriplesfromthesampleofTokovinin(2014),
0.2
rejectingsystemsinwhicheitherthemassesororbitalperiodsare
unknown.
0.0
For ‘2+2’ systems, the distributions generated for the sim-
ulations approximate the observed distributions from Tokovinin binary
0.8
(2014), except for small values of a , i.e. a ∼ 10−1AU. This
2 2
is because there is an enhancement of short orbital periods in
0.6
the observations which is believed to be due to LK oscilla- F
D
tions induced by the tertiary star combined with tidal dissipation C
0.4
(Fabrycky&Tremaine 2007). Note that for the ‘2+2’ configura-
tion,tidaldissipationisincludedthestellarbinary(orbit2),hence
0.2
thelattereffectistakenintoaccountinthesimulations.
Forthe‘3+1’systems,thesampleddistributionsaredistinctly
0.0
differentfromtheobservations,especiallyregardinga2.Thisisdue 10−2 10−1 100 101 102 103 104 105
totheimposedrequirementofstabilityofthesystemwiththeadded log10(ai/AU)
planet orbiting one of the stars in the inner binary of the stellar
triple, pushing a to larger values, and, subsequently, a as well.
2 3
Presumably, this is also reflected in the true distributions of the
semimajoraxesofstellartripleswithembeddedplanets,whichare
Figure3.Theinitialdistributionsofthesemimajoraxesforthethreeconfig-
sofarverypoorlyconstrainedgiventhesmallnumberofobserved
urations(cf.Fig.1).Thickblue(thingreen)solidlinesindicatethesampled
systems.
distributionsofa2 (a3);eachrowcorrespondstoadifferentconfiguration
Lastly, in the case of stellar binary systems (cf. the bottom (notethatforastellarbinary,or3bodies,onlya2 applies). Dashedlines
rowofFig.3),thereisasimilarcutoffina2imposedbydynamical show distributions from observed F- and G-type stellar triples from the
stability. sampleofTokovinin(2014),selecting systemsinwhichboththemasses
andorbitalperiodsareknown.
4.2 Stoppingconditions
The integrations were stopped if one of the following conditions (iv) Theorbitoftheplanetarounditshoststarbecamedynami-
wasmet. callyunstablebecauseofperturbationsbytheotherstars,evaluated
usingthecriterionofHolman&Wiegert(1999).
(i) Thesystemreachedanageof10Gyr.
(v) Thestellarorbitsbecamedynamicallyunstableaccordingto
(ii) AHJwasformedandnofurtherevolutionisexpectedwithin
thecriterionofMardling&Aarseth(2001).Notethatthisonlyap-
aHubbletime,i.e.P <10dande <10−3.
1 1 pliestostellartriples.
(iii) Theinnermostplanetwastidallydisruptedbyitshoststar,
(vi) Inthe‘2+2’case,starscollidedinorbit2.Inprinciple,the
i.e.r =a (1−e )<r,wherer isgivenby
p,1 1 1 t t collisionremnant couldsubsequently secularlyexcitetheorbitof
m 1/3 theplanetproducingaHJ.However,theprobabilityofstellarcolli-
rt =ηR1 m0! . (2) sionissmall(seebelow).
1
(vii) Theruntimeofthesimulationexceeded12CPUhrs(im-
Here,ηisadimensionlessparameter;throughout,weassumedη= posedforpracticalreasons).
2.7(Guillochonetal.2011).
MNRAS000,1–14(2017)
HJs intriples 7
fnomigration fHJ fTD fDI,planet fcol,stars fruntimeexceeded
tend/Gyr tend/Gyr tend/Gyr tend/Gyr tend/Gyr tend/Gyr
1.37×10−2 2+2 0.782 0.771 0.787 0.040 0.046 0.038 0.155 0.158 0.153 0.000 0.000 0.000 0.015 0.015 0.015 0.002 0.004 0.003
1.37×10−2 3+1 0.324 0.300 0.338 0.039 0.045 0.036 0.399 0.403 0.393 0.226 0.228 0.222 0.000 0.000 0.000 0.010 0.023 0.009
1.37×10−2 binary 0.817 0.805 0.820 0.033 0.037 0.031 0.145 0.146 0.143 0.001 0.001 0.001 0.000 0.000 0.000 0.005 0.013 0.006
1.37×10−1 2+2 0.788 0.772 0.793 0.015 0.021 0.016 0.174 0.182 0.169 0.000 0.000 0.000 0.013 0.013 0.013 0.002 0.005 0.003
1.37×10−1 3+1 0.326 0.303 0.340 0.019 0.023 0.016 0.423 0.429 0.415 0.217 0.218 0.213 0.000 0.000 0.000 0.014 0.026 0.014
1.37×10−1 binary 0.839 0.827 0.840 0.013 0.015 0.012 0.143 0.144 0.141 0.000 0.000 0.000 0.000 0.000 0.000 0.005 0.014 0.006
1.37×100 2+2 0.796 0.781 0.802 0.003 0.005 0.003 0.179 0.186 0.172 0.000 0.000 0.000 0.015 0.015 0.015 0.002 0.007 0.003
1.37×100 3+1 0.332 0.305 0.346 0.002 0.004 0.002 0.444 0.450 0.433 0.208 0.210 0.202 0.000 0.000 0.000 0.012 0.028 0.014
1.37×100 binary 0.828 0.820 0.831 0.002 0.003 0.002 0.164 0.167 0.162 0.000 0.000 0.000 0.000 0.000 0.000 0.005 0.011 0.005
Table2. OutcomesoftheNMC=4000MonteCarlorealizationsforvariouscombinationsoftheplanetaryviscoustime-scaletV,1(inunitsofyr,roundedto
twodecimalplaces),andthethreehierarchicalconfigurations(HCs;cf.Fig.1):aplanetorbitingthetertiarystarinastellartriple(‘2+2’),aplanetorbiting
oneofthestarsintheinnerbinaryofastellartriple(‘3+1’),andaplanetorbitingastarinanS-typeorbitinastellarbinary(‘binary’).
4.3 Results atshortdistancesasorbit2isdriventohigheccentricitybyorbit
3.Asmentioned above, thecollisionremnant couldsubsequently
4.3.1 Outcomefractions
secularlyexcitetheorbitoftheplanet,producing aHJ.However,
the probability of stellar collision is small and therefore also the
In Table2, fractions are given of several pathways inthe simula-
contributiontoformingHJs.
tions.Wedistinguishbetweennomigration,i.e.theplanet’ssemi-
Inafewpercentofsystems,theCPUtimelimitwasexceeded
major axis not changing significantly, the formation of a HJ, the
andthesimulationswerestopped(cf.thelastcolumnofTable2).
planetbeingtidallydisrupted(‘TD’),theplanetbecomingdynam-
Thesesystemsareunlikely toproduce HJsifthestopping condi-
ically unstable (‘DI, planet’), stars colliding in orbit 2 (this ap-
pliesonlytothe‘2+2’ cases), andtheexceeding of theCPUrun tionhadnotbeenimposed;thisisdiscussedinmoredetailinSec-
tion5.1.
time.Eachrowcorrespondstoaspecifichierarchicalconfiguration
(‘HC’)andviscoustime-scaleoftheplanet,t .Thefractionsare
V,1
givenafter5and10Gyrofevolution,andbypickingarandomtime
between0.1and10GyrforeachoftheN =4000simulations. 4.3.2 HJorbitalperioddistribution
MC
Table2showsthattheHJfractionsbetweenthevariouscon- In Fig.4, we show the orbital period distributions of the short-
figurations are very similar. Nevertheless, there is a systematic period planets formed in our simulations for each configuration.
trend,forallviscoustime-scales,thattheHJfractionsforthestel- Inthetopthreepanels,resultsarecombinedfromthedifferentval-
lar triple ‘2+2’ and ‘3+1’ configurations are higher compared to uesoft .Redsolid,bluedashedandgreendottedlinesshowthe
V,1
the stellar binary configuration. This confirms our expectation of distributionsfromthesimulationsafter5Gyr,10Gyrandaftera
Section1,althoughtheenhancementisonlysmall,notexceedinga randomtime(‘t ’).Blackcrosses witherrorbarsindicatetheob-
x
fewpercentrelativetotheNMC=4000sampledsystems. servations from Santerneetal. (2016), and the orbital periods of
Astrikingdifferencebetweenthe‘3+1’andtheotherconfigu- thethreeHJsinstellartriplesknowntodate,WASP-12,HAT-8-P
rationsisthehighfractionoftidallydisruptedsystems.Thiscanbe andKELT-4Ab,areindicatedwiththeverticalblackdottedlines.
attributedtosmallervaluesofa2insimulationswiththe‘3+1’con- For reference, we also include with black open circles the distri-
figurationcomparedtothe‘outer’orbitsoforbit1inotherconfig- butionfromFig.23of Andersonetal. (2016)for M = 1M and
p J
urations(cf.Fig.3),whichresultsinastrongerexcitationoftheec- χ=100,whereχ≡10τ /sandτ isthetidaltimelagoftheplanet
1 1
centricityoftheorbitoftheplanet,orbit1,byorbit2(seealsoSec- (cf. Table 1 of the latter paper). With k = 0.25 (cf. Table1),
AM,1
tion4.3.3).Anotherdifferenceofthe‘3+1’configurationcompared m = 1M and R = 1R, χ = 100 or τ = 10s corresponds to
1 J 1 J 1
totheothers,isalargefractionofsystemsinwhichtheplanetbe- aviscoustime-scaleof≈0.082yr.Distributionsarenormalizedto
comesdynamicallyunstable,ofupto∼0.2.Thiscanbeattributed unittotalarea.Inthebottompanel,thecumulativedistributionsare
to enhanced eccentricity of orbit 2 excited by LK oscillations of comparedforthedifferentconfigurationsat10Gyr,andfordiffer-
orbit 3.Thereareanumber ofoutcomes of suchadynamical in- entviscoustimes-scalest shownwithdifferentlinethicknesses.
V,1
stability;thiswasinvestigatedfor asimilar situation(inthiscase Thedistributionsoftheorbitalperiodsareverysimilarforthe
withaP-typecircumbinaryplanet,ratherthananS-typeplanet)us- different configurations. Compared to the case of a stellar binary
ingN-bodyintegrationsbyHamersetal.2016b.Inthelatterwork, andcombining thedifferent t ,theKS Dand pvaluesare D ≈
V,1
it wasfound that ejection of the planet, which ismuch lessmas- 0.10and p ≈ 0.29forthe‘2+2’configurationand D ≈ 0.08and
sivethanthestars,ismostlikely.Inprinciple,planetscouldalsobe p ≈ 0.58forthe‘3+1’configuration.Thesedistributionsarealso
tidallycapturedbystarsandformingHJsinthismanner(tidalef- similar to those of Andersonetal. (2016), although the latter are
fectswerenottakenintoaccountinHamersetal.2016b),although truncatedatasmallerorbitalperiod.
weexpectthatthisisnotverylikely.Evenifafractionof0.1ofun- Thedependence ontheviscoustime-scalet ,shown inthe
V,1
stablesystemswouldleadtotidalcapture,theenhancementofthe bottompanelofFig.4,iseasilyunderstoodbynotingthatstronger
HJfractionwouldatmostbeafewpercent.Nevertheless,calcu- tidesresult inlargerstallingseparationsof theHJ(e.g.Petrovich
latingthecontributionofHJsfromdynamicalinstabilitytriggered 2015a). Another HJ property that depends significantly on t is
V,1
bysecular evolutioninthesesystems isaninteresting avenuefor theformationtime(cf.Section4.3.4).Otherpropertiessuchasthe
futurework. initialsemimajoraxesandeccentricitiesareweaklydependent on
Collisions between stars occur only in the ‘2+2’ configura- t , and below the corresponding results are shown for all three
V,1
tion,inwhichthestarslabeledm andm canapproacheachother viscoustime-scalescombined.
2 3
MNRAS000,1–14(2017)
8 Hamers
and≈2.99d,respectively)lieatthepeakofthesimulationsaround
3d.
The simulations fail to produce WJs, i.e. planets with peri-
55GGyyrr 1100GGyyrr ttxx
101 odsbetween10and100d,inanymeaningfulnumbers.Thenum-
2+2 berofWJs(outof4000systems)at5Gyr,10Gyrandarandom
timet are 0, 0 and 0 for the ‘2+2’ configuration, 1, 1 and 2 for
x
100 the ‘3+1’ configuration, and 1, 1, and 2 for the binary configu-
F
D ration, respectively. These few systems appear as low bumps in
P10−1 the orbital period distribution between 10 and 100 days. This is
incontrast toobservations, whichshow amuchlargernumber of
planetsinthisperiodregimecomparedtoHJs.Thisresultissim-
101 ilartopreviousstudiesofhigh-emigrationinstellarbinaries(e.g.
3+1 Petrovich&Tremaine 2016; Antoninietal. 2016), and in multi-
100 planetsystems(Hamersetal.2016a).
F
D
P
10−1 4.3.3 Orbitalpropertiesofthestellartriplearrangedbyoutcome
In Fig5, we show the distributions of the initial semimajor axes,
101 arranged by various outcomes in the simulations. As mentioned
binary above,thesedistributionsfortheHJsareweaklydependentontV,1
and the results in Fig5 are combined for the three viscous time-
100 scales. Thick (thin) lines apply to orbit 2 (3). We distinguish be-
F
D tweennomigration(blackdotted),HJformation(redsolid),plan-
P
10−1 etarytidaldisruption(yellowdashed)anddynamicalinstabilityof
theplanet(bluedot-dashed).
Several interestingconstraintsonthesemimajor axesarere-
vealed inFig.5. For ‘2+2’ systems, HJsare only formed if a .
2
10Gyr 102AU, and a needs to be & 20AU and . 103AU. These lim-
0.8 3
itscanbeunderstood asfollows.Thelowerlimitona coincides
3
F0.6 withthelowestsampledvalueofa3(cf.thethinredsolidandblack
D
C0.4 dotted lines in the top panel of Fig.5), and which do not lead to
2+2 tidaldisruption.Thelowestsampledvalueofa arisesfromthere-
3
0.2 3+1 quirementofdynamicalstabilityofthesystem(cf.Section4.1and
binary
Fig.3).Theupperlimitona forHJsystemsarisesfromthewell-
3
0.0 0.5 1.0 1.5 2.0 known phenomenon that LK cycles are suppressed if the ‘outer’
log10(P1/d) orbit (here, orbit 3) is wide compared to the ‘inner’ orbit (here,
orbit 1), inwhich case precession due toGR, tidal bulges and/or
rotationquenchesLKoscillations(e.g.Naozetal.2013).
This upper limit on a , a . 103AU, translates to an up-
3 3
per limit on a because of the requirement of dynamical stabil-
Figure4.Theorbitalperioddistributions oftheHJsformedinthesimu- 2
ity.Atypicalratioofa /a requiredfordynamicalstabilityinour
lations foreachconfiguration (firstthreepanels).Inthetopthreepanels, 3 2
redsolid,bluedashedandgreendottedlinesshowthedistributions from simulationsis∼ 6(assumingthecriterionofMardling&Aarseth
thesimulationsafter5Gyr,10Gyrandafterarandomtime(‘tx’),combin- 2001).Thisimpliesanupperlimitona2of∼103/6AU≈167AU,
ingresultsfromthethreeviscoustime-scalestV,1.Blackcrosseswitherror andwhichisindeed roughlyconsistent withtheupper limita2 .
barsindicate theobservations fromSanterneetal.(2016),andtheorbital 102AUfoundinthesimulations.
periodsofthethreeHJsinstellartriplesknowntodate,WASP-12,HAT-8- Similarlimitsapplytothetidaldisruptionsystemsinthe‘2+2’
PandKELT-4Ab,areindicatedwiththeverticalblackdottedlines.Black configuration.Mostofthesesystemshaveaninitiala smallercom-
3
opencircles showthedistribution fromFig.23ofAndersonetal.(2016)
paredtotheHJsystems:forverysmalla ,closetodynamicalin-
form1 =1MJandχ=100.Distributionsarenormalizedtounittotalarea. stability with respect the planet (orbit 1),3strong secular dynami-
Inthebottompanel,thecumulativedistributionsarecomparedforthedif-
cal interactions excite very high eccentricities in orbit 1, leading
ferentconfigurationsat10Gyr,andforthethreedifferenttV,1shownwith
differentlinethicknesses(decreasingtV,1withincreasinglinethickness). totidaldisruptionoftheplanet.Theupperlimita3 . 5×102AU
islower compared toHJssystems, implyingalower valueof the
upper limit on a , in this case a . 50AU. Similarly to the HJ
2 2
systems,theupperlimitsona anda areroughlyconsistentwith
3 2
CompanionsofHJsintheobservedsampleofSanterneetal. requiringdynamicalstabilityoforbits2and3,respectively.
(2016)arepoorlyconstrained,andsoitisuncleartowhichextent Considering‘3+1’systems,a forHJsystemsislargercom-
2
theHJorbitalperioddistributionisdifferentforsinglestarscom- paredtothe‘2+2’configuration,whichcanbeascribedtothedif-
paredtohigher-ordersystems.TheorbitalperiodsofthethreeHJs ferenthierarchy,i.e.a ≫a isalwaysrequiredfordynamicalsta-
2 1
observed in triple systems so far, indicated with the black verti- bilityfor‘3+1’systems,whereasthisisnotthecasefor‘2+2’sys-
caldottedlinesinFig.4,areconsistent withthesimulations.The tems (for the latter, systems with similar a and a are typically
1 2
periodofWASP-12(P ≈1.09d)liesatthelowendofthesimula- dynamicallystableunlessa issmall).Thetypicalvaluesofa for
1 3 2
tions,whereastheperiodsofHAT-8-PandKELT-4Ab(P ≈ 3.07 HJsin‘3+1’systemsareverysimilartothoseofa forHJsin‘2+2’
1 3
MNRAS000,1–14(2017)
HJs intriples 9
nnoommiiggrraattiioonn HHJJ TTDD DDII,,pp
nnoommiiggrraattiioonn HHJJ TTDD DDII,,pp
2+2
2+2
0.8
0.8
0.6 0.6
F F
D D
C C
0.4 0.4
0.2
0.2
0.0
0.0
3+1 −4 −2 0 2 4 6 8
0.8 log10(R2+2)
0.6
F 3+1
D 0.8
C
0.4
0.6
0.2
F
D
C
0.0 0.4
binary
0.8 0.2
0.6
DF 0.−012 −10 −8 −6 −4 −2 0 2 4
C0.4 log10(R3+1)
0.2
0.0 Figure6. Thedistributions ofR2+2 (cf. equation 1)andR3+1 (cf. equa-
10−2 10−1 100 101 102 103 104 105 tion3)shownfor‘2+2’and‘3+1’systemsinthetopandbottompanels,
log10(ai/AU) respectively,andarrangedbytheoutcomesinthesimulations(combining
resultsforthethreevaluesoftV,1).Theblacksolidlinesshowtheinitial
distributionsofR2+2 andR3+1,andthethickblackdashedlinesshowthe
initialdistributionsforthesystemsinwhichtheruntimewasexceeded(cf.
Section5.1).
Figure5.Thedistributionsoftheinitialsemimajoraxes,arrangedbyvari-
ousoutcomesinthesimulations,andcombinedforthethreevaluesoftV,1.
Eachpanelcorrespondstoadifferentconfiguration;thick(thin)linesapply
Inadditiontothesemimajoraxes,aquantityofinterestisthe
toorbit2(3).Blackdottedlines:nomigration;redsolidlines:HJforma-
ratioofLKtime-scalesappliedtodifferentorbits,withitsdefinition
tion;yellowdashedlines:planetarytidaldisruption;bluedot-dashedlines:
dependingonthehierarchy.For‘2+2’systems,therelevantratio,
dynamicalinstabilityoftheplanet.Theinitialsemimajoraxisdistributions
ofallsampledsystemswereshownwiththesolidlinesinFig.3. R2+2 ≡ PLK,13/PLK,23,wasintroducedinSection3(cf.equation1).
Asmentionedthere,particularlyhigheccentricities,andtherefore
HJsandtidaldisruptions,areexpectedintheregimeR ∼ 1.In
2+2
thetoppanelofFig.6,thedistributionofR isshownfor‘2+2’
systems.Thesesimilarlimitscanagainbeexplainedbytherequire- 2+2
systems,arrangedbytheoutcomesinthesimulations(combining
mentofdynamicalstabilityandthesuppressionofLKoscillations
resultsforthethreevaluesoft ).HJssystemsindeedshowapref-
for large ‘outer’ orbit semimajor axes (i.e. in the ‘3+1’ case, the V,1
erenceforR ∼ 1comparedtonon-migratingsystems.Fortidal
outerorbitisorbit2,andinthe‘2+2’case,theouterorbitisorbit 2+2
disruptionsystems,thedistributioniseven moreconcentrated to-
3).
wardsR ∼1,consistentwiththenotionthatmoreviolenteccen-
For ‘3+1’ planetary tidal disruption systems, a is smallest, 2+2
2
tricityexcitationismorelikelytoresultintidaldisruption.
witha . 5×102AU.For systems inwhichtheplanet becomes
2 For‘3+1’systems,therelevantLKtime-scaleratioisassoci-
dynamicallyunstable,a liesroughlyinbetweenthetypicalvalues
2
atedwiththeorbitalpairs(1,2)and(2,3)(Hamersetal.2015),and
fortidaldisruptionandHJformation.Theseinstabilitiesaredriven
isgivenby
byseculareccentricityexcitationoforbit2bythetorqueoforbit3,
i.e.instabilitywouldnothaveoccurredintheabsenceofthefourth P a3 3/2 m +m 1/2 m 1−e2 3/2
body. R ≡ LK,12 ≈ 2 0 1 3 2 .
3+1 P a a2! m +m +m ! m 1−e2!
The implications of these constraints on a2 and a3 in triple LK,23 1 3 0 1 2 2 3
(3)
systemsarediscussedinSection5.3.
MNRAS000,1–14(2017)
10 Hamers
As shown by Hamersetal. (2015), complex eccentricity oscilla-
tionsandpotentiallyhigheccentricitiesareexpectedifR ∼ 1.
3+1
InthebottompanelofFig.6,weshowthedistributionofR for
3+1 nnoommiiggrraattiioonn HHJJ TTDD DDII,,pp
the‘3+1’configuration,arrangedbytheoutcomes.TheHJsystems
indeeddisplayapreferenceforlargervaluesofR closetounity
3+1
2+2
(median value of log R ∼ 0) compared to the non-migrating
10 3+1 0.8
systems(medianvalueoflog R ∼−3).
10 3+1
Thetidallydisruptedsystems,ontheotherhand,showadistri-
0.6
butionofR whichisbiasedtowardsmallvaluesofR ,seem- F
3+1 3+1 D
ingly contradicting the above expectation. However, as already C
0.4
mentioned in Section4.3.1, the fraction of tidally disrupted plan-
ets in the ‘3+1’ configuration is large owing to the small value
0.2
of a compared to a in the ‘2+2’ configuration, and a in the
2 3 2
binary configuration (cf. Fig.3). As shown in Fig.5, a for the
2 0.0
tidaldisruptionsystemsisindeedstronglybiasedtosmallvalues,
10AU . a . 5×102AU,inwhichcaseaveryhigheccentricity 3+1
2 0.8
canbeinducedinorbit1regardlessoforbit3,leadingtotidaldis-
ruption.Suchsmallvaluesofa correspondtosmallvaluesofR
2 3+1 0.6
(cf.equation3). F
D
InFig7,theinitialdistributionsoftheeccentricitiesareshown C
0.4
arrangedbythevariousoutcomes,similarlytoFig.5.Asmightbe
expected,thereissomepreferenceforHJandtidaldisruptionsys-
0.2
temstohavelargerinitiale (for‘2+2’systems)ande (for‘3+1’
3 2
systems). The planetary dynamical instability outcome shows a
0.0
strongpreferenceforhighvaluesofe .Ife ishighinitially,onlya
2 2
smallchangeofe duetotheseculartorqueoforbit3issufficient binary
2 0.8
todriveinstabilityoftheplanet.
0.6
F
D
4.3.4 HJformationtimes C
0.4
The distributions of the ‘end times’ associated with various out-
comesofthesimulationsareshowninFig.8,eachrowcorrespond- 0.2
ingtoadifferentconfiguration.Here,‘endtime’correspondstothe
timeofHJformationortimeofdisruptionforthetidallydisrupted 0.0
0.0 0.2 0.4 0.6 0.8 1.0
planets.Resultsareshownseparatelyforthethreeassumedvalues
e
i
oft withdifferentlinethicknesses(decreasingt withincreas-
V,1 V,1
inglinethickness).
Asexpected,theshortervaluesoft (strongertides)leadto
V,1
shorterHJformationtimes.SimilarlytothedistributionoftheHJ
orbitalperiods,theHJformationtimesarenotsignificantlydiffer- Figure7.SimilartoFig.5nowshowingtheinitialdistributionsoftheec-
ent between the three configurations. The red open circles show centricities(combiningresultsforthethreevaluesoftV,1).Thick(thin)lines
data for simulations in stellar binaries from the second panel of showthedistributions fore2 (e3).Notethattheinitialeccentricities were
sampledfromRayleighdistributions,forwhichthecumulativedistribution
Fig. 22 (m = 1M) by Andersonetal. (2016), and which are
1 J ∝1−exp(−βe2).
roughly consistent with our simulations. Tidal disruptions, how- i
ever,occur earlierinour simulationsof stellarbinariescompared
toAndersonetal.(2016)(cf.theyellowopencirclesinFig.8),and
Inthecaseofastellarbinary,therearetwodistinct,butbroad,
thismaybebecauseofdifferentassumptionsontheorbitalparam-
peaks in the obliquity distribution near ∼ 50◦ and ∼ 100◦. Such
eters.Therearenolargedifferencesinthetidaldisruptiontimedis-
peaksintheobliquitydistributionarewellknowntoariseinstel-
tributionsbetweenthedifferentconfigurations.
larbinaries(Fabrycky&Tremaine2007;Naoz&Fabrycky2014;
Andersonetal.2016;seeStorchetal.2016foradetailedstudyon
theoriginofthebimodel distribution).Thereappearstobeadis-
4.3.5 Stellarobliquities
crepancywithAndersonetal.(2016),whofindpeaksnearvalues
InFig.9,weshowthedistributionsofthestellarobliquity,i.e.the of ∼ 25◦ and ∼ 130◦. This may be due various reasons, includ-
anglebetweenthespinoftheplanet-hostingstarandtheorbitofthe ing the initially shorter spin period of P = 2.3d rather than
s,⋆
planet,for thenon-migrating (blackdottedlines)and HJsystems 10 d for the host star, magnetic braking, which was included in
(redsolidlines).Resultsarecombinedforthethreevaluesoft . Andersonetal.(2016)but not inthiswork(magnetic brakingre-
V,1
Data from Fig. 24 of Andersonetal. (2016) for m = 1M and sultsinthespindownofthestar,which,duetospin-orbitcoupling,
1 J
χ=100areshownwithblackopencircles.Additionally,weshow cansignificantlychangethefinalobliquity,Storchetal.2016),or
with black stars observational data adopted from Lithwick&Wu differentt .
V,1
(2014).Theinitialobliquitiesinthesimulationswereassumedto Forstellartriples,theobliquitydistributionsseemmarginally
bezero. lesspeakedcomparedtothecaseofastellarbinary.Inthebottom
MNRAS000,1–14(2017)