The formation efficiency of close-in planets via Lidov-Kozai migration: analytic calculations

0.6 MB·

by  Diego J. Muñoz

Mon.Not.R.Astron.Soc.000,1–??(2016) Printed25January2016 (MNLATEXstylefilev2.2)
The formation efficiency of close-in planets via Lidov-Kozai
migration: analytic calculations
Diego J. Mun˜oz1(cid:63), Dong Lai1 and Bin Liu2
1CornellCenterforAstrophysicsandPlanetaryScience,DepartmentofAstronomy,CornellUniversity,Ithaca,NY14853,USA
2CenterforAstrophysics,UniversityofScienceandTechnologyofChina,Hefei,Anhui230026,People’sRepublicofChina
6 25January2016
1
0
2 ABSTRACT
n Lidov-Kozai oscillations of planets in stellar binaries, combined with tidal dissipation, can
a lead to the formation of hot Jupiters (HJs) or tidal disruption of planets. Recent population
J synthesis studies have found that the fraction of systems resulting in HJs (F ) depends
HJ

1 strongly on the planet mass, host stellar type and tidal dissipation strength, while the total
2 migration fraction F = F +F (including both HJ formation and tidal disruption)
mig HJ dis
exhibitsmuchweakerdependence.WepresentananalyticalmethodforcalculatingF and
HJ
] F in the Lidov-Kozai migration scenario. The key ingredient of our method is to deter-
P mig
mine the critical initial planet-binary inclination angle that drives the planet to reach suffi-
E
cientlylargeeccentricityforefficienttidaldissipationordisruption.Thiscalculationincludes
.
h theeffectsofoctupolepotentialandshort-rangeforcesontheplanet.Ouranalyticalmethod
p reproducestheresultingplanetmigration/disruptionfractionsfrompopulationsynthesis,and
- canbeeasilyimplementedforvariousplanet,stellar/companiontypes,andfordifferentdistri-
o
butionsofinitialplanetarysemi-majoraxes,binaryseparationsandeccentricities.Weextend
r
t ourcalculationstoplanetsinthesuper-Earthmassrangeanddiscusstheconditionsforsuch
s
a planetstosurviveLidov-Kozaimigrationandformclose-inrockyplanets.
[
Key words: binaries:general – planets: dynamical evolution and stability – star: planetary

1 system
v
4
1
8
5 1 INTRODUCTION tidaldissipationatpericentertoshrinktheplanetsemi-majoraxis
0 downto(cid:46)0.1AU.Thus,suchmechanismsaresometimesreferred
The occurrence of “hot Jupiters” (HJs) – gas giants with periods
. tocollectivelyas“high-emigration”.
1 (cid:46) 5 days – is estimated to be around 1% for FGK stars (Marcy
0 Planet migration following eccentricity excitation through
etal.2005;Wrightetal.2012;Howardetal.2012;Fressinetal.
6 Lidov-Kozai oscillations (Lidov 1962; Kozai 1962) induced by
2013;Petigura,Howard&Marcy2013).Itiscommonlybelieved
1 a stellar companion1 – often called Lidov-Kozai (LK) migra-
thattheseplanetsformedatlargerseparationsfromtheirhoststars
: tion – has been investigated by different researchers via Monte
v (presumablybeyondtheiceline)andsubsequently“migrated”to
Carlo experiments (Wu, Murray & Ramsahai 2007; Fabrycky &
i semi-majoraxesoflessthan0.1AU.Theknownmigrationmech-
X Tremaine 2007; Naoz, Farr & Rasio 2012; Petrovich 2015b; An-
anisms can be (i) “disc mediated” due to planet-disc interaction
r derson,Storch&Lai2016).Thelateststudieshaveconcludedthat
a (e.g.,Goldreich&Tremaine1980;Lin,Bodenheimer&Richard- at most 20% of observed HJs can be explained by LK migration
son 1996), (ii) “planet mediated”, including strong planet-planet
in stellar binaries. Similarly, observational evidence (Dawson &
scatteringsandvariousformsofsecularinteractionsamongmulti-
Murray-Clay 2013; Dawson, Murray-Clay & Johnson 2015) also
pleplanets(e.g.Rasio&Ford1996;Chatterjeeetal.2008;Juric´
indicatesthatthepopulationofHJsisunlikelytobeexplainedbya
&Tremaine2008;Nagasawa,Ida&Bessho2008;Wu&Lithwick
singlemigrationmechanism.
2011;Beauge´&Nesvorny´ 2012;Petrovich2015a),or(iii)“binary
TheoccurrencerateofHJsproducedbyLKmigrationinstel-
mediated”,i.e.,secularinteractionswithadistantstellarcompan-
larbinariescanbecomputedvia
ion(Wu&Murray2003;Wu,Murray&Ramsahai2007;Fabrycky
&Tremaine2007;Naoz,Farr&Rasio2012;Correiaetal.2011; RLHKJ =Fb×Fp×FHLJK, (1)
Storch,Anderson&Lai2014;Petrovich2015b;Anderson,Storch
whereF isfractionofstarshavingawidebinarycompanion,F is
&Lai2016).Excludingdisc-drivenmigration,thesemechanisms b p
requirethemigratingplanettoreachhigheccentricitiesinorderfor
1 LKoscillationsinducedbyaplanetarycompanioncanplayanimportant
roleinsomeplanet-mediatedhigh-emigrationscenarios.Inthispaper,we
(cid:63) E-mail:[email protected] focusonLKoscillationsinducedbyastellar-masscompanion.
(cid:13)c 2016RAS
2 Mun˜oz,Lai&Liu
thefractionofsolar-typestarshostinga“regular”giantplanetata semi-majoraxisandeccentricity),M ,M ,M (massesofthe
p ∗ out
distanceofafewAU,andFLKistheformationfraction/efficiency planet,hoststarandouterbinarycompanion),andtheplanet’sra-
HJ
of HJs (i.e., the fraction of systems that result in HJs for a given diusR anditsinternalstructure(Lovenumber).
p
distributions of binaries and “regular” planets). Optimistic values InLiu,Mun˜oz&Lai(2015)(hereafterLML),wehavestud-
ofF ∼20%(forbinaryseparationsbetween100AUand1000AU; iedthemaximumeccentricityinLKoscillationsincludingtheoc-
b
seeRaghavanetal.2010)andF ∼15%(Petrovich2015b)require tupole potential of the binary companion and the effect of short-
p
FLK∼30%–40% to explain the observed HJ occurrence rate (∼ rangeforces(associatedwithGeneralRelativity,thetidalbulgeand
HJ
1%)usingLKmigrationalone. rotationalbulgeoftheplanet).Herewesummarizethekeyresults
In a recent work, Anderson, Storch & Lai (2016) (hereafter neededforthispaper.
ASL)conductedacomprehensivepopulationsynthesisstudyofHJ The strength of the octupole potential from the binary com-
formationfromLKoscillationsinstellarbinaries,includingallrel- panionrelativetothequadrupolepotentialischaracterizedbythe
evantphysicaleffects(theoctupolepotentialfromthebinarycom- dimensionlessratio
panion, mutual precession of the host stellar spin axis and planet a e
(cid:15) ≡ 0 out . (2)
orbitalaxis,tidaldissipation,andstellarspin-downduetomagnetic Oct a 1−e2
out out
braking).Unlikepreviousworks,ASLconsideredarangeofplanet
When(cid:15) →0,theoctupoleeffectcanbeneglected.Inthislimit,
masses(0.3−5M )andinitialsemi-majoraxes(a =1−5AU), Oct
J 0 theplanet’sdimensionlessangularmomentum(projectedalongthe
differentpropertiesforthehoststar,andvaryingtidaldissipation √
binary axis) j = 1−e2cosi is conserved, resulting in regu-
strengths. They found that the HJ formation efficiency depends z
lar LK oscillations in eccentricity and inclination. Together with
stronglyonplanetmassandstellartype,withFLK = 1−4%(de-
HJ theconservationofenergy,themaximumeccentricitye canbe
pendingontidaldissipationstrength)forM =1M ,andlarger max
p J calculated analytically as a function of i (and other parameters)
(upto8%)formoremassiveplanets.Theproductionefficiencyof 0
(Fabrycky & Tremaine 2007). Assuming that the planet has zero
“hot Saturns” (M = 0.3M ) is much lower, because most mi-
p J initialeccentricity,e isdeterminedby(LML)
gratingplanetsaretidallydisrupted.Remarkably,ASLfoundthat max
thefractionofmigratingsystems,i.e.thosethatresultineitherHJ (cid:18) 1 (cid:19) (cid:15) (cid:18)1+3e2 + 3e4 (cid:19)
ifsorrmouagtiholnyocorntisdtaanltd,ihsaruvpintigonli,ttFlemLvKiagri≡atioFnHLwJKit+hpFladLniKsetm(cid:39)as1s1,−st1e4ll%ar (cid:15)GR(cid:15) jm(cid:18)in −1 1 +(cid:19)T1i5de9e2 (cid:18) mjam9xin 58 max −(cid:19)1 (3)
typeandtidaldissipationstrength.ASLprovidedaqualitativeex- + Rot −1 = max j2 − cos2i ,
planationofhowsucharobustFLK maycomeabout(seeSections 3 jm3in 8 jm2in min 3 0
mig √
3.4and5.4.1ofthatwork).Notethat(Petrovich2015b)considered wherej ≡ 1−e2 ,andthedimensionlessquantities
min max
M = 1M aroundsolar-typestarsandfoundasimilarFLK as
p J HJ
AobStLai,nbeudtFammLKiugc≈hl2ar7g−er2d9i%sr.uTpthieondiffrfaecrteinocne(FardiLsiKses∼fro2m5%th)e,afancdttthhuast (cid:15)GR ≡ 3GM∗2aa3o40uct2(M1−ouet2out)3/2 (4a)
s(Pumetreodvtihchat2t0h1e5bbi)nparlayceecdcaelnltprilcaintyetsdiisntirtiibaulltyioanteax0te=nd5sAupUtaon0d.9a5s-, (cid:15)Tide ≡ 15M∗2a3ouat(81M−Me2out)3/2k2pRp5 (4b)
whereas ASL considered a = 1−5 AU and a maximum binary 0 p out
0 M a3 (1−e2 )3/2k Ω2R5
eccentricityof0.8(seeSection6.2ofASLforfurtherdiscussion). (cid:15) ≡ ∗ out out qp p p , (4c)
In this paper, following on the work of ASL, we present an Rot Ga50MpMout
analytic study that explains (i) how the LK migration fraction of representtherelativestrengthsofapsidalprecessionsduetoGen-
close-inplanetsarelargelydeterminedbygeometricalconsidera- eral Relativity (GR), tidal bulge, and rotational distortion of the
tions,and(ii)howtheplanetsurvivalanddisruptionfractionsde- planet.Herek istheLovenumber,k theapsidalmotioncon-
2p qp
pendonthepropertiesoftheplanetsandhoststars.Weoutlinethe stantandΩ therotationrateoftheplanet.Inthe“pure”LKlimit
p
necessary steps that allow for simple calculations of FmLKig, FHLJK ((cid:15)GR = (cid:15)Tide = (cid:15)Rot = 0),Equation(3)reducestothestandard
andFdLiKs undervariousconditionsandassumptions(e.g.,thedis- expressionemax = (cid:112)1−(5/3)cos2i0.Fortypicalplanetaryro-
tributions of initialplanetary semi-major axes, binary separations tationrates,thecontributionfromtherotationalbulge((cid:15) )canbe
Rot
andeccentricities,themassesofplanetandhoststar).Ouranalytic neglectedcomparedtothetidalterm((cid:15) )(seeLML)
Tide
calculationsagreewiththeresultsofpopulationsynthesisstudies. Themaximumvalueofe isachievedwheni = 90◦ in
max 0
Inaddition,weextendourcalculationstoplanetsinthesuper-Earth Equation(3),andisreferredtoasthe“limitingeccentricity”e .
lim
massrange. Whene ≈1,thelimitingeccentricitysatisfies(Eq.53ofLML)
lim
(cid:15) 7 (cid:15) 9
GR + Tide (cid:39) . (5)
(1−e2 )1/2 24(1−e2 )9/2 8
lim lim
2 LIDOV-KOZAIOSCILLATIONSWITHOCTUPOLE
Inthepresenceoffiniteoctupolepotential((cid:15) (cid:54)= 0),j is
EFFECTANDSHORT-RANGEFORCES:MAXIMUM Oct z
nolongeraconstant,butaslowlyvaryingquantity(Ford,Kozinsky
ECCENTRICITY
& Rasio 2000; Lithwick & Naoz 2011; Katz, Dong & Malhotra
ThesuccessofLKmigrationdependsonthemaximumeccentric- 2011),withdj /dt∝(cid:15) .Forsufficientlylargeinitialinclinations
z Oct
itye thatcanbeachievedbytheplanetintheLKcycles.This (i >i ),theslowevolutionofj cantakeittoextremelysmall
max 0 lim z
eccentricityneedstobehighenoughsuchthatthepericentersepa- valuesandevenmakeitcrosszero(i.e.i → 90◦,seee.g.,Katz,
rationbetweentheplanetanditshoststarissufficientlysmallfor Dong & Malhotra 2011). As j becomes small, e can reach very
z
tidaldissipationintheplanettobeefficient(seeSection3.1).The highvalues.
maximumeccentricitydependsontheinitialinclinationi between AsdescribedbyLML,theeffectoftheoctupoleistoincrease
0
theplanetaryandbinaryorbitalaxes,andonvarioussystemparam- therangeofpossibleinitialanglesi forwhiche canbereached.
0 lim
eters:a (initialplanetarysemi-majoraxis),a ande (binary TocapturetheeffectdescribedbyLML,wecanapproximatethe
0 out out
(cid:13)c 2016RAS,MNRAS000,1–??
Lidov-Kozaimigrationfractions 3
90 ofthecriticalinclinationanglefororbitalflip–thisanalyticresult
region of extreme eccentricity Liu, Muñoz & Lai (2015) isvalidonlyfor(cid:15) (cid:28)1(seeFig.1).
oct
We note that, in general, the octupole-active window of ex-
80
tremeeccentricityisreachedonlywhenω (theargumentofperi-
center) of the planet is circulating (as discussed by Katz, Dong
70 & Malhotra 2011). However, since planets commonly start with
e (cid:39) 0, this distinction is irrelevant, as planets at e (cid:39) 0 lie very
] moderate eccentricity
◦ closetotheseparatrixbetweenthelibratingandcirculatingregions
[
i060 ofphasespace.
i († )
lim oct
50
3 DERIVATIONOFMIGRATIONFRACTIONS
Katz et al (2011) -- analytic
40 3.1 Requiredpericenterdistanceformigrationordisruption
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 In order to migrate, a distant planet must reach small pericenter
†
oct distance, r = a (1−e), such that tidal dissipation can reduce
p 0
Figure 1. The behavior of LK oscillations with the octupole and short- theplanet’ssemi-majoraxiswithinafewGyr.Theeffectiveorbital
range-force(SRF)effectscanbedividedinto“moderate”eccentricityevo- decayrateduringLKoscillationscanbeestimatedfromtheorbital
lutionand“extreme”eccentricityevolution.Theboundarybetweenthese decayrateduetotidaldissipationate = e multipliedbythe
max
two regions is a function of (cid:15)Oct (the dimensionless octupole strength) fraction of the time the planet spends in the high-e phase (e ∼
andi0 (theinitialinclinationangle).Inthe(cid:15)Oct-i0 plane,thisboundary e ),giving(seeSection3.2andEq.32ofASL)
max
is represented by the critical initial angle ilim((cid:15)Oct) above which plan-
(cid:12) (cid:12)
aiethlrtiseemctr(aea(cid:15)kdnOencrceutfa)rrcvoflhematdthtteehepneislcinmotusuimtttihtenoergiaficeatcctloicninengsnttetafrgunircntaicvtttiyaiolonuenselim(o≈Efq(L4uE8iauq◦t,ui.oMaFntiouo7rn˜n)co.o5zFm)&o.prTaL(cid:15)hraOiesicod(tn2a,0t(cid:38)at1h5pe0)ob.ia0lnnu5tdes, td1ec ≡(cid:39)(cid:12)(cid:12)(cid:12)6a1kddat∆(cid:12)(cid:12)(cid:12)tTidMe,L∗K(cid:18)Rp(cid:19)5n2f1(emax) (8)
curveshowstheanalyticresultforthecriticali0 fororbitalflipwithout 2p LMp a0 jm14in
SRF,asobtainedbyKatz,Dong&Malhotra(2011).
GM2 R5
(cid:39)2.8k ∆t ∗ p
2p L M a r7
p 0 p,min
maximumeccentricitywith2
where∆tisthelagtime,thepericenterdistanceatmaximumec-
 e(mQauxad)(i0) i0 <ilim((cid:15)Oct) centricityisdefinedasrp,min ≡a0(1−emax)andtheexpression
emax(i0,(cid:15)Oct)(cid:39) (6) f1(emax)isreplacedbyitsapproximatevalue3861/64.Inorder
 elim i0 (cid:62)ilim((cid:15)Oct) tohavemigrationwithin∼ 109 years(i.e.,tdec (cid:46) 109 years),we
requirer (cid:46)r ,with
p,min p,mig
where e(Quad)(i ) is the solution to Equation (3) and e is the
esolilmutiiosnrmetoaacxEhqeudat(0ifoonr((cid:15)5O)c,tan→dili0m, iitsitsherecqriutiirceadlatnhgatleilaibmolvi→mew9h0ic◦h, rp,mig ≡2.4×10−2(cid:18)RRpJ(cid:19)57 AU
such that the quadrupole approximation is recovered). Note that
Equation(6)neglectsthe“transitionregion”fori0 justbelowilim ×(cid:18)χ(cid:19)1/7(cid:18)Mp(cid:19)−17(cid:18) a0 (cid:19)−17(cid:18)k2p (cid:19)17(cid:18) M∗ (cid:19)27 ,
inwhichtheoctupolepotentialmakesemaxlargerthane(mQauxad)but 10 MJ 1AU 0.37 1M(cid:12)
not as large as e . The numerical calculations of LML showed (9)
lim
thatwhentidaldistortion((cid:15) )dominatestheSRFs,thetransition
Tide whereχ≡∆t/(0.1s)isadimensionlesstidalenhancementfactor.
regionisnarrow.
We see that, except for planet radius, the dependence of the
The critical inclination angle i , which defines the
lim criticalpericenterdistancer onthephysicalparametersofthe
“octupole-active” window, depends on (cid:15) . Figure 1 shows the p,mig
Oct systemisweak,andthusitrepresentsaveryrobust(albeitnotan
numericallycomputedvaluesofofi (LML).Thesecanbewell
lim exact) estimate of how close planets must get to their host stars
approximatedbythefittingformula
inordertomigrateeffectively(seeASL).Forthispericentertobe
cos2i =A(cid:18)(cid:15)Oct(cid:19)+B(cid:18)(cid:15)Oct(cid:19)2+C(cid:18)(cid:15)Oct(cid:19)3+D(cid:18)(cid:15)Oct(cid:19)4 reached,aplanetinitiallylocatedata0needstoachieveamaximum
lim 0.1 0.1 0.1 0.1 eccentricityof
(7) r
for(cid:15)Oct (cid:54) 0.05withA = 0.26,B = −0.536,C = 12.05and emig ≡1− pa,m0ig (10)
D=−16.78.For(cid:15) >0.05,thedependenceon(cid:15) flattensoff
Oct oct viaLKoscillations.Notethat,sincethefinalsemi-majoraxisa
atcos2i =0.45ori =48◦(Fig.1).Notethat,asshownby F
lim lim oftheplanetaftertidaldecayisapproximatelya (cid:39) r (1+
LML,theoctupole-active’windowisequivalenttotherangeofin- F p,mig
e ) (cid:39) 2r ,Equation(9)indicatesthatitisdifficulttoform
clinationssusceptibletoorbital“flips”inLKcycleswithout SRFs. max p,mig
HJswitha (cid:38)0.06AUviaLKmigration(Petrovich2015b;ASL),
Katz,Dong&Malhotra(2011)hasobtainedananalyticexpression F
unless the tidal dissipation parameter χ is much larger than the
canonicalvalueof10.
2 Equation(6)isrestrictedtoi0 (cid:54) 90◦.However,notethatthesystem Equation(9)setsthemaximumpericenterseparationthatal-
issymmetricaround90◦,thusthelimitingeccentricityisalsoreachedfor lowsforLKmigration.Thereisalsoaminimumpericentersepa-
90◦<i0<180−ilim. rationthatpermits“successfullymigrated”planets.Forpericenter
(cid:13)c 2016RAS,MNRAS000,1–??
4 Mun˜oz,Lai&Liu
(cid:113)
separationssmallerthanthetidalradius withj = 1−e2 andinwhichtherotationaldistortionef-
mig mig
(cid:18)M (cid:19)1/3 fectinEquation(3)hasbeenignoredforconsistency.Notethatthe
rp,dis =ηRp M∗ conditionemig <elimisequivalenttoC(emig)>0.Similarly,the
p criticaldisruptionanglei(Quad)isgivenbyEquation(17)withe
=1.3×10−2AU(cid:18)Rp(cid:19)(cid:18)Mp(cid:19)−1/3(cid:18)M∗(cid:19)1/3 . replacingemig. 0,dis dis
RJ MJ M(cid:12) In the first panel of Fig. 2, we show fmLKig, fdLiKs and fHLJK
(11) for a 0.3M planet started at different initial semi-major axis a
J 0
and a binary companion with a = 200 AU and e = 0.
(whereη = 2.7;Guillochon,Ramirez-Ruiz&Lin2011),planets out out
We see that, for large a , SRFs are unimportant for determining
willbedisruptedandthereforeunabletoturnintoHJs.Asbefore, 0
e around e , thus the shape of the curve is simply fLK =
forthisspecificpericentertobereached,aplanetinitiallylocated max mig mig
(cid:113) (cid:112)
ata0needstoachieveLKmaximumeccentricityof cosi0,mig = 3/5(1−e2mig) ≈ (6/5)(rp,mig/a0).Forsmall
e ≡1− rp,dis . (12) a0, elim is smaller than the required emig, making the migration
dis a fractionrapidlygotozero.
0
Notethat,ifr >r ,allmigration-inducingLKoscillations
p,dis p,mig
willresultintidaldisruptions.Thisconditionimposesalowerlimit
intheplanetmassthatpermitstheformationofclose-inplanets: 3.2.2 Includingtheoctupoleeffect
Mp >0.04MJ(cid:18)RRp(cid:19)32(cid:18)1χ0(cid:19)−34(cid:18)1MM∗ (cid:19)14(cid:18)1aA0U(cid:19)34(cid:18)0k.23p7(cid:19)−43 .eWccheenntrtihceityocatcuhpioevleedefifnecLtKisciyncclleusdceadn(b(cid:15)Oecatpp(cid:54)=rox0i)m, athteedmbayxEimquuma-
J (cid:12)
(13) tion(6).Foremig >elim,nomigrationispossibleandwestillhave
ThisimpliesthatSaturn-massplanetscanrarelymigratesuccess- cosi0,mig = 0.Foremig < elim,sinceanon-negligible(cid:15)Oct can
fullyviaLKoscillationswithoutfirstbeingdisrupted(seeASL). expandtherangeofinclinationanglesthatproduceemig,thecriti-
calanglei isthelowestpossibleanglethatresultsine ,i.e.,
0,mig mig
i =min(i(Quad),i ).Thus,
0,mig 0,mig lim
3.2 Migrationanddisruptionfractions
(cid:40) (cid:20) (cid:21)
Having defined the eccentricities required for migration and tidal cos2i = max C(emig),cos2ilim((cid:15)Oct) , ifemig(cid:54)elim
0,mig
disruption,wecannowcomputethefractionsofmigrationandtidal
0 , ife >e
mig lim
disruptionforgivena ,a ande (plustheotherfixedparame-
0 out out (19)
tersofthesystemssuchasMp,M∗,Mout,Rpandχ).Toachieve where C(e ) is given by Equation (19) and cos2i ((cid:15) )
mig lim Oct
e ,theinitialplanetaryinclinationangle(relativetothebinary)
mig is given by Equation (7). An analogous expression applies to
mustbesufficientlylarge,i.e.,i0,mig < i0 < 180◦−i0,mig.For cos2i .
0,dis
anisotropicdistributionoftheinitialplanetaryangularmomentum WeillustratetheeffectoftheoctupoletermonfLK andfLK
mig dis
vectors,thosethatcanleadtoemig orlargeroccupyasolidangle inthethreelastpanelsofFig.2.Forallpanels,a (cid:112)1−e2 =
4πcosi .Thusthemigrationfractionis out out
0,mig 200AU,whiletheeccentricityisvariedfrom0to0.8.Ase isin-
out
fmLKig(a0,aout,eout)=cosi0,mig. (14) creased,thevalueof(cid:15)Octforagivenplanetsemi-majoraxisa0also
grows.Goingfrome = 0toe = 0.8illustratesthegradual
out out
Similarly,thedisruptionfractionisgivenby transitionfromcos2i =C(e )foralla (purequadrupole)
0,mig mig 0
fLK(a ,a ,e )=cosi , (15) tocos2i0,mig = cos2ilim((cid:15)Oct)foralla0 (entirelydominatedby
dis 0 out out 0,dis theoctupoleeffect).NotethattheHJfractionfLK =fLK −fLK
HJ mig dis
wherei0,disisthecriticalinclinationthatleadstoemax =edis.The isalwaysanarrowdistributionboundedbyatthe“no-migration”
fractionofHJformation(migrationwithoutdisruption)issimply limit (e < e ) at small a and by the tidal disruption limit
lim mig 0
(e >e )atlargera .Thewidthandheightofthispeakisonly
fLK(a ,a ,e )=fLK−fLK =cosi −cosi . (16) max dis 0
HJ 0 out out mig dis 0,mig 0,dis mildlyaffectedbytheoctupolecontribution.Ontheotherhand,the
fractionoftidallydisruptedplanetsfLK isincreasedsignificantly
dis
forlargere .Thus,themaineffecttheoctupoleistoincreasethe
3.2.1 Migration/disruptionfractioninquadrupole-ordertheory out
numberofdisruptedplanets,ratherthanthenumberofsuccessfully
Foremig >elim,nomigrationispossibleandwehavecosi0,mig = migratedones(Petrovich2015b;ASL).
0.Foremig < elim,toquadrupole-order,thecriticalangleformi- Fig.3showsthesameexamplesasFig.2,exceptforaplanet
grationi0,migcanbecomputedanalyticallybydirectlysolvingfor massof3MJ.Althoughthelowerlimitfornon-zeromigrationis
cosi0inEquation(3)withemax =emig.Thus almostunchanged(seeEq.47ofASL),theboundarybetweensuc-
(cid:26) C(e ) if e <e cessfulmigrationandtidaldisruptionoccursatalargera0 (edis is
cos2i(Quad) = mig mig lim (17) largerformoremassiveplanets,makingdisruptionmoredifficult;
0,mig 0 if e >e
mig lim seeEquation(13).Thus,alargerfractionofthesemassiveplanets
where cansurviveLKoscillationsandbecomeHJs.Themass-dependent
(cid:18) (cid:19) boundarybetweendisruptionandsuccessfulmigrationexplainsthe
3 8 (cid:15)
C(emig)= 5jm2ig− 15e2GR jmig−jm2ig lack of “hot Saturns” in observations (ASL). Note that, as in the
mig previousexample,themostnoticeableeffectoftheoctupoleterms
− 8 (cid:15)Tide(cid:18)1+3e2mig+ 38e4mig −j2 (cid:19) –whichdifferentiatethelastthreepanelsfromthefirst–istoin-
225 e2 j2 mig creasetherateoftidaldisruptions,nottosignificantlyincreasethe
mig mig
(18) numberofHJs.
(cid:13)c 2016RAS,MNRAS000,1–??
Lidov-Kozaimigrationfractions 5
† † †
oct oct oct
0.002 0.004 0.006 0.005 0.010 0.015 0.02 0.04
f a =200AU f a =204AU f a =218AU f a =333AU
100 % mig out mig out mig out mig out
fHJ eout=0 fHJ eout=0.2 fHJ eout=0.4 fHJ eout=0.8
LK fdis fdis fdis fdis
f
n
o
ti
c
a
fr 10 %
n
o
ti
a
r
mig Mp =0.3MJ
R =1R
p J
1 %
1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7
a [AU] a [AU] a [AU] a [AU]
0 0 0 0
Figure2.MigrationfractionfLK (Equation14,inblack),tidaldisruptionfractionfLK(inred)andsuccessfulclose-inplanetformationfractionfLK =
mig dis HJ
fmLKig−fdLiKs (inblue)foragasgiantofmass0.3MJandtidaldissipationparameterχ=10.Thefirstpanelshowsthefractionsforabinarystellarcompanion
withaout =200AUandeout =0,forwhichtheoctupoletermsinthebinarypotentialareidenticallyzero.Thesecondtofourthpanelsshowincrements
ineout(0.2,0.4and0.8)whilekeepingaout(1−e2out)1/2constant,implyingamonotonicincreaseoftheoctupolestrength(cid:15)Oct.Iftheeffectsofoctupole
potentialwereneglected,allfourpanelswouldbeidentical.Upperhorizontalaxisshows(cid:15)Oct,whichisdirectlyproportionaltoa0forfixedaoutandeout.
Themaineffectoftheoctupolepotentialistoincreasethetidaldisruptionfraction.Theareaunderthebluecurveis,fromlefttoright,10%,7.8%,5.1%and
5.3%.
† † †
oct oct oct
0.002 0.004 0.006 0.005 0.010 0.015 0.02 0.04
f a =200AU f a =204AU f a =218AU f a =333AU
100 % mig out mig out mig out mig out
fHJ eout=0 fHJ eout=0.2 fHJ eout=0.4 fHJ eout=0.8
LK fdis fdis fdis fdis
f
n
o
ti
c
a
fr 10 %
n
o
ti
a
r
mig Mp =3MJ
R =1R
p J
1 %
1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7
a [AU] a [AU] a [AU] a [AU]
0 0 0 0
Figure3.SameasFig.2butforaplanettentimesmoremassive(Mp=3MJ).Generaltrendsareanalogoustothelessmassiveexample(thetotalmigration
curvefLK inblackisessentiallyunchanged).Inthiscase,however,planetsaremuchmorelikelytosurviveLKmigration,becausetheboundarybetween
mig
HJformationandtidaldisruptiontakesplaceathighereccentricity(Equation13).Theareaunderthebluecurveis,fromlefttoright,20%,9.2%,7.7%and
13%.
4 TOTALMIGRATIONFRACTIONSASAFUNCTION Forconcreteness,weadopttheintegrationlimitsof[0,0.8]fore
out
OFPLANETMASS and[2.0,3.0]forlog a (ASL),althoughotherchoicesarepos-
10 out
sible.WecomputethisintegralusingtheMonteCarlomethod,uni-
Intheprevioussection,wedescribedhowtocalculatethemigration
formlysamplinge andlog a .
fractionfLK anddisruptionfractionfLKforagivensetoforbital out 10 out
mig dis
parameters a , a and e (Eqs. 14-15). We now consider the
0 out out
migrationanddisruptionfractionsintegratingoverarangeofval- 4.1 Gasgiants
uesina ande :
out out
Fig. 4 shows FLK(a ) and FLK(a ) for different planet masses
(cid:90) (cid:90) dis 0 HJ 0
FLK(a )= fLK(a ,a ,e ) andtidaldissipationstrengths.Forthefiducialdissipationparame-
mig 0 mig 0 out out
ter(χ=10),theHJfractionofJupiter-massobjectsis1−3%(for
(20)
d2N a0 = 1−5 AU). For Saturn-mass objects, this fraction becomes
× dloga de .
dlogaoutdeout out out is much smaller for a wide range of a0. For smaller χ, the “hot
(cid:13)c 2016RAS,MNRAS000,1–??
6 Mun˜oz,Lai&Liu
100 % FHJ Fdis χ=1 Table1.TheHJformationfractionFHLKJ,tidaldisruptionfractionFdLiKs
) Mp Mp=0.3MJ R =1R andtotalmigrationfractionFmLKigforgasgiantswiththreedifferentmasses
a0 Mp Mp=0.5MJ p J andthreedifferentdissipationstrengths.
K( Mp Mp=1MJ
L
on F 10 % MMpp MMpp==35MMJJ χ=1 χ=10 χ=100
ti
ac Mp(MJ) 0.3 1 3 0.3 1 3 0.3 1 3
r
f
n FLK(%) 0 0.8 1.8 1.2 2.1 2.8 2.6 3.1 3.6
o HJ
ti 1 % FLK(%) 12.4 11.6 10.8 12.4 11.6 10.8 12.4 11.6 10.8
a dis
r
g
mi FmLKig(%) 12.4 12.4 12.6 13.6 13.7 13.6 15 14.7 14.4
0.1 %
100 % a0[AU] χ=10 χwhilekeepingMpfixeddoesnotchangeFdLiKs,butonlyFmLKig,as
r growsslightlywithstrongerdissipation.
p,mig
) R =1R
0 p J ThesetrendsareinagreementwiththeextensiveMonteCarlo
a
K( experiments of ASL. In that work, the computed total migration
L
F fractions(HJsplustidaldisruptions)dependveryweaklyonχ.In
n 10 % order to compare directly with the results of ASL, we integrate
o
ti FLK(a ) over a range of planet semi-major axes. Assuming that
c mig 0
ra a0isdistributeduniformlybetweena0,inneranda0,outer,wehave:
f
gration 1 % FmLKig = a(cid:90)0,outerFmLKig(a0)ddaN0da0 . (21)
mi a0,inner
Usingthevaluesa =1AUanda =5AU,wecompute
0,inner 0,outer
0.1 % the integrated migration fractions FLK, FLK and FLK for three
mig dis HJ
100 % a0[AU] χ=100 planetmasses(Mp =0.3,1.0and3MJ)andthreedissipationpa-
rameters(χ=1,10and100).Theresults(seeTable1)agreewith
) R =1R
a0 p J the findings of ASL. In particular, the total migration fraction is
(
K confirmedtobeveryrobustandtolieintherangebetween12%
L
n F 10 % and15%.
o
ti
c
a
r 4.1.1 Otherstudies
f
n
o DespitetherobustnessoftheHJmigrationfractionfoundbyASL,
ati 1 % this number is not in perfect agreement with the results of other
r
g recent works, such as Naoz, Farr & Rasio (2012) and Petrovich
mi
(2015b) (both considered only M = 1 M ). The main differ-
p J
encewithNaoz,Farr&Rasio(2012)isinthechoiceofηinEqua-
0.1 %
tion(11).Ontheotherhand,Petrovich(2015b)usedη = 2.7,the
1 2 3 4 5 6 7
a [AU] sameasASL.However,thesetwolastworksdifferintheproperties
0
oftheassumedunderlyingpopulationofcompanions(seeSection
Figure4. IntegratedHJformationfractionFHLJK(a0)andtidaldisruption 6.2ofASLforadiscussion).Here,we(andASL)haveassumedthe
fractionFdLiKs(a0)(seeEq.20)forarangeofplanetmassesMpanddimen- binaryorbitstofollowuniformdistributionsineout andlogaout,
sionlesstidaldissipationstrengthsχ.PlanetmassesareMp =0.3,0.5,1, with e ∈ [0,0.8] and loga ∈ [2,3]. Following one of the
out out
3and5MJandallplanetradiiaretakentobeRp=RJ.Thethreepanels examples of Petrovich (2015b), we choose e ∈ [0,0.95] and
correspondtoχ=1,10,100. out
loga ∈ [2,3.2], and find FLK = 1.2%, FLK = 21.2% for
out HJ dis
atotalofFLK = 22.4%,inagreementwiththeresultsofMonte
mig
Carloexperiments.
planet”fractionofSaturn-massobjectsis(cid:28)1%,whileonlyplan-
etswithmassesabove3M cansuccessfullyturnintohotplanets.
J 4.2 Rockyplanets
Itisonlyintheextremedissipationregime(χ = 100)thatallgas
giantsacrosstheexploredmassrangecanturnintohotplanetsat The procedure outlined in the previous section can be applied to
ratesofafewpercent.Thisbehaviorcanbeeasilyunderstoodfrom rockyplanetsbysimplemodificationoftheplanetmass-radiusre-
thedefinitionsofrp,mig andrp,dis (Eqs.9and11).Foragivenχ, lation, Love number k2p and dimensionless tidal strength χ. For
varyingtheplanetmassbyafactorofafewdoesnotchangesig- SolarSystemrockyplanets∆t ≈ 600s(Lambeck1977;Neron
L
nificantlythetotalmigrationfraction(FmLKig+FdLiKs)becauserp,mig de Surgy & Laskar 1997), i.e., χ = 6000, and k2p ≈ 0.3 (Yo-
isessentiallyunchanged.However,therelativeimportanceofFLK der 1995). We take the generic mass-radius relation from Seager
mig
andFLKischangedasr dependsonM .Similarly,changing et al. (2007) for an “Earth-like” composition and calculate FLK
dis p,dis p mig
(cid:13)c 2016RAS,MNRAS000,1–??
Lidov-Kozaimigrationfractions 7
andFLK asabove.Thesuccessfulmigrationrateof“hotplanets”
is,anadloisgoustoHJs,FLK =FLK −FLK. 100 % FhFp HL JKFFdidLsiK s χ=6000
entroIcnkFyigp.la5n,ewtse,srahnogwhinpFghLipnKm(imnasibsglfureo)madni2sdMF⊕dLiKsto(1in5Mred⊕).fTorhediifnfeter-- (a)0 MMMppp MMMppp===251MM0M⊕⊕⊕
gratedhotplanetfractionFhLpK(fora0between1and5AU)isbe- LKF Mp Mp=15M⊕
tween1%and1.5%.Thus,theseplanetsaremorelikelytosurvive n 10 %
LKmigrationthanSaturn-massplanets.Thereasonforthisisthat tio
c
thereisenoughroombetweenrp,mig andrp,dis(Eqs.9and11)to ra
accomodatetheplanetsintosuccessfullymigratedorbits.Rewrit- n f
ingrp,mig andrp,dis inunitsmoresuitableforrockyplanets,we tio 1 %
a
have r
g
(cid:18) (cid:19)5 mi
r =2.47×10−2 Rp 7 AU
p,mig 1.2R
⊕
0.1 %
×(cid:18) χ (cid:19)1/7(cid:18) Mp (cid:19)−17(cid:18) a0 (cid:19)−17(cid:18)k2p(cid:19)71(cid:18) M∗ (cid:19)27 1 2 3 a0[AU]4 5 6 7
6000 2M 1AU 0.3 1M
⊕ (cid:12)
(22) Figure 5. Integrated LK “hot planet” formation fraction FhLpK(a0) and
tidal disruption fraction FdLiKs(a0) as in Fig. 4 but for rocky planets.
and Planet masses are Mp=2M⊕, 5M⊕, 10M⊕ and 15M⊕, with cor-
rp,dis =7.6×10−3AU(cid:18)1.R2Rp⊕(cid:19)(cid:18)2MMp⊕(cid:19)−1/3(cid:18)1MM∗(cid:12)(cid:19)1/3 . artwensdepeon2nRd1i⊕nga.nrdWadh5iienAofUin(,StetegharegaetigrnlgoebtaFallhL.piKn2t0(ea0g07ra))teRdanpdf=ra1Fc.t2diLoRiKsn⊕s(a,a01r)e.,5oRrve⊕esrp,ea1c0t.i8vRebl⊕ey-,
(23) (FLK,FLK)=(1.08%,12.9%),(1.23%,12.9%),(1.35%,12.8%)and
hp dis
Interestingly,rp,migisveryclosetothevalueobtainedforaJupiter- (1.43%,12.7%). If, instead of χ = 6000 we use χ = 600, the hot
massplanet;however,r issmaller,sincethehigherdensityof planet fractions are reduced by half: (FLK,FLK)= (0.62%,12.9%),
p,dis hp dis
rockyplanetspushestheRochelimitclosertothestar.Thus,pro- (0.74%,12.9%),(0.85%,12.8%)and(0.92%,12.7%).
vided χ is high enough (in this case 6000), some small but non-
negligible fraction of rocky planets can survive LK oscillations
anoccurrencerateof∼ 0.5%intheKeplercatalog(e.g.Sanchis-
andmigrateintoclose-inorbits.Notethat,for2M rockyplanets,
⊕ Ojeda et al. 2014), and the occurrence rate of rocky planets with
Eqs.(22)and(23)indicatethattheLKmigratedhotsuper-Earths
slightlylongerperiods(afewdays)isappreciablyhigher(Fressin
shouldresidewithinanorbitalperiodrangeof0.7to4days.
etal.2013).Whilemanyoftheseareexpectedtobemembersof
Note that the dimensionless tidal dissipation parameter χ is
multiple-planetsystems(Sanchis-Ojedaetal.2014),somemaybe
highlyuncertain.TidaldissipationintheEarthisdominatedbyits
true“singles”(Steffen&Farr2013).LKmigrationofrockyplanets
oceans (e.g., Peale 1999, and references therein), which does not
maycontributetothispopulationofclose-inplanets.
necessarilyapplytomodelsofmigratingsuper-Earthswithuncer-
tainamountsofgasenvelopes.Ifweassumeasmallertidaldissipa-
tionstrength(χ=600),thehotplanetformationratesarereduced
byafactoroftwo(seethecaptionofFig.5).Suchsmallsuccess 5 SUMMARY
rates in producing close-in rocky planets are in rough agreement
We have derived a simple analytical method for calculating mi-
with the N-body experiments of Plavchan, Chen & Pohl (2015),
gration fractions of close-in planets via Lidov-Kozai oscillations
who implemented a constant-phase-lag (or constant tidal quality
in stellar binaries. An important ingredient of the method is the
factorQ)modeltostudyLKmigrationofsmallplanetsinthebi-
relationship between the maximum eccentricity e that can be
narysystemαCenB.Inthatstudy,only1−3integrationsoutof max
achievedinLKoscillationsasafunctionoftheinitialplanet-binary
∼600endedupinclose-inplanets.
inclination(Equation6)–theeffectsofshort-rangeforcesandoc-
Ifwewritetheoccurrencerateofhotrockyplanetsformedvia
tupole potential play an essential role. Our method can be easily
LKmigrationas(seeEquation1)Rrocky =F ×Frocky×FLK,
hp b p hp implemented for various planet and stellar/companion types, and
andassumethattheplanetbearingfractionofrockyplanetswithin
consistsofthefollowingsteps:
a few AU is Frocky∼100% (Petigura, Howard & Marcy 2013;
p
Fressinetal.2013),then,withFb ∼ 20%andFhLpK ∼ 1%(see • Forgiveninitialplanetsemi-majoraxisa0 andbinaryorbital
Fig. 5), we find Rrhopcky ∼ 0.2%. Thus, an intriguing possibility parameters(aout andeout),calculatethe“migrationeccentricity”
isthatofLKmigrationplayingaroleintheoriginofshort-period emig =1−rp,mig/a0,withrp,mig(therequiredpericenterdistance
rockyplanets,especiallyforsingle-planetsystems3.Observation- inorderfortidaldissipationtoinduceefficientorbitaldecay)given
ally,ultra-short-period((cid:46) 2days)rockyplanetsarefoundtohave byEquation(9).
• Knowinge andtheoctupoleparameter(cid:15) ,calculatethe
mig oct
criticalplanet-binaryinclinationangleformigration,i ,using
0,mig
Equation(19).
3 ForsystemscontainingmultipleplanetswithafewAUsemi-majoraxes,
• Assumingisotropicdistributionofthebinaryorientations,the
LKoscillationscanbesuppressedbymutualplanet-planetinteractionsif
migrationfractionisfLK = cosi .Thetidaldisruptionfrac-
theplanetmassistoolarge.FromEq.(7)ofMun˜oz&Lai(2015),thecrit- mig 0,mig
tionissimilarlycalculatedfromfLK = cosi andtheHJ(hot
ical“shieldingmass”isabout3M⊕fora0 ∼3AUandastellarcompan- dis 0,dis
ionlocatedat150AU(seealsoMartin,Mazeh&Fabrycky2015;Hamers, planet)formationfractionisfHLJK =fmLKig−fdLiKs.
Perets&PortegiesZwart2016).Forsmallerplanetmasses,LKoscillation • Foradistributionofa0,aout andeout,theintegratedmigra-
mayoperateforeachplanetindependently. tion/disruptionfractionscanbeobtainedusingEqs.20and21.
(cid:13)c 2016RAS,MNRAS000,1–??
8 Mun˜oz,Lai&Liu
Theresultsofourcalculationsareingoodagreementwiththe Martin D. V., Mazeh T., Fabrycky D. C., 2015, MNRAS, 453,
recentpopulationstudiesofHJformationviaLKmigrationinstel- 3554
larbinaries(Petrovich2015b;Anderson,Storch&Lai2016).They Mun˜ozD.J.,LaiD.,2015,ProceedingsoftheNationalAcademy
canbeeasilymodifiedtoreflectdifferenttypesofplanets/binaries ofScience,112,9264
anddistributionsofthebinary’sorbitalelements.Thisagreement NagasawaM.,IdaS.,BesshoT.,2008,ApJ,678,498
confirmsthatthetotalmigrationfractionofplanets(thesumofHJ NaozS.,FarrW.M.,RasioF.A.,2012,ApJ,754,L36
andtidaldisruptionfractions)viatheLKmechanismisprimarily NerondeSurgyO.,LaskarJ.,1997,A&A,318,975
determined by the initial geometry of the hierarchical triple, and PealeS.J.,1999,ARA&A,37,533
dependsweaklyonthepropertiesoftheplanet(Anderson,Storch PetiguraE.A.,HowardA.W.,MarcyG.W.,2013,Proceedings
& Lai 2016). The production fraction of close-in planets, on the oftheNationalAcademyofScience,110,19273
other hand, does depend on the internal properties of the planet PetrovichC.,2015a,ApJ,805,75
(e.g., mass-radius relation and the strength of dissipation), and it PetrovichC.,2015b,ApJ,799,27
decreases with decreasing mass in the case of gas giants, result- PlavchanP.,ChenX.,PohlG.,2015,ApJ,805,174
inginanaturalabsenceof“hotSaturns”.Inaddition,wefindthat RaghavanD.etal.,2010,ApJS,190,1
rocky planets, provided they are sufficiently dissipative, may mi- RasioF.A.,FordE.B.,1996,Science,274,954
gratemoreeffectivelythanSaturn-massplanets.Itispossiblethat Sanchis-OjedaR.,RappaportS.,WinnJ.N.,KotsonM.C.,Levine
somefractionoftheobservedshort-periodrockyplanetsarepro- A.,ElMellahI.,2014,ApJ,787,47
ducedviaLKmigrationinstellarbinaries. SeagerS.,KuchnerM.,Hier-MajumderC.A.,MilitzerB.,2007,
ApJ,669,1279
SteffenJ.H.,FarrW.M.,2013,ApJ,774,L12
StorchN.I.,AndersonK.R.,LaiD.,2014,Science,345,1317,
ACKNOWLEDGEMENTS
preprint(arXiv:1409.3247)
WethankKassandraAndersonforusefuldiscussionsandCristo´bal WrightJ.T.,MarcyG.W.,HowardA.W.,JohnsonJ.A.,Morton
Petrovich for comments on the manuscript. This work has been T.D.,FischerD.A.,2012,ApJ,753,160
supported in part by NSF grant AST-1211061, and NASA grants WuY.,LithwickY.,2011,ApJ,735,109
NNX14AG94GandNNX14AP31G. WuY.,MurrayN.,2003,ApJ,589,605
WuY.,MurrayN.W.,RamsahaiJ.M.,2007,ApJ,670,820
YoderC.F.,1995,inGlobalEarthPhysics:AHandbookofPhys-
icalConstants,AhrensT.J.,ed.,p.1
REFERENCES
AndersonK.R.,StorchN.I.,LaiD.,2016(ASL),MNRAS,456,
3671
Beauge´C.,Nesvorny´D.,2012,ApJ,751,119
ChatterjeeS.,FordE.B.,MatsumuraS.,RasioF.A.,2008,ApJ,
686,580
CorreiaA.C.M.,LaskarJ.,FaragoF.,Boue´ G.,2011,Celestial
MechanicsandDynamicalAstronomy,111,105
DawsonR.I.,Murray-ClayR.A.,2013,ApJ,767,L24
DawsonR.I.,Murray-ClayR.A.,JohnsonJ.A.,2015,ApJ,798,
66
FabryckyD.,TremaineS.,2007,ApJ,669,1298
FordE.B.,KozinskyB.,RasioF.A.,2000,ApJ,535,385
FressinF.etal.,2013,ApJ,766,81
GoldreichP.,TremaineS.,1980,ApJ,241,425
GuillochonJ.,Ramirez-RuizE.,LinD.,2011,ApJ,732,74
HamersA.S.,PeretsH.B.,PortegiesZwartS.F.,2016,MNRAS,
455,3180
HowardA.W.etal.,2012,ApJS,201,15
Juric´M.,TremaineS.,2008,ApJ,686,603
Katz B., Dong S., Malhotra R., 2011, Physical Review Letters,
107,181101
KozaiY.,1962,AJ,67,591
LambeckK.,1977,PhilosophicalTransactionsoftheRoyalSoci-
etyofLondonSeriesA,287,545
LidovM.L.,1962,P&SS,9,719
Lin D. N. C., Bodenheimer P., Richardson D. C., 1996, Nature,
380,606
LithwickY.,NaozS.,2011,ApJ,742,94
LiuB.,Mun˜ozD.J.,LaiD.,2015(LML),MNRAS,447,751
Marcy G., Butler R. P., Fischer D., Vogt S., Wright J. T., Tin-
neyC.G.,JonesH.R.A.,2005,ProgressofTheoreticalPhysics
Supplement,158,24
(cid:13)c 2016RAS,MNRAS000,1–??