Analyses on a Relativistic Hierarchical Resonance with the Hamiltonian Approach

0.82 MB·English

by  Naoki Seto

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed16January2013 (MNLATEXstylefilev2.2)
Analyses on a Relativistic Hierarchical Resonance with the
Hamiltonian Approach
Naoki Seto
3 DepartmentofPhysics,KyotoUniversity,Kyoto606-8502,Japan
1
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2
16January2013

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a
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ABSTRACT

4
1 We study dynamical evolution of a resonant triple system formed by an inner EMRI
and an additionalouterMBH. The relevantresonantstate (λ2 −̟1 ∼ const) is supported
]
P by the relativistic apsidal precession of the inner EMRI, and, unlike standard mean motion
E resonances,thetriplesystemcanhaveahierarchicalorbitalconfiguration(butdifferentfrom
theKozaiprocess).Inordertoanalyzethisunusualresonantsystem,weextendtheso-called
.
h Hamiltonianapproach,andderiveamappingfromtheEMRI-MBHtriplesystemtoasimple
p
one-dimensionalHamiltonian.Withthederivedmapping,wemakeanalyticalpredictionsfor
-
o characteristicquantitiesof the resonance,such as the captureprobability,and find that they
r reasonablyagreewithnumericalsimulationsuptomoderateeccentricities.
t
s
Keywords: gravitationalwaves—binaries:close
a
[

1
v
5 1 INTRODUCTION tionaltotheinnerandoutermeananomaliesnearlycancel(Peale
3 1986; MD). Consequently, the resonant EMRI-MBH system can
1 Inthesolarsystem,orbitalresonancesarebroadlyobservedatvar- haveahierarchicalorbitalconfigurationandthemassesofthetwo
3 ious spatial scales (Peale 1986; Murray & Dermott 2000 (here- MBHscanbecomparable.Thesepropertiesareremarkablydiffer-
1. afterMD)). Forexample, PlutoandNeptune haveorbitalperiods ent from the standard MMRs where two orbital periods (equiva-
0 of3:2andtheirorbitalstabilityissustainedbythissimplerelation. lently,twosemimajoraxes)arecomparablebutthemassesof the
3 The resonant states with such commensurable orbital periods are central body must be much larger than other ones due to orbital
1 termed mean motion resonances (MMRs), and have been identi- stability(Gladman1993).
: fiedalsoamongextrasolarplanetarysystems(Lissaueretal.2011;
v Inthispaper,wefocusouranalysistotheresonantdynamics
Petrovich,Malhotra,&Tremaine2012).
i of the dominant mode λ ̟ const, paying special atten-
X In a recent paper (Seto 2012), triple system formed by an tiontodependenceonthe2in−nere1cc∼entricity.Tothisend,weutilize
r EMRIandanadditionaloutermassiveblackhole(MBH)wasnu- theso-calledHamiltonianapproachincelestialmechanics(Sinclair
a
merically studied, using the post-Newtonian (PN) approximation 1972;Yoder1979;Henrard1982;Henrard&Lamaitre1983;Peale
(see Fig.1forthe orbital configuration). Here “EMRI”stands for 1986;MD).ThismethodhasbeenappliedforthestandardMMRs.
“extreme-mass-ratioinspiral”andrepresentsaninspiral ofacom- Its basic prescription is to extract the essential dynamical degree
pact object (CO) around a MBH (see Gairetal. (2004) for de- of freedom from the complicated original triple system and map
tail).Thenumericalsimulationswereperformedmainlyfromsmall thetriplesystemdowntoasimpleone-dimensionalsystemwhose
initial eccentricities, and two resonant states were identified with dynamicsisdescribedbyarescaledHamiltonian(moreprecisely,
λ2 ̟1 constand3λ2 ̟1 2Ω1 const.Hereλ2repre- in a two-dimensional phase space with a canonical variable and
− ∼ − − ∼
sentsthemeananomalyoftheouterMBHaroundthecentralMBH. itsconjugatemomentum).Ourresonanceisanunusualmeanmo-
Theangles̟1andΩ1 arethelongitudesofthepericenterandthe tionresonance,butcertainlyclassifiedasaneccentricity-typeres-
ascendingnodeoftheCO. onance. Therefore, theimportant dynamical parameterswould be
Seto(2012)alsodiscussedastronomicalaspectsforthetriples, theinnereccentricityandtheresonanceangleλ ̟ const.
2 1
− ∼
including prospects for gravitational wave and electromagnetic Aroundtheresonance, otherparametersapproximately behaveas
waveobservations.Theexpectednumbersofresonantcaptures(not cyclicvariablesorconstants(seee.g.MD).
thecaptureprobabilitiesattheresonantencounters) were roughly So far, various characteristic behaviours of the standard
estimatedandthemodeλ2 ̟1 constturnedouttooccupythe MMRs have been successfully explained with the Hamilto-
− ∼
majorityofthecaptureevents. nian approach, taking advantage of basic principles on ana-
The two resonant states are induced by the relativistic apsi- lytical mechanics, such as conservation of adiabatic invariants
dalprecessionoftheEMRIanddonotdependontheinnermean (Borderies&Goldreich1984;Peale1986;MD).Inthispaper,we
anomalyλ ,unlikethestandardMMRsinwhichtwotermspropor- are primarily interested in whether we can suitably extend the
1
(cid:13)c 0000RAS
2 N. Seto
Hamiltonianapproach forourunusual resonant state λ2−̟1 ∼ (cid:2) l
const.Ifitworkswell,wecaneasilymakeastrophysicalarguments 2
ontheresonantEMRI-MBHsystemswithoutusingcostlynumeri-
calsimulations,and,furthermore,wecanbetterunderstandtheef-
v
ficientanalyticalapproachitselfinaperspectivedifferentfromthe (cid:1) 1
traditionalanalysesforthestandardMMRs.
Inthispaper,byappropriatelyhandlingtheeffectsoftherela- (cid:3) (cid:3)(cid:2)(cid:8)(cid:2)(cid:3)(cid:2)(cid:6)(cid:5)(cid:2)(cid:9)(cid:10)(cid:4)(cid:3)(cid:2)(cid:5)(cid:7)(cid:4)(cid:11)(cid:6)
tivisticapsidalprecession,wederivethemappingfromtheEMRI- (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:7)(cid:2)(cid:3)
MBH triple system to the simple Hamiltonians whose forms are
identicaltothoseusedforanalyzingthestandardMMRs.Wethen
makeanalyticalpredictionsonthedynamicalevolutionofthehier-
archicaltriplesaroundtheresonantencounters.Wecomparethese
predictions with numerical simulations and confirm good agree-
Figure1. Thecoplanartriplesystemcomposedbytwomassiveblackholes
mentsforcertainrangeoftheeccentricityeoftheinnerEMRI. (MBHs;0and2)andacompactobject(CO;1).TheMBH0andtheCO1
Thispaperisorganizedasfollows.In§2wesummarizebasic formaninnerEMRI.Wemeasuretheangularposition̟1ofthepericenter
notations,brieflydescribeournumericalscheme,andprovidesome ofthe CO with respect tothe fixed reference direction. Theouter MBH
ofrepresentativenumericalresultsaroundtheresonantencounters. hasanearlycircularorbitanditsangularpositionisgiven byλ2.Wecan
In 3 we discuss the relativistic apsidal precession. Later, its de- similarlydefinethevariable λ1 fortheCO,butitdoesnotappearinour
pen§dence on the inner eccentricity e plays a critical role for the resonantvariableλ2−̟1∼const.
overallstructureofthemapping.In 4,wecomparethestrengths
§
ofthefirst-orderterm( ecos(λ ̟ ))andthesecond-orderone
2 1
∝ −
(m∝apep2incgosm2e(nλt2io−ne̟d1a)b)ovfoer,boyurerxetseonndainngtstthaetep.rIenvi§o5u,swsetuddeierisvdeotnhee case m0 m1 m2 a1in α8 ddτδ¯ D
forthestandardMMRs.Inthenextthreesections,usingthederived I 0.90 0 0.1 50 0.017 0.00168 24.7
mapping, wemakeanalytical predictionsontheresonant dynam- II 0.98 0 0.02 30 0.027 0.0289 68.0
icsandextensivelycomparethemwithnumericalsimulations.The III 0.999 0 0.001 20 0.035 1.43 890
capturerateisexaminedin 6.In 7,wediscussthegapoftheec-
§ §
centricityobservedatafailureofresonantcapture.In 8,westudy Table1.Themodelparametersadoptedinournumericalsimulations.The
§
resonant encounters for relatively inclined orbits. We summarize initialoutersemimajoraxisa2inisadjustedtoyieldaresonantencounter.
thispaperin 9. Theoutereccentricity isinitially setate2in ≃ 0anditstaysatasmall
§ value.Theleftthreecolumnsshowthebasicparameterscharacterisingthe
resonant dynamics, and they are evaluated for coplanar orbits. The ratio
α/8isthecriticaleccentricityfortheshiftoftheresonantangleλ2−̟1
2 EVOLUTIONOFTHESYSTEM (discussedin§4).Thetransversevelocitydδ¯/dτ andthecoefficientDare
definedin§5,andevaluated withrespecttotheHamiltonian (49)forthe
OurtriplesystemiscomposedbytwoMBHswithmassesm0,m2 secondorderresonance.
and a CO of m ( m ,m ). The two components m and m
1 0 2 0 1
≪
formaninnerEMRI andthethirdone m isrotating outsidethe
2
EMRI(seeFig.1).Fortheorbitalelementsofthetriple,wefollow H . But this term is time consuming and less important for our
2
thepositions of m1 and m2 relativeto thecentral MBH m0 and resonance.Wethusdropithere.
determinethe(instantaneous)semimajoraxesalandeccentricities The equations of motions for thepositions xl and momenta
el (l = 1,2). Since we only handle triples with nearly circular pl ofthethreemassesml (l = 0,1,2)areobtainedbytakingap-
outerorbitandtheoutereccentricity e2( 1)isnotimportantin propriatepartialderivativesoftheHamiltonian.AsinSeto(2012),
≪
twheismpaaipnelry,wstuedpyuctoep1la=naeroforbristiamlcpolincfiitgyuoraftnioontast,iaosn.shEoxwcenpitnfForig§.81,, owfethueseCtOhewneitwhmvariable1s(lin≡clupdli/nmglthteotpersotppearrltyichlaenldimleitthmem=oti0o)n.
1 1
anddefinethemeananomaliesλlandthelongitudesofpericenters Theseequations are≪integratedbyaRunge-Kuttamethod withan
̟l(l=1,2),followingthestandardconvention(MD).Below,we adaptive step sizecontrol (Presset al. 1996, and see alsoSeto&
usethegeometricalunitG=c=M =1(M m0+m1+m2: Muto2011fordetail).
≡
thetotalmass). In Table 1, we present the model parameters of our numer-
Fornumericalevolutionofthesystem,weusethethree-body ical simulations. Since dependence of the resonant dynamics on
ADMHamiltonianHTBinthepost-Newtonianformalism,andne- theinnereccentricity eisourcentralissue,wesystematicallyan-
glecteffectsofspins.TheHamiltonianisexpandedas alyzeitforcommonlyarrangedsetsofparameterssuchasmasses
(m ,m ,m ) and the initial inner semimajor axis a . Among
H =H +H +H (1) 0 1 2 1in
TB N 1 2.5 thethreemodelslistedinTable1,wemainlyusemodelsIandII,
(Scha¨fer 1987; Jaranowski&Scha¨fer 1997; Lousto&Nakano targetingcomparableMBHs,andmodelIIIisstudiedforaspecific
2008; Arunetal. 2009; Galaviz&Bruegmann 2010) (see also purposein 4.
§
Moore1993).HereH istheNewtonianterm,andH isthe1PN In Figs.2-4, wepresent samples of typical orbital evolutions
N 1
term, namely the leading order relativistic correction. The 2.5PN ofmodelIaroundtheresonantencounters.Wesettheinitialouter
termH isthefirstdissipativetermcausedbygravitationalradia- distancea sothatthesystemtransversestheresonantcondition
2.5 2in
tionreaction,andinvokestheorbitaldecayofthesystem.InEq.(1), duetotheradiationalorbitaldecay.Throughoutthispaper,weuse
weputthesubscript“TB”representing“three-body”todistinguish theoutersemimajoraxis a toshow thetime.Thisvariable a is
2 2
therescaledHamiltonianHintroducedin 5. monotonicallydecreasingfromitsinitialvaluea .
2in
§
Inthepreviouspaper(Seto2012),weincludedthe2PNterm InFig.2,weshowtheresultsfromaninitialinnereccentricity
(cid:13)c 0000RAS,MNRAS000,000–000
Analyseson a RelativisticHierarchicalResonancewith theHamiltonianApproach 3
6 0.4 6
Dn5 Dn5 0.7
dia4 0.3 dia4 0.6
a a
@rv123 e0.2 @rv123 e0.5
Λ-21 0.1 Λ-21 0.4
0 0.0 0 0.3
300 310 320 330 340 350 360 300 310 320 330 340 350 360 220 240 260 280 300 220 240 260 280 300
a2 a2 a2 a2
53
50.2
50.0 52
a149.8 a151
49.6
50
49.4
300 310 320 330 340 350 360 49
a2 220 240 2a620 280 300
Figure2.EvolutionofacoplanarEMRI-MBHtriplesystemfromaninner
Figure3.Evolutionofacoplanartriplefromalargeinnereccentricitye∼
eccentricitye∼0.1(modelI).Thehorizontalaxisa2isthesemimajoraxis 0.57(modelI).TheinnerEMRIisresonantlycapturedbytheouterMBH
otofgthraevoituatteironMalBwHavme2emanisdsdioenc.reTahseesthfrreoempiatnseilnsitsihaolwvatlhueerae2sionna=nt3v6ar0iadbulee arounda2∼280.Thelibrationamplitudeislarge.
λ2−̟1(theupperleftone),theinnereccentricitye(theupperrightone,in
modulo2π)andtheinnersemimajoraxisa1(thebottomone).Thissystem
6
encounterstheresonancearounda2 ≃340,andthecaptureissuccessful. Dn5 0.7
Theinnereccentricityestartstoincreaseafterward. dia4 0.6
a
@rv123 e0.5
ein ≃0.1.Thetestparticlem1isresonantlycapturedbytheouter Λ-21 0.4
MBH binaryatthetimea 340, corresponding totheratioof
2 0 0.3
∼ 220 240 260 280 300 220 240 260 280 300
orbitalperiodsat 17.Incidentally,theinnereccentricityestarts
∼ a2 a2
togrowandtheinneraxisa1decreasesveryslowly.Theresonant 53
variableλ ̟ soonlocalizesaround π/2.
2− 1 ∼ 52
FortherunshowninFig.3,wesetalargerinitialeccentricity
e 0.57. The test particle m is captured into the resonance a151
in 1
≃
arounda2 280.IncontrasttoFig.2,thecombinationλ2 ̟1 50
∼ −
nowhasalargelibrationalamplitudewithasmallexcludedregion
49
aroundπ.Again,aftertheresonantcapture,theinnereccentricitye 220 240 260 280 300
a2
increasesandthesemimajoraxisdecreases.
InFig.4,theinitialinnereccentricity e 0.57 iscloseto Figure4. Evolution ofacoplanar triple fromsimilar orbital parameters
in
∼
that in Fig.3. But the initial orbital phases are different between (e.g.ein∼0.57)asFig.3butwithadifferentrelativeorbitalphase(model
Figs.3 and 4. While evolutions in Figs.3 and 4 are similar down I).Theresonant capture is unsuccessful. Wecan observe gaps oforbital
to a 280, their subsequent profiles are completely different. parametersattheresonantencounterarounda2∼280.
2
∼
Aroundthecriticalepocha 280,theinnereccentricityeshows
2
∼
a large gap in Fig.4, but the EMRI is not captured into the reso-
nseamnciem,aajosrianxdiiscaat1edalbsoyhtahsearosmtaatilnlggavpa,rbiaubtlteheλf2ol−low̟i1n.gTThiseseinrannedr t̟anR, tihsethlaergleosntgsitautedlelitoefotfheSaptuerrinceanttethreodfistthaencrein∼glet20aRndS.λHTeries
relation(Murray&Dermott2000,butnowforacoplanarsystem) the mean anomaly of Titan. The apsidal precession ̟˙R of the
holdsnearlysmoothlyarounda 280; ringlet is mainly driven by the multiple moments of Saturn (e.g.
2
∼ itsquadrupolemoment;J =1.6 10−2).Thisringlethasafinite
2
a2 + a1(1 e2) const. (2) eccentricityeR ∼(2.6±1.4)×1×0−4andaradialwidth∼20km.
2a1 a2 − ≃ Another potentially interesting example is the Earth-Moon
r
system. In its early history, the system might be resonantly af-
Thisrelationconnectsthegapsforeanda inFig.4.
1
fectedbytheSunasanouterperturber,throughthestate λ
For the standard MMRs, it is well known that the capture ̟ const(Touma&Wisdom1998;C´uk&Stewart2S0u1n2)−.
becomes a stochastic process when we increase the eccentricity Moon∼
of the perturbed mass (Borderies&Goldreich 1984; Peale 1986;
Malhotra1988;Dermott,Malhotra,&Murray1988;MD).Inaddi-
tion,theeccentricityshowsagapifthecaptureisfailed.Thesein-
3 RELATIVISTICAPSIDALPRECESSION
terestingcharactersaresuccessfullyexplainedbytheHamiltonian
approach.Forourunusualresonance,wemakedetailedanalysison Asdemonstratedintheprevioussection,ourresonantstateischar-
theseissueslaterin 6and7. acterizedbythefollowingrelationbetweenthetwoangularparam-
§
OurmaintargetsinthispaperaretheEMRI-MBHtriplesys- etersλ and̟
2 1
tems in the resonant state λ ̟ const. But it would be
2 1
worth mentioning that a related−resona∼nt structure was identified η λ2 ̟1 const. (3)
≡ − ∼
intheringofSaturn(Porcoetal.1984).TheTitanringlethasthe
Takingthetimederivativeofthisrelation,wehave
semimajoraxisof1.29R (R 6 104km:theradiusofSat-
S S
∼ ×
urn) and is in the resonant state λ ̟ const with Ti- n ̟˙ (4)
T R 2 1
− ∼ ∼
(cid:13)c 0000RAS,MNRAS000,000–000
4 N. Seto
on average (thedot˙representing the timederivative). Here n is relativisticone. Thedistantouterbody m alsohasa1PNeffect
l 2
theangularfrequencyoftheobjectm (l=1,2)aroundthecentral fortheprecession̟˙ (seethe1PNinteractionterminNaozetal.
l 1
MBHm ,andevaluatedwithKepler’sthirdlawas 2012).ButitsmagnitudeisO(α2.5)timessmallerthanEq.(9),and
0
notimportantfortheprecession̟˙ . Wehereafterput
1/2 1
m
n1 = a30 (5) ̟˙1=̟˙1r, (15)
(cid:18) 1 (cid:19)
m +m 1/2 a 3/2 asalreadyassumedtoderiveEq.(12).
n = 0 2 =m−1/2n 1 (6)
2 a3 0 1 a
(cid:18) 2 (cid:19) (cid:16) 2(cid:17)
form1 m0+m2=1.Tocharacterizethehierarchyoftheinner 4 COMPARISONBETWEENTHEFIRSTANDSECOND
≪
andouterorbits,weintroducethefactorαas ORDERRESONANCES
a
α 1 1. (7) Thegravitationalinteractionbetweentheinnerandouterorbitsof
≡ a2 ≪
atriplesystemhasbeenperturbativelyanalyzewiththedisturbing
Thentheouterfrequencyisroughlygivenas
function (MD). For our resonant state η const in a coplanar
∼
n n α3/2 (8) configuration,therelevantelementofthedisturbingfunctionisex-
2 1
∼ pandedas
form =O(1).
0
Next we discuss the apsidal precession rate ̟˙1 of the inner ζ =e1C1,0cosη+e2C2,0cos2η, (16)
EMRI.AsiswellknownforMercury,relativisticcorrectiongener-
wherewetakethetermsuptotheorderO(e2).ThefunctionsC
atestheprecessionwiththerate 1,0
andC dependonthehierarchyparameterα 1oftheorbital
2,0
≪
3m3/2 3pn configuration.Theyareexplicitlygivenas
̟˙ = 0 = 1 (9)
1r a15/2(1−e2) 1−e2 C1,0 = 32α− α2∂α +1 b(10/)2(α)=−1156α3 (17)
atthe1PNorder(Landau&Lifshitz1971).Here,inordertoexplic-
(cid:16) (cid:17)
itlyshowtherelativisticeffects,weadditionallydefinedthepost- 3 3α∂ 3α2∂2 15 105
C = + α + α b(2)(α)= α2+ α4(18)
NewtonianparameterpoftheEMRIas 2,0 4 4 8 1/2 8 64
(cid:18) (cid:19)
m
p 0. (10) withtheLaplacecoefficientsb(j)(α).InEq.(17)thefirstterm2α/3
≡ a1 i
istheindirectpartandiscanceled bythe O(α)termofitsdirect
Inthispaper,weonlydealwiththeregime p ≪ 1wherethePN part.Asaresult,thefunctionC1,0 α3hasastrongerdependence
frameworkworkswell.Therelativisticprecession(9)dependson ∝
ontheparameterα( 1)thanthecounterpart C α2.Actu-
theeccentricityeas (1 e2)−1.Asweseelaterin 5,thisdepen- ≪ 2,0 ∝
∝ − § ally,thesecond-orderone C2,0 hasthelowestpowerofαamong
dencebecomesparticularlyimportantforourunusualresonance.
theresonanttermsintheformcosNη withN > 1.Wehereafter
FromEqs.(4)(6)and(9),weobtainthefollowingrelationfor
neglecttheterm α4inEq.(18)andput
theonsetoftheresonance ∝
15 e
13pe2 =m0−1/2α3/2 (11) ζ =−16α3e cosη−2αcos2η . (19)
− This expressio(cid:16)n shows that the se(cid:17)cond order term can dominate
orequivalently
thefirstorderone even atasmall eccentricity e α,due tothe
∼
a =3−2/3a5/3(1 e2)2/3m−1. (12) hierarchyofthesystemα 1.
2 1 − 0 ≪
Interestingly,thecompetitionofthetwotermscanbedirectly
The expression for e = 0 was studied in the previous paper
observed as a shift of the mean angle η of libration, during the
(Seto 2012, see also Hirata 2011) and we have the relation be-
resonantamplificationoftheinnereccentricity e.Wenowdiscuss
tween the PN parameter p and the orbital hierarchy parameter α
thisinsomedetail.Forsimplicity,weassumethatthedissipative
asp α3/2/3.Fortheeccentriccasesshownintheprevioussec-
∼ evolutionisnegligibleduringonelibrationperiod.
tion,Eq.(12)providesa 360forFig.2anda 285forFigs.3
2≃ 2≃ First,thesystemaround theresonance canbeeffectivelyre-
and4,reasonablyreproducingthedependence ontheeccentricity
duced to one dimensional system (with the variable η and its
e. conjugatemomentum e2,see 5fordetail).Theeffec∝tiveHamil-
Eq.(11)isobtainedbyneglectinginfluenceofthedistantouter ∝ §
tonianhastheresonantterm ζ,andthevariableηappearsonly
MBHm andassumingthattheprecessionrate ̟˙ isdominated ∝
2 1 inthisterm.Then,fromthecanonicalequation,weshouldhave
bytherelativisticeffect̟˙ .HereweevaluatetheNewtoniansec-
1r
ular contribution ̟˙ due to m . For moderate eccentricity and de2 ∂ζ
1N 2 =0 (20)
inclination,theseculareffect̟˙ isestimatedas(MD) dt ∝ ∂η
1N
̟˙1N = 43ma32ma131//22. (13) aet=theeee,qwuieliabsrsiuomciapteoitnhte(ceoer,rηeesp).oTndhiunsg,feoqrugiliivberniuemquainligblreiuηmeavsaltuhee
2 0 minimumofthefollowingpotentialV( ζ)
∝
Then,atthecriticaldistance(12),wehave
2e
V cosη+ e cos2η. (21)
̟˙1N = 9m0m2 1 (14) ≡− α
̟˙1r 4 a1 ≪ TheshapeofthispotentialisshowninFig.5forrepresentativeval-
with p = m /a 1 and m < 1. Therefore, the Newtonian uesoftheratioe /α.Thepositionsofthepotentialminimaqualita-
0 1 2 e
≪
contributionfortheprecession̟˙ wouldbemuchsmallerthanthe tivelychangeatthecriticalvaluee =α/8.InTable1,wepresent
1 e
(cid:13)c 0000RAS,MNRAS000,000–000
Analyseson a RelativisticHierarchicalResonancewith theHamiltonianApproach 5
itsvalueformodelsI-III.Ate < α/8,thepotential V isdomi-
e 2
natedbythefirstordertermandwehavetheequilibriumangle
e (cid:144)Α=1
ηe =0, (ee<α/8). (22) 1 e
When increasing e beyond the critical value α/8, the angle η
e e
startstomoveas V 0 0.4
1(cid:144)8
η = arccos(α/8e ), (e >α/8). (23)
e ± e e -1
0
We have η π/2 for e α/8, dominated by the second
e e
≃ ± ≫
orderterminEq.(21). -2
Nowweexamineoursimplemodel(22)and(23)fortheequi-
-3 -2 -1 0 1 2 3
libriumresonant angle, by using numerical simulations. In Fig.6,
Η=Λ -v
weshowtheevolutionoforbitalparametersformodelIII.Owing 2 1
toitssmalloutermass m ,theforcedeccentricityissmallatthe
2 Figure5. TheeffectivepotentialV definedinEq.(21).Itsshapesareplot-
earlystage,andthismodelallowsustomakeasuitabledemonstra-
tedforrepresentativevaluesofee/α.Theequilibriumangleηewouldbe
tionforthepresentanalysis.
theminimumofthepotential.Foree/α < 1/8,thepotentialhasthesin-
The EMRI is resonantly captured by the outer MBH binary gleminimumatη = 0,dominatedbythefirstorderterm.Incontrast,for
arounda2 ∼55,anditseccentricityestartstoincreaseafterward. ee/α > 1/8,wehavetwominimaatη 6= 0,reflectingthesecondorder
Herethecriticaleccentricityfortheonsetoftheshiftoftheequilib- component.Forlargeree/α,theminimalpointsapproach±π/2.
riumangleisα/8 0.035.Sincethelibrationcanberegardedas
∼
acirculationaroundtheequilibriumpoint(e ,η ),themeanvalue
e e 6
ofthelibrationwouldbeclosetotheequilibriumpoint(ee,ηe),at Dn5 0.20
a
leastforasmalllibrationamplitude.Todirectlyshowtheshiftof di4 0.15
a
tcthheeentaarnincagiltlyyetieηc,ealwtmheooprdoloeultg(th2h2eth)ceaonmrdebs(oi2nn3aa)tn.iotnam(ep,lηifi)ciantiFonig.o7fttohgeetihnenrerweicth- @rΛ-v21123 e00..0150
0 0.00
Fig.7showsthat,eventhoughthedissipativetimescaleisnot 50 52 54 56 58 50 52 54 56 58
sufficientlylongcomparedwiththelibrationperiod,thesimplean- a2 a2
20.0
alyticalpredictionshowsagoodagreementwiththenumericalone.
19.9
ForalargerlibrationamplitudeasinFig.3,thepotentialwallofV 19.8
around η = 0 (see Fig.5) iseasily crossed over, and the angle η a119.7
movesaroundabroadregion,leavingasmallexcludedregimenear 19.6
η=π. 19.5
19.4
Note that also in Fig.2, the angle η finally localizes around 50 52 54 56 58
π/2,asinthecaseofFig.6.Butthisshouldberegardedasamere a2
coincidence.Laterin 6and7,wedealwithalargesampleofnu- Figure6. Evolutionoftheorbital elements ofthecoplanar innerEMRI
§
mericalsimulationsformodelsIandII.Amongthem,thereareno formodelIII.Duetothesmalloutermassm2,theforcedcomponentscan
definiteasymmetriesforthepreferenceofthetwopotentialminima besuppressed,comparedwithcasesIandII.Wehaveasmallinitialinner
atη=π/2and π/2(equivalently3π/2). eccentricitye∼0.0065.TheEMRIiscapturedbytheouterMBHbinary
Insummary−,duetothehierarchyoftheorbitswithα 1,the arounda2 ∼55.Wecanclearlyobservetheshiftoftheequilibriumangle
second order term e2α2cos2η could become more im≪portant ηe.
∝
thanthefirstorderone eα3cosη,evenforasmalleccentricity
∝
e α.Wecanobservetheresultantshiftofthemean(equilibrium) 3
∼
angleη duringtheresonantamplificationoftheinnereccentricity
e
D 2
e. n
a
di 1
a
r
@
1 0
v
5 HAMILTONIANAPPROACH
- -1
2
In this section, we apply the Hamiltonian approach for the reso- Λ
= -2
nantdynamicsoftheEMRI-MBHtriplesystemswithλ ̟ Η
2 1
− ∼
const. By taking appropriate set of conjugate variables, the dy- -3
namics around the resonant encounter can be reduced to a sim- 0.00 0.05 0.10 0.15 0.20
pleone dimensional system (Sinclair1972; Yoder 1979; Henrard e
1982; Henrard&Lamaitre 1983; Peale 1986; MD). Forthe stan-
dardMMRssuchas2:1or3:1resonances,thisapproachsuccess- Figure7. Correspondencebetweentheinnereccentricity eandthereso-
fullyexplainscharacteristicphenomenon around theresonant en- nantangle η = λ2 −̟1.Thepoints areobtained fromthe runshown
inFig.6.Theredcurvesaretheanalyticalpredictions(22)and(23)forthe
counters(Borderies&Goldreich1984;Peale1986;MD).Ouraim
equilibriumpoints(ee,ηe)withthetransitioneccentricityatα/8=0.035.
hereistoextenditforourunusualresonance.Asdetaileddescrip-
tionsoftheapproachforthestandardMMRscanbefoundinthe
literatureandmanyofthemaresharedwithourresonance,itwould
beunfruitfultolengthilyexpoundalltheinvolvedsteps.Werather
(cid:13)c 0000RAS,MNRAS000,000–000
6 N. Seto
followthecomprehensiveformulationgivenin 8.8ofMDandex- Thequadraticterm Θ2playsacriticalroleforourresonance,as
§ ∝ 1
plainthemodificationsnecessaryforourspecificresonance. weseeinthenextsubsection.Thistermoriginatesfromthedepen-
dence̟˙ (1 e2)−1.
1r
∝ −
One might has an impression that the present derivation for
Eq.(32)isphenomenological, asitisconstructedtoreproducethe
5.1 SimplifiedHamiltonianforSecondOrderResonance
desired precession rate ̟˙ . But wecan actually derive theterm
1r
Basedontheresultsintheprevioussection,weanalyzethesecond (proportionaltoe2+3e4/4+ )identicaltoXinEq.(32),start-
···
orderresonancewiththeresonantvariable ingdirectlyfromthe1PNHamiltonianH inEq.(1)(seeEq.(28)
1
inNaozetal.2012).Wetooktheaboveroutetoelucidatethemod-
θ =jλ +(2 j)λ 2̟ , (24)
1 2 − 1− 1 ificationrelativetothetypicalanalysisforthestandardMMRs.
identical to Eq.(8.78) of MD with k = 2. While we are mainly WiththeexplicitformoftherelativisticcorrectionXinhand,
interestedinthespecificcasej = 2,wedonotfixtheparameter wecannextapplythestandardargumentsinMDtoderiveasim-
j at this stage, in order to enable a simple comparison with the plifiedHamiltonianforMMRs.Aftersomecalculations(e.g.intro-
standardsecond-orderMMRscorrespondingtoj >2. ducing the new conjugate variables θ1′ = θ1/2 and Γ ≡ 2Θ1),
As explained in MD, the variable θ has the conjugate mo- we have the following Hamiltonian (corresponding to Eq.(8.102)
1
mentumΘ definedby ofMD)
1
m † =α¯Γ+β¯Γ2+2ǫ¯Γcos2θ′. (34)
Θ1 = 21√m0a1 1− 1−e2 (25) HHerethecoefficientsα¯,β¯and¯ǫ1aregivenas
m (cid:16) pe4 (cid:17)
≃ 41√m0a1 e2+ 4 +··· . (26) α¯= (j−2)n1−jn2+6pn1, (35)
(cid:18) (cid:19) 2
Weshouldnoticethatthismomentumisdirectlyrelatedtotheinner
3 (j 2)2 j2 p
epcecrteunrtbraictiivtyeeexapsaΘns1io∝nse,2a.sIsnumouirnganealytic1a.lstudiesbelow,wemake β¯= 8(cid:18) m−1a21 + m2a22(cid:19)+3m1a21, (36)
≪
notedAamsong),mthueltkiepyleetleermmesnitnfothreouHraumniultsounailarnes(o8n.9a8n)ceofisMthDe(fdoel-- ¯ǫ=Cj,j−2n1mm20α. (37)
H
lowingone InEqs.(35)and(36),thetermsproportionaltothePNparameterp
kΘ ̟˙ X (27) clearlyshowtherelativisticcorrections.ThefactorCj,j−2 =C2,0
− 1 sec ≡ forj = 2wasalreadygiveninEq.(18).Intheright-handsideof
with k = 2 for the present analysis. Here, the notation ̟˙ in Eq.(36),thefirstparenthesisappearsinthestandardMMRsandhas
sec
MD represents the secular precession rate of the inner pericenter itsoriginintheKeplariantermsinthetriplesystem(seeMD).Its
andisidenticaltotherelativisticapsidalprecession̟˙1r underour secondterm(∝p)isduetothequadraticterm∝Θ21inthesecular
prescriptionin 3(hereafterusing̟˙ insteadof̟˙ ). correctionXfortherelativisticapsidalprecession.
1r sec
§
Withrespecttothecanonicalequation Wefurthermaketransformationofvariablesasfollows
ddθt1 = ∂∂ΘH, (28) Φ= Γ2βǫ¯¯ = Θǫ¯1β¯, τ =2ǫ¯t (38)
1
θ′ (ǫ¯<0)
thetermXinthetotalHamiltonian hasaroletoprovidethesec- φ= 1 , (39)
H θ′ +π (ǫ¯>0)
ularcontribution−k̟˙1rforthetimederivativedθ1/dt.Therefore, n 1
weshouldhavetheequationbelow andfinallyobtaintherescaledHamiltonian
∂X H =Φ2+δ¯Φ+Φcos2φ (40)
= k̟˙ . (29)
∂Θ1 − 1r withthesingleparameterδ¯definedby
Meanwhile,asgiveninEq.(9),therelativisticprecessionrate̟˙
1r α¯
hasthefollowingformat1PNorder δ¯= . (41)
2ǫ¯
̟˙ = 3pn1 3pn (1+e2), (30) Theassociatedcanonicalequationsarewrittenas
1r 1 e2 ≃ 1
− dΦ ∂H dφ ∂H
andtherate̟˙1ritselfdependsontheconjugatemomentumΘ1 dτ =−∂φ, dτ = ∂Φ. (42)
∝
e2.ThuswehavethefollowingperturbativesolutionXforEq.(29)
The rescaled Hamiltonian (40) is slightly different from the
e2 relatedexpression(8.116)inMD,butidenticaltothoseinQuillen
X = 3kpn Θ 1+ (31)
− 1 1 2 (2006)andMustill&Wyatt(2011).Weadoptthepresentform,in
(cid:18) (cid:19) ordertousethesetworeferenceslateranddiscusswhetherevolu-
= 3kpn Θ 1+ 2Θ1 (32) tionoftheparameterδ¯canberegardedasadiabaticforourresonant
− 1 1 m1√m0a1 dynamics.
(cid:18) (cid:19)
expandedintermsofthemomentumΘ,insteadoftheeccentricity Roughlyspeaking,thisparameterδ¯representsaneffectivedis-
e.Notethatthissolutionisdifferentfromthenaiveexpression(27) tancetotheresonance.DuetotheGWemission,theorbitsofthe
thatisperturbativelyexpandedas EMRI-MBH triple system decay gradually, and the parameter δ¯
variesaccordingly.
kΘ ̟˙ = 3kpn Θ 1+ 4Θ1 . (33) Wenowestimatethetransitionratedδ¯/dτ.First,apartshortly
− 1 sec − 1 1 m1√m0a1 fromthetriplesystems,weconsiderasimplebinarywithasemi-
(cid:18) (cid:19)
(cid:13)c 0000RAS,MNRAS000,000–000
Analyseson a RelativisticHierarchicalResonancewith theHamiltonianApproach 7
majoraxisa,aneccentricityeandmassesm,m′.Itsorbitaldecay Φ= 2pm0a32e2 (48)
rateda/dtbyGWemissionisgivenas(Peters1964) 5m2a31
da = 64mm′(m+m′) 1+ 73e2+ 37e4 . (43) without depending on m1. As mentioned earlier, the post-
dt − 5 a3(1 e2)7/2 24 96 Newtonian terminEq.(47) comes fromthequadratic termof the
− (cid:16) (cid:17) momentumΦinEq.(32)andintrinsicallyfromthedependenceof
Next,forourtriples,weassumethat,beforetheresonanten-
theprecessionrateontheeccentricityasshowninEq.(9).
counters, the EMRI and MBH binary independently evolve with
Now we derive formulae specifically for j = 2 with the
Eq.(43).Thenweobtain
rescaledHamiltonian
ddα¯t = 458 m1m05/2a−113/2{(j−(12)a1e2+)71/20m0} H =Φ2+δ¯Φ+Φcos2φ. (49)
h1+ 73e2+ 37e4 jm− m a−11/2 (44) Thevariableφisrelatedtotheoriginalresonantangleη≡λ2−̟2
× 24 96 − 0 2 2 as
(cid:16) (cid:17) i
fore2 =0.Withthescaledtimeτ,wecanobtainthetransitionrate φ=η+π. (50)
as
After some algebra with Eq.(12), the principal quantities for the
dδ¯ 1 dα¯ dt 1 dα¯
= = . (45) rescaledHamiltonianaregivenbytheoriginalparametersas
dτ 2ǫ¯dt dτ 4ǫ¯2 dt
Φ=De2 (51)
ForagivenEMRI-MBHtriplearoundtheresonantencounter,
we can now analyze its evolution through the one-dimensional D 2a1 (52)
rescaled Hamiltonian (40). The information of the original triple ≡ 45m0m2
systemisconvertedto(i)thenewvariables(φ,Φ),(ii)theparam-
eterδ¯and(iii)itstimederivativedδ¯/dτ.Inpractice,thismapping dδ¯ = 512m30/2 . (53)
can be made with Eqs.(35)-(39) and (44)-(45). In the next sub- dτ −31/3375m2a113/6
section, we concretely study the relation in the test particle limit
Notethatthetransitspeeddδ¯/dτ isnearlyconstantaroundtheres-
m 0.But,here,wederivearesultvalidalsoform =0.
1 → 16 onantencounter,andweomittheexpressionforthetimedependent
Torealizeacapture(i.e.transitionofφfromrotationtolibra- parameterδ¯itself.
tion)withEq.(42),theresonanceshouldbecrossedinthedirection
InTable.1,weprovidethetransitspeeddδ¯/dτ aswellasthe
dδ¯/dτ < 0 (Peal 1986; MD). In the cases of standard MMRs,
coefficientD.Theformerisanusefulmeasuretodiscusstheadi-
thiscorresponds torelativelyapproaching orbits. Forexample, to
abaticity of the timeevolution of the parameter δ¯at the resonant
becapturedintothe3:2resonance,theratiooftheorbitalperiods
encounter.
shouldchangeinthedirectionof1.6 1.5not1.4 1.5.With
Eqs.(12)and(44)forj =2,theinequ→alitydδ¯/dτ <0→isrewritten Forcomparison, including only thefirst order resonant term
einEq.(19),wederivetherelevantexpressionsforthetestpar-
as ∝
ticle limit. In Appendix A, we summarize the results. Again, we
a 5 73 37 m m 1/2 have a regular mapping between the momentum Φ and the inner
1 > (1 e2)−5/2 1+ e2+ e4 0 1 .(46)
a2 r3 − 24 96 m2 eccentricitye,duetothePNcorrection.
(cid:16) (cid:17)(cid:16) (cid:17) ForaNewtonianapsidalprecessioninducedbymultiplemo-
Forthespecificcasee=0,thisexpressionagreeswiththatderived
mentsofmasses,theprecessionrateate 1generallyhascorrec-
andexaminedinSeto(2012).Notethatourlabels(0,1,2)forthe tionfortheeccentricityestartingfrome2≪(Sterne1939).Therefore,
threemassesaredifferentfromthoseinSeto(2012).
themappingtoacorrespondingrescaledHamiltonianbecomesreg-
ular, asfor therelativisticone discussed above. Forexample, the
precessionrateofatestparticleduetothequadrupolemomentJ
5.2 TestParticleLimit 2
ofthecentralbodyisexpandedas̟˙ J (1+2e2+ ).
2
∝ ···
HerewediscussthemappingbetweentheEMRI-MBHtriplesys- Inthenextthreesections,usingthemappingfromtheEMRI-
temandthesimplifiedHamiltoniansystem(40)inthetestparticle MBHtriplesystem,wemakequantitativepredictionsonthereso-
limitm1 0.Inthislimit,wecaneasilycontroltherelativeor- nantdynamicsandcomparethemwithnumericalsimulations.Be-
→
bitalevolutionsofthetriplesysteminnumericalsimulations,and, low,welimitouranalysistothetestparticlelimitm 0.
1
→
furthermore, the role of the post-Newtonian corrections becomes
transparent.
FromEqs.(26)(36)(37)and(38),theinnereccentricityeisre-
6 CAPTURERATE
latedtothemomentumΦas
m a2 (j 2)2 m j2 8p Inthissection,westudywhethertheanalyticalmodelbasedonthe
Φ= 0 1 − + 1 + e2. (47) rescaledHamiltoniancanreproducethecapturerateestimatedfrom
20m α3 a2 m a2 a2
2 (cid:20) 1 2 2 1(cid:21) numericalsimulations.
Inthisrelation,wepayourattentiontothedependenceofthemass For this comparison, we obtained the capture rate from the
parameterm .Wecanputp = 0inthetraditionalanalysisofthe numericalsideinthefollowingmanner.First,formodelsIandII,
1
standardMMRswithj >2(seeEq.(8.109)ofMD).However,for wetookvarious( 13)innerinitialeccentricitiese between 0
in
∼ ∼
ourunusual one withj = 2, themapping (47) becomes singular and 0.7.Foreacheccentricity,weassignedaninitialouterradius
∼
Φ m e2inthelimitm 0,iftherelativisticeffectisdropped a largerthanEq.(12)toassurearesonantencounter,andmade
1 1 2in
∝ →
withp=0.Therefore,interestingly,theregularityofthemapping 20runsstartingfromrandomlydistributedrelativeorbitalphases.
(47)ismaintainedbythepost-Newtoniancorrection( p)forour Therefore,thetotalnumberoftherunsis 2 13 20 500.We
∝ ∼ × × ∼
resonancewithj =2as judgearunasaresonantlycapturedeventwhentheangleη=λ
2
−
(cid:13)c 0000RAS,MNRAS000,000–000
8 N. Seto
1.0 I 0.4
II
P 0.8 0.3
e
at
r 0.6
ure á efin 0.2
pt 0.4
a I
c 0.2 0.1
á i=0
0.0
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
ein ein
1.0 II
Figure9. GapsoftheeccentricitieseofinnerEMRIswithfailedcaptures
P 0.8 (asshowninFig.4).Withthesolidcurves,weplottheanalyticcorrespon-
rate 0.6 rdeesnocneasnbteetnwceoeunnttehrefgoarpmpoeddeelcscIeanntrdicIiIt.ieTshbeecfiorrcele(seainre)tahnedmaeftaenre(eccfeinnt)ritchie-
re tiesofthenumericalrunsaccompaniedbythegaps.
u
pt 0.4
a áá
c
0.2
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 corresponding to e = Φ /D = 0.20 (model I) and 0.12
in cr
(model II)that aremuch largerthan α/8 = O(10−2)(seeTable
e
in p
1). The capture occurs at 100% for Φ < Φ . Meanwhile, for
0 in cr
Figure 8. Resonant capture rate of an inner EMRI by a coplanar outer Φin >Φcr,thecapturebecomesastochasticprocesswiththerate
MBH.Theresultsaregivenasfunctionsoftheinnereccentricity ein for P2definedimplicitlybyEqs.(B2)and(B3).
modelsIandII.ThesolidcurvesaretheanalyticalpredictionsP2forthe InFig.8,withthesolidcurves,wepresenttheanalyticalrates
second order term ∝ cos2η andthe dashed curves are P1 forthe first- P2.Atein . 0.4,theyshowreasonable agreementswiththenu-
orderterm∝ cosη. Thecirclesarenumericalresultsforminimumdura- mericalones.But,atlargereccentricities e & 0.5,wehavesig-
in
tiona2/10,andtheadditionalopensquaresarefora2/20(shownonlyif nificant discrepancies. This is not surprising, since we made, at
different).Eachofthemisestimatedfrom20runs.
least,various approximations, validonly for e2 1. Forrefer-
in ≪
ence,wealsoshowtherateP expectedforthefistorderresonance
1
̟ hasasingleexcludedrange∆ηlargerthanπ/10forminimum (see B.2),butitpoorlyfitsthenumericaldata,asexpected. Note
1 §
duration a /10 (and also a /20 for comparison) in terms of the alsothat,forhighereccentricities,thenumericalresultsareaffected
2 2
decayingouterradiusa (seee.g.Figs.2and3). AnMMRisoften bytheappliedconditionsforidentificationoftheresonances.
2
identified0withasharpconcentrationofaresonanceangle,asfound As we explained earlier, the critical eccentricity Φcr/D
inFig.2.But,here,weemployedthecriteriafor∆ηtohandlethe characterisestheoverallshapeofthecapturerate.Herewe should
p
resonant statewithlargelibrationamplitudes, asdemonstrated in comment on its magnitude for the standard second-order MMRs
Fig.3. (withthe variable (24) at j > 2). Forthese resonances, dynami-
ofcaTpthuerne,dfeovreenatcshaimniotinaglethceceanstsroicciitayteedin2,0wreuncosuanntdedrtohuegnhulymobber- Tcahlesntawbeilictyanofshoorwbitassrecqaluiinrgesbmeh2av≪iourmD0 (=assΦum/ein2gm11/<m2ma2n)d.
∝
tainedthecapturerate.Weprovide thenumerical resultsin Fig.8 obtainthecriticaleccentricity Φ /D 1muchsmallerthan
cr
≪
withthebluecircles forminimumdurationa /10.Uptomoderate ourhierarchicalonewithm /m = O(1)(asinmodelI).There-
2 0 2
p
eccentricity,weobtainthesameresultsfortheminimumduration fore,forthestandardsecond-orderMMRs,perturbativeexpansion
a /20.Onlywhentheyaredifferent,weaddedthelatterwiththe oftheeccentricityismoreeffectiveintheregimewherethe reso-
2
opensquares..AsshowninFigs.2-4,theeccentricitiesoftheEM- nant capture is probable (e.g. P > 0.1), unlike our hierarchical
2
RIsarealwaysoscillatingtosomeextent.Tohandlethis,wetook onewithlarger Φ /D.Wecanmakesimilarargumentsforthe
cr
timeaveragedeccentricityforeachrunatitsearlystage,andsub- first-orderresonances.
sequently2evaluatedthemeanvalueamongthe20runs.Theinitial FortheanaplyticalpredictionsinAppendixB,wefullyusethe
eccentricitiesinFig.8aremadeupinthisway. arguments based on the adiabatic invariant that is conserved for
Next we analytically estimate the capture rate through the atransitspeed dδ¯/dτ muchsmallerthanthelibrationfrequency
rescaled Hamiltonian (49), following Borderies & Goldreich (Landau&Lifs|hitz196|9).Toexaminetheimpactsoffinitenessof
(1984). In Appendix B, we briefly describe their results. As we dδ¯/dτ on theresonant dynamics, Quillen(2006) andMustill &
havealreadydiscussed,thesecondorderresonanceisrelevantfor |Wyatt(2|011)numericallystudieddependenceofthecapturerateon
oursystems(unlesseinislessthanO(10−2),seeTable1).There- thetransitionspeed dδ¯/dτ .Theirresults(seee.g.Fig.2inMustill
fore,wemainlyusetheirresultsfork=2(see B1). &Wyatt2011)indic|atetha|ttheadiabaticapproximationwouldbe
§
Foraninitialeccentricityein,theanalyticalrateP2isobtained efficient for dδ¯/dτ < 0.1. As shown in Table.1, two models I
throughtheprojectedinitialmomentum andIIwells|atisfyth|iscriteria.Foracoplanar EMRI-MBHtriple
Φ =De2 (54) of comparable MBHs (m2/m0 = O(1)) with converging orbits
in in dδ¯/dτ < 0,wegenerallyhave dδ¯/dτ 0.1, unlessthetarget
for the re2scaled Hamiltonian. There is a critical value Φ = 1, EMRIishighlyrelativistic. | | ≪
cr
(cid:13)c 0000RAS,MNRAS000,000–000
Analyseson a RelativisticHierarchicalResonancewith theHamiltonianApproach 9
7 GAPOFINNERECCENTRICITY 6 0.4
Dn5
It is well known that, for the standard MMRs, the eccentricity dia4 0.3
a
of a perturbed body shows a gap when the resonance is encoun- @r3 e0.2
teredbutcaptureresultsinfailure(e.g.Peale1986; MD;seealso -v12 0.1
Amaro-Seoaneetal.2012).Thisphenomena iswellexplainedby Λ21
0 0.0
theHamiltonianapproachintheassociatedphasespace,asarapid 260 280 300 320 340 360 300 310 320 330 340 350 360
changeofrotationalmotionattheseparatrixcrossing.InFig.4,we a2 a2
8
canobserveasimilargapoftheeccentricityfortherunwithouta 50.2 7
resonantcapture. 50.0 Dee6
InordertofurtherexaminevalidityofourHamiltonianmodel a149.8 degr5
extendedfortheunusualresonance,weanalyzethecorrespondence @
49.6 i4
ofthetwoeccentricities;e (beforetheencounter) and e (af-
in fin
49.4 3
ter the encounter). We derive an analytical prediction using the 300 310 320 330 340 350 360 300 310 320 330 340 350 360
rescaled second-order Hamiltonian (49) and compare it with the a2 a2
numericalsimulationsdonein 6. Figure10.EvolutionofanEMRI-MBHtriplesystemfromaninnereccen-
§
We first discuss the analytical approach in which the corre- tricity e = 0.10(modelI).Thissystemhasinitial parameters similarto
spondencebetweenthetwoeccentricitiesisequivalenttotherela- Fig.2,butwithaninitialrelativeinclination i ∼ 5.8◦.TheEMRIiscap-
tionbetweentheinitialmomentumΦin = De2in andthefinalone turedintoresonancearound a2 ≃ 340.Theinclination angleiiswithin
Φ = De2 bothfarfromtheresonantencounter.Foragiven 5.7◦<i<6.0◦.
fin fin
initialmomentum Φ , theparameter δ¯at theseparatrixcrossing
in
isgiveninasomewhatcomplicatedformasinEqs.(B3)and(B4).
But,becauseofasimpleexpressionforadefineintegral,wehave 6 0.60
Dn5 0.55
thefollowingconciserelationbetweenthetwomomenta(Malhotra dia4 0.50
1Φ988+) Φ = δ¯. (55) @ra-v123 e000...344505
in fin − Λ21 0.30
Notethattheseparatrixrelevantforouranalysisisformed atδ¯< 0 260 280 300 320 0.25 260 280 300 320
1wherethecaptureratebecomeslessthanunity.InEq.(55),we a2 a2
−haveΦ =1andΦ =0forδ¯= 1. 0.65
in fin − 50.0 0.60
Wecannowobtainthedesiredcorrespondence Φin Φfin 0.55
(eq1u).ivIanleFnitgly.9,eiwne→shoewfinth)istharnoaulgyhticthaelcinotrerervsepnoinndgenpcaerafmore→tmeroδ¯d(e<ls a149.5 cosi00..4550
− 49.0 0.40
IandIIwiththesolidcurves. 0.35
Wealsoanalyzethesamplesofthenumericalsimulationsde- 260 280 300 320 0.30 260 280 300 320
scribedintheprevioussection.InFig.9,thenumericalresultsare a2 a2
presented with the filled circles. In numerical data, formation of
Figure11.EvolutionofanEMRI-MBHtriplesystemfromaninnereccen-
agapcanbe easilyidentifiedasainstantaneous event, compared tricitye ∼ 0.39andrelativeinclination i ∼ 60◦ (modelI).Thissystem
with continuation of a resonant state. We can observe small sys- encounterstheresonancearounda2 ≃305wheretheinnereccentricity e
tematic deviations between the analytical and numerical results. showslargervariation∆e∼0.2comparedwith∆cosi∼0.03.
But,asawhole,thesimpleanalyticalpredictionsshowreasonable
agreementswiththenumericalonesthatwereobtainedafterrather
complicateddynamicalevolutions. iispresentedinageometricformcosi.Wecanobserveoscillation
ofcosi.But,aroundtheresonantencounter a 305,itsampli-
2
∼
tudeismuchsmallerthanthatoftheeccentricitye.
In the analytical Hamiltonian approach, we need to clarify
8 INCLINEDORBITS
how the resonant interaction depends on the inclination angle.
Sofarwehavediscussedtheresonantdynamicsforcoplanarorbits. Here,based on theabove numerical demonstrations, wemake an
Inthissection,weextendourstudytoinclinedorbits.Weusethe approximation that the inclination angle i is constant around the
parameteriastherelativeinclinationanglebetweentheinnerand resonant encounter. In the previous case for coplanar orbits, the
outerorbits. disturbingfunctionhasthefollowingsecond-orderresonanceterm
InFig.10weprovideanumericalexampleforinclinedorbital
15α2e2
configurations.Thistriplesystemhasasmallinitialinclinationan- ζ = cos2η. (56)
glei=5.8◦,butitsinitialsemimajoraxesa ,a andeccentricity 8
1 2
eareclosetothoseinFig.2.Wefindthattheoverallevolution of Hereweneglectedsubleadingcontributionsofo(α2e2).Incelestial
thethreequantitiesλ ̟ ,eanda aresimilartoFig.2. mechanics,theeffectsoftheinclinationonthedisturbingfunction
2 1 1
−
Note that the inclination angle i stays nearly at a constant areoftenhandledperturbativelywiththeexpansionparameters
≡
value.ThisisinaccordwiththesimplifiedHamiltonianapproach sin[i/2].But,here,weareinterestedinhighlyinclinedorbitswith
thathasonlytwodynamicallyimportantvariableseandλ ̟ s=O(1),wellbeyondtheperturbativeregimes 1.
2 1
− ≪
inthepresenteccentricityresonance. We should notice that the factor ǫ¯for the resonant term in
InFig.11,weshowtheresultsfromalargerinclinationangle theHamiltonian(34)iscontributedbyallthetermsproportionalto
i=60◦.Inthelowerrightpanel,evolutionoftheinclinationangle cos2η amongthedisturbingfunction.Fortunately,wecanreadily
(cid:13)c 0000RAS,MNRAS000,000–000
10 N. Seto
collectthetermsatthelowestorderα2e2cos2ηasfollows; 1.0 I
15α2e2
ζ = (1 2s2+s4)cos2η (57) P 0.8
8 − e
15α2e2 rat 0.6
= cos4[i/2]cos2η. (58) e
8 ur
pt 0.4
Therefore,withrespecttotheoriginalHamiltonian † given a
inEq.(34)andattheorderO(α2e2)oftheresonantinteraHction,we c 0.2 á á
just need tomultiply thefactor cos4(i/2) totheparameter ǫ¯that
0.0
wasdefinedinEq.(37)forthecoplanarsystem.Underthepresent
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
approximation i = const, this is basically what we need to do
fordealingwiththeinclined orbits.Accordingly, fortherescaled ein
Hamiltonian(49)inthetestparticlelimit(see 5.2),thecoefficients 1.0 II
DinEq.(52)andthetransitionspeeddδ¯/dτ a§regivenas
P 0.8
2a
D= 45m0m2c1os4[i/2], (59) rate 0.6
e
r
dδ¯ = 512m93/2 . (60) ptu 0.4 á
dτ 31/3375m2a113/6cos8[i/2] ca 0.2 á áá
Withtheseexpressions,itisstraightforwardtoapplytheprevious ááá
analyticalmethodsin 6and7toinclinedorbits. 0.0
§ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Now,wecomparetheseanalyticalpredictionswithnumerical
simulationsfortriplesystems.Below,wefixtheinclinationangle e
in
atarelativelylargevalue i = 60◦ andperformedalargenumber 0
of simulations for two models I and II. Even with the strong de- Figure12. Resonantcapture rateofaninnerEMRIwiththeinclination
pendenceontheinclination 1/cos8[i/2] = (4/3)4 = 3.2,the anglei = 60◦.ThesolidcurvesaregivenbytheHamiltonian approach
transitionspeedsdδ¯/dτ arel∝essthan0.1bothwithmodelsIandII withthesecondordermode.Thecirclesareresultsfromnumericalresults
(seeTable1),andtheadiabaticapproximationwouldbestilleffec- (eachfrom20runs)withtheminimumdurationa2/10oftheresonantstate,
tiveforanalyzingresonantdynamics. andtheadditionalopensquaresarefora2/20(shownonlyifdifferent).
Note also that due to the relativistic apsidal precession, the
Kozai process (Kozai 1962; Lidov 1962) does not work here
0.4
(Holman,Touma,&Tremaine1997;Blaes,Lee,&Socrates2002; II
Seto2012). Even for m m , the characteristicfrequency of
2 1
KozaiprocessisO(α3n ),∼whilethe1PNprecessionfrequencyof 0.3
1
theinner EMRIis 3pn . Whenasystem encounters ourreso-
1 I
nance,wehaveα3/∼2 = 3p ≪ 1(seeEq.(11))andtheKozaipro- efin 0.2 0
cessissuppressedbythe1PNprecessioneffect(seealsoNaozet
al.2012).Notethatthesemimajoraxesofthesystemsshownhere 0.1
arenotconstantofmotion,andtheorbitalaveragingassociatewith i=60°
the Kozai mechanism cannot be applied here. These systems lay
0.0
belowthestabilitycriterionpresentedinLithwick&Naoz2011. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
As for the coplanar orbits, we present the capture rates
e
(Fig.12)andthegapsoftheeccentricities(Fig.13).Theanalytical in
predictionsbasedontheHamiltonianapproachshowgoodagree-
Figure13.Correspondences oftheeccentricities aroundtheresonant en-
mentswiththenumericalresultsate.0.4.
counters.TheresultsaregivenformodelsIandII,asinFig.8,butnowfor
Thefavourableresultsinthissectioncouldberegardedasad- inclinedorbitswithi=60◦.
ditionalsupportsforvalidityofourHamiltonianapproachextended
fortherelativisticresonance.
2
Asapreliminaryanalysis, in 4, wediscussed thedominant
§
orderoftheresonant interactionforourstate λ ̟ const.
2 1
− ∼
Duetotheorbitalhierarchy,dependenciesontheparameter α(
9 SUMMARY ≪
1)playacriticalroletoassesstherelevantterms,andthesecond-
Wehavestudieddynamicsoftheresonantstateλ ̟ const orderone α2e2(e:theeccentricityoftheEMRI)couldbecome
2 1
foratriplesystemcomposedbyanEMRI(CO+MB−H)an∼danad- more impo∝rtant than the first-orderone eα3. This result isre-
∝
ditional outer MBH. Thisresonant stateissupported by the rela- markablydifferentfromthestandardMMRsforwhichtheparam-
tivisticapsidalprecessionoftheinnerEMRI,anddoesnodepend eterα=O(1)islessimportant.
on its mean anomaly λ . As a result, the two orbits can become In 5, we derive the mapping from the resonant triple sys-
1
§
hierarchical with α a /a 1, and then the two masses of tems to the rescaled one-dimensional Hamiltonian for the state
1 2
≡ ≪
theMBHsm andm canbecomecomparable,incontrasttothe λ ̟ const. We basically followed the framework of the
0 2 2 1
− ∼
standardMMRswherewehavem ,m m duetodynamical Hamiltonianapproachexplainedintheliterature,butpayedspecial
1 2 0
stabilityoforbitswithα=O(1). ≪ attention2tothetermassociatedwiththerelativisticapsidalpreces-
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