Modified evolution of stellar binaries from supermassive black hole binaries

5.2 MB·

by  Bin Liu

Mon.Not.R.Astron.Soc.000,1–12(2014) Printed18January2017 (MNLATEXstylefilev2.2)
Modified evolution of stellar binaries from supermassive
black hole binaries
Bin Liu1,2,3, Yi-Han Wang2,3 and Ye-Fei Yuan2,3
1 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China
2 Department of Astronomy, University of Science and Technology of China, Hefei, Anhui 230026, China
3 CAS Key Laboratory for Research in Galaxies and Cosmology, Hefei, Anhui 230026, China
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18January2017
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n ABSTRACT
a The evolution of main sequence binaries resided in the galactic centre is influenced a
J
lot by the central super massive black hole (SMBH). Due to this perturbation, the

7 stars in a dense environment are likely to experience mergers or collisions through
1
secular or non-secular interactions. In this work, we study the dynamics of the stellar

binariesatgalacticcenter,perturbedbyanotherdistantSMBH.Geometrically,sucha
]
A four-bodysystemissupposedtobedecomposedintotheinnertriple(SMBH-star-star)
and the outer triple (SMBH-stellar binary-SMBH). We survey the parameter space
G
anddeterminethecriteriaanalyticallyforthestellarmergersandthetidaldisruption
.
h events (TDEs). For a relative distant and equal masses SMBH binary, the stars have
p more opportunities to merge as a result from the Lidov-Kozai(LK) oscillations in
- the inner triple. With a sample of tight stellar binaries, our numerical experiments
o
revealthatasignificantfractionofthebinaries,∼70%,experiencemergereventually.
r
t WhereasthemajorityofthestellarTDEsarelikelytooccuratacloseperiapsestothe
s
SMBH,inducedbytheouterKozaieffect.Thetidaldisruptionsarefoundnumerically
a
[ as many as ∼ 10 per cent for a close SMBH binary that is enhanced significantly
than the one without the external SMBH. These effects require the outer perturber
1 to have an inclined orbit ((cid:62) 40◦) relatively to the inner orbital plane and may lead
v
to a burst of the extremely astronomical events associated with the detection of the
0
SMBH binary.
8
5
Key words: black hole physics - (stars: ) binaries:close
4
0
.
1
0
7 1 INTRODUCTION the time-scale shorter than the secular times-cale has been
1 studied extensively: (i) when the stellar binaries undergo
: The supermassive black hole (SMBH) resided in the galac- low angular momentum orbits, the stars are considered to
v
tic centre is ubiquitous (e.g., Kormendy & Ho 2013). At be unbound to each other by the strong tidal field of the
i
X present, in the inner zone of our Milky Way, within ∼ 1 SMBH. As a result, one component is captured by the
r pc, the stars are observed in a dense stellar environments, SMBH, while the other is ejected with high velocity on the
a where the gravitational potential is dominated by the cen- orderof500−1000kms−1,whichmayberesponsibleforthe
tralSMBH(e.g.,Alexander2005).Sincethehighdensityof hypervelocitystars(HVSs)(e.g.,Hills1988;Yu&Tremaine
starsinthegalacticcentre,themajorityofstarsinthefield, 2003; Antonini et al. 2010); (ii) due to the high density of
are believed to reside in binaries or higher multiplicity sys- the stellar distribution at the galactic center, stars in a bi-
tems(e.g.,Raghavanetal.2010;Evans2011).Furthermore, naryapproachingtheSMBHcanbetidallydisruptedifthe
these binaries are gravitational bound to the SMBH and relativedistanceisontheorderofthetidaldisruptedradius
effectively form hierarchical triple systems with the SMBH. ofasinglestar.Theaccretionofthestellardebrisresultsin
a strong flare in the electromagnetic radiation, leading to a
Therefore, it is interesting to take account a close en-
tidal disruption event (TDE) (e.g., Hills 1975; Rees 1988;
counter between a stellar binary and an SMBH. When the
Phinney 1989; Gezari et al. 2012; Mandel & Levin 2015);
centreofmassofthebinaryapproachestheSMBHwiththe
(iii) similarly, the interaction between the SMBH and the
pericentre distance below the tidal break-up of the binary,
stellar binary may also lead to a physical collision between
both the angular momentum and the orbital energy of the
twocomponentsofthebinary(e.g.,Ginsburg&Loeb2007).
binary are exchanged with those of the SMBH. Recently,
theevolutionofthebinaryorbitingaroundtheSMBHwith Iftheorbitalplaneofthestellarbinaryisinclinedwith
(cid:13)c 2014RAS
2 Liu, Wang & Yuan
the external plane of the SMBH over 40◦, the stellar eccen-
tricity and inclination will experience periodic oscillations
on the secular time-scale, known as Lidov-Kozai (LK) os-
cillations (e.g., Lidov 1962; Kozai 1962). In addition to the
leading order (quadrupole) approximation, it has been rec-
ognizedthathigh-order(octupole)interactionsbetweenthe
inner and outer orbits can lead to more abundant dynami-
calbehavioursinhierarchicaltriples(e.g.,Harrington1968;
Marchal et al. 1990; Ford et al. 2000b; Naoz et al. 2011;
Liu, Mun˜oz, & Lai 2015). The effects of LK oscillations in
theformationandevolutionofvariousastrophysicalsystems
havebeenextensivelystudiedinrecentyears(e.g.,Eggleton
&Kiseleva–Eggleton2001;Fabrycky&Tremaine2007;Hol-
man, Touma, & Tremaine 1997; Innanen et al. 1997; Wu &
Murray 2003; Storch, Anderson, & Lai 2014; Liu, Lai, &
Yuan 2015; Anderson, Storch, & Lai 2016). In particular,
in the environment of SMBH, the formation and merger of
blackholebinariesatthecentresglobularclustersorgalax-
ies is increased significantly (e.g., Blaes et al. 2002; Miller
& Hamilton 2002; Wen 2003; Antonini, Murray & Mikkola
2014), as well as the production of Type Ia supernova from
white-dwarf binary mergers (e.g., Thompson 2011; Prodan,
Murray, & Thompson 2013) or direct collisions (e.g., Katz
& Dong 2012; Kushnir et al. 2013). Also, an enhanced rate
of stellar TDEs is studied in the SMBH binary, due to the
eccentric LK mechanism (e.g., Li et al. 2015).
Themotionofthebinariesinthegalacticcentreisvar-
iedbytheinfluenceoftheSMBH.Thisprocessissupposed
tobeacceleratedbyanothermassiveperturberoutsidethis
Figure1.Theconfigurationofthefour-bodysystemwestudied.
triplesystem.Inthispaper,weconsiderthestellarbinaries
orbiting around an SMBH, perturbed by a distant SMBH. BinaryAcomprisestwostars(m0 andm1);BinaryBconsistsof
the Binary A and an SMBH (m2), while the outer SMBH (m3)
As expected, the evolution of the stellar binary is modified
orbits the centre of mass of Binary B, constituting Binary C.
by the existence of the forth body, re-populating the stars
Here,a1,2,3 isthesemimajoraxes,e1,2,3 istheeccentricitiesand
around the first SMBH. Different from the simple three- r01,2,3 istheseparationbetweentwocomponentsofeachbinary,
body system, the existence of the second SMBH allows us respectively.
toconstructtwotriplesystems.Iftheratioofdistancesbe-
tween the stars and the SMBH is sufficient small, and the
massofstarisnegligiblecomparedtotheSMBH,thestellar
binary can be treated as a test mass particle. So, in ad- major axis. Our results and conclusions are summarized in
dition to the stellar binary-SMBH system, the outer triple Section 5.
system (SMBH-stellar binary-SMBH) is formed. Obviously,
when the inner triple has the mutual inclination lied in the
range 40◦ ∼ 140◦, the LK oscillations occur in this three
body system. The eccentricity of the stellar binary can be
excited, leading to a merger between two components. In-
2 STELLAR BINARY SURROUND SMBH
versely, when the Kozai effect of the outer triple come into
BINARY
play, the stellar binary will have a eccentric orbit moving
aroundtheSMBH.Iftheperiastronissmallsufficiently,the We study the merger and the tidal break-up of the stellar
binary may be tidal break-up after the close passage, pro- binaries surrounding the SMBH binary. First, as shown in
ducing TDE or HVS. Therefore, the dynamics in the latter the upper panel of Fig. 1, we introduce the masses of the
caseisexpectedtoberichandinteresting,andwewillfocus stars by m and m (stellar binary components), m (the
0 1 2
on this regime in this work. first SMBH), and m (the second SMBH). Then, for the
3
Our paper is organized as follows. In Section 2, we in- orbitalparametersweusea asthesemimajoraxes,e
1,2,3 1,2,3
troducethegeometryofthefour-bodysystem.Throughthe as the eccentricities and r as the separations between
01,2,3
comparison of different time-scales, we characterize the pa- two components of each binary. Subscript “1,2,3” denote
rameter space where the outer LK oscillation is dominated. three binaries (Binary A, B and C, respectively), as shown
InSection3,wecarryouttheN-bodysimulationstoexplore inthelowerpanelofFig.1.Basedontheconfiguration,this
the evolution of the stellar binary approaching the SMBH. four-body system can be decomposed into two three-body
Furthermore, considering an illustrative example, we study systems:the“innertriple”isthestellarbinary-SMBHtriple
the dynamics of the four-body system and obtain the rates system,consistingofm −m −m ;whilethe“outertriple”
0 1 2
ofdifferenteventsinSection4.Wealsotracethevariations isconsistedofthestellarbinary-SMBHbinarysystem,i.e.,
of the rates when the SMBH binary has a different semi- m −(m +m )−m .
2 0 1 3
(cid:13)c 2014RAS,MNRAS000,1–12
Stellar binaries surrounding SMBH binary 3
2.1 Time Scales the galaxy to evolve into equilibrium, leading to dynamical
relaxation. Supposing the equal mass stars have distribu-
In order to identify the parameter space where Kozai effect
tionwithisotropicvelocity,thetime-scaleisdefinedas(e.g.,
isimportant,weconsiderthedifferentsourcesofapsidalpre-
Spitzer 1987)
cessioninthegalacticnuclei.Fortheinnertriplesystem,the
stellarbinariesareimmersedinadenseenvironment.Other
(cid:18) σ (cid:19)(cid:18)lnΛ(cid:19)−1
gravitationalprocessmaycomeintoplayandaffecttheevo- T =1.6×1010yr
lution of the stellar binary, even change the distribution. REL 306 kms−1 15
(5)
Binariesorbitedbyahighlyinclinedperturberwillun- (cid:18) ρ (cid:19)−1
× .
dergoLKoscillations.Intheinnertriple,thecorresponding 2.1×106M(cid:12)
time-scale at the approximation of quadrupole level is de-
scribed as (e.g., Kozai 1962; Lidov 1962)
Inthedenseenvironment,suchasthestellarcluster,the
1 (cid:18)m +m (cid:19)(cid:18)a (cid:19)3 angularmomentumoftheorbitofthestellarbinaryaround
TK,B =n 0m 1 a2 (1−e22)3/2 SMBHmaychangeboththemagnitudeandtheorientation
1 2 1
by the resonant relaxation. Here, we take into account the
=1.3×106yr(cid:18) m2 (cid:19)−1(cid:18)m0+m1(cid:19)1/2 (1) time-scale due to the scalar resonant relaxation (related to
1×106M(cid:12) 1M(cid:12) magnitude) as (e.g., Rauch & Tremaine 1996)
(cid:18) a (cid:19)3(cid:18) a (cid:19)−3/2
× 0.12pc 1a1u (1−e22)3/2, T =4.6×108yr(cid:18) m2 (cid:19)1/2(cid:18) a2 (cid:19)3/2, (6)
(cid:112) RRS 1×106M(cid:12) 0.1pc
where n = G(m +m )/a3 is the mean motion of the
1 0 1 1
stellar binary.
andthevectorresonantrelaxation(relevantfororientation)
ComparedwiththeKozaitime-scale,somerelatedtime-
as (e.g., Merritt et al. 2011)
scales are reviewed as follows. First, to date, we have only
little understanding about the distribution of the main se-
quence stars resided in the galactic centre from observa- T =3.8×106yr(cid:18) m2 (cid:19)1/2(cid:18) a2 (cid:19)3/2
tion (e.g., Duquennoy & Mayor 1991; Willems & Kolb RRV 1×106M(cid:12) 0.1pc
(7)
2004), whereas most of the theoretical studies are focused (cid:18) N (cid:19)−1/2
on the close binary (i.e., X-ray binary and gravitational × ,
6000
wave source) and the distribution predicted has a strong
dependence on the models. For simplicity, in the following
whereN isthenumberofthestarsinteriortothesemimajor
text, we take an assumption that the stellar density profile
axis a .
is described as a power law and normalized at 1 pc (e.g., 2
If the stellar binary separation at pericentre is suffi-
Alexander 2005; Scho¨del, Merritt, & Eckart 2009; Antonini
cientlysmall,theadditionalforces,suchasgeneralrelativity
& Perets 2012b) as
(GR) may overcome the tidal torque exerted by the outer
(cid:18) r (cid:19)−α(cid:20) (cid:18) r (cid:19)2(cid:21)(α−1.8)/2 binary, suppressing the excitation of the eccentricity of the
ρ(r)=ρ 1+ , (2)
0 r r stellar binary (e.g., Blaes et al. 2002; Naoz et al. 2013a).
0 0
The time-scale of the precession of the argument of periap-
whereρ0 =5.2×105M(cid:12)pc−3,r0 =0.5pc.Theconfiguration siscausedbythefirstorderpostNewtonian(PN)correction
of stars observed at the galactic nuclei is given when the in the inner and outer orbit is
inner slope index α = 0.5. Here, we assume α to be 1.8,
representativeoftherelaxeddistributionofstarsdominated 2πc2 a5/2(1−e2)
by a point-mass potential. TGR,B1 = 3G3/2(m1 +m )31/2
0 1
Near the galactic center, the kinetic energy of the field
starsmaybelargerthanthatofthestellarbinary,wherethe =3.4×107yr(cid:18) a1 (cid:19)5/2(cid:18)m0+m1(cid:19)−3/2 (8)
binarymightevaporateduetothesufficientencounterswith 1au 1M(cid:12)
thesurroundingstars.Theassociatedtime-scaleisgivenby ×(1−e2),
1
(e.g., Binney & Tremaine 1987)
(cid:18)m +m (cid:19)(cid:18) σ (cid:19) and
T =1.6×107yr 0 1
EV 1M(cid:12) 306 kms−1
×(cid:18) a1 (cid:19)−1(cid:18)lnΛ(cid:19)−1(cid:18) ρ (cid:19)−1, (3) TGR,B2 = 32Gπ3c/22(m a+52/m2(1+−me22))3/2
1au 15 2.1×106M(cid:12) 0 1 2
(cid:18) a (cid:19)5/2(cid:18)m +m +m (cid:19)−3/2
where lnΛ ∼ ln[m /(m +m )] is the Coulomb logarithm. =2.1×109yr 2 0 1 2
2 0 1 0.1pc 1×106M(cid:12)
Here,theone-dimensionalvelocitydispersion,σ,issatisfied
with the Jeans equation ×(1−e22),
(9)
ρ(r)σ(r)2 =G(cid:90) ∞ ρ(r(cid:48))(cid:2)m +M (r(cid:48))(cid:3)dr(cid:48), (4)
r(cid:48)2 2 (cid:63)
r respectively.Inparticular,theinteractionbetweentheinner
with M (r(cid:48)) the mass of the stars within the radius r. and outer binaries at 1 PN order can affect the long-term
(cid:63)
Theencountersbetweenthestellarstarsmayalsopush evolution of the triple. The related time-scale is given by
(cid:13)c 2014RAS,MNRAS000,1–12
4 Liu, Wang & Yuan
Kozai effect is completely quenched by the GR effect for
the much tighter orbit a = 0.05au with fixed e . But the
1 2
boundary of Kozai-dominated region of the inner triple is
shifted to ∼ 0.4pc for the larger a . Typically, compared
1
with other time-scales, there is an upper limit on a for
2
a fixed a . In fact, the observed stellar disc also has the
1
radial extended from ∼ 0.04pc to ∼ 4pc (Alexander 2005;
Prodan, Antonini, & Perets 2015). Thus, we identify the
parameter space with moderately small a in the following
2
test, irrespective of other dynamics.
2.2 Parameter space of outer Kozai Oscillations
As shown in Fig. 2, the increased eccentricity of Binary B
woulddecreasetheKozaitime-scaleatthequadrupolelevel,
and does not affect others. That is to say, the highly ec-
centric orbit has more opportunities to affect the evolution
of the stellar binary near the galactic nuclei, which can be
produced by the scattering between stars (e.g., Rauch &
Tremaine 1996; Perets, Hopman, & Alexander 2007) or by
theKozaioscillationsinducedbyamassiveanddistantper-
turber, such as another SMBH.
The LK mechanism in the outer triple operates if the
following criteria are satisfied:
(i)Thestellarbinary(BinaryA)staysintheHillsphere
oftheinnerSMBH(m )inorderforthemtoremainbound
2
to it
(cid:18) m (cid:19)1/3
R =a (1−e ) 2 >a (1+e ). (11)
Hill 3 3 3m 2 2
3
(ii) The configuration of Binary C should be satisfied
with the hierarchical condition
a e
ε = 2 3 <0.1, (12)
oct a 1−e2
3 3
which is always valid when e =0.
3
(iii)TheconditionsofparametersofBinaryCaresub-
ject to the stability criterion of Mardling & Aarseth (2001)
Figure 2.Differenttime-scalesassociatedwiththegalacticcen-
treandtheLKoscillations(equations3-10)asafunctionofa2,for a3 >2.8(cid:18)1+ m3 (cid:19)2/5(1+e3)2/5. (13)
anillustrativeexampleofm0=2M(cid:12),m1=1M(cid:12),m2=106M(cid:12) a2 m0+m1+m2 (1−e3)6/5
ande1,0=0.001.Intheupperpanel,a1isassumedtobe0.05au,
(iv)ThequadrupoleKozaitime-scaleoftheoutertriple
andinthelowerpanela1 equalsto1au.
1 (cid:18)m +m +m (cid:19)(cid:18)a (cid:19)3
T = 0 1 2 3 (1−e2)3/2, (14)
(e.g., Naoz et al. 2013a; Will 2014b,a) K,C n2 m3 a2 3
16 c2 a3 (1−e2)3/2 (m +m )3/2 shouldbesmallerthanthetime-scaleof1PNcorrectionsin
TGR,B3 = 9 G3/2a1/22e (1−e22)1/2(m2+m0 m 1+m2)m Binary B
1 1 0 0 1 1 2
=2.5×1014yr(cid:18)0.a12pc(cid:19)3(cid:18)1aa1u(cid:19)−1/2(cid:18)m10M+(cid:12)m1(cid:19)3/2 TGR,C1 =TGR,B2 = 32Gπ3c/22(m0a+52/m2(11+−me222))3/2, (15)
(cid:18)m2+m m +m2(cid:19)−1(cid:18) m (cid:19)−1 and the Newtonian precession time-scale associated with
× 0 0 1 1 2 the precession of the stellar binary due to the gravitational
1M(cid:12) 1×106M(cid:12)
potential of stars around the first SMBH (e.g., Kocsis &
(1−e2)3/2
× 2 . Tremaine 2011; Li et al. 2015)
e (1−e2)1/2
Figure 2 s1hows a1comparison of the time-scales for(t1h0e) TNP =2π(cid:18)πmn22e2(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)0πM(cid:63)(r)cosψdψ(cid:12)(cid:12)(cid:12)(cid:12)(cid:19)−1, (16)
illustrativeexample:m0 =2M(cid:12),m1 =1M(cid:12),m2 =106M(cid:12) where n2 =(cid:112)G(m0+m1+m2)/a32 is the mean motion of
and e =0.001. In the upper panel, the stellar binary has Binary B and r =r(ψ)=a (1−e2)/(1+e cosψ) in Kep-
1,0 2 2 2
the small semimajor axis as a = 0.05au (requiring to be lerian motion.
1
survival after many orbits around the SMBH in the further TomaketheKozaieffectprominentintheoutertriple,
simulations),whilea =1auinthelowerpanel.At(cid:38)0.01pc, it is required to compare the double Kozai time-scales of
1
(cid:13)c 2014RAS,MNRAS000,1–12
Stellar binaries surrounding SMBH binary 5
Figure 3. The a2−a3 parameter space where the outer LK oscillation is dominated. Here, we choose the parameters as m0 =2M(cid:12),
m1 =1M(cid:12), m2 =m3 =106M(cid:12) and e1,0 =e3,0 =0.001. Different values of a1 and e2 are included as labeled. The parameter space
boundedbythestabilitycriterion,TGR=TK,CandTNP=TK,C,correspondstotheKozaieffectallowedintheoutertriple(redregion).
ButitmaydominatethedynamicsofthesystemwiththeparametersabovethecurveofTK,B=TK,C.Thedashedlineofa2=0.01pc
markedinthelowerpanelsgivetheupperlimitona2,abovewhichtheinnerLKoscillationsarequenchedbyotherprocessesrelatedin
thegalacticnuclei(seeFig.2).
the inner and outer triples. If the oscillating period driven highly elliptical bound orbit (even a parabola), whose peri-
by the inner triple is much shorter and the one from the centreliesclosetotheSMBH(m ).Thatmeans,thestellar
2
outer, the Kozai mechanism may play an important role in binary might undergo tidal separation and TDEs or HVSs
the inner triple (Binary A and an SMBH). The eccentric- would occur in succession.
ity of Binary A can be driven to approach unit, leading to
the merger event between m and m . Meanwhile, the per- Considering the previous illustrative example, we in-
0 1
turbation from the second SMBH can be neglected (e2 will clude a second SMBH (m3 = 106M(cid:12)) and start the explo-
not change so much). Conversely, the evolution of the four- rationina2−a3 parameterspace(asshowninFig.3).The
body system will be dominated over by the LK oscillations initial eccentricities of the binaries are chosen to be zero
in the outer triple (Binary B and the second SMBH) if the (e1,0 =e3,0 =0), and two semimajor axes of the stellar bi-
mutual inclination lies in the range of 40◦ −140◦ and the nary are considered to include a1 = 0.05au and a1 = 1au
aboveconditionsaresatisfied.ItisnotedthattheinnerLK (upperandlowerpanels).Here,thedistantunitispc,which
oscillationsarestillallowedtooccurwhenT >T ,but makesalargeroomforthecompetitionofthedoubleKozai
K,B K,C
the dynamics of the system is dominated by the outer one. oscillations. In addition, we zoom in the small-scale area in
Therefore,e2canbeexcitedfasterthane1,makingBinaryA the unit of AU. To test the dependence on e2, we assume
e =0(left-handpanels)ande =0.95(right-handpanels),
2 2
(cid:13)c 2014RAS,MNRAS000,1–12
6 Liu, Wang & Yuan
respectively. As seen, the outer LK oscillations are allowed
to occur in the red region, bounded by the criteria (equa-
tions13-16).However,thecorrespondingparameterspaceis
dividedintotwopartsbythecurveofT =T ,andleft
K,B K,C
side denotes T > T while opposite in the right side.
K,B K,C
The outer Kozai dominated region is reduced into a nar-
rowerspace(deepredregion).Ontheotherhand,requiring
the inner Kozai time-scale to be shorter than GR effects,
the compact stellar binary (a = 0.05au) cannot be so far
1
awayfromSMBH,andtheregionofT <T shouldbe
K,B K,C
limited at a ∼0.01 pc. It is also required to note that the
2
parameterspacenearbythecurveofT =T isgoingto
K,B K,C
be governed by the inner triple with high e . In all, we find
2
that the outer Kozai dominated region becomes relatively
larger with a small a , although the physical processes are
1
the same.
3 BASIC MODEL OF TIDAL BREAK-UP OF
BINARIES Figure 4. The illustration of the motion of the stellar binary
with a highly elliptical orbit, whose pericentre is close to the
Beforerunningthesimulationsofthefour-bodysystemsys-
SMBH. When the angular momentum of the stellar binary is
tematically, we explore the close periapse approach of a bi- sufficient small, entering ‘Tidal Loss Cone’, the centre of mass
naryinahighlyeccentricorbitaroundtheSMBH.Asshown ofthebinarycrossesthetidalseparatedradius(r )andtheor-
bt
inFig.4,whenr2,theseparationbetweenthecentreofthe bit becomes unstable, resulting unbound stars. Since the orbital
mass of Binary A and the SMBH, is below the tidal sepa- energyisexchangedduringtheclosepassage,thesubsequentevo-
rated radius of the binary: lution may be one of the mergers, TDE (accompanied by HVS)
orsurviving.
(cid:16) m (cid:17)1/3
r ∼a 2 , (17)
bt 1 m +m
0 1
the binary is considered to be broken apart by the tidal
field.Sincetheorbitalenergyisexchanged,thedynamicsof In this section, for simplicity, we place the stellar bi-
the inner three-body may become chaotic during the close nary (m0 = 2M(cid:12), m1 = 1M(cid:12) and a1 = 0.05au) at the
passagebythefirstSMBH.Followingthetidalseparationof apastronoftheouterorbit(m2 =106M(cid:12))initially,outside
thebinary,bothstarsremainclosetotheoriginalKeplerian thebinarytidalseparatedradius.Theinitialorbitalphaseis
orbits of the binary around the SMBH, separately. Then, fixed, i.e, the arguments of periapse of the two orbits defer
the stars will either scatter off or undergo individual TDE nπ (n is a integer). But the eccentricity e ∈ (0,1), semi-
2
immediately.Here,weintroducethetidalradiusofeachstar major axis a ∈ (1,250)au and the inclination between the
2
as orbital planes of Binaries A and B i ∈(0◦,180◦) are dis-
tot
(cid:16)m (cid:17)1/3 tributedrandomly.Werun5×104 integrationsandexplore
r =3R 2 (α=0,1), (18)
αt α m what happened to the stellar binary during the first close
α
passagebytheSMBH.Wedistinguishfiveoutcomesforthe
where R is the stellar radius and the coefficient 3 refer to
α evolution of the stellar binaries: merger, single TDE (may
the case where the star will experience accumulated heat-
be accompanied by HVS), double TDE and uneventful fly-
ing to be disrupted outside the “pure” tidal radius (Li &
bys. We identify merger with a (1−e ) < R +R . When
Loeb2013).Weassumeamass-radiusrelation(e.g.,Hansen, 1 1 1 2
the distance between m and m (α=1,2) is smaller than
Kawaler, & Trimble 2004) as 2 α
r ,werefertothisasTDE.Additionally,ifTDEoccurson
αt
(cid:18)m (cid:19)3/4 eitherofthecomponentsofthestellarbinary,theothermay
Rα =R(cid:12) M(cid:12)α , (19) beejectedandrunoutofthegravitationalpotentialwellof
the SMBH. When the velocity of this ejected star is higher
where R(cid:12) is radius of the sun. than500km/s,werecordaHVShere.Forthefifthcategory,
welabelthestarsneitherdisruptednormergerafteraclose
approach as uneventful fly-bys.
3.1 Numerical setup
Inthiswork,allthedynamicsarecarriedoutbyusingac++
codewithexplicitFrogLeapingAlgorithm.Themethodin-
cludes the adaptive step with error determined by the rel-
3.2 Statistical Results
ative position that set to be 10−9 to trace the motion of
binaries accurately in arbitrarily mass ratios. To test our As shown in the upper panel in Fig. 5, we present the des-
(cid:112)
code, we reproduce Fig. 10 in Naoz et al. (2013b) and find tiniesofstarsfromnumericalexperimentsin a (1−e2)−
2 2
perfect agreement. The code can be downloaded freely at a phasespace,whichisassociatedwiththeorbitalangular
2
https://github.com/wyh102324/SMBHbinary momentum L and the orbital energy E of Binary
Bin,B Bin,B
(cid:13)c 2014RAS,MNRAS000,1–12
Stellar binaries surrounding SMBH binary 7
when e becomes larger. We find that the stellar binary
2
has more opportunities to be survival with lower e . Mean-
2
while, compared to the uneventful fly-bys, the numbers of
the mergers are rare but distributed below a critical value
of the angular momentum (e (cid:38) 0.6). Most of them locate
2
outsidethe‘TidalLossCone’(r <r ).Althoughaclose
per bt
passage by the SMBH has provided numerous mergers, the
following simulations will reveal that the Kozai mechanism
produces more after many orbits around the SMBH.
Whenthesystementerstheso-called‘TidalLossCone’,
itispossibleforeither(orboth)ofthetwostarstobetidal
disrupted. Due to the fact that the tidal radius of m is
0
largerthanm ’s,thesingleTDEsrecordedinthesesimula-
1
tions are all from m . In particular, when e (cid:38)0.98, 13.8%
0 2
ofthetotalsingleTDEsareaccompaniedbytheproduction
of HVSs of m .
1
(cid:112)
However,when a (1−e2)issufficientlysmallandthe
2 2
periapsis separation r (cid:46) 0.3r , the stellar binary ap-
per bt
proaches the SMBH on a nearly radial orbit. In this case,
both of the stars loss the angular momentum, falling into
individual tidal radius (equation 18), ending up with the
double TDE (e.g., Mandel & Levin 2015)
The lower panel shows the numbers of two types of
TDEs of the stellar binaries as a function of i , which
tot
corresponds to the initial inclinations between two orbital
planes. We find that, single or double TDE does not have
significant dependence on the orbital orientation.
4 NUMERICAL EXPLORATION ON THE
FOUR-BODY SYSTEM
Figure5indicatesthatthedestiniesofthestellarbinariesare
Figure 5. Upper panel: different endings of the stellar binaries
determined by the initial parameters (i.e., L ,E ).
withoneorbitpassingclosetoanSMBHareshowninthestellar Bin,B Bin,B
Exactly,theexistenceofadistantSMBHmayinducetheLK
angular momentum-energy phase space. Lower panel: the num-
oscillationsintheoutertriplesothatthestellarbinarycan
bers of the single and double TDEs distributed in the initial in-
clinations. approach to the first SMBH in a variety of possible orbits.
Due to the close encounter with the SMBH, the motion of
the system is failed to be described by the orbital averaged
B, respectively
analysis. In this section, we try to explore the evolution of
Gm (m +m ) the stellar binary in the four-body system by carrying out
E ∼− 2 0 1 ,
Bin,B 2a N-body integrations, and discuss the process that can re-
2
m (m +m ) (cid:113) populate the binaries.
L ∼ 2 0 1 G(m +m +m )a (1−e2).
Bin,B m +m +m 0 1 2 2 2
0 1 2
(20)
4.1 Numerical setup
Among our 5×104 simulations, 80.7 per cent are sur-
vival,2.2percentaremergers,2.7percentaresingleTDEs In our illustrative example: Binary A is consisted of two
onthemoremassivestar(m0)and14.4percentaredouble starsasm0 =2M(cid:12) andm1 =1M(cid:12) withatightsemimajor
TDEs. axis a = 0.05au. Since the distrubuton of stars surround-
1
IntheupperpanelofFig.5,theblackcurveprovidesa ing SMBH binary remains unclear to date, and the current
boundary of the eccentricity of Binary B, below which the numerical results have model dependence inevitability. For
phase space is indicated to have large e . The black dashed simplicity,wefocusonthebinaryclosetotheSMBHinthe
2
curvegivenbyr =r ≡a (1−e )showsthatthestellar allowed parameter space (shown in Fig. 2 and 3). The dis-
bt per 2 2
binary approaches the SMBH with the pericentre distance tance between the centre of mass of the stellar binary and
as large as the tidal separated radius. Here, the destinies of the first SMBH is assumed to be distributed uniformly at
the binary are colour coded (as labelled). During the first a short range of a ∼ (1au,150au). The secondary SMBH
2
close encounter, the orbital energy changes, giving chances has mass of m3 = 106M(cid:12) and the separation of SMBH
for the binary to be break-up. We find that the fates of the is chosen to be a = 400au. The inner and outer triples
3
stars depend sensitively on the angular momentum of the are setup by uniform mutual inclinations (i and i )
AB BC
orbit rather than the energy. between 0◦ and 180◦, and the eccentricities are set to be
For any values of a , the angular momentum decreases e =e =e =0 initially.
2 1,0 2,0 3,0
(cid:13)c 2014RAS,MNRAS000,1–12
8 Liu, Wang & Yuan
Figure 6. The final distributions of different endings of the stellar binaries for our illustrative example: m0 = 2M(cid:12), m1 = 1M(cid:12),
m2=m3=106M(cid:12),a1=0.05au,a3=400auande1,0=e2,0=e3,0=0.001,withtheinitialuniformdistributionsina2,iAB andiBC.
Thetoprightpanelshowsthedistributionsofthetimeinterval(betweentwosingleTDEs)forthetotaldoubleTDEs.
In particular, it is required that each component of Bi- tidaldisruptionatthefirstencounterbytheSMBHandwill
nary A is outside its own tidal disrupted radius r at the be fully disrupted at the following encounters if the star is
αt
beginning. We run 2000 Monte-Carlo N-body integrations still bound to the SMBH. Additionally, we do not include
forthetime-scale∼10(T +T ),withoutinvolvingthe the fate of the massive star produced by the merger, and if
K,B K,C
short range forces and the energy dissipation. Three out- either of the stars is captured by the secondary SMBH we
comes are distinguished for the evolution of the stellar bi- do not make a record as well.
naries: merger, TDE and uneventful fly-bys. Using the cri-
terions listed in Section 3.1, we also identify the double
TDEifthetimeintervalbetweentwoTDEsisshorterthan 4.2 Statistical Results
∼ 0.01/n especially (where the two stars are expected to
2 Figure 6 shows the final distributions of our N-body sim-
be tidal disrupted sequentially). Otherwise, we refer to the
ulations with a = 400au. Most of the events occur within
single TDE, depending on either of m and m is tidal dis- 3
0 1 ∼(T +T ) and the different endings of the stellar bi-
rupted first. Since we define that each set of initial param- K,B K,C
nariesaremarkedbythe‘rainbow’colours.Outofall,2000
eters has only one corresponding result, when TDE occurs
stellar binaries between a = 1au and 150au, 58.2% are
on m or m , the motion of the other is not recorded. Ac- 2
0 1 mergers, 7.9% are TDEs (totally) and 33.9% are survival.
tually, the surviving companion may experience the partial
Notethathalfoftheeventsofthesingletidaldisruptionare
(cid:13)c 2014RAS,MNRAS000,1–12
Stellar binaries surrounding SMBH binary 9
Figure 7. The time evolution of the orbital elements for several cases chosen in Fig. 6. From the top to bottom, the orbital elements
aretheeccentricitiesofBinaryAandB(e1 ande2),andtheseparationsbetweenm0,m1 andm2 (r01,r02 andr12).Here,Rα andrαt
(α=0,1)aretheradiusandtidaldisruptedradiusofeachstar,respectively.Fromthelefttoright,theoutcomesofthestellarbinary
arethe“typical”merger,doubleTDEandsingleTDEonm0.
Figure 8.SimilartoFig.7,butsome‘special’caseswiththemerger,singleTDEanddoubleTDE(fromlefttoright).
accompaniedbytheproductionofHVSs(notlabeled).Since So, the distribution of mergers in the final inclinations has
the definition of the double TDE is not clear, we plot the a strong corresponding peak around 90◦. The fraction of
distribution of the time interval between two single TDEs mergers with the inclination i ∼ (40◦,140◦) is ∼ 70%
AB
aswell,whichareallsatisfiedwith<0.01/n .Itseemsthat of the total mergers, while the TDEs still have a thermal
2
most of the secondary TDE occurs in tens of minutes later distribution (see the lower left panel in Fig. 6). Similarly,
after the first TDE. the outer triple with high inclinations will also undergo the
On the basis of the above analysis, the stellar binaries LKoscillations,makingtheangularmomentumLBin,B suf-
with close distance from SMBH are more likely to merge ficientsmall.ThestellarbinariesgetclosetotheSMBHand
(TK,B < TK,C). If the stars surrounding SMBH are trun- TDE occurs, which with iBC ∼ (40◦,140◦), accounting for
cated at a larger semimajor axis (a ), T will approach ∼73percentofallpossibleinclinations(seethelowerright
2 K,B
T (equations 1, 14), the stars have more opportunities panel in Fig. 6). However, it is noted that the number dis-
K,C
to be tidal disrupted and the merger rate will decrease (see tribution in the semimajor axis a2 shows a secondary peak
the upper left panel in Fig. 6). Although the initial inclina- at the larger a2, and there are still some merger events and
tions have an approximate isotropic distribution, the Kozai TDEs as shown out of the LK dominated region (0◦−40◦
effect plays a major role with high inclinations, the eccen- or 140◦−180◦). We will explore these ‘unexpected’ distri-
tricities of the stellar binaries can be driven to be large. butions in more detail further.
(cid:13)c 2014RAS,MNRAS000,1–12
10 Liu, Wang & Yuan
Figure7showstheexamplesoftheevolutionofdifferent
final endings from our N-body simulations, which presents
thecorrespondingorbitalelementsastheeccentricitiesofBi-
nariesAandB(e ande ),theseparationsbetweenm ,m
1 2 0 1
andm (r ,r andr )indifferentcolumns.Asexpected
2 01 02 12
in the parameter space (Fig. 3), the representative stellar
mergers occur for T < T and TDEs are inserted in
K,B K,C
the region of T > T , where the LK oscillations dom-
K,B K,C
inate the inner or outer triples, respectively. Noted that,
in the middle panel, the angular momentum of the stellar
binary get a “kick” during the sufficient close passage by
the SMBH, exchanging the orbital energy. The eccentricity
rapidly jumps to a high value. In the subsequent evolution,
the binary components are still bound to each other until
the second periapsis passage. After that, the stars will be
separated, leading TDEs or experience clean collision. This
has also been pointed out by Antonini et al. (2010).
However,wealsofindthatthemergersareabletooccur
eveniftheKozaieffectoperatesintheoutertripleinitially.
As shown in the left-hand panel of Fig. 8, for the relatively
Figure9.Thevariationsoftheratesofthestellarmergers,TDEs
distant stars, the pericentre distance of them can be closer
anduneventfuleventsasafunctionofdifferentsemimajoraxisof
by the outer Kozai oscillations. Therefore, the inner Kozai theSMBHbinary.
effect comes into play with high e and the eccentricity of
2
thestellarbinaryisexcitedinasmooth-likeway.Thisisalso
associatedwiththesecondarypeakofthedistributionina find that the enhanced rate of mergers achieves its maxi-
2
inthetopleftpanelofFig.6.Inaddition,wefindsuchatype mumata ∼700au,thatis∼1.5timeshigherthantheone
3
of systems, where one component of the binary undergoes withoutthesecondSMBH,whereastherateofTDEshasa
tidal disruption and the surviving companion is still bound peak around a ∼ 400au. As we discussed in the compari-
3
to the SMBH in a highly eccentric orbit after a close en- sonbetweentheKozaitime-scalesfromtheinnerandouter
counter. But, because of the existence of the outer SMBH, three-body (Fig. 3), for a given range of a (1au−150au),
2
the star may experience the Kozai oscillation induced by TDEs are predicted to be the most of the products when
thisnewlyproducedtriplesystem(SMBH-star-SMBH),ex- a ∼ 550au. On the other hand, the maximum probabil-
3
changing energy at periastron, and TDE occurs eventually ity of occurrence of a merger event is supposed to occur
(Lietal.2015)(asshowninthemiddlepanelofFig.8).Since with a ∼ 1000au. Note that all the rates are significantly
3
the time interval between two TDEs is so long that beyond decreased when a ∼ 2000au, and the affect of the outer
3
∼ 0.01/n , we identify this event as the single TDE rather SMBH is negligible typically. Of course, because the gap in
2
than the double TDE. Finally, to check the “unexpected” a we have chosen is large, the extreme values of the rates
3
distributionwechooseseveralcasestoexplorethetimeevo- arenottracedaccurately.However,theoverallconclusionis
lution of the orbital elements (as shown in the right-hand held qualitatively.
panelofFig.8).Insuchanexample,thesystemisappeared
to be very unstable. As a result, the quantities are vary-
ing irregularly. There are still some other non-hierarchical
5 SUMMARY AND DISCUSSION
systemsthatcontributethedistributionsintheinclinations
near 0◦ or 180◦. SMBHbinaryisexpectedtobetheby-productofthemerg-
ing galaxies. In this work, we consider the binary main se-
quencestarsorbitaroundtheSMBH,perturbedbyadistant
4.3 Efficiency of Double Kozai Oscillations
SMBH. Due to the existence of the secondary SMBH, the
The Kozai effect in the outer triple is no longer to be im- evolution of the stellar binary varies a lot through LK os-
portantifthesecondarySMBHisveryfaraway.Therefore, cillations or the chaotic interactions. The rates of different
theevolutionofthestellarbinaryisdeterminedbythefirst final endings of the main sequence stars are affirmed to be
SMBH completely.Inversely,if m becomes closer, theout- enhanced significantly, compared to the case without the
3
come of the main sequence binary is influenced by both of outer SMBH. Due to the enhanced merger rate of the stel-
theLKoscillations.Inordertodetectthesignificanceofthe lar binaries, the population of the massive stars (OB star)
externalSMBH,wetunethesemimajoraxisa graduallyin will be modified (Perets & Fabrycky 2009). Similar process
3
this section and find out what happens to the rates. Again, couldalsotakeplaceforthecompactobjects,increasingthe
we use the same initial parameters and distributions as the formationofTypeIasupernova,gammarayburstandgrav-
previous sections. For each a , we carry out 2000 integra- itationalwavesources.Contrarily,theincreasedratesofthe
3
tions for the time-scale ∼10(T +T ). eventscanpredicttheexistenceoftheexternalSMBH,that
K,B K,C
Figure 9 depicts the rates of different events as a func- isdifficulttobedetectedasmentionedinChenetal.(2009)
tion of a . In comparison to the mergers, TDEs have a rel- and Li et al. (2015).
3
atively small rate. However, it is a natural consequence of Comparing the time-scales of various processes related
theLKoscillationinducedbythesecondarySMBH.Wealso to the galactic center, we point out a non-neglected regime
(cid:13)c 2014RAS,MNRAS000,1–12