Busting Up Binaries: Encounters Between Compact Binaries and a Supermassive Black Hole

3.1 MB·English

by  Eric Addison

Draftversion February2,2015
PreprinttypesetusingLATEXstyleemulateapjv.2/16/10
BUSTING UP BINARIES: ENCOUNTERS BETWEEN COMPACT BINARIES AND A SUPERMASSIVE
BLACK HOLE
Eric Addison
DepartmentofPhysics,UtahStateUniversity,Logan, UT84322, USA.
Pablo Laguna
5 CenterforRelativisticAstrophysicsandSchoolofPhysics,
1 SchoolofPhysics,GeorgiaInstituteofTechnology, Atlanta,GA30332, USA.
0
2
Shane L. Larson

n Center forInterdisciplinaryExplorationandResearchinAstrophysics,
a NorthwesternUniversity,Evanston, IL60208,USA
J andDepartmentofAstronomy,AdlerPlanetarium,Chicago,IL60605USA.
Draft versionFebruary 2, 2015
6
2
ABSTRACT

] Giventhe stellardensitynearthe galacticcenter,closeencountersbetweencompactobjectbinaries
E and the supermassive black hole are a plausible occurrence. We present results from a numerical
H study of close to 13 million such encounters. Consistent with previous studies, we corroborate that,
for binary systems tidally disrupted by the black hole, the component of the binary remaining bound
.
h totheholehaseccentricity 0.97andcircularizesdramaticallybythetimeitenterstheclassicalLISA
p band. Our results also show∼that the population of surviving binaries merits attention. These binary
-
systems experience perturbations to their internal orbital parameters with potentially interesting
o
r observationalconsequences. Weinvestigatedtheregionsofparameterspaceforsurvivalandestimated
t the distribution of orbital parameters post-encounter. We found that surviving binaries harden and
s
a their eccentricity increases, thus accelerating their merger due gravitational radiation emission and
[ increasing the predicted merger rates by up to 1%.

1
v 1. INTRODUCTION than 75% of O-type and 70% of B-type stars have some
6 numberofcompanions(Raghavan et al.2010). Nearthe
Observations of gravitational waves (GWs) will allow
5
galacticcenter,thedensityofstarsgrowsverylargecom-
8 us to probe dynamical astrophysical systems in regimes
pared to field conditions, with density estimates up to
07 wGhWersefrsotrmonsgtegllraarvimtyaspslainytsearakcetyionroslew.itIhnagasluapcteircmnauscslievie, 108M⊙ pc−3 within the inner 0.1pc, (Alexander 2005).
Giventhis information,itisreasonabletoexpecttheex-
. black hole (SMBH) will be excellent probes of the prop-
1 istence of compact object (CO) binaries in the galactic
erties of the SMBH and the stellar population itself. It
0 center, and in fact it has been observed by way of X-
is thus important to investigate the types of encounters
5 ray transients that CO binaries exist near the GC, and
that should be expected as well as the strength and fre-
1 are more abundant within the inner 1 pc (Muno et al.
quency of GWs emitted by these events.
: 2005). It has been estimated that as many as 20,000
v In the present work, we will focus on encounters
∼
i between stellar mass binaries and a galactic SMBH. stellarmassBHbinarieshavesegregatedwithintheinner
X 1 pc of the Milky Way galactic center (O’Leary et al.
Previous studies have focused on the hyper-velocity
≈
r star (HVS) produced by the disruption of the binary 2009).
a With the existence of CO binaries near the galac-
and stellar collisions (Hills 1988; Gualandris et al. 2005;
tic center, it is reasonable to expect that some number
Antonini et al. 2010). Miller et al. (2005) investigated
of them may interact directly with the SMBH. Many
the eccentricity of the bound component created by the
knownmain sequence starsexist inbound orbits around
binary disruption, namely the extreme mass-ratio inspi-
the SMBH (Ghez et al. 2005, 2008, 2009), suggesting
ral (EMRI) left behind. More recently, binary interac-
that the same could be true for COs and CO binaries.
tions with SMBH have been explored in the context of
Such interactions have implications for GW campaigns
theGWemissionsintheLIGObandfrombinariesdriven
of observations, for example ground based observato-
to merger by Kozai resonance (Antonini & Perets 2012;
ries (e.g. LIGO) searching for compact binary coales-
Antonini et al. 2014).
cences (CBCs). Proposed space based interferometers
The strongest GW sources will involve compact, de-
(e.g. LISA) will be sensitive to EMRIs which can be
generate stellar remnants that can survive close encoun-
created by the tidal disruption of CO binaries near the
ters with the central black hole, namely neutron stars
SMBH.
and stellar mass black hole (BH)s. The fraction of field
In the present study, we corroborate and extend the
stars in binaries varies by stellar type, but for O- and
resultsofMiller et al.(2005)fortheformationofEMRIs
B-type stars,which are the stellar types massive enough
by tidal disruption to arbitrary binary orientations. In
to form neutron stars or BHs, it is estimated that more
2
addition,wefocusourattentiononthosebinarysystems proximate v 0. The total energy thus become
3
≈
thatsurvivetheencounterwiththeSMBH.Inparticular,
2
we investigatewhether those binariesare able to survive E = 1M V2 GmiM• +E . (5)
andfacesubsequentencountersbeforemergingfromGW 2 b b − r b
i
emissionormergefaster thanthosenotexperiencingen- Xi=1
counters with SMBH, and discuss the implication of the where V is the velocity of the center-of-mass of the bi-
b
encounters on predicted CBC rates. We concentrate on nary relativity to the SMBH. Since for the incoming bi-
initially circular, equal mass (m1 = m2 m), CO bina- nary R r, with R denoting the distance of the center-
riesapproachingtheSMBHinaparaboli≡corbit. Wevary of-mass≫of the binary to the SMBH, we can rewrite the
the orientation of the orbital angular momentum of the total energy as
CO binary relativeto the angularmomentum of the CO
binary orbiting the SMBH. We also vary the pericenter E 1M V2 GMbM• +E =E +E . (6)
distance r between the CO binary and the SMBH. Our ≈ 2 b b − R b cm b
p
study explores a greater volume of the parameter space;
As mentioned before, we inject the binaries in parabolic
Antonini & Perets (2012) focused on binaries bound to
orbits; therefore, the initial center of mass energy of the
theSMBHatafixedorientation,andMiller et al.(2005)
binary with the black hole is E = 0. Therefore, the
cm,0
explored hyperbolic encounters with coplanar binaries.
totalenergyisgivenbytheinitialinternalbindingenergy
Considering a larger region of parameter space comes
of the binary, E =E <0.
b,0
at the expense of integration sophistication, though the
The effect of the encounter is to re-distribute the en-
time and distance scales involved in our simulation sug-
ergy available, i.e. E , among the three bodies. If the
b,0
gestthatrelativisticeffectswillnotplayasignificantrole
binary survives, E <0. Given that
b
for the majority of the parameter range, and Newtonian
gravitationalforces should suffice. E =Ecm+Eb =Eb,0 <0, (7)
The paper is organized as follows: Section 2 outlines
the CO binary after the encounter could be bound
the outcomes of the three body encounters based on en-
(E <0) or unbound (E >0) to the SMBH.
ergy arguments. In Section 3, we describe the setup of cm cm
On the other hand, if the CO binary does not survive
the encounters. Parameter probability distributions and
(E > 0), the separation r of its components will grow;
an estimate of the tidal radius are given in section 4. b
thus, one can neglect in Eq. (1)
TheanalysisofthedisruptedbinariesisfoundinSection
5. In Section 6, we discuss the results of those binaries Gm m
1 2
that survived the encounter with the SMBH. Section 7 r r ≈0, (8)
2 1
includes a discussion of the changes in the lifetime of | − |
the CO binary due to GW emission as a result of the and rewrite the total energy as
encounter with the hole as well as consequences to GW
2 2
detection rates for CO binaries. The paper ends with E = 1m v2 GmiM• = E , (9)
conclusions in Section 8. 2 i i − r i
i=1(cid:18) i (cid:19) i=1
X X
2. ENERGETICSATAGLANCE wherewehaveusedagainthatr =0andv 0. There-
3 3
≈
The total energy for a three-body system can be writ- fore, for CO binaries that are disrupted
ten as
E =E +E =E <0. (10)
1 2 b,0
3 2 3
E = 1 m v2 Gmimj (1) The possible outcomes in this situation are both COs
2 i i − ri rj bound to the SMBH (E1,E2 < 0), or one bound and
Xi=1 Xi=1j=Xi+1| − | the other unbound (Ei < 0 < Ej). Clearly, the case in
In the barycentric frame, with Mb = m1+m2 the mass which bound components are unbound (0 < E1,E2) is
of the binary and M• = m3 the mass of the SMBH, the not possible.
total energy reads WewillrefertoaCObinaryinaboundorbitarounda
SMBHasabinaryextrememass-ratioinspiral(BEMRI),
E = 21MMbb+MM••V2−i=21 |Grim−iMr3•| +Eb. (2) aecnlnacdsosuewsni:tllerclwasitshifythtehSeMafBteHrminattohoonfetohfethbeinfaorllyowfrionmg fothuer
X
Above, DB: Disrupted binary.
•
1 GµM Long BEMRI: Survived binary bound to SMBH
E = µv2 b (3) •
b 2 − r with τgw >P•.
Short BEMRI: Survived binary bound to SMBH
vis=thve intevrnaalndenµer=gymofmth/eMbi.naFruyrtwheitrhmorre=, r1 −r2, • with τgw <P•.
1− 2 1 2 b SU: Survived binary unbound to the hole.
•
2
1
V=v m v (4) Inthisclassification,τgw referstothe binarymergerlife-
3 i i
− M time from GW emission, the so-called Peters lifetime
b
i=1
X (Peters 1964), and P• is the period of the bound binary
is the relative velocity between the binary and the around the SMBH. One of the main motivations of our
SMBH. Since M• Mb, we can set r3 = 0 and ap- study is to investigate how the probability of a binary
≫
3
falling into one of these categories could alter the pre- Thefollowingparametersinthethree-bodysystemare
dicted CBC. kept fixed: SMBH mass (M• = 106M⊙), CO binary
ThefateofaCObinaryencounteringaSMBHdepends masses (m1 = m2 = m = 10M⊙), initial binary ec-
on its penetration factor β. The penetration factor is centricity (e = 0), and initial binary semi-major axis
0
definedastheratioβ rt/rp,wherert isthetidalradius (a0 = 10R⊙ = 0.047 AU). With these parameters, the
≡
andr the distance ofclosestapproachto the SMBH.In CO binary has a period
p
analogy with the common definition of tidal radius for
the stellar disruption of stars by massive BHs, we define 2πa3/2
P = 0
the tidal radius rt as b,0 √GMb
r M• 1/3a , (11) =7.16 104s a0 3/2 Mb −1/2 ,(14)
t ≡(cid:18)Mb(cid:19) 0 × (cid:18)10R⊙(cid:19) (cid:18)20M⊙(cid:19)
where a is the initial semi-major axis of the CO bi- and tidal radius of the three-body system is
0
nary. The radius r is an approximation to the distance
t −1/3
where within the tidal forces by the SMBH exceed the r =173r M−2/3 Mb a0
self-bindingenergyofthebinary. Theexactdistancewill t g 6 20M⊙ 10R⊙
(cid:18) (cid:19) (cid:18) (cid:19)
depend on the orbital parameters and orientation of the
−1/3
binary (Hamilton & Burns 1991, 1992). One of the ob- =1.73AUM1/3 Mb a0 .(15)
jectives of our study is to provide a statistical estimate 6 20M⊙ 0.047AU
of the tidal radius from our set of encounters. (cid:18) (cid:19) (cid:16) (cid:17)
As mentioned before, the effect of the encounter is a where M6 M•/106M⊙ and rg =GM•/c2 is the gravi-
≡
redistribution of energy and angular momentum in the tational radius.
system. As a consequence,orbitalparametersin the CO The parameters we vary are the CO binary orbital in-
binary, such as eccentricity and semi-major axis are af- clinationι,theinitiallongitudeoftheascendingnodeΩ0,
fected. Regarding eccentricity, Heggie & Rasio (1996) thebinaryinitialphaseθ0,andthepericentricdistancerp
provided an analytic estimate of this perturbation for a viathepenetrationfactorβ. Thus,oursimulationsofCO
circular binary system in parabolic orbit around a third binaryencounterswithaSMBHspanafour-dimensional
body. For the case of M• ≫Mb, the perturbation reads sppaarcaemwetitehrsrpaancdeo{mβv,ιa,luΩe0s,θta0k}.inWguensiafmorpmledtihstisripbaurtaiomnestienr
δe=3√2π Mb 1/4 2rp 3/4 pβh−a1s∈e θ[0.,35w,e5]t,ackoesι20∈0[−ev1e,n1l]y, asnpdacΩed0 ∈val[u0,e2sπb].etFwoerenth0e
(cid:18)M•(cid:19) (cid:18) a0 (cid:19) and 2π.0The distribution in β−1 implies that the values
2 2M 1/2 r 3/2 of the pericentric distance are uniformly distributed in
b p
exp (ι,φ) (12)
r [0.865,8.65]AU. Furthermore,since the duration of
× "−3(cid:18) M• (cid:19) (cid:18)a0(cid:19) #F thte∈encounter is of order
where 2πr3/2
(ι,φ)=cos2 ι[cos4 ι + 4sin4 ι Tp=√GMp • =β−3/2Pb,0, (16)
F 2 2 9 2
4 ι ι the range of parameters we consider for β−1 imply that
+ cos2 sin2 cosφ1/2] T [0.1,3]P . Moreover,in the nomenclature of Heg-
3 2 2 p ∈ b,0
gie, our binary system is “soft” since
withιtheinclinationoftheCObinaryrelativetotheor-
bitaroundthehole,andφdependingontheinitialphase 1M V2 E (17)
of the binary and the longitude of the ascending node. 2 b p ≫− b,0
Inserting the definition of tidal radius, Eq. 12 becomes
which can be seen from
δe=6√π21/4β−3/4exp 2√2β−3/2 (ι,φ) (13) 1M V2 = GMbM• = GMb2 M• 2/3β
"− 3 #F 2 b p rp a0 (cid:18)Mb(cid:19)
This expressionwill be comparedagainstour simulation GMb2 = E . (18)
results in section 6.1. Since δe vanishes as inclination ≫ 4a − b,0
0
ι π and β 0, we expect to see stronger agreement
be→tween simul→ation results and Eq. 13 for the low incli- Also, since rt ∼173rg, even for the deepest penetration
encounters with β = 2, the pericentric distance will be
nation and large pericenter encounters.
r = 87r . Therefore, it is perfectly safe to ignore gen-
p g
3. BINARYENCOUNTERSETUP eral relativistic effects in all our encounters. Similarly,
given that at pericentric passage
The CO binary is injected in a parabolic orbit around
the SMBH. The integration runs until either the binary v2 GM• 1
istidallydisruptedbytheSMBHandanamountoftime , (19)
c2 ∼ c2r ∼ 87
equal to the initial time passes, or the center of mass p
of the binary has reached a true anomaly of Θ = Θ , post-Newtonian corrections to the orbital motion of the
0
−
where Θ < 0 is the initial true anomaly of the orbit of binary are at the level of a percent and thus will be ig-
0
the binary’s center of mass about the SMBH. nored. However, in analyzing the the aftermath of an
4
1 2 parameters β,ι,Ω0,θ0 . Forexample,f(β−1 DB)isthe
probability{distribution}function for the para|meter β−1
0.8
1.5 for disrupted binaries, i.e. the probability that a dis-
0.6 rupted binary had a particular value of β−1 (solid curve
f f 1
0.4 in upper left panel of Figure 1). We use for this purpose
0.5 the technique of kernel density estimation (KDE). KDE
0.2
is a non-parametric method for estimating probability
0 0
densitiesinwhichakernelfunctionK isconvolvedwitha
1 2 3 4 5 −1 0 1
β−1 cos ι collection of Dirac delta functions. KDE asymptotically
convergestothe truedistributionfasterthanhistograms
0.3 0.3 (Scott 1979). The KDE probability distribution f of a
parameter x is computed from
0.2 0.2
N N
1 1
f f f(x)= K(x) δ(x x )= K(x x ) (21)
n n
0.1 0.1 N ∗ − N −
n=0 n=0
X X
0 0 whereN isthenumber ofdatapointsinthe sample. We
0 2 4 6 0 2 4 6 useaGaussiankernelK(x)=(2πh2)−1/2exp x2/(2h2)
Ω θ
0 0 with a variance of h2 = [parameter range]/100, chosen
(cid:2) (cid:3)
Fig. 1.— KDE probability distribution function f as computed to produce distributions that retain structure while not
from Eq. 21 for each of the {β,ι,Ω0,θ0} parameters. The pdf f being over-smoothed. Figure 1 shows the resulting nor-
representstheprobabilitythatabinarywithaparticularend-state malized KDE probability distributions f for each of the
began with a particular value of a certain parameter. The lines-
β,ι,Ω ,θ parameters obtained from Eq. 21 .
classcorrespondenceis: soliddenotesclassDB,dot-dashclasslong { 0 0}
BEMRI,dashclassshortBEMRI,andfinedashclassSU. We can draw from these probability distributions sev-
eralconclusions about the nature of the end-state of the
encounter, we will take into account the Peters lifetime
binary relative to the input parameters, as well as the
τ of the binary, if survived, or if the EMRI if the bi-
gw predictivepowerofthe individualparameters. Itisclear
nary is disrupted. For the CO binary systems that we
from the panel of the β−1 probability distribution (top
consider,the Peters lifetime due to the emissionof GWs
leftpanelinFigure1)thattheencountersyieldingbinary
in the absence of the SMBH is (Peters 1964)
disruption(i.e. DB-type,black,dot-dashline)occurpri-
a 4 M −3 marilyforsmallvaluesofβ−1orlargepenetrationfactors
τgw,0 =0.95 108yr 0 b . (20) β. This should be obvious: closer passes translates into
× (cid:18)10R⊙(cid:19) (cid:18)20M⊙(cid:19) stronger tidal forces and higher likelihood of disruption.
The present study consists of N 13 million individ- Additionally, regarding the parameter ι, prograde bina-
c
ual simulations, resulting in: 2.1 m≈illion simulations of ries (defined here as those with inclination ι < π/2) are
DB type encounters,1.7 million yielding long BEMRI, 2 moreprobabletodisruptthanretrogradebinaries(those
million producing short BEMRIs and 7.1 million of the with ι>π/2). This is so because, in general, retrograde
SU type. We integrate the Newtonian equations of mo- binaries are more likely to survive and become bound to
tion for the three-body system using Burlish-Stoer ex- the SMBH, implying that while the BH orbit loses en-
trapolation with a leap-frog integrator as described in ergy,thesurvivingretrogradebinariesmustgainanequal
Mikkola & Tanikawa (1999). We increase the accuracy amount energy, though not generally enough to disrupt.
of the integration by the use of the CHAIN concept ala Notice that the probability distribution for the initial
Mikkola & Aarseth (1993). The coordinate system used orbital phase θ0 is flat. That is, the value of θ0 does
forthenumericalintegrationarebarycentriccoordinates not play a role on the outcome of the encounter. The
with the three body center of mass at the origin. The distributions in Figure 2 are over all the simulations in
orbitalangularmomentumoftheCObinary–SMBHsys- the parameter space. The flat distribution in θ0 illus-
tem is aligned with the z-axis. That is, the plane of the tratesthat there is no strongcorrelationbetweenθ0 and
CO binary center of mass orbiting the SMBH is the xy- disruption across the entire parameter space of interest;
plane. We inject the CO binary in the first quadrant the combination of other parameters has far greater in-
(x,y > 0) at a distance 200rp and orient the orbit of fluence. Clearly if all parameters except θ0 were fixed,
the CO binary about SMBH such that pericentric dis- some binaries would be disrupted, but there is no con-
tance r occurs along the x-axis, specifically at y = 0 sistent trend. Also interesting is that the probability
p
and x<0. Conservation of energy and angular momen- of longitude of the ascending node Ω0 is relatively flat.
tum is checked at every time step, and simulations are Therefore,wewill focus ourattentiononthe parameters
halted and rejected if either quantity deviates from the β and ι since they have the largest effect on the binary
initial value by one part in 106. end state.
Thefactthatβ andιarethemostrelevantparameters
4. PARAMETERPROBABILITYDISTRIPUTIONSAND can also be seen in Figure 2 where we show the fraction
TIDALRADIUS ofsurvivingbinariesforeachparameter. Consistentwith
As already mentioned, we have classified the encoun- Figure 1, the top panels in Figure 2 show that the sur-
ters into four types: DBs, long and short BEMRIs and vival probability depends more strongly on β−1 and ι.
SUs. Next, for eachof these outcome types, we estimate Thetoprightpanelshowsagainthatprograde(ι<π/2)
the probability distribution as a function of the varying binaries in general more likely to be disrupted than ret-
5
−3
1 1 x 10
1 4
on on 3.5
cti 0.5 cti 0.5
Fra Fra 0.5 3
2.5
ι
0 1 2 3 4 5 −01 0 1 os 0 2
β−1 cos ι c
1.5
1 1 −0.5 1
0.5
n n
actio 0.5 actio 0.5 −1 0.5 1 1.5 2 2.5 0
Fr Fr β−1
0 0 Fig.3.—2D histogram ofdisruptions vs. β−1 and cosι, shown
0 2 Ω 4 6 0 2 θ 4 6 asfractionoftotal disruptions. Thewhitelinedenotes Eq.22.
0 0
Fig. 2.— Fraction of survivingbinaries as a function of each in
parameterforeachparameter{β,ι,Ω0,θ0}.
n
o 0.1
rograde(ι>π/2)binaries. Similarly,it is clearfromthe cti
top left panel that the probability of disruption goes to a
zero for β−1 & 2.1. Also consistent with Figure 1, the Fr 0.05
bottom panels in Figure 2 show that Ω0 and θ0 have a 0
−300 −200 −100 0 100 200 300
very weak influence on the survival ratio. Therefore, we
Energy/|E |
can assume with confidence that for practical purposes b0
the parameter space is two-dimensional, namely β,ι . 0.08
{ }
Thedependenceofthebinarysurvivalonιimpliesthat
n 0.06
thetidalradiusdoesnotonlydependonβbutalsoonthe o
inclination of the binary. This is not surprising because cti 0.04
a
the forcesfromthe SMBHresponsible for disrupting the Fr 0.02
binary are those projected along the semi-major axis of
0
thebinary. Figure3depictsatwo-dimensionalhistogram 0.9 0.95 1 1.05 1.1
of tidal disruptions as a function of cosι and β−1. The Eccentricity
solid line is the condition obtained by setting in Eq. 12
0.08
the eccentricity perturbation to δe = 1 with φ = 0 to
maximize the effect. Namely, n 0.06
o
cti 0.04
−1(ι,0)=6√π21/4β−3/4exp 2√2β−3/2 (22) Fra 0.02
F "− 3 #
0
0 50 100 150
Notice that this analytic approximation bounds the re- Semi−major Axis (AU)
gion containing disruptions well for β−1 & 1, but does
notdosowellforβ−1 <1,wherethedeepestpenetration Fig. 4.— Top panel: Histogram of the binding energy Ei from
encounters are located. Eq.25normalizedtotheinitialbindingenergyofthebinaryEb,0.
Middlepanel: Histogramofthecorrespondingeccentricity. Bottom
Panel: HistogramofresultingEMRIsemi-majoraxis. Histograms
5. DISRUPTEDBINARIES shownasfractionoftotal disruptions.
As stated in the introduction, CO binaries disrupted
by the SMBH are of interest as both sources of EMRIs
and HVSs. In this section, we investigate the channels
for EMRI formation and use previous HVS results as a
corresponding eccentricities (bottom panel). The sym-
check for our simulation accuracy.
metry of the histograms gives the impression when a bi-
We recall that after a binary is disrupted, the total
nary is disrupted the outcome is invariable one compo-
energyofthesystemisapproximatelygivenbyE =E +
1 nent bound to the SMBH, i.e. an EMRI, and the other
E =E ,whereE isthe initialbinding energyofthe
2 b,0 b,0 component unbound to the hole, namely a HVS. This is
CO binary and
indeedthecaseformostofthedisruptions( 99.9967%).
∼
Ei = 12mivi2− GmriiM• , (23) EtHhoowu<egvh0e,rr,aansriednc(te∼heE0o1.u0+0tc3Eo3m2%e=),isEfotbrw,0ow<hEi0Mch,RtbhIsoe.rtehMaEore1reco<avsee0rs,,ananold--
2
with i=1,2. tice from the bottom panel in Figure 4 that the average
Figure4showsthecombinedhistogramsofthenormal- eccentricity for the bound (unbound) component of the
izedenergiesE1/Eb,0 andE2/Eb,0 (toppanel)andthe disrupted binary is e− 0.97 (e+ 1.3). This can be
| | | | ≈ ≈
6
understood from the definition of eccentricity
10
e2± =1+ G22Em±3±LM2±•2 , (24) 0 km/s) 68
0
where we approximate 0
1 4
E± ≈±GM•m±a0rp−2 (25) v (∞ 2
and 0
L2± ≈L2cm =2GM•m2±rp. (26) 0.4 0.6 0.8 1 1.2β 1.4 1.6 1.8 2 2.2
Thus,
a a M 1/3
e2 1= 4 0 = 4β 0 = 4β b
±
− ± rp ± rt ± (cid:18)M•(cid:19)
n 0.1
e±≈1±0.05β(cid:18)20MMb⊙(cid:19)1/3M6−1/3. (27) Fractio 0.05
Furthermore, from Eq. 25 and E = GM2/(4a ), we
b,0 | b 0 |
have that
0
E± 74β2M1/3 Mb −1/3 , (28) 0 2 v4 (1000 km6/s) 8 10
|Eb,0| ≈± 6 (cid:18)20M⊙(cid:19) ∞
consistent with the histograms in the top panel of Fig- Fig.5.— Top panel: Grey dots denoted the ejection velocity
v∞ as a function of β for all the unbound stars from disrupted
ure 4.
binaries. Starsshowtheaverages over100evenlyspacedintervals
inβ. Solidlineshows the model prediction fromEq. 31. Bottom
5.1. Ejected Hyper-velocity Stars
panel: distributionofv∞ forallejectedcomponents.
Hypervelocity stars are stars with velocities of the or-
der of hundreds or thousands of km s−1, which may with D =79.37β−1.
exceed the escape velocity of our galaxy. These stars InthetoppanelinFigure5weshowwithgreydotsthe
were predicted by Hills (1988) as the result of binary ejection velocity v∞ as a function of β−1 for all the un-
disruption in the galactic center, and discovered obser- bound stars from disrupted binaries. In the same panel,
vationally nearly two decades later (Brown et al. 2005; stars show the averages over 100 β bins. The solid line
Edelmann et al. 2005). showsthemodelpredictionfromEq. 31. Theagreement
The production of HVS from the tidal disruption of of our average velocities with those from Bromley et al.
binary systems by a SMBH has been discussed by Hills (2006) is evident. However, it should be noted that the
(1988); Yu & Tremaine (2003); Antonini et al. (2010); range of possible velocities can vary significantly from
Bromley et al.(2006). Ourstudyisconsistentwiththose this average, with the largest values roughly double the
results. From Eq. 25 and E =m v2 /2, one has that analytic prediction and nearly 59% of ejections in our
i i ∞ simulations exceeding the predicted value.
v∞2 ≈2GM•a0rp−2 5.2. Extreme-Mass-Ratio Inspirals
≈2Gβ2M•a0rt−2 ThetraditionalformationchannelofanEMRIiswhen
≈2Gβ2M•1/3a−01Mb2/3, (29) two-body relaxation not only bounds a star to a SMBH
butalsobringsitintotheGW-emissioninspirilingregime
which yields the following asymptotic velocity approxi-
(Amaro-Seoane et al. 2007). The pericentric distance to
mation of a HVS
capture a star bound to a SMBH with semi-major axis
v∞≈5,312 km/s ·β ·M61/6 ac and form an EMRI is
×(cid:16)0.04a70AU(cid:17)−1/2(cid:18)20MMb⊙(cid:19)1/3 . (30) rc ≈3rgM6−4/7(cid:18)10mM⊙(cid:19)2/7(cid:18)0.0a5cpc(cid:19)2/7 . (33)
Rescaled to our case, the numerical study by
Since the star needs to penetrate so deeply into the the
Bromley et al. (2006) found that
neighborhoodof the SMBH, despite circularizationfrom
v∞≈4,468 km/s ·g(D) ·M61/6 GcaWnt eemcciesnstiorinc,ittyh,eeEM0R.5I ar0r.i9v.es to merger with signifi-
a −1/2 M 1/3 The disruptionof∼a bina−ry by a SMBH providesanal-
0 b
(31)
× 0.047AU 20M⊙ ternative channel since the disruption will always leave
(cid:16) (cid:17) (cid:18) (cid:19) at least one component bound to the SMBH. The cap-
where ture distance in this situation is much larger. It is not
g(D)=0.774+0.0204D 6.23 10−4D2 rc but the tidal radius rt 173rg (see Eq. 15). Being
∼
− × bound to the SMBH does not necessarily translates into
+7.62 10−6D3 4.24 10−8D4 becoming an EMRI. The condition for a star to qual-
× − ×
+8.62 10−11D5. (32) ify as an EMRI is that the timescale τ for its orbital
gw
×
7
−4
x 10

0.99 4.5
0.98 4
3.5
0.97
3
RI 0.96
M 2.5
E
e
0.95 2
0.94 1.5
1
0.93
0.5
0.92
0
20 40 60 80 100 120 140
a (AU)
EMRI
Fig.6.—2Dhistogram ofpotential EMRIcandidates asafunctionofeccentricity ei andsemi-majoraxisai,shownasfractionoftotal
disruptions. White lines are lines of constant Peters lifetimewith values τgw =10α years, where fromleft to right α= 5.0, 5.5, 6.0, 6.5,
7.0,7.5,8.0and8.5, respectively. Redlinesshowlinesofconstant β−1 withβ−1=0.35,0.5, 0.75,1.0,1.251.51.75, and2.0fromleftto
right.
decay by GW emission is sufficiently shorter than the will have a typical eccentricities of e 0.97 and semi-
0
≈
relaxation time t of the stars in the galactic center majoraxisa 45AU;thus,c 2.3 AUinEq.35. For
rlx 0 0
≈ ≈
(Amaro-Seoane et al. 2007): a space-based interferometer like LISA, the low end of
the sensitivity window is f 10−4 Hz. If we approx-
σ 3 gw ∼
trlx≈1.8×108yr 100kms−1 imate fgw ≈ 2f with f−1 = P = 2πa3/2/√GM• the
Keplerian orbital period, we have that when the EMRI
10M⊙ (cid:16)106M⊙pc−3(cid:17) enters the sensitivity band of a LISA-like detector it has
(34)
× m m¯ n a semi-major axis of
(cid:18) (cid:19)(cid:18) (cid:19)
where σ is the local velocity dispersion, n is the local −2/3
number density of stars, and m¯ is the average stellar a.74r fgw M−2/3. (36)
g 10−4Hz 6
mass. (cid:18) (cid:19)
Figure 6 displays a 2D histogram of (a,e) for EMRI FromEq.35,onehasthatthecorrespondingeccentricity
candidates from the disruption of binaries. White lines is e . 0.15. By the time the EMRI merges, i.e. a r ,
g
arelinesofconstantPeterslifetimewithvaluesτgw =10α itwillhaveeccentricitye 10−4 andemitGWsat∼dom-
years,where from left to right α= 5.0, 5.5, 6.0, 6.5, 7.0, ∼
inant frequencies of f 60 mHz. This is clear from
gw
7.5, 8.0 and 8.5, respectively. Red lines show lines of ≈
Figure 7 where we plot the eccentricity of the EMRIs
constant β−1 with β−1 = 0.35, 0.5, 0.75, 1.0, 1.25 1.5
as a function of the GW frequency f . We approxi-
gw
1.75, and 2.0 from left to right. It is clear from this
histogramthat mostof the candidates qualify as EMRIs mate fgw ≈ 2f with f−1 = P = 2πa3/2/√GM• the
Keplerian orbital period and a from (Peters 1964). Fig-
since τ t 108 yr. Also interesting is that the
gw ≪ rlx ≈ ure 7 showsthat in the sensitive LISA band (0.1 mHz.
encounters with deepest penetration factors, i.e. small
β−1, yield EMRIs with larger eccentricity. fGW . 100 mHz), EMRIs can have average eccentricity
e¯ 0.1. At the most sensitive frequencies, however, at
OurresultsareconsistentwiththosefromMiller et al. ≤
f .10mHz,theseEMRIswillhavetypicaleccentric-
(2005) in the following respect. Because their “capture” GW
ities e<0.01, in agreement with Miller et al. (2005).
radius is much larger than in the traditional scenario,
i.e. rt ∼ 170rg instead of rc ∼ fewrg, the EMRI from 6. SURVIVINGBINARIES
binarydisruptionswill havecircularizeddramaticallyby
We now turn the attention to the population of bina-
the time they merger or they enter the sensitive band
of a LISA-like detector. One can see this using Peters ries that survive the encounter with the SMBH. These
binariesmake up the majorityof the encounterswe sim-
(1964)
ulated given the range of values of β we considered. In
c e12/19 121 870/2299 these scenarios, the energy exchange is simpler: energy
a(e)= 0 1+ e2 (35) is either drainedfrom the orbitofthe binary aroundthe
(1 e2) 304
− (cid:18) (cid:19) BH and donated to the CO binary orbit, or vice versa
with c determined by the initial condition a=a when (see Eq. 6). In general terms, we expect the binary to
0 0
e=e . FromFigure6,wehavethatat“birth”theEMRI softenandbecomeboundtotheSMBHortotightenand
0
8
0 0
10 10
y
cit
−1 ntri
10 cce 10−2
e
−2 0 1 2 3 4 5 6 7
y 10 θ
cit 0
entri 100
cc −3
e 10 y
cit
ntri
e
−4 c
10 ec −1
10
0 1 2 3 4 5 6 7
−5 θ
10 0
−5 −4 −3 −2 −1
10 10 10 10 10 Fig.8.— Resulting eccentricity of surviving binaries as a func-
frequency f
gw tion of θ0. Upper panel: the dots show cases with parameters
(β−1,ι,Ω0)=(0.77,123◦,46◦),squareswith(1.9,101◦,312◦),and
Fig. 7.—EMRIeccentricity atfundamental frequency fgw. Ec- diamonds with (4.6,49◦,152◦). Points where the curve jumps
centricity for individual EMRIs areshown ingray, average shown above 1 indicate disrupted binaries. Lower panel: dots with
inblack. parameters (β−1,ι,Ω0) = (0.57,177◦,255◦), and squares with
(1.7,79◦,308◦).
remainunbound. Specifically, the survivingbinaries will
belong to one of the following types: binaries bound to
0.05 1
the SMBH with a Peters lifetime τ longer that their
gw
orbital period P• about the hole (i.e. long BEMRIs); 0.045 0.9
binaries also bound to the SMBH but that merge be- 0.04 0.8
fore completing an orbit around the hole, τgw < P• (i.e. 0.035 0.7 F
shortBEMRIs);andbinariesthatareunboundfromthe D
SMBH. We will analyse the distributions of orbital pa- on 0.03 0.6 y C
rameters(eccentricityandsemi-majoraxis)ofthesurviv- cti 0.025 0.5 cit
ingbinaries,withthe goalofdeterminingthe netchange Fra 0.02 0.4 entri
to merger lifetime and whether this leads to detectable cc
0.015 0.3 E
changes in the CBC rate observed by detectors such as
LIGO. But before doing so, we will briefly revisit the 0.01 0.2
effect of θ0 and Ω0. 0.005 0.1
In Section 4, we estimated probability distributions of
0 0
the parameters β,ι,θ ,Ω for each of the encounter −7 −6 −5 −4 −3 −2 −1 0
0 0
types. We observ{ed that th}e distributions for the phase log(Eccentricity)
ofthe binaryθ andthe longitude ofthe ascendingnode Fig.9.— Fraction of surviving binaries as a function of post-
0
Ω are to a good approximation flat. Thus, the param- encounter eccentricity as well as with a solid line the cumulative
0 distributionfunction(CDF)oftheeccentricityvalues.
eters β and ι have a more dominant role on the out-
come. This does not mean that choices of θ and Ω
0 0 In Figure 9, we show the fraction of surviving bina-
do not influence the end state. Figure 8 displays the
ries as a function of post-encounter eccentricity as well
eccentricity of the CO binary after the encounter with
asthe cumulative distribution function (CDF) of the ec-
the SMBH as a function of θ . In the upper panel,
we show with dots cases with p0arameters (β−1,ι,Ω ) = centricityvalues. Notethatthemajorityofthesurviving
(0.77,123◦,46◦), squares with (1.9,101◦,312◦), and0 di- binaries are relatively unperturbed in eccentricity, with
amonds with (4.6,49◦,152◦). Points where the curve the 65% quantile lying at approximately e ≈ 0.109 and
the 90% quantile at e 0.553.
jumps above 1 indicate disrupted binaries. Notice as ≈
Figure 10 show how the post-encounter eccentricity
expected that as the penetration factor increase, so the
depends on β−1 and ι. The surviving binary acquires
eccentricity gained by the binary as well as its depen-
eccentricity for larger penetration factors (smaller β−1)
dence of the initial binary phase. In the lower panel in
Figure 8 we show with circles parameters (β−1,ι,Ω ) = and prograde orbits (cosι > 0). Contours of constant
(0.57,177◦,255◦) and with squares have (1.7,79◦,3008◦). loge=−5.5,−5.0,−4.5,...,0fromrighttoleftareshown
inwhite. Blacklinesarethecorrespondingestimatesob-
Noticethatbinariesmakingacloserpass(circles)gener-
tained from Eq. 13 averaged over φ. Notice that the
ally suffers areducedperturbationto eccentricity due to
contour for loge = 0 is not present for the simulation
the nearly retrograde orbit.
data. All simulation values have been averaged over θ .
0
As expected, we see better agreement between the pre-
6.1. Post-Encounter Binary Eccentricity
diction from Eq. 13 and simulation results for large β−1
9
that E = E +E = E < 0, an increase in binary
1 0 cm b b,0
energy E must be compensated by a decrease in the
b
0.8 −0.5 BH orbit energy Ecm, resulting in capture of the binary.
0.6 −1 Therefore the points in Figure 13 where a/a0 >1 corre-
spond to binaries which become bound, while a/a < 1
−1.5 0
0.4 indicates binaries that remain unbound.
−2
0.2 7. PETERSLIFETIMEANDCBCRATES
ιos 0 −2.5 Asmentionedbefore,thePeterslifetimeisanestimate
c −3
of the time that a binary system with eccentricity e and
−0.2
−3.5 semi-majoraxis a takes to merge as it looses energy due
−0.4 −4 toemissionofGWs(Peters1964). AspresentedinEqua-
−0.6 −4.5 tion 20, the CO binary system in our study, with initial
semi-major-axis a0 = 10R⊙ and vanishing eccentricity
−0.8 −5 (i.e. circular orbit), has a Peters lifetime of τ 108
gw,0
−1 −5.5 years. Figure 14 shows the Peters lifetime τgw no≈rmal-
1 2 3 4 5 ized to τ for all of the surviving binaries. Notice the
β−1 gw,0
similarityofthemapswiththoseforthesemi-majoraxis
in Figure 12.
Fig.10.—Resultingeccentricitylogeforsurvivingbinariesplot-
ted as an intensity map vs. β−1 and ι. Contours of constant Unbound binaries that survive the encounter with the
loge=−5.5,−5.0,−4.5,...,0areshownfromrighttoleftinwhite SMBH,whatwecallSUbinaries,couldinprinciplehave
andfortheanalyticestimatefromEq.13inblack. ashorter(τ <τ )orlonger(τ >τ )Peterslife-
gw gw,0 gw gw,0
time. The last situation, however, is not possible given
and small ι since that is the regime where Eq. 13 was
our initial setup. An unbound binary requires increas-
derived. The small overestimation of the prediction at
ing the energy of its center of mass E . This will come
small inclination is consistent with what was found in cm
at the expense of decreasing its binding energy E since
Heggie & Rasio (1996) for an equal mass binary. b
E = E +E = E < 0, which in turn requires de-
Finally, Figure 11 shows gray scale maps of the post- cm b b,0
creasingitssemi-majoraxis. Adecreaseinitssemi-major
encounter loge against pairs of parameters β−1,ι, and
axistranslatesintoashorterPeterslifetime. Thus,allof
Ω , averagedoverθ . Noticeable are the different effects
0 0 the unbound binaries in our study will have accelerated
of the input parameters. As β−1 increases, the final ec-
merger times due to the SMBH
centricity decreases toward zero, however even binaries
with β−1 =3 can receive a non-negligible amount of ec- Werecallthatashort EMRIisoneforwhichτgw <P•.
Thatis,thebinaryisexpectedtomergebeforereturning
centricity, e 0.2, after the encounter. Inclination plays
∼ to the hole. On the other hand, a long BEMRI is one in
asmallerbutimportantroleindeterminingtheallowable
range of final eccentricities, with eccentricity decreasing which τgw > P•. These BEMRIs have typically highly
ellipticalorbits with averageeccentricity e 1 10−5
as the binary orbit approaches retrograde with respect cm
≈ −
and long periods. Therefore,
to the SMBH orbit. As stated before, the longitude of
the ascending node Ω0 plays no predictable role in the 2πa3/2 2πr3/2
perturbed eccentricity. P•= cm = p (1 ecm)−3/2
√GM• √GM• −
6.2. Post-Encounter Binary Semi-major Axis 2πr3/2
=β−3/2(1 e )−3/2 t
Semi-major axis of the surviving binaries is shown as cm
− √GM•
a histogramin Figure 12. The semi-majoraxis has been
normalized to its initial value. It is clear that the semi- =β−3/2(1 e )−3/22πa30/2
majoraxisis generallynotperturbedtoa strongdegree, − cm √GM
b
withtheoverallmeanbeing a/a =1.03. Ifthesample
h 0i =7.2 104yrβ−3/2
size is restricted to binaries with final semi-major axes
×
intherange0.5<a/a0 <1.5,whichaccountsfor>97% 1 ecm −3/2 Pb,0
of the data and removes the outliers, then the mean is − (37)
× 10−5 7.16 104s
a/a0 =0.988 and standard deviation of 0.084. (cid:18) (cid:19) (cid:18) × (cid:19)
h i
Figure13issimilartoFigure11butfora/a0. Itcanbe The CO binary in a long BEMRI could potentially sur-
seen here that for β−1 < 2 the binary can suffer a large viveseveralorbitsaroundtheSMBH,withitsorbitbeing
change to the semi-major axis, while above this value perturbed by each passage. We will investigate the fate
the change is small. Inclination has the general effect of of BEMRI systems will be the focus of a future study.
tightening the binary for ι < π/2, and loosening it for Independently of being inanBEMRI orunbound, CO
ι > π/2. There is some variation in the extent of semi- binaries that after an encounter with a SMBH are af-
majoraxisperturbationasafunctionofΩ , particularly fected in sucha way that τ <τ have particular in-
0 gw gw,0
when the inclination is around ι π/2 or at low values terest. They couldpotentially increasethe rates ofCBC
of β−1, with minimum change at≈Ω 0 or π. to be detected by LIGO. We construct a simple formula
0
Sincethebindingenergyofthebina≈ryisrelatedtothe to obtain a first estimate of this effect as
semi-major as E a−1, one can from Figure 13 iden-
tifyregionsofpabra∝m−eterspacewherethebindingenergy ECBC =[Γ∗fb∗NG(Dh)]∗(ET ∗fL), (38)
of the binary increases or decreases. Furthermore, given where istheenhancementtothecurrentpredicted
CBC
E
10
1 0
6 6
5 5 −1
0.5
4 4 −2
ιcos 0 Ω0 3 Ω0 3 −3
2 2
−0.5 −4
1 1
−5
−1 0 0
1 2 3 4 5 1 2 3 4 5 −1 −0.5 0 0.5 1
β−1 β−1 cos ι
Fig.11.—Grayscalemapsofpost-encounterlogeasafunctionofcombinationsofinputparameters. Allvalueshavebeenaveragedover
θ0.
0.1 stars, fb = 0.5. The number of MWEGs observable
by aLIGO with a BH-BH merger horizon distance
0.08 of D = 2187 Mpc is given as (Abadie et al. (2010);
h
Kalogeraet al. (2001))
on 0.06
acti N = 4π Dh 3 0.0116, (39)
Fr 0.04 G 3 Mpc (2.26)3
(cid:18) (cid:19)
0.02 which gives a value of N 4.4 107 MWEGs.
G
≈ ×
Fromoursimulationresults,wefindthatbinariesfrom
0 the f categories have a mean Peters lifetime of τ˜
0.4 0.6 0.8 1 1.2 1.4 1.6 L ≈
Semi−Major Axis ratio a/a 0.84τ0 giving T 0.16 and LIGO fraction fL 0.68.
0 E ≈ ≈
Combining these factors, we compute an estimated en-
Fig.12.—Histogramofthebinarysemi-majoraxisdistribution hancement to the CBC rate of 0.25 yr−1. The
CBC
after SMBH encounter. The horizontal axis has been stretched E ≈
predicted rate of expected BH-BH mergers has been es-
so that the few outliers did not dominate the plot scale, and the
vertical axis has been truncated to show detail with the central timated to lie between 0.4 MWEG−1 Myr−1 < ΓBH <
binhaving a true value of ∼0.69. It isclear that the majorityof 30 MWEG−1 Myr−1 for realistic to optimistic scenarios
encounter resultinverysmallchangeinsemi-majoraxis.
Kalogeraet al. (2007); Abadie et al. (2010). This corre-
CBC rate R , i.e., Rˆ = R + . The sponds to an estimated merger rate within the aLIGO
CBC CBC CBC ECBC volume of 20 yr−1 <N˙ <1300 yr−1, which our es-
first three terms in Eq. 38 are observational quantities BH
∼
timatedrateenhancement maychangebyasmuch
taken from the literature. Γ is the estimated encounter CBC
E
as 1%. This enhancementwill likelybe difficult to de-
rate of stellar mass objects / per SMBH per year, f is
b ≈
tectwithsmallobservationcatalogsandgiventheuncer-
the binaryfraction, andN (D )is the number ofMilky
G h
tainty in the estimated merger rates, but could become
Way equivalent galaxies (MWEGs) observable by a GW
noticeable over long observation times.
detectorwithhorizondistance,D . Theproductofthese
h
factors is the rate of binary encounters with a SMBH in 8. SUMMARYANDCONCLUSIONS
the observable volume of a GW detector.
In this paper, we have presented the results of 13
The next two factors come from our simulation ∼
millionindividualsimulationsofparabolicencountersbe-
results. The single binary enhancement =(1 τ˜/τ )
Eτ − 0 tween a CO binary and a galactic center SMBH while
is the percent difference between the old merger
varying the orientation of the binary and its distance of
lifetime, τ , and the mean new lifetime, τ˜, and
0 closest approach to the SMBH.
f is the fraction of binaries from our simulations
L Tidal disruption of the binary occurs with about 16%
that result in guaranteed LIGO sources after the
probability giventhe full range of parameterscoveredin
SMBH encounter. Encounter rates in galaxies with a
this study. Consistent with previous work in this area,
106M⊙ SMBH, estimates for the single CO capture this sort of disruption can create HVS which can escape
rate range from Γ 5 10−9 yr−1 MWEG−1 to as from the SMBH with high speed. We also explored dis-
high as Γ 10−∼6 yr−×1 MWEG−1 (Hils & Bender ruptionas aformationmechanismforEMRIs,whichare
∼
(1995); Sigurdsson & Rees (1997); Ivanov (2002); of interest to space-based GW detection missions, and
Hopman & Alexander (2005); Merritt et al. (2011)); found that the EMRIs formed in this way will gener-
we use the higher side of these estimates, allyhaveveryloweccentricitywhentheyenterthe LISA
Γ = 10−7 yr−1 MWEG−1, as the pericenter dis- band. Thisworkshowsthatconsideringthe fullrangeof
tances in our simulations reach much larger values than possible orientations gives a broader range of formation
the single star capture radius. The binary fraction, f , eccentricities than previous estimates have predicted.
b
near the galactic center is not a well determined value, Surviving binaries can either become bound to the
however, in the absence of better knowledge, we take it SMBH after the encounter and become a BEMRI or re-
to be roughly the same as the binary fraction of field main unbound from the hole. Among them, there is a