Interaction of massive black hole binaries with their stellar environment: II. Loss-cone depletion and binary orbital decay

0.27 MB·English

by  A. Sesana

ApJ,in press
PreprinttypesetusingLATEXstyleemulateapjv.12/14/05
INTERACTION OF MASSIVE BLACK HOLE BINARIES WITH THEIR STELLAR ENVIRONMENT: II.
LOSS-CONE DEPLETION AND BINARY ORBITAL DECAY
Alberto Sesana1, Francesco Haardt1, & Piero Madau2,3
ApJ, in press
ABSTRACT
Westudythe long-termevolutionofmassiveblackholebinaries(MBHBs)atthe centersofgalaxies
usingdetailedscatteringexperimentstosolvethefullthree-bodyproblem. Ambientstarsdrawnfrom
a isotropic Maxwellian distribution unbound to the binary are ejected by the gravitational slingshot.
We construct a minimal, hybrid model for the depletion of the loss cone and the orbital decay of
7
the binary, and show that secondary slingshots – stars returning on small impact parameter orbits
0
0 to have a second super-elastic scattering with the MBHB – may considerably help the shrinking of
2 the pair in the case of large binary mass ratios. In the absence of loss-cone refilling by two-body
relaxation or other processes, the mass ejected before the stalling of a MBHB is half the binary
n
a reduced mass. About 50% of the ejected stars are expelled in a “burst” lasting ∼104yrM61/4, where
J M6 isthebinarymassinunitsof106M⊙. Thelossconeiscompletelyemptiedinafew bulgecrossing

2 timescales, 107yrM1/4. Even in the absence of two-body relaxation or gas dynamical processes,
2 unequal ma∼ss and/or e6ccentric binaries with M & 0.1 can shrink to the gravitational wave emission
6
regimeinlessthanaHubbletime,andaretherefore“safe”targetsfortheplannedLaserInterferometer
3
Space Antenna (LISA).
v
Subject headings: black hole physics – methods: numerical – stellar dynamics
5
6
2
1. INTRODUCTION al. 2005). The coalescence rate of such “LISA MBHBs”
2
depends, however, on the efficiency with which stellar
1 It is now widely accepted that the formation and evo-
and gas dynamical processes can drive wide pairs to the
6 lution of galaxies and massive black holes (MBHs) are
0 strongly linked: MBHs are ubiquitous in the nuclei of GW emission stage.
/ nearby galaxies, and a tight correlation is observed be- Following the merger of two halo+MBH systems of
h
comparable mass (“major mergers”), it is understood
p tween hole mass and the stellar mass of the surrounding
thatdynamicalfrictionwilldraginthesatellitehalo(and
- spheroid or bulge (e.g. Magorrian et al. 1998; Gebhardt
o et al. 2000; Ferrarese & Merritt 2000; Haring & Rix itsMBH) towardthe centerofthe moremassiveprogen-
r 2004). IfMBHswerealsocommoninthepast(asimplied itor (see, e.g., Kazantzidis et al. 2005): this will lead to
st by the notion that distant galaxies harbor active nuclei the formation of a bound MBH binary in the violently
a relaxed core of the newly merged stellar system. As the
for a shortperiod of their life), and if their host galaxies
: binary separation decays, the effectiveness of dynami-
v experiencemultiplemergersduringtheirlifetime, asdic-
i tated by cold dark matter (CDM) hierarchical cosmolo- cal friction slowly declines because distant stars perturb
X the binary’s center of mass but not its semi-major axis
gies, then close MBH binaries (MBHBs) will inevitably
r formin largenumbersduring cosmic history (Begelman, (Begelman et al. 1980). The bound pair then hardens
a by capturing stars passing in its immediate vicinity and
Blandford, & Rees 1980). Observations with the Chan-
ejecting them at much higher velocities (gravitational
dra satellite have indeed revealed two active MBHs in
slingshot). It is this phase that is considered the bot-
the nucleus of NGC 6240 (Komossa et al. 2003), and a
tleneck of a MBHB’s path to coalescence, as there is a
MBHB is inferred in the radio core of 3C 66B (Sudou
finite supply of stars on intersecting orbits and the bi-
et al. 2003). The VLBA discovery in the radio galaxy
nary may “hung up” before the back-reactionfrom GW
0402+379ofaMBHBsystemwithaprojectedseparation
emission becomes important. This has become known
of just 7.3 pc has recently been reported by Rodriguez
as the “final parsec problem” (Milosavljevic & Merritt
etal.(2006). MBHpairsthatareable to coalescein less
2003,hereafter MM03).
than a Hubble time will give origin to the loudest grav-
While the final approach to coalescence of binary
itational wave (GW) events in the universe. In particu-
MBHsisstillnotwellunderstood,severalcomputational
lar,alow-frequencyspaceinterferometerliketheplanned
toolshavebeendevelopedtotackletheproblemathand.
Laser Interferometer Space Antenna (LISA) is expected
The orbital decay rate depends on severalparameters of
tohavethesensitivitytodetectnearlyallMBHBsinthe
mass range 104 107 M⊙ that happen to merge at any the guest binary (mass, mass ratio, orbital separation,
− and eccentricity), and on the stellar distribution func-
redshift during the mission operation phase (Sesana et
tion of the host galaxy bulge. In early treatments (e.g.
1DipartimentodiFisicaeMatematica,Universita´dell’Insubria, Mikkola&Valtonen1992;Quinlan1996,hereafterQ96),
viaValleggio11,22100Como,Italy. the stellar ejection rate and the rate of change of the
2DepartmentofAstronomy&Astrophysics,UniversityofCali- binarysemi-majoraxisandeccentricitywerederivedvia
fornia,1156HighStreet,SantaCruz,CA95064.
three-bodyscatteringexperimentsinafixedstellarback-
3Max-Planck-Institut fuer Astrophysik, Karl-Schwarzschild-
Strasse1,85740GarchingbeiMuenchen, Germany. ground. The assumption of a fixed background breaks
2 Sesana, Haardt, & Madau
down once the binary has ejected most of the stars on groundofstarsofmassm∗. Inthecaseofalightintruder
intersecting orbits, and the extraction of energy and an- with m∗ M2, the problem is greatly simplified by set-
≪
gular momentum from the binary can continue only if tingthe centerofmassofthe binaryatrestatthe origin
new stars can diffuse into low-angular momentum or- of the coordinate system. It is then convenient to de-
bits (refilling the binary’s phase-space “loss cone”), or fine an approximate dimensionless energy change C and
via gas processes (Escala et al. 2004; Dotti, Colpi, & angular momentum change B in a single binary-star in-
Haardt 2006). Hybrid approaches in which the rate co- teraction as (Hills 1983)
efficients derived from numerical experiments in a fixed
M ∆E a∆E∗
backgroundarecoupledwithamodelforloss-conerepop- C = = , (1)
ulation have been used, e.g., by Yu (2002) and MM03, 2m∗ E Gµ
whilethelimitingcaseinwhichthelossconeisconstantly and
refilled but the central stellar density decreases due to M ∆Lz M ∆Lz∗
B = = . (2)
mass ejection has been studied in a cosmological con- −m∗ Lz µ Lz
text by Volonteri, Haardt, & Madau (2003) and Volon-
Here ∆E/E is the fractional increase (decrease if neg-
teri,Madau,&Haardt(2003). Afullyself-consistent,N-
ative) in the orbital specific binding energy E =
body approachto theevolutionofMBHBs,whileclearly
GM/(2a), ∆L /L is the fractional change in orbital
desirable, is limited today to N . 106 particles, cor- − z z
Qreuspinolnadnin&gHteornaqmuiassts19re9s7o;lMutiiloonsaovfljmev∗i/cM&M∼er1r0i−tt32(0e0.g1.; ∆spEec∗ifiacndan∆guLlza∗ramreomtheenctourmreLspzo=ndingGcMhaan(g1e−sfeo2r),thwehiinle-
p
Hemsendorf,Sigurdsson,&Spurzem2002;Aarseth2003; teractingstar. ThequantitiesB andC areoforderunity
Chatterjee, Hernquist, & Loeb 2003; Makino & Funato andcanbederivedbythree-bodyscatteringexperiments
2004; Berczik, Merritt, & Spurzem 2005). Such per- that treatthe star-binaryencounters one at a time (Hut
formance figures are not sufficient to reproduce central &Bahcall1983;Q96). Foreachencounteronesolvesnine
bulges,evenoffaintgalaxies;andthesmallparticlenum- coupled, second-order,differential equations supplied by
bers cause an artificialenhancement of star-starscatter- 18initialconditions. Theinitialconditionsdefineapoint
ings and of the Brownian motion of the binary, leading inanine-dimensionalparameterspacerepresentedbythe
to a spurious refilling of the loss cone. mass ratio q = M2/M1 of the binary, its eccentricity e,
Thisisthesecondpaperinaseriesaimedatadetailed the mass of the incoming field star, its asymptotic ini-
study of the interaction of MBHBs with their stellar en- tial speed v, its impact parameter at infinity b, and four
vironment. In Sesana, Haardt & Madau (2006, here- angles describing the initial direction of the impact, its
after Paper I), three-body scattering experiments were initial orientation, and the initial binary phase. A sig-
performed to study the ejection of hypervelocity stars nificant star-binary energy exchange (i.e. characterized
(HVSs) by MBHBs in a fixed stellar background. In bya dimensionlessenergychangeC >1)occursonlyfor
this paper, we use a hybrid approach to investigate the v < V M /M, where V = GM/a is the binary or-
c 2 c
orbital decay and shrinking of MBHBs in time-evolving bitalvelocity(the relativevelocityofthetwoholesifthe
p p
stellar cusps. Numerically derived rates of stellar ejec- binary is circular, see e.g. Saslaw, Valtonen, & Aarseth
tions stars are coupled to an extension of the analytical 1974;Mikkola & Valtonen 1992).
formulationof loss-cone dynamics given by MM03. This A set of 24 scattering experiments was performed for
methodallowsustosimultaneouslyfollowtheorbitalde- different binary mass ratios and initial eccentricities,
cayofthepairaswellasthetime evolutionofthestellar each run tracking the orbital evolution of 4 106 stars.
×
distributionfunction. MBHBsareembeddedinthedeep The binary evolution in an isotropic stellar background
potential wells ofgalaxy bulges,so when the binary first ofdensityρandone-dimensionalvelocitydispersionσ at
becomes “hard” only a few stars acquire a kick velocity infinity is determined by three dimensionless quantities
large enough to escape the host. The bulge behaves as (Q96): the hardening rate
acollisionlesssystem,andmanyejectedstarswillreturn
σ d 1
to the central region on nearly unperturbed, small im- H = , (3)
pact parameter orbits, and will undergo a second super- Gρdt a
(cid:18) (cid:19)
elastic scattering with the binary, as first discussed by
the massejectionrate(M isthe stellarmassejectedby
ej
MM03. Under the assumption of a spherical potential,
the binary)
we quantify the role of these “secondary slingshots” in 1 dM
ej
determining the hardening of the pair. The plan of the J = , (4)
M dln(1/a)
paper is as follows. In 2 we describe our hybrid model
§
for the orbital evolution of MBHBs in a time evolving and the eccentricity growth rate
stellar density profile. The shrinking and coalescence of
de
the binary is discussed in 3. K = . (5)
§ dln(1/a)
2. HARDENINGINATIME-EVOLVINGBACKGROUND
The hardening rate H is approximatively constant for
2.1. Scattering experiments
separations smaller than the “hardening radius”,
Our hybrid method relies on the large number of out-
GM
putsfromthe suiteofthree-bodyscatteringexperiments a<a = 2 (6)
presented in Paper I. In the following, we briefly sum- h 4σ2
marize the basic theory. Consider a binary of mass (Q96). Thebinaryisassumedtobeembeddedinabulge
M = M +M = M (1+q) (M M ), reduced mass of mass M , radius R , and stellar density profile ap-
1 2 1 2 1 B B
≤
µ=M M /M,andsemimajoraxisa,orbitinginaback- proximated by a singular isothermal sphere (SIS). Stars
1 2
Massive black hole binaries 3
binary separation. The degree of anisotropy is smaller
forunequalmassbinariesandlargerforstarswithhigher
kickvelocities. EccentricMBHBsproduceamorepromi-
nent tail of high-velocity stars and break axisymmetry,
ejecting HVSs along a broad jet perpendicular to the
semimajoraxis. Thejettwo-sidednessdecreaseswithin-
creasing binary mass ratio, while the jet opening-angle
increases with decreasing kick velocity and orbital sepa-
ration.
2.2. Loss-cone time evolution
In the absence of loss-cone refilling by two-body re-
laxation or other processes, the supply of stars that can
interact with the black hole pair is limited. Analytic ex-
pressions for non-equilibrium loss-cone dynamics based
on the evolution of the stellar distribution function as a
result of repeated ejections have been given in MM03.
Hereweadoptahybridapproachinstead,combiningthe
results of scattering experiments with an extension of
MM03’s study.
Fig. 1.— Velocity diagram of scattered stars in three different
speed ranges: 3Vc <V <3.2Vc (black vectors), 4Vc <V <4.5Vc 2.2.1. Stellar content
(greenvectors),andV >5Vc(redvectors). Upperpanel: longitude
diagram of scattered stars. Each vector length is proportional to The stellar content of the loss cone can be esti-
the modulus of the star’s total velocity (not to the velocity pro-
mated from simple geometrical considerations. When
jected into the xy plane). The blue ellipse shows the counter-
clockwiseorbitofthelighterblackholeofthebinary. Lowerpanel: the MBHB separation is a < ah, only a small fraction
latitudediagramofscattered stars. of bulge stars have low-angu∼lar momentum trajectories
are counted as “ejected” from the bulge if, after three- with pericenter distance rp < a. In a spherical velocity
body scattering, their velocity V far away from the bi- distribution, the fraction of trajectories originating at r
nary is greater than the escape velocity from the ra- and crossing a sphere of radius a < r around the center
dius of influence of the binary, r GM/(2σ2). The is
inf
SIS potential is φ(r) = 2σ2[ln(GM≡ /2σ2r) +1] (for a 2
r <RB =GMB/2σ2), an−d the escapeBspeed from rinf is Θ(r)="1−r1− r #. (9)
then (cid:16) (cid:17)
The stellarmasswithinthe geometricallosscone isthen
v 2φ(r )
esc ≡ − inf (7) a RB
=p2σ [ln(MB/M)+1]=5.5σ, M∗ = 4πr2ρ(r)dr+ 4πr2ρ(r)Θ(r)dr. (10)
Z0 Za
wherethesecondeqpualitycomesfromtheadoptedbulge- For a SIS ρ(r)=σ2/(2πGr2), and equation (10) is read-
black hole mass relation M =0.0014MB (Haring & Rix ily integrated to yield in the limit a RB
2004). Stars that do not acquire a kick velocity large ≪
πσ2 π a
enough to escape the host bulge, i.e. with M∗ a= M2. (11)
≃ G 4 a
(cid:18) h(cid:19)
V <v = 2φ(R ) 2φ(r )
ret B − inf (8) The abovescheme is oversimplified,as it assumesstel-
=p2σ ln(MB/M)= 5.1σ, lar trajectories to be straight lines. The gravitational
field of the stellar mass distribution increases the net
are allowed multiple intepractions with the binary. They numberofdistantstarswithpericenterdistancesr <a.
p
can return to the central regions on small impact pa-
Consider a star at distance r > a from the binary mov-
rameter orbits and undergo a second super-elastic scat-
ing with random velocity v. For an SIS, conservation of
tering (“secondary slingshots”). Secondary scatterings
energy gives:
are not allowed for stars ejected with 5.1σ <V <5.5σ,
v2 =v2+4σ2ln(r /r), (12)
sinceevenasmalldeviationfromsphericity∼ofthe∼galaxy p p
gravitationalpotentialwouldmakethemmisstheshrink- where v is the star’s velocity at pericenter. If b is the
p
ing MBHB on their return to the center. Note that, impactparameteratdistancer,angularmomentumcon-
even if they were able to undergo another interaction servation yields
with the binary, such stars would not contribute sig- b2 =r2[1+4(σ2/v2)ln(r/r )]. (13)
nificantly to binary hardening as long as the condition p p
The second integral on the right-hand-side of equation
V > V M /M = 2σ a /a is satisfied, since in this
c 2 h
(10) can then be rewritten as
case the star-binary energy exchange would be negligi-
p p
ble. RB
As shown in Figure 1 and discussed in details in Pa- 4πr2ρ(r)Θ(r)
per I, three-body interactions create a subpopulation of Za
∞
HVSs on nearly radialorbits, with a spatial distribution 4πv2f(v)[1+4(σ2/v2)ln(r/a)]dv dr,
that is initially highly flattened in the inspiral plane of
(cid:26)Z0 (cid:27)
the binary, but becomes more isotropic with decreasing (14)
4 Sesana, Haardt, & Madau
where f(v) is the stellar velocity distribution. For a
Maxwellian,the aboveequationcanbesimplifiedbyset-
tingv = v =√3σinequation(13): onecanthendefine
h i
a Θ-factor that includes gravitationalfocusing
a 2 4 r
Θ(r) Θ(r) 3.1 1 1 1+ ln .
→ ≃ " −r −(cid:16)r(cid:17) # (cid:20) 3 (cid:16)a(cid:17)(cid:21)
(15)
Numerical integration of equation (10) finally yields for
the stellar mass in the loss cone
8.2σ2 a
M∗ a 2 M2. (16)
≃ G ≃ a
(cid:18) h(cid:19)
Note that a fraction 0.5M of the mass contained in
2
∼
the loss cone when the binary becomes hard (a = a ,
h
M∗ 2M2) lies within ah. Let t = 0 be the time at
≃
which the binary separation is a=a . The number flux
h
of stars into the geometrical loss cone, i.e. the flux of
stars with r >a at t = 0 that interact with the binary
h
at a later time t, is Fig. 2.— Decay of binary separation a (in units of ah) as
a function of time [in units of the bulge crossing time P(0) =
2√3σ3 1.32×107 yrM1/4 for a SIS]. The curves, from bottom to top,
6
Θ, (17) are for q = 1/243,1/81,1/27,1/9,1/3,1. The q = 1 case is com-
F ≃ Gm∗πa2h paredto the analytical estimate (long-dashed line) of MM03, and
to an N-body simulation (short-dashed line) of MM03 performed
where Θ=Θ(√3σt+ah). with18,000starsinitiallyinthelosscone,andthestellarpotential
replaced by a smooth component to prevent relaxation. We set
2.2.2. Energy exchange M =0.0014MB andusetheM−σ relation(eq.24).
Let us denote with the total binding energy of the MM03’s analysis can be expanded to account for the
MBHB, =GM M /E2a. The total energy transfer rate effect of three-body scatterings when the MBHB first
1 2
from theEbinary to stars in the loss cone can be written becomes hard at separation a = ah. Stars with r < ah
as at t=0 will interact with the binary within a timescale
t a /(√3σ). Substitution of t in equation (18),
ch h ch
d ∆E∗(a)M∗(a) follo∼wed by simple algebra, leads to the expression
E , (18)
dt ≃ t
ch
a q √3σ
h
where t is a characteristic interaction timescale and =1+C t. (21)
ch a 1+q a
∆E∗(a) Gµ/aisthecharacteristicspecificenergygain (cid:18) (cid:19) h !
∼
of stars as a consequence of the gravitational slingshot.
Afurthercontributiontotheshrinkingisassociatedwith
MM03 have written equation (18) in the case of re-
stars having r >a at t=0 that are bound to enter the
turning stars, i.e. kicked stars that do not escape the h
loss cone at later times. This population has total mass
host bulge and can have a secondary super-elastic in-
1.5M , and its contribution to the orbital decay is
teraction with the MBHB. Returning stars have energy ∼ 2
given by
tEch(ac)an∼bφe(ridinefn)t+ifi∆edEw∗(itah),thanedtytphieciarlirnatdeiraalcptieorniotdimofesstcaarles dE ∆E∗(a)m∗ πa2. (22)
dt ≃ F
in an SIS potential,
A straightforwardsubstitution gives
t P(E)=P(0)exp(E/2σ2)
ch
∼
=P(0)(M/MB)exp(cid:20)(cid:18)12+Cq(cid:19)(cid:16)aah(cid:17)−1(1(cid:21)9.) ddt(cid:16)aah(cid:17)=C(cid:18)1+q q(cid:19) √a3σ! Θ(√3σt+ah). (23)
The above equation holds for a bulge crossing time, t <
Here,P(0)=√πGMB/2σ3 isoforderthebulgecrossing RB/√3σ, and must be solved numerically.
time, and the average dimensionless energy change C
is of order unity and nearly independent of a for a < 2.2.3. Orbital decay
ah. As noted by MM03, in this case M∗(a)∆E∗(a) Thesimpleanalyticalformulationdescribedabovecan
a1a−1 const. From equation (16) simple calculation∝s be refined using results from our scattering experiments
∼
lead to (PaperI).Thisallowsustofollowatthesametimeboth
a a 1+q q (t t ) the orbital decay of the binary and the evolution of the
h h + ln 1+8C2 − 1 , (20) distributionfunctionofinteractingstars. The procedure
a ≃ a 2C (1+q)2 P
1 (cid:20) 1 (cid:21) is the following. We first isolate, from the initial distri-
where a is the binary separation at time t = t when bution of kicked stars, the new loss cone, i.e. the subset
1 1
secondary slingshots start, and P is the period of stars of stars returning to the center on orbits with r < a ,
1 p 1
with energy E(a ). where a is the binary separation at the end of the first
1 1
Massive black hole binaries 5
binary. Thelowerpanelclearlyillustrateshowsuccessive
TABLE 1
Binaryhardening: theimpactof returningstars interactions of stars returning on quasi radial orbits can
reduce the final binary separation by an extra factor of
q ah/a1 ah/af order 2, i.e. af a1/2. The progressiveemptying of the
1 2.81 4.41 loss cone is sket∼ched in the upper panel, where we plot
1/3 2.19 3.07
the (differential) mass in stars approaching the binary
1/9 1.64 2.09
with a given periastron. After the first interaction only
1/27 1.29 1.49
few stars are kicked out from the bulge. The loss cone,
1/81 1.12 1.19
1/243 1.04 1.06 while substantially hotter, remains nearly full and only
gets progressively depleted as secondary slingshots take
Note. —Binaryshrinkingfactors. ah/a1isthebinaryshrinking
after the first interaction only, while ah/af take into account for place. Table 1 quantifies the role of returning stars for
subsequentreejections uptothefourthinteraction. different values of the binary mass ratio q: returning
starscanincreasetheshrinkingoftheMBHBbyasmuch
interaction with the stellar background. Then we com- as a factor of 2, and play a larger role for equal mass
putethehardeningrateH byaveragingH (v)(provided binaries. This is because binary-to-starenergy exchange
1
byourscatteringexperiments)overthevelocitydistribu- is significant only for V . √qVc (see Paper I). After
tion function of suchstars,which are allowedto interact the firstbinary-starinteraction,the stellar population is
with the binary for the timescale in equation(19), again heated up and stars have on average V √qVc. In the
∼
averagedoverthestellarvelocitydistribution. Aftereach case q =1, the binary shrinks by a significant factor, Vc
step,the velocitydistributionfunctionofreturningstars increases, and most of the returning stars have V < V :
c
is updated, and so is the timescale of the following in- thehardeningprocessisstillefficient. Bycontrast,∼when
teraction. We iterate the process until the loss cone is q 1, V does not increase appreciably after the first
c
≪
emptied. Convergence to the final stalling separation is interaction, returning stars have V √qVc, and binary
∼
usually obtained after & 4 iterations. The mathemati- hardening stops.
cal details of the numerical procedure are given in the
Appendix. 3. DISCUSSION
In order to specify the two parameters defining the Undertheassumedcriterionforstellarejection,wecan
SIS, the Haring & Rix (2004) bulge-black hole mass re- compute M , the mass of stars expelled with V > v .
ej esc
lation was complemented by the M σ relation (Fer- An example of the effects of the slingshot mechanismon
−
rarese & Merrit 2000; Gebhardt et al. 2000) proposed the stellar population is shown for an equal mass circu-
by Tremaine et al. 2002: lar binary in Figure 4, where the initial (t = 0) velocity
distributionofinteractingstarsiscomparedtothedistri-
σ70 =0.84M61/4, (24) bution after loss-cone depletion. As already mentioned
intheprevioussection,afterthefirstinteractionwiththe
where σ is the stellar velocity dispersion in units of 70
70
km/s, and M6 is the MBHB mass in units of 106 M⊙. binary a large subset of kicked stars still lies in the loss
cone of the shrinking binary, has velocities v <v , and
Using equation (24), the hardening radius can then be ret
is potentially avaliable for further interactions. These
written as
are the stars we termed “returning”. While scattering
a 0.32 pc M1/2 q , (25) with the binary increases the stellar velocity thus reduc-
h ≃ 2,6 1+q ing the energy exchanged in secondary interactions, it
r
moves kicked stars on more radial orbits thus reducing
where M2,6 M2/106M⊙. The binary separation as a their impact parameter as well. Our calculations show
≡
function of time is shown in Figure 2, where it is also
thatthe highvelocitytailofthe distributiondepends on
compared to the results of an N-body simulation and
theMBHBeccentricitye,althoughtheeffectofchanging
to an analytical prescription, both presented in MM03
e is small for small values of q. In this case, fewer stars
(their Fig. 6). The agreement with the simulation is arekickedoutcomparedto the caseq .1,but athigher
fairly good. Our results, in terms of the final separation
velocities on the average. In general, both a small mass
achievedas a function of q, are perfectly consistent with
ratio and a high eccentricity increase the tail of HVSs.
the stalling radii estimated by Merritt (2006).
Integratingthe curvesinFigure 4 overvelocity givesthe
Figure 2 shows how the rate of orbital decay de-
massofinteractingstars: thisis 2M forthecaseshown
clines after a few bulge crossing times P(0) = 1.32 ≃
(q = 1), 1.2M for q = 1/3, and 0.6M for q = 1/27
∼ ×
107 yr M1/4: this is due to the decreasing supply of low (allassum≃ing e=0; we checkedtha≃t eccentricity playsa
6
angular momentum stars from the outer regions of the negligible role).
bulge, once the stars in the central cusp have interacted Figure5depictstheejectedmassM normalizedtoM
ej
with the binary. Note that equal mass binaries shrink (leftscale)andtoM (rightscale),asafunctionofq. Our
2
more than unequal binaries: this is both because of the resultsshowthatM /M 0.5µ/M =0.5q/(1+q)2,i.e.,
ej
∼
scaling of the stellar mass available for the interaction M /M 0.5/(1+q), both ratios being independent of
ej 2
∼
andoftheenergyexchangedduringatypicalthree-body thetotalbinarymass. Therateofstellarmassejectionis
encounter. It is easy to see that d(1/a)/dlnt q. Ec- showninFigure6asafunctionoftime. Afraction.50%
∝
centricityplaysamarginalroleintheorbitalevolutionof of the expelled stars is ejected in a initial burst lasting
the MBHB. For a given q, the orbital shrinking is larger a /σ, this fraction being smaller for smaller binary
h
∼
by at most 10% for highly eccentric binaries. mass ratios. The burst is associated to the ejection of
Figure 3 shows an example of the role of secondary those stars already present within the geometrical loss
slingshotsonorbitalshrinkingforanequalmasscircular cone when the binary first becomes hard. Note that, for
6 Sesana, Haardt, & Madau
as M1/4). Wemustthencomparea totheseparation
f
∝
at which the orbital decay timescale from GW emission,
5c5 a4
t =
GW 256G3M M MF(e)
1 2
−1 4
MM M a
0.25Gyr 1 2 F(e)−1
≈ (cid:18)1018.3 M⊙3(cid:19) (cid:18)0.001pc(cid:19)
(26)
(Peters 1964), is shorter than, say, 1 Gyr. Here, to 4th
order in e,
73 37
F(e)=(1 e2)−7/2 1+ e2+ e4 . (27)
− 24 96
(cid:18) (cid:19)
Inverting equation (26), one can define the separation
a at which the binary will coalesce in a given time t,
GW
256G3 1/4
a = tM M MF(e)
GW 5c5 1 2
(cid:20) (cid:21) (28)
Fig. 3.— Upper panel: evolution of the loss-cone population in 1/4
MM M
termsof the(differential) stellarmassthat approaches the binary 0.0014pc 1 2 F(e)1/4t1/4,
fwroitmhatogpivteonbpoetrtioamst:rolnosrsp-c.oTnheinpolpinuel:atiinointiaalftloersstchoen1e.stS,o2lnidd,lin3reds,, ≈ (cid:18)1018.3 M⊙3(cid:19) 9
and4thinteraction. Lower panel: binaryseparationasafunction where t t/1Gyr. Using equations (25) and (28),
9
of time. From bottom to top, the curves depict the shrinking as- one finds t≡hat a /a M−1/4q3/4 (see also MM03,
sociated withonly the firstone, two, three, and fourinteractions, f GW
∝
respectively. Anequalmass,circularbinaryisassumed. eq.90),i.e. the moremassivethe binaryandthe smaller
the binary mass ratio, the smaller the factor the binary
must shrink to reach the GW emission regime. Eccen-
tricity plays a double role. For a given binary mass and
mass ratio, on one hand the hardening rate slightly in-
creases with increasing eccentricity (. 20% from e = 0
to e = 0.9), leading to a smaller a ; on the other hand,
f
from equations (27) and (28), a is larger for larger
GW
e thus reducing the a -a gap. An important effect,
h GW
includedinourcalculations,isthattheeccentricitytypi-
callyincreasesduringthebinary-starinteraction,though
the functional form of F(e) is such that the effect is sig-
nificant only for binaries with e & 0.6 already at a . In
h
otherwords,forMBHBs withinitially low eccentricities,
theincreaseofeduringthegravitationalslingshotaffects
only weakly the final a /a ratio.
f GW
It is interesting to compare the total mass actually
ejected prior to complete loss-cone depletion with the
stellar mass that must be expelled in order to reacha fi-
nalorbitalseparationwheret (a )=1Gyr,i.e. where
GW f
GW emission leads to coalescence within 1 Gyr. An ex-
ample is shown in Figure 5. The shaded areas define
Fig. 4.—Stellarvelocitydistributionforanequalmass,circular such mass (in units of M) for a 106 M⊙ and a 109 M⊙
binary, at different stages of binary hardening. The vertical lines
MBHB, where the top boundary assumes e=0 and the
tmoaprtkovbroettt=om5,.1dσist(rwibeurteicoanllotfhsatatrvseisnct=he5s.5hσri)n.kDinagslhoesdsclionnees:befrfoorme bottom e = 0.9. Note how the e = 0.9,M = 106 M⊙
the1st,2nd,3rdand4thiteration. Forclarity,theinitiallosscone lowerboundarypracticallycoincideswiththee=0,M =
distribution is marked with a thicker line. Thin solid lines: from 109 M⊙ upper one. The figure clearly shows how, even
toptobottom, distributionofstarsthat havereceived1,2, 3and
inthe absence ofother mechanismdrivingorbitaldecay,
4 kicks. Thick solid line: final stellar velocity distribution after
loss-conedepletioniscompleted. pairs involving genuinely supermassive holes should not
stall, while for lighter binaries both a small mass ratio
small q, mass ejection is already significant at a a , anda largeeccentricityareprobablyrequiredfor coales-
h
≃
as in this case the binary orbital velocity is V &v for cence to take place.
c esc
a.a . Using our hybrid model, we can also sample the
h
Is the amount of ejected mass sufficient to shrink the (M ,q,e) 3-D space, compute the separation a and the
1 f
MBHB orbit down to the GW-dominated regime? To eccentricity e at a , then fold the calculated values of
f
answerthis question,we startdefining the “finalsepara- a and e into equation(26), and finally comparet to
f GW
tion” a as the separation reached by the binary before the Hubble time at two reference redshifts, z = 1 and
f
completeloss-conedepletion,i.e. afterafewbulgecross- z = 5. In Figure 7, binaries that will coalesce within a
ing time ( 107 yrs, weakly depending on binary mass then Hubble time after loss-cone depletion populate the
∼
Massive black hole binaries 7
Fig. 5.—EjectedstellarmassMejnormalizedtothetotalbinary Fig. 7.— M1−q plane. The vertical shaded area shows LISA
mass M (left scale, solid points), and to the mass of the lighter potential targets with S/N >5. Thediagonal shaded areaonthe
binary member M2 (right scale, empty points), as a function of lower right corner marks binaries that will coalesce within a then
binarymassratio. Thecurvesarepolynomialinterpolations. Note Hubbletimeafterloss-conedepletion. Ineachpanel,theassumed
thattheratiosMej/M andMej/M2donotdependontheabsolute redshiftandeccentricity oftheMBHBsarelabeled.
valueofM,andarenearlyindependent on e. Upper, dark shaded
area: themass(normalizedtoM)aM =106M⊙ binaryneedsto
eject to reach a final separation af such that tGW =1 Gyr. Top It is important to remark, at this stage, that our cal-
and bottom boundaries assume e = 0 and e = 0.9, respectively. culations are meant to define a minimal model for the
Lower, light shaded area: samebutforaM =109M⊙ binary. evolutionofMBHBs,andthatseveralothermechanisms
mayhelptheorbitaldecayandwidentherangeofpoten-
tialLISAtargets. First,wehaveassumedallthestarsin
the loss cone to be unbound to the MBHs. In a realistic
case, each MBH will bind stars inside its radius of influ-
ence r : the star binding energy can be extracted by
inf
slingshot,henceenhancingbinaryhardening. Thiseffect
is not expected to be important for equal mass binaries
as,inthiscase,a r ,andonlyasmallfractionofin-
h inf
∼
teracting stars will be bound to the binary. Indeed, our
results match well the numerical simulations of MM03
(Fig. 2). For lower mass ratios, however, it is a r
h inf
≪
andmoststarsintheloss-coneareactuallyboundtothe
binary. A forthcoming paper will be devotedto an anal-
ysis of three-body scattering experiments for a MBHB
with bound stars, providing a more realistic model for
the case q 1.
≪
Second,eveninsphericalstellarbulges,loss-conerefill-
ingduetotwo-bodyrelaxation(Yu2002;MM03)andthe
wanderingofthe blackhole pairin the nucleus (Quinlan
& Hernquist 1997; Chatterjee et al. 2003) could both
increase the amount of stellar mass interacting with the
Fig. 6.—Ejectedstellarmassperunitlogarithmictimeinterval,
MBHB. The two-body relaxation timescale is such that
as a function of time. A circular binary is assumed. Solid line:
q = 1. Long-dashed line: q = 1/3. Short-dashed line: q = 1/9. loss-conerefillingisprobablyimportantforM .106M⊙.
Dot-dashed line: q=1/27. TheBrownianmotiontimescaleisofthe orderof15Gyr
fora106M⊙ binary,andscalesasM5/4. Itis thenlikely
diagonally-shaded area in the M q plane, while the that the two effects considered here may affect orbital
1
−
vertically-shaded area marks MBHBs that, if driven to decay only for light binaries, helping them to cover the
coalescence by z = 1 or z = 5, would be resolved by residualgapbetweena anda andleadinglightbina-
f GW
LISA with a signal-to-noise ratio S/N > 5 (see Sesana ries to coalesce within a then Hubble time even at high
etal.2005andreferencesthereinfordetails). Theregion redshifts. On the other hand, their contribution to the
ofoverlapselectsunequalmass,highlyeccentricMBHBs shrinking of supermassive binaries with M & 106M⊙ is
withM >105 thatcanshrinkdowntotheGWemission probably negligible.
regime in∼less than an Hubble time, and that are “safe” If the stellar bulge is not spherical, but axisymmetric,
targets for LISA even in the pessimistic case, treated stars on highly eccentric orbits are typically centrophilic
here,ofstellarslingshots+loss-conedepletionwithnore- (Touma&Tremaine1997;Magorrian&Tremaine1999).
filling. Inthiscase,thelossconeissubstitutedbya“losswedge”
8 Sesana, Haardt, & Madau
(see Yu 2002 for a detailed discussion). The stellar con- diskdrasticallyincreasesdynamicalfriction,reducingthe
tent of suchwedge is largerthan that of the correspond- timescale on which MBHs can reach the center of the
ing loss cone, and depends on the degree of flattening ǫ bulge (Escala et al. 2004; Dotti et al. 2006). On par-
of the stellar distribution. Typically, for a galaxy with sec scales, torques induced by the disk can drive the bi-
ǫ=0.3,the stellarcontentofthe losswedgeis aorderof nary to decay on a timescale of order the gas accretion
magnitudelargerthanthestellarcontentofthelosscone time (Ivanov et al. 1999; Armitage & Natarajan 2002).
were the bulge spherical (Yu 2002). Note that Faber et It is important to point out that the interaction with a
al. (1997) estimate an average ǫ = 0.36 for a sample of gaseousdisktypicallycircularizesthebinaryorbit(Dotti
galaxies,leading to the conclusion that the hardening of etal.2006),hencemaximizingthea a gap. Ifthisis
h GW
−
a MBHB in such a potential might be much faster than the case,the slingshotdrivencoalescencewouldbe more
our“spherical”estimate. We alsorecallthattriaxialpo- difficult to achieve.
tentials drive many bulge stars on chaotic orbits, many
ofthemcentrophilic: onethenexpectsanincreaseinthe
number of interacting stars similar to that produced by
axisymmetric potentials (Merritt & Poon 2004; Berczik Support to this work was provided by the Italian MIUR
et al. 2006). grant PRIN 2004 (A.S. and F.H.), and by NASA grants
Finally,MBHBorbitalevolutioncanalso,atleastpar- NAG5-11513 and NNG04GK85G (P.M.). P.M. also ac-
tially, be driven by drag in a gaseous nuclear disk. The knowledges support from the Alexander von Humboldt
role of gas is, basically, twofold. On 100 pc scales, the Foundation.
APPENDIX
NUMERICAL INTEGRATION OF THE MBHB ORBITAL DECAY
The average hardening rate for a Maxwellian stellar velocity distribution f(v,σ) = (2πσ2)−3/2 exp( v2/2σ2) is
−
given by
∞
σ
H(σ) f(v,σ) H (v)4πv2dv, (A1)
1
≡ v
Z0
where
∞
H (v) 8π C xdx (A2)
1
≡ h i
Z0
isthedimensionlesshardeningrateifallstarshavethesamevelocityv,x b/ 2GMa/v2 isthedimensionlessimpact
≡
parameter, and the energy exchange C is averagedover the orbital angular variables (Paper I; Q96). An expression
h i p
analogous to equation (A1) relates the thermally-averagedeccentricity growth rate K(σ) to K (v):
1
∞
(1 e2) B C xdx
K1(v) − 0 h∞ − i , (A3)
≡ 2e C xdx
R 0 h i
where B C is the mean angular momentum minus enerRgy exchange. For a binary with given mass ratio and
h − i
eccentricity, the quantities C and B, and thus H and K , are only function of the ratio v/V v√a, where V is the
1 1 c c
binary circular velocity. Given an incoming velocity v, we record the bivariate distribution h∝(V,b′ v) of stars with
1
ejectionspeedsintheintervalV, V +dV,leavingthebinarywithan“exit”impactparameterintheint|ervalb′, b′+db′.
The distribution function is normalized so that
∞ ∞
h (V,b′ v)dV db′ =1. (A4)
1
|
Z0 Z0
The subscript “1” indicates that the scattering experiments are performed for a binary at separation a=1.
The interaction of the MBHB with stars in the loss cone is assumed to take place in discrete steps. The binary
first interacts with a givenpopulation of stars and shrinks accordingly. We then isolate the returning sub-population,
which becomes the input for the next step, and so on. Consider the binary at separationa , interacting with a stellar
i
population of mass M∗,i and (normalized) velocity distribution fi(v). The orbit decays according to the differential
equation
d 1 Gρ
= H(a), (A5)
dt a <v >
(cid:18) (cid:19)
where
∞ <v >
H(a) = f (v) H (v√a)dv, (A6)
i 1
v
Z0
and ρ is the stellar density. Straightforwardintegration of equation (A5) gives
<v > ai da′
t(a) = , (A7)
Gρ a′2H(a′)
Za
Massive black hole binaries 9
the time the orbit needs to shrink from a to a. The solution above is not physically meaningful, as the time variable
i
involveddepends on the particular value assumed for ρ. Stated more directly, to solve equation (A5) we need to set a
realistic pace at which star-binary interactions occurs. This can be done in two steps.
First, we write the stellar mass M∗ that will interact with the binary in a given time t as
M∗(t)=k∗t, (A8)
where
∞
k∗ ≡ fi(v)πb2max(v)ρvdv, (A9)
Z0
and the maximum allowed impact parameter is
2GM
b2 (v)=a2 1+ . (A10)
max i a v2
(cid:18) i (cid:19)
Equation (A8) is valid as long as M∗(t) M∗,i. Now we simply write t =M∗/k∗, and substitute into equation (A7),
≤
which now gives M∗(a), the stellar mass interacting with the binary as the orbit shrinks from ai to a. Note that in
M∗(a) term ρ cancels out, i.e., the interacting mass is independent of any pre–assigned value for the stellar density.
We can now numerically invert the equation for M∗(a), obtaining a(M∗), i.e., the binary separation as a function
of the interacting mass. We define the final separation af a(M∗,i). The very same procedure can be applied for
≡
the evolution of the binary eccentricity e, resulting in a function describing e as a function of the interacting mass,
e=e(M∗).
We need now to relate M∗ to physical time. The stellar mass interacting with the binary per unit time is
dM∗ dM∗dv dv
= = fi(v)M∗,i , (A11)
dt dv dt dt
where the term dv/dt can be computed considering the typical interaction time for stars with velocity v in a SIS
potential, i.e., t(v) = P(0)exp[(v2 v2 )/4σ2]. Straightforward algebra yields dv/dt as a function of t. Equation
− ret
(A11) can be then integrated, and the resulting M∗(t) finally substituted into a(M∗) to give the time evolution of the
binary separation a(t).
We can now compute the distribution of stars that, after the interaction, have velocities V < v , and are then
ret
available for a further encounter with the MBHB. The bivariate distribution h (V,b′ v) can be extracted from the
a
h (V,b′ v) distributions recorded in the scattering experiments. In the three-body pr|oblem integration, the binary
1
|
mass and separationare taken as unity. This set-up allows to rescale the velocities and trajectories of kicked stars for
any given physical value of the binary mass and separation. It can be shown that
1 V
h (V,b′ v)= h ,b′av√a , (A12)
a 1
| √a × √a |
(cid:18) (cid:19)
where the prefactor 1/√a normalizes the distribution according to equation (A4). The normalized distribution of
scattered stars must be averagedover the input velocity v and the separation a, and can be written as
h(V,b′)= aaif 0∞fi(a,v)(dM∗/da)ha(V,b′|v)dvda . (A13)
0∞ 0∞RaaifR0∞fi(a,v)(dM∗/da)ha(V,b′|v)dvdadV db′
Here, the distribution of stars withRvelRocitRy beRtween v and v +dv that are going to approach the binary within a
distance <a is
f (v)πb2(v,a)
i
f (v,a)= , (A14)
i f (v)πb2 (v)dv
i max
where
R
2GM
b2(v,a)=a2 1+ , (A15)
av2
(cid:18) (cid:19)
andbmax(v)isgivenbyequation(A10). ThechangeoftheinteractingmasswithbinaryseparationdM∗/daisobtained
by differentiation of the function M∗(a), obtained above.
Finally, the (normalized) velocity distribution of returning stars f (v) can be computed as
r
af h(V,b′)db′
f (v)= 0 , (A16)
r vret af h(V,b′)db′dV
0 R 0
and the mass avaiable for the subsequent interactioRn M∗R,r is given by
vret b(v,af)h(V,b′)db′dV
M∗,r =M∗,i 0 ∞0 ∞h(V,b′)db′dV . (A17)
R 0R 0
Thenumericalprocedurecanbetheniterated,consideringthebinaryatastartingseparationa =a ,interactingwith
R R i f
astellarpopulationwhose(normalized)velocitydistributionisf (v)=f (v),allowingatotalmassofinteractingstars
i r
M∗,i =M∗,r. For the first interaction only, we assume that the stars already within the binary separation interact on
a time scale a /σ, along the lines discussed in Section 2.2.2.
h
∼
10 Sesana, Haardt, & Madau
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