Massive Black Hole Binaries from Collisional Runaways

Abstract

Recent theoretical work has solidified the viability of the collisional runaway scenario in young dense star clusters for the formation of very massive stars (VMSs), which may be precursors to intermediate-mass black holes (IMBHs). We present first results from a numerical study of the collisional runaway process in dense star clusters containing primordial binaries. Stellar collisions during binary scattering encounters offer an alternate channel for runaway growth, somewhat independent of direct collisions between single stars. We find that clusters with binary fractions >~10% yield two VMSs via collisional runaways, presenting the exotic possibility of forming IMBH--IMBH binaries in star clusters. We discuss the implications for gravitational wave observations, and the impact on cluster structure. Comment: 4 pages with emulateapj. To appear in ApJ letters. Minor revisions to include the changes suggested by the referee

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arXiv:astro-ph/0512642v2 11 Feb 2006
To appear in ApJ
Preprint typeset using LATEX style emulateapj v. 6/22/04
MASSIVE BLACK HOLE BINARIES FROM COLLISIONAL RUNAWAYS
M. Atakan G¨ urkan1, John M. Fregeau2, and Frederic A. Rasio3
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
To appear in ApJ
ABSTRACT
Recent theoretical work has solidified the viability of the collisional runaway scenario in young dense
star clusters for the formation of very massive stars (VMSs), which may be precursors to intermediate-
mass black holes (IMBHs). We present first results from a numerical study of the collisional runaway
process in dense star clusters containing primordial binaries. Stellar collisions during binary scattering
encounters offer an alternate channel for runaway growth, somewhat independent of direct collisions
between single stars. We find that clusters with binary fractions ?10% yield two VMSs via collisional
runaways, presenting the exotic possibility of forming IMBH–IMBH binaries in star clusters. We
discuss the implications for gravitational wave observations, and the impact on cluster structure.
Subject headings: stellar dynamics — globular clusters: general — intermediate-mass black holes —
methods: n-body simulations
1. INTRODUCTION
Observations hinting at the possible existence of
intermediate-mass black holes (IMBHs) have mounted
in recent years. Ultra-luminous X-ray sources—point
X-ray sources with inferred luminosities ?1039ergs−1—
may be explained by sub-Eddington accretion onto BHs
more massive than the maximum mass of ∼10M⊙ ex-
pected via core collapse in main sequence stars, although
viable alternative explanations exist (Miller & Colbert
2004).Similarly, the cuspy core velocity dispersion
profiles of the globular clusters M15 and G1 may also
be explained by the dynamical influence of a central
IMBH (van der Marel et al. 2002; Gerssen et al. 2002;
Gebhardt et al. 2005), although theoretical work sug-
gests that the observations of M15 may be equally-well
explained by a collection of compact stellar remnants in
the cores of the clusters (Baumgardt et al. 2003).
At least three distinct IMBH formation mechanisms
have been discussed in the literature.
possibly simplest, is core collapse of a massive Pop III
star (Madau & Rees 2001). The very low metallicity of
Pop III stars (Z ? 10−5Z⊙) allows much larger main-
sequence stars to form, limits mass loss during stellar
evolution, and increases the fraction of mass retained
in the final BH (for stars more massive than ∼250M⊙)
(Fryer et al. 2001; Miller & Colbert 2004). The second
is the successive merging of stellar mass BHs via dy-
namical interactions, which may occur in star clusters
that do not reach deep core collapse before ∼3Myr,
when the most massive cluster stars have become
BHs (Kulkarni et al. 1993; Sigurdsson & Hernquist 1993;
Portegies Zwart & McMillan 2000; Miller & Hamilton
2002; O’Leary et al. 2006).
inefficient—in terms of the amount of mass added to
the growing BH per BH ejected from the cluster—
requiring BH seeds ?500M⊙ to create 103M⊙ IMBHs
(G¨ ultekin et al. 2004), even when aided by the Kozai
The first, and
The process is relatively
1ato.gurkan@gmail.com Current address: Foundations Devel-
opment, Sabancı University, 34956˙Istanbul, Turkey.
2fregeau@alum.mit.edu
3rasio@northwestern.edu
mechanism (Miller & Hamilton 2002) or gravitational
wave losses during close approaches (G¨ ultekin et al.
2005).Introducing a mass spectrum for the BHs de-
creases the required seed mass, although growth is still
rare (O’Leary et al. 2006).
merging of main-sequence stars via direct physical colli-
sions to form a very massive star (VMS), which may then
collapse to form an IMBH (Portegies Zwart et al. 1999;
Ebisuzaki et al. 2001; Portegies Zwart & McMillan 2002;
G¨ urkan et al. 2004). Recent work shows that runaway
growth of a VMS occurs generically in clusters with deep
core collapse times shorter than ∼3Myr (Freitag et al.
2005a).
Withtheexception
(Portegies Zwart et al. 2004), all simulations of runaway
collisional growth in clusters have ignored the effects
of primordial binaries, which are known to exist in
clusters in dynamically significant numbers (Hut et al.
1992). Indeed, some numerical results suggest that the
primordial binary fraction (fb) may have to be nearly
100% to explain the currently observed binary fractions
in cluster cores (Ivanova et al. 2005). Primordial bina-
ries are an important piece of the runaway collisional
growth puzzle, since they introduce two effects which
may strongly affect the process.
binaries generate energy via dynamical
interactions in cluster cores, supporting the core against
deep collapse and limiting the maximum stellar density
attainable, and hence limiting the direct stellar collision
rate (Heggie & Hut 2003). On the other hand, stellar
collisions are much more likely in dynamical interactions
of binaries, since the interactions are typically resonant
(Bacon et al. 1996; Fregeau et al. 2004). Since these two
effects of primordial binaries act in opposite senses with
respect to the collision rate, it is not clear a priori how
they affect the collisional runaway scenario.
Before appealing to numerical methods, however, one
can gain insight into the effects of primordial binaries
by considering the coagulation equation (Lee 1993, 2000;
Malyshkin & Goodman 2001)—a simplification of which
is presented in Freitag et al. (2005b)—which describes
the evolution of a spectrum of masses due to mergers.
The third is the runaway
ofonesimulation
On the one hand,
scattering

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2 G¨ urkan, et al.
For growth to occur in a runaway fashion, the coagula-
tion equation requires that the cross section for collisions
with the runaway object scales sufficiently rapidly with
its mass: Scoll ∝ Mη, with η > 1. For single–single
star collisions in star clusters in which the central ve-
locity dispersion is less than the escape speed from the
surface of a typical star (so that the cross section is domi-
nated by gravitational focusing), this corresponds to the
constraint R ∝ Mαwith α > 0 on the main-sequence
mass–radius relationship, which is satisfied by main-
sequence stars of any mass or metallicity. With some ap-
proximations, the coagulation equation analysis (Heggie
1975; Hut & Bahcall 1983; Sigurdsson & Phinney 1993;
G¨ ultekin et al. 2005) can be applied to collisions occur-
ring in binary scattering interactions. From Fig. 4 of
G¨ ultekin et al. (2005), the cross section for close ap-
proach distances of rmin in binary–single scattering en-
counters scales as (Scoll/πa2)(v∞/vc)2∝ (rmin/a)γ, with
0.3 ? γ ? 1, where a is the binary semimajor axis, v∞is
the relative velocity between the binary and single star
at infinity, and vcis the critical velocity (see eq. [12] of
G¨ ultekin et al. 2005). Using the radius of the runaway
star for rmin, R ∝ Mβwith 0.5 ? β ? 1 for the scaling
of the mass–radius relation, and assuming that the bind-
ing energy of the binary is roughly preserved during the
encounter so that a ∝ M, we find:
Scoll∝ M2+γ(β−1),
with 0.3 ? γ ? 1 and 0.5 ? β ? 1. The minimum of the
exponent is ≈1.5. Thus according to the coagulation
equation, collisions induced in binary–single scattering
interactions should yield runaway growth of a VMS. This
result says nothing about the rate of growth of the run-
away object. In other words, it is still not clear whether
binary interactions will limit the cluster core density such
that the runaway timescale is longer than the massive
star main-sequence lifetime of ≈3Myr, in which case the
process would be halted.
Assuming that the cluster core density reached is high
enough for the runaway to proceed, it would appear
that a single binary is sufficient for a binary interaction-
induced runaway to occur. This is, of course, not the
case, since binary scattering interactions tend to destroy
binaries. Thus we expect that for sufficiently low fb,
the runaway will be primarily mediated by single–single
collisions. For sufficiently large fb, the runaway will be
primarily mediated by binary–binary interactions (the
analysis above assumes binary–single interactions—for
binary–binary the value of γ will be smaller, still allowing
a runaway by eq. [1]). For intermediate fbit is possible
that a binary interaction induced-runaway could proceed
until the core binary population is sufficiently depleted
that the cluster’s core collapses. If the first runaway is
far enough from the center of the cluster when the core
collapses, it is possible that a second runaway will be
formed during core collapse, mediated by single–single
collisions. The exotic possibility of forming two VMSs
in a cluster, and thus two IMBHs, is a tantalizing one,
with implications for gravitational wave observations and
cluster dynamics.
In this letter we present first results from a study of the
runaway collisional scenario for the formation of VMSs in
young dense clusters with primordial binaries. We briefly
describe our numerical method, present results showing
(1)
0.1%
0.3%
1%
2%
5%
10%
20%
30%
50%
70%
90%
99%
0.01
0.1
1
Lagrange Radii [rh(0)]
0.001
0.01
0.1
1
Lagrange Radii [pc]
−0.500.511.52
t [Myr]
00.050.1
t/trc(0)
100
1000
mass [M⊙]
Fig. 1.— Evolution of the cluster Lagrange radii (lower panel)
in units of the initial cluster half-mass radius (left axis) and in
parsecs (right axis), and mass of the (single) runaway for a cluster
with 106objects and fb= 0.05. Time is shown both in Myr and
in units of the central relaxation time.
the growth of two runaways, and discuss the implications.
2. NUMERICAL ANALYSIS
We use our Monte Carlo cluster code to simulate the
evolution of young star clusters with primordial bina-
ries (Joshi et al. 2000; Fregeau et al. 2003; G¨ urkan et al.
2004). This code uses the H´ enon method for two-body
relaxation, and incorporates the Fewbody N-body inte-
grator to perform dynamical scattering encounters of bi-
naries. Stellar collisions are handled by assuming that
stars whose surfaces touch merge with no mass loss, an
assumption that has been shown to be valid for clusters
with low velocity dispersion, such as globulars or young
dense clusters (Freitag et al. 2005b,a). Collisions are al-
lowed to occur directly in single–single star encounters,
and during binary interactions (Fregeau et al. 2004).
For initial conditions we assume a W0= 3 King model
density profile for the cluster, with no initial mass seg-
regation. All stars are main-sequence stars with masses
in the range 0.2 < M/M⊙ < 120, distributed accord-
ing to a Salpeter mass function. Simulations show that
the primary condition required for a runaway is that the
core collapse time is shorter than the main-sequence life-
time for the most massive stars, ∼3Myr (Freitag et al.
2005a); and that for clusters of single stars with a
wide mass spectrum, the core collapse time is always
tcc ≈ 0.15trc(0), (where trc(0) is the initial relaxation
time in the core), independent of cluster mass, size, or
density profile (G¨ urkan et al. 2004). This allows us to
set the central density of the cluster such that the pre-
dicted core collapse time is either less or greater than

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Massive Black Hole Binaries from Collisional Runaways3
Fig. 2.— Merger trees for the three most massive stars at the end
of the simulation for the model presented in Figure 1. Red shows
the most massive, intermediate green the second most, and blue
the third most. For each collision with the massive star, a symbol
is plotted at its mass at the time of the collision with lines drawn
connecting to the symbols representing the star participating in
the merger. A triangle represents a single–single collision, a square
a collision in a binary–single interaction, a pentagon a collision
in a binary–binary interaction, and a circle a star that has not
yet undergone a collision. As an example, the three intermediate
green points at t ≈ 0.2Myr represent a collision in a binary–single
interaction of a ≈100M⊙ star with a ≈0.3M⊙ star, each having
never undergone a collision previously.
clearly just one VMS, created primarily in single–single collisions.
For this model there is
3Myr. We perform simulations with binary fractions4
up to 0.2. The binary population is created from a clus-
ter of single stars by adding secondary companions to
randomly chosen cluster stars, with the secondary mass
chosen uniformly in the binary mass ratio. The binary
binding energy is distributed uniformly in the logarithm,
truncated at high energy so that the binary members do
not make contact at pericenter, and truncated at low en-
ergy so that the orbital speed of the lightest member in
the binary is larger than the local stellar velocity disper-
sion (see Fregeau et al. 2005). The eccentricity is chosen
according to a thermal distribution, truncated at large e
so that the binary members do not make contact at peri-
center. We use either N = 5 × 105or 106total cluster
objects, finding no difference in the runaway results be-
tween the two. We run all simulations until 3Myr. Our
criterion for a runaway is that its final mass is ?500M⊙.
In our runs the final masses of the VMSs are always at
least twice this value.
Figure 1 shows the evolution of the cluster Lagrange
radii and the mass of the (single) runaway as a function
of time for a cluster with tcc < 3Myr and fb = 0.05.
The evolution is typical for models with fb ? 0.1 and
tcc < 3Myr in that: 1) the energy generated in bi-
nary interactions is not sufficient to postpone core col-
lapse beyond 3Myr, and 2) there is a single runaway.
In the model shown in Figure 1, the runaway grows to
≈2800M⊙ before the cluster begins to expand in re-
sponse to the energy generated in collisions with the run-
away.
We follow the membership of the collision products
4The binary fraction is defined such that fbis the fraction of
objects in the cluster that are binaries, an object being either a
binary or a single star.
Fig. 3.— Same as Figure 2 for the same model but with fb=
0.1.A binary interaction-induced collisional runaway begins at
t ≈ 1.5Myr and proceeds to ≈1.3 × 103M⊙. A direct collision-
induced runaway begins at t ≈ 2.3Myr, yielding a second runaway,
of mass ≈2.5 × 103M⊙. The third most massive star at the end
of the simulation has only grown to ≈400M⊙.
in binaries during the collisions and binary interactions.
This allows us to produce merger trees to track the con-
tributions to the formation of VMSs. Figure 2 shows
the merger trees for the three most massive stars at the
end of the simulation. There is clearly only one runaway
for this model, and it grows primarily via direct, single–
single collisions. Figure 3 shows the same as Figure 2
for the same model but with fb = 0.1. In this model
there are two VMSs at the end of the simulation. A
binary interaction-induced collisional runaway begins at
t ≈ 1.5Myr and proceeds to ≈1.3 × 103M⊙. A direct
collision-induced runaway begins at t ≈ 2.3Myr, yield-
ing a second runaway, of mass ≈2.5×103M⊙. The third
most massive star at the end of the simulation has only
grown to ≈400M⊙. The evolution is typical for models
with fb? 0.1 and tcc< 3Myr in that there are two run-
aways: one via direct collisions, and one via collisions in
binary interactions.
Although we have not yet conducted a full param-
eter space survey, we have mapped a significant slice
in fb–trc(0) space, from fb = 0.02 to 0.2 and from
tcc/(3Myr) = 0.16 to 28. Without exception, and in-
dependent of fb, models with tcc= 0.15trc(0) > 3Myr
show no runaways (7 models), while those with tcc =
0.15trc(0) < 3Myr show either one or two (10 models).
This is in agreement with the results for clusters with
no primordial binaries, presumably since binary fractions
smaller than 20% are not sufficient to postpone core col-
lapse beyond 3Myr. Of the models that show runaways,
those with fb < 0.1 always yield only single runaways
(4 models), while those with fb≥ 0.1 always yield dou-
ble runaways (6 models). No models ever produce more
than two runaways. Due to the computational cost in-
volved, we have not yet explored the fb> 0.2 region of
parameter space.
The numerical results presented here clearly agree with
the analytical arguments made above on the runaway
nature of binary interaction-induced collisions. Binary
interaction-induced runaways occur in the outskirts or
just outside the core (typically a few percents of a parsec
for our models). This is a region where densities, in

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4 G¨ urkan, et al.
particular the binary densities, are high enough to start
a runaway but the relaxation time is long enough that
massive objects do not rapidly sink to the center. The
rapid collapse near the center leads to an expansion of
these regions, further increasing the relaxation time and
prevents a prompt merger of the VMSs.
Finally, we note the apparent disagreement between
our results and those of Portegies Zwart et al. (2004),
who performed two direct N-body integrations of mod-
els of the cluster MGG-11 with fb = 0.1 and found
only single runaways. Our results predict that whenever
fb? 0.1, and the cluster central relaxation time is short
enough so that a runaway is expected in a corresponding
cluster with only single stars, a double runaway should
result. The models of Portegies Zwart et al. (2004) are
extremely centrally concentrated, with W0 = 12, while
our models all use W0= 3. From the discussion above,
it is clear that if the density profile is sufficiently steep,
the binary interaction-induced runaway will take place
close enough to the center so that it will seed the subse-
quent direct collision-induced runaway, yielding just one
runaway. In addition, the condition fb ? 0.1 for dou-
ble runaways is only approximate. Since the models of
Portegies Zwart et al. (2004) are right on this boundary,
it is possible that a small difference in the initial condi-
tions has prevented a double runaway in their models.
3. DISCUSSION
Although the process is somewhat uncertain, it is likely
that the VMSs formed in young clusters via collisional
runaways will undergo core-collapse supernovae and be-
come IMBHs on a timescale of ∼1Myr (Freitag et al.
2005a). Our results show no evidence of the VMSs merg-
ing prior to becoming IMBHs. After their separate for-
mation, the two IMBHs will quickly exchange into a
common binary via dynamical interactions. The IMBH–
IMBH binary will then shrink via dynamical friction due
to the cluster stars, to the point at which the stellar mass
enclosed in the binary is comparable to the binary mass.
This occurs on a timescale ∼tr?m?/MIMBH, where tris
the local relaxation time and ?m? is the local average
stellar mass. Since ?m?/MIMBH ? 10−2, this timescale
is likely to be ?10Myr. The binary will then shrink
via dynamical encounters with cluster stars, the rate of
which is governed by loss-cone physics. The timescale
for the binary to shrink to the point at which it merges
quickly via gravitational radiation energy loss is likely to
be ?1Gyr (Yu & Tremaine 2003; Miller 2005).
Although IMBH–IMBH binaries do not merge in the
LISA band—the gravitational wave frequency at merger
is ∼1Hz—they do represent bright sources that take at
least ∼106yr to cross the LISA band. Their inspiral
(chirp) signals should be easily detectable by LISA out to
a few tens of Mpc. Thus the number of detectable IMBH
binary sources may be quite large, since most clusters are
probably born with fb? 0.1, and any cluster with mass
?106M⊙and central relaxation time ?20Myr will lead
to a double runaway.
Significant core rotation (with rotational speed com-
parable to the local velocity dispersion) is observed
in the clusters M15, ω Cen, 47 Tuc, and G1 (e.g.
van der Marel et al. 2002; Gebhardt et al. 2005). This
rotation suggests the presence of a core angular mo-
mentum source, such as an IMBH binary (Mapelli et al.
2005). Similarly, observations of a millisecond pulsar in
the halo of NGC 6752, and two others in the core with
high negative spin derivatives, hint at the existence of an
IMBH binary in the core (Colpi et al. 2003). Since the
IMBH–IMBH binaries formed via collisional runaways
will merge within ∼1Gyr after formation, any angular
momentum imparted to the cluster by the IMBH–IMBH
binary will quickly diffuse out of the core on a core relax-
ation time. An alternate mechanism must be at work in
creating the core rotation seen in some globular clusters
today.
We thank Marc Freitag, Shane Larson, and Cole Miller
for many fruitful discussions, and an anonymous referee
for comments which improved the paper. Some of the
simulations were performed on the Tungsten cluster at
the National Center for Supercomputing Applications.
MAG thanks Sabancı University for their hospitality dur-
ing the finalization of this letter. This work was sup-
ported by NASA grant NNG04G176G.
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