Run–away formation of massive black holes
1
2
2
in dense star clusters? Marc Freitag , Atakan Gürkan , Frederic Rasio
1freitag@ari.uni-heidelberg.de
2ato,rasio@northwestern.edu
Astronomisches Rechen-Institut, Heidelberg, Germany
Department of Physics and Astronomy, Northwestern University, USA
Summary
We investigate the conditions under which a dense stellar cluster may undergo a phase of collisional run-away leading to the formation of a very massive star (VMS,
with M∗ 100 M), a possible progenitor for an intermediate-mass black hole. In particular, we have established that systems with a realistically broad stellar mass
function (0.2 − 120 M) undergo core-collapse driven by mass-segregation in just ∼15 % of the central core collapse. When we allow for collisions between stars, we
find that growth of a VMS star occurs in all cases for which core collapse (driven by relaxation or collisions) takes place before ∼ 100 M stars evolve off the main
sequence, i.e. within 3 Myrs, even in proto-galactic nucleus models with a velocity dispersion in excess of 100 km s −1.
1
Numerical method
We simulate the evolution of dense stellar clusters using Monte Carlo (MC) codes 2;3;10;11. Based on the
ideas of Hénon 8, they are ideal compromises, in terms of physical realism and computational efficiency, between direct N -body simulations, which, being extremely computer-intensive, are still limited to a few 10 5 stars,
and methods that treat the stellar cluster as a continuum (Fokker-Planck integrations and gaseous models)
which do not allow realistic account of many processes (stellar collisions, role of an arbitrary mass or velocity
distribution. . . ). The MC code methods are based on the assumptions of spherical symmetry, dynamical
equilibrium and diffusive 2-body relaxation. They allow simulations with a few millions particles to be carried out on a single-processor PC. Various prescriptions can be used for the outcome of stellar collisions
(mass end energy losses), including inter/extrapolation form the results of some 15 000 SPH simulations 4.
Mergers of two stars simulated with SPH hydro code.
2
Fast core collapse
Our Monte Carlo simulations have shown that for a broad mass function (Salpeter or Kroupa, typically), core
collapse, driven by mass-segregation, occurs very quickly, in order 15 % of the initial central relaxation time 6.
Collisional histories for simulations of clusters with W0 = 3, N∗ = Npart = 3 × 105. Left: Rh ' 0.4 pc. Right: Rh ' 0.03 pc. We
represent the evolution of the five particles that have experienced the largest number of collisions. The top panel represents
the radius (distance from the cluster’s centre) where the collision occurred. The bottom panel shows the evolution of the
mass.
Evolution of the central region of a cluster of stars. Left: initial configuration. Right: during core collapse, at the moment massive
stars start colliding with each other at the centre. All the stars within a slice containing the centre are depicted. For clarity,
their radii are highly magnified. The white circles represent spheres containing 1, 3 and 10 % of the total cluster mass (from the
centre). Note how the massive, large stars concentrate to the centre.
Conditions for quick core collapse.
This diagram shows
which cluster masses and radii
will lead to core collapse in less
than 3 Myrs, i.e.
before the
most massive stars evolve of
the MS, for King models with
W0 = 3 and W0 = 8 and a
Salpeter 0.2−120 M IMF. Solid
black lines correspond to the
condition Tcc =3 Myrs.
Below this line, core collapse time
is shorter, above it is longer.
Dashed lines indicate where the
collision time for a 120 M star
is 3 Myrs. Below this line, one
may expect collisional effects to
be important before core collapse. The arrows show which
decrease in Mcl or Rh leads to a
shortening of Tcc by a factor of 2.
We show the position in the
(Mcl, Rh) plane of a variety of observed clusters 1;7;9;12;13. If clusters are born as concentrated as
W0 = 8, a significant fraction
of them should go through core
collapse in less than 3 Myrs and
hence enter a phase of run-away
collisions.
3
Formation of a very massive stars
In recent MC simulations, we have introduced collisions between single MS stars 5;15. We observe that, provided core collapse occurs within less than ∼ 3 Myrs, the cluster always enters a run-away phase in which a
star more massive than 1000 M grows through repeated mergers.
Merger trees for the same simulations. We follow the growth of the run-away star to ∼ 2000 M . In the left case, the cluster is
not initially collisional (the collision time for a 120 M star is much larger than its MS life-time) and most collisions occur in deep
collapse and feature stars of mass 70 − 120 M that have segregated to the centre. Most of them have are not themselves
merger products. In the right case, the cluster is initially collisional and, although the run-away also occurs in core collapse,
most stars contributing to it have experienced earlier collisions. Note that the largest fraction of the mass comes from stars
near the top end of the IMF, around 100 M.
4
Open questions
• Stellar dynamics
– Role of binaries: stop collapse and/or foster collisions? (John Fregeau, NU)
– “Loss-cone” effects for collision with the VMS.
– Minimum number of stars in the core for run-away (suggested by recent N −body work 14).
– Growth of IMBH to larger masses (through tidal disruptions, collisions, stellar winds 3; see poster by Pau
Amaro-Seoane)
• Hydrodynamics
– Collisions featuring VMS. Object is kept out of thermal equilibrium by frequent collisions. Is there a “transparency problem”?
• Stellar evolution
– Role of pre-MS phase and gas in young clusters.
– Stability and evolution of bombarded VMS.
– End product of VMS evolution: an IMBH, really?
Acknowledgements.
The work of MF is funded by the Sonderforschungsbereich (SFB) 439 ‘Galaxies in the Young Universe’ (subproject A5) of the Germ
Science Foundation (DFG) at the University of Heidelberg. The work of AG and FR is supported by NASA ATP Grant NAG5-12044 a
NSF Grant AST-0206276 to Northwestern University.
References
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astro-ph/0403621
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