Star dynamics of dense star systems
1
2
3
with a massive BH
Pau Amaro-Seoane , Marc Freitag , Rainer Spurzem
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2
pau@ari.uni-heidelberg.de
Astronomisches Rechen-Institut, Heidelberg, Germany
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Abstract
We have performed calculations to follow the joint evolution of a spherical star cluster with a central BH making feasible anisotropy in order
to check for the reliability of our numerical method. Here we present the study of the growth of the central BH due to star accretion at
its tidal disruption radius thanks to a diffusion model to treat loss-cone physics. The core collapse is studied in detail in a self-consistent
manner, as well as the post-collapse evolution of the surrounding stellar cluster. The results are in good agreement with classical literature
about this subject. The current new version of the program enables the analysis of the effects of an a (discretised) stellar mass spectrum
and stellar evolution. Using it, we present also more realistic models of dense clusters and give a description of mass segregation in these
systems with and without a central BH.
1
A numerical approach to the problem
The anisotropic model solves numerically moment equations of the full Fokker-Planck equation with
Boltzmann-Vlasov terms on the left- and interaction (collisional) terms on the right-hand side of the equations 1;2;7. This approach allows us to study the most important physical processes that are present in the
evolution of a spherical cluster, like self-gravity, two-body relaxation, stellar evolution, collisions, binary stars
etc and, undoubtedly, the interaction with a central BH and the role of a mass spectrum. The cluster is modelled like a self-gravitating, conducting gas sphere. In this method, all quantities of interest are accessible as
smooth functions of the radius and time. This “gaseous model” has the advantage of being much faster to run
than competing ones (N-body or Monte Carlo) and of providing smooth data free of numerical noise.
2
First step: Single mass star clusters with a massive BH
We followed the evolution of a one-star component stellar cluster with a so-called “seed BH” at its centre. We
consider two possible configurations for the stellar system; one of a total mass of Mtot * 105M + and another of
106M + . For the initial BH mass, we have chosen ,.- 0 / * 50M + and 500M + and we model it as a Plummer of
RPl * 1pc.
The cluster evolves during its pre-collapse phase up to a maximum central density from which the energy
input due to star accretion near the tidal radius becomes sufficient in order to halt and reverse the core collapse. Immediately afterwards, the post-collapse evolution starts. At the beginning of the re-expansion phase,
the BH significantly grows to several 103 solar masses. Thereon, a slow further expansion and growth of the
BH follow.
In Fig. 1, we follow the evolution of the mass of a central BH in a globular cluster of 10 5 stars of 1 M + . The
left panel shows the mass of the BH as function of time. On the right panel, we present the accretion rate on
to the BH, i.e. its growth rate. For bh - 0 / * 50 M + , the early cluster’s evolution is unaffected by the presence
of the BH which starts growing suddenly at the moment of deep core collapse, around T 0 14 1 5 Trh - 0 / .
From this first figure, we can see that the differences between the cases a, b and c are nearly negligible after
core collapse. In general, the structure of the cluster at late times is nearly independent of ,.- 0 / and xb (the
stellar structure parameter). From these plots we can infer that this occurs since core collapse leads to higher
densities if the initial BH mass is smaller and thus the integrated accreted stellar mass increases.
Figure 2: Here we show in the evolution of a stellar cluster of 15 components (in colours); m is the mass (in M 2 ) of the stars in
each component and f m the corresponding fraction of the total mass. In the upper box we have the density profile, where the solid
black line represents the total density; below, we have the average total mass for the system. We show different moments of the
system (T 8 0 corresponds to the initial model, which duly shows no mass segregation): As time elapses, at T 8 5 ; 30 < 10 9 2 Trh 5 0 7 ,
we observe how mass segregation has fragmented the initial configuration; the heavy components have sunken into the central
regions of the stellar cluster and, thus, risen the mean average mass. The outer parts of the system start losing their heavy stars
quickly and, consequently, their density profile retrogresses. This becomes more acute for later times at T 8 6 ; 75 < 10 9 2 Trh 5 0 7 , as
the right plots of Fig. 2 show. In these plots and, more markedly in the right panel of density profile, we can observe a depletion
at intermediate radius
4
Idem with BH
In this final section we extend our analysis to systems for which we use an evolved mass function of an age
of about 10 Gyr. We consider a mass spectrum with stellar remnants. We put at the centre a seed BH whose
initial mass is of 50 M + . The initial model for the cluster is a Plummer sphere with a Plummer radius R Pl * 1
pc. The total number of stars in the system is = cl * 106. We employ a Kroupa IMF 5;6 with Mzams1 from 0 1 1 to
120 M + with the turn-off mass of 1 M + . And with the following component’s distribution:
> Main sequence stars of 0 1 1 ? 1 M + ( @ 7 components)
> White dwarves of @ 0 1 6 M + (1 component)
> Neutron stars of @ 1 1 4 M + (1 component)
> Stellar black holes of @ 10M + (1 component)
If mMS is the mass of a MS star, we have defined the following mass ranges for the evolution into compact
remnants:
White dwarves in the range of 1 A mMS B M + C 8
Neutron stars for masses 8 A mMS B M + C 30
Stellar black holes for bigger masses, D 30M +
The presence of a small fraction of stellar remnants may greatly affect the evolution of the cluster and growth
of the BH because they segregate to the centre from which they expel MS stars but, being compact cannot be
tidally disrupted. This kind of evolution is shown in the following figure.
Figure 1: Evolution of the mass of a central BH in a globular cluster of 105 stars of 1 M 2 . We considered three cases. In case
a (solid line), the initial BH mass is 3 465 0 78 50 M 2 and xb 8 1, case b (dashes) has the same initial BH mass but xb 8 2 while
case c (dash-dot) corresponds to 3 465 0 7!8 500 M 2 and xb 8 1. An accretion efficiency of εeff 8 1 is assumed. The left panel shows
the mass of the BH as function of time and right panel the accretion rate on to the BH. At late times, the mass of the central BH
increases like 3 ˙4 ∝ T 9 1 : 2 as predicted by simple scaling arguments (see text)
3
Star clusters with a broad mass spectrum without central BH
So as to be able to interpret observations of young stellar clusters extending to a larger number of mass
components, it is of paramount relevance to understand the physics behind clusters without a central BH first.
It has been shown that for a cluster with a realistic IMF, equipartition cannot be reached, for the most massive
stars build a subsystem in the cluster’s centre as the process of segregation goes on thanks to the kinetic
energy transfer to the light mass components until the cluster undergoes core collapse 3;4;8.
Whereas the case in which the BH ensconces itself at the centre of the host cluster is more attractive from
the dynamical point of view, one should study, in a first step, more simple models.
In this section we want, thus, to go a step further and evaluate stellar clusters with a broad mass function
(MF hereafter). We study those clusters for which the relaxation time is short enough, because this will lead
the most massive stars to the centre of the system due to mass segregation before they have time to leave
the main sequence (MS). In this scenario, we can consider that stellar evolution plays no role; stars did not
have time to start evolving.
1
The zero age main sequence (ZAMS) corresponds to the position of stars in the Hertzsprung-Russell diagram where stars begin hydrogen fusion.
Figure 3: In this figure we plot the density profiles of the system before and after the post-collapse phase. We can see
that the slope of ρ ∝ R 9 7 E 4 on account of the cusp of stellar BHs that has formed around the central BH. We can see how the
different components redistribute in the process and how the BH dominates the dynamics at the centre.
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