## Dynamical evolution of idealised star cluster models

##### Abstract

This thesis is concerned with the dynamical evolution of globular star clusters modelled as
the classical gravitational N-body problem. The models in this thesis are idealised in order to
allow the detailed study of particular dynamical aspects of the cluster evolution. Examples of
properties which tend to be omitted are stellar evolution, primordial binaries and the effect of
an external tidal gravitational field. The methods used in this thesis are gas models, N-body
models and physical arguments.
One of the main topics in this thesis is gravothermal oscillations in multicomponent star
clusters. The evolution of one-component globular clusters, systems with equal particle masses,
is quite well understood. However, the evolution of more realistic globular clusters, with a
range of particle masses, is a much more complicated matter. The condition for the on-set
of gravothermal oscillations in a one-component system is simply that the number of stars
is greater than a certain number ( ≈7000). In a multi-component system the relationship
between the number of stars at which the gravothermal oscillations first appear and the stellar
mass distribution of a cluster is a complex one. In order to investigate this phenomenon
two different types of multi-component systems were studied: two-component systems (the
simplest approximation of a mass spectrum, Chapter 2) and ten-component systems (which
were realisations of continuous power law IMFs, Chapter 3). In both cases the critical number
of stars at which gravothermal oscillations first appear are found empirically for a range of stellar
mass distributions. The nature of the oscillations themselves are investigated and it is shown
that the oscillations can be understood by focusing on the behaviour of the heavier stars within
the cluster. A parameter Nef (de nined Mtot/mmax where Mtot is the total mass and mmax
is the maximum stellar mass) acts as an approximate stability boundary for multicomponent
systems.The stability boundary was found to be at Nef ~- 12000.
In this Chapter 4, globular star clusters which contain a sub-system of stellar-mass black
holes (BH) are investigated. This is done by considering two-component models, as these
are the simplest approximation of more realistic multi-mass systems, where one component
represents the BH population and the other represents all the other stars. These systems are
found to undergo a long phase of evolution where the centre of the system is dominated by a
BH sub-system. After mass segregation has driven most of the BH into a compact sub-system,
the evolution of the BH sub-system is found to be in
uenced by the cluster in which it is
contained. The BH sub-system evolves in such a way as to satisfy the energy demands of the
whole cluster, just as the core of a one component system must satisfies the energy demands of
the whole cluster. The BH sub-system is found to exist for a significant amount of time. It takes
approximately 10trh;i, where trh;i is the initial half-mass relaxation time, from the formation
of the compact BH sub-system up until the time when 90% of the sub-system total mass is lost
(which is of order 103 times the half-mass relaxation time of the BH sub-system at its time of
formation). Based on theoretical arguments the rate of mass loss from the BH sub-system (M2)
is predicted to be (βζM)/(αtrh): where M is the total mass, trh is the half-mass relaxation time,
and α, β, ζ are three dimensionless parameters. (see Section 4.3 for details). An interesting
consequence of this is that the rate of mass loss from the BH sub-system is approximately
independent of the stellar mass ratio (m2/m1) and the total mass ratio (M2/M1) (in the range
m2/m1 ≥ 10 and M2/M1 ≈ 10-2, where m1, m2 are the masses of individual low-mass and
high-mass particles respectively, and M1, M2 are the corresponding total mass). The theory is
found to be in reasonable agreement with most of the results of a series of N-body simulations,
and all of the models if the value of ζ is suitable adjusted. Predictions based on theoretical
arguments are also made about the structure of BH sub-systems. Other aspects of the evolution
are also considered such as the conditions for the onset of gravothermal oscillation.
The final chapter (Chapter 5) of the thesis contains some concluding comments as well as a
discussion on some possible future projects, for which the results in this thesis would be useful.