## Homologous evolution in the post-collapse expansion of globular clusters

##### Abstract

We examine the evolution of globular star clusters, modelled as spherically symmetric
stellar systems, using various techniques. Such clusters possess a central region of
approximately uniform density which is referred to as the core. We concentrate our
analysis on the evolution of the cluster after the core has undergone core collapse;
a process where its radius decreases and its density increases. After this collapse,
the system as a whole can expand in a self-similar fashion (homologous post-collapse
evolution) which has long been thought to be due to gravitational interactions between
different populations of single stars and binary stars in the core. We confirm this
assumption by constructing a simple analytical model which combines much of the
theoretical knowledge of previous research in the field. This model consists of two
stellar populations, each defined by the mass of the individual stars, and a separate
core. Our simple model is itself constructed from two simpler models – a twocomponent
model without a core and a single mass model with a core – and takes
into account the main gravitational interactions thought to drive the post-collapse
evolution.
To ensure that no important mechanisms have been neglected in our simple model,
we will compare it with an N-body simulation. We compute our N-body models with
NBODY6 (using a GPU version for large N). When we compare the N-body model with
the simple model, we find qualitative agreement between them for most cases. Even
though some mechanisms (e.g. escape of stars) are neglected in our simple model, we
find that both models show homologous post-collapse evolution.
We also review the homologous post-collapse Fokker-Planck model in the case of
equal stellar masses derived by H´enon (1961) with the intention of extending this for
the two-component case. We present our numerical solutions for H´enon’s model and
find that our numerical solutions are in satisfactory agreement with the results shown
in this paper. When we extend this work for a general two-component model (i.e.
with no restriction on the number of heavier stars), we find that a homologous solution
cannot be found with this approach. By contrast, we suggest that it would be possible
to find a homologous two-component solution by extending the one-component solution
published later by H´enon (1965), which differs from the earlier model by neglecting the
external tidal field of the parent galaxy. Much of the work shown in this thesis would
be relevant for such future study.