

2.3 Globular cluster
evolution
An overview of the evolution of globular clusters can be found in
Hut et al. [83
], Meylan and
Heggie [107
], and
Meylan [106]. We summarize
here the aspects of globular cluster evolution that are relevant to
the formation and concentration of relativistic binaries. The
formation of globular clusters is not well understood [54] and the details of
the initial mass function (IMF) are an ongoing field of star
cluster studies. Although the IMF is expected to be flatter at the
low-mass end (see [94] for a
discussion of the local IMF), many theorists assume an IMF that
follows a standard Salpeter form:
Once the stars form out of the initial molecular cloud, the system
will undergo violent relaxation as the protocluster first begins to
collapse. The effect of this is that the stars are distributed
widely in position and velocity, with a distribution that is
independent of the stellar mass. Thus, the more massive stars will
have more kinetic energy. These stars will then lose their kinetic
energy to the less massive stars through stellar encounters,
leading towards equipartition of energy. Through virialization,
this tends to concentrate the more massive stars in the center of
the cluster. The process of mass segregation for stars of mass
occurs on a timescale given by
.
The higher concentration of stars in the center
of the cluster increases the probability of an encounter, which, in
turn, decreases the relaxation time. Thus, the relaxation time
given in Equation (1) is an average over
the whole cluster. The local relaxation time of the cluster is
given in Meylan and Heggie [107] and can be
described by:
where
is the local mass density,
is the
mass-weighted mean square velocity of the stars,
is the mean stellar mass, and
(although a choice of
may be more
appropriate in the presence of a mass spectrum [57]). Note that in
the central regions of the cluster, the value of
is much lower than the average relaxation time. This
means that in the core of the cluster, where the more massive stars
have concentrated, there are more encounters between these stars.
The concentration of massive stars in the core of
the cluster will occur within a few relaxation times,
. This time is longer than the lifetime
of low metallicity stars with
[142].
Consequently, these stars will have evolved into carbon-oxygen (CO)
and oxygen-neon (ONe) white dwarfs, neutron stars, and black holes.
After a few more relaxation times, the average mass of a star in
the globular cluster will be around
and these
degenerate objects will once again be the more massive objects in
the cluster, despite having lost most of their mass during their
evolution. Thus, the population in the core of the cluster will be
enhanced in degenerate objects. Any binaries in the cluster that
have a gravitational binding energy significantly greater than the
average kinetic energy of a cluster star will act effectively as
single objects with masses equal to their total mass. These
objects, too, will segregate to the central regions of the globular
cluster [161]. The core
will then be overabundant in binaries and degenerate objects.
The core would undergo what is known as core
collapse within a few tens of relaxation times unless these
binaries release some of their binding energy to the cluster. In
core collapse, the central density increases to infinity as the
core radius shrinks to zero. An example of core collapse can be
seen in the comparison of two cluster evolution simulations shown
in Figure 4 [89
]. Note the core
collapse when the inner radius containing 1% of the total mass
dramatically shrinks after
. Since
these evolution syntheses are single-mass, Plummer models without
binary interactions, the actual time of core collapse is not
representative of a real globular cluster.
The static description of the structure of globular
clusters using King-Michie or Plummer models provides a framework
for describing the environment of relativistic binaries and their
progenitors in globular clusters. The short-term interactions
between stars and degenerate objects can be analyzed in the
presence of this background. Over longer time scales (comparable to
), the dynamical evolution of the distribution
function as well as population changes due to stellar evolution can
alter the overall structure of the globular cluster. We will
discuss the dynamical evolution and its impact on relativistic
binaries in Section 5.
Before moving on to the dynamical models and
population syntheses of relativistic binaries, we will first look
at the observational evidence for these objects in globular
clusters.

