There are also important characteristic time
scales that govern the dynamics of globular clusters. These are the
crossing time , the relaxation time
, and the evaporation time
. The crossing
time is the typical time required for a star in the cluster to
travel the characteristic size
of the cluster (typically
taken to be the half-mass radius). Thus,
, where
is a typical velocity (
10 km/s). The
relaxation time is the typical time for gravitational interactions
with other stars in the cluster to remove the history of a star’s
original velocity. This amounts to the timerequired for
gravitational encounters to alter the star’s velocity by an amount
comparable to its original velocity. Since the relaxation time is
related to the number and strength of the gravitational encounters
of a typical cluster star, it is related to the number of stars in
the cluster and the average energy of the stars in the cluster.
Thus, it can be shown that the mean relaxation time for a cluster
is [19
, 116]
The evaporation time for a cluster is the time
required for the cluster to dissolve through the gradual loss of
stars that gain sufficient velocity through encounters to escape
its gravitational potential. In the absence of stellar evolution
and tidal interactions with the galaxy, the evaporation time can be
estimated by assuming that a fraction of the stars in the
cluster are evaporated every relaxation time. Thus, the rate of
loss is
. The value of
can be determined by noting
that the escape speed
at a point
is related to the
gravitational potential
at that point by
. Consequently, the mean-square escape speed in a
cluster with density
is
The characteristic time scales differ
significantly from each other: . When discussing stellar interactions during a given
epoch of globular cluster evolution, it is possible to describe the
background structure of the globular cluster in terms of a static
model. These models describe the structure of the cluster in terms
of a distribution function
that can be thought of as
providing a probability of finding a star at a particular location
in phase-space. The static models are valid over time scales which
are shorter than the relaxation time so that gravitational
interactions do not have time to significantly alter the
distribution function. We can therefore assume
. The structure of the globular cluster is then
determined by the collisionless Boltzmann equation,
The solutions to Equation (5) are often described
in terms of the relative energy per unit mass
with the relative potential defined as
. The constant
is chosen so that there are
no stars with relative energy less than 0 (i.e.
for
and
for
). A simple class of
solutions to Equation (5
),