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
(
). 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 time required 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 [24
, 169]
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 of globular clusters 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
),
![]() |
http://www.livingreviews.org/lrr-2006-2 |
© Max Planck Society and the author(s)
Problems/comments to |