**2.6. Velocity distribution of galaxies**

The existence of the morphological sequence of clusters from irregular to regular clusters (Section 2.5) suggests that the regular clusters may have undergone some sort of dynamical relaxation. The nature of this relaxed distribution is examined in the next two sections through the distribution of cluster galaxy velocities and positions.

The redshifts in Section 2.2 are
determined from the mean radial
velocity of galaxies in a cluster; in fact, the radial velocities of
individual
galaxies are distributed around this mean. It has been conventional to
characterize this distribution by the dispersion
_{r} of radial
velocities about the mean

(2.4) |

where *v*_{r} is the radial velocity, which is the
component of the galaxy velocity
along the line-of-sight. The dispersion completely characterizes the
radial velocity distribution function if it is Gaussian:

(2.5) |

Here, *p*(*v*_{r})*dv*_{r} is the
probability that an individual cluster galaxy has a radial
velocity in the range *v*_{r} to *v*_{r} +
*dv*_{r}. While the Gaussian distribution has
usually been adopted simply for convenience, statistical tests reveal
that it is a consistent fit to the observed total distribution function
in many clusters, at least if high velocities (|*v*_{r} -
<*v*_{r}>| >
3_{r}) are
excluded
(Yahil and Vidal, 1977).
However, the velocity dispersion in a given cluster
generally decreases with distance from the cluster center; in
Coma and
Perseus the decline is about a factor of two from
the center to the outer edge
(Rood *et al.*,
1972;
Kent and Gunn, 1982;
Kent and Sargent, 1983).
Moreover, the velocity dispersion can differ in different clumps of an
irregular cluster showing subclustering
(Geller and Beers, 1982;
Bothun *et al.*,
1983).

A Gaussian distribution for a single component of the velocity obtains for a
system of non-identical particles in thermodynamic equilibrium, in which
case we identify the velocity dispersion with
_{r}
(*kT/m*)^{1/2}, where *T* is
the galaxy 'temperature' and *m* the galaxy mass. While the Gaussian
velocity distribution found in clusters suggests that they are at least
partially relaxed systems, they are not fully relaxed to thermodynamic
equilibrium. In thermodynamic equilibrium, all components of the cluster
would have equal temperatures; what is observed is that it is the velocity
dispersion (not temperature) which is nearly independent of galaxy mass and
position
(Rood *et al.*,
1972;
Kent and Gunn, 1982;
Kent and Sargent, 1983).

Some variation of the velocity dispersion with galaxy mass or cluster
position is observed
(Rood *et al.*,
1972;
Kent and Gunn, 1982;
Kent and Sargent,
1983).
In the Coma and Perseus clusters, the most luminous galaxies have a
somewhat smaller velocity dispersion than the less luminous galaxies,
and the velocity
dispersion decreases with increasing projected distance from the cluster
center. In
fact, the latter effect must necessarily occur if clusters are finite, bound
systems. Then, the velocities of bound galaxies at any point in the cluster
cannot exceed the escape velocity at that position. As the escape velocity
decreases with increasing distance from the cluster center, the velocity
dispersion must also decrease with projected distance from the cluster
center.

The observed mean galaxy velocity <*v*_{r}> will
depend on the projected
position in the cluster if the cluster is rotating. The projected shapes of
many clusters are substantially flattened. Of course, this is true of the L
(line) and F (flat) clusters (Section 2.5);
however, many regular (cD and B) clusters are also significantly flattened
(Abell, 1965;
Dressler, 1981).
If these clusters are actually oblate in shape due to significant
rotational support, the variation in <*v*_{r}> across
the cluster would be expected to be comparable to
_{r}. In fact,
such large velocity gradients are not observed
(Rood *et al.*,
1972;
Gregory and Tifft 1976;
Schipper and King, 1978;
Dressler, 1981);
apparently, the flattening of clusters is not due to rotation.

Useful compilations of velocity dispersions for clusters have been given by
Hintzen and Scott
(1979),
Danese *et al.*
(1980),
and particularly
Noonan (1981).