**2.9. Dynamics of galaxies in clusters**

In Sections 2.5, 2.6, and 2.7, evidence was presented indicating that the velocity and spatial distribution of galaxies in regular, compact clusters are in a relaxed, quasistationary state. In this section, the nature of the relaxation processes of galaxies in clusters will be briefly discussed.

If clusters result from the gravitational growth of initially small perturbations, we expect the initial state of material in the protocluster to be irregular. The relaxation of the cluster thus involves the spatial motion of the galaxies, and a lower limit to the relaxation time is the crossing time

(2.26) |

where the radial velocity dispersion
_{r} has been
substituted for the radial
velocity *v*_{r}. An upper limit on the age of the cluster
is the Hubble time (age of the universe)

(2.27) |

where the numerical coefficient depends on the cosmological model. Thus the
outer parts of a cluster or surrounding supercluster for which *r*
10 Mpc cannot
possibly have relaxed and are expected to be irregular, as is observed
(Sections 2.7 and
2.11.2).

The phase-space distribution of galaxies in the central parts of
spherical regular
clusters can be represented as *f*(**r**, **v**)
exp(- /
_{r}^{2}),
where
1/2 *v*^{2} +
(*r*) is the
energy per unit mass in the cluster
(Section 2.7). This is very similar
to a Maxwell-Boltzmann distribution, except that it is the energy per
unit mass (velocity dispersion) which is determined, and not the energy
(temperature). We first consider the possibility that clusters have relaxed
thermodynamically.

Thermodynamic equilibrium could result from elastic collisions between galaxies, and would be expected if the time scale for energy transfer in such collisions were shorter than the age of the cluster or the time-scale for the loss of kinetic energy through dissipative processes. The most important elastic two-body collisions in a cluster are gravitational. The resulting relaxation times have been calculated analytically by Chandrasekhar (1942) and Spitzer and Harm (1958), and numerically through N-body simulations, including a realistic luminosity function for galaxies by White (1976c, 1977b) and Farouki and Salpeter (1982). A useful characteristic time-scale (0.94 times the Spitzer 'reference time') is

(2.28) |

for the relaxation of a galaxy of mass *m* in a background field of
galaxies of mass *m*_{f} and number density
*n*_{f}.
is the ratio of
maximum to minimum impact
parameters of collisions which contribute to the relaxation; the maximum
impact
parameter is on the order of half the gravitation radius of the cluster
*R*_{G}
(equation 2.21), while the minimum impact parameter is roughly the larger of
the galaxy radius *r*_{g} or the 'turning radius'
*Gm*_{f} /
3_{r}^{2}
(White, 1976b).
Thus

(2.29) |

Usually, the second value applies. Thus, for *r*_{g}
20 kpc,
*R*_{G}
1 Mpc,
ln
3.

In order to give a lower limit to the relaxation time, we assume for the
moment that all the mass in a cluster (including the missing mass
(Section 2.8)) is bound to individual
galaxies; we later show this is
unlikely to be the case. Then, we define an average density
<>
3*M*_{tot}
/ (4
*R*_{G}^{3}),
where *M*_{tot} is the total mass of the cluster. If we
assume a Schechter luminosity function (equation 2.3), and a fixed
galaxy mass-to-light ratio, we find

(2.30) |

where *m*^{*} is the characteristic galaxy mass
(corresponding to *L*^{*}),
*N*^{*} is the
characteristic galaxy number (richness), *t*_{cr} is the
cluster crossing time (equation 2.26), and
(*r*) is
the total cluster density at *r*. Equation (2.30) is a
lower limit because it assumes all the cluster mass is bound to individual
galaxies. Typically,
*R*_{G}
1 Mpc, _{r}
1000 km/s,
*N*^{*}
1000, and ln
3,
giving *t*_{cr}(*R*_{G})
10^{9} yr and
*t*_{rel}(*m*^{*},
*R*_{G})
3 × 10^{11} yr. This is much
longer than a Hubble time (equation 2.27); it is therefore unlikely that the
apparently relaxed state of regular clusters results from two-body
collisions. However, two-body relaxation processes can affect the more
massive galaxies (*m* >> *m*^{*}) near
the cluster center
((*r*)
>> <>),
as is discussed below.