**5.4. Transport processes**

Processes that redistribute energy, momentum, or heavy elements within the intracluster gas will now be reviewed.

**5.4.1 Mean free paths and equilibration time
scales**

The mean free paths of electrons and ions in a plasma without a magnetic
field are determined by Coulomb collisions. As in the stellar dynamical
case, it is important to include distant as well as nearby
collisions. The mean free path
_{e}
for an electron to suffer an energy exchanging collision with another
electron is given by
(Spitzer, 1956)

(5.32) |

where *T*_{e} is the electron temperature,
*n*_{e} is the electron number density, and
is the ratio of
largest to smallest impact parameters for the collisions. For
*T*_{e}
4 ×
10^{5} K, this Coulomb logarithm is

(5.33) |

which is nearly independent of density or temperature. Equation (5.32)
assumes that the electrons have a Maxwellian velocity distribution at
the electron temperature. The equivalent mean free path of ions
_{i} is given
by the same formula, replacing the electron temperature and density with
the ion temperature
*T*_{i} and density, dividing by the ion charge to the
fourth power, and slightly
increasing ln . In
the discussion that follows the ions will generally be assumed
to be protons, and the diffusion of heavy elements will be discussed in
Section 5.4.5 below. Numerically,

(5.34) |

assuming that *T*_{e} = *T*_{i} =
*T*_{g}.

In general, these mean free paths are shorter than the length scales of interest in clusters ( 1 Mpc), and the intracluster medium can be treated as a collisional fluid, satisfying the hydrodynamic equations. Note that the mean free paths are comparable to the size of a galaxy, however, and in the interaction between intracluster gas and individual galaxies the gas may be nearly collisionless, as mentioned previously (Section 5.3.4).

If a homogeneous plasma is created in a state in which the particle
distribution is non-Maxwellian, elastic collisions will cause it to
relax to a
Maxwellian distribution on a time scale determined by the mean free paths
(Spitzer, 1956,
1978).
Electrons will achieve this equilibration (isotropic
Maxwellian velocity distribution characterized by the electron
temperature) on a time scale set roughly by *t*_{eq}(*e,
e*)
_{e}
/ <*v*_{e}>_{rms}, where the denominator is the
rms electron velocity,

(5.35) |

The time scale for protons to equilibrate among themselves is
*t*_{eq}(*p, p*)
(*m*_{p}
/ *m*_{e})^{1/2} *t*_{eq}(*e, e*),
or roughly 43 times longer than the value in
equation (5.35). Following this time, the protons and ions would each have
Maxwellian distributions, but generally at different temperatures. The
time scale for the electrons and ions to reach equipartition
*T*_{e} = *T*_{i} is
*t*_{eq}(*p, e*)
(*m*_{p}
/ *m*_{e})*t*_{eq}(*e, e*), or roughly
1870 times the value in equation (5.35).
For heavier ions, the time scales for equilibration are generally at least
this short if the ions are nearly fully stripped, because the increased
charge more than makes up for the increased mass. For
*T*_{g}
10^{8} K and
*n*_{e}
10^{-3} cm^{-3}, the longest equilibration time scale is
only *t*_{eq}(*p, e*)
6 ×
10^{8} yr.
Since this is shorter than the age of the cluster or the cooling time, the
intracluster plasma can generally be characterized by a single kinetic
temperature *T*_{g}. Under some circumstances, plasma
instabilities may bring
about a more rapid equilibration than collisions
(McKee and Cowie, 1977).