**5.4.2. Thermal conduction**

In a plasma with a gradient in the electron temperature, heat is
conducted down
the temperature gradient. If the scale length of the temperature gradient
*l*_{T}
*T*_{e} / |
*T*_{e}| is much longer than the mean free path of
electrons _{e}, then the heat flux is given by

(5.36) |

where the thermal conductivity for a hydrogen plasma is (Spitzer, 1956)

(5.37) |

Because of the inverse dependence on the particle mass, thermal conduction is primarily due to electrons. This equation includes a correction for the self-consistent electric field set up by the diffusing electrons. If the very weak dependence of ln on density is ignored, then is independent of density but depends very strongly on temperature.

If the scale length of the thermal gradient *l*_{T} is
comparable to or less than the mean free path of electrons, then
equation (5.36) overestimates the heat flux,
since it would imply that the electrons are diffusing at a speed greater
than their
average thermal speed. Under these circumstances the conduction is said to
'saturate', and the heat flux approaches a limiting value
*Q*_{sat}.
Cowie and McKee (1977)
calculate this saturated heat flux by assuming that the electrons have a
Maxwellian distribution, and an infinitely steep temperature gradient,
and that
the correction for a self-consistent electric field is the same as in
the unsaturated case. They find

(5.38) |

A general expression for the heat flux, which interpolates between the two limits of equations (5.36) and (5.38), is then

(5.39) |

The mean free path of electrons in the intracluster gas (equation 5.34) is typically small compared to the cluster dimensions, and heat conduction within the intracluster gas itself is probably unsaturated. However, the mean free path is comparable to the size of a galaxy, and saturated heat conduction may be important in evaporation from or accretion to galaxies (Sections 5.7 and 5.9).

Within the intracluster medium, thermal conduction will act to transport heat from hot to cold regions and, in the absence of any competing effect, to make the temperature spatially constant (isothermal). Assuming equal ion and electron temperatures, the temperature in a Lagrangian element of the intracluster gas will vary as

(5.40) |

where _{g}
is the gas density. It is useful to define a conduction time scale as
*t*_{cond}
-(*d* ln *T*_{e} / *dt*)^{-1}, which is
on the order of |*t*_{cond}|
(*n*_{e}
*l*_{T}^{2}*k*) /
. As a
specific example, consider a cluster in which the gas is hydrostatic,
adiabatic
(isentropic), and extends to very large distances but is not in contact
with any intercluster gas; such models are discussed in some detail in
Section 5.5.2 below.
If the cluster potential is given by the analytic King form (equation
5.59 below),
the gas is assumed to cool isobarically (at constant pressure), and the
variation of ln
is ignored, then the conduction time at a radius *r* is given by

(5.41) |

where _{0} and
_{go}
are the conductivity and gas density at the cluster center,
and *r*_{c} is the cluster core radius. The function
*g*(*x*) is

(5.42) |

where *f*(*x*)
(*r*) /
_{0} is the ratio of
the cluster potential to its central value
and is given by equation (5.59) below. The function *g*(*x*)
is plotted in Figure 36.
Because conduction only transports heat, the average temperature of the
gas is not changed; in the inner parts the gas is cooled and in the
outer parts the gas is
heated. However, because the X-ray emission is proportional to the
square of the
density, temperatures determined from X-ray spectra are mainly affected
by the innermost gas and are lowered by conduction. As is clear from
Figure 36, heat conduction is most effective in
the cluster core, and |*t*_{cond}|
increases very rapidly with radius. Since *g*(0) = 1, the central
value to the conduction time scale *t*_{cond}(0) is given
by the first term in equation (5.41), or

(5.43) |

where *n*_{0} is the central proton density, and solar
abundances have been
assumed. Thus heat conduction may be relatively effective in the core of a
cluster. At radii *r*
2*r*_{c}, the conduction time is typically a factor of
100 longer,
and conduction is only marginally effective in the outer parts of the
cluster. The
conduction time may be increased further by the presence of a magnetic
field in the cluster.