  On the basis of the universal gas fraction in clusters of galaxies, we
estimate that the effective thermal conductivity required to balance radiative
cooling in the cores, where the gas temperature is 3-10keV, is about one tenth
of the Spitzer rate. This confirms that thermal conduction can be important for
the energy balance provided that it is not highly suppressed by magnetic fields
in the gas. We determine the global effective conductivity in a sample of 29
clusters using published X-ray data on the inferred cooling rates and show that
most lie between one and one tenth of the Spitzer rate. More work on the
profiles in cooling flow clusters is required to test the conduction hypothesis
further. We examine the possibility that conduction operates during galaxy
formation, and show that it provides a simple explanation for the upper-mass
cutoff in galaxy masses.
