  Chandra and XMM-Newton observations have confirmed the presence of large
temperature gradients within the cores of many relaxed clusters of galaxies.
Here we investigate whether thermal conduction operating over those gradients
can supply sufficient heat to offset radiative cooling. Narayan & Medvedev
(2001) and Gruzinov (2002) have noted, using published results on cluster
temperatures, that conduction within a factor of a few of the Spitzer rate is
sufficient to balance bremsstrahlung cooling. From a detailed study of the
temperature and emission measure profiles of Abell 2199 and Abell 1835, we find
that the heat flux required by conduction is consistent with or below the rate
predicted by Spitzer in the outer regions of the core. Conduction may therefore
explain the lack of observational evidence for large mass cooling rates
inferred from arguments based simply on radiative cooling, provided that
conductivity is suppressed by no more than a factor of three below the full
Spitzer rate. To stem cooling in the cluster centre, however, would necessitate
conductivity values at least a factor of two larger than the Spitzer values,
which we consider implausible. This may provide an explanation for the observed
star formation and optical nebulosities in cluster cores. The solution is
likely to be time dependent. We briefly discuss the possible origin of the
cooler gas and the implications for massive galaxies.
