  We determine the critical size l_crit of a filament of cold (T~10^4 K) gas
that is in radiative equilibrium with X-ray emitting gas at temperatures
T_out~10^6 - 10^8 K. Filaments smaller than l_crit will be rapidly evaporated,
while longer ones will induce the condensation of the ambient medium. At fixed
pressure P, l_crit increases as T_out^(11/4), while at fixed T_out it scales as
1/P. It scales as f^(1/2), where f is the factor by which the magnetic field
depresses the thermal conductivity below Spitzer's benchmark value. For
plausible values of f, l_crit is similar to the lengths of observed filaments.
In a cluster such as Perseus, the value of l_crit increases by over an order of
magnitude between the centre and a radius of 100 kpc. If the spectrum of seed
filament lengths l is strongly falling with l, as is natural, then these
results explain why filaments are only seen within a few kiloparsecs of the
centres of clusters, and are not seen in clusters that have no cooling flow. We
calculate the differential emission measure as a function of temperature for
the interface between filaments and ambient gas of various temperatures. We
discuss the implications of our results for the origin of the galaxy luminosity
function.
