Cold filaments in galaxy clusters: effects of heat conduction

We determine the critical size l(crit) of a filament of cold (T similar to 10(4) K) gas that is in radiative equilibrium with X-ray emitting gas at temperatures T-out similar to 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 P-1. 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.