Thermal instability in clusters of galaxies with conduction

We consider a model of galaxy clusters in which the hot gas is in hydrostatic equilibrium and maintains energy balance between radiative cooling and heating by thermal conduction. We analyze the thermal stability of the gas using a Lagrangian perturbation analysis. For thermal conductivity at the level of similar to 20% - 40% of Spitzer conductivity, consistent with previous estimates for cluster gas, we find that the growth rate of the most unstable global radial mode is similar to 6-9 times lower than the growth rate of local isobaric modes at the cluster center in the absence of conduction. The growth time in typical clusters is similar to 2-5 Gyr, which is comparable to the time since the last major merger episode, when the gas was presumably well mixed. Thus, we suggest that thermal instability is not dynamically significant in clusters, provided that there is an adequate level of thermal conduction. On the other hand, if the heating of the gas is not the result of thermal conduction or any other diffusive process such as turbulent mixing, then the thermal instability has a growth time under a gigayear in the central regions of the cluster and is a serious threat to equilibrium. We also analyze local nonradial modes and show that the Lagrangian technique leads to the same dispersion relation as the Eulerian approach, provided that clusters are initially in strict thermal equilibrium. Because cluster gas is convectively stable, nonradial modes always have a smaller growth rate than equivalent radial modes.