An analytic model of gravitational collapse induced by radiative cooling: Instability scale, density profile, and mass infall rate

We present an analytic description of the spherically symmetric gravitational collapse of radiatively cooling gas clouds, which illustrates the mechanism by which radiative cooling induces gravitational instability at a characteristic mass scale determined by the microphysics of the gas. The approach is based on developing the density-temperature relationship of the gas into a full dynamical model. We convert the density-temperature relationship into a barotropic equation of state, based on which we develop a refined instability criterion and calculate the density and velocity profiles of the gas. From these quantities, we determine the time-dependent mass infall rate on to the centre of the cloud. This approach distinguishes the rapid, quasi-equilibrium contraction of a cooling gas core to high central densities from the legitimate instability this contraction establishes in the envelope. We explicate the model in the context of a primordial mini-halo cooled by molecular hydrogen, and then provide two further examples: a delayed collapse with hydrogen deuteride cooling and the collapse of an atomic-cooling halo. In all three cases, we show that our results agree well with full hydrodynamical treatments.