Photodissociation of H2 in protogalaxies: modelling self-shielding in three-dimensional simulations

The ability of primordial gas to cool in protogalactic haloes exposed to ultraviolet (UV) radiation is critically dependent on the self-shielding of H2. We perform radiative transfer calculations of LW line photons, post-processing outputs from three-dimensional adaptive mesh refinement simulations of haloes with Tvir? 104 K at z similar to 10. We calculate the optically thick photodissociation rate numerically, including the effects of density, temperature and velocity gradients in the gas, as well as line overlap and shielding of H2 by H i, over a large number of sightlines. In low-density regions (n ? 104 cm-3) the dissociation rates exceed those obtained using most previous approximations by more than an order of magnitude; the correction is smaller at higher densities. We trace the origin of the deviations primarily to inaccuracies of (i) the most common fitting formula for the suppression of the dissociation rate, from Draine and Bertoldi and (ii) estimates for the effective shielding column density from local properties of the gas. The combined effects of gas temperature and velocity gradients are comparatively less important, typically altering the spherically averaged rate only by a factor of ?2. We present a simple modification to the Draine & Bertoldi fitting formula for the optically thick rate which improves agreement with our numerical results to within similar to 15 per cent, and can be adopted in future simulations. We find that estimates for the effective shielding column can be improved by using the local Sobolev length. Our correction to the H2 self-shielding reduces the critical LW flux to suppress H2 cooling in Tvir? 104 K haloes by an order of magnitude; this increases the number of such haloes in which supermassive (M similar to 105 M?) black holes may have formed.