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

The ability of primordial gas to cool in proto-galactic haloes exposed to Lyman-Werner (LW) radiation is critically dependent on the self-shielding of H_2. We perform radiative transfer calculations of LW line photons, post-processing outputs from three-dimensional adaptive mesh refinement (AMR) simulations of haloes with T_vir > 10^4 K at redshifts around z=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 H_2 by HI, over a large number of sight-lines. In low-density regions (n<10^4 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 (Draine & Bertoldi 1996) for the suppression of the dissociation rate 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 less than two. We present a simple modification to the DB96 fitting formula for the optically thick rate which improves agreement with our numerical results to within approx. 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 H_2 self-shielding reduces the critical LW flux to suppress H_2-cooling in T_vir>10^4 K haloes by an order of magnitude; this increases the number of such haloes in which supermassive (approx. M=10^5 M_sun) black holes may have formed.