Relativistic Ionization Fronts

We derive the equations for the propagation of relativistic ionization fronts (I-fronts) in both static and moving gases, including the cosmologically-expanding intergalactic medium (IGM). We focus on the supersonic R-type phase that occurs right after a source turns on, and we compare the nonrelativistic and relativistic solutions for several important cases. Relativistic corrections can be significant up until the light-crossing time of the equilibrium Strömgren sphere. When q, the ratio of this lightcrossing time and the recombination time, exceeds unity, the time for the expanding I-front to reach the Strömgren radius is delayed by a factor of q. For a static medium, we obtain exact analytical solutions and apply them to the illustrative problems of an O star in a molecular cloud and a starburst in a high-redshift cosmological halo. Relativistic corrections can be important at early times when the H II regions are small, as well as at later times, if a density gradient causes the I-front to accelerate. For the cosmologically-expanding IGM, we derive an analytical solution in the case of a steady source and a constant clumping factor. Here relativistic corrections are significant for short-lived, highly-luminous sources like QSOs at the end of reionization (z ≃ 6), but negligible for weaker or higher-redshift sources. Finally, we numerically calculate the evolution of relativistic I-fronts in the presence of small-scale structure and infall, for a large galaxy undergoing a starburst and a luminous, high-redshift QSO. For such strong and short-lived (z ∼ 7) sources, the relativistic corrections are quite significant, and small-scale structure can decrease the size of the H II region by up to an additional ∼ 25%. However, most of the IGM was ionized by smaller, higher redshift sources. Thus the impact of relativistic corrections on global reionization is small and can usually be neglected. Subject headings: H II regions—ISM: bubbles—ISM: galaxies: halos—galaxies: high-redshift— intergalactic medium—cosmology: theory