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). For the supersonic R-type phase that occurs right after a source turns on, relativistic corrections can be significant up until the light crossing time of the equilibrium Stromgren sphere. When q, the ratio of this light-crossing time to the recombination time, exceeds unity, the time for the expanding I-front to reach the Stromgren 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 both at early times when the H (II) regions are small and at later times if a density gradient causes the I-front to accelerate. An analytical solution is also derived for a steady source in a cosmologically expanding IGM. Here relativistic corrections are significant for short-lived, highly luminous sources such as QSOs at the end of reionization (z similar or equal to 6) but negligible for weaker or higher redshift sources. Finally, we numerically solve the equations for relativistic, cosmological I-fronts in the presence of evolving small-scale structure, for a large-galaxy starburst and a luminous QSO. For such strong and short-lived sources at z similar to 7, relativistic corrections are quite significant, and small-scale structure can decrease the size of the H (II) region by up to an additional similar to 25%. However, most of the IGM was ionized by smaller, higher redshift sources. Thus, the effect of relativistic corrections on global reionization is small and can usually be neglected.