First and second type self-similar solutions of implosions and explosions containing ultrarelativistic shocks

We derive self-similar solutions for ultrarelativistic shock waves propagating into cold material of power law density profile in radius ρ^∝r^-k. We treat both implosions and explosions in three geometries: planar, cylindrical, and spherical. For spherical explosions these are the first type solutions of Blandford and McKee for k<4; they are the second type solutions found by Best and Sari for k>5-√3/4. In addition we find new, hollow (with evacuated interior), first type solutions that may be applicable for 4<k<17/4. This "sequence" with increasing k of first type solutions, hollow first type solutions, and then second type solutions is reminiscent of the nonrelativistic sequence. However, although in the nonrelativistic case there is a range of k which corresponds to a "gap" - a range in k with neither first nor second type solution which separates the hollow first type solutions and the second type solutions, here there is an "overlap": a range of k for which current considerations allow for both hollow first and second type solutions. Further understanding is needed to determine which of the two solutions apply in this overlap regime. We provide similar exploration for the other geometries and for imploding configurations. Interestingly, we find a gap for imploding spherical shocks and exploding planar shocks and an overlap for imploding planar solutions. Cylindrical configurations have no hollow solutions and exhibit a direct transition from first type to second type solutions, without a gap or an overlap region.