Second-type self-similar solutions to the ultrarelativistic strong explosion problem

The well-known Blandford–McKee solution describes the ultrarelativistic flow in a spherical blast wave enclosed by a strong shock. It is valid when the density of the external medium into which the shock propagates varies with the distance r from the origin as $r^{\mathrm{-k}},$ for $\text{k<4.}$ These are first-type self-similar solutions in which the shock Lorentz factor Γ varies as ${\Gamma }^2{\mathrm{\propto t}}^{\mathrm{-m}},$ where $\text{m=3−k}$ to ensure energy conservation. New second-type self-similar solutions, valid for $\text{k>5−}\sqrt{\text{3/4}}\text{≈4.13,}$ are presented. In these solutions Γ varies as ${\Gamma }^2{\mathrm{\propto t}}^{\mathrm{-m}}$ with $\text{m=(3−2}\sqrt{3}\text{)k−4(5−3}\sqrt{3})$ so that the shock accelerates and the fraction of the flow energy contained in the vicinity of the shock decreases with time. The new solutions are shown to be in excellent agreement with numerical simulations of the flow equations. It is proved that no second-type self-similar solutions exist for $\text{k<5−}\sqrt{\text{3/4}}\text{≈4.13.}$