= Spherical blast wave in a power-law ambient density
{title2=$R\propto(E/C)^{1/(5-\beta)}t^{2/(5-\beta)}$}
For an <energy>-conserving strong spherical <shock wave> expanding into <mass density> $Cr^{-\beta}$, $0\leq\beta<3$, <dimensional analysis> gives the displayed radius law. Neglect ambient <pressure>, radiation losses and gravity in this blast-dominated approximation. Write $u=\dot R U(\xi)$, $\rho=CR^{-\beta}D(\xi)$ and $p=CR^{-\beta}\dot R^2P(\xi)$, with $\xi=r/R$. The radial <continuity equation>, momentum equation and adiabatic <pressure> equation reduce to
$$
(U-\xi)D'+D(U'+2U/\xi-\beta)=0,\qquad
(U-\xi)U'+\frac{\beta-3}{2}U=-P'/D,
$$
$$
(U-\xi)P'+[\gamma(U'+2U/\xi)-3]P=0.
$$
The <Strong-shock Rankine-Hugoniot conditions> give $U(1)=P(1)=2/(\gamma+1)$ and $D(1)=(\gamma+1)/(\gamma-1)$. The explosion <energy> fixes the radius coefficient through $E=4\pi CR^{3-\beta}\dot R^2\int_0^1[DU^2/2+P/(\gamma-1)]\xi^2d\xi$.
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