Sedov-Taylor blast wave ($R_s=\xi (E{t}^2/{\rho }_0{)}^{1/5}$)

A self-similar energy-conserving strong spherical explosion into a uniform ambient density has $R_s=\xi (E{t}^2/{\rho }_0{)}^{1/5}$ and $v_s=2R_s/(5t)$. The dimensionless coefficient depends on the equation of state. Radiative losses, nonuniform media or continuous energy injection alter the similarity solution.

A monatomic strong shock has $T_d=3\bar{m}{V}_s^2/(16k_B)$. Together with the Sedov-Taylor blast wave radius this gives $T_d\propto t^{-6/5}$. Extrapolating to ambient temperature or sonic speed estimates when the cold-upstream assumption fails, but the strong-shock law is already becoming inaccurate there. Radiative cooling can end the energy-conserving stage earlier; its transition age additionally needs a cooling function and composition.