= Solution
The <conservation of energy> to use here is the total <kinetic energy> plus <internal energy> inside the shock, not just the internal energy of a uniform-pressure core. With no radiative loss and negligible initial ambient energy,
$$
\dot Et=4\pi\int_0^{R(t)}\left(\frac12\rho u^2+\frac{p}{\gamma-1}\right)r^2\,dr.
$$
Substitute the <adiabatic superbubble similarity solution> and define the finite positive dimensionless integral
$$
I=\int_0^1\left(\frac12f(\eta)h(\eta)^2+\frac{g(\eta)}{\gamma-1}\right)\eta^2\,d\eta.
$$
Then $\dot Et=4\pi\rho_1\dot R^2R^3I$. Since $\dot R=3R/(5t)$ and $R^5=\alpha^5(\dot E/\rho_1)t^3$,
$$
\dot Et=\frac{36\pi}{25}\alpha^5\dot Et\,I,
\qquad\boxed{\alpha=\left(\frac{25}{36\pi I}\right)^{1/5}.}
$$
This is the <energy normalization of a continuously driven spherical shock>. It fixes $\alpha$ once the similarity profiles are determined. If the profiles describe only the shocked ambient layer, the energy of the interior hot bubble must also be included before using this normalization. A radiatively cooled shell has a different energy budget; its familiar numerical coefficient should not be substituted into this loss-free model.
Back to article page
