= Energy normalization of a continuously driven spherical shock
If the similarity profiles account for all kinetic and internal energy inside an adiabatic strong shock, let
$$
I=\int_0^1\left(\frac12fh^2+\frac{g}{\gamma-1}\right)\eta^2\,d\eta.
$$
The injected energy $\dot Et$ equals $4\pi\rho_1\dot R^2R^3I$. Consequently <conservation of energy> fixes
$$
\alpha=\left(\frac{25}{36\pi I}\right)^{1/5}.
$$
An interior region not included in the profiles must have its energy added separately; a radiatively cooled shell changes this energy budget.
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