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For point injection of constant power, the outward energy flux through a sphere tending to the source must tend to $\dot{E}$. In the adiabatic superbubble similarity solution, this requiresImposing finite regular central pressure and density with $h=O(\eta )$ would instead give zero flux and fail to supply the prescribed power. A resolved source region or an inner contact-boundary model can replace the point-source condition.

Substitution of the adiabatic superbubble similarity solution into spherical mass conservation gives

The dimensions of the injected power and ambient mass density are $[\dot{E}]=M{L}^2{T}^{-3}$ and $[{\rho }_1]=M{L}^{-3}$. Requiring ${\dot{E}}^a{\rho }_1^b{t}^c$ to have the dimensions of length givesThus dimensional analysis gives the adiabatic superbubble similarity solutionThe dimensionless coefficient $\alpha$ can depend on $\gamma$ and the specified inner structure, but dimensions alone cannot determine it. Constant power differs from a fixed-energy explosion because the energy available at time $t$ is proportional to $t$. This scaling assumes that ambient pressure, gravity, radiative losses and the finite injection radius introduce no relevant competing scale.

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,Substitute the adiabatic superbubble similarity solution and define the finite positive dimensionless integralThen $\dot{E}t=4\pi {\rho }_1{\dot{R}}^2{R}^{3}I$. Since $\dot{R}=3R/(5t)$ and $R^5={\alpha }^5(\dot{E}/{\rho }_1)t^3$,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.