 Solution

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.