= Solution
The dimensions of the injected power and ambient <mass density> are $[\dot E]=ML^2T^{-3}$ and $[\rho_1]=ML^{-3}$. Requiring $\dot E^a\rho_1^bt^c$ to have the dimensions of length gives
$$
a+b=0,\qquad2a-3b=1,\qquad-3a+c=0.
$$
Thus <dimensional analysis> gives the <adiabatic superbubble similarity solution>
$$
\boxed{a=\frac15,\quad b=-\frac15,\quad c=\frac35,\qquad R(t)=\alpha\left(\frac{\dot E}{\rho_1}\right)^{1/5}t^{3/5},\qquad\dot R=\frac{3R}{5t}.}
$$
The 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.
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