is independent of $\zeta$. This means, however, that the observer-frame density of scatterers, given by $\rho = \Gamma\rho' \simeq \zeta\,{g}$, can be made arbitrarily large (as $g$ is independent of $\zeta$ in the self-similar limit), which is a nonsensical result.

Motivated by this reasoning, we conclude that the comoving density of scatterers is unlikely to be independent of $\zeta$. On the contrary, a more reasonable approximation is that the *observer*-frame density of scatterers varies self-similarly. Investigating equations and , we see that the $\rho'$-dependent quantity that enters both is the combination $1+4e'/(3\rho')$. Our expectation that the observer-frame density varies self-similarly then prompts the assumption

$$1+\frac{4}{3}\frac{e'}{\rho'} = \frac{4}{3}\mu\,\Gamma\,g(\xi) \label{rhosim},$$

where $\mu \equiv e'/\rho'_0$, $\rho'_0$ being the density of scatterers in the ambient medium. (Note that we could have simply let $\rho'\Gamma = g(\xi)$, but equation , which is merely a change of variables from the initial assignment $\rho'\Gamma = g(\xi)$, will allow equations and to be written in a more compact form.)

We can determine the value of $m$, which controls how rapidly the flow decelerates due to entrainment, by integrating the momentum flux, $\dot{P} \simeq e'\Gamma^2v_z^2$ for a radiation-dominated system, over the entire boundary layer, and requiring that the result 