\tau/\delta\tau$; $\delta\tau$ is the characteristic boundary layer thickness in terms of $\tau$, which, from equations and in the previous subsection, is given by $\delta\tau \simeq \rho'\kappa\,\delta{y}$, or

$$\delta\tau^2 \simeq \frac{\rho'\kappa{z}}{\Gamma_jv_j}.$$

We are most interested in the behavior of the properties of the outflow when the velocities are relativistic, as this is the limit that is most applicable to sources of astronomical interest, and in order for the self-similar nature of our solutions to be upheld, the dependence of our self-similar functions on the bulk properties of the outflow, such as the $z$-dependence of the jet Lorentz factor along the axis, should be minimal. A fully self-similar solution is likely impossible here, as the speed of light plays a role in setting a finite scale factor for our solutions and becomes problematic when we try to connect the ultra- and non-relativistic regions of the outflow. We will show, however, that the non-self-similarity of our solutions only affects a small region of the outflow.

In paper I, we showed that the self-similarity of the comoving density of scatterers, $\rho'$, was approximately satisfied, i.e., we could find solutions with $\rho' \sim g(\xi)$. For the case at hand, then, this assumption implies that $g$ 