equation , we find $f_{\xi}(\infty) = 0$. Finally, the density of scatterers should approach that of the ambient medium far from the jet center. Equation then gives $g(\infty) = 1+3/(4\mu)$.

As we noted, the truly self-similar limit of equations and is obtained by setting $S = \ln(\Gamma{v_z})$, but we encounter an issue with this scaling when we consider the boundary conditions on our flow in the ambient medium. In particular, we expect the outflow velocity to approach zero for $\xi \gg 1$; when the velocity becomes subrelativistic, however, $\ln(\Gamma{v_z}) \rightarrow -\infty$, when in actuality the $v_z \ll 1$ limit of equation is $S \simeq \ln(1+\Gamma{v_z}) \simeq \Gamma{v_z} \simeq 0$. Thus, we see that taking the ultrarelativistic limit of equations and will not result in the solutions matching the correct boundary conditions far from the jet center.

To correct this problem, we will allow the function $S$ to take on its full form and treat $\zeta$ as a constant, which will break the self-similarity of our solutions. However, since our boundary conditions do not depend on the value of $\zeta$, the functions $f$ and $g$ themselves should be largely independent of that parameter. We therefore expect the assumption of self-similarity to 