functions for different $\zeta$ are indistinguishable for $\xi \ll 1$ and $\xi \gg 1$, meaning that the self-similarity is nearly exact in those regions. In between those limits, where the flow is trans-relativistic, the deviations from self-similarity are apparent, though they remain small. Keeping the full form of $S$ in equations and therefore preserves well the self-similarity of our solutions and provides a [reasonable]{} interpolation between the relativistic and non-relativistic regions of the flow.

The solution for $g$ approaches infinity as we near the axis, which can be seen by investigating the small-$\xi$ behavior of equation . Specifically, letting $f \simeq \xi$, $f_{\xi} \simeq 1$, $\Gamma \simeq \zeta$, and $v_z \simeq 1$, equation can be written as

$$\xi{g_{\xi}}+\frac{1}{2}g \simeq -2(f_{\xi\xi,0})^2,$$

where $f_{\xi\xi,0}$ is the second derivative of $f$ evaluated at $\xi = 0$. This equation can be integrated, and we find

$$g \simeq \frac{C}{\sqrt{\xi}}-(f_{\xi\xi,0})^2,$$

$C$ being a constant of integration. Since it is roughly proportional to $1/\rho'$, the value of $g$ cannot be negative, meaning that $C$ must be greater than zero. Therefore, the asymptotic, $\xi \ll 1$ behavior of $g$ is $g \sim 1/\sqrt{\xi}$.

In addition to the mass density of scatterers and the velocity, we can also calculate 