0$.

Solutions
---------

In this section we plot solutions for the outflow velocity, the density of scatterers and the density of photons for various values of $\mu$ and $\zeta$. As was mentioned in the previous subsection, the physical self-similar variable against which we would like to plot our solutions is given by $\alpha$. However, as is apparent from equation , the definition of $\alpha$ depends on $\zeta$. Therefore, if we are comparing, for example, the outflow velocity of two systems with differing $\zeta$, we must incorporate the $\zeta$ dependence in $\alpha$ so that the range of physical space that we consider for each solution is the same. For this reason, in this section we will plot our solutions as functions of the variable

$$\tilde\alpha \equiv \frac{\alpha}{\zeta} = y\sqrt{\frac{9}{2}\frac{\rho'_0\kappa}{z_0}}\bigg{(}\frac{z}{z_0}\bigg{)}^{-1/2} \label{alphatilde}.$$

![The normalized $z$-component of the four-velocity ($f_{\xi}$) for $\mu = 1$ and $\zeta = 4$, 10, and 25 (the solid, blue curve, the dot-dashed, purple curve, and the dashed, red curve, respectively), which, for $z \simeq z_0$, correspond to $\Gamma_j = 4, 10, $ and 25. We see that the width of the boundary layer is nearly unchanged as we alter the value of $\zeta$. []{data-label="fig:fpplotsgamjet"}](Figure3.eps){width="3.5in"}

![The normalized fluid-frame density of 