‘explosion’. Consequently we expand the transition functions in the Newtonian potential difference $ - 2M(1/R - 1/R_0)$, which is a small quantity at the beginning of the expansion. Case (3) is trivial. For case (2) we approximate $\mathrm{d}U/\mathrm{d}u$ around the ‘ignition’, $R = R_0$. For case (1) we have to consider two points at which the ray crosses the shell. The effect from the entrance of the ray into the shell will be larger than the effect from the exit of the ray. Thus we need to approximate $\mathrm{d}V/\mathrm{d}v$ around $\gamma_0$ and $\mathrm{d}U/\mathrm{d}u$ around $\gamma(R(U=0)) =: \gamma_1$. Around $\gamma_0$ we include terms up to linear order, whereas around $\gamma_1$ we keep only the leading term. These expansions yield $$\begin{aligned}
\frac{{\mathrm{d}}U}{{\mathrm{d}}u} \bigg|_{U>-U_0} 
    &\approx& c_0 - c_0' \frac{2M}{R(U)}\; \; \mbox{around} \; \; \gamma_0\\ 
\frac{{\mathrm{d}}U}{{\mathrm{d}}u} \bigg|_{U>-U_0} 
    &\approx& c_1 \; \; \mbox{around} \; \; \gamma_1\\ 
\frac{{\mathrm{d}}V}{{\mathrm{d}}v} \bigg|_{V>0} 
    &\approx& d_0 - d_0' \frac{2M}{R(V)} 
                          \; \; \mbox{around} \; \; \gamma_0\;\;,\end{aligned}$$ where $$\begin{aligned}
c_0  &:=& \frac{{\mathrm{d}}U}{{\mathrm{d}}u}\bigg|_{\gamma_0} +
          c_0'(1-\gamma_0)\ , \qquad 
c_0'  :=  \frac{\partial}{\partial\gamma}\,
          \frac{{\mathrm{d}}U}{{\mathrm{d}}u}\bigg|_{\gamma_0}\ ,\\
\label{c1}
c_1  &:=& \frac{{\mathrm{d}}U}{{\mathrm{d}}u}\bigg|_{\gamma_1}\ ,\\ 
\label{d}
d_0   &:=& \frac{{\mathrm{d}}V}{{\mathrm{d}}v}\bigg|_{\gamma_0} +
           d_0' (1-\gamma_0)\ , \qquad
d_0' := \frac{\partial}{\partial\gamma}\,
        \frac{{\mathrm{d}}V}{{\mathrm{d}}v}\bigg|_{\gamma_0}\ .\end{aligned}$$ As a consequence $$\begin{aligned}
\label{dudU}
\frac{{\mathrm{d}}u}{{\mathrm{d}}U} &\approx& 
\frac{R_0(1-\nu)+\nu(U+U_0)}{c_0[R_0(1-\nu)+\nu(U+U_0)]-c_0' 2M(1-\nu)} \\
\label{dvdV}
\frac{{\mathrm{d}}v}{{\mathrm{d}}V}&\approx& 
\frac{R_0(1+\nu)+\nu 