$$\label{polytropic}
\gamma = \cases {\begin{array}{lll}
1       &                         & \rho \leq 1.0 \times 10^{-13} \cr
7/5     & \ 1.0 \times 10^{-13}  < \hspace{-6pt} & \rho \leq 5.7 \times 10^{-8} 
\cr
1.15    & \ 5.7 \times 10^{-8\ } < \hspace{-6pt} & \rho < 1.0 \times 10^{-3} \cr
5/3     &                         & \rho > 1.0 \times 10^{-3} \cr
\end{array}}$$ where the densities are in ${\rm g\ cm}^{-3}$ (see Figure \[bate.tmr\]). The values of $\gamma$ and the transition densities were derived from Tohline [@Tohline82]. The variable value of $\gamma$ mimics the following behaviour of the gas. The collapse is isothermal ($\gamma=1$) until the gas becomes optically thick to infrared radiation at $\rho \approx 10^{-13} {\rm \ g \ cm}^{-3}$, beyond which $\gamma=7/5$ (appropriate for a diatomic gas). When the gas reaches a temperature of $\approx$ 2000 K ($\rho = 5.7 \times 10^{-8} {\rm \ g \ cm}^{-3}$), molecular hydrogen begins to dissociate and the temperature only slowly increases with density. In this phase $\gamma = 1.15$ is used to model [*both*]{} the decreasing mean molecular weight and the slow increase of temperature with density, the latter of which has an effective $\gamma \approx 1.10$. Finally, when the gas has fully dissociated ($\rho \approx 10^{-3} {\rm 