Thermal energy conservation

It is enlightening to construct from Eqs.(2.22) an equation for the energy per volume unit $e=(p_{\rm r}+2p_{\rm t})/2$ which, in the case of an isotropic gas ( $p_{\rm r}=p_{\rm t}$) is $e=3p/2$. For this aim we take, for instance, equation (2.37) and in the term $2p_{\rm r}\partial u_{\star}/\partial r$ we include now a source for radiation pressure, $2(p_{\rm r}+p_{\rm rad})\partial u_{\star}/\partial r$ and we divide everything by $e$ so that we get the logarithmic variables. The resulting equation is

$$\frac{\partial \ln e}{\partial t}+(u_{\rm g}+3v_{\rm g}) \fr... ... \frac{1}{e}\bigg( \frac{\delta e}{\delta t} \bigg)_{\rm coll}$$ (62)

The interaction terms for this equation are

$$\bigg( \frac{\delta e}{\delta t} \bigg)_{\rm drag}=X_{\rm drag}(\sigma_{\rm r}^2+\sigma_{\rm t}^2 +(u_{\star}-u_{\rm g})^2)$$ (63)

$$\bigg( \frac{\delta e}{\delta t} \bigg)_{\rm coll}=\frac{1}{... ...)^2 \epsilon+(u_{\star}-u_{\rm g})^2 -\xi \sigma_{\rm coll}^2)$$ (64)