Radiative Hydrodynamics

The basic equation for gas which is affected by the radiation is as follows:

$$\frac{d \rho}{d t}+\rho\mbox{\boldmath${\nabla\cdot v}$}=0,$$ (D.1)

$$\rho\frac{d \mbox{\boldmath${v}$}}{d t}=\rho\nabla\phi - \nabla \psi + \frac{\chi_{F0}}{c}\mbox{\boldmath${F}$}_0,$$ (D.2)

$$\rho\frac{d \epsilon}{d t}+p\mbox{\boldmath${\nabla\cdot v}$}=h_{\rm CR}\rho +c\chi_{E0}E_0-4\pi \chi_{P0}B,$$ (D.3)

$$\nabla^2\psi=4\pi G \rho,$$ (D.4)

$$\epsilon = \frac{p}{(\gamma-1)\rho},$$ (D.5)

where $\chi$, $h_{\rm CR}$, and $B$ are the absorption coefficient, the cosmic-ray heating rate perunit mass, and teh Planck function $B=\sigma T^4/\pi$. In equation (D.2), the term $\frac{\chi_{F0}}{c}\mbox{\boldmath${F}$}_0$ represents the acceleration of gas due to the photon pressure. In equation (D.3), $h_{\rm CR}\rho$, $c\chi_{E0}E_0$ and $4\pi \chi_{P0}B$ represent, respectively the heating due to the CR particles, heating due to the absorption of radiation and cooling die to the emission. The frequence-integrated radiation energy density and radiation flux are defined as

$$E_0=\int_0^\infty d\nu_0\int d\Omega I(\nu,\mbox{\boldmath${n}$}),$$ (D.6)

$$\mbox{\boldmath${F}$}_0=\int_0^\infty d\nu_0\int d\Omega I(\nu,\mbox{\boldmath${n}$})\mbox{\boldmath${n}$},$$ (D.7)

where $I(\nu,\mbox{\boldmath${n}$})$ denotes the specific intensity of radiation at frequency $\nu$ along the direction vector $n$. These equation could be solved if the the radiation transfer is solved. The frequency-averaged absorption coefficients are defined as follows:

$$\chi_{F_0}=\left(\int d\nu \chi(\nu)\mbox{\boldmath${F}$}_0(\nu)\right)/\mbox{\boldmath${F}$}_0,$$ (D.8)

$$\chi_{E_0}=\left(\int d\nu \chi(\nu)E_0(\nu)\right)/E_0,$$ (D.9)

$$\chi_{P_0}=\left(\int d\nu \chi(\nu)B(\nu)\right)/B,$$ (D.10)

While the equation for the radiation transfer is basically as follows:

$$\frac{1}{c}\frac{\partial I(\nu,\mbox{\boldmath${r}$},\mbox{... ...$},\mbox{\boldmath${n}$})-S(\nu,\mbox{\boldmath${r}$})\right],$$ (D.11)

where $S(\nu,\mbox{\boldmath${r}$})$ represent the source term.