Radiative Hydrodynamics

The basic equation for gas which is affected by the radiation is as follows:
    $\displaystyle \frac{d \rho}{d t}+\rho\mbox{\boldmath${\nabla\cdot v}$}=0,$ (D.1)
    $\displaystyle \rho\frac{d \mbox{\boldmath${v}$}}{d t}=\rho\nabla\phi - \nabla \psi + \frac{\chi_{F0}}{c}\mbox{\boldmath${F}$}_0,$ (D.2)
    $\displaystyle \rho\frac{d \epsilon}{d t}+p\mbox{\boldmath${\nabla\cdot v}$}=h_{\rm CR}\rho
+c\chi_{E0}E_0-4\pi \chi_{P0}B,$ (D.3)
    $\displaystyle \nabla^2\psi=4\pi G \rho,$ (D.4)
    $\displaystyle \epsilon = \frac{p}{(\gamma-1)\rho},$ (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
$\displaystyle E_0=\int_0^\infty d\nu_0\int d\Omega I(\nu,\mbox{\boldmath${n}$}),$     (D.6)
$\displaystyle \mbox{\boldmath${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:
$\displaystyle \chi_{F_0}=\left(\int d\nu \chi(\nu)\mbox{\boldmath${F}$}_0(\nu)\right)/\mbox{\boldmath${F}$}_0,$     (D.8)
$\displaystyle \chi_{E_0}=\left(\int d\nu \chi(\nu)E_0(\nu)\right)/E_0,$     (D.9)
$\displaystyle \chi_{P_0}=\left(\int d\nu \chi(\nu)B(\nu)\right)/B,$     (D.10)

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

\frac{1}{c}\frac{\partial I(\nu,\mbox{\boldmath${r}$},\mbox{...
\end{displaymath} (D.11)

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

Kohji Tomisaka 2009-12-10