The basic equation for gas which is affected by the radiation is as follows:
 (D.1) (D.2) (D.3) (D.4) (D.5)

where , , and are the absorption coefficient, the cosmic-ray heating rate perunit mass, and teh Planck function . In equation (D.2), the term represents the acceleration of gas due to the photon pressure. In equation (D.3), , and 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
 (D.6) (D.7)

where denotes the specific intensity of radiation at frequency along the direction vector . These equation could be solved if the the radiation transfer is solved. The frequency-averaged absorption coefficients are defined as follows:
 (D.8) (D.9) (D.10)

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

 (D.11)

where represent the source term.

Kohji Tomisaka 2012-10-03