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) |