# 輻射轉移

## 定義

${\displaystyle dE_{\nu }=I_{\nu }(\mathbf {r} ,{\hat {\mathbf {n} }},t)\cos \theta \ d\nu \,da\,d\Omega \,dt}$

## 輻射轉移方程式

${\displaystyle {\frac {1}{c}}{\frac {\partial }{\partial t}}I_{\nu }+{\hat {\Omega }}\cdot \nabla I_{\nu }+(k_{\nu ,s}+k_{\nu ,a})I_{\nu }=j_{\nu }+{\frac {1}{4\pi c}}k_{\nu ,s}\int _{\Omega }I_{\nu }d\Omega }$

## 輻射轉移方程式的解

${\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau (s_{0},s)}+\int _{s_{0}}^{s}j_{\nu }(s')e^{-\tau (s',s)}\,ds'}$

${\displaystyle \tau (s_{1},s_{2})\ {\stackrel {\mathrm {def} }{=}}\ \int _{s_{1}}^{s_{2}}\alpha _{\nu }(s)\,ds}$

### 局部熱力學平衡

${\displaystyle {\frac {j_{\nu }}{\alpha _{\nu }}}=B_{\nu }(T)}$

${\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau (s_{0},s)}+\int _{s_{0}}^{s}B_{\nu }(T(s'))\alpha _{\nu }(s')e^{-\tau (s',s)}\,ds'}$

### 愛丁頓近似

${\displaystyle I_{\nu }(\mu ,z)=a(z)+\mu b(z)}$

${\displaystyle J_{\nu }={\frac {1}{2}}\int _{-1}^{1}I_{\nu }d\mu =a}$
${\displaystyle H_{\nu }={\frac {1}{2}}\int _{-1}^{1}\mu I_{\nu }d\mu ={\frac {b}{3}}}$
${\displaystyle K_{\nu }={\frac {1}{2}}\int _{-1}^{1}\mu ^{2}I_{\nu }d\mu ={\frac {a}{3}}}$

${\displaystyle \mu {\frac {dI_{\nu }}{dz}}=-\alpha _{\nu }(I_{\nu }-B_{\nu })+\sigma _{\nu }(J_{\nu }-I_{\nu })}$

${\displaystyle {\frac {dH_{\nu }}{dz}}=\alpha _{\nu }(B_{\nu }-J_{\nu })}$

${\displaystyle {\frac {dK_{\nu }}{dz}}=-(\alpha _{\nu }+\sigma _{\nu })H_{\nu }}$

${\displaystyle {\frac {d^{2}J_{\nu }}{dz^{2}}}=3\alpha _{\nu }(\alpha _{\nu }+\sigma _{\nu })(J_{\nu }-B_{\nu })}$

## 延伸閱讀

• Jacqueline Lenoble. Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing. 1985: 583. ISBN 0-12-451451-0.
• George B. Rybicki, Alan P. Lightman. Radiative Processes in Astrophysics. Wiley-Interscience. 1985. ISBN 0-471-82759-2.
• G. E. Thomas and K. Stamnes. Radiative Transfer in the Atmosphere and Ocean. Cambridge University Press. 1999. ISBN 0-521-40124-0.

## 參考資料

1. ^ S. Chandrasekhar. Radiative Transfer. Dover Publications Inc. 1960: 393. ISBN 0-486-60590-6.
2. ^ Jacqueline Lenoble. Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing. 1985: 583. ISBN 0-12-451451-0.