# 輻射轉移

## 定義

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

## 輻射轉移方程式

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

## 輻射轉移方程式的解

$I_\nu(s)=I_\nu(s_0)e^{-\tau(s_0,s)}+\int_{s_0}^s j_\nu(s') e^{-\tau(s',s)}\,ds'$

$\tau(s_1,s_2) \ \stackrel{\mathrm{def}}{=}\ \int_{s_1}^{s_2} \alpha_\nu(s)\,ds$

### 局部熱力學平衡

$\frac{j_\nu}{\alpha_\nu}=B_\nu(T)$

$I_\nu(s)=I_\nu(s_0)e^{-\tau(s_0,s)}+\int_{s_0}^s B_\nu(T(s'))\alpha_\nu(s') e^{-\tau(s',s)}\,ds'$

### 愛丁頓近似

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

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

$\mu \frac{dI_\nu}{dz}=- \alpha_\nu (I_\nu-B_\nu) + \sigma_{\nu}(J_\nu -I_\nu)$

$\frac{dH_\nu}{dz}=\alpha_\nu (B_\nu-J_\nu)$

$\frac{dK_\nu}{dz}=-(\alpha_\nu+\sigma_\nu)H_\nu$

$\frac{d^2J_\nu}{dz^2}=3\alpha_\nu(\alpha_\nu+\sigma_\nu)(J_\nu-B_\nu)$

## 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.
• , Barbara Weibel-Mihalas. Foundations of Radiation Hydrodynamics. Dover Publications, Inc. 1984. ISBN 0-486-40925-2.
• 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.