Changes between Version 14 and Version 15 of u/erica/scratch3
- Timestamp:
- 04/04/16 11:46:02 (9 years ago)
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u/erica/scratch3
v14 v15 20 20 The value of [[latex($\lambda$)]] controls whether the radiation is diffusing in the free-streaming limit ([[latex($\lambda \rightarrow 0$)]]), i.e. at the speed of light, or is diffusing as it would in the optically thick limit ([[latex($\lambda \rightarrow \frac {1}{3}$)]]). The FLD approximation does well at these two limits, but not in between. Let's examine these two limits in more detail and then consider why it doesn't perform as well in between. Here is the functional form of [[latex($\lambda$)]], 21 21 22 [[latex($\ lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})$)]]22 [[latex($\boxed{\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})}$)]] 23 23 24 24 where, … … 26 26 [[latex($R=|\frac{\nabla E}{\kappa_R \rho E}|$)]] 27 27 28 Note that (see bottom of page for more detail), the mean free path is given by: 29 30 [[latex($l_\nu=\frac{1}{\rho \kappa_\nu}$)]] 31 32 and that, 33 34 [[latex($\frac{\nabla E}{E}\approx -\frac{1}{h}$)]] 35 36 where h is the ''scale height''. Thus, we can interpret R as the ratio of the ''mean free path'' to the '' 'radiation scale height' '': 37 38 [[latex($\boxed{R\approx \frac{l}{h}}$)]] 39 28 40 Graphically, we have: 29 41 30 42 [[Image(fld.png, 35%)]] 31 43 44 So we see that for small [[latex($R$)]], [[latex($\lambda \approx e^{-R}$)]], and for large [[latex($R$)]], [[latex($\lambda\approx 1/R$)]]. 45 32 46 For what follows, it will be helpful to quickly recall the following. Opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by, 33 47 … … 121 135 122 136 for 10 K gas. 137 138 == Review of relation between opacity, optical depth, and mean free path == 139 140 For what follows, it will be helpful to quickly recall the following. Opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by, 141 142 [[latex($\kappa \rho=\alpha (cm^{-1})$)]] 143 144 (where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]]) 145 146 and the optical depth is defined by, 147 148 [[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]] 149 150 When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is optically thick, and when [[latex($\tau<1$)]], optically thin. 151 152 The mean optical depth of an absorbing material can be shown to =1, and so in terms of the mean free path ([[latex($l_\nu$)]]) we have: 153 154 [[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]] 155 156 or 157 158 [[latex($l_\nu=\frac{1}{\rho \kappa}$)]]