Changes between Version 14 and Version 15 of u/erica/scratch3


Ignore:
Timestamp:
04/04/16 11:46:02 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/scratch3

    v14 v15  
    2020The 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($\lambda$)]],
    2121
    22 [[latex($\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})$)]]
     22[[latex($\boxed{\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})}$)]]
    2323
    2424where,
     
    2626[[latex($R=|\frac{\nabla E}{\kappa_R \rho E}|$)]]
    2727
     28Note 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
     32and that,
     33
     34[[latex($\frac{\nabla E}{E}\approx -\frac{1}{h}$)]]
     35
     36where 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
    2840Graphically, we have:
    2941
    3042[[Image(fld.png, 35%)]]
    3143
     44So we see that for small [[latex($R$)]], [[latex($\lambda \approx e^{-R}$)]], and for large [[latex($R$)]], [[latex($\lambda\approx 1/R$)]].
     45 
    3246For what follows, it will be helpful to quickly recall the following.  Opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by,
    3347
     
    121135
    122136for 10 K gas.
     137
     138== Review of relation between opacity, optical depth, and mean free path ==
     139
     140For 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
     146and the optical depth is defined by,
     147
     148[[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
     149
     150When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is optically thick, and when [[latex($\tau<1$)]], optically thin.
     151
     152The 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
     156or
     157
     158[[latex($l_\nu=\frac{1}{\rho \kappa}$)]]