Changes between Version 13 and Version 14 of u/erica/scratch3


Ignore:
Timestamp:
04/03/16 16:33:01 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/scratch3

    v13 v14  
    3030[[Image(fld.png, 35%)]]
    3131
     32For what follows, it will be helpful to quickly recall the following.  Opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by,
     33
     34[[latex($\kappa \rho=\alpha (cm^{-1})$)]]
     35
     36(where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]])
     37
     38and the optical depth is defined by,
     39
     40[[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
     41
     42When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is optically thick, and when [[latex($\tau<1$)]], optically thin.
     43
     44The 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:
     45
     46[[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]]
     47
     48or
     49
     50[[latex($l_\nu=\frac{1}{\rho \kappa}$)]]
     51
     52Thus, given the opacity and the density of the material, one can compute the mean free path.
     53
     54
    3255== Optically thick limit ==
    3356
    34 From the graph above, we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. How to interpret this? R is essentially the ratio of the optical depth [[latex($\tau=1/\kappa$)]], to the radiation's scale height, [[latex($L^{-1}=\frac{\nabla E}{E}$)]]. Thus, as [[latex($R\rightarrow 0$)]], we have:
     57From the graph above, we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. How to interpret this? R is essentially the ratio of the mean free path [[latex($l=1/\kappa \rho$)]], to the radiation's scale height, [[latex($h^{-1}=\frac{\nabla E}{E}$)]] (shown below). Thus, as [[latex($R\rightarrow 0$)]], we have:
    3558
    36 [[latex($\boxed{\frac{\tau}{L}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]]
     59[[latex($\boxed{R=\frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]]
     60
     61(note there is no frequency dependence now in the mean free path, as [[latex($K_R$)]] integrated over frequency space to give an 'average' opacity).
    3762
    3863That is, the radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime.
     
    4873In the other limit, [[latex($R\rightarrow \infty$)]], we have:
    4974
    50 [[latex($\boxed{\frac{\tau}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
     75[[latex($\boxed{R=\frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
    5176
    5277That is, a photon travels an infinite distance before it interacts with another particle -- i.e. we are in the free-streaming limit.
     
    95120[[latex($\kappa_R=0.23~\frac{cm^2}{g}$)]]
    96121
    97 for 10 K gas. Recall that opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by,
    98 
    99 [[latex($\kappa \rho=\alpha (cm^{-1})$)]]
    100 
    101 (where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]])
    102 
    103 The optical depth is then defined by,
    104 
    105 [[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
    106 
    107 When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is said to be optically thick, and when [[latex($\tau<1$)]], 'optically thin'.
    108 
    109 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:
    110 
    111 [[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]]
    112 
    113 or
    114 
    115 [[latex($l_\nu=\frac{1}{\rho \kappa}$)]]
     122for 10 K gas.