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

Timestamp:

04/03/16 16:33:01 (10 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/scratch3

@@ -30 +30 @@
 [[Image(fld.png, 35%)]]
 
+For what follows, it will be helpful to quickly recall the following.  Opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by,
+
+[[latex($\kappa \rho=\alpha (cm^{-1})$)]]
+
+(where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]])
+
+and the optical depth is defined by, 
+
+[[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
+
+When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is optically thick, and when [[latex($\tau<1$)]], optically thin. 
+
+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:
+
+[[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]] 
+
+or
+
+[[latex($l_\nu=\frac{1}{\rho \kappa}$)]]
+
+Thus, given the opacity and the density of the material, one can compute the mean free path. 
+
+
 == Optically thick limit == 
 
-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:
+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 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:
 
-[[latex($\boxed{\frac{\tau}{L}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]] 
+[[latex($\boxed{R=\frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]] 
+
+(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).
 
 That is, the radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime.
@@ -48 +73 @@
 In the other limit, [[latex($R\rightarrow \infty$)]], we have: 
 
-[[latex($\boxed{\frac{\tau}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
+[[latex($\boxed{R=\frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
 
 That is, a photon travels an infinite distance before it interacts with another particle -- i.e. we are in the free-streaming limit.
@@ -95 +120 @@
 [[latex($\kappa_R=0.23~\frac{cm^2}{g}$)]]
 
-for 10 K gas. Recall that opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by,
-
-[[latex($\kappa \rho=\alpha (cm^{-1})$)]]
-
-(where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]])
-
-The optical depth is then defined by, 
-
-[[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
-
-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'. 
-
-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:
-
-[[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]] 
-
-or
-
-[[latex($l_\nu=\frac{1}{\rho \kappa}$)]]
+for 10 K gas.