| 32 | 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 | |
| 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 | |
| 38 | and the optical depth is defined by, |
| 39 | |
| 40 | [[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]] |
| 41 | |
| 42 | When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is optically thick, and when [[latex($\tau<1$)]], optically thin. |
| 43 | |
| 44 | 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: |
| 45 | |
| 46 | [[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]] |
| 47 | |
| 48 | or |
| 49 | |
| 50 | [[latex($l_\nu=\frac{1}{\rho \kappa}$)]] |
| 51 | |
| 52 | Thus, given the opacity and the density of the material, one can compute the mean free path. |
| 53 | |
| 54 | |
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: |
| 57 | 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: |
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($l_\nu=\frac{1}{\rho \kappa}$)]] |
| 122 | for 10 K gas. |