26 | | (For L, the radiation is leaving from the center of the volume, so is going approximately 1 half the length). I am not completely sure on [[latex($\lambda$)]], but from Offner's paper it should have units of, |
27 | | |
28 | | [[latex($\lambda=\frac{1}{R^2}$)]] |
29 | | |
30 | | where R has units of: |
31 | | |
32 | | [[latex($R=\frac{1}{L}\frac{E}{\kappa_R \rho E}$)]] |
33 | | |
34 | | and so I gather an estimate for lambda might be: |
35 | | |
36 | | [[latex($\lambda = (L \kappa_R \rho)^2 $)]] |
37 | | |
38 | | which using our values gives: |
39 | | |
40 | | [[latex($\lambda=4e-6$)]] |
| 26 | (For L, the radiation is leaving from the center of the volume, so is going approximately 1 half the length). |
| 27 | |
| 28 | Let's now take a closer look at the flux limiter. |
| 29 | |
| 30 | |
| 31 | == A closer look at the 'Flux Limiter' == |
| 32 | |
| 33 | [[latex($\lambda$)]] is the 'flux limiter'. It comes into the equations in the diffusion term: |
| 34 | |
| 35 | [[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)$)]] |
| 36 | |
| 37 | This is because instead of solving the following conservation equation, |
| 38 | |
| 39 | [[latex($\frac{\partial E}{\partial t} =- \nabla \cdot F$)]] |
| 40 | |
| 41 | (which would require an additional equation for F), we make the 'flux-limited' approximation that, |
| 42 | |
| 43 | [[latex($F=-\frac{c \lambda}{\kappa_R \rho}\nabla E$)]] |
| 44 | |
| 45 | This then turns the conservation equation into a diffusion equation: |
| 46 | |
| 47 | [[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)$)]] |
| 48 | |
| 49 | |
| 50 | The value of [[latex($\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}$)]]). The FLD approximation does well at these two limits, but not in between. Here is the functional form of [[latex($\lambda$)]], |
| 51 | |
| 52 | [[latex($\boxed{\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})}$)]] |
| 53 | |
| 54 | where, |
| 55 | |
| 56 | [[latex($R=|\frac{\nabla E}{\kappa_R \rho E}|$)]] |
| 57 | |
| 58 | Note that (see bottom of page for more detail), the mean free path is given by: |
| 59 | |
| 60 | [[latex($l_\nu=\frac{1}{\rho \kappa_\nu}$)]] |
| 61 | |
| 62 | (i.e. given the opacity and the density of the material, one can easily compute the mean free path), and, |
| 63 | |
| 64 | [[latex($\frac{\nabla E}{E}\approx -\frac{1}{h}$)]] |
| 65 | |
| 66 | where h is the ''scale height''. Thus, we can interpret R as the ratio of the ''mean free path (l)'' to the '' 'radiation scale height (h)' '': |
| 67 | |
| 68 | [[latex($\boxed{R\approx \frac{l}{h}}$)]] |
| 69 | |
| 70 | (note there is no frequency dependence now in the mean free path, as [[latex($K_R$)]] integrated over frequency space to give an 'average' opacity). |
| 71 | |
| 72 | Graphically, we have: |
| 73 | |
| 74 | [[Image(fld.png, 35%)]] |
| 75 | |
| 76 | So we see that for small [[latex($R$)]], [[latex($\lambda \approx e^{-R}$)]], and for large [[latex($R$)]], [[latex($\lambda\approx 1/R$)]]. Let's examine these two limits closer. |
| 77 | |
| 78 | |
| 79 | === Optically thick limit === |
| 80 | |
| 81 | From the graph above, we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. Thus, we have: |
| 82 | |
| 83 | [[latex($\boxed{R \approx \frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]] |
| 84 | |
| 85 | |
| 86 | That is, the mean free path is << scale height for the radiation, i.e. we're in the optically thick regime. |
| 87 | |
| 88 | The rad diffusion equation in this limit becomes, |
| 89 | |
| 90 | [[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)$)]] |
| 91 | |
| 92 | which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer. |
| 93 | |
| 94 | === Free streaming limit === |
| 95 | |
| 96 | In the other limit, [[latex($R\rightarrow \infty$)]], we have: |
| 97 | |
| 98 | [[latex($\boxed{R \approx \frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]] |
| 99 | |
| 100 | That is, the mean free path >> scale height, and we are in the free-streaming limit. |
| 101 | |
| 102 | In this case, the rad diffusion equation becomes: |
| 103 | |
| 104 | [[latex($\frac{\partial E}{\partial t} = 0$)]] |
| 105 | |
| 106 | That is, the radiation diffuses instantly through the grid. Recall, how this radiative energy ''couples'' to the gas is given by the coupling term in the radiation equation, not shown here. |
| 107 | |
| 108 | == Estimating the flux limiter for a 1 solar mass core, with r=.05 pc == |
| 109 | |
| 110 | Estimating [[latex($\lambda(R)$)]] starts with approximating R. |
| 111 | |
| 112 | Note that, |
| 113 | |
| 114 | [[latex($\frac{\nabla E}{E}=\frac{d(\ln E)}{dx}$)]] |
| 115 | |
| 116 | so if we make the approximation, |
| 117 | |
| 118 | [[latex($\frac{\nabla E}{E}\approx -\frac{1}{h}$)]] |
| 119 | |
| 120 | (by dimensional arguments and assuming the gradient is negative), we have: |
| 121 | |
| 122 | [[latex($\frac{d(\ln E)}{dx}=-\frac{1}{h}$)]] |
| 123 | |
| 124 | which integrates to: |
| 125 | |
| 126 | [[latex($\ln E = \ln E_0 - \frac{x}{h}$)]] |
| 127 | |
| 128 | or, |
| 129 | |
| 130 | [[latex($E = E_0 e^{ - \frac{x}{h}}$)]] |
| 131 | |
| 132 | From this equation, it is clear that h is the scale-height. Thus, we have, |
| 133 | |
| 134 | [[latex($R \approx |-\frac{1}{h \kappa_R\rho}| \approx \frac{1}{h \kappa_R\rho} = \frac{l}{h}$)]] |
| 135 | |
| 136 | By setting h to be the distance between the sink and box radius (h=r), ''we imagine it as the scale height for the radiation. '' |
| 137 | |
| 138 | Now, using our parameters from above, the mean free path is: |
| 139 | |
| 140 | [[latex($\boxed{l\approx 22 ~pc}$)]] |
| 141 | |
| 142 | Given the radius of the box is r=.05 pc, we have, |
| 143 | |
| 144 | [[latex($\boxed{R\approx \frac{22}{.05} =440}$)]] |
| 145 | |
| 146 | This moves us into the 'free-streaming' regime on the [[latex($\lambda(R)$)]] curve, and thus we can approximate [[latex($\lambda\approx 1/R$)]]. Thus, we have: |
| 147 | |
| 148 | [[latex($\boxed{\lambda\approx .002}$)]] |
| 149 | |
| 150 | == Back to the diffusion time estimate == |