Changes between Version 15 and Version 16 of u/erica/scratch3


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Timestamp:
04/04/16 12:10:37 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/scratch3

    v15 v16  
    1818
    1919
    20 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. 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($\lambda$)]],
     20The 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($\lambda$)]],
    2121
    2222[[latex($\boxed{\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})}$)]]
    … …  
    3030[[latex($l_\nu=\frac{1}{\rho \kappa_\nu}$)]]
    3131
    32 and that,
     32(i.e. given the opacity and the density of the material, one can easily compute the mean free path), and,
    3333
    3434[[latex($\frac{\nabla E}{E}\approx -\frac{1}{h}$)]]
    3535
    36 where h is the ''scale height''. Thus, we can interpret R as the ratio of the ''mean free path'' to the '' 'radiation scale height' '':
     36where 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)' '':
    3737
    3838[[latex($\boxed{R\approx \frac{l}{h}}$)]]
     39
     40(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).
    3941
    4042Graphically, we have:
    … …  
    4244[[Image(fld.png, 35%)]]
    4345
    44 So we see that for small [[latex($R$)]], [[latex($\lambda \approx e^{-R}$)]], and for large [[latex($R$)]], [[latex($\lambda\approx 1/R$)]].
     46So 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.
    4547 
    46 For what follows, it will be helpful to quickly recall the following.  Opacity ([[latex($\kappa$)]]) is related to the absorption coefficient ([[latex($\alpha$)]]) by,
    47 
    48 [[latex($\kappa \rho=\alpha (cm^{-1})$)]]
    49 
    50 (where the absorption coefficient reduces the intensity of the ray by [[latex($dI_\nu=-\alpha_\nu I_\nu ds$)]])
    51 
    52 and the optical depth is defined by,
    53 
    54 [[latex($\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'$)]]
    55 
    56 When [[latex($\tau>1$)]] (integrated along a typical path through the medium), the material is optically thick, and when [[latex($\tau<1$)]], optically thin.
    57 
    58 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:
    59 
    60 [[latex($\bar{\tau_\nu}=\alpha_\nu l_\nu = 1$)]]
    61 
    62 or
    63 
    64 [[latex($l_\nu=\frac{1}{\rho \kappa}$)]]
    65 
    66 Thus, given the opacity and the density of the material, one can compute the mean free path.
    67 
    6848
    6949== Optically thick limit ==
    7050
    71 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:
     51From the graph above, we see that as [[latex($R\rightarrow 0$)]], [[latex($\lambda\rightarrow \frac{1}{3}$)]]. Thus, we have:
    7252
    73 [[latex($\boxed{R=\frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]]
     53[[latex($\boxed{R \approx \frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}$)]]
    7454
    75 (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).
    7655
    77 That is, the radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime.
     56That is, the mean free path is << scale height for the radiation, i.e. we're in the optically thick regime.
    7857
    79 The rad diffusion equation then becomes,
     58The rad diffusion equation in this limit becomes,
    8059
    8160[[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)$)]]
    … …  
    8766In the other limit, [[latex($R\rightarrow \infty$)]], we have:
    8867
    89 [[latex($\boxed{R=\frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
     68[[latex($\boxed{R \approx \frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
    9069
    91 That is, a photon travels an infinite distance before it interacts with another particle -- i.e. we are in the free-streaming limit.
     70That is, the mean free path >> scale height, and we are in the free-streaming limit.
    9271
    9372In this case, the rad diffusion equation becomes:
    … …  
    9776That 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.
    9877
    99 == Estimating the diffusion time ==
     78== Estimating the diffusion time for a 1 solar mass cloud that has r=.05 pc ==
    10079
    101 Estimating [[latex($\lambda$)]]...
    102 
    103 
    104 Depending on the value of R, we will get different values of [[latex($\lambda$)]]. So, we start by approximating R.
     80Estimating [[latex($\lambda(R)$)]] starts with approximating R.
    10581
    10682Note that,
    … …  
    124100[[latex($E = E_0 e^{ - \frac{x}{h}}$)]]
    125101
    126 From this equation, it is clear that h is the scale-height. Thus, by writing R as:
     102From this equation, it is clear that h is the scale-height. Thus, we have,
    127103
    128 [[latex($R \approx |-\frac{1}{h \kappa_R\rho}| \approx \frac{1}{h \kappa_R\rho}$)]]
     104[[latex($R \approx |-\frac{1}{h \kappa_R\rho}| \approx \frac{1}{h \kappa_R\rho} = \frac{l}{h}$)]]
    129105
    130 we can set h to be the distance between the sink and box side (h=L), and imagine it as the scale height for the radiation.
     106By setting h to be the distance between the sink and box side (h=L), we imagine it as the scale height for the radiation.
    131107
    132108Now, Offner et al. 2009 state that the Rosseland specific opacity is best fit by
    … …  
    134110[[latex($\kappa_R=0.23~\frac{cm^2}{g}$)]]
    135111
    136 for 10 K gas.
     112for 10 K gas. Using our parameters above, gives the mean free path as:
     113
     114[[latex($l\approx 22 ~pc$)]]
     115
     116Given the radius of the box is r=.05 pc, we have,
     117
     118[[latex($R\approx \frac{22}{.05} =440$)]]
     119
     120This 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:
     121
     122[[latex($\boxed{\lambda\approx .002}$)]]
     123
    137124
    138125== Review of relation between opacity, optical depth, and mean free path ==