Changes between Version 20 and Version 21 of u/erica/radtimescales


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Timestamp:
04/04/16 12:26:04 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/radtimescales

    v20 v21  
    1 '''Diffusion time estimate'''
     1= Diffusion time estimate for a 1 solar mass core, of r=.05 pc =
    22
    33This is an estimate for how long it should take radiation to diffuse through gas. The longer this timescale, the more confined the radiation is -- and thus, the more it can act as a source of heating.
     
    1111[[latex($\boxed{t_{diff}\approx \frac{L^2}{c}\frac{\kappa_R \rho}{\lambda}}$)]]
    1212
    13 L is the size of our system, c is speed of light, kappa_R is the rosseland specific mean opacity, rho is density of the system, and lamba is a dimensionless parameter that seems to do with the length scale of gradients in radiative energy.
     13L is the size of our system, c is speed of light, kappa_R is the rosseland specific mean opacity, rho is density of the system, and lambda is the dimensionless 'flux limiter' that we will look at below.
    1414
    1515Offner et al '09 gives,
     
    2424|| [[latex($\kappa_R$)]]|| .23 ||
    2525
    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$)]]
     26(For L, the radiation is leaving from the center of the volume, so is going approximately 1 half the length).
     27
     28Let's now take a closer look at the flux limiter.
     29
     30
     31== A closer look at the 'Flux Limiter' ==
     32
     33[[latex($\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)$)]]
     36
     37This is because instead of solving the following conservation equation,
     38
     39[[latex($\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$)]]
     44
     45This then turns the conservation equation into a diffusion equation:
     46
     47[[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)$)]]
     48
     49
     50The value of [[latex($\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})}$)]]
     53
     54where,
     55
     56[[latex($R=|\frac{\nabla E}{\kappa_R \rho E}|$)]]
     57
     58Note that (see bottom of page for more detail), the mean free path is given  by:
     59
     60[[latex($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}$)]]
     65
     66where h is the ''scale height''. Thus, we can interpret R as the ratio of the ''mean free path (l)'' to the '' 'radiation scale height (h)' '':
     67
     68[[latex($\boxed{R\approx \frac{l}{h}}$)]]
     69
     70(note there is no frequency dependence now in the mean free path, as [[latex($K_R$)]] integrated over frequency space to give an 'average' opacity).
     71
     72Graphically, we have:
     73
     74[[Image(fld.png, 35%)]]
     75
     76So we see that for small [[latex($R$)]], [[latex($\lambda \approx e^{-R}$)]], and for large [[latex($R$)]], [[latex($\lambda\approx 1/R$)]]. Let's examine these two limits closer.
     77 
     78
     79=== Optically thick limit ===
     80
     81From the graph above, we see that as [[latex($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}}$)]]
     84
     85
     86That is, the mean free path is << scale height for the radiation, i.e. we're in the optically thick regime.
     87
     88The rad diffusion equation in this limit becomes,
     89
     90[[latex($\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)$)]]
     91
     92which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer. 
     93
     94=== Free streaming limit ===
     95
     96In the other limit, [[latex($R\rightarrow \infty$)]], we have:
     97
     98[[latex($\boxed{R \approx \frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}$)]]
     99
     100That is, the mean free path >> scale height, and we are in the free-streaming limit.
     101
     102In this case, the rad diffusion equation becomes:
     103
     104[[latex($\frac{\partial E}{\partial t} = 0$)]]
     105
     106That is, the radiation diffuses instantly through the grid. Recall, how this radiative energy ''couples'' to the gas is given by the coupling term in the radiation equation, not shown here.
     107
     108== Estimating the flux limiter for a 1 solar mass core, with r=.05 pc ==
     109
     110Estimating [[latex($\lambda(R)$)]] starts with approximating R.
     111
     112Note that,
     113
     114[[latex($\frac{\nabla E}{E}=\frac{d(\ln E)}{dx}$)]]
     115
     116so if we make the approximation,
     117
     118[[latex($\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}$)]]
     123
     124which integrates to:
     125
     126[[latex($\ln E = \ln E_0 - \frac{x}{h}$)]]
     127
     128or,
     129
     130[[latex($E = E_0 e^{ - \frac{x}{h}}$)]]
     131
     132From this equation, it is clear that h is the scale-height. Thus, we have,
     133
     134[[latex($R \approx |-\frac{1}{h \kappa_R\rho}| \approx \frac{1}{h \kappa_R\rho} = \frac{l}{h}$)]]
     135
     136By setting h to be the distance between the sink and box radius (h=r), ''we imagine it as the scale height for the radiation. ''
     137
     138Now, using our parameters from above, the mean free path is:
     139
     140[[latex($\boxed{l\approx 22 ~pc}$)]]
     141
     142Given the radius of the box is r=.05 pc, we have,
     143
     144[[latex($\boxed{R\approx \frac{22}{.05} =440}$)]]
     145
     146This moves us into the 'free-streaming' regime on the [[latex($\lambda(R)$)]] curve, and thus we can approximate [[latex($\lambda\approx 1/R$)]]. Thus, we have:
     147
     148[[latex($\boxed{\lambda\approx .002}$)]]
     149
     150== Back to the diffusion time estimate ==
    41151
    42152Using all of these values in the formula above for the diffusion time gives,
    43153
    44 [[latex($\boxed{t_{diff}\approx 3.75e+9 ~s}$)]]
    45 
    46 or ~118 years.
    47 
    48 I am not sure on this because lambda is not well constrained, and you can get very different estimates based on what you choose lambda to be (i.e tdiff = 4 hours when lambda =1, tdiff=87 days when lambda = .002, etc).
     154[[latex($\boxed{t_{diff}\approx 87 ~days}$)]]
     155
    49156
    50157Compare this time to the 'free streaming limit':
    51158
    52 [[latex($\boxed{t_{fs}=\frac{L}{c}\approx\frac{1.5e+17}{3e+10}=.5e+7 ~s}$)]]
    53 
    54 or 57 days.   
     159[[latex($\boxed{t_{fs}=\frac{L}{c}\approx\frac{1.5e+17}{3e+10}= 57 ~days}$)]]   
    55160
    56161''' Coupling time estimate '''