A closer look at the 'Flux Limiter'

\lambda is the 'flux limiter'. It comes into the equations in the diffusion term:

\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)

This is because instead of solving the following conservation equation,

\frac{\partial E}{\partial t} =- \nabla \cdot F

(which would require an additional equation for F), we make the 'flux-limited' approximation that,

F=-\frac{c \lambda}{\kappa_R \rho}\nabla E

This then turns the conservation equation into a diffusion equation:

\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)

The value of \lambda controls whether the radiation is diffusing in the free-streaming limit (\lambda \rightarrow 0), i.e. at the speed of light, or is diffusing as it would in the optically thick limit (\lambda \rightarrow \frac {1}{3}). The FLD approximation does well at these two limits, but not in between. Here is the functional form of \lambda,

\boxed{\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})}

where,

R=|\frac{\nabla E}{\kappa_R \rho E}|

Note that (see bottom of page for more detail), the mean free path is given by:

l_\nu=\frac{1}{\rho \kappa_\nu}

(i.e. given the opacity and the density of the material, one can easily compute the mean free path), and,

\frac{\nabla E}{E}\approx -\frac{1}{h}

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)' :

\boxed{R\approx \frac{l}{h}}

(note there is no frequency dependence now in the mean free path, as K_R integrated over frequency space to give an 'average' opacity).

Graphically, we have:

So we see that for small R, \lambda \approx e^{-R}, and for large R, \lambda\approx 1/R. Let's examine these two limits closer.

Optically thick limit

From the graph above, we see that as R\rightarrow 0, \lambda\rightarrow \frac{1}{3}. Thus, we have:

\boxed{R \approx \frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}

That is, the mean free path is << scale height for the radiation, i.e. we're in the optically thick regime.

The rad diffusion equation in this limit becomes,

\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)

which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer.

Free streaming limit

In the other limit, R\rightarrow \infty, we have:

\boxed{R \approx \frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}

That is, the mean free path >> scale height, and we are in the free-streaming limit.

In this case, the rad diffusion equation becomes:

\frac{\partial E}{\partial t} = 0

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.

Estimating the diffusion time for a 1 solar mass cloud that has r=.05 pc

Estimating \lambda(R) starts with approximating R.

Note that,

\frac{\nabla E}{E}=\frac{d(\ln E)}{dx}

so if we make the approximation,

\frac{\nabla E}{E}\approx -\frac{1}{h}

(by dimensional arguments and assuming the gradient is negative), we have:

\frac{d(\ln E)}{dx}=-\frac{1}{h}

which integrates to:

\ln E = \ln E_0 - \frac{x}{h}

or,

E = E_0 e^{ - \frac{x}{h}}

From this equation, it is clear that h is the scale-height. Thus, we have,

R \approx |-\frac{1}{h \kappa_R\rho}| \approx \frac{1}{h \kappa_R\rho} = \frac{l}{h}

By setting h to be the distance between the sink and box side (h=L), we imagine it as the scale height for the radiation.

Now, Offner et al. 2009 state that the Rosseland specific opacity is best fit by

\kappa_R=0.23~\frac{cm^2}{g}

for 10 K gas. Using our parameters above, gives the mean free path as:

l\approx 22 ~pc

Given the radius of the box is r=.05 pc, we have,

R\approx \frac{22}{.05} =440

This moves us into the 'free-streaming' regime on the \lambda(R) curve, and thus we can approximate \lambda\approx 1/R. Thus, we have:

\boxed{\lambda\approx .002}

Review of relation between opacity, optical depth, and mean free path

For what follows, it will be helpful to quickly recall the following. Opacity (\kappa) is related to the absorption coefficient (\alpha) by,

\kappa \rho=\alpha (cm^{-1})

(where the absorption coefficient reduces the intensity of the ray by dI_\nu=-\alpha_\nu I_\nu ds)

and the optical depth is defined by,

\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'

When \tau>1 (integrated along a typical path through the medium), the material is optically thick, and when \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 (l_\nu) we have:

\bar{\tau_\nu}=\alpha_\nu l_\nu = 1

or

l_\nu=\frac{1}{\rho \kappa}