Estimating Timescales for Radiation Diffusion

So this is an execerise in thinking dimensionally and in order of magnitudes.

Consider the equation for radiation energy conservation ignoring a bunch of terms that should be of lower order

The second term on the right hand side tells us about the diffusion of radiation and the right hand term tells us about coupling with matter.

Lets consider the diffusion term to get an estimate of the timescale,

We do this terms of dimensions (L = length, T= time etc). Radiation energy density E is, for example

So the diffusion term reads

which has units of energy of time as would be expected. To get a timescale for radiative diffusion we can write things in terms of the scales of the problem and

or

Note drops out here

Now we just have to find values of the Rosseland opacity relevant to something like a dusty in falling protostellar envelope. I am seeing the Rosseland opacity being given in units of cm2/g which means we will need to multiply by a density scale in the above expression i.e.

where is the value we get from tables and is our density scale.

Here is a paper I found that gives some useful opacity information

​http://www.aanda.org/articles/aa/full/2003/41/aa3802/aa3802.right.html

Check out this table from the paper

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