Consider the random walk of photons generated at a point source of luminosity L

Question

Consider the random walk of photons generated at a point source of
luminosity L at the centre of a uniform sphere of radius R. Assume that isotropic
scattering occurs, where l is the mean free path. Also assume that there is a small
probability p << 1 that the photon is absorbed each time it scatters and that p does not
depend on wavelength.

How many times does a photon scatter before reaching the surface of the sphere
on average, assuming that it is not absorbed? Assuming that the photon moves at
the speed of light between scattering events, how long does this take?

Derive an expression for the luminosity of the sphere. [Do not assume thermal
equilibrium.]

For the first question is it as simple as knowing the R = lN plugging into the volume equation of a sphere, but where does the speed of light come into play? I got nothing for second half.

Explanation / Answer

In addition to emission and absorption, the photons can also be scattered in the medium. A volume elements emits photons due to the scattering with the rate completely dependent on the amount of radiation falling on the element. Let us consider a simple case of coherent (or elastic, monochromatic, i.e. with no frequency shift) isotropic (i.e. scattering probability is the same in all directions) scattering. Often the ’absorption’ coe?cient for scattering is denoted s (do not mix it up with cross-section or Stefan-Boltzmann constant!).

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