# How to derive equation for time it takes photons to diffuse through the Sun

I am wanting to use the Rosseland radiative heat flux equation to find the time it takes for photons to diffuse through the sun. The answer I am wanting to derive is:

$$\tau_D~\frac{\rho \bar C_p R^2_{sun}}{\kappa\rho}$$

I see a lot of solutions that have $R^2$ in the answer. Anyone know how to arrive at a solution that is dependent on the variables I have in the equation?

I have:

$$\kappa = \frac{\lambda \nu \rho C_v}{3}$$

where $\kappa$ is the thermal conductivity.

Also, in a diffusive random walk process, a particle will move an average distance d, where

$$N=(\frac{d}{\lambda})^2$$

• Define all variables: $ν$, $λ$, $\bar C_p$ (why the bar?). Also, your last sentence looks unfinished. – L. Levrel Apr 21 '16 at 8:11
• I have a feeling that photons are absorbed after not so long distances and so it would be better to describe it as heat diffusion. If they were not absorbed we should see millions-kelvin radiation from the core, rather than thousands-kelvin radiation from the surface. – Nanite Apr 21 '16 at 8:51
• @L.Levrel lambda is mfp, Cp for the bar is I guess just an average over the distance the photons have to move. – Jackson Hart Apr 21 '16 at 15:13
• And $ν$ is? Please, edit your Q rather than answering in comments – L. Levrel Apr 21 '16 at 15:28
• @Nanite: yes, photons are in thermal equilibrium with the plasma (which is opaque to photons), so those coming out the Sun are at the photosphere temperature. But you can look at absorption-emission as inelastic diffusion of a single photon. – L. Levrel Apr 21 '16 at 19:18