Diffusion Timescale of a Photon in the Sun

In this post I'll be doing #1, parts a-d from this week's worksheet on photon diffusion and random walks. Parts a through c essentially build up to the question in d, which is :

What is the diffusion timescale for a photon moving from the center of the sun to the surface? The scattering cross section for electron scattering is 7x10^(-25) cm^2, and you can assume pure hydrogen for the Sun's interior. Be careful about the mass of material through which the photon travels.

Let's start with random walks. Photons in the Sun do not travel radially outwards continuously - they will bump into hydrogen atoms and bounce off them, essentially bouncing back and forth in random directions. Contrary to intuition, this does result in a net radial displacement.

Consider the above diagrams, with N random displacement vectors starting with l1 (only 5 are shown). We can write the net radial displacement of N steps of this random walk as follows:

If we find the dot product of each side of this equation with itself, we get:

This simplifies to:

Since each step is in a random direction, the part of the right hand side in the second parentheses averages out to zero. In addition, if we assume an average step length of l, we get:

What is the average velocity of a photon over this distance D? We can find this by taking the average radial distance, D, divided by the total time it takes for the photon to travel it. A photon travels at the speed of light c, and the total distance it traverses is Nl, the number of steps times the average step size. So, we have:

Now let's express this diffusion velocity in terms of other parameters. Let p be the mass density of absorbers (the atoms that the photon bounces off of), and K be the absorption coefficient (the cross-sectional area of absorbers per unit mass). We can see using dimensional analysis that:

Finally, we're ready to calculate the diffusion timescale for a photon moving from the center of the sun to the surface (i.e. how long does it take for a photon to travel from the center of the sun to the surface?). We have:

We know the scattering cross section of electrons, and we're assuming pure hydrogen for the Sun's interior. Assuming constant density, we can rewrite K and p as:

where m_H is the mass of a hydrogen atom

Thus, we have:

So:

The actual timescale is supposed to be in the tens of thousands - my error may have come from assuming a constant density, or in neglecting other processes in the sun. However, 3000 years is within an order of magnitude, so as an estimate, the timescale is roughly accurate.