A drunk photon randomly walks into a bar

a. The dot product of all the random path lengths in the random walk determines the distance D squared.


If the angle between two consecutive vectors is greater than 90 degrees, the dot product is negative, and if less it is positive (at 90 degrees it equals 0). Therefore, although our sum involves dot products between vectors pointing at different directions as well as identical vectors, we can say that the sum of all dots of different vectors is 0. We are assuming we have many vectors (the pathlength of a photon is small compared to D, and scattering is frequent), so we have a Gaussian distribution of dots of different vectors centered around 0 (dot of two vectors at 90 degrees to each other) so we have as many positive dot products as negative and they average to 0. The dot of a vector with itself is just its magnitude squared, so our sum reduces to:
 

so the average number of steps n of a drunk photon is:


b. The photon travels a total distance of n * l. It travels each pathlength l at the speed of light c, so it takes time t = l/c to traverse one pathlength. There are n pathlength ys, so it takes total time n*l/c to go from core to surface. This covers a displacement D, so the photons average velocity is:

v (diffusion) =

c. The mean free path l has units of cm; mass density rho has units g/cm^3, absorption coefficient \kappa has units of cm^2/g. From inspection we see that rho*/kappa ~ 1/l to be dimensionally correct. Daniel asked why we had cross sectional area of observers rather than volume (since by introducing a constant with desired units the equation could still be dimensionally correct). 

We can answer this, and show the pathlength is equal not merely proportional to the rhs as follows:
Consider a cube with face area A and edge length l. Assume it has N particle density each with cross sectional area c, so the total cross sectional area is NAlc (assuming particles don't overlap).
The probability of  a photon going through and hitting a particle is P=NAlc/A = Nlc (equivalently the number of particles in that volume). The particle has moved a distance l, so the average distance between collisions (pathlength) is l/Nlc = 1/Nc. We can convert number density to mass density and cross sectional area to cross-sectional area per gram and maintain consistent units.

d. There is a flux of photons moving through concentric shells of the sun. We have a difference in energy density since there is a flux into a radius of disk r to a disk of radius r + \delta r, and flux varies inversely with radius so it decreases for outer shells. Thus the inner shells have a high energy density \delta u since flux F equals luminosity L over surface area, and energy density equals F over the speed of light c.



e. Since u= aT^4,



and the change in flux F is proportional to the opacity (total cross-section per volume, or the inverse pathlength) times the flux and the distance traveled, so




There is a negative sign since the flux decreases with radius due to absorption.
Taking the differential limits of both equations and equating we have



The worksheet equation has a coefficient of 3, which might arise from some quantum effects we don't have the background to consider.