The path of a photon from the core to the convective zone, resembles a random-walk process. Photons traveling through the core and radiative zone initially collide with an electron every fraction of a centimeter (less often as they move outward) producing a change in direction. It can take millions of years for these photons to travel from the sun's core to the solar surface. Photons that were generated in the sun during the time when cave men roamed the Earth are just now arriving here.
Random Walk Statistics For simplicity, a 1D random walk is described below. These formulas are easily generalized to 3D. Imagine a photon constrained to move along one dimension. After colliding with an electron, the photon will either move forward or backward along the line a distance equal to the average mean free path (average distance between collisions). Within the sun's radiative zone a photon moves on average l=1 cm between collisions. The photon is equally likely to move in any direction following the collision. After N jumps each of length, l, the average displacement of the photon from the origin is where m is an integer lying between -N<m<N. If R=pN is the number of steps to the right and L=qN=(1-p)N is the number of steps to the left where N=L+R, p is the probability of a step to the right and q=1-p is the probability of a step to the left, then: A fundamental assumption is that each step is independent of all others. So the probability of any sequence of L steps to the left and R steps to the right is simply But there are many different ways to move R steps to the right and L steps to the left. The number of distinct possibilities is given by: Thus the probability of taking R steps to the right and L=N-R steps to the left out of a total of N steps is given by: This distribution of probabilities is called a binomial distribution. Click here for a graphical display of an N=20 binomial probability distribution with p=q=0.5. For this distribution it can be shown that the mean displacement of the photon from the origin and the dispersion (which measures the scatter of values about the mean) are given by: Click here for a graphical representation of these quantities.

Use this 1D random walk java applet over a few dozen trials to see how closely the above statistical formula for the dispersion matches the actual displacement from the origin. For this program select a path length and an initial location of the particle so that the particle is in the center with equal path lengths on either side. This most closely represents the problem of a photon at the center of the sun. For example: path length =21 and particle position= 11. When the particle hits the ends of the path (analogous to the surface of the sun) it explodes. Make a table showing the trial number, your calculation of the predicted number of collisions to get to the edge of the path, and the actual number of collisions.

At the sun, the mean free path for a photon in the radiative zone is approximately 1 cm. Use the statistical formulas for the random walk problem above to determine how many "collisions" a typical photon undergoes before reaching the surface (radius=7E10 cm). Assume a photon takes little or no time to be absorbed and reemitted during each "collision" and that all of its time is spent in travelling between collision points. Using the above, estimate how long (in years) it takes for the photon to move from the core, where it originated, to the sun's surface? Compare this to estimates of the time for a photon to reach the base of the convective zone. As you will see, being absorbed and re-emitted adds considerable time to the process.

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