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Simulation of Random Walk

Is There an Edge to the Sun?
The visible edge of the Sun is called the photosphere. It is a difficult concept for students to grasp. It might be thought that this edge is very solid—it is not.

Some teachers use the analogy of how light comes off of a cloud. The light bounces around a cloud. We see the last bounce which is the light that bounces off of particles close to edge and then out. The edge doesn’t have to be dense or even denser than the rest of the cloud. It is just the surface of last scattering.

For the Sun, a photon is generated at the center and makes its way to the surface. It may take up to several million years to get to the surface, and the form of the energy may change from X-ray to visible wavelengths. When the photon leaves the Sun, it takes eight minutes to get to us at the speed of light.

What Is a Random Walk?
Most electromagnetic energy gets out of the Sun in a very round about way which depends on random motion. The path a photon takes is unpredictable but we can gain an understanding through models of how it works.

The photons that we see as sunlight actually took a long time to leave the interior of the Sun. They did what physicists call a random walk. It's kind of like trying to make your way through a crowded subway station. You head off in one direction, get bounced in another direction, then still another, and so on. The problem is that, unlike the person in a subway station, the photons (particles of light) don't know which way they want to go. They are jostled about in scattering collisions with particles (mostly electrons). It's like being blindfolded in a crowded subway station. Eventually you will work your way to the door by chance and leave. But how many steps will it take you to go a certain distance?

A demonstration that can be used in class illustrates that one can predict quite a bit about a random or unpredictable path.

  1. Pick a point in the middle of a room where you can move 10 steps in any direction. Place a marker at that center starting point. With a piece of string, or if you can, mark with chalk on the floor a circle that is 10 steps away from the center. The radius of the circle should be ten student-sized steps.
  2. One student acts as the photon and we see how many steps it takes to travel ten paces, if the person is moving in a random direction on each step. After the instructor has defined a circle around the student ten paces in radius he or she should now entertain predictions from the class. Write the predictions on the board.
  3. The question is, how many steps will it take for the student to get outside the circle if they move one step in a random direction each time? At best it will take a minimum of ten steps, if they moved the same way each time, which is unlikely. We will use a little compass wheel or spinner to tell him/her which direction to take each step. If that is not available, drop a pencil with the eraser down (or a cheap pen) on the ground and notice which way the tip is pointed. Use the orientation like a compass.
  4. Don’t have the student move in the direction the tip points but rather at the angle (relative to north, or better yet to the front of the room). To do this, have students determine the angle relative to the front of the room and specify a number (e.g. 137 degrees). Then have the student in the middle of the room rotate to exactly that angle (as verified by the class) and then take one step forward. Again verify that the step is exactly one-tenth of the distance to the circle.

It turns out that to go outside the circle that is ten paces in radius will require (10)2 or about a hundred different paces, if each pace is in a random direction. Stated differently, a random walk of N steps, on the average, will move you a distance of N1/2 steps away. So, it should take about 100 steps to get across a circle ten steps in radius.

For the Sun, radiation leaves the center, where it was generated, by a random walk process. A photon moves about 1 cm before it has a collision and is randomly redirected. Since the radius of the Sun is 1011 (100 billion) cm, a photon must travel about 1022 cm before it will make its way to the surface. Since the photon travels at the speed of light (3 X 1010 cm/sec), it will take a minimum of 104 (a thousand) years for a photon to emerge. A much more detailed calculations give a time range of a few hundred thousand to 107 (ten million) years, since the mean free path of a photon is not always one centimeter throughout the Sun.

This exercise implies that changes deep in the Sun will take a considerable amount of time to be reflected in the light that we see. This is true. Neutrinos, which are not scattered in the same way as photons (neutrinos go right through matter without interacting), pass directly out of the Sun at the speed of light. They reflect very rapidly any changes in the interior of the Sun. They are also very hard to detect, since nothing absorbs them well. A great many neutrinos pass through our body every second without any interaction or effect.

To conclude, ask the students to identify other random processes that have some element of predictability. A common example is traffic lights—we don’t know how many traffic signals there'll be, but we can approximate how many red lights we'll hit in going from one place to another (about half).

Related to chapter 2 in the print guide.
Related Materials

For an overview of the Electromagnetic Spectrum, see the Electromagnetic Spectrum Chart.

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