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11.13 Collisions and Opacity

Why does the Sun appear to have a sharp edge if it is made of gas? Why does a cloud appear to have an edge? In each case, there is no sharp boundary. In the Sun, the density of hot gas increases smoothly right as you go down through the region we see as the edge. In a cloud, the density of air inside the cloud is not very much higher than it is outside the region we see as the edge. In both cases, the answer lies in the way in which light interacts with particles. This is also a problem of contrast, and the edge is also the point at which our eye can detect either the Sun or a cloud against the background.

 

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The Sun as seen by the SOHO Satellite. Click here for original source URL

 

Opacity, or optical depth, is the degree to which a material allows light to pass through it. If it transmits all background light, it is transparent and the optical depth is zero. If it transmits no light, it is opaque and the optical depth is a large number. Everyday examples are obvious — glass is transparent, iron is opaque; water is transparent, milk is opaque. A gas can have opacity too. We cannot see to the center of the Sun, so we know it has a non-zero opacity.

In a very dense gas, like the hot core of the Sun, photons travel only a very short distance before colliding with a particle. The rate of progress of the photon is proportional to the square root of the number of collisions it suffers. Imagine you were standing blindfolded at the center of a large circular park. You could leave the park just by walking purposefully in any direction. Now imagine there are people standing randomly across the grass. As you tried to leave, you would collide with one person, get disoriented, move off in another direction, collide with another person, move off in yet another direction, and so on. It would take you a lot longer to get out of the park!

Motion that results from a sequence of randomly-directed collisions is called a random walk, analogous to the lurching movement of someone who is totally dizzy trying to find their way home. After 100 direct steps, you would of course be 100 steps away from the center of the grassy area. If you collided with 25 people, however, your rate of progress would be √25 = 5 times slower, so 100 steps would only take you 100/5 = 20 steps away from the center. If you had the misfortune to bump into 100 people, your rate of progress would be √100 = 10 times slower. Your 100 steps would only take you 10 steps from the center. In probability theory, the distance traveled per second will be inversely proportional to the square root of the number of collisions per second. And since the number of collisions depends on the density, the distance traveled per second is inversely proportional to the square root of the density.

If x is the distance a photon travels between collisions, and it suffers N collisions, the distance the photon travels in any one direction is:

d = x√N

So it takes 100 steps to travel a distance 10x, 10,000 steps to travel a distance 100x, and a million steps to travel a distance 1000x. Diffusion of radiation is a slow and inefficient way to transport energy, although stars like the Sun move energy by convection too.

In the center of the Sun, the density is so high that photons diffuse much slower than they would traveling through empty space, even though photons travel at the speed of light between collisions. As the photon works its way out of the Sun's core, it loses energy in its collisions, gradually changing from an X-ray photon to an ultraviolet photon. At every point in its slow journey, the photon is in equilibrium with its surroundings, so its wavelength corresponds by Wien’s Law to the temperature of the gas surrounding it.

How much more slowly does a photon move at the Sun's core than at its surface. (Remember that at any time the photon is traveling at the speed of light — we are concerned here with how fast a photon makes its way outwards given its many interactions.) The ratio of speeds is given by the inverse square root of the ratio of densities. The Sun's density is 158,000 kg/m3 at the center and 0.001 kg/m3 at the edge. Photons thus travel (158,000/0.001)1/2 = 12,600 times slower in the core than they do when they leave the Sun. It takes about 50,000 years for radiation to get from the Sun's core to the surface!

Diffusion caused by collisions or interactions relates to the reduction in the intensity of light as it travels through a gas. The change in intensity of a beam of light as it passes through a gas is:

I = I0e

In this equation I0 is the starting intensity, I is the final intensity and the Greek letter τ is the opacity or optical depth. We can relate this to our previous discussion by noting that the opacity is the square root of the number of collisions, so:

I = I0e-√N

A reduction in light intensity occurs for scattering of the photons or absorption of the photons. (Even if a photon is absorbed and raises an electron's energy level, it will eventually be re-emitted in a random direction.) With a gas thin enough that no collisions occur, I = I0 and the light intensity is not diminished. After 5 collisions, N = 5, and e-√5 = 0.11, so the light intensity is diminished by nearly 90%. After 10 collisions, N = 10, and e-√10 = 0.04. After 15 collisions, N = 15 and e-√15 = 0.02. You can see that the light is rapidly extinguished as the opacity increases.

At some point in its journey from the center of the Sun, radiation travels through rarified gas. Photons will reach a region where the density is so low that they are unlikely to suffer any collisions at all. At this point, they travel freely in a straight line to the Earth. Astronomers call this region the Sun's photosphere; it occurs at a temperature of 5770 Kelvin, which has thermal radiation in the form of visible light. We see it as the Sun's edge. Inside this region, light bounces around and the Sun is opaque. Beyond this region, the Sun is transparent, light travels directly, and we have the appearance of a surface. The Sun still has material beyond this point, but except during solar eclipses, we can't see this material against the background sky. 

The principle of opacity also applies to a cloud. There is water vapor throughout the atmosphere, and in some places the concentration is higher than in others. Inside a cloud, the concentration is high enough that light bounces around from one water droplet to another and cannot travel freely; this region is opaque. The edge of the cloud just defines the region where, on average, light will not collide with a water droplet and can travel directly to our eyes. To take another example, consider a glass of water. Water is transparent, but if you slowly add drops of milk, it will become opaque. Milk contains large, fat molecules that scatter light. As you add more milk, the number of collisions between photons and large molecules increases, and the opacity increases.Objects are often partially opaque, and some light can travel through them for some distance. Fog is a good example of this. You may be able to see your hand faded but still in front of your face but not be able to see a tree 10 feet away from you.

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Example of a random walk in two dimensions. Click here for original source URL.