Approximate Mass-Luminosity Relationship

As a first step in understanding the luminosity of stars, we will assume that energy is transported through a star only by radiation; that is, by photons of light diffusing through successive layers of the star. The net radiation flow through any given radius is equal to the surface area at that radius, multiplied by the amount of radiation traveling through that surface per unit area. We can write this as

where we introduce F(r) as the radiation flow through a unit of area at radius r within the star; this quantity is called the flux.

How can we evaluate F? Without going into detail, we can deduce how it depends on various fundamental stellar quantities. A net flow of photons is possible only if there is a difference between the density of photons at one radius of the star, and at a larger radius. Specifically, photons will flow from a region where they are more dense to a region where they are less dense. How quickly the photons will flow depends upon how rapidly the density of photons changes with radius. However, even if there is such a gradient in photon density, why would the photons not simply stream outward at the speed of light? They cannot do so because of the enormous quantity of very dense matter that they must traverse. Photons collide with the particles of matter on their journey through the interior. At each collision, the photon is absorbed and re-emitted, and this slows it to an effective speed considerably less than its speed in a vacuum, just as it is much more difficult to walk through a bustling crowd of people, than along an empty sidewalk. Hence the photon flux will be decreased by increasing matter density. Combining these two ideas, that photons travel from higher photon density to lower photon density, and that photons are slowed by matter collisions, gives the approximate relationship

where we use the symbol for the energy density of the photons. (Photons of all energies, not just gamma rays, are usually symbolized by the Greek letter .)

We next need to relate the energy density of the photons to one of our stellar variables. To a good approximation, we can use the blackbody-radiation rules given in Chapter 4; the energy density of photons, which is immediately related to the intensity of the light, is thus proportional to the fourth power of the temperature, . The rate at which the photons leak out will be determined, then, by the change in the fourth power of temperature with radius. Since the temperature goes down as one moves out through the star, and the photons move toward the cooler regions, the radiation will eventually leak out of the star. From these general considerations, we can say that the radiation flux is roughly given by

Next we approximate the temperature difference in equation (5.13) by the difference between the core temperature and the surface temperature, and we will make the approximation that the surface temperature is zero. This may seem drastic, but the surface temperature of a star is very much lower than its core temperature, so it is actually not a bad first estimate. Then, as before, we set , where R is the radius of the star. Further, let us approximate the mass density by the total mass divided by the total volume of the star. Now remember from equation (5.10) that , and use equation (5.11) for the total luminosity to get

Carrying out the division, we finally obtain

It may not be immediately obvious, but this result is an amazing thing. Admittedly, our derivation was rather crude, but an analysis based upon some simple physical considerations has shown that we can expect the luminosity of star to go up roughly as the mass cubed. This relationship comes quite close to describing the observed relationship between luminosity and mass for hydrogen-burning stars.