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**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.

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Copyright © 1997 John F. Hawley