the
M-L and L-T_{e} relations |

Without even fully solving the homologous equations of stellar structure, we can deduce the gradients of the mass-luminosity and luminosity-effective temperature relations for main-sequence stars.

We have assumed that the luminosity of any point inside a star depends on some power of

At the surface of the star (

Since

Hence we can see that the homologous stellar models predict a mass-luminosity relation in which the luminosity is proportional to the

Turning now to the luminosity-effective temperature relation, these two quantities are related to the radius of a star through the well-known relation:

Hence, we can write:

Combining this equation with the relations

Since

We have already proved that

This shows that stars lie on a straight line of gradient 4a

We must now see if the predicted gradients of the mass-luminosity and luminosity-effective temperature relations are in agreement with the observed values. In order to prove that this is so, we must solve the 5 algebraic equations for

We have already stated that a general solution to these equations is too complicated to give here, but it is possible to write down solutions for special values of ,

In our discussions of stellar opacity, we found that one approximation to the opacity law which appears to work well at intermediate temperatures is given by = 1 and

=

A reasonable approximation to the rate of energy generation by the proton-proton chain is given by

=

Substituting = 1,

We now have 5 algebraic equations with 5 unknowns, so it is a simple matter to obtain an exact solution by eliminating each of the

Substituting these values into the mass-luminosity and luminosity-effective temperature relations derived above, we obtain:

The observed mass-luminosity law is not a simple power law but, if the central part of the curve (corresponding to stars of about solar mass) is approximated by a power law, it has an exponent of approximately 5, in good agreement with the value of 5.5 predicted by the above homologous solution; see figure 10.4 in Bohm-Vitense. Similarly, the lower part of the main-sequence on the observed luminosity-effective temperature (or HR) diagram is well represented by the power-law exponent of 4.1 predicted by the above homologous solution; see figure 10.5 in Bohm-Vitense. We have therefore verified the observed mass-luminosity relation of main-sequence stars and the slope of the main-sequence on the HR diagram and hence fulfilled one of our initial aims when we discussed the observed properties of stars at the start of this lecture course.