## The Mass Luminosity Relation

The mass-luminosity or M-L relation between stellar luminosities and masses is the most fundamental stellar property. It was first derived analytically by Eddington [168]. Nowadays, this relation is better established by numerical models, see for example Fig. 25.6 or Table 25.4. However, the expression of the radiative flux enables us to obtain a first estimate of this relation. Let us write (3.15) as follows:

Making the following estimates for a star of radius R and central temperature Tc: -dT4/dr - Tc4/R, Lr - L and r - R/2, one gets

3 KQ

For the density, we take the average density Q = 3M/(4 nR3) and for the central temperature, we adopt the order of magnitude of the internal temperature (1.26) T — (pmu/k)(GM/R), where x is here the average mean molecular weight. We obtain for the luminosity r (Gmu\A X4M3

where we have skipped the numerical factors of the order of unity. This fundamental relation shows several remarkable properties:

- The luminosity grows with the third power of the mass, this is indeed a good average for stars from 1 to 100 M0 in the hydrogen-burning phase. As an example, a star of 100 M0 has a luminosity of about 106 L0.

- A star with a higher mean molecular weight x is overluminous with respect to a star of the same mass with a lower x. Typical examples of such overluminous stars are Wolf-Rayet stars (Fig. 27.15), which are evolved bare cores left over from initially massive stars.

- In (3.25), there is nothing related to the nuclear energy production and it is a most striking result: the luminosity L of a star does not depend on its nuclear reactions, but on its opacity. This means that if the stellar opacity permits the photons to escape from the star, they do it. As a consequence, the star must produce in a way or another the energy that is going out. This is done by stellar contraction and nuclear reactions. The nuclear reactions feed the stars in energy and allow them to shine for very long times, but the value of the luminosity is determined by the opacity.

We have derived the above result by considering that the transport of energy is done only by radiative transfer. There are other mechanisms of transport (e.g., convection, neutrino emission). However, radiative transfer is always present and during the main phases of evolution (H- and He-burning phases) it largely dominates, so the above relation is meaningful.

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