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IV. Improvement of Classical Estimation

The last figure, giving the life of Sun, seems to be exaggerated when regarded from the perspective of the laws of conservation of energy. We don’t have means to verify its truth, but we are aware that it has a shortcoming: most certainly the density inside the Sun is not constant! There is no way to circumvent this shortcoming but only by evaluating the density itself. Knowing the potential energy, an evaluation of density can be made via Poisson equation for potential. Using equation (13) the density then is

(16)

It is so normalized as to give the current mass of Sun at its current radius. This approach cannot be used in the original Newtonian theory of gravitation, because the potential is a solution of the Laplace equation: the density should thus be always zero! While this fact is quite true in the outer space of a source of gravitation, it is not true inside that source, this is why the classical theory had to adopt a constant density in order to evaluate the age of Sun.

     Using expression (16) for the density, we have for the mass m(r) of a spherical shell and for the mass M(r) of the corresponding internal core the following equations:

(17)

So, instead of (14) we have

(18)

The integral can be performed and gives the closed form result:

(19)

This time, with the help of formula (19), we can find the following: assuming again the rock ages of 4 billion years for instance, gives the fact that the light comes from a region inside Sun having a radius about 7.4% from the Sun’s current radius, therefore a volume of about 0.04% from the Sun’s current volume. This region is, again, continuously decreasing inside the Sun, reaching a radius of 1% from the current radius of Sun in about 121 billion years. This may be taken as an estimation of the life span of the Sun. The figure is a little closer to the current thinking!

     Current thinking started after the discovery of the possibility of fusion of the nuclei, with the subsequent liberation of energy, as predicted by Einstein formula. The first work that we know about, estimating the life of stars along this line of thought, is that of Condon (Condon, 1925). It gives the life of stars of the order of magnitude of our figure here or bigger. Later on, Bethe (Bethe, 1939) took into consideration detailed cycles of thermonuclear reactions, estimating the life of stars around the figure we gave above. Zahn (Zahn, 1964) gives exactly the figure we presented above.

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