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II.The Classical Way of Calculating Sun’s Energy and Age

In estimating the gravitational energy stored in the Sun, one starts from the expression of the Newtonian force of gravitation. In vector form this force is

(1)

Here G is the classical constant of gravitation (in international units G = 6.673.10^-11 m³kg^-1s^-2); M is the attractive mass, in our case the Sun; m is the attracted mass, whatever this may be, situated in a position marked by the vector  with respect to Sun. One can see that the force vector is oriented against the position vector, therefore it represents an attractive force, as it should indeed be. The elementary work performed by this force in order to move the mass m towards the attracting center is

(2)

If m is carried by gravitation between R₂ and R₁, with R₂ > R₁, then the work performed by the gravitation force is

(3)

In particular if the mass m is brought from infinity to the point R, it has an energy due to gravitation as given by

(4)

Up to this point the theory refers to point particles, i.e. bodies so remote from each other that they can be considered of negligible dimensions when compared with the distance between them.

     Now, the formula (4) can be used in the interior of Sun, with the following assumptions: (a) the Sun is perfectly spherical and (b) the density of Sun’s matter is constant. In this case M is considered as the mass of an inner core of radius R, and m is considered as the mass of a spherical shell surrounding this inner core. Thus W(R) from equation (4) is the energy due to gravitation of the inner core acting upon the shell. If we sum this energy over the whole volume of the Sun, then we can find the whole content of the gravitational energy of the Sun. Let’s do the exercise: we have

(5)

where ρ is the density of Sun. Thus, the gravitational energy of a inner core of radius R attracting a shell of thickness dR, will be given by (4) as

(6)

and the summation of all these energies over the whole volume of Sun can be replaced by an integral from zero the external radius of Sun, say R₀. We thus have

(7)

where M₀ is the current mass of Sun. This equation gives the classical result regarding the energetic content of the Sun due to gravitation. With M₀ = 2·10³⁰ kilogram, R₀ = 6.96·10⁸ meter and the value above of G, this gives

(8)

Now here comes the interesting part: if we accept that (c) this whole energy is spent exclusively on throwing energy in the surrounding space, then we can have a rough estimate of the age of Sun, for we have an estimate of the luminosity of Sun. This is about 4·10²⁶ Watt, so that the life of Sun up to this point in time would be about

(9)

This is a figure which, when compared with the age of rocks on Earth, must be rejected as an acceptable age of Sun: this age has obviously to be greater than the age of rocks!

 

 

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