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III. A Critique of the Classical Calculations

What is striking in this flat rejection is the fact that nobody ever questioned the correctness of the above classical argument: it is considered correct by default as it stands, therefore the classical theory must be abandoned because gives wrong results. It’s like Classical Mechanics is a Bible, whose truth should be eternal as it is. We’ll try here to unlock that image and open the classical argument to a dialectical criticism.

     First of all it is unfair, and indeed unnatural, to allow to any other theory the licence that the energy of the Sun comes from a limited region in the center of the Sun, while denying it for the classical theory of gravitation. To wit, if we use equation (7) in order to calculate the radius of an inner core of the Sun which, assuming that it is the source of the Sun’s energy, would give us an age of, say, four billion years, we find the value of about 3174 Kilometers. This would mean that the matter of the outer shell of the Sun above 3174 Kilometers squeezes by gravitation the matter of the inner core, and this last one releases energy which we perceive as light and heat. Isn’t this perfectly natural?

     Two main objection might be rightfully raised against this reasoning. One of them is the lack of physical understanding of the process by which the gravitational energy of squeezing is transformed into light and heat. This objection brought the modern theories based upon fusion to attention of the scientific community. This might not be a serious objection, for if the theory proves to be right, it can work simply based on the law of conservation of energy, without being necessary to know the details of the physical process that produces the observable energy. There is, however, another, more serious objection: there is no guarantee that in the conditions from the interior of the Sun the Newtonian force of gravitation is maintained, to say nothing of the fact that the density inside the Sun is surely not a constant. These are shortcomings of the classical theory of gravitation, and they are to be addressed in the first place, if we are to give a correct estimation of energy to be used by the law of conservation.

     In order to do this, the first thing coming naturally to our mind is the fact that the Newtonian force of gravitation might be considered an approximation of the reality, valid whenever the real celestial bodies are approximated by points. As known, this only can be done when the distances between the centers of the bodies are large enough, so the dimension of the bodies can be neglected in comparison with that distance. As a matter of fact this is how the force of gravitation was discovered in the first place. This property can be easily put into equation in order to reveal the possible general form of the force of gravitation in more general conditions. We will go simply along the following line of reasoning.

     The Newtonian force (1) can be derived from a potential energy by

(10)

In other words, this is the energy represented by the work of the force of gravitation in bringing the mass m from infinity to distance r from the center of attraction. If the force of gravitation is an approximation, this energy is an approximation too. So if we want to know just what approximation is the Newtonian force, we have to see what approximation is this potential energy, and this can be easily seen. Indeed, let’s put W(r) from equation (10) in the form

(11)

making thus obvious the fact that the Newtonian potential energy may be the first approximation of a function which reduces to a linear homogeneous form in the limit of small values of R₀/r. There are obviously many such energy functions but, for a momentarily undisclosed reason we choose a closed form logarithmic function

(12)

When the argument is very small, the logarithmic function reduces to linear function, so that in the limit of large r we have the relation (11), therefore the Newtonian case. However, now we can accept (12) as valid everywhere a priori, even in the interior of the Sun or Earth. The force coming out of this potential energy through equation (10) is

(13)

and obviously generalizes the Newtonian force.

     The main advantage of using the force (13) rests upon the fact that the solar radius can be used as an experimental parameter just like the mass of Sun, gravitational constant and such. The expression (13) is like a theoretical extension of our knowledge inside the unknown region of Sun, based on the knowledge we gained due to our position in the Universe. Thus we can say that this knowledge involves three fundamental constants: the mass of the Sun, the radius of the Sun and the gravitational constant.

     We can now carry on the classical calculations leading to the value (8) of the solar energy in exactly the same way as before. Thus, we have

(14)

where ξ ≡ R/R₀. Performing the integration gives

(15)

Now we can use this expression in conjunction with the Earth’s rock age, in order to estimate a region inside Sun from which the energy originates. Assuming 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 66% from the Sun’s current radius, therefore a volume of about 30% from the Sun’s volume. This region is continuously decreasing inside the Sun, reaching a radius of 1% from the current radius of Sun in about 10¹⁰ billion years. Consequently this is the time interval after which the Sun loses practically all the possibility of giving away any energy. In other words, this time interval is the life span of the Sun.

 

 

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