Suppose, with Helmholtz, that the sun originally existed as a nebulous mass, filling the entire space presently occupied by the solar system and extending into space indefinitely beyond the outermost planet. The total amount of work in foot-pounds performed by gravitation in the condensation of this mass to an orb of the sun’s present size can be found by means of the following formula given by Helmholtz,[202]
Work of condensation = 3/5 × r2M2/Rm × g
M is the mass of the sun, m the mass of the earth, R the sun’s radius, and r the earth’s radius. Taking M = 4230 × 1027 lbs., m = 11,920 × 1021 lbs., R = 2,328,500,000 feet, and r = 20,889,272 feet; we have then for the total amount of work performed by gravitation in foot-pounds,
Work = 3/5 × (20,889,272·5)2 × (4230 × 1027)2/2,328,500,000 × 11,920 × 1021
= 168,790 × 1036 foot-pounds.
The amount of heat thus produced by gravitation would suffice for nearly 20,237,500 years.
These calculations are based upon the assumption that the density of the sun is uniform throughout. But it is highly probable that the sun’s density increases towards the centre, in which case the amount of work performed by gravitation would be somewhat more than the above.
Some confusion has arisen in reference to this subject by the introduction of the question of the amount of the sun’s specific heat. If we simply consider the sun as an incandescent body in the process of cooling, the question of the amount of the sun’s specific heat is of the utmost importance; because the absolute amount of heat which the sun is capable of giving out depends wholly upon his temperature and specific heat. In this case three things only are required: (1), the sun’s mass; (2), temperature of the mass; (3), specific heat of the mass. But if we are considering what is the absolute amount of heat which could have been given out by the sun on the hypothesis that gravitation, either according to the meteoric theory suggested by Meyer or according to the contraction theory advocated by Helmholtz, is the only source of his heat, then we have nothing whatever to do with any inquiries regarding the specific heat of the sun. This is evident because the absolute amount of work which gravitation can perform in the pulling of the particles of the sun’s mass together, is wholly independent of the specific heat of those particles. Consequently, the amount of energy in the form of heat thus imparted to the particles by gravity must also be wholly independent of specific heat. That is to say, the amount of heat imparted to a particle will be the same whatever may be its specific heat.
Even supposing we limit the geological history of our globe to 100 millions of years, it is nevertheless evident that gravitation will not account for the supply of the sun’s heat during so long a period. There must be some other source of much more importance than gravitation. What other source of energy greater than that of gravitation can there be? It is singular that the opinion should have become so common even among physicists, that there is no other conceivable source than gravitation from which a greater amount of heat could have been derived.
The Origin and Chief Source of the Sun’s Heat.—According to the foregoing theories regarding the source of the sun’s heat, it is assumed that the matter composing the sun, when it existed in space as a nebulous mass, was not originally possessed of temperature, but that the temperature was given to it as the mass became condensed under the force of gravitation. It is supposed that the heat given out was simply the heat of condensation. But it is quite conceivable that the nebulous mass might have been possessed of an original store of heat previous to condensation.