With this substitution we find the expression for the foot-pounds of work corresponding to the contraction of the nebula from infinity to a sphere of radius a to be,
⅗ · 27 a M = 16 a M very nearly.
Hence we have the following fundamental theorem due to Helmholtz, which is the basis of the theory of sun heat.
If the sun he regarded as a homogeneous sphere of mass M pounds and radius a feet, then the foot-pounds of energy rendered available for sun heat by the contraction of the solar material from, an infinite distance is 16 a M.
§ 4. Evaluation of the Sun Heat Given Out in Contraction.
The number of foot-pounds of work given out in the contraction from infinity is 16 a M. As 772 foot-pounds are equal to one unit of heat, i.e. to the quantity of heat necessary to raise 1 lb. of water 1° Fahrenheit, we see that 772 M is the work required to raise a mass of water equal to the mass of the sun through 1° Fahrenheit. Hence the number of globes of water, each equal to the sun in mass, which would be raised 1° Fahrenheit by the total heat arising from the contraction, is
(16 a)/772,
but a, the radius of the sun in feet, is 2,280,000,000, and hence we have the following theorem:—
The energy liberated in the contraction of the sun from infinity to its present dimensions would, if turned into heat, suffice to raise 47,000,000 globes of water, each having the same mass as the sun, through 1° Fahr.
It is found by experiment that 1 lb. of good coal may develop 14,000 units of heat, and is therefore equivalent to 14,000 × 772 foot-pounds of work. A mass of coal equal to the sun would therefore (granted oxygen enough) be equivalent to 14,000 × 772 × M foot-pounds of work. But we have