Page:Popular Science Monthly Volume 58.djvu/150
142 POPULAR SCIENCE MONTHLY.
energy generated by the fall of the superficial portions towards the center is sufficient to double the absolute amount of heat. Whether this will be the case depends on a variety of circumstances; the mass of the whole body, and the capacity of its substance for heat. If we are to proceed with mathematical rigor, it is, therefore, necessary to determine in any given case whether this condition is fulfilled. Let us suppose that in any particular case the mass is so small or the capacity for heat so considerable that the temperature is not doubled by the contraction. Then the contraction will go on further and further, until the mass becomes a solid. But in this case let us reverse the process. The body being supposed nearly in a state of equilibrium in position A, let the elastic force be slightly in excess. Then the gas will expand. In order that it be reduced to a state of equilibrium by expansion, its tempera- ture must diminish according to the same law that it would increase if it contracted. When its diameter doubles, its temperature should be re- duced to one half or less by the expansion, in order that the equilibrium shall subsist. But, in the case supposed, the temperature is not reduced so much as this. Hence, it is too high for equilibrium by a still greater amount and the expansion must go on indefinitely. Thus, in the case supposed, the hypothetical equilibrium of the body is unstable. In other words, no such body is possible.
This conclusion is of fundamental importance. It shows that the possible mass of a star must have an inferior limit, depending on the quantity of matter it contains, its elasticity under given circumstances and its capacity for heat. It is certain that any small mass of gas, taken into celestial space and left to itself, would not be kept together by the mutual attraction of its parts, but would merely expand into in- definite space. Probably this might be true of the earth, if it were gaseous. The computation would not be a difficult one to make, but I have not made it.
In what precedes, we have supposed a single mass to contract. But our study of the relations of temperature and pressure in the two masses assumes no relationship between them, except that of equality. Let us now consider any two gaseous bodies, A and B, and suppose that the body B, instead of having the same mass as that of A, is another body with a different mass.
Since the mass, B, may be of various sizes, according to the amount of attraction it has undergone, let us begin by supposing it to have the same volume as A, but twice the mass of A. We have then to inquire what must be its temperature in order that it may be in equilibrium. We have first to inquire into the hydrostatic pressure at any point of the interior. Referring once more to a figure like either of those in Fig. 2, a spherical shell like C D will now in the case of the more mass- ive body have double the mass of the corresponding shell of A. The