The Earth's Beginning

by Robert S. Ball

To follow the fortunes of a system of bodies, large or small, starting under any arbitrary conditions at the commencement, and then abandoned to their mutual attractions, is a problem for the mathematician. It certainly presents to him questions of very great difficulty, and many of these he has to confess are insoluble; there are, however, certain important laws which must be obeyed in all the vicissitudes of the motion. There are certain theorems known to the mathematician which apply to such a system, and it is these theorems which afford us most interesting and instructive information. I am well aware that the subject upon which I am about to enter is not a very easy one, but its importance is such that I must make the effort to explain it.

Let me commence by describing what is meant when we speak of the energy of a system. Take, first, the case of merely two bodies, and let us suppose that they were initially at rest. The energy of a system of this very simple type is represented by the quantity of work which could be done by allowing these two bodies to come together. If, instead of being in the beginning simply at rest, the bodies had each been in motion, the energy of the system would be correspondingly greater. The energy of a moving body, or its capacity of doing work in virtue of its movement, is proportional jointly to its mass and to the square of its velocity. The energy of the two moving bodies will therefore be represented by three parts; first, there will be that due to their distance apart; secondly, there will be that due to the velocity of one of them; and, thirdly, there is that due to the velocity of the other. In the case of a number of bodies, the energy will consist in the first place of a part which is due to the separation of the bodies, and measured by the quantity of work that would be produced if, in obedience to their mutual attraction, all the bodies were allowed to come together into one mass. In the second place, the bodies are to be supposed to have been originally started with certain velocities, and the energy of each of the bodies, in virtue of its motion, is to be measured by the product of one-half its mass into the square of its velocity. The total energy of the system consists, therefore, of the sum of the parts due to the velocities of the bodies, and that which is due to their mutual separation.

If the bodies could really be perfectly rigid, unyielding masses, so that they have no movements analogous to tides, and if their movements be such that collisions will not take place among them, then the laws of mechanics tell us that the quantity of energy in that system will remain for ever unaltered. The velocities of the particles may vary, and the mutual distances of the particles may vary, but those variations will be always conducted, subject to the fundamental condition that if we multiply the square of the velocity of each body by one-half its mass, and add all those quantities together, and if we increase the sum thus obtained by the quantity of energy equivalent to the separation of the particles, the total amount thus obtained is constant. This is the fundamental law of mechanics known as the conservation of energy.

For such material systems as the universe presents to us, the conservation of energy, in the sense in which I have here expressed it, will not be maintained; for the necessary conditions cannot be fulfilled. Let us suppose that the incessant movements of the bodies in the system, rushing about under the influence of their mutual attractions, has at last been productive of a collision between two of the bodies. We have already explained in Chapter VI. how in the collision of two masses the energy which they possess in virtue of their movements may be to a large extent transformed into heat; there is consequently an immediate increase in the temperature of the bodies concerned, and then follows the operation of that fundamental law of heat, by which the excess of heat so arising will be radiated away. Some of it will, no doubt, be intercepted by falling on other bodies in the system, and the amount that might be thus possibly retained would, of course, not be lost to the system. The bodies of the solar system at least are so widely scattered, that the greater part of the heat would certainly escape into space, and the corresponding quantity of energy would be totally lost to the system. We may generally assume that a collision among the bodies would be most certainly productive of a loss of energy from the system.

No doubt collisions can hardly be expected to occur in a system consisting of large, isolated bodies like the planets. Even in any system of solid bodies collisions may be presumed to be infrequent in comparison with the numbers of the bodies. But if, instead of a system of few bodies of large mass, we have a gas or nebula composed of innumerable atoms or molecules, the collisions would be by no means infrequent, and every collision, in so far as it led to the production of heat, would be productive of loss of energy by radiation from the system.

It should also be added that, even independently of actual collisions, there is, and must be, loss of energy in the system from other causes. There are no absolutely rigid bodies known in nature, for the hardest mineral or the toughest steel must yield to some extent when large forces are applied to it, and as the bodies in the system are not mere points or particles of inconsiderable dimensions, they will experience stresses something like those to which our earth is subjected in that action of the moon and sun which produces the tides. In consequence of the influences of each body on the rest, there will be certain relative changes in the parts of each body; there will be, as it were, tidal movements in their liquid parts and even in their solid substance. These tides will produce friction, and this will produce heat. This heat will be radiated from the system, but the heat radiated corresponds to a certain amount of energy; the energy is therefore lost to the system, so that even without actual collisions we still find that energy must be gradually lost to the system.

Thus we have been conducted to an important conclusion, which may be stated in the following way. Let there be any system of bodies, subject to their mutual attractions, and sufficiently isolated from the disturbing influence of all bodies which do not belong to the system, then the original energy with which that system is started must be undergoing a continual decline. It must at least decline until such a condition of the system has been reached that collisions are no longer possible and that tidal influences have ceased. These conditions might be fulfilled if all the bodies of the system coalesced into a single mass.

As illustrations of the systems we are now considering, we may take the sun and planets as a whole. A spiral nebula is a system in the present sense, while perhaps the grandest illustration of all is provided by the Milky Way.

It will be noted that we may have a system which is isolated so far as our present argument is concerned, even while it forms a part of another system of a higher order of magnitude. For instance, Saturn with his rings and satellites is sufficiently isolated from the rest of the solar system and the rest of the universe, to enable us to trace the consequences of the gradual decline of energy in his attendant system. The solar system in which Saturn appears merely as a unit, is itself sufficiently isolated from the stars in the Milky Way to permit us to study the decline of energy in the solar system, without considering the action of those stars.

This general law of the decline of energy in an isolated system, is supplemented by another law often known as the conservation of moment of momentum. It may at first seem difficult to grasp the notion which this law involves. The effort is, however, worth making, for the law in question is of fundamental importance in the study of the mechanics of the universe. In the Appendix will be found an investigation by elementary geometry of the important mechanical principles which are involved in this subject.