The Earth's Beginning

by Robert S. Ball

But among all possible planes there is one of special significance in its relation to the system. It is called the “principal plane,” and it is characterised by the fact that the sum (with due attention to sign) of the areas described each second by the feet of the perpendiculars, multiplied into the masses of the corresponding particle, is greater than the like magnitude for any other plane, and is thus a maximum. For all planes parallel to this principal plane, the result will be, of course, the same; it is the direction of the plane and not its absolute situation that is material. We thus see that while this remarkable quantity is constant in any plane, for all time, yet the actual value of that constant depends upon the aspect of the plane; for some planes it is zero, for others the constant has intermediate values, and there is one plane for which the constant is a maximum. This is the principal plane, and a knowledge of it is of vital importance in endeavouring to understand the nebular theory. Nor are the principles under consideration limited only to a system consisting of sun and planets; they apply, with suitable modifications, to many other celestial systems as well.

The instructive character of this dynamical principle will be seen when we deduce its consequences. The term “moment of momentum” of a particle, with reference to a certain point in a plane, expresses double the product of the rate at which the area is described by the foot of the perpendicular to this plane, multiplied by the mass of the particle. The moment of momentum of the system, with reference to the principal plane, is a maximum in comparison with all other planes; that moment of momentum retains precisely the same value throughout all time, from the first instant the system was started onwards. And it retains this value, no matter what changes or disturbances may happen in the system, provided only that the influence of external forces is withheld. Subject to this condition, the transformations of the system may be any whatever. The several bodies may be forced into wide changes of their orbits, so that there may even be collisions among them; yet, notwithstanding those collisions, and notwithstanding the violent alterations which may be thus produced in the movements of the bodies, the moment of momentum will not alter. No matter what tides may be produced, even if those tides be so great as to produce disruption in the masses and force the orbits to change their character radically, yet the moment of momentum will be conserved without alteration.

It is essential to notice the fundamental difference between the principle which has been called the conservation of energy in the system, and the conservation of moment of momentum. We have pointed out that when collisions take place, part of the energy due to motion is transformed into heat, and energy in that form admits of radiation through space, and thus becomes lost to the system, with the result that the total energy declines. Even without actual collision, we have shown how certain effects of tides, or other consequences of friction, necessarily involve the squandering of energy with which the system was originally endowed. A system started with a certain endowment of energy may conserve that energy indefinitely, if all such actions as collisions or frictions are absent. If collisions or frictions are present the system will gradually dissipate energy. Our interpretation of the future of such a system must always take account of this fundamental fact.

It is, of course, conceivable that the moment of momentum with which a system was originally endowed might have happened to be zero. A system of particles could be so constructed and so started on their movements that their moment of momentum with regard to a certain plane should be zero. It might happen that the moment of momentum of the system with regard to a second plane, perpendicular to the former one, should be also zero; and, finally, that the moment of momentum of the system with regard to a third plane perpendicular to each of the other two, should be also zero. If these three conditions were found to prevail at the commencement, they would prevail throughout the movement, and, more generally still, we may state that in such circumstances the moment of momentum of the system would be zero about any plane whatever. There would be no principal plane in such a system. We thus note that though it is inconceivable that a group of mutually attracting bodies should be started into movement without a suitable endowment of energy, it is yet quite conceivable that a system could be started without having any moment of momentum. And if at the beginning the system had no moment of momentum, then no matter what may be the future vicissitudes of its motion, no moment of momentum can ever be acquired by it to all eternity, so long as the interference of external forces is excluded.

But having said this much as to the conceivability of the initiation of a system with no moment of momentum, we now hasten to add that, so far as Nature is actually concerned, this bare possibility may be set aside as one which is infinitely improbable. Nature does not do things which are infinitely improbable, and, therefore, we may affirm that all material systems, with which we shall have to deal, do possess moment of momentum. However the system may have originated, whatever may have been the actions of forces by which it was brought into being, we may feel assured that the system received at its initiation some endowment of moment of momentum, as well as of energy. Hence we may conclude that every such system as is presented to us in the infinite variety of Nature, must stand in intimate relation to some particular plane, being that which is known as the principal plane of moment of momentum. In our effort to interpret Nature, the physical importance of this fact can hardly be over-estimated.

In a future chapter we shall make some attempt to sketch the natural operations by which individual systems have been started on their careers. Postponing, then, such questions, we propose to deal now with the phenomena which the principles of dynamics declare must accompany the evolution of a system under the action of the exclusive attraction of the various parts of that system for each other. The system commences its career with a certain endowment of energy, with a certain endowment of moment of momentum, and with a certain principal plane to which that moment of momentum is specially related. In the course of the evolution through which, in myriads of ages, the system is destined to pass, the energy that it contains will undergo vast loss by dissipation. On the other hand, the moment of momentum will never vary, and the position of the principal plane will remain the same for all time. We have to consider what features, connected with the evolution, may be attributed to the operation of these dynamical laws. We have, in fact, to deduce the consequences which seem to follow from the fact that, in consequence of collisions, and in consequence of friction, an isolated system in space must gradually part with its initial store of energy, but that, notwithstanding any collisions and any friction, the total moment of momentum of the system suffers no abatement.

As the system advances in development, we have to deal with a gradual decline in the ratio of the original store of energy to the original store of moment of momentum. And hence we must expect that a system will ultimately tend towards a form in which, while preserving its moment of momentum, it shall do so with such a distribution of the bodies of which it consists as shall be compatible with a diminishing quantity of energy. It is not hard to see that in the course of ages this tends, as one consequence, to make the movements of each of the bodies in the system ultimately approximate to movements in a plane.

Let us, for simplicity, begin with the case of three attracting particles, A, B and C. Let B be started in any direction in the plane L, and let A be started in an orbit round it, and in the same plane L. Now let C be started into motion, in any direction, from some point also in L. It is certain that the sum of the areas projected parallel to any plane, which are described in a second by these three bodies, must be constant, each of the areas being, as usual, multiplied by the mass of the corresponding body. Let us specially consider the plane L in which the motions of A and B already lie. It is on this plane that the area described by C has to be projected. The essential point now to remember is that the projected area is less than the actual area. It is plain that if C has to describe a certain projected area in a certain time, the velocity with which C has to move must be greater when C starts off at an inclination to the plane than would have been necessary if C had started in the plane, other things being the same. Thus we see that, if the three bodies were all moving in the same plane, they could, speaking generally, maintain more easily the requisite description of areas, that is, the requisite moment of momentum with smaller velocities than if they were moving in directions which were not so regulated; that is to say, the moment of momentum can be kept up with less energy when the particles move in the same plane.

In a more general manner we see that any system in which the bodies are moving in the same plane will, for equal moment of momentum, require less energy than it would have done had the bodies been moving in directions which were not limited to a plane. Thus we are led to the conclusion that the ultimate result of the collisions and the friction and the tides, which are caused by the action of one particle on another, is to make the movements tend towards the same plane.

In this dynamical principle we have in all probability a physical explanation of that remarkable characteristic of celestial movements to which we have referred. The solar system possesses less energy in proportion to its moment of momentum than it would require to have if the orbits of the important planets, instead of lying practically in the same plane, were inclined at various angles. Whatever may have been the original disposition of the materials forming the solar system, they must once have contained much more energy than they have at present. The moment of momentum in the principal plane, at the beginning, was not, however, different from the moment of momentum that the system now possesses. As the energy of the system gradually declined, the system has gradually been compelled to adjust itself in such a manner that, with the reduced quantity of energy, the requisite moment of momentum shall still be preserved. This is the reason why, in the course of the myriads of ages during which the solar system has been acquiring its present form, the movements have gradually become nearly conformed to a plane.