The Earth's Beginning

by Robert S. Ball

Whatever may have been the origin of the primæval nebula, and whatever may have been the forces concerned in its production we may feel confident that it was not originally at rest. We do not indeed know any object which is at rest. Not one of the heavenly bodies is at rest, nothing on earth is at rest, for even the molecules of rigid matter are in rapid motion. Rest seems unknown in the universe. It would be, therefore, infinitely improbable that a primæval nebula, whatever may have been the agency by which it was started on that career which we are considering, was initially in a condition of absolute rest. We assume without hesitation that the nebula was to some extent in motion, and we may feel assured that the motions were of a highly complicated description. It is fortunate for us that our argument does not require us to know the precise character of the movements, as such knowledge would obviously be quite unattainable. We can, however, invoke the laws of mechanics as an unerring guide. They will tell us not indeed everything about those motions, but they will set forth certain characteristics which the movements must have had, and these characteristics suffice for our argument.

To illustrate the important principle on which we are now entering I must mention the famous problem of three bodies which has engaged the attention of the greatest mathematicians. Let there be a body A, and another B, and another C. We shall suppose that these bodies are so small that they may be regarded merely as points in comparison with the distances by which they are separated. We shall suppose that they are all moving in the same plane, and we shall suppose that each of them attracts the others, but that except these attractions there are no other forces in the system. To discover all about the motions of these bodies is so difficult a problem that mathematicians have never been able to solve it. But though we are not able to solve the problem completely, we can learn something with regard to it.

We represent by arrows in Fig. [36] the directions in which A, B, and C are moving at the moment. We choose any point O in the plane, and for simplicity we have so drawn the figure that A, B, and C are forces tending to turn round O in the same direction. The velocity of a body multiplied into its mass is termed the momentum of the body. Draw the perpendicular from O to the direction in which the body A is moving, then the product of this perpendicular and the momentum of A is called the moment of momentum of A around O. In like manner we form the moment of momentum of B and C, and if we add them together we obtain the total moment of momentum of the system.

We can now give expression to a great discovery which mathematicians have made. No matter how complicated may be the movements of A, B, and C; no matter to what extent these particles approximate or how widely they separate; no matter what changes may occur in their velocities, or even what actual collision may take place, the sum of the moments of momentum must remain for ever unaltered. This most important principle in dynamics is known as the conservation of moment of momentum.

Though I have only mentioned three particles, yet the same principle will be true for any number. If it should happen that any of them are turning round O in the opposite direction, then their moments of momentum are to be taken as negative. In this case we add the moments tending in one direction together; and then subtract all the opposite moments. The remainder is the quantity which remains constant.

Fig. 36.—To illustrate Moment of
Momentum.

We may state this principle in a somewhat different manner as follows: Let us consider a multitude of particles in a plane; let them be severally started in any directions in the plane, and then be abandoned to their mutual attractions, it being understood that there are no forces produced by bodies external to the system; if we then choose any point in the plane, and measure the areas described round that point by the several moving bodies in one second, and if we multiply each of, those areas by the mass of the corresponding body, then, if all the bodies are moving in the same direction round the point, the sum of the quantities so obtained is constant. It will be the same a hundred or a thousand years hence as it is at the present moment, or as it was a hundred or a thousand years ago. If any of the particles had been turning round the point in the opposite direction, then the products belonging to such particles are to be subtracted from the others instead of added.

We have now to express in a still more general manner the important principle that is here involved. Let us consider any system of attracting particles, no matter what their masses or whether their movements be restricted to a plane or not. Let us start them into motion in any directions and with any initial velocities, and then abandon them to the influence of their mutual attractions, withholding at the same time the interference of any forces from bodies exterior to the system. Draw any plane whatever, and let fall perpendiculars upon this plane from the different particles of the system. It will be obvious that as the particles move the feet of the perpendiculars must move in correspondence with the particles from which the perpendiculars were let fall. We may regard the foot of every perpendicular as the actual position of a moving point, and it can be proved that if the mass of each particle be multiplied into the area which the foot of its perpendicular describes in a second round any point in the plane, and then be added to the similar products from all the other particles, only observing the proper precautions as to sign, the sum will remain constant, i.e., in any other second the total quantity arrived at will be exactly the same. This is a general law of dynamics. It is not a law of merely approximate truth, it is a law true with absolute accuracy during unlimited periods of time.

The actual value of the constant will depend both on the system and on the plane. For a given system the constant will differ for the different planes which may be drawn, and there will be some planes in which that sum will be zero. In other words, in those planes the areas described by the feet of the perpendiculars, multiplied by the masses of the particles which are moving in one way, will be precisely equal to the similar sum obtained from the particles moving in the opposite direction.