Towards a Unified Cosmology

The Problem

So long as no external couple is applied, any change in the moment of inertia of the system is accompanied by a corresponding change in its angular velocity. Provided the mass remains constant, the moment of inertia is proportional to the radius of gyration. In a system to which no couple is applied, and of which the mass is constant, the radius of gyration varies, therefore, in inverse ratio to the angular velocity. This is why contraction of a star under the influence of its own weight is accompanied by an increase in its rate of spin.

The angular momentum of a star, such as the Sun, is considerable. We know from the principle of conservation of angular momentum that this large quantity has been imparted to the Sun at some time during its past history and that a couple must have been applied to do this. What was the couple? How did it act? When was it applied?

The same questions can also be asked concerning the enormous angular momentum of the galaxies and, if they can be answered for the one, the same kind of answer should hold, broadly speaking, for the other. So it will be convenient to discuss the problem mainly for stars. It is more hopeful to look for the couple during the star's earliest history than later, so attention will be directed towards the incipient star.

One might for a moment be tempted to think that a non-uniform distribution of the substance of the incipient star would suffice to generate angular momentum during the process of contraction from a tenuous gas into the compact star, for any asymmetry would cause the average movement of the particles that were falling inward to be off-centre. Their force of impact with other particles would produce a couple and this would lead to a swirl about the centre of gravity of the star. An accumulation of such swirls would amount to rotation provided they were all about the same axis and in the same sense of rotation.

But they would not all be like that. There would be swirls in all directions, left hand and right hand ones. These would collide and lead only to turbulence. A swirl in one direction could never preponderate over the others, even to a small extent, however irregular the shape of the incipient star was, so long as movement of particles was random. If the component parts are at rest relative to each other when the contraction begins, a mere process of contraction cannot generate the least spin of the star about its axis.

This follows quite simply from the principle of conservation of angular momentum. The forces that lead to contraction are forces between the component parts of the incipient star and the principle asserts that these cannot generate any angular momentum. This was zero when the contraction began and so it must remain zero unless an external couple changes it.

It might, of course, be argued that the component parts were never at rest relative to each other, and this is probably true. One may expect some thermal agitation, some collisions between molecules. But a moment's reflection shows that these cannot constitute angular momentum.

The movement of a particle has two components, one along a line that joins it to the centre of gravity of the incipient star; it is a radial component. The other is at right-angles to this line, a tangential component. Only the tangential components could contribute to angular momentum and to do this there would have to be more components in one plane and direction of movement than in the others. But with random movement of particles in thermal agitation this does not happen. For thermal agitation means random movement of a very large number of very small particles. On probability considerations it is easy to show that the vector sum of all the momenta must be zero to a very close approximation.

Can one then justify the assumption that large masses of gas move in solid phalanx in a particular direction? Suppose there were a limited number of such phalanxes and the movements within each were not random. Each phalanx would have a tangential component of its velocity and momentum. If the vector sum of the momenta were taken, there would be a residual vector in a particular direction. With a random distribution of phalanxes it would be small if there were many of them, but not negligible if there were not many. Such a residual vector would amount to angular momentum.

But the difficulty is in finding a reason for the phalanxes. Why should a large number of particles move together in the same direction? Where is one to look for a co-ordinating force, for the guiding principle that would preclude random movement of the individual particles? By this line of reasoning, angular momentum would be accounted for only by the assumption of something that acted like a cosmic drill sergeant. But such an assumption would be at least equally difficult to explain. There does not seem to be any tenable alternative to attributing the rotation of stars and galaxies to an initiating couple, and it is very difficult indeed to account for this or to understand how it can act. Facile explanations have been provided for the rotation of stars, it is true, and sometimes with the backing of high authority. But they are based on flimsy reasoning which collapses at the first breath of criticism. Let us consider the problem in the appropriate terms, namely those of simple mechanics.

The Conditions Needed to Cause Continued Rotation

1. An external couple must be applied to the system.
2. The system must have such asymmetry as enables the couple to act on it. Thus the slot in the head of a screw allows a screw-driver to act on the screw. The asymmetrical structure of a compass needle, with N at one end and S at the other, enables a bar magnet brought close to the compass needle to exert a rotating couple. In consequence the needle turns until one pole points towards the bar magnet. A crank provides a similar asymmetry and enables the force exerted by steam on the piston of an engine to impart a couple to the flywheel that is rotated.
3. The external couple must either rotate with the system to which it is applied or cease to operate after rotation has been initiated. The first requirement is met by a screwdriver. This rotates with the screw. The second requirement would be provided if a bar magnet brought near to a compass needle were quickly withdrawn after rotation had been initiated. Provided the needle were shielded from the Earth's magnetic field, it would continue by virtue of its inertia. But when the bar magnet is not withdrawn, the needle comes to rest in a position in which it points towards the magnet This is a position of minimum potential energy, of equilibrium. A technical name for it is 'dead centre'. For rotation to be continuous, the applied force must be such that it fails to hold the rotating system at dead center.

Accounting for Asymmetry

The biggest unbalanced force would come from a neighbouring star. In particular, two components of an incipient double star must exert a gravitational force on each other. This must greatly exceed the gravitational force exerted by more distant stars.

To meet the second condition, the incipient star must itself be of irregular shape. But it has been shown in Chapter 27 that the incipient star would be of such a shape. It would be a collection of incipient concentrations with an irregular distribution. Only at a later stage of development would the star be pulled together by its own gravitational field into a homogeneous sphere. For dynamic purposes, the structure of the star could be resolved into a dumb-bell shape. The two concentrations that were equidistant from the geometric centre would be the equivalent of two halves of the star's mass.

Fig. 5. Rotation of incipient concentrations under the mutual effects of asymmetrical structure, intermittent gravitational pulses and gradual disappearance of asymmetry under contraction

Two neighbouring stars, when represented by their equivalent dumbbells, could have, at a given moment, one of the two relative positions shown in Fig. 5 at A and B. The nearer mass would be drawn towards the neighbouring star more strongly than the further one. The consequence would be that gas particles in the nearer mass would acquire a greater velocity towards the neighbouring star than those in the further mass. If the dumb-bells were rigid structures, they would rotate until they lay on a common axis, provided the attracting force were continuous. In fact they are not rigid structures. Each gas particle acquires a velocity towards the neighbouring star as well as a velocity towards the geometrical centre. This leads to a component of its movement at right angles to the above-mentioned radius, r. But the component of particles in the concentration nearest to the neighbouring star predominates. If the component is maintained, collisions between gas particles will lead to a net movement around the geometrical centre in the direction shown by the arrows.

How Dead Centre is Passed

This fact makes it possible for the irregular rotating system to pass dead centre. True, the pulses do not diminish in intensity and they increase in frequency as each of the stars grows. So one might think that the couple that initiates rotation would increase and hold the dumb-bells all the more firmly to dead centre when it was reached. But there are two ways of reducing a couple.

A couple is the product of a force and the projected length of a lever arm. The couple may be eliminated by removing the force, as happens when one quickly withdraws the bar magnet from the vicinity of a compass needle. But the couple may also be eliminated by eliminating the asymmetry of the system. This would happen to the compass needle if its magnetism were to cease.

The lever arm of the incipient star is represented by the average distance between the parts represented by the halves of a dumb-bell and the geometric centre. As the star contracts, this distance decreases; and, therewith, the capacity of an external force to produce a couple decreases too.

Thus each successive pulse coming from the neighbouring star finds less irregularity to 'get hold of', as it were. It is during the intervals between pulses that the star is enabled to pull itself into homogeneity. By the time the position that would be dead centre is reached, there is a less pronounced dead centre — for the star has become more symmetrical.

Galaxies do not have companions capable of exerting couples. But it will be remembered that, during the first stage of growth, the core is surrounded by the enormous spokes. These are irregularly spaced and the core has developed around an astronomical summit of irregular shape. The requisite asymmetry, both external and internal, is there to initiate rotation of the core.