28.1: The Problem
It is known that stars rotate about an axis. So do galaxies. Why?
The answer must conform to the principle of conservation of angular
momentum and it is this that makes it difficult to find an answer. The
angular momentum of a system is the product of its moment of inertia
and its angular velocity. The principle of conservation of angular momentum states that the value of this product is not changed by any forces that
act only between the component parts of the system. To change the
angular momentum it is necessary to apply a couple from outside.
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
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.
28.2: The Conditions needed to cause Continued Rotation
For continued rotation to occur in a system it is axiomatic that the
following three conditions must be met:
(a) An external couple must be applied to the system.
(b) 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.
(c) 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.
In order to explain the rotation of stars and galaxies one must find
circumstances that meet these conditions.
28.3: Accounting for Asymmetry
To meet the first condition the incipient star must be subjected to
preponderating force from one particular direction. It would not rotate
if subjected to forces from many directions of which the vector sum wasl
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 dis-
appearance 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
28.4: How Dead Centre is Passed
It has been said above that the dumb-bells would align themselves along
a common axis if the attracting force were continuous; and it would be
continuous according to the traditional theory of gravitation. But it is not
continuous according to the new theory. While the incipient stars are
still very tenuous, the pulses of gravitation between them are intermittent.
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 geo-
metric 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 is there, both external and internal, to initiate rotation of the core.
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