19.1: Whence the Semblance of Rigidity?
Our galaxy rotates about its axis. So do all the spiral nebulae. Their great
arms wheel in unison like platoons of soldiers on the parade ground,
though not with the same angular velocity as that of the core. This
presents us with a problem that has, I am afraid, not been appreciated.
The nebulae are mixtures of a diffuse gas and stars. The distance between
a star and its nearest neighbours is usually many light years. How is it that
this flimsy mixture turns around a common axis as though it were a
It is surprising that there has not been a more assiduous search for an
answer to this question. It seems usually to be ignored, perhaps because it
has been regarded as unanswerable; it can hardly be thought that the
answer is known already. So far as I know, only two explanations have
ever yet been offered and less tenable ones are hard to imagine.
One is that magnetic forces act between the component parts and are
powerful enough to lend the property of rigidity to the tenuous structure.
I have already shown in Chapter 17 that an hypothesis that attributes large
scale unidirectional movement to magnetic forces does not stand the test
of quantitative thinking. The other explanation so far given is that tides
raised by the stars on each other bring them all into line. I do not think
there is any likelihood that astronomers will have seriously entertained
this strange theory, but others may not feel qualified to judge it, and so it is
worthwhile to point out that, like the hypothesis of magnetic forces, it
fails badly by the test of quantitative thinking.
It is true enough that tides do impart a small measure of rigidity to an
astronomical system. They tend to bring parts that are moving relatively to
each other into unison. The mechanism by which this happens is rather
complicated but it is well-known to experts and so does not need elaboration. It will suffice to mention that tides have caused the moon always to
present the same face to the earth, as if the two were connected by a rigid
rod. Tides around our estuaries are also slowing down the earth's rotation,
and if they persist for long enough, they will similarly cause the earth
always to present the same face to the sun.
Tides raised by the sun, the planets, and their satellites on each other
are also tending to cause the equator and the ecliptic for each of these
bodies to coincide. They are tending to cause all the planets to rotate with
the same angular velocity about the sun, just as the arms of the spiral
nebulae rotate with the same angular velocity about a common axis.
But tides within the solar system are as yet far from achieving this;
they have not yet caused any of the planets, except the smallest and
nearest to the sun, always to present the same face to the sun. Still less
have they brought all the planets into a common ecliptic or caused the
year to last the same time for all of them. And yet the tides within the solar
system are powerful compared with those that can be raised by one star
on another one several light years distant. If tides have not caused all
planets to have the same solar year, they can certainly not cause all fixed
stars to have the same galactic year. The notion that tides could bring
stars into alignment is as absurd as thinking that a fly could deflect a
19.2: Uniform Rotation as the Consequence of Impacts
I have said in Section 18.2 that it is difficult to understand by what
process a celestial body, be it a star or a galaxy, can acquire the observed
angular momentum. But whatever the explanation may be, I know of only
one way by which uniform rotation could be imparted to all the com-
ponents of so flimsy a structure as a cloud of gas, and this is by successive
impacts of molecule on molecule. But as appropriate impacts are not
being made on the stars now, they must have been made on the substance
forming the stars at some time in the past.
Rotation can only be imparted to the molecules that are captured from
outside because these have sufficient momentum to fall, not only on to, but
into the core. If the captured molecules were to rest on the surface they
would not be entrained. That a spherically shaped cloud such as the core
will rotate as a whole is only to be expected provided always that there is
a source of angular momentum and that the core grows from the centre
outwards, and this is ensured by the outward movement of the reversal
zone and is accentuated by the small difference between equations (12e)
in Section 12.4, which prevents the cloud from forming simultaneously over a large volume.
19.3: The Part Played by the Spokes
Let us consider what the cloud looks like at the moment when the
reversal zone has reached the base of the spokes. There is a central core,
which is becoming denser from molecules that fall into it as well as from
shrinking - two distinct causes. Around this core there is a potential
gradient that slopes towards the centre and down which hydrogen is
falling. It only contains matter that is, as it were, in transit; and has,
because of its extreme tenuousness, been called the depleted region. This
is bounded by the expanding reversal zone and beyond this lie the spokes.
It is obvious that there can be no collisions between molecules in the
rotating core and those in the motionless spokes; there is a great gap
The structure so far inferred does not resemble the spiral nebulae. To
become like them it will have to change greatly. I think that it must do so
and in the following manner.
When the reversal zone reaches the base of the spokes, masses of
hydrogen outside this zone pour into it in the few places where there are
spokes. We have to picture half a dozen or so well separated places from
which hydrogen enters the domain of the core. It is as though the core
were surrounded by a far distant outer shell with this number of openings
through which gas is pouring into the interior. Gas continues to do so as
long as, first the base of each spoke and then successively more distant
parts, are traversed by the reversal zone and find themselves attracted
towards the core.
Here it is necessary, however, to guard against exaggerating the
analogy to a terrestrial system. Gas would only pour into a steel vessel if
the pressure was greater outside than inside; and if this happened the gas
would, after entry, diffuse almost instantly throughout the whole volume;
the pressure and density would even out very quickly. But in our astronomical system it is gravity and not pressure that causes the substance of
the spokes to pour inwards. The pressure and density considered in
a radial direction are far from uniform. Under its own weight the core is
dense at the centre and tenuous at the surface. Above this surface there is
the depleted region through which the gas is falling.
While falling across this region, molecules from the spokes are being
accelerated. By the time they reach the surface of the core they must have
attained a substantial momentum and must penetrate deeply. They are
entrained by molecules that already have a tangential velocity in the
direction of rotation.
This shows that the process that introduces rotation is different from
what one might at first sight suppose. The substance of which the spokes
are formed does not take part in any rotatory movement at the place far out
in space where they form; it only does so after it has fallen a long way
down into the rotating core.
19.4: A Tentative Explanation of the Spiral Shape of the Arms
In a structure built to terrestrial dimensions the substance of each
spoke would quickly spread out all around the core. But we are here
concerned with a structure so vast that light itself takes a very long time to
traverse a small fraction of its span. The rate at which gases diffuse is very
slow compared with the velocity of propagation of light, even at atmospheric pressure. In the extragalactic cloud the collisions by which diffusion
is effected are comparatively rare, so the diffusion rate is correspondingly
slow. For practical purposes the effect of diffusion on the bulk movement
of gas must be negligible compared with the effects of gravitation and
This means that to all intents and purposes the substance of the
spokes stays, after falling, where it has penetrated, deep within the body of
the core. Here, as we have already seen, there is a local thickening.
While these falling molecules are being entrained they slow down those
that entrain them. Each place into which the substance of a spoke is falling
is therefore not only denser than the rest of the core but also rotating more
slowly. In front of each thickening the gas of the core moves away, leaving
the region there more tenuous. Behind the local thickening, on the other
hand, the faster molecules pile up, causing the thickening to extend
backwards. This thickening must therefore have a well-defined leading
edge, in front of which there is a near vacuum, and behind which the
thickening extends for some distance and tails off slowly.
As time goes on more substance from the spokes pours into the core;
but as the core is moving forwards the later substance always reaches it at a
place further back. There is, in other words, an increasing angular displacement between the leading edge of the thickening and the place at
which each successive part of the spoke arrives, and as the further gas falls
on to piled-up substance it penetrates less deeply. It consequently finds
itself in a place that is both tangentially behind the previous part and
radially further from the centre. Thus does the leading edge of the thickening acquire a spiral form.
The spokes must not, it has now become clear, be thought of as the
embryonic arms of the nebula. The spokes do not form where the arms
will form but much further from the centre of the cloud; and they have
quite a different shape. The spokes are really a motionless store from which
the rotating core acquires the substance that is to be fashioned into the
spiral arms. But the spokes provide only a part of the arms. The remainder
consists of gas from the core itself, which piles up behind the thickening
where substance from the spokes is falling into the cloud.
The comparatively small dense ball that is observed at the centre of
the spiral nebula must not, moreover, be identified with the part of the early
cloud that I have called its core. This latter extends over and beyond the
whole volume that will eventually be occupied by the nebula. The ball at
the centre is but the inner compressed part of the core.
The function of a store will be served by each spoke whether it lies in
the plane of rotation or not. Should the axis of the core be so inclined
that four spokes happen to be in this plane and two along the axis, the
resulting nebula will have about four spiral arms, rarely more, and they
will be of about equal sizes and equally spaced. The other two spokes will
probably be absorbed into the central ball without forming arms. But with
a different inclination of the axis there may be a different number of arms,
and these will be less uniform both in size and position. They will, however, always occur in the plane of rotation and be flattened out by
19.5. The Extent of the Cloud on Completion of the First Stage
We are now able to form some rough and tentative conclusion concerning the span of the cloud at the time when it reaches its maximum
extent and the first stage of growth is completed. Some time before this
moment the lower parts of the spokes have been falling down into the
rotating core. Now, at the end of the first stage, it is the very tips that do so.
The core has been shrinking and so the tips of the spokes must have a
very long way to fall; one must consequently conclude that the radius of
the core is small compared with the distance from the centre to the tip
of a spoke. But it has been found that the arms of the spiral nebulae must
have been formed wholly within the rotating core, so this core cannot be
less voluminous than the nebula that will eventually appear. Indeed it
may be more voluminous; for it probably goes on shrinking long after the
spiral arms have been formed.
The reason for this supposition is that the density of the cloud must
increase by a large factor during the second stage of growth. Consequently
its angular velocity must decrease and with this the distance from the
centre at which gravitational and centrifugal forces balance. From these
considerations we have to conclude that the span across the spokes reaches
a value several times greater than the diameter of the nebula that will
There is no apparent reason why the shape of the spiral arms should
change very rapidly. The substance of which they are formed is subjected
to two opposed radial forces, namely gravitation towards the centre and
centrifugal force away from the centre. One should expect a balance
between these to be reached in the course of time. For there is some reason
to believe that the mass of the nebula will eventually fluctuate around a
limiting value, as will be explained in Appendix B.
It seems possible that the spiral arms may become narrower and
lose their clear outline with time. Each has its own gravitational field and
one might expect substance at its fringe to be attracted in a tangential
direction towards this. Given time enough one should expect further
departures from the original shape and a less and less regular structure.
The structureless nebulae might be degenerate spiral ones. But elliptical
nebulae would still remain unexplained. They may even make it necessary
for the theory presented here to be revised.
19.6 Why Stars?
In one important feature the model that has just been inferred is quite
unlike actuality. It consists entirely of diffuse hydrogen, whereas nebulae
consist of a mixture of diffuse hydrogen and stars. Must the model be
abandoned for this reason, or can one infer stars, too, only on the basis of
the Hypothesis of Symmetrical Impermanence?
This question worried me for some thirty years and restrained me from
previously publishing a cosmological model based on Symmetrical Impermanence. But the question has got to be faced seriously and with a
proper sense of scientific responsibility.
Rather surprisingly this does not seem to have been done yet any more
than the question seems to have been asked seriously why the flimsy
nebulae rotate as though they were viscous structures. To both questions
only untenable answers seem to have been suggested.
One rather surprising explanation that has been offered to account for
the occurrence of discrete stars is that, from considerations of probability,
one might expect some lack of uniformity in any extragalactic cloud. There
would be regions of higher and regions of lower density. The regions of
higher density would have centres of gravity of their own and each would
attract hydrogen to itself and away from the more powerful, but also more
distant, centre of gravity of the whole cloud.
This theory is probably not seriously entertained by any astronomer
or astrophysicist, but others may accept it and so it is worthwhile to men-
tion that it is as untenable as the above-mentioned theories about the effect
of magnetic forces or tides in giving rigidity to the nebulae. Like those,
it does not pass the test of quantitative criticism.
This theory postulates local, or parochial, centres of gravity in addition
to the common centre for the whole cloud. It claims that the parochial
centres are powerful enough to compete with the common centre. For this
to happen the potential gradient near each parochial centre must be
reversed, the slope must be away from the common centre and towards
the parochial one. But a moment's thought shows that, according to
the accepted views about gravitation, the density of the gas around the
parochial centre would then have to be enormous.
The following calculation would not be precise if an actual situation
had to be evaluated, but is near enough to the truth to convey a correct
sense of proportion.
Consider a point at a short distance D1 from the parochial centre and
at a large distance D2 from the common one. Let the density around the
parochial centre be σ1 and the average density for the whole cloud have the
lesser value σ2. For the sake of simplicity let the mathematics be applied
to spherical regions around the parochial centres. The masses that are in
competition at this point are then
m1 = (4/3) π D13 σ1 and m2 = (4/3) π D23 σ2
For the gradient to reverse we must have
( D1 / D2 )2 = m1 / m2 = ( D1 / D2 )3 ( σ1 / σ2 )
It follows from this that
σ1 / σ2 = D2 / D1 ...... (19a)
The number of stars in a galaxy is perhaps 109 or more. This means
that each star must draw its substance from a region that is but a tiny
fraction of the whole volume. Hence D1, the radius of each of these small
volumes, must be very small indeed compared with the average distance
from the parochial centre to the common centre. It follows from equation
(19a) that the local density around the parochial centre must also be an
enormously high multiple of the average density of the whole nebula if
this local centre is to be effective in competition with the common centre.
On probability considerations, however, one is entitled to expect only
small local departures from the average density. Once again quantitative
thinking defeats a theory of superficial attractiveness.
Those who find this theory acceptable have probably been misled by
the analogy to a terrestrial cloud of water vapour. This breaks up into
droplets of water, each very small in comparison with the whole cloud.
Hence, the argument by analogy may have run, an extragalactic cloud
would break up into droplets each as big as a star, but small in comparison
with the whole cloud. But one must not forget that this is no analogy.
The gravitational field of a terrestrial cloud is negligible; there is no
tendency for molecules of water vapour to move towards a common centre
of force. The forces that cause droplets to form are, moreover, not gravitational; they are electrostatic and free from competition from a common
centre. Compared with gravitational forces they are also, per unit mass,
very powerful. The occurrence of these parochial centres of attraction is
well understood and does not depend on variations in local gas density.
Nevertheless, it does not seem possible to account for the condensation
of discrete stars out of a diffuse cloud unless one can postulate parochial
centres towards which molecules of hydrogen are attracted with a force
sufficient to overcome the gravitational force that the whole cloud exerts.
So the question arises whether such parochial centres can occur in a gas
that is virtually of uniform density.
At the moment it must seem as though such powerful centres could
only be postulated on the basis of some ad hoc hypothesis. But it will
appear in due course that this is not necessary. Such centres do follow,
surprising though it may seem at present, as a logical consequence of the
Hypothesis of Symmetrical Impermanence.
Before this can be made clear, however, another inference will have to
be drawn from Symmetrical Impermanence. It concerns gravitation and
will be the object of Part Four of this essay.
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