13.1: The Problem Defined
In the last chapter it was shown that, if matter originates continuously,
as is asserted in (A3), a cloud must occasionally form in extragalactic
space. But this in itself does not go far towards proving that a cosmological
model based on (A3) resembles actuality. To prove this one must do much
more: One must show that in such a model the undifferentiated accumulation of hydrogen that I have called a 'cloud' will evolve into the kind of
structure to which the name galaxy has been given. The simple accumulation of hydrogen must acquire a mass of the same order of magnitude as is
possessed by the observed galaxies; its distance from its nearest neighbours
must be of the same order of magnitude; it must behave in a similar
manner; it must have a similar volume, density, shape, detailed structure.
Why, one is led to ask, should all this happen?
13.2: The Variables that Influence Growth
There is a substantial risk of drawing wrong inferences when constructing a cosmological model on the basis of any selected hypotheses, and one
reason for this is that a large variety of circumstances must influence the
evolution of an extragalactic cloud, and that most of them interact with
each other in complicated ways. Hence great care must be taken to con-
sider all these circumstances and to assess their effects. It will help towards
the understanding of what is to follow if they are listed here. Let it be
pointed out, therefore, that what happens to a cloud after it has begun
to form on an astronomical summit must depend on:
(a) The rate of origin of new matter in the region, which is at present
(b) The rate of extinction of matter in the region, whichis also unknown.
(c) The known expansion of space, which causes the gravitational
pull of distant nebulae on the substance of the cloud to decrease with time.
(d) Changes, if any, in the masses of neighbouring distant galaxies,
which must occasion changes in the gravitational pull that they exert on
the substances of the cloud.
(e) Changes in the density of the cloud.
(f) The gravitational field of the cloud itself, which must fundamentally
influence the distribution of forces in the region where the cloud is, revers-
ing the local potential gradient and causing particles to fall towards the
cloud that previously fell towards a distant galaxy.
(g) Irregularities in the potential gradient around an astronomical
(h) Shrinking of the cloud under the influence of its own gravitational
(i) Rotation of the cloud, which introduces centrifugal forces.
The length of the above list serves as a warning against premature
treatment of the subject in mathematical terms. It is at the stage of
qualitative consideration that one is best able to ensure that none of the
interacting circumstances is overlooked, and so I shall here, as in Chapter
12, restrict mathematical treatment to the unavoidable minimum. The
cosmological model can be checked for its detailed quantitative resemblance
to actuality more easily after, than before, its broad qualitative features
have been inferred. The small amount of mathematics introduced here will
therefore be no more than is needed in order to test whether certain inferred
quantities are of the right order of magnitude.
13.3: The Cloud's Income and Loss Account
If the inferred cosmological model is to resemble actuality, one thing
that must certainly happen to it is that the incipient cloud must grow in
extent and density during its transition from cloud to galaxy, and for this
to happen it must receive more particles than it loses. Let us consider both
the cloud's sources of income and its sources of loss. They can be thought
of as analogous to profit and loss in accountancy; but here income and
loss are more precise terms. The sources and amounts of both differ
greatly according to whether the Hypothesis of Asymmetrical or of Symmetrical Impermanence is used as a basis.
Sources of Income: According to both hypotheses there are two sources
of income. The first is origins within the region occupied by the cloud, and
the second is existing matter that the cloud attracts from its surroundings
by virtue of its own gravitational field. According to both hypotheses the
first source of income is constant per unit volume. (This is probably obvious, but the reason will be fully given in Chapter 15.) But the amount of
income from origins is not the same in the two hypotheses. According to
Asymmetrical Impermanence it is always the 500 atoms of hydrogen per
cubic kilometre per year estimated by McCrea. But according to Symmetrical Impermanence this number is the difference between the rate of
origin and the average rate of extinctions per unit volume for the whole
universe. The gross rate of origins is greater and the net income depends on
the local rate of extinctions, as will be explained in Chapter 14.
Remembering the two distinct sources of income, one can speak of
growth by the origin of new matter and growth by the capture of existing
matter. While the cloud is still small and tenuous it has little mass and
cannot attract matter towards itself in competition with the more massive
neighbouring galaxies; so growth by capture cannot occur. On the contrary, the cloud does not even retain all the particles that originate within it.
Some of these are attracted away towards the neighbouring galaxies; in
the metaphor of an astronomical landscape they fall away down the slope.
But in so far as origins cause the cloud to become more massive, it competes more and more successfully with the distant galaxies and growth
by capture occurs eventually and then at an increasing rate. Growth of
an extragalactic cloud thus occurs in two successive stages. During the
first it depends entirely on the excess of origins over extinctions in the
region occupied by the cloud. At the second it depends to an increasing
extent on capture of particles from beyond its fringe.
Sources of Loss: Two sources of loss are postulated by Symmetrical
Impermanence: loss by falling down the slope and loss by extinction. But
only one source of loss is postulated by the Hypothesis of Asymmetrical
Impermanence, i.e. the gravitational fields of the neighbouring galaxies,
which are the cause of the potential gradient down which some of the
particles in the cloud fall during the first stage of growth. When the second
stage has begun, this source of loss ceases and there are the two sources of
income: the 500 atoms of hydrogen per cubic kilometre per year and the
particles captured from outside the cloud.
13.4: Income and Loss During the First Stage of Growth
Now 500 atoms of hydrogen is not much. It would not go far towards
fattening a microbe. Even if the 500 were all retained in the cloud from the
moment when it began to form, it would take about four million million
years before the cloud gained one molecule of hydrogen per cubic centimetre, and this number would still leave the cloud very tenuous.
How minute a quantity 500 atoms of hydrogen is can be appreciated
from another consideration. Some advocates of hypothesis (A2) have been
claiming recently that the universe was created about seven thousand
million years ago. A cloud that was acquiring 500 atoms of hydrogen per
cubic kilometre per year during this span of time would now have a
density such that a volume equal to that of the earth would weigh about
But in our model the rate at which the cloud gains mass is much less
than even this very low value. For only a small fraction of the newly
originating atoms are retained in it. When the cloud is just beginning to
form, nearly the whole lot of them are lost, for the potential gradient is
such that the number of particles falling down the slope is just, but only
just, less than the number that originate in the region. In this model the
periods of time taken by a volume equal to that of the earth to increase its
mass by one gramme is much more than seven thousand million years.
It is not until the gradient has become nearly flat that most of the atoms
are retained and that this maximum rate of growth is achieved.
In this model the rate of flattening is, moreover, slow, for the potential
gradient is subjected to two opposing tendencies. The first is the expansion
of space. As the neighbouring nebulae, which are the cause of the potential
gradient, move further away, the force per unit mass with which they attract
particles out of the region diminishes, which is another way of saying that
the metaphorical slope flattens.
But for the model based on Asymmetrical Impermanence there is an
opposing tendency. It will shortly be shown that in this model the retreating
galaxies are themselves rapidly becoming more massive. Hence the
flattening effect of increasing distance is at least partly offset by the
steepening effect of the growing masses of the galaxies. It does not seem
certain that the potential gradient decreases at all in this model; but if it
does it must happen very slowly. However, it will be shown in Appendix B
that this opposing tendency need not operate for the model based on
There is yet another circumstance that retards the growth of the cloud
during the first stage. This is the fact that the loss rate is a direct function
of the density, as can be inferred from equation (12a) in Section 12.4. For
every gradient there is a limiting density above which the rate of loss down
the slope exceeds the rate of income. The cloud cannot become more dense
until the gradient has become flatter.
The cumulative effect of these various retarding factors is that, in our
model the first stage of growth is exceedingly slow. The cloud will, however, become significantly more massive during the second stage of
growth, when it captures matter from outside. This will cause its density
to increase: and the density will also increase as the cloud shrinks under
its own gravitational field. One must expect the cloud to be much more
extensive at first than the galaxy into which it evolves.
For the model based on Symmetrical Impermanence the rate of income
during the first stage of growth is greater by an amount at present unknown.
The 500 atoms of hydrogen per cubic kilometre per year (or whatever the
correct figure may be) are a net and not a gross rate. They are the average
excess of origins over extinctions for the whole of the observable universe
and are not relevant to any particular locality.
The region where the cloud has just begun to form has been depleted
of matter until that moment, and so there cannot be many particles to
become extinct in the region and the local net rate of origins must be very
nearly equal to the gross rate. Hence the model based on Symmetrical
Impermanence permits a more rapid growth during the first stage. One
could only assess the significance of this distinction if one knew the gross
rates of origins and extinctions.
13.5: End of First Stage
The moment when the first stage of growth is ended and the second
stage begins can be defined precisely in terms of the fringe of the cloud.
This is the surface around the incipient cloud at which the income from
new origins balances the loss from extinctions and from falling away down
the slope. (See page 102.) It is the boundary within which the density of
the cloud increases and it is associated everywhere with a specific potential
gradient, which I have called the critical one. The end of the first stage is
the moment when particles just beyond the fringe of the cloud reverse
their direction and fall towards, instead of away from, the cloud. This
happens at the moment when the potential gradient at the fringe reverses,
when the gravitational pull exerted by the cloud at its fringe just suffices
to cancel the pull exerted by a neighbouring galaxy. It is the moment
when the astronomical reversal zone of the cloud coincides with its fringe.
After the cloud has become massive enough for this to happen, no more
particles fall out of it.
In the metaphor of an astronomical landscape, the zone at which the
potential gradient reverses is, it will be remembered, a ridge from which
the ground slopes to either side. This ridge begins as the lip of a crater at
the astronomical summit and moves gradually outwards as the mass of
the cloud increases. The diagrams in Fig. 3 will help to make the course
of events clear.
Fig. 3. the two stages of growth of an incipient galaxy. Above: First stage
Below: Second stage.
At first, when the cloud has only just begun, the crater is small. There
is cloud on each side of its lip, as is shown in the diagram that represents
the first stage. Particles do not fall out of this part of the cloud that is
within the crater; but they do fall away from the outer part. But as the lip
moves outwards, becoming more like a mountain ridge, less and less of the
cloud is outside it and the loss rate from falling down the slope decreases.
The moment when the ridge, or reversal zone, reaches the fringe of the
cloud is the moment when loss from this source comes to an end and the
second stage of growth begins. As the cloud's mass increases still further,
this zone moves out beyond the fringe. The situation is then as shown in
the diagram for the second stage. There is a growing region surrounding
the cloud from which particles fall towards it, and its mass increases by
capture and not only by origins within it.
Should the cloud eventually acquire a mass equal to that of a neighbouring galaxy, the boundary will be half way between the cloud and this
galaxy; it will, as mentioned already, be analogous to a mountain pass over
which one must travel in order to go from the cloud to the galaxy. A model
that resembles reality must be one in which the reversal zone will eventually
arrive at approximately such a half-way position, and this depends in turn
on the rate at which the slope around the summit flattens. It will be shown
later, when the subject is treated quantitatively, that by astronomical
standards this rate is very rapid.
In this model the incipient cloud remains tenuous and spreads quickly
until the product of its density and its volume suffices to attract particles
that are just outside it and that would, before that moment, have been
attracted towards the neighbouring nebula. Thereafter its volume may
decrease by shrinking, but its mass per unit volume must increase at a
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