Why, one is led to ask, should all this happen?
(a) The rate of origin of new matter in the region, which is at present unknown.
(b) The rate of extinction of matter in the region, which is 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, reversing 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 summit.
(h) Shrinking of the cloud under the influence of its own gravitational field.
(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; 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, rather 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.
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; 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 stage, it depends entirely on the excess of origins over extinctions in the region occupied by the cloud. In the second stage, it increasingly captures particles from beyond its fringe.
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 one gramme.
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 this maximum rate of growth is achieved.
In this model, moreover, the rate of flattening is 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. This 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 Symmetrical Impermanence.
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 Chapter 12. 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, as there cannot be many particles to become extinct in the region, 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 assess the significance of this distinction only if one knew the gross rates of origins and extinctions.
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.
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 in turn depends on the rate at which the slope around the summit flattens. It will be shown later, when the subject is treated quantitatively, that this rate is very rapid by astronomical standards.
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 greater rate.