|English and Mathematics
The Time between Successive Generations of Nebulae
The Rate at which an Extragalactic Cloud Grows in Extent
The Rate of Growth of the Cloud's Domain
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However, the dangers of over-indulgence in formula spinning are avoided if mathematics is treated, wherever possible, as a language into which thoughts may be translated only after they have first been expressed in the language of words. The use of mathematics in this way is indeed disciplinary, helpful, and sometimes indispensable.
It is disciplinary in the sense in which it is always disciplinary to translate a statement from one language into another; this is the best way of revealing shoddy thinking. It is helpful because some statements are of such a nature that it is almost, if not quite, impossible to express them in the language of words; it is reasonable to ask to be spared the ordeal of attempting to do so. It is indispensable because quantitative conclusions can be reached only with the help of algebraic expressions.
Such considerations demand now that the comparatively light task be undertaken of seeking algebraic symbols for some of the concepts that have been discussed in previous chapters. Only the simplest mathematics is needed at this stage of the inquiry. But the field of cosmology that is being studied here is complicated and so far almost unexplored. Once clarity is achieved concerning its broad outline, the remaining problems will lie wholly within the mathematician's domain.
Let D be a given distance in space. The rate at which this increases is
Let D = D0 at the moment when one puts t = 0, and the above expression becomes
H is, according to a recent estimate, 185 kilometres per second per megaparsec, which is the same as 5.84 x 109 kilometres per year per megaparsec and has the dimension of reciprocal time. One megaparsec is 3.084 x 109 kilometres, from which it follows that
When the distance doubles, one puts D/D0 = 2 and obtains ln2 = Ht, from which t = 3.66 x l09 years.
This means that a new nebula begins to form adjacent to a predecessor about every three-and-a-half thousand million years.
This is the time during which the lip of a crater on an astronomical summit goes through the complete evolution of moving outwards, becoming a ridge far from any concentration, and flattening sufficiently to attain at its summit the critical gradient at which a new cloud can begin to form.
For the sake of simplicity let us assume that the cloud is forming on a pass between two nebulae of equal masses m. Although the cloud forms on a summit and not on a pass, this incorrect assumption will serve to illustrate the method of calculation and I hope that it will not introduce a wrong order of magnitude. Let Dc0 be the distance from the pass to one of the nebulae at the moment when the cloud just begins at the very top of the pass, and let Dcr be the distance at a moment when the fringe of the cloud reaches a distance r from the pass. Let Ec be the critical potential gradient at which the rate of origins just balances the rate of loss from particles falling away down the slope.
We want to find the time that it takes for the fringe of the cloud to reach a certain fraction of the distance to the nearest galaxy. Let this fraction be a, so that r = aDcr.
Consider equations 12e and 12f. lf (12f) were correct, Ec would be reached at distance r at the same time as at every other distance; it would be reached for the separation Dco. But the correct equation is (12e) and for this Ec is not reached until the separation has become Dcr. This gives two expressions for E namely:
When one replaces r by aDcr and equates these expressions, one obtains
From this one can calculate that when r is 1 per cent of Dcr the ratio Dcr/Dc0 is 1.000067, and when r is 5 per cent of Dcr, it is 1.00167.
If when t = 0, one substitutes Dc0 for D in equation (15a), one can write
These figures are over-estimates, and probably considerable ones, for they do not allow for the flattening of the slope that is occasioned by the mass of the cloud itself. When this is taken into consideration, one must arrive at shorter, and probably substantially shorter, times, particularly for the 5 per cent distance.
The estimate does not allow, either, for any change in the mass m of the neighbouring galaxies. The conclusion has been reached in the last section that this is growing only slowly when cloud formation begins. It will be shown in Appendix B that it may be dwindling later. For the condition that is being considered, it must be near the turning point, and so neglect of change of mass is not likely to introduce a big error.
It has been found herewith that, compared with the three-and-a-half thousand million years that are available for the cloud to acquire its final mass, the time that it needs to become rather extensive, though it is also still very tenuous, is quite short.
The orders of magnitude in the model based on Symmetrical Impermanence seem to fit.
As the mass of the cloud increases, this reversal zone moves outwards; the crater grows in depth and diameter. It continues to do so until the reversal zone coincides with the fringe of the cloud. The first stage of growth then ends and the second stage begins.
We thus have to picture two simultaneous movements, both proceeding radially outwards. One is the movement of the fringe of the cloud, the other that of its reversal zone. This begins like the lip of a crater; at a later stage it envelops the whole cloud, and it eventually extends far out into space, as a range of passes and summits.
The reversal zone can form only after the very tenuous incipient cloud has become fairly extensive, and even then it is no more than a small shell right at the centre of the cloud. To extend beyond the cloud, it must grow outwards more quickly than the fringe does. Some simple mathematics shows that this happens.
In practice the cloud's shape must be very irregular, but for the sake of simplicity a spherical shape will be assumed here. This simplification may lead to quantities that would need a fairly large correction factor, but it suffices to give an idea of the relative movements of the fringe and the boundary.
Let the distance from the centre of the cloud to the reversal zone at any moment be D1, and let the distance from the neighbouring galaxy to the same point on the reversal zone be D2. At the same moment, let the mass of the cloud within its domain be m1 and the mass within the domain of the neighbouring galaxy be m2. By the inverse square law
Too much importance must not be given to the above equations. They do little more than provide reassurance that in the model based on Symmetrical Impermanence, the time must arrive rather soon in the history of an extragalactic cloud when it ceases to grow in extent — when, indeed, it begins to shrink under its own gravitational field, and when at the same time it grows in mass by capture of hydrogen from without.