TOWARDS A UNIFIED COSMOLOGY
by Reginald O. Kapp
PART III - THE ORIGIN AND EVOLUTION OF GALAXIES
Chapter 12 - How Might a Galaxy Begin?
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It happens to all of us occasionally that we reach a wrong conclusion through neglect of the one or the other of these aspects. Sometimes our mathematical reasoning is quite sound and we err by wrongly interpreting the symbols, while at other times we are quite clear as to what we mean but we fail to notice that our purely qualitative reasoning leads to a result of the wrong order of magnitude. Only mathematical analysis can correct such an error. For these reasons an investigation is often best conducted in two stages, of which the first is qualitative and the second quantitative. The non-mathematical investigator cannot follow beyond the first stage, and this may on occasion leave him quite unenlightened; for there are recondite fields of study where the mathematical expressions cannot be translated into ordinary language. But it is hardly so here. I have planned to limit this inquiry almost entirely to the first, qualitative, stage and shall introduce only that bare minimum of mathematics that is needed to give precision to the reasoning. This minimum is unavoidable when one seeks to discover whether the quantities involved, the forces, the distances, the velocities, are of the order of magnitude from which one would infer a cosmological model that resembles actuality. To help the non-mathematical investigator of the subject I shall precede the mathematical treatment by a short qualitative discussion. To speak thus is to present a two-dimensional analogy to a three-dimensional reality and is in this sense imprecise. But the three-dimensional reality is difficult to visualize and this difficulty may lead the investigator into error. On the other hand, I do not think that it can cause any serious misapprehension if I use the language here of the two-dimensional analogy, so I propose to do this. Molecules of hydrogen in extragalactic space are always on a metaphorical slope, such as corresponds to this analogy, unless they happen to be precisely on the top of an astronomical summit. These molecules are therefore always falling in one direction or another. Where galaxies on opposite sides of a molecule are attracting it in opposite directions it is moved only by the difference between these opposed forces, forces that are themselves very weak. Hence the acceleration that the molecule experiences is small; but it is finite, except at the mathematical point called the summit. While molecules are continuously falling down the slopes around a summit, no molecules are falling on to this region, for there is no greater height from which they can fall. For a cosmological model based on (Al) or (A2) the mountain would be entirely depleted of matter. Not necessarily so for a model based on (A3). Although hydrogen cannot fall on to an astronomical summit from anywhere else it does originate there as everywhere else. It depends on the gradient whether the gas near the summit is becoming more dense or more tenuous. If the rate at which particles fall down the slope is greater than the rate at which new matter originates there, the region will be almost entirely depleted of gas; such as is left there at any moment will consist entirely of particles that have recently originated and are in process of falling out of the region. But there must be one particular potential gradient at which the rate at which particles fall away equals the rate of new origins. This can be called the critical gradient. In any region where the gradient is below the critical value, a cloud will form of which the density will be a function of the gradient. Such a region must occur around an astronomical summit provided this is flat enough. In an expanding universe astronomical summits must become flatter and flatter as the neighbouring galaxies recede further and further from them. It can therefore only be a matter of time for the critical gradient around an astronomical summit to be reached. When, in other words, neighbouring galaxies have been removed to a certain distance, which can be called the critical one, cloud formation must begin on the astronomical summit that lies between them. As the distance between the neighbouring galaxies further increases and the gradient around the summit becomes still flatter, the cloud must then become increasingly more dense. The critical distance depends, of course, on the masses of the neighbouring galaxies, but it will eventually be reached for all of them provided only that the masses of the retreating galaxies do not increase at a rate sufficient to prevent this from happening. To say that a particle now falls towards the cloud that previously fell away from it is, in metaphorical language, to say that the gravitational slope has been reversed. A space traveller who was in the cloud would have to climb against a potential gradient to get away from it. In other words, the newly formed cloud acquires an astronomical reversal zone of its own. So long as the cloud is small and very tenuous the distance from the centre of the cloud to its astronomical reversal zone, and therewith the domain of the cloud, must also be small. One may almost picture the astronomical summit with a cloud on it as having the shape of a volcano, i.e. as a mountain top with a slight crater-like depression. The shape of this must conform to that of the cloud and grow with it so that, while it is the local topography that first moulds the cloud, the cloud eventually moulds the topography. It does so with increasing vigour, causing the depression to become ever deeper and steeper and thereby exerting an increasing attraction on the gas that surrounds it. The difference between the landscape before the cloud begins and after the cloud has grown has some resemblance to the difference between a photographic negative and positive. The effect of the growing cloud is that peaks may become craters and eventually develop into deep wells, shoulders become valleys, valleys become ridges. As the reversal of the potential gradient, the astronomical reversal zone, travels out into distant space, what began as the lip of a crater comes to be more analogous to an extensive ridge. As I have said before, the astronomical landscape is a changeful one. Should, in the course of time, the cloud acquire a mass equal to that of the older and more distant galaxies, this ridge will be half-way between, two concentrations: the cloud and the older galaxy. The point of lowest potential along this is on a straight line connecting neighbouring galaxies and is analogous to a mountain pass, as has been explained in Chapter 10. Particles will fall away from this either in one direction or the other, according to the side of the pass on which they occur. From the top of the pass a space traveller could either descend to one of the concentrations or move at right angles to the direction of descent along the ridge, where he would be ascending a gentle slope until he reached a new astronomical summit. Let distance be regarded as positive when measured in a direction away from an astronomical summit. The acceleration A of a particle is then also positive in that direction. The potential decreases with increasing distance from the summit, so the potential gradient E around the summit is negative. Numerically the potential gradient and the acceleration of a particle at a given point are equal, so one can write A = -E. There are particular conditions for which the rate at which particles fall away, down the gradient around an astronomical summit, equals the net rate at which particles originate within the region around the summit, the net rate of origins being the excess rate of origins over that of extinctions in the region. I shall call these 'critical conditions' and define them by the suffix c. Thus Ac= -Ec is the critical average acceleration experienced at the critical gradient Ec. The critical distance from the summit at which this occurs is rc and the critical average velocity with which particles reach the surface of a sphere with radius rc and centre at the astronomical summit is vc. Let n be the net rate of origins within the sphere per unit volume and time and let N be the density of gas at the surface of the sphere expressed as number of particles present there per unit volume. The number of particles that originate within the sphere in unit time is (4/3)πnrc3 and the number that cross the boundary in unit time is 4 π Nrc2vc. To assume that the summit is a smooth dome, lacking topographical features, would be incorrect. But no mistake is made if one considers conditions for one particular radial direction. For that direction the con- ditions are the same as if the summit were dome shaped, as implied by the above expressions.
The critical conditions are defined as those for which the content of
the sphere remains constant,
which gives:
from which
An expression is needed for Vc in terms of rc and the shape of the slope. Consider a particle that originates at distance r from the summit. Let its velocity be vs, when it reaches the surface of the sphere. By definition
A = dv / dt = dv /dr . dr / dt If the astronomical summit has the shape of a parabola, one can put φ(r)=kr and the above equation becomes r∫rc k r dr = 1 / 2k(rc2 - r2) = 1 / 2 vs2 ...... (12b) This expression holds for any particle that originates at distance r from the summit. To obtain the average value of vs which is vc, one must obtain the average value of r. This is obtained by considering a shell with its centre at the summit and radius r, smaller than rc. Let its thickness be dr. The volume occupied by this shell is 4πr2dr and the net number of particles that originate within the thickness of the shell is 4πnr2dr. Each particle reaches the surface of the sphere after falling through the distance rc - r. The total number of particles that fall in unit time multiplied by the distance through which each falls is 4πn 0∫ rc r2(rc - r) dr = ( 1/3 )nπ rc4
The total number of particles originating within the sphere in unit time is (rc - rm) = {(1/3)π n rc4} / {(4/3)π n rc3} = rc / 4
Where rm is the mean value. From which: rm = (3/4)rc
In equation (12b), r = rm when vs = vc . Inserting these values one obtains vc = √ (7k) (rc / 4) ...... (12c) When this value is inserted in equation (12a) one obtains
n/N = (3/2)√ (7k) This shows that for a parabolic slope the density, N, depends only on the value of k and is independent of the radius. A parabolic slope is therefore a region in which the gas density is uniform and increases as the slope flattens and k decreases. It will be shown below that the actual slope around an astronomical summit is nearly, but not quite, parabolic. The calculation is somewhat complicated for the general case, but it will suffice for the present purpose to consider a gradient that lies on a straight line between two concentrations having equal masses, m; to consider, in other words, a pass instead of a summit. The error made by doing so is probably not negligible by any means. But the simplification of pretending that a summit is just like a pass facilitates understanding of the broad outline of the problem. The gradient at a pass is E = Gm[1 / (D + r) 2 – 1 / (D – r)2] Where D is the distance from the astronomical summit to one of the centres of mass. The above formula can be written more simply
When r is small compared with D this becomes nearly Putting 4 G m / D3 = k gives A= -E = kr which is the expression for the parabola defined above. This means that, when k has dropped to the value at which a cloud can begin to form at an astronomical summit it has nearly dropped to the same critical value at a little distance from the summit. As space expands and k decreases further, it soon will acquire the critical value there. Hence the cloud around an astronomical summit is of nearly, but not quite, uniform density. It is a little denser at the centre than at its fringe; and it grows outwards rather rapidly. The difference between equation (12e) and (12f) is a measure of the rate of spread. This is illustrated by the diagram in Fig. 2. It should be noted that k is inversely proportional to the cube of the distance from a galaxy, which means that, in expanding space, the flattening of the parabola around an astronomical summit is rather rapid.
This theme should not be left without a hint as to how to arrive at the
equation that must replace (12e) when it is necessary to express accurately
the gradient around an astronomical summit. Equation (12e) has been
obtained by superimposing the potential gradients of two equal masses
The true equation must represent the superposition of all masses neai
enough to have an appreciable effect. Their distribution is in three dimen-
sions, which makes the calculation complicated. But none of them is as
near to a summit as to the nearest pass. As those on opposite sides of a
pass have potential gradients of opposite sign and so largely cancel eaci
other, so do masses on opposite sides of a summit. Hence the gradient  Fig. 2. Shape of an astronomical pass around a summit is flatter than that in a corresponding position near a pass. One may expect this to be expressible by the addition of a term dependent on r/D. This would be done if equation (12e) took some such form as E = - { G m D r / (D2 - r2)2 } φ ( r/D) ...... (12g) Equation (12g) can hardly be of the correct form, but it serves to suggest the way in which a number of masses in three dimensional arrangement. might be expected to influence the potential gradient. As the factor φ (r / D) must reduce the value of E for it to mark the change from a pass to a summit and as r is much less than D, the function must be a positive one. In other words the function must vary directly with r / D. The fringe of the cloud has been mentioned above and needs to be precisely defined. This choice of term should not mislead anyone into picturing a sharp outline between a region of high and one of low mass densities. Our imaginary space traveller would not observe any frontier on passing through the fringe. Nevertheless it can be defined quite precisely in words. The fringe of a cloud is a shell of indefinite thinness that surrounds the centre of the cloud and is so placed that the rate at which matter falls down the potential gradient balances the rate of origins. On one side of the fringe the rate of falling exceeds the rate of origins; the region becomes increasingly depleted. On the other, where the gradient is flatter, the rate of origins exceeds the rate of falling away. If N is the number of particles per unit volume, the fringe is the shell for which dN / dt = 0. |