|Whence the Semblance of Rigidity?
Uniform Rotation from Impacts
The Part Played by the Spokes
A Tentative Explanation of the Spiral Shape of the Arms
The Extent of the Cloud on Completion of the First Stage
A False Analogy?
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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 suggestion 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 cannon ball.
Rotation can be imparted to the molecules that are captured from outside only 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; this is ensured by the outward movement of the reversal zone and is accentuated by the small difference between equations (12e) and (12f), which prevents the cloud from forming simultaneously over a large volume.
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 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.
It is necessary here, however, to guard against exaggerating the analogy to a terrestrial system. Gas would pour into a steel vessel only if the pressure was greater outside than inside; and if this happened the gas would, after entry, diffuse almost instantly throughout the whole volume; hence 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 does so only after it has fallen a long way down into the rotating core.
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 or not it lies in the plane of rotation. 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 centrifugal force.
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 eventually result.
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: 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, as will be explained in Appendix B, there is some reason to believe that the mass of the nebula will eventually fluctuate around a limiting value.
It seems possible that the spiral arms may with time become narrower and lose their clear outline. 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.
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 mention 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
It follows from this that
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.
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.
At the moment, it must seem as though such powerful centres could be postulated only on the basis of some ad hoc hypothesis; but it will appear in due course that this is not necessary. Surprising though it may seem at present, such centres do follow 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.