Testing the Practicability of the Hollow Sphere Theory. Retrograde Motions, Positions, Densities, Masses, etc. etc., Considered.

Before going any farther it will be convenient to try to find out whether the solar system could have been constructed from a hollow nebula such as we have been describing gradually contracting as the matter for the formation of one planet after another was abandoned until—as we have put it—the nebula could abandon no more matter, and finally resolved itself into the sun. For this purpose we may suppose it to have been condensed and contracted until its extreme diameter was 6,600,000,000 miles; the same as we supposed it to have been, when we began the analysis of the nebular hypothesis. We will not now, however, suppose it then to have contained the whole of the cosmic matter out of which the system was formed, as we did before; because we have seen as we have come along that a very considerable part of that matter must have been left behind, almost from the moment that contraction commenced. We have already given the reasons for this in describing the domains of the sun; and, leaving the peaks out of account altogether for the present, we will only deal with the regions of what we have called the main body.

Although we have fixed a limit beyond which the neighbouring stars could not draw off any cosmic matter from the domains of the sun, that does not mean to say that their attractive powers would cease at that limit; because we have had to acknowledge that each one of them continues, even now, to exert its attractive power up to the very centre of the sun. They would still have power to counteract, in some measure, the sun's attraction of the matter of the nebula towards his centre, and the result would follow that there would be one or more, even many, fragments of the main body which would be left more or less behind, and in varied forms, when the more central part had contracted to the dimensions to which we have now reduced the nebula—all much the same as we have already said a few pages back.

When the nebula was 6,600,000,000 miles in diameter its volume would be 150,533²⁴ cubic miles—as we have seen at page 87—the half of which is 75,266²⁴ cubic miles, corresponding to a diameter of 5,238,332,000 miles, or radius of 2,619,166,000 miles. Now, according to our theory, it would be at this distance from the centre that the greatest density and activity of the nebulous matter would be, where we have just been showing how a movement of rotation could be generated, and where, in consequence, its motive power, so to speak, originated and existed. Here we find by dividing 5,238,332,000 by 6,600,000,000 that the region of greatest density in such a nebula would be at 0·7937 of its diameter. In our calculations about the earth, as it is, the proportion was found to be 0·7939, but the densities of the outer layers were empirically arranged by us; and, besides, almost the whole of the mass was supposed to be solid matter, so that no accurate result could be expected from that operation. There also we found that the inner surface of the hollow shell was at 0·5479 of the whole diameter, which we may adopt for the nebula we are about to deal with, as that dimension may be varied considerably—so may the other also—without in any way vitiating our theory.

Having found these proportions, which can only be considered as distantly approximate, let us go back to the 9 nebulÆ—excluding the final solar one—into which we supposed the original nebula to have been divided—in the analysis just alluded to—and see how the regions of greatest density in them would correspond to the orbits of the planets formed out of them. This examination requires a good deal of calculation and accompanying description, which it might be found tiresome to follow, and would really answer no good end were it written out; so we shall suppose it to be made and the results obtained from the calculations to be represented in the form of Table IX., where they can be seen at a glance almost, and compared without much trouble. This arrangement will also furnish a readier means of reference for the remarks we shall have to make on, and the information obtained from, the examination. And we have still to add that the extreme diameters of the 9 nebulÆ are the same as those we used for the analysis; as also, that we make use of only the first of the proportions just cited, viz., 0·7937, it being the only one required for determining the positions of the regions of greatest density in the nebulÆ.

TABLE IX..— Dimensions of the nine NebulÆ, with their Diameters and Regions of greatest Density compared with the Diameters of the Orbits of the Planets formed from them.

From the table we see that the region of greatest density of our original nebula was at 6·26 per cent. within the distance of Neptune's orbit from the sun, a state of matters which precludes the idea of condensation during, at least, a great part of the act of abandoning the ring for the formation of that planet. But it will be remembered that we gave it the diameter of 6,600,000,000 miles without assigning any adequate reason for doing so, and, we can say with truth, with the idea, more than anything else, of not increasing the almost unimaginable tenuity of the matter composing the nebula; and the position of Neptune in the system is so peculiar compared with the other planets, that it cannot be properly used as a standard for any kind of inquiry. The result obtained above can therefore be of no use for the investigation we have undertaken. Not only so, but the almost similar result in the case of Uranus is also rendered useless from the same cause, in which we find that the region of greatest density of the nebula is only 1·92 per cent. beyond the orbit of the planet. If the mean distance from the sun of Neptune's orbit had been what was used by Leverrier in the calculations which led to his discovery, namely, 36·152 radii of the earth's orbit, the region of greatest density of the Uranian nebula would have been 14·48 per cent. beyond his orbit, as may be seen from the addition to Table IX., in finding which we have used the same system as in all our work.

In the next four nebulÆ of the table—including the one we introduced to represent the Asteroids—we see that their regions of greatest density are respectively 19·58, 12·47, 13·56 and 12·63 per cent. farther out from the centre of the sun than the orbits of the planets formed from them. Here, then, we see a very apparent approach of uniformity, and can say with much reason that planets could certainly be formed out of the matter abandoned, through centrifugal force, by hollow nebulÆ similar in construction to what we have demonstrated that of the original nebula to have been; each of them occupying the position corresponding to its orbit.

Following these come the Earth and Venus nebulÆ. In the former, the region of greatest density almost coincides with the orbit of the planet, being only 0·15 per cent. beyond it, instead of something like 12 per cent. as it ought to be to conform with the four preceding cases; and in the latter it is 5·25 per cent. within the orbit of the planet to be made from it. But in this case we have to note that the orbit of Venus is 3·33 per cent. beyond the position pointed out for it by Bode's law, and that it is the only one of the whole number of planets whose orbit is farther removed from the sun than the distance assigned to it by that law. Also we see from our reversal of Bode's law, that the rates of acceleration of rotation for these two planets are 1·880 for the earth and 1·626 for Venus, instead of the average of 2·5896 of the four preceding planets; that the density of Venus is less than that of the Earth, instead of being greater as it is successively in all the other planets from Saturn inwards; and we may add that the diameters are nearly equal. All showing that influences had been at work in the formation of these two planets, different to those in the preceding four; and that until we know what these influences have been, we cannot account for any anomalies produced by them. Neither are we called upon to consider that our theory is destroyed by these anomalies, any more than it can be by the anomaly in the case of Neptune's position.

Lastly, we have in Mercury the region of greatest density of his nebula at 13·55 per cent. beyond his orbit, and the rate of acceleration of revolution over Venus 2·5543 times, both of which conform fairly well with the same noted facts; in relation to Mars, the Asteroids, Jupiter, Saturn, and, we may add, Uranus. But, in justice, we must not omit to add that there may be some error in the excess of 13·55 per cent. in the distance from the sun beyond his (Mercury's) orbit, arising from the fact that there may have been some difference from what we made it to be, in the line of separation between his nebula and that of Venus; and also that we had to guess at the line of separation between his and the residuary nebula. Moreover, it has to be taken into account that his orbit is 3·22 per cent. within the position assigned to it by Bode's law.

From the Table IX., and an examination of it, we learn that out of the 9 nebulÆ into which we divided the original one, in the analysis of the nebular hypothesis, we have five—four of which are consecutive—which may have been almost of the same construction, and not far from the same proportions; that the original nebula cannot, for reasons assigned, be looked upon as either similar, or the reverse, to the five just classed; that one, the Uranian, is practically similar to the five, and might be exactly similar could the anomaly in the position of Neptune be explained; and that the remaining two, the Earth and Venus nebulÆ, seem to show that they have been abandoned in a manner different from the others. Perhaps we may be able, later on, and in a different way, to give a reasonable explanation of the anomalies in the positions occupied by Neptune, the Earth, and Venus, and also of the peculiarities of their dimensions. So far, we believe we are justified in concluding that out of the 9 nebulÆ, 6 may really be considered as supporting our theory, and the remaining 3 as, in all probability, capable of being shown to be, at least, not opposed to it. To this we may add that on several occasions we have stated our opinion, that the divisions between the nebulÆ we have established, could not have taken place at the half-distance between the orbits of any two planets, but much nearer to the outer one. It is evident, then, that if we had made the divisions at any distance farther out, say at three-fourths of that distance from the inner orbit, the extreme diameter of each one of the nebulÆ would have been just so much greater, the region of greatest density farther out from the centre of the sun, and even that of Neptune would have been beyond his orbit. All this could be done, yet but it would serve no good purpose, as will be seen presently; and we might be accused of cooking our data in order to produce a result favourable to our theory.

We have made the foregoing examination because, when we began our work, the general idea was that, according to the nebular hypothesis, the material for the formation of each planet was abandoned by the ideal nebula in a distinct and separate mass from any other—we are not at all sure, however, that this was Laplace's idea. This, we found out, could not be the case when we attempted to give some sort of separate or distinct form to the matter out of which Neptune was supposed to have been formed; and when we became convinced that all the matter abandoned by the nebula, from first to last, must have been thrown off in one continuous and, most probably, uninterrupted sheet. This, of course, makes us think of how the division of the sheet into separate rings was brought about, for there must have been absolute separation between them, otherwise separate planets could not have been made out of the sheet; and the only explanation that can be given is, that it must have depended on the quantity of matter that was abandoned, in nearly equal times, at different periods of the operation; for the areolar law precludes the idea of there having been very rapid changes in the rate of rotation of the nebula, and certainly of its decrease at any period as long as condensation and contraction went on. Whereas, although the sheet thrown off may have been continuous, we have no reason to suppose that it was of constant volume or density from beginning to end of the operation; in fact, we have already seen that its density was constantly increasing, and have suggested, in the reversal of Bode's law, that the differences in dimensions and densities of the planets have arisen, from irregularity in the quantities of matter abandoned from time to time. This irregularity could only arise from the mode of construction of the nebula, and from the forms it assumed during condensation, as we shall attempt to show in due time. Meanwhile we can conclude that the region of greatest density in any of our nebulÆ had no influence whatever on the position of the orbit of the planet that was formed out of it.

We have shown, very clearly we believe, at page 109, from quotations—at second hand—from his own exposition of his hypothesis, that Laplace considered that condensation could only take place at the surface, or in the atmosphere as he called it, of his nebula, on account of its being possible only after radiation into space of part of its excessive heat; and that consequently there could be no acceleration of rotation in the nebula, due to the areolar law, except where there was condensation. On the other hand, in our cold hollow-sphere nebula, condensation could only take place at the region of greatest density, or greatest mass, which must be always very much nearer to the surface than to the centre; so that in both cases, equally, the abandoning of matter under the influence of centrifugal force would be virtually the same, and no further remarks are called for, on our part, on that head.

Neither is it necessary for us to show how planets could be formed out of the rings abandoned by their respective nebulÆ, for everybody seems to agree that when they broke up, the fragments could not do otherwise than form themselves into small nebulÆ, which in the course of time condensed into planets. M. Faye's explanations are good for that.

With respect to their motions of rotation being direct or retrograde, we have seen, at page 116, and following, that Laplace's description of how the former motion could be brought about is mechanically correct; and, at page 121, that he did not consider that the direction of revolution of a ring necessarily demands that the rotation of a planet formed from it should be in the same direction. As already said, he has shown how direct rotation could be produced, and we have no doubt that he could have shown how retrograde rotation could also be produced, had he found it to be at all necessary. Be that as it may, however, it is a very simple matter to show how, following our method of construction of the primitive nebula, the retrograde rotation of Uranus and Neptune could, or rather must, have been determined.

It will be remembered that when we were "getting up" the original nebula in the domains of the sun, whose form we described as well as our limited means would admit of, we said that when the cosmic matter contained in them began to contract, not only the parts contained in the peaks and promontories would soon be left behind, and come in at a slower rate, but also large masses of the outer part of the main body, especially of what was on the sides opposite to the deep hollows made in the domains by the most powerful of the sun's neighbours, in the form of fragments, crescents, and parts of hollow segments. Let us now, then, suppose the operation of planet-making to have advanced so far that the whole nebula was rotating on its axis, and abandoning matter through centrifugal force, from its equatorial regions in a continuous sheet, as we have said several times that it must have done, and that the matter destined for Neptune and Uranus has not only been abandoned, but divided into two distinct rings—a supposition made in this case only for facility of description. Then some of the matter which had been left behind, but still being gradually drawn in, would be almost totally intercepted in the equatorial regions of the nebula by these two rings, and would fall in greater quantity upon their outer edges than anywhere else, more especially in the case of the outer one. These adventitious additions would come in without any angular, or tangential, movement whatever, because rotary motion was not yet established in them, and would retard the revolutionary movement of the rings—in decreasing degree from their outer to their inner edges—while acquiring angular motion themselves; and would also intensify the original difference in revolutionary motion already existing at these edges. At the same time these additions of extraneous matter would seriously impede the contraction of the rings in the radial direction on account of their volume, but would have little or no effect on contraction in the circumferential direction; the consequence of which would be that they would break up before friction, and the mutual collisions of their particles, had time to produce a uniform revolving motion throughout their whole breadth; that is, while their inner edges would be still revolving with more rapid velocities than the outer ones; and the rotary motions of the planets derived from them would be retrograde, according to M. Faye's demonstration—or that of any other who has taken the trouble to think over the matter. And we may add that this mode of reasoning, applied with a little more detail, will very fully account for the rotation of Neptune being more decidedly retrograde than that of Uranus, because the quantity of matter so deposited on the outer flat ring in this process would unquestionably be greater than on the inner one, and consequently the difference of velocity between the outer and inner edges of the two rings also greatest on the outer one.

We take it to be unnecessary even to say that, the revolution of the satellites of these two planets being retrograde and anomalous, the rotation of their principals must be retrograde and anomalous also.

Before going any farther we have something to say about the anomalous position of the orbit of Neptune, which is certainly not the position sought for by M. Leverrier; in fact, the elements employed by him in his calculations to discover a perturbing planet—whose existence may be said to have been known—are so different from the elements of the one actually discovered, that there would be nothing out of reason in saying that Neptune is not the perturber that was sought for, but only an instalment of the perturbing force. It may raise a storm in some quarters to say so, but the fact remains the same, or it must be confessed that mathematics is a more elastic science than it professes to be. He has not the power of attraction required to produce the perturbations in the movements of Uranus which gave rise to the search for an outer planet. M. Leverrier made his calculations under the belief that a planet of 1/9300th part of the mass of the sun was required to produce the perturbations that had been observed in the orbital motion of Uranus; whereas the planet discovered has only 1/20,000th of that mass—not one-half of what was required. On the other hand, the semi-axis major of the orbit of the planet discovered is found to be 30·037 instead of 36·154 (Bode's law measures) used for the search; which greater proximity to the sun, it is true, increases its power of attraction 1·449 times, but as its mass is only 0·465 per cent. of what was expected, the attractive force would amount to less than 0·68 per cent. of what was required. Then the question comes to be, Where did the wanting 0·32 per cent. of attractive force come from? And the answer is that some astronomers have been searching for another planet to make up the weight, with more or less diligence, ever since the deficiency came to be recognised. But all that we want to have to do with the question is to suggest a very plausible reason for the anomalous position of the orbit of Neptune.

If there is another planet beyond Neptune, the ring (perhaps the rings) out of which he and the others were made, must have been much greater in breadth than what we have assigned to it at page 88, viz. 1,010,000,000 miles; perhaps even one-half more, as may be deduced from the addition made to Table IX., and what we have said in connection with the semi-axis major adopted for the sought-for planet, by M. Leverrier in his calculations. Now, that a ring of such enormous breadth should have held together in one piece, until it finally broke up through condensation and contraction, requires an extraordinary effort of imagination, after seeing what has taken place with the rings of Saturn; even the breadth of 954,000,000 miles appropriated to the Uranian ring (see page 90) demands an elastic imagination to conceive its holding together; so that the outer ring of the system may very well have been divided into two, as we have said at page 134, and two not very unequal planets made out of it—one into Neptune, and the other into one as far beyond M. Leverrier's adopted distance of 36·154, and of such mass as would make up the missing 0·32 per cent. of deficient attractive power. No doubt the outer ring may have broken up into several planets, or even into a swarm of asteroids, but we prefer to think of only two planets; because it seems to us that to draw Uranus into the position he occupied when Neptune was discovered, the two planets must have been operating in conjunction; an idea that is not so easily entertained when there are several planets, or a host of asteroids, to be taken into account.

We have already discussed, at page 115, the mode of formation of the sheet of matter abandoned by the nebula, its posterior division into separate rings, and how the part of these rings from Saturn inwards could revolve themselves into planets having direct motion, so it is not necessary to go over the same ground again, merely because we are dealing with a hollow nebula instead of one full of cosmic matter to the centre.

We have also shown, at page 119, that the nebula must have been somewhat in the form of a cylinder terminated at each end by what may be looked upon as a segment of a sphere, although it would more probably be an almost shapeless mass of cosmic matter, because the greater part of it would be very slowly brought under the influence of centrifugal force as it fell in from the polar directions; and again, a few pages back, that almost all the matter coming in from its equatorial regions—even what might be called its tropical regions—would be intercepted before it could reach the Saturnian nebula. Likewise, at page 137, when examining Bode's law reversed, we have seen a limit set to the acceleration of the movement of revolution in the planets of the system as they approached the centre, because any acceleration beyond a certain limit, clearly marked out, would of necessity be within the nebula itself, and the rate of revolution would be less than that of the sun on its axis at the present day. This may be used as an argument against the nebular hypothesis, but we think we have shown in the same Chapter VII. that this is not the case. But we have still to try to account for the repeated rises and falls in density in the planets from Neptune to Mercury, or even farther; which operation causes us to bring forward, first of all, a new idea as to what the form of the nebula would come to be.

Fig. 2.