CHAPTER XVII
THE EFFECT OF COMMENSURATE PERIODS
The Asteroids and Saturn’s Rings
Ever inquiring, ever fertile, his mind turned to seek the explanation of divers astronomical phenomena. In 1912, for example, under the title “Precession and the Pyramids,” we find him discussing in the Popular Science Monthly the pyramid of Cheops as an astronomical observatory, with its relation to the position of the star then nearest to the North Pole, its lines of light and shadow, in a great gallery constructed with the object of recording the exact changes in the seasons.
But leaving aside these lesser interests, and the unbroken systematic observation of the planets, his attention in the later years of his life was chiefly occupied by two subjects, not unconnected, but which may be described separately. They are, first, the influence over each other’s position and orbits of two bodies, both revolving about a far larger one; and, second, the search for an outer planet beyond the path of Neptune. Each of these studies involved the use of mathematics with expanding series of equations which no one had better attempt to follow unless he is fresh and fluent in such forms of expression. For accurate and quantitative results they are absolutely essential, but an impression of what he was striving to do may be given without them.
Two bodies revolving about a common centre at different distances, and therefore different rates of revolution, will sometimes be on the same side of the central body, and thus nearer together; sometimes on opposite sides, when they will be much farther apart. Now it is clear that the attraction of gravity, being inversely as the square of the distance, will be greatest when they are nearest together; and if this happens at the same point in their orbits every time they approach each other the effect will be cumulative, and in the aggregate much larger than if they approach at different parts of their orbits and hence pull each other sometimes in one direction and sometimes in another. To use a homely, and not altogether apt, illustration: If a man, starting from his front door, walk every day across his front lawn in the same track he will soon make a beaten path and wear the grass away. If, instead, he walk by this path only every other day and on the alternate days by another, he will make two paths, neither of which will be so much worn. If he walk by three tracks in succession the paths will be still less worn; and if he never walk twice in the same place the effect on the grass will be imperceptible.
Now, if the period taken by the outer body to complete its orbit be just twice as long as that taken by the inner, they will not come close together again until the outer one has gone round once to the inner one’s twice, and they will always approach at the same point in their orbits. Hence the effects on each other will be greatest. If the outer one take just two turns while the inner takes three they will approach again only at the same point, but less frequently; so that the pull will be always the same, but repeated less often. This will be clearly true whenever the rates of the revolution differ by unity: e.g., 1 to 2, 2 to 3, 3 to 4, 4 to 5, etc.
Take another case where the periods differ by two; for example, where the inner body revolves about the central one three times while the outer one does so once; in that case the inner one will catch up with the outer when the latter has completed half a revolution and the inner one and a half; and again when the outer has completed one whole revolution and the inner three. In this case there will be two strong pulls on opposite sides of the orbits, and, as these pulls are not the same, the total effect will be less than if there were only one pull in one direction. This is true whenever the periods of revolution differ by two, e.g., 1 to 3, 3 to 5, 5 to 7. If the periods differ by three the two bodies will approach three times,—once at the starting point, then one third way round, and again two thirds way round, before they reach the starting point; three different pulls clearly less effective.
In cases like these, where the two bodies approach in only a limited number of places in their orbits the two periods of revolution are called commensurate, because their ratio is expressed by a simple fraction. The effect is greater as the number of such places in the orbit is less, and as the number of revolutions before they approach is less. But it is clearly greater than when the two bodies approach always at different places in their orbits, never again where they have done so before. This is when the two periods are incommensurate, so that their ratio cannot be expressed by any vulgar fraction. One other point must be noticed. The commensurate orbit, and hence the distance from the Sun, and the period of revolution, of the smaller and therefore most affected body, may not be far from a distance where the orbits would be incommensurate. To take the most completely incommensurate ratio known to science, that of the diameter of a circle to the circumference, which has been carried out to seven hundred decimal places without repetition of the figures. This is expressed by the decimal fraction .314159 etc. and yet this differs from the simple commensurate 1/3 or .333333 etc. by only about five per cent.; so that a smaller body may have to be pulled by the larger, only a very short way before it reaches a point where it will be seriously affected no more.
The idea that commensurateness affects the mutual attraction of bodies, and hence the perturbations in their orbits, especially of the smaller one, was not new; but Percival carried it farther, and to a greater degree of accuracy, by observation, by mathematics and in its applications. The most obvious example of its effects lay in the influence of Jupiter upon the distribution of the asteroids, that almost innumerable collection of small bodies revolving about the Sun between the orbits of Jupiter and Mars, of which some six hundred had been discovered. These are so small, compared with Jupiter, that, not only individually but in the aggregate, their influence upon it may be disregarded, and only its effect upon them be considered. In its immediate neighborhood the commensurate periods, Percival points out, come so close together (100 to 101, 99 to 100, etc.) that although occasions of approach would be infrequent they would be enough in time to disturb any bodies so near, until the planet had cleared out everything in its vicinity that did not, by revolving around it, become its own satellite.
Farther off Jupiter’s commensurate zones are less frequent, but where they occur the fragments revolving about the Sun would be so perturbed by the attraction of the planet as to be displaced, mainly, as Percival points out, to the sunward side. This has made gaps bare of such fragments, and between them incommensurate spaces where they could move freely in their solar orbits. Here they might have gathered in a nucleus and, collecting other fragments to it, form a small planet, were it not that the gaps were frequent enough to prevent nuclei of sufficient size arising anywhere. Thus the asteroids remained a host of little bodies revolving about the Sun, with gaps in their ranks—as he puts it “embryos of planets destined never to be born.”
The [upper diagram] in the plate opposite [page 166] shows the distribution and relative densities of the asteroids, with the gaps at the commensurate points. The plate is taken from his “Memoir on Saturn’s Rings,”[35] and brings us to another study of commensurate periods with quite a different set of bodies obeying the same law. Indeed, among the planets observed at Flagstaff not the least interesting was Saturn, and its greatest peculiarity was its rings.