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3. The Limits of Perihelion Distance
It is sufficiently obvious that whenever the perihelion distance of a planet or comet is less than the sun's radius, a collision must occur as the moving body approaches the focus of its path. The great comet of 1843 passed so near the sun as almost to graze its surface. With a perihelion distance but very slightly less, it would have been precipitated into the sun and incorporated with its mass. In former epochs, when the dimensions of the sun were much greater than at present, this falling of comets into the central orb of the system must have been a comparatively frequent occurrence. Again, if Mercury's orbit had its present eccentricity when the radius of the solar spheroid was twenty-nine million miles, the planet at its nearest approach to the centre of its motion must have passed through the outer strata of the central body. In such case a lessening of the planet's mean distance would be a necessary consequence. We thus see that in the formation of the solar system the eccentricity of an asteroidal orbit could not increase beyond a moderate limit without the planet's return to the solar mass. The bearing of these views on the arrangement of the minor planets will appear in what follows.
4. Was the Asteroid Zone originally Stable?—Distribution of the Members in Space
One of the most interesting discoveries of the eighteenth century was Lagrange's law securing the stability of the solar system. This celebrated theorem, however, is not to be understood in an absolute or unlimited sense. It makes no provision against the effect of a resisting medium, or against the entrance of cosmic matter from without. It does not secure the stability of all periodic comets nor of the meteor streams revolving about the sun. In the early stages of the system's development the matter moving in unstable orbits may have been, and probably was, much more abundant than at present. But even now, are we justified in concluding that all known asteroids have stable orbits? For the major planets the secular variations of eccentricity have been calculated, but for the orbits between Mars and Jupiter these limits are unknown. With an eccentricity of 0.252 (less than that of many asteroids), the distance of Hilda's aphelion would be greater than that of Jupiter's perihelion. It seems possible, therefore, that certain minor planets may have their orbits much changed by Jupiter's disturbing influence.8
Whoever looks at a table of asteroids arranged in their order of discovery will find only a perplexing mass of figures. Whether we regard their distances, their inclinations, or the forms of their orbits, the elements of the members are without any obvious connection. Nor is the confusion lessened when the orbits are drawn and presented to the eye. In fact, the crossing and recrossing of so many ellipses of various forms merely increase the entanglement. But can no order be traced in all this complexity? Are there no breaks or vacant spaces within the zone's extreme limits? Has Jupiter's influence been effective in fixing the position and arrangement of the cluster? Such are some of the questions demanding our attention. If "the universe is a book written for man's reading," patient study may resolve the problem contained in these mysterious leaves.
Simultaneously with the discovery of new members in the cluster of minor planets, near the middle of the century, occurred the resolution of the great nebula in Orion. This startling achievement by Lord Rosse's telescope was the signal for the abandonment of the nebular hypothesis by many of its former advocates. To the present writer, however, the partial resolution of a single nebula seemed hardly a sufficient reason for its summary rejection. The question then arose whether any probable test of Laplace's theory could be found in the solar system itself. The train of thought was somewhat as follows: Several new members have been found in the zone of asteroids; its dimensions have been greatly extended, so that we can now assign no definite limits either to the ring itself or to the number of its planets; if the nebular hypothesis be true, the sun, after Jupiter's separation, extended successively to the various decreasing distances of the several asteroids; the eccentricities of these bodies are generally greater than those of the old planets; this difference is probably due to the disturbing force of Jupiter; the zone includes several distances at which the periods of asteroids would be commensurable with that of Jupiter; in such case the conjunctions of the minor with the major planet would occur in the same parts of its path, the disturbing effects would accumulate, and the eccentricity would become very marked; such bodies in perihelion would return to the sun, and hence blanks or chasms would be formed in particular parts of the zone. On the other hand, if the nebular hypothesis was not true, the occurrence of these gaps was not to be expected. Having thus pointed out a prospective test of the theory, it was announced with some hesitation that those parts of the asteroid zone in which a simple relation of commensurability would obtain between the period of a minor planet and that of Jupiter are distinguished as gaps or chasms similar to the interval in Saturn's ring.
The existence of these blanks was thus predicted in theory before it was established as a fact of observation. When the law was first publicly stated in 1866, but ten asteroids had been found with distances greater than three times that of the earth. The number of such now known is sixty-five. For more than a score of years the progress of discovery has been watched with lively interest, and the one hundred and eighty new members of the group have been found moving in harmony with this law of distribution.9
COMMENSURABILITY OF PERIODS
When we say that an asteroid's period is commensurable with that of Jupiter, we mean that a certain whole number of the former is equal to another whole number of the latter. For instance, if a minor planet completes two revolutions to Jupiter's one, or five to Jupiter's two, the periods are commensurable. It must be remarked, however, that Jupiter's effectiveness in disturbing the motion of a minor planet depends on the order of commensurability. Thus, if the ratio of the less to the greater period is expressed by the fraction 1⁄2, where the difference between the numerator and the denominator is one, the commensurability is of the first order; 1⁄3 is of the second; 2⁄5, of the third, etc. The difference between the terms of the ratio indicates the frequency of conjunctions while Jupiter is completing the number of revolutions expressed by the numerator. The distance 3.277, corresponding to the ratio 1⁄2, is the only case of the first order in the entire ring; those of the second order, answering to 1⁄3 and 3⁄5, are 2.50 and 3.70. These orders of commensurability may be thus arranged in a tabular form, the radius of the earth's orbit being the unit of distance:
Do these parts of the ring present discontinuities? and, if so, can they be ascribed to a chance distribution? Let us consider them in order.
I.—The Distance 3.277
At this distance an asteroid's conjunctions with Jupiter would all occur at the same place, and its perturbations would be there repeated at intervals equal to Jupiter's period (11.86 y.). Now, when the asteroids are arranged in the order of their mean distances (as in Table II.) this part of the zone presents a wide chasm. The space between 3.218 and 3.376 remains, hitherto a perfect blank, while the adjacent portions of equal breadth, interior and exterior, contain fifty-four minor planets. The probability that this distribution is not the result of chance is more than three hundred billions to one.
The breadth of this chasm is one-twentieth part of its distance from the sun, or one-eleventh part of the breadth of the entire zone.
II.—The Second Order of Commensurability.—The Distances 2.50 and 3.70
At the former of these distances an asteroid's period would be one-third of Jupiter's, and at the latter, three-fifths. That part of the zone included between the distances 2.30 and 2.70 contains one hundred and ten intervals, exclusive of the maximum at the critical distance 2.50. This gap—between Thetis and Hestia—is not only much greater than any other of this number, but is more than sixteen times greater than their average. The distance 3.70 falls in the wide hiatus interior to the orbit of Ismene.
III.—Chasms corresponding to the Third Order.—The Distances 2.82, 3.58, and 3.80
As the order of commensurability becomes less simple, the corresponding breaks in the zone are less distinctly marked. In the present case conjunctions with Jupiter would occur at angular intervals of 120°. The gaps, however, are still easily perceptible. Between the distances 2.765 and 2.808 we find twenty minor planets. In the next exterior space of equal breadth, containing the distance 2.82, there is but one. This is No. 188, Menippe, whose elements are still somewhat uncertain. The space between 2.851 and 2.894—that is, the part of equal extent immediately beyond the gap—contains thirteen asteroids. The distances 3.58 and 3.80 are in the chasm between Andromache and Ismene.
IV.—The Distances 2.95, 3.51, 10 and 3.85, corresponding to the Fourth Order of Commensurability
The first of these distances is in the interval between Psyche and Clytemnestra; the second and third, in that exterior to Andromache.
The nine cases considered are the only ones in which the conjunctions with Jupiter would occur at less than five points of an asteroid's orbit. Higher orders of commensurability may perhaps be neglected. It will be seen, however, that the distances 2.25, 2.70, 3.03, and 3.23, corresponding to the ratios of the fifth order, 2⁄7, 3⁄8, 4⁄9, and 6/11, still afford traces of Jupiter's influence. The first is in the interval between Augusta and Feronia; the last falls in the same gap with 3.277; and the second and third are in breaks less distinctly marked. It may also be worthy of notice that the rather wide interval between Prymno and Victoria is where ten periods of a minor planet would be equal to three of Jupiter. The distance of Medusa is somewhat uncertain.
