Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 6

GAPS IN THE DISTRIBUTION OF ASTEROIDS

by YUSUKE HAGIHARA

SMITHSONIAN INSTITUTION

Washington, D.C.

1961

Publications of the Astrophysical Observatory

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Contents

Publications of the Astrophysical Observatory

Gaps in the Distribution of Asteroids

References

Gaps in the Distribution of Asteroids

By YUSUKE HAGIHARA1

When we plot the mean motions of asteroids, we observe several gaps around the values commensurable with the mean motion of Jupiter (Kirkwood, 1867), and an apparent clustering at values that are not exactly commensurable. These observations suggest that when an asteroid acquires a mean motion exactly commensurable with that of Jupiter, its motion becomes unstable and the asteroid quickly shifts into a nearby, noncommensurable orbit.

Let us consider a system consisting of an asteroid and Jupiter, both moving around the sun. According to the usual theory of perturbations, we determine the motion by the method of the variation of the elements. When we perform a first integration of the equations of motion, neglecting the perturbation due to Jupiter, we obtain an orbit that is a Keplerian ellipse with six constants of integration, called the elements of the asteroid. When we take the perturbation of Jupiter into account, the elements are no longer constant and are determined by a system of differential equations expressing the variation of the elements E as functions of the perturbation, in the form,

Here n is the mean motion, є the mean longitude at the epoch, Dij a function of the elements, Bij also a function of the elements, i and j are integers, positive, negative, or zero, and the letters with primes represent the corresponding quantities for Jupiter. The series is arranged in an integral power series of small quantities such as the eccentricities of the orbits of the asteroid and Jupiter and the inclination of the asteroidal orbit to that of Jupiter. In the first approximation we usually integrate by neglecting the variation of the elements on the right-hand side of such equations, because the right-hand side is multiplied by the mass of Jupiter, which we consider as a small quantity of the first order. By such a formal integration, t being the only variable, we obtain the equation,

In this way the solution is expressed in the form of a trigonometric series in which each term represents a periodic inequality with the period . Thus the variation of the elements is expressed formally as a sum of periodic terms and the orbit is considered to be stable in a certain sense.

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1 Smithsonian Astrophysical Observatory.

However, if i=j=0, then the corresponding term is expressed as a linear function of t and the term is called secular. It can be shown from Poisson’s theorem that there is no secular term in the expression for the semimajor axes in the first-order perturbation. If in+jn'=0 while i≠j≠0, then the corresponding term is called critical. If in+jn'≠0 but is very small, then the corresponding term has a small divisor in+jn' in the denominator, and becomes very large compared with the remaining terms. This circumstance would spoil the formal convergence of the series obtained by formal integration of the differential equations. Actually such terms (called the “great inequalities”) exist, corresponding to a small divisor of the form 2n—5n' in the mutual perturbation of Jupiter and Saturn, and to a smaller divisor n—2n' for Uranus and Neptune. In such cases the usual theory of perturbation fails and a different method, such as that of Hansen (1857) or of Newcomb (1874, 1891), is employed for the actual computation of the perturbation.

For the motion of an asteroid with very small values of in + jn' an entirely new theory should be worked out, such as that of Glydén (1893), of Delaunay (1860, 1867), or a periodic solution due to Poincaré (1890, 1892).

It was once thought that the existence of a small divisor or the close commensurability of the mean motions would endanger the stability of the motion, and that the gaps in the distribution of the mean motions of asteroids must result from the presence of such small divisors. It should be emphasized that the series obtained in this manner is only a formal solution and that it does not by any means represent the true solution of the equations because it is not uniformly convergent, as was proved by Poincaré (1890, 1893). The problem of stability is closely related to the question of the form of the solution, which is written in its uniformly convergent form. Hence, with the ordinary formal expansion used in the perturbation theory, we cannot draw any definite conclusion as to the behavior of the motion over a comparatively long interval of time. Thus we cannot attempt to explain the occurrence of the gaps in question on the basis of such a nonconvergent series. As a matter of fact, the observations show (see fig. 1) that asteroids actually do exist with mean motions nearly commensurable with that of Jupiter, in the ratios 3 to 2 for the Hilda group, 4 to 3 for the Thule group, and 1 to 1 for the Trojan group. On the other hand, no asteroids have been found with mean motions close to the commensurable values in the ratios 2 to 1 for the Hecuba group, 3 to 1 for the Hestia group, 5 to 2, and 7 to 3. Hence we cannot accept the explanation that the gaps result from the presence of small divisors.

Hirayama (1918, 1928; see also Hagihara, 1928a, 1928b, 1928c, 1928d, 1928e) once postulated the existence of a resisting medium to explain the gaps, but later withdrew the suggestion. The families of asteroids that Hirayama (1922, 1927, 1933) discovered, which Brouwer (1950, 1951) has recently enlarged, have no direct relation to the gaps.

Similar gaps seem to exist in the rings of Saturn, as shown in figure 2. It might be thought that the occurrence of Cassini’s division, the so-called Encke’s division, and other smaller irregularities in the brightness distribution over the ring is related to the fact that the particles constituting the ring around Saturn have mean motions commensurable with that of Mimas or possibly of Enceladus. In addition to Cassini’s division corresponding to the ratio 2 to 1 of the mean motion of a particle with that of Mimas, and Encke’s division corresponding to the ratio 5 to 3 of the mean motions, Lowell (1915) reported several gaps in Saturn’s rings corresponding to 3 to 1, 8 to 3, 5 to 2, 7 to 3, 9 to 4, 11 to 5 of Mimas, and to 4 to 1, 5 to 2, 7 to 3 of Enceladus. However, recent observations by Lyot at the Pic du Midi Observatory, and by others with big telescopes in the United States (Kuiper, 1960, and personal communication), show that some of the reported gaps, such as those corresponding to 5 to 2, 9 to 4 and 11 to 5 of Mimas, do not exist, and that even Encke’s “division” is only an irregularity in the distribution of brightness.

Brown (1928) believed that the mean motion of an asteroid changes quickly if it once becomes nearly commensurable with that of Jupiter, and that it makes a libration about the exact value, like the pendulum near its stable equilibrium position. Thus the mean time of sojourn at the nearly commensurable value is much shorter, statistically, than that spent at noncommensurable values. Brown (1932) has therefore studied in detail the resonance phenomena in simplified dynamical systems such as a coupled pair of pendulums, but has not applied the results to the actual motion of the asteroids. This theory does not, however, explain the cases of the Thule, Hilda and Trojan groups. It has recently been proved by Roy and Ovendon (1954, 1955), from the theory of probability, that the existence of the gaps is not due to chance.

Wilkens (1933) discussed the case of multiple commensurability among the mean motions of an asteroid, of Jupiter, and of Saturn (Okyay, 1935; Urban, 1935). But according to Hagihara’s (1928a, 1928b, 1928c, 1928d, 1930, 1940) study of the secular perturbation of higher degrees, supplemented by the work of Kozai (1954), this multiple commensurability has little to do with the stability of the motion, at least in the formal aspect of the problem as treated in the ordinary theory of perturbations.

FIGURE 1.—Distribution of the mean motion of asteroids.

FIGURE 2.—Saturn’s rings according to Lowell (1915).

While some of the asteroids apparently tend to avoid such commensurable values for their mean motions, several satellites, for example the asteroids of the Trojan, the Hilda, and the Thule groups, have mean motions that are mutually in almost exactly commensurable ratios. The first three Galilean satellites of Jupiter have for their daily mean motions the values  so that  and, even more remarkable,  (de Sitter, 1908a, 1908b, 1909, 1918, 1928). In Saturn’s system the mean motions of Mimas and Tethys are in the ratio 2 to 1, those of Enceladus and Dione in the ratio 2 to 1, and those of Titan and Hyperion 4 to 3.

The question naturally arises whether these gaps in the distribution of the mean motions of asteroids or of satellites can be reasonably explained by purely gravitational arguments. The recent cosmogonical theory of Alfvén (1954), explaining the gaps in asteroidal distribution and in Saturn’s system on the basis of magnetohydrodynamical theory, is still far from being convincing evidence for or against the occurrence of the gaps (see also Kuiper, 1950, 1953a, 1953b, 1956a, 1956b, 1956c, 1956d, 1957; Rabe, 1954, 1956, 1957; Kerr and Whipple, 1951, 1954). In any case, we must first find the answer on purely gravitational grounds, before we consider the cause of the gaps on a cosmogonical basis.

Saturn has several pairs of satellites whose mean motions are commensurable. For such a pair, the configuration of the perisaturniums is such that their motions are stable. For Titan and Hyperion the conjunction occurs always at the aposaturnium of Hyperion (Newcomb, 1891; Woltjer, 1918, 1928; Brouwer, 1924). For Enceladus and Dione the conjunction occurs at the perisaturnium of Enceladus. For Mimas and Tethys the conjunction occurs at the node instead of at the perisaturnium. The conjunction of the two satellites occurs near the midpoint between their two ascending nodes on Saturn’s equator. The actual configurations of these pairs of satellites are known to be stable.

It might be questioned whether the stability of the motion is affected when we take into account other disturbing bodies, such as the other satellites and the sun, and the effect of the figure of the planet. Usually such effects are not considered when the stability of the motion is discussed for the three-body problem (Brown, 1901). For the Trojan asteroids, Brown (1923, 1925) has proved that the action of the planets other than Jupiter never increases the amplitudes of the long-period libration terms without limit. Hagihara (1927) has discussed the stability of a satellite system composed of several satellites, two of which have nearly commensurable mean motions, by considering the effects of the solar perturbation and the figure of the planet. Starting with a set of intermediary nonperiodic orbits with moving pericenters for the pair of satellites, he discussed the differential equations for the disturbed motion on the basis of Poincaré’s (1890, 1892) theory of characteristic exponents, and proves that the motion is stable if the positions of the pericenters of the two satellites are suitably chosen.

The most interesting case is that of the three Galilean satellites of Jupiter, I, II, and III. De Sitter (1908a, 1908b, 1909, 1918, 1928) has adopted as the intermediary orbits for the three satellites a periodic solution in which all the three satellites have the same perijove motion. He also proved the stability of this solution by Poincaré’s method of characteristic exponents. The perijove of the second satellite lies opposite those of the first and third satellites. When the first and third satellites are at a certain epoch in conjunction, and the second satellite is in opposition, all in their apojoves, then the motion is unstable. But when the first and third satellites are in conjunction and the second satellite is in opposition, all in their perijoves at a certain epoch, then the motion is stable. This case of stable motions is what we actually observe in nature.

The situation is entirely different for the gaps in Saturn’s rings and in the asteroidal distribution. Goldsbrough (1921, 1922, 1924) has discussed the stability of the motion of small bodies with mean motions commensurable with that of Mimas (Pendse, 1933, 1935, 1937). He studied the motion of small bodies situated at the vertices of an equilateral polygon having Saturn at its center and rotating around Saturn. This configuration is what was called a central figure by Dziobek (1900; see also Andoyer, 1906; Meyer, 1933; Hölder, 1929). Goldsbrough considered a small deviation of the bodies from this equilibrium position and discussed the solution, from the point of view of stability, of the differential equations of the motion when disturbed by Mimas; he proved the instability of the motion around the ratios of the mean motions in the principal commensurable values corresponding to Cassini’s and Encke’s divisions. Hagihara (1940) has studied the system of differential equations for the disturbed motion with parametric representation, employing Peano-Baker’s matrix method (Baker, 1916). From the theory of nonlinear integrodifferential equations, Lichtenstein (1923, 1924, 1933, 1932, 1931) has proved the existence of periodic solutions and found that the deviation from the equilibrium position propagated in a wave form.

The motion of a “characteristic” asteroid, having a mean motion nearly commensurable with that of Jupiter, has been studied theoretically on the basis of the corresponding periodic solution (Hill, 1902a, 1902b; Poincaré, 1890, 1892, 1902a, 1902b; Andoyer, 1903; Schwarzschild, 1903; Wilkens, 1913; Heinrich, 1912, 1922, 1925, and others). Poincaré (1890, 1892) has proved the existence of periodic solutions for a continuous set of values of a parameter μ in the neighborhood of μ=0 for the case in which a generating periodic solution exists for μ=0. Consider two planets of small mass, revolving around the Sun. Suppose that the generating periodic solution for the motion of the two planets is such that their orbits around the Sun have nonzero eccentricities but perihelia with a special configuration of position, mean motions that are commensurable, a ratio of eccentricities that is related to the ratio of the semimajor axes; suppose further that the nearby periodic solutions are such that their perihelia are immovable; then a continuous set of periodic solutions exists with a continuous set of values of μ differing slightly from zero, which Poincaré called periodic solutions of the second sort.

Another approach to the character of the motion, when there is a small divisor in + jn' considers the corresponding periodic term with a long-periodic variation called libration. Brown (1911a, 1911b, 1911c, 1912; Brown and Shook, 1933) has studied the libration in asteroidal motion by his own ingenious method and Hagihara (1944) has based his discussion on Delaunay’s method used in the theory of perturbation. In view of the approximate character of such analytical theories, Hirayama (1918, 1928; Wilkens, 1927, 1930; Hirayama and Akiyama, 1937a, 1937b; Ura and Takenouchi, 1951; Kozai, 1952; 1953) undertook the computation of the libration by the method of special perturbation and verified the analytical theory of Brown (1912).

These methods have shown that the presence of small divisors in the ordinary perturbation theory does not make the situation worse as to the stability of the motion. On the contrary, the commensurability of the mean motions strengthens the durability and the tendency to keep the existing configuration, provided that the perihelia are suitably located to produce a stable configuration.

We may now summarize the main conclusion of the present discussion. The stability of a satellite system is not weakened but strengthened by the presence of commensurable mean motions, provided that the relative configuration is the one which is stable. But, if the system contains many members, and the positions of the pericenters and the longitudes of the satellites in their orbits are arbitrary, as for Saturn’s rings and, presumably, the asteroidal rings, then gaps could occur in the distribution of the mean motions, because of the accumulated effects of disturbing actions by the neighboring small masses passing close by. In fact, there are very few members in the Hilda group of asteroids and only one member in the Thule group. Until a detailed precise theory is worked out on the motion of such characteristic asteroids, the present point of view (Hagihara, 1957) may serve as a tentative theoretical explanation of the occurrence of gaps in the distribution of the mean motions of asteroids and the gaps in Saturn’s rings. We await new evidence such as that to be provided by the photometric study of asteroids by Kuiper and his colleagues, and the infrared study of Saturn’s rings by Whipple and his colleagues, which will help in clearing up this difficult problem.

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