Smithsonian

Contributions to Astrophysics

VOLUME 5, NUMBER 6

GAPS IN THE DISTRIBUTION

OF ASTEROIDS

by YUSUKE HAGIHARA

SMITHSONIAN INSTITUTION

Washington, D.C.

1961
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Washington 25, D.C. - Price 20 cents
Gaps in the Distribution of Asteroids

By YUSUKE HAGIHARA

 1

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,

^ = S Bti S cos (i\ +j\' +DtJ), (1)

\'=n't+€f.

Here n is the mean motion, « the mean longitude at the epoch, Da a function of the elements,

Bf} 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 righthand side of such equations, because the righthand 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,

^ sm fc+jX'+A,). (2)

1

 Smithsonian Astrophysical Observatory.

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 2ir/(in-\-jn'). 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.

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'^Q 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.

59
60 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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 Glyd6n

(1893), of Delaunay (1860, 1867), or a periodic

solution due to PoincarS (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 Poincare (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.
GAPS IN THE DISTRIBUTION OF ASTEROIDS 61

• 1 = •

ro|eo -

u>| = •

*lf - •

io|i n .

- IS
62 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

SATURN S RINGS

RADII IN SECONDS OF ARC AT DISTANCE 9.5388 ASTRONOMIC UNITS

12 13 14 15 16 17 18 19

1305 13.60 14.04 14.74 15.32 16.98 1856 | I9.S9

.5.691 | | 18.97 |

DISTANCES AT WHICH PERIODS ARE COMMENSURATE WITH THOSE OF MIMAS AND ENCELADUS

I I I I I I I I I II I

I I 1 E 3. 2. 3. 4 5 I 2 E i LE

4" 34857 9 II 2 55 ?

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 7i!=203?48895528, w2=101?37472396,

7ia=50?31760833, so that %—2n2=0?73950736,

Tia—2n3=0?73950730, and, even more remarkable, nx—3n2+2n3=0?00000006 (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 Alfve'n

(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 Poincare"'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.
NO. 6 GAPS IN THE DISTRIBUTION OF ASTEROIDS 63

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

Poincare'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; Holder, 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; Poincare",

1890, 1892, 1902a, 1902b; Andoyer, 1903;

Schwarzschild, 1903; Wilkens, 1913; Heinrich,

1912, 1922, 1925, and others). Poincare" (1890,

1892) has proved the existenee of periodic

solutions for a continuous set of values of a

parameter n in the neighborhood of n=0 for

the case in which a generating periodic solution

exists for n=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 p differing slightly from zero,

which Poincare" 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 bis discussion

on Delaunay s method used in the theory of

perturbation. In view of the approyimate

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.
64 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

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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