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...o...b..t being the unit of distance:

+--------+----------------+-----------+

Order.

Ratio.

Distance.

+--------+----------------+-----------+

First

1/2

3.277

Second

1/3, 3/5

{ 2.50

{ 3.70

{ 2.82

Third

2/5, 4/7, 5/8

{ 3.58

{ 3.80

{ 2.95

Fourth

3/7, 5/9, 7/11

{ 3.51

{ 3.85

+--------+----------------+-----------+

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 s.p.a.ce 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 s.p.a.ce of equal breadth, containing the distance 2.82, there is but one. This is No. 188, Menippe, whose elements are still somewhat uncertain. The s.p.a.ce 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...o...b..t. 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.

The FACT of the existence of well-defined gaps in the designated parts of the ring has been clearly established. But the theory of probability applied in a single instance gives, as we have seen, but one chance in 300,000,000,000 that the distribution is accidental. This improbability is increased many millions of times when we include all the gaps corresponding to simple cases of commensurability. We conclude, therefore, that those discontinuities cannot be referred to a chance arrangement. What, then, was their physical cause? and what has become of the eliminated asteroids?

What was said in regard to the limits of perihelion distance may suggest a possible answer to these interesting questions. The doctrine of the sun's gradual contraction is now accepted by a majority of astronomers.

According to this theory the solar radius at an epoch not relatively remote was twice what it is at present. At anterior stages it was 0.4, 1.0, 2.0,[11] etc. At the first mentioned the comets of 1843 and 1668, as well as several others, could not have been moving in their present orbits, since in perihelion they must have plunged into the sun. At the second, Encke's comet and all others with perihelia within Mercury's...o...b..t would have shared a similar fate. At the last named all asteroids with perihelion distances less than two would have been re-incorporated with the central ma.s.s. As the least distance of aethra is but 1.587, its...o...b..t could not have had its present form and dimensions when the radius of the solar nebula was equal to the aphelion distance of Mars (1.665).

It is easy to see, therefore, that in those parts of the ring where Jupiter would produce extraordinary disturbance the formation of chasms would be very highly probable.

5. Relations between certain Adjacent Orbits.

The distances, periods, inclinations, and eccentricities of Hilda and Ismene, the outermost pair of the group, are very nearly identical. It is a remarkable fact, however, that the longitudes of their perihelia differ by almost exactly 180. Did they separate at nearly the same time from opposite sides of the solar nebula? Other adjacent pairs having a striking similarity between their orbital elements are Sirona and Ceres, Fides and Maia, Fortuna and Eurynome, and perhaps a few others. Such coincidences can hardly be accidental. Original asteroids, soon after their detachment from the central body, may have been separated by the sun's unequal attraction on their parts. Such divisions have occurred in the world of comets, why not also in the cl.u.s.ter of minor planets?

6. The Eccentricities.

The least eccentric orbit in the group is that of Philomela (196); the most eccentric that of aethra (132). Comparing these with the orbit of the second comet of 1867 we have

The eccentricity of Philomela = 0.01 " " " aethra = 0.38 " " " Comet II. 1867 (ret. in 1885) = 0.41

The orbit of aethra, it is seen, more nearly resembles the last than the first. It might perhaps be called the connecting-link between planetary and cometary orbits.

The average eccentricity of the two hundred and sixty-eight asteroids whose orbits have been calculated is 0.1569. As with the orbits of the old planets, the eccentricities vary within moderate limits, some increasing, others diminishing. The average, however, will probably remain very nearly the same. An inspection of the table shows that while but one orbit is less eccentric than the earth's, sixty-nine depart more from the circular form than the orbit of Mercury. These eccentricities seem to indicate that the forms of the asteroidal orbits were influenced by special causes. It may be worthy of remark that the eccentricity does not appear to vary with the distance from the sun, being nearly the same for the interior members of the zone as for the exterior.

7. The Inclinations.

The inclinations in Table II. are thus distributed:

From 0 to 4 70 " 4 to 8 83 " 8 to 12 59 " 12 to 16 32 " 16 to 20 8 " 20 to 24 8 " 24 to 28 7 " 28 to 32 0 above 32 1

One hundred and fifty-four, considerably more than half, have inclinations between 3 and 11, and the mean of the whole number is about 8,--slightly greater than the inclination of Mercury, or that of the plane of the sun's equator. The smallest inclination, that of Ma.s.salia, is 0 41', and the largest, that of Pallas, is about 35.

Sixteen minor planets, or six per cent. of the whole number, have inclinations exceeding 20. Does any relation obtain between high inclinations and great eccentricities? These elements in the cases named above are as follows:

+------------+--------------+--------------+

Asteroid.

Inclination.

Eccentricity.

+------------+--------------+--------------+

Pallas

34 42'

0.238

Istria

26 30

0.353

Euphrosyne

26 29

0.228

Anna

25 24

0.263

Gallia

25 21

0.185

aethra

25 0

0.380

Eukrate

24 57

0.236

Eva

24 25

0.347

Niobe

23 19

0.173

Eunice

23 17

0.129

Electra

22 55

0.208

Idunna

22 31

0.164

Phocea

21 35

0.255

Artemis

21 31

0.175

Bertha

20 59

0.085

Henrietta

20 47

0.260

+------------+--------------+--------------+

This comparison shows the most inclined orbits to be also very eccentric; Bertha and Eunice being the only exceptions in the foregoing list. On the other hand, however, we find over fifty asteroids with eccentricities exceeding 0.20 whose inclinations are not extraordinary.

The dependence of the phenomena on a common cause can, therefore, hardly be admitted. At least, the forces which produced the great eccentricity failed in a majority of cases to cause high inclinations.

8. Longitudes of the Perihelia.

The perihelia of the asteroidal orbits are very unequally distributed; one hundred and thirty-six--a majority of the whole number determined--being within the 120 from longitude 290 50' to 59 50'.

The maximum occurs between 30 and 60, where thirty-five perihelia are found in 30 of longitude.

9. Distribution of the Ascending Nodes.

An inspection of the column containing the longitudes of the ascending nodes, in Table II., indicates two well-marked maxima, each extending about sixty degrees, in opposite parts of the heavens.

I. From 310 to 10, containing 61 ascending nodes.

II. " 120 to 180, " 59 " "

--- Making in 120 120 " "

A uniform distribution would give 89. An arc of 84--from 46 to 130--contains the ascending nodes of all the old planets. This arc, it will be noticed, is not coincident with either of the maxima found for the asteroids.

10. The Periods.

Since, according to Kepler's third law, the periods of planets depend upon their mean distances, the cl.u.s.tering tendency found in the latter must obtain also in the former. This marked irregularity in the order of periods is seen below.

Between 1100 and 1200 days 6 periods.

" 1200 " 1300 " 7 "

" 1300 " 1400 " 43 "

" 1400 " 1500 " 13 "

" 1500 " 1600 " 46 "