In Table II. the unit of column a is the earth's mean distance from
the sun, or ninety-three million miles. On this scale the breadth of the
zone is 1.8196. Or, if we estimate the breadth from the perihelion of
Æthra (1.612) to the aphelion of Andromache (4.726), it is 3.114,—more
than three times the radius of the earth's orbit. A very remarkable
characteristic of the group is the interlacing or intertwining of
orbits. "One fact," says D'Arrest, "seems above all to confirm the idea
of an intimate relation between all the minor planets; it is, that if
their orbits are figured under the form of material rings, these rings
will be found so entangled that it would be possible, by means of one
among them taken at hazard, to lift up all the rest."[6] Our present
knowledge of this wide and complicated cluster is the result of a vast
amount, not only of observations, but also of mathematical labor. In
view, however, of the perturbations of these bodies by the larger
planets, and especially by Jupiter, it is easy to see that the
discussion of their motions must present a field of investigation
practically boundless.
While the known minor planets were but few in number the theory of
Olbers in regard to their origin seemed highly probable; it has,
however, been completely disproved by more recent discoveries. The
breadth of the zone being now greater than the distance of Mars from the
sun, it is no more probable that the asteroids were produced by the
disruption of a single planet than that Mercury, Venus, the earth, and
Mars originated in a similar manner.
In taking a general view of the solar system we cannot fail to be struck
by the remarkable fact that Jupiter, whose mass is much greater than
that of all other planets united, should be immediately succeeded by a
region so nearly destitute of matter as the zone of asteroids. Leverrier
inferred from the motion of Mars's perihelion that the mass of Jupiter
is at least twelve hundred times greater than that of all the planets in
the asteroid ring. The fact is suggestive of Jupiter's dominating energy
in the evolution of the asteroid system. We find also something
analogous among the satellites of Jupiter, Saturn, and Uranus. Jupiter's
third satellite, the largest of the number, is nearly four times greater
than the second. Immediately within the orbit of Titan, the largest
satellite of Saturn, occurs a wide hiatus, and the volume of the next
interior satellite is to that of Titan in the ratio of one to
twenty-one. In the Uranian system the widest interval between adjacent
orbits is just within the orbit of the bright satellite, Titania.
The foregoing facts suggest the inquiry, What effect would be produced
by a large planet on interior masses abandoned by a central spheroid? As
the phenomena in all instances would be of the same nature, we will
consider a single case,—that of Jupiter and the asteroids.
The powerful mass of the exterior body would produce great perturbations
of the neighboring small planets abandoned at the solar equator. The
disturbed orbits, in some cases, would thus attain considerable
eccentricity, so that the matter moving in them would, in perihelion, be
brought in contact with the equatorial parts of the central body, and
thus become reunited with it.[7] The extreme rarity of the zone between
Mars and Jupiter, regarded as a single ring, is thus accounted for in
accordance with known dynamical laws.
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.
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]
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:
| Order. | Ratio. | Distance. |
|---|
| First | 1⁄2 | 3.277 |
| Second | 1⁄3, 3⁄5 |
⎧
⎨
⎩2.50
3.70 |
| Third | 2⁄5, 4⁄7, 5⁄8 |
⎧
⎨
⎩2.82
3.58
3.80 |
| Fourth | 3⁄7, 5⁄9, 7⁄11 |
⎧
⎨
⎩2.95
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 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.
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
orbit 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 mass. As the least distance of Æthra is but 1.587, its
orbit 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.
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 cluster of
minor planets?
The least eccentric orbit in the group is that of Philomela (196); the
most eccentric that of Æthra (132). Comparing these with the orbit of
the second comet of 1867 we have
| The | eccentricity | of | Philomela = 0.01 |
| " | " | " | Æthra = 0.38 |
| " | " | " | Comet II. 1867 (ret. in 1885) = 0.41 |
The orbit of Æthra, 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.
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
Massalia, 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 |
2630 |
0.353 |
| Euphrosyne |
2629 |
0.228 |
| Anna |
2524 |
0.263 |
| Gallia |
2521 |
0.185 |
| Æthra |
250 |
0.380 |
| Eukrate |
2457 |
0.236 |
| Eva |
2425 |
0.347 |
| Niobe |
2319 |
0.173 |
| Eunice |
2317 |
0.129 |
| Electra |
2255 |
0.208 |
| Idunna |
2231 |
0.164 |
| Phocea |
2135 |
0.255 |
| Artemis |
2131 |
0.175 |
| Bertha |
2059 |
0.085 |
| Henrietta |
2047 |
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.
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.
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.
Since, according to Kepler's third law, the periods of planets depend
upon their mean distances, the clustering 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 | " |
| " | 1600 | " | 1700 | " | 54 | " |
| " | 1700 | " | 1800 | " | 20 | " |
| " | 1800 | " | 1900 | " | 13 | " |
| " | 1900 | " | 2000 | " | 19 | " |
| " | 2000 | " | 2100 | " | 33 | " |
| " | 2100 | " | 2200 | " | 2 | " |
| " | 2200 | " | 2300 | " | 2 | " |
| " | 2300 | " | 2400 | " | 8 | " |
| " | 2400 | " | 2800 | " | 0 | " |
| " | 2800 | " | 2900 | " | 2 | " |
The period of Hilda (153) is more than two and a half times that of
Medusa (149). This is greater than the ratio of Saturn's period to that
of Jupiter. The maximum observed between 2000 and 2100 days corresponds
to the space immediately interior to chasm I. on a previous page, that
between 1300 and 1400 to the space interior to the second, and that
between 1500 and 1700 to the part of the zone within the fourth gap. The
table presents quite numerous instances of approximate equality; in
forty-three cases the periods differing less than twenty-four hours. It
is impossible to say, however, whether any two of these periods are
exactly equal. In cases of a very close approach two asteroids,
notwithstanding their small mass, may exert upon each other quite
sensible perturbations.
But four minor planets had been discovered when Laplace issued his last
edition of the "Système du Monde." The author, in his celebrated seventh
note in the second volume of that work, explained the origin of these
bodies by assuming that the primitive ring from which they were formed,
instead of collecting into a single sphere, as in the case of the major
planets, broke up into four distinct masses. But the form and extent of
the cluster as now known, as well as the observed facts bearing on the
constitution of Saturn's ring, seem to require a modification of
Laplace's theory. Throughout the greater part of the interval between
Mars and Jupiter an almost continuous succession of small planetary
masses—not nebulous rings—appears to have been abandoned at the solar
equator. The entire cluster, distributed throughout a space whose outer
radius exceeds the inner by more than two hundred millions of miles,
could not have originated, as supposed by Laplace, in a single nebulous
zone the different parts of which revolved with the same angular
velocity. The following considerations may furnish a suggestion in
regard to the mode in which these bodies were separated from the equator
of the solar nebula.
(a) The perihelion distance of Jupiter is 4.950, while the aphelion
distance of Hilda is 4.623. If, therefore, the sun once extended to the
latter, the central attraction of its mass on an equatorial particle was
but five times greater than Jupiter's perihelion influence on the same.
It is easy to see, then, that this "giant planet" would produce enormous
tidal elevations in the solar mass.
(b) The centrifugal force would be greatest at the crest of this tidal
wave.
(c) Three periods of solar revolution were then about equal to two
periods of Jupiter. The disturbing influence of the planet would
therefore be increased at each conjunction with this protuberance. The
ultimate separation (not of a ring but) of a planetary mass would be the
probable result of these combined and accumulating forces.
Observations of some minor planets have indicated a variation of their
apparent magnitudes. Frigga, discovered by Dr. Peters in 1862, was
observed at the next opposition in 1864; but after this it could not be
found till 1868, when it was picked up by Professor Tietjen. From the
latter date its light seems again to have diminished, as all efforts to
re-observe it were unsuccessful till 1879. According to Dr. Peters, the
change in brightness during the period of observation in that year was
greater than that due to its varying distance. No explanation of such
changes has yet been offered. It has been justly remarked, however, that
"the length of the period of the fluctuation does not allow of our
connecting it with the rotation of the planet."
At the meeting of the American Association for the Advancement of
Science in 1884, Professor Mark W. Harrington, of Ann Arbor, Michigan,
presented a paper in which the elements of the asteroid system were
considered on the principle of averages. Two hundred and thirty orbits,
all that had then been determined, were employed in the discussion.
Professor Harrington supposes two planes to intersect the ecliptic at
right angles; one passing through the equinoxes and the other through
the solstices. These planes will intersect the asteroidal orbits, each
in four points, and "the mean intersection at each solstice and equinox
may be considered a point in the average orbit."
In 1883 the Royal Academy of Denmark offered its gold medal for a
statistical examination of the orbits of the small planets considered as
parts of a ring around the sun. The prize was awarded in 1885 to M.
Svedstrup, of Copenhagen. The results obtained by these astronomers
severally are as follows:
| | Harrington. | Svedstrup. |
|---|
| Longitude of perihelion |
14°39´ |
101°48´ |
| Longitude of ascending node |
11356 |
13327 |
| Inclination |
10 |
66 |
| Eccentricity |
0.0448 |
0.0281 |
| Mean distance |
2.7010 |
2.6435 |
These elements, with the exception of the first, are in reasonable
harmony.
Did comets originate within the solar system, or do they enter it from
without? Laplace assigned them an extraneous origin, and his view is
adopted by many eminent astronomers. With all due respect to the
authority of great names, the present writer has not wholly abandoned
the theory that some comets of short period are specially related to the
minor planets. According to M. Lehmann-Filhès, the eccentricity of the
third comet of 1884, before its last close approach to Jupiter, was only
0.2787.[12] This is exceeded by that of twelve known minor planets. Its
mean distance before this great perturbation was about 4.61, and six of
its periods were nearly equal to five of Jupiter's,—a commensurability
of the first order. According to Hind and Krueger, the great
transformation of its orbit by Jupiter's influence occurred in May,
1875. It had previously been an asteroid too remote to be seen even in
perihelion. This body was discovered by M. Wolf, at Heidelberg,
September 17, 1884. Its present period is about six and one-half years.
The perihelion distance of the comet 1867 II. at its return in 1885 was
2.073; its aphelion is 4.897; so that its entire path, like those of the
asteroids, is included between the orbits of Mars and Jupiter. Its
eccentricity, as we have seen, is little greater than that of Æthra, and
its period, inclination, and longitude of the ascending node are
approximately the same with those of Sylvia, the eighty-seventh minor
planet. In short, this comet may be regarded as an asteroid whose
elements have been considerably modified by perturbation.
It has been stated that the gap at the distance 3.277 is the only one
corresponding to the first order of commensurability. The distance
3.9683, where an asteroid's period would be two-thirds of Jupiter's, is
immediately beyond the outer limit of the cluster as at present known;
the mean distance of Hilda being 3.9523. The discovery of new members
beyond this limit is by no means improbable. Should a minor planet at
the mean distance 3.9683 attain an eccentricity of 0.3—and this is less
than that of eleven now known—its aphelion would be more remote than
the perihelion of Jupiter. Such an orbit might not be stable. Its form
and extent might be greatly changed after the manner of Lexell's comet.
Two well-known comets, Faye's and Denning's, have periods approximately
equal to two-thirds of Jupiter's. In like manner the periods of
D'Arrest's and Biela's comets correspond to the hiatus at 3.51, and that
of 1867 II. to that at 3.277.
Of the thirteen telescopic comets whose periods correspond to mean
distances within the asteroid zone, all have direct motion; all have
inclinations similar to those of the minor planets; and their
eccentricities are generally less than those of other known comets. Have
these facts any significance in regard to their origin?
