Popular Science Monthly/Volume 39/June 1891/Questions Concerning the Minor Planets
By M. F. F. TISSERAND, of the Institute of France.
KEPLER, having found a break in the continuity of the mean distances of the planets from the sun, boldly filled it by supposing a new planet between Mars and Jupiter. The publication of Bode's empiric law in 1772 helped confirm the ideas of Kepler, and fixed the distance of the hypothetical planet at 2·8 times that of the earth. A new authority was given to this conclusion after the discovery of Uranus by William Herschel in 1781. The calculations of Lexell and Laplace showed in fact that Uranus's distance might have been furnished in advance, with a near approach to exactness, by Bode's law. At a conference. held in Gotha in 1796, Lalande and De Zach proposed to search for the unknown planet, and to divide the labor among twenty-four astronomers, each of whom should examine an hour of the zodiac.
On the first day of this century — that is, January 1, 1801 — Piazzi discovered at Palermo a star which he took at first for a little comet and observed several times till the 11th of February following, when illness stopped his work. Bode was the first to recognize that the star could not be a comet, and thought that in Ceres Piazzi had found the planet suspected by Kepler. When Piazzi had become well again, he did not know where to look for Ceres. It was to be sought for toward the end of the year, after coming out from the glare of the sun, but no data were at hand for determining its position except the geocentric arc of 3° which it had traversed during the forty days it had been under observation.
Here Gauss came to the rescue; he was then twenty-four years of age, and had had little or nothing to do with astronomical calculations, having been occupied chiefly with the higher arithmetic. In less than a month he invented an admirable method for calculating the elements of the elliptical orbit of Ceres and an ephemeris, by means of which Olbers found the star again on the first day of January, 1802. The mean distance of Ceres from the sun is 2·77. It corresponds exactly with Bode's law, and fills the gap, but with a very modest planet, having a diameter of only about 200 miles. 196 THE POPULAR SCIENCE MONTHLY.
Olbers's discovery, on the 28th of March, 1802, of a second planet, Pallas, revolving around the sun at the same mean dis- tance as Ceres, presented the question under a new aspect. Gauss's calculation showed that Ceres and Pallas might, in time, come to pass very near each other on the line of intersection described as A B of the planes of their several orbits. Olbers was thus led to think that the two little bodies might be fragments of a greater planet which had been broken up by an internal commo- tion. If this were the case, there would probably be other frag- ments, the orbits of which would pass the line A B, so that by watching the two points A and B, where this line strikes the celestial sphere, chances might occur of seeing the fragments of the primitive planet pass. Eventually Harding found Juno near A in 1804, and Olbers discovered Vesta near B in 1807.
Further researches, carried on by Olbers till 1816, brought no result ; and it was not till 1845 that a fifth body, still smaller than the other four, was discovered by Encke. After this time, dis- coveries became frequent and regular, till now 299 are known.* But the size of the new stars keeps getting smaller : the first four were between the sixth and the eighth magnitudes ; the two dis- covered by Encke were of the ninth ; and those which are now discovered from time to time seldom exceed the thirteenth mag- nitude. William Herschel, in consideration of the small size of these bodies, and hardly regarding them as sufficient to fill the place of a planet, thought it more fitting to call them asteroids than planets.
- A survey of the orbits of the asteroids as a whole will help us to gain clearer ideas respecting them, and may bring out a few simple relations that will cast some light on the origin of the bod- ies. The supposition of Olbers is not sustained. Prof. Newcomb, having studied the orbits of the first forty asteroids, found that, as they move to-day, they are far from passing in the same line. Even the hypothesis that the geometrical condition supposed may once have existed, but has been changed by the perturbations caused by the attractions of the other planets, is contradicted by the calculations.
But this hypothesis, though it must be abandoned, has the credit of having provoked the discovery of Juno and Vesta, and of having suggested Lagrange's theory of the origin of comets, to which M. Faye has added a number of curious speculations, and by the aid of which we may, perhaps, some day find an expla- nation of the origin of meteors. The smallest mean distance from the sun of the known asteroids, 149, is 2'13 (times the mean distance of the earth) ; the greatest, that of 279, 4*26 ; the corre-
- Since increased to 308.
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sponding periods of rotation are 3"11 and 8*81 years. They both revolve, therefore, on either hand, outside of the limit, 2'8, as- signed "by Bode's law. If we take account of the eccentricities, 131 comes within the distance 1*31 from the sun, and 175 goes as far as 4'73 from it. The asteroids, therefore, perform their move- ments within a very extended zone, and the ensemble of their positions forms a kind of ring, the breadth of which is more than three times the distance of the earth from the sun. A comparison of the eccentricities shows a mean of 0*15, much higher than the corresponding mean, 0'8G, of the older planets. This, too, indicates that there are notable differences in the conditions of their forma- tion. The difference is still more striking in the inclinations of their orbits. The mean of the inclinations is 8, a little higher than that of Mercury and that of the equator of the sun. But of two hundred and ninety-three asteroids, there are seventeen that have inclinations higher than 20. When their mean distances are also considered, these seventeen seem to arrange themselves in two groups, around the distances 2*75 and 3'15 respectively ; but this appears to be only because asteroids are more numerous in those two regions. It is also noteworthy that the much inclined orbits are usually also very eccentric, but the converse does not hold. A great eccentricity does not seem to involve of necessity a great inclination.
The question may arise whether the asteroids may not all at first have been placed in orbits of slight eccentricity and slightly inclined to the plane of the ecliptic, and their eccentricities and inclinations then have increased considerably at least those of some of them under the influence of perturbations. The re- searches of Lagrange and Laplace have shown that the eccen- tricities and inclinations of the old planets could vary under the influence of their mutual attractions only within narrow limits. But this result is established only for determined distances of the planets from the sun. Is it sure in advance for other inter- mediate positions, and particularly for the space in which the asteroids move ? Leverrier asked this question, and made the curious remark on the subject that there exists, between Jupiter and the sun, a region in which, if we place a small mass, in an orbit but little inclined to that of Jupiter, that mass will leave its primary orbit and attain large inclinations to the plane of the orbit of Jupiter and to that of Saturn. It is remarkable that this position is at nearly twice the distance of the earth from the sun that is, near the interior edge of the zone in which the minor planets are found. This fact, interesting as it is in itself, does not explain the large inclinations that have been determined at the distances 2"15 and 3'15, which are very different from those in- dicated by Leverrier as those at which a certain amount of in-
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clination may be produced. I made a similar research a few years ago concerning eccentricities, and found a region of in- stability corresponding with a still smaller distance from the sun, 1*83. The perturbations caused by Jupiter and Saturn must there- fore be regarded as insufficient to explain the considerable values of the eccentricities and inclinations of so many asteroids. These values were never small, and consequently the conditions in which Laplace's nebula existed were not the same at the times of the formation of the asteroids as they were when the old planets were fixed. An interesting cosmological question is presented here, and the accumulation of new discoveries of asteroids can only facilitate its solution.
The distribution of the asteroids, according to their mean dis- tances from the sun, or (which amounts to the same thing) accord- ing to their mean diurnal motions,* offers some curious facts. A table showing this factor for all the asteroids but three presents the striking feature of accumulations of minor planets about the mean motions 640", 780*, and 815", with which correspond the mean distances 3'13, 275, and 2*67. Two principal voids may also be recognized, about 600" and 900", or the mean distances 3*27 and 2*50 from the sun. The mean diurnal motion of Jupiter is 299*42" or very nearly 300". The voids that have been pointed out thus correspond with regions where the mean motion of the planet would be exactly double or triple that of Jupiter. There are other less well-defined voids, in which the relation of the mean motions to that of Jupiter, instead of being 2 or 3, would be represented by one of the fractions f , f , f , and . Prof. Kirkwood first brought out this fact in 1866, and generalized it by saying that the parts of the zone of the asteroids in which exists a simple relation of commensurability between the period of revolution of a minor planet and that of Jupiter are represented by gaps like the inter- vals between the rings of Saturn. "We remark in addition that the gaps are less sharply marked than in the case of the rings of Saturn, in that after a void the number of asteroids does not increase suddenly, but gradually, till it regains its normal value.
Can the voids be accounted for by the theory of perturbations ? We should have a very simple explanation if we could show that two planets, the durations of whose revolutions are in a simple commensurable relation, exist for that reason in an eminently un- stable condition which they are liable to abandon at any moment. If these conditions are realized, the usual theory of perturbations defaults ; but we can not conclude that instability would result from it. Recent calculations seem rather to lead to opposite con-
��* The mean diurnal motion is the quotient of the division of the number of seconds (1,290,000) in the circumference by the number of days of the planet's revolution.
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elusions. Gauss, communicating to Bessel in 1812 the discovery of the ratio of ^ between the mean motions of Pallas and of Jupiter, said that that value ought to be realized more and more exactly" under the influence of the attraction of Jupiter, in the same way as the motions of translation and rotation of the moon are equalized. Newcomb has expressed the same opinion, to which he was led by his studies of the system of Saturn, saying that while one would think that, in the case of movements ab- solutely commensurable, perturbations would not fail to grow be- yond limits to the point of compromising the stability of the system, the consequence is not a necessary one; there would probably be only oscillations more or less irregular, but equi- librium would be re-established incessantly. The labors of M. Gylde'n and my own personal researches tend to the same con- clusion.
It is therefore probable that, if the voids had not existed in the beginning, further perturbations by Jupiter would not have been sufficient to produce them ; they without doubt already ex- isted, immediately on the formation of the asteroids, and give another reason for considering the question they present as of primary interest from a cosmogonical point of view. It is of no less interest in the matter of the celestial mechanism, for it cor- responds, as we have said, to a case in which the old methods de- fault, and which has instigated the most interesting studies of the period. Laplace has already considered it in his theory of the satellites of Jupiter, but the asteroids present it to us under conditions which make its solution still more difficult.
The minor planets situated at the outer limit of the ring are interesting from several points of view. Some of them furnish a kind of transition between the asteroids and some of the peri- odical comets. Thus, the orbit of 175 is very like that of Tem- pers periodical comet. The . distinction between planets and comets, founded on the dissimilarity of their orbits, vanishes here. We have nothing left to distinguish them but their phys- ical aspect. The asteroids of which we are speaking, being near Jupiter, are always very distant from the earth. They must, therefore, appear small in proportion to their dimensions. It is possible that, by seeking with a strong enough glass, we shall find others ; and some of these may come in to corroborate the resemblances with the periodical comets. Planet 279, discov- ered two years ago by M. Palisa, is one of the most remarkable of the group of which we have just spoken. In 1912 it will come with the distance 1 of Jupiter, and will continue there for a considerable time. The attraction of Jupiter will then be more than one fiftieth as great as that of the sun. The calcula- tion of the perturbations promises to be interesting and difficult,
�� � and will make it possible to deduce the mass of Jupiter with great precision. The planets at the lower limit of the ring may, in case their orbits are very eccentric, come very near the earth even within a distance of 0*7 in which case it will be possible to determine their parallax accurately by observing it from two distant stations, as we do the moon. Thence we can deduce the parallax of the sun ; and this is one of the best methods within our reach of obtaining this fundamental element in astronomy.
We have said that it is impossible to connect all the asteroids with the rupture of a single planet ; but we can form groups of two planets whose orbits present curious resemblances which do not seem due to chance only. The most interesting group is formed of the planets 37 and 66. Their orbits are nearly equal ellipses, situated almost in the same plane, and differing only in the orientation of the major axes. This almost complete iden- tity of four of the five elements, which exists now, and will, ac- cording to the calculations, be maintained, can not be accidental. Many facts of this kind will not be needed to illustrate to us the origin and formation of the asteroids. Maia (66) was lost for fifteen years, till it was found again by the help of M. Schulhof's calculations. There are other similar groups, and more will probably be found.
I trust that this notice will show a rich harvest of interest- ing facts in prospective. To speak only of acquired results, it may be recollected that Gauss composed one of his finest works for the purpose of recovering Ceres. It was in seeking for a quick way of verifying Leverrier's numerical calculations upon the great inequality of Pallas that the illustrious Cauchy wrote admirable memoirs, from which great advantages are now de- rived for the theories of the older planets and for the most deli- cate points in the theory of the moon. It would be unjust, too, to forget the excellent labors of Hansen and Gylddn, which had the same origin.
The asteroids have also been the occasion of important advances in observation. The search for them has trained observers of the first order. The purpose of following them with greater facil- ity has led to the construction of powerful instruments, among which is the great meridian circle of the Observatory of Paris. The maps of the heavens and catalogues of stars have through them been made more nearly perfect. Among these the eclip- tic maps of MM. Henry deserve special mention. When these as- tronomers undertook to construct the maps across the milky way they were dismayed by the immensity of the work, and in- voked the aid of photography. The remarkable results they ob- tained have served as the point of departure for the enterprise of the photographic map of the sky. The Astro-Photographic Congress at Paris, in 1887, decided to photograph, the whole sky down to stars of the fourteenth magnitude. Could this enterprise have borne all the fruits it has if the planets of the thirteenth magnitude had been let pass unperceived?
For all these reasons, I think that the search for minor planets ought to be continued. It demands, indeed, considerable work in calculation; but that can be divided among several scientific establishments. The Bureau of Longitudes is disposed to do its part in the matter. — Translated for the Popular Science Monthly from the Revue Scientifique.