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For if this supposition be true, the planet would almost exactly have filled the gap between Mars and Jupiter, where, according to an empirical formula, much in vogue at that time, an unknown planet had been long suspected. Indeed a society of German astronomers had been already forined, to search for this suspected member of our solar system.

As a corroboration of this hypothesis, he referred to the circumstance that both Pallas and Ceres seemed to vary considerably in magnitude, which he explained by the conjecture that these bodies were not round, but of very irregular figure.

“This idea,” he wrote to Zach,* “has at least one great advantage over some other hypotheses, that it can be soon tested. For if it is true, we shall be able to find still more fragments of the shattered planet, and the easier still, because all those fragments, which describe an elliptical orbit around the sun, must pass the descending node of Pallas upon the orbit of Ceres."

The discovery of Juno soon after, and not far from the apparent place of this node, seemed to afford a strong confirmation of Olbers's hypothesis, and Zach immediately begant to consider it a tested and confirmed theory. A simple calculation gives however the following results:

True anomaly of Ceres. In 88 of Pallas on the Ceres-orbit, - - 220° 9' 56.6 66 66 66 Juno 66 66 66 66 . - 242 2 18 3 In 8 of Ceres on the Pallas-orbit, - - 249° 32' 36.0 “ * “ Juno “ “ “ “ - - 232 33 1.9

In October, 1804, Olbers wrotef Zach again that the distance between the two nodes on the Ceres-orbit (the calculations of Gauss gave 24° at that time) was in no wise discordant with his hypothesis ; that as a necessary consequence of the very different inclinations to the plane of Jupiter's orbit, the motion of their lines of nodes produced by Jupiter's attraction must be very different from the motion of their apsidal lines, which would result from the same attraction; that still farther, inasmuch as these orbits have nearly equal major-axes, but very unequal excentricities, they must have cut one another at some former time in their node upon the Ceres-orbit. Indeed if we assume according to the determinations of Oriani, the annual motion of the aphelion for Pallas = 106":1, and for Ceres = 120“.9, and consider the nodes as sidereally at rest, and the inclinations constant, it results that a section of the Ceres and Pallas orbits in the above mentioned node, must have taken place 7463 years before, and in 282 years again occur. In the descending node the same would happen in 925 years.

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Later, after Gauss had computed the secular variations of Ceres and Pallas, Encke, at his suggestion, made farther investigations for the purpose of determining whether the distance of the two orbits at the node, were on the increase or decrease. The result of his computations was, that the orbits are approaching one another.

Encke found* the following radius-vectors.

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6 According to this,” to use the wordst of Gauss, “a section in the node would actually take place about the year 3397, which may be considered, at any rate, as an approximation to the truth.

To be sure a section must also at some former period have occurred; but from the progression of the numbers in the third and sixth columns, we can at least conclude that this can only have been many thousands of years before. If we therefore adopt the hypothesis of Dr. Olbers concerning the origin of the new planets, the occurrence must have taken place at an epoch, for us at present immeasurably long before the times to which history reaches back.”

It is also evident from the foregoing table, that the distance between the two orbits in the descending node upon the Pallas-orbit, is at present on the decrease.

At any rate we are justified in concluding, without any farther computation of the secular variations, that at the last time that a section of the Pallas and Ceres orbits took place, neither of the nodes of Juno coincided with the node of Pallas. Although the subsequent discovery of five more asteroids has most certainly confirmed the conjecture of Olbers, that still more similar bodies would be found, it has nevertheless almost immeasurably multiplied the difficulties in our way ;-if indeed it has not rendered it absolutely impossible to assign a period, by computation of the secular variations of the apsidal and nodal lines of these eight orbits, when at the same time all the nodal lines upon one of the orbits coincided, and all the radius vectors were equal.

In this place belongs, perhaps, the remark, that as far as we are yet able to determine the orbit of Flora, the aphelion of this planet falls within the perihelion distance of Ceres.

On the other hand, it must be mentioned that all the nodes upon the Ceres-orbit fall within a single quadrant.

The following table gives the distances of the several nodes upon the orbit of Ceres from one another. These distances are

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Juno,
Vesta,

Flora,

reckoned in heliocentric arcs upon the orbit, and counted from the Hebe-node, inasmuch as the latter lies nearest to the perihelion of Ceres, corresponding to a true anomaly of only 350 20'.

Distances from the node of

Hebe on the Ceres-orbit. Pallas,

4° 49' 26 42

43 31 Astræa,

47 55

50 18 Iris,

77 46 These interesting considerations seemed to me to make it worth while to make still more accurate and extensive investigations concerning the relative position of the asteroidal orbits. I have, therefore, for every pair of the eight known to us, i. e., for twenty-eight combinations, calculated the radius vectors in each node.

The elements of which I have made use are, for the four older asteroids, the osculating elements for the epoch nearest to the 1st January, 1848, which Dr. Bremiker in Berlin has computed. I am indebted for them to the kindness of Professor Gauss.

For the four newly discovered, I have selected those elements which, of all known to me, satisfy all observations best. These are for Astræa and for Hebe, the orbits* of Hrn. D’Arrest in Berlin; for Iris, the excellent orbit of Prof. Goldschmidt in Göttingen; and for Flora, an orbit lately published by myself.

The longitudes are referred to the mean Equinox of Jan. 1, 1848. The notation I have used, is that of the Theor. Motus.

Elements of the Orbils of the Asteroids.

Epoche. M 18 - 8 i I Pallas, 1848, March 4.0 24 54 234 31 26 35-1 142 42 12-334 34 31:1 13 54 48.9 768-8858 Hebe, 1847, July 10-0 274 54 2:

6123 36 42:) 133 40 44.8 14 44 25-311 31 11:4 942:3754 Juno, 1847, July 9.5 253 6 21 116 35 40170 53 52.0 13 2 39.314 42 19.6 812-7012 Ceres, 1848, March 12-0 21 4 0:5 293 28 147 80 47 17.910 37 13.1 4 24 56.81 770.9866 Vesta, 1847, April 4.0 310 46 14.7212 16 15.4 103 22 1:3 7 8 30 3 5 7 21.5 977.9481 Flora, 11848, Jan. 1:0 35 53 32:0 77 26 49.1 110 18 50:8 5 52 55.9 9 1 36.9 1086.1100 Iris, 1847, Sept.. 1.C 298 16 372 218 18 35 6.259 45 19 6 5 28 10:9 13 20 50 1 963-4498 Astræa, 1847, March 16.0 63 30 49 3 6 031:4 141 29 29 2 5 19 17.1 10 49 55 6 857-8493

2.105304

I subjoin the following table of subsidiary quantities, because, as far as my knowledge extends, they are no where else to be found together. Per. Dist.

Period of rev. Aph. Dist.

in sidereal days. 3.438312

1685.55
1.936886
2.903557

1375.25
1.993166
3-349363

1594.68
2-553746
2.979793

1680.96
2150345
2.571997

1325.22
1.856244
2.547126

1193-25
1.834262
2.935335

1345-16 Astræa, .

2-092456
3.060943

1510.75
* Ast. Nachr., xxv, p. 321 ; xxvi, p. 147.

Pallas, .
Hebe,
Juno,
Ceres,
Vesta,
Flora,
Iris, .

From these elements I have employed the following table, where for the sake of simplicity and in order to have a definite rule, each orbit is referred to those other orbits whose inclination is inferior.

Upon the orbit of Astrea.
Ascending Node.

Descending Node.
Rad. veci. Rad. vect. Astrwa. Rad. vect. , Rad. vect. Asiræa.
Pallas-orbit, 2:306615

2.180479 3.005974

2-90203 Hebe . . 2.599472 2:092602 2.100824

3.060630 Juno . . 3.033050 2.807325 2:125046 2:230187 Ceres . . 2.775450 2.442815

2.725737 2.530106 Vesta 2.562481 2:407127

2.157039 2.569565 Flora . . 1.862868 2.480134

2.534758 2.491281 Iris . . 2.083886 2.993750 2:463162 2.125061

Pallas-orbit,
Hebe
Juno
Ceres
Vesta
Flora

Upon the orbit of Iris.
Ascending Node.

Descending Node.
Rad. vect. | Rad. vect. Iris. | Rad. vect. ī Rad. vect. Iris.
2.214686 2.585784 3.181677 2.003507
2:480293 2:193029 2.185698 2.326314
2.537652 2:420336

2.461760 2:115556 2 669673 1.914261 2.863094 2:751336 2.386301 2.427211 2299979 2:110331 2.003169 1.989275 2:314213 2-609881

Pallas-orbit, .
Hebe
Juno
Ceres
Vesta .

Upon the orbit of Flora.
Ascending Node.

Descending Node.
Rad. vect. 1 Rad. vect. Flora. Rad. vect. Rad. vect. Flora.
2.340702
2-477113

2 953261 1.895266
2.743165
2.341091

2.015498 1.982454
3.140343 2:529811 2.075370 1.865549
2.766548 1.872111 2:734377 2.517845
2 5648791 1.958076

2:155:343 2377464

Pallas-orbit,
Hebe.
Juno
Ceres.

Upon the orbit of Vesta.
Ascending Node.

Descending Node.
Rad. veci. | Rad. vect. Vesta. | Rad. vect. Rad. vect. Vesta.
2:361500 2 266614 2.920807 2:423312
2.801046 2335006

1.983355

2.349731 3.1985:36 2.210443 2.050712

2490982 2.791893 2-549792 2.710061 2.166114

Upon the orbit of Ceres.
Ascending Node.

Descending Node.
Rad. veci. | Rad. vect. Ceres. Rad. vect. 1 Rad. vect. Cercs.
2.409084
2.597537

2.8511533 2.022309 2.889707 2.587842

1.943098

2.934678 3-255644 2.654538 2.027906 2.853377

Pallas-orbit,
Hebe.
Juno

Upon the orbit of Juno.
Ascending Node.

Descending Node.
Rad. vect. Rad. vect. Juno. Rad. vect. Rad. vect. Juno.
2:278408 2.854559 3.058776 2.222411
2.120485 1 2:025683 12.5699831 3261389

Pallas-orbit,
Hebe

Upon the orbit of Hebe.
Ascending Node.

Descending Node.
Rad. vect. | Rad. vect. Hebe. Rad. vect. 7 Rad. vect. Hebe.
2.432568 2.901680 2.818945 1 .937722

Pallas-orbit,

Of these twenty-eight combinations there are eighteen cases in which the orbits lock into one another, like the links of a chain, and ten where the one orbit is entirely inclosed within the other.

Linked into one another are the following orbits:
Hebe and Astræa. Hebe and Iris. Ceres and Juno.
Juno "
Vesta 16 "

Pallas " Hebe.
Vesta 6
Flora 1 "

Vesta 16 "
Flora"
Pallas " Juno.

Ceres 16 "
Iris 16 16 Vesta 66 66

Pallas “ Flora. Pallas “ Iris. Hebe * " Vesta 16 Entirely included are the following orbits :

Flora in that of Hebe.
Iris and Flora in that of Juno.
Astræa and Vesta in that of Pallas.

Iris, Flora and Pallas, and consequently, also Astræa and Vesta in that of Ceres.

I add a second table, similar to the foregoing, which shows the so-called longitude in the orbit for each of the nodes. The first two columns give the longitude of the ascending node of the first named orbit upon the second. The third contains the mutual inclinations of the two.

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