صور الصفحة
PDF
النشر الإلكتروني

VII. On the Extension of Bode's Empirical Law of

the Distances of the Planets from the Sun, to the Distances of the Satellites from their respective Primaries.

By J. CHALLIS, M. A.

FELLOW OF TRINITY COLLEGE, AND OF THE CAMBRIDGE
PHILOSOPHICAL SOCIETY.

[Read Dec. 8, 1828.]

=

1. MORE than half a century has elapsed since Bode of Berlin discovered a singular law of the mean distances of the planets from the Sun, according to which if 4= Mercury's distance, 4 + 3 will = Venus's, 4+2.3 = Earth's, 4 + 3. 22 Mars's, &c. No one, I believe, has ever suggested an existing cause of this physical fact; the theory of universal gravitation points to no such law of the distances at which several small bodies will perform revolutions about a much larger; it has in consequence been customary to ascribe the law of Bode to the original arrangement of the planets at the time they were first set in motion. If however it be owing to a constantly operating cause, a similar phenomenon ought always to be observed under similar circumstances:-the satellites, which revolve round their primaries just as the latter revolve round the Sun, ought to obey the same law of distances. Should this be found to be the case it would afford some reason to think that the cause of the phenomenon is not incidental but permanent. In endeavouring to ascertain' whether the satellites

observe a like law, I have met with success, the more surprising, that, though easily attainable, it had not been anticipated. Before I state the result of my enquiry, I will enunciate the law in more express terms. It is this:

When several small bodies revolve round a much larger in orbits nearly circular, their mean distances observe with more or less accuracy, the following progression:

a, a + b, a +rb, a + r2b, &c.

It is to be observed that the differences between the true distances and those assigned by this progression, are in several instances very considerable in the system of the planets, the only one in which it has hitherto been recognized.

2. The distances of Jupiter's satellites from his centre are proportional to 60485, 96235, 153502, 269983. These distances, diminished by the least, leave remainders 35750, 93017, 209498. The ratio of the second to the first is 2.60, of the third to the second, 2.25. Half their sum = 2.42, which differs from either about one-fourteenth of its own value. Let a=60485, a+b=96235, and r=2.42.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small]

The coincidence of the true and empirical values is as near as happens with respect to the planets, and sufficiently exact to warrant the assertion that the law of distances is true for the satellites of Jupiter.

It is well known that the mean motion of the first satellite + twice that of the third = three times that of the second. Now Laplace has shewn, (Mec. Cel. Liv. ii. cap. 8.) that if the primitive mean motions of these satellites were near this proportion,

their mutual action must in time have brought about an accurate conformity to it. The law before us would arrange them nearly as they are arranged, and thus cause the mean motions to be nearly such as they actually are. The conclusion of Laplace therefore makes it probable that the deviations from the law of distances are produced, in part at least, by the mutual actions of the satellites, and so far connects the phenomenon with gravitation.

3. Proceeding now to the satellites of Saturn and subtracting from their mean distances the least mean distance, the remainders will be found proportional to 95, 193, 347, 623, 1873, 6101. The ratios of every two taken consecutively are 2.03, 1.80, 1.79, 3, 3.25. These ratios, seem to indicate a twofold series: the three first are not much different from each other, but different from the two last, which again do not much differ from each other. The mean of the three values 2.03, 1.80, 1.79, is 1.87; and the mean of 3.00, 3.25 is 3.12. Let a=335, a+b=430, r=1,87, r'=3.12.

[blocks in formation]

It appears by this that the first, second, third, fourth, and fifth are arranged according to one series; the first, fifth, sixth, and seventh according to another. The recurrence of the first and fifth in the two series is remarkable.

If we have rightly inferred in the preceding Article that the derangements of the law of distances are connected with gravitation, may we not ascribe the singular interruption of the law

in this instance to the action of the enormous ring of Saturn? The arrangement which would have taken place but for the ring, appears from the passage of the value of r from 2.03 to 1.79, to be partially disordered, then entirely broken after the fifth satellite and it is worth observing how the remaining two, which are considerably more distant from Saturn than the others, tend to comply with the law. If our conjecture be admitted, and the anomaly be rightly ascribed to an existing cause, it is natural to suppose that the cause of the law itself is existing.

4. Should the preceding instances be deemed insufficient to establish the law of distances, no doubt I think will be entertained of its reality when the satellites of Uranus have been discussed. Their mean distances are as 1312, 1720, 1984, 2275, 4551, 9101. Subtracting from each of these the least mean distance, the results are 408, 672, 963, 3239, 7789. The ratios of every two taken consecutively are 1.65, 1.43, 3.36, 2.41. Of these the first and second differ by a quantity not greater than has happened in other instances: the remaining two require particular consideration. The cube root of 3.36 is 1.50, and the square root of 2.41 is 1.55. These quantities, being near each other and not very different from the other two ratios, prove that the mean distances diminished by the least mean distance are terms of a geometric series nearly. The mean value of the common ratio is 1.53. The great distance of Uranus, and the apparent smallness of his satellites, which have never been seen but by the most powerful telescopes, leave us at liberty to suspect that there are others besides those already discovered. We know how the law of Bode rendered probable the existence of a planet between Mars and Jupiter. The same law, extending to the satellites of Uranus, authorizes the conjecture that there are two between the fourth and fifth, and one between the fifth and sixth, making in all nine.

Suppose a = 1312, a + b=1720, r=1.53.

Then a

a+b

[blocks in formation]
[ocr errors]

= 1720...............1720...............
a + rb = 1936...............1984..............+ 48

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

It is unnecessary to go through the same process for the planets. I will only observe that the ratios of consecutive distances diminished by the least distance, are in order, 1.823, 1.854, 2.066, 2.051, 1.900, 2.054, and the mean value = 1.958.

5. The foregoing investigation has answered the purpose intended, which was to shew the general coincidence of the arrangement of the satellites with an empirical law. When a law, recommending itself by its simplicity, has been established by observation, and is found to be attended with considerable deviations, we are naturally led to enquire whether the deviations themselves are subject to any law. The following process may in some degree answer this end in regard to the law before us. As any three distances would suffice to determine values of a, b, and r, and different values of these quantities would be obtained from every three that may be fixed upon, I have combined the equations in all the ways they admit of, and taken the mean of the different values. Substituting the mean values in the progression a, a + b, a + rb, &c. and comparing the distances thus obtained with the true distances, if any of the differences compared with the distances should be found much greater or much less than the others, we shall be directed to some peculiarities.

« السابقةمتابعة »