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at that precise time, to sheer accident I* What share accident may have had in the successful issue of the calculations, we presume the reader, after what has been said, will have little difficulty in satisfying himself. As regards the time when the discovery was made, much has also been attributed to fortunate coincidence. The following considerations will, we apprehend, completely dissipate this idea, if still lingering in the mind of any one at all conversant with the subject. The period of Uranus being 84*0140 years, and that of Neptune 164*6181, their synodic revolution (art. 418.), or the interval between two successive conjunctions, is 171*58 years. The late conjunction having taken place about the beginning of 1822; that next preceding must have happened in 1649, or more than 40 years before the first recorded observation of Uranus in 1690, to say nothing of its discovery as a planet. In 1690, then, it must have been effectually out of reach of any perturbative influence worth considering, and so it remained during the whole interval from thence to 1800. From that time the effect of perturbation began to become sensible, about 1805 prominent, and in 1820 had nearly reached its maximum. At this epoch an alarm was sounded. The maximum was not attained, — the event, so important to astronomy, was still in progress of developement,—when the fact (any thing rather than a striking one) was noticed, and made matter of complaint. But the time for discussing its cause with any prospect of success was not yet come. Every thing turns upon the precise determination of the epoch of the maximum, when the perturbing and perturbed planet were in conjunction, and upon the law of increase and diminution of the perturbation itself on either side of that point. Now it is always difficult to assign the time of the occurrence of a maximum by observations liable to errors bearing a ratio far from inconsiderable to the whole quantity observed. Until the lapse of some years from 1822 it would have been impossible to have fixed that epoch with any certainty, and as respects the law of degradation and total arc of longitude over which the sensible perturbations extend, we are hardly yet arrived at a period when this can be said to be completely determinable from observation alone. In all this we see nothing of accident, unless it be accidental that an event which must have happened between 1781 and 1953, actually happened in 1822; and that we live in an age when astronomy has reached that perfection, and its cultivators exercise that vigilance which neither permit such an event, nor its scientific importance, to pass unnoticed. The blossom had been watched with interest in its developement, and the fruit was gathered in the very moment of maturity.*

* These doubts seem to have originated partly in the great disagreement between the predicted and real elements of Neptune, partly in the near (/xmiWy precise) commensurability of the mean motions of Neptune and Uranus. We conceive them however to be founded in a total misconception of the nature of the problem, which was not, from such obviously uncertain indications as the observed discordances could give, to determine as astronomical quantities the axis, excentricity and mass of the disturbing planet; but practically to discover where to look for it: when, if once found, these elements would be far better ascertained. To do this, any axis, excentricity, perihelion, and mass, however wide of the truth, which would represent, even roughly the amount, but with tolerable correctness the direction of the disturbing force during the very moderate interval when the departures from theory were really considerable, would equally serve their purposes; and with an excentricity, mass, and perihelion disposable, it is obvious that any assumption of the axis between the limits 30 and 38, nay, even with a much wider inferior limit, would serve the purpose. In his attempt to assign an inferior limit to the axis, and in the value so assigned, M. Leverrier, it must be admitted, was not successful. Mr. Adams, on the other hand, influenced by no considerations of the kind which appear to have weighed with his brother geometer, fixed ultimately (as we have seen) on an axis not very egregiously wrong. Still it were to be wished, for the satisfaction of all parties, that some one would undertake the problem de novo, employing formulae not liable to the passage through infinity, which, technically speaking, hampers or may be supposed to hamper the continuous application of the usual perturbation,* 1 formula when cases of commensurability occur

* The student who may wish to see the perturbations of Uranus produced by Neptune, as computed from a knowledge of the elements and mass of that planet, such as we now know to be pretty near the truth, will find them stated at length from the calculations of Mr. Walker, (of Washington, U. S.) in the "Proceedings of the American Academy of Arts and Sciences,'' vol. i. p. 334. et seq. On examining the comparisons of the results of Mr. Walker's formula; with those of Mr. Adams's theory in p. 342, he will perhaps be surprised at the enormous difference between the actions of Neptune and Mr. Adams's "hypothetical planet" on the longitude of Uranus. This is easily explained. Mr. Adams's perturbations are deviations from Bouvard's orbit of Uranus, as it stood immediately previous to the late conjunction. Mr. Walker's are the deviations from a mean or undisturbed orbit freed from the influence of the long inequality resulting from the near commensurability of the motions.

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PART III.

OF SIDEREAL ASTRONOMY.

CHAPTER XV.

OF THE FIXED STABS.— THED3 CLASSIFICATION BY MAGNITUDES.— PHOTOMETRIC SCALE OF MAGNITUDES. — CONVENTIONAL OR VULGAR SCALE. — PHOTOMETRIC COMPARISON OF STARS DISTRIBUTION OF STARS OVER THE HEAVENS OF THE MILKY WAY OR

GALAXY.—ITS SUPPOSED FORM THAT OF A FLAT STRATUM PARTIALLY SUBDIVIDED. ITS VISIBLE COURSE AMONG THE CONSTELLATIONS. ITS INTERNAL STRUCTURE. — ITS APPARENTLY INDEFINITE EXTENT IN CERTAIN DIRECTIONS. OF THE DISTANCE OF

THE FIXED STARS. — THEIR ANNUAL PARALLAX. — PARALLACTIC UNIT OF SIDEREAL DISTANCE.—EFFECT OF PARALLAX ANALOGOUS TO THAT OF ABERRATION.— HOW DISTINGUISHED FROM IT. DETECTION OF PARALLAX BY MERIDIONAL OBSERVATIONS. — HENDERSON'S APPLICATION TO a. CENTAURI. — BY DIFFERENTIAL OBSERVATIONS. DISCOVERIES OF BESSEL AND STRUVE. — LIST OP

STARS IN WHICH PARALLAX HAS BEEN DETECTED. OF THE REAL

MAGNITUDES OF THE STARS. COMPARISON OF THEIR LIGHTS

WITH THAT OF THE SUN.

(777.) Besides the bodies we have described in the foregoing chapters, the heavens present us with an innumerable multitude of other objects, which are called generally by the name of stars. Though comprehending individuals differing from each other, not merely in brightness, but in many other essential points, they all agree in one attribute, — a high degree of permanence as to apparent relative situation. This has procured them the title of "fixed stars;" an expression which is to be understood in a comparative and not an absolute sense, it being certain that many, and probable that all, are in a state of motion, although too slow to be perceptible unless by means of very delicate observations, continued during a long series of years.

(778.) Astronomers are in the habit of distinguishing the stars into classes, according to their apparent brightness. These are termed magnitudes. The brightest stars are said to be of the first magnitude; those which fall so far short of the first degree of brightness as to make a strongly marked distinction are classed in the second; and so on down to the sixth or seventh, which comprise the smallest stars visible to the naked eye, in the clearest and darkest night Beyond these, however, telescopes continue the range of visibility, and magnitudes from the 8th down to the 16th are familiar to those who are in the practice of using powerful instruments; nor does there seem the least reason to assign a limit to this progression; every increase in the dimensions and power of instruments, which successive improvements in optical science have attained, having brought into view multitudes innumerable of objects invisible before; so that, for any thing experience has hitherto taught us, the number of the stars may be really infinite, in the only sense in which we can assign a meaning to the word.

(779.) This classification into magnitudes, however, it must be observed, is entirely arbitrary. Of a multitude of bright objects, differing probably, intrinsically, both in size and in splendour, and arranged at unequal distances from us, one must of necessity appear the brightest, one next below it, and so on. An order of succession (relative, of course, to our local situation among them) must exist, and it is a matter of absolute indifference, where, in that infinite progression downwards, from the one brightest to the invisible, we choose to draw our lines of demarcation. All this is a matter of pure convention. Usage, however, has established such a convention; and though it is impossible to determine exactly, or a priori, where one magnitude ends and the next begins, and although different observers have differed in their magnitudes, yet, on the whole, astronomers have restricted their first magnitude to about 23 or 24 principal stars; their second to 50 or 60 next inferior; their third to about 200 yet smaller, and Bo on; the numbers increasing very rapidly as we descend in the scale of brightness, the whole number of stars already registered down to the seventh magnitude, inclusive, amounting to from 12000 to 15000.

(780.) As we do not see the actual disc of a star, but judge only of its brightness by the total impression made upon the eye, the apparent "magnitude " of any star will, it is evident, depend, 1st, on the star's distance from us; 2d, on the absolute magnitude of its illuminated surface; 3d, on the intrinsic brightness of that surface. Now, as we know nothing, or next to nothing, of any of these data, and have every reason for believing that each of them may differ in different individuals, in the proportion of many millions to one, it is clear that we are not to expect much satisfaction in any conclusions we may draw from numerical statements of the number of individuals which have been arranged in our artificial classes antecedent to any general or definite principle of arrangement. In fact, astronomers have not yet agreed upon any principle by which the magnitudes may be photometrically classed a priori, whether for example a scale of brightnesses decreasing in geometrical progression should be adopted, each term being one half of the preceding, or one third, or any other ratio, or whether it would not be preferable to adopt a scale decreasing as the squares of the terms of an harmonic progression, i. e. according to the series 1, i> Tv» Jt> &0, ^ne former would be a purely photometric scale, and would have the apparent advantage that the light of a star of any magnitude would bear a fixed proportion to that of the magnitude next above it, an advantage, however, merely apparent, as it is certain, from many optical facts, that the unaided eye forms very different judgments of the proportions existing between bright lights, and those between feeble ones. The latter scale involves a physical idea, that of supposing the scale of magnitudes to correspond to tho appearance of a first magnitude standard star, removed successively to twice, three times, &c its original or standard distance. Such a scale, which would make tho nominal magnitude a sort of index to the presumable or average dis

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