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SOLAR PARALLAX
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See also:SOLAR See also:PARALLAX . :The problem of the distance of the See also:sun has always been regarded as the fundamental one of See also:celestial measurement. The difficulties in the way of solving it are very See also:great, and up to the See also:present See also:time the best authorities are not agreed as to the result, the effect of See also:half a See also:century of See also:research having been merely to reduce the uncertainty within continually narrower limits. The mutations of See also:opinion on the subject during the last fifty years have been remarkable. Up to about the See also:middle of the 19th century it was supposed that transits of See also:Venus across the disk of the sun afforded the most trustworthy method of making the determination in question; and when See also:Encke in 1824 published his classic discussion of the transits of 1761 and 1769, it was supposed that we must wait until the transits of 1874 and 1882 had been observed and discussed before any further See also:light would he thrown on the subject. The parallax 8.5776" found by Encke was therefore accepted without question, and was employed in the Nautical See also:Almanac from 1834 to 1869. Doubt was first thrown on the accuracy of this number by an announcement from See also:Hansen in 1862 that the observed parallactic inequality of the See also:moon was irreconcilable with the accepted value of the solar parallax, and indicated the much larger value 8.97". This result was soon apparently confirmed by several other researches founded both on theory and observation, and so strong did the See also:evidence appear to be that the value 8.95" was used in the Nautical Almanac from r87o to 1881. The most remarkable feature of the discussion since 1862 is that the successive See also:examinations of the subject have led to a continually diminishing value, so that at the present time it seems possible that the actual parallax of the sun is almost as near to the old value of Encke as to that which first replaced it. The value of 8-848", determined by S. See also:Newcomb, was used from 1882 to 1900; and since then the value 8.8o" has been employed, having been adopted at a See also:Paris See also:conference in 1896.1 1 Five fundamentally different methods of determining the distance of the sun have been See also:work-ed out and applied. They are as follows: I. That of See also:direct measurement.—From the See also:measures of the parallax of either Venus or See also:Mars the parallax of the sun can 1 R. S. See also:Ball, Spherical See also:Astronomy, p. 303. be immediately derived, because the ratios of distances in the solar See also:system are known with the last degree of methods of precision. Transits of Venus and observations of Determine. various kinds on Mars are all to be included in this See also:Lion• class. II. The second method is in principle extremely See also:simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the sun to the See also:earth. The velocity is now well determined; the difficulty is to determine the time of passage. IV. The See also:fourth method is through the parallactic inequality in the moon's See also:motion. For the relation of this inequality to the solar parallax see MOON. V. The fifth method consists in observing the displacement in the direction of the sun, or of one of the nearer See also:planets, due to the motion of the earth See also:round the See also:common centre of gravity of the earth and moon. It requires a precise knowledge of the moon's See also:mass. The uncertainty of this mass impairs the accuracy of the method. I. To begin with the results of the first method. The transits of Venus observed in 1874 and 1882 might be expected to hold a leading See also:place in the discussion. No purely T astronomical enterprise was ever carried out on so Ven See also:sus. . of large a See also:scale or at so great an See also:expenditure of See also:money and labour as was devoted to the observations of these transits, and for several years before their occurrence the astronomers of every leading nation were busy in discussing methods of observation and working out the multifarious details necessary to their successful application. In the preceding century reliance was placed entirely on the observed moments at which Venus entered upon or See also:left the See also:limb of the sun, but in 1874 it was possible to determine the relative positions of Venus and the sun during the whole course of the transit. Two methods were devised. One was to use a See also:heliometer to measure the distance between the limbs of Venus and the sun during the whole time that the See also:planet was seen projected on the solar disk, and the other was to take photographs of the sun during the See also:period of the transit and subsequently measure the negatives. The Germans laid the greatest stress on measures with the heliometer; the Americans, See also:English, and See also:French on the photo-graphic method. These four nations sent out well-equipped expeditions to various quarters of the globe, both in 1874 and 1882, to make the required observations; but when the results were discussed they were found to be extremely unsatisfactory. It had been supposed that, with the greatly improved telescopes of See also:modern times, contact observations could be made with much greater precision than in 1761 and 1769, yet, for some See also:reason which it is not easy to explain completely, the modern observations were but little better than the older ones. Discrepancies difficult to See also:account for were found among the estimates of even the best observers. The photographs led to no more definite result than the observations of contacts, except perhaps those taken by the Americans, who had adopted a more See also:complete system than the Europeans; but even these were by no means satisfactory. Nor did the measures made by the Germans with heliometers come out any better. By the See also:American photographs the distances between the centres of Venus and the sun, and the angles between the See also:line adjoining the centres and the See also:meridian, could be separately measured and a See also:separate result for the parallax derived from each. The results were: Transit of 1874: Distances ; See also:par. = 8.888 ". Pos. angles; „ =8.873”. Distances; ,, =8.873" Transit of 1882: Pos. angles; „ =8-772'. The See also:German measures with the heliometer gave apparently concordant results, as follows: Transit of 1874: par. = 8.876 Transit of 1882: ,, =8.879". The combined result from both these methods is 8.857", while the See also:combination of all the contact observations made by all the parties gave the much smaller result, 8.794". Had the See also:internal contacts alone been used, which many astronomers would have considered the proper course, the result would have been 8' 776". In 1877 See also:Sir See also:David Gill organized an expedition to the See also:island of See also:Ascension to observe the parallax of Mars with the heliometer. By measurements giving the position of Mars among Planetary See also:Para!laxeathe neighbouring stars in the See also:morning and evening, . the effect of parallax could be obtained as well as by observing from two different stations; in fact the rotation of the earth carried the observer himself round a parallel of See also:latitude, so that the comparison of his own morning and evening observations could be used as if they had been made at different stations. The result was 8.78". The failure of the method based on transits of Venus led to an See also:international effort carried out on the initiative of Sir David Gill to measure the parallax by observations on those See also:minor planets which approach nearest the earth. The See also:scheme of observations was organized on an extended scale. The three bodies chosen for observation were: See also:Victoria (See also:June to to Aug. 26, 1889); See also:Iris (Oct. 12 to Dec. to, 1888); and See also:Sappho (See also:Sept. 18 to Oct. 25, 1888). The distances of these bodies at the times of opposition were somewhat less than unity, though more than twice as great as that of Mars in 1877. The See also:drawback of greater distance was, however, in Gill's opinion, more than compensated by the accuracy with which the observations could be made. The See also:instruments used were heliometers, the construction and use of which had been greatly improved, largely through the efforts of Gill himself. The planets in question appeared in the See also:telescope as See also:star-like See also:objects which could be compared with the stars with much greater accuracy than a planetary disk like that of Mars, the apparent See also:form of which was changed by its varying phase, due to the different directions of the sun's See also:illumination. These observations were worked up and discussed by Gill with great elaboration in the See also:Annals of the Cape See also:Observatory, vols. vi. and vii. The results were for the solar parallax a: From Victoria, 7 = 8.8oi"*0.006". Sappho, 7 = 8.798"to•oI I ". „ Iris, Y = 8.812"to•oo9”. The See also:general mean result was 8.8o2". From the meridian observations of the same planets made for the purpose of controlling the elements of motion of the planets Auwers found tr=8.8o6". In 1898 the remarkable minor planet See also:Eros was discovered, which, on those rare occasions when in opposition near See also:perihelion, would approach the earth to a distance of o•16. On these occasions the actual parallax would be six times greater than that of the sun, and could therefore be measured with much greater precision than in the See also:case of any other planet. Such an approach had occurred in 1892, but the planet was not then discovered. At the opposition of 1900-1901 the minimum distance was o•32, much less than that of any other planet. See also:Advantage was taken of the occasion to make photographic measures for parallax at various points of the earth on a very large scale. Owing to the difficulties inherent in determining the position of so faint an See also:object among a great number of stars, the results have taken about ten years to work out. The photographic right ascensions gave the values 8.8o" + 0.007" ± 0'0027" (Hinks) and 8.8o" + o•oo67" =I- 0.0025" (Perrine); the micrometric observations gave the value 8.8e6"±o•0o4 (Rinks).' II. The velocity of light (q.v.) has been measured with all the precision necessary for the purpose. The latest result is 299,86o kilometres per second, with a probable See also:error of perhaps 30 kilometres—that is, about the ten-thousandth See also:part of the quantity itself. This degree of precision is far beyond any we 1 Mon. Not. R.A.S. (May 1909,) p. 544; ibid. (June 1910), p. 588.can See also:hope to reach in the solar parallax. The other See also:element which enters into See also:consideration is the time required for light to pass from the sun to the earth. Here no such precision can be attained. Both direct and indirect methods are available. The direct method consists in observing the times of some momentary or rapidly varying celestial phenomenon, as it. appears when seen from opposite points of the earth's See also:orbit, The only phenomena of the sort available are eclipses of See also:Jupiter's satellites, especially of the first. Unfortunately these eclipses are not sudden but slowly changing phenomena, so that they cannot be observed without an error of at least several seconds, and not infrequently important fractions of a See also:minute. As the entire time required for light to pass over the See also:radius of the earth's orbit is only about 500 seconds, this error is fatal to the method. The indirect method is based upon the observed See also:constant of See also:aberration or the displacement of the stars due to the earth's motion. The minuteness of this displacement, about 2o•5o", makes its precise determination an extremely difficult See also:matter. The most careful determinations are affected by systematic errors arising from those diurnal and See also:annual changes of temperature, the effect of which cannot be wholly eliminated in astronomical observation; and the recently discovered variation of latitude has introduced a new element of uncertainty into the determination. In consequence of it, the values formerly found were systematically too small by an amount which even now it is difficult to estimate with precision. See also:Struve's classic number, universally accepted during the second half of the x9th century, was 20.445". Serious doubt was first See also:cast upon its accuracy by the observations of Nyren with the same See also:instrument during the years 188o-1882, but on a much larger number of stars. His result, from his observations alone, was 20.52"; and taking into account the other Puikowa results, he concluded the most probable value to be 2o•492". In 1895 See also:Chandler, from a general discussion of all the observations, derived the value of 20.50". Since then, two elaborate See also:series of observations made with the See also:zenith telescope for the purpose of determining the variation of latitude and the constant of aberration have been carried on by See also:Professor C. L. Doolittle at the See also:Flower Observatory near See also:Philadelphia, and Professor J. K. See also:Rees and his assistants at the observatory of See also:Columbia University, New See also:York. Each of these See also:works is self-consistent and seemingly trustworthy, but there is a difference between the two which it is difficult to account for. Rees's result is 2o•47"; Doolittle's, from 2o•46" to 2o•56". This last value agrees very closely with a determination made by Gill at the Cape of See also:Good Hope, and most other See also:recent determinations give values exceeding 20.50". On the whole it is probable that the value exceeds 20.5o"; and so far as the results of direct observation are concerned may, for the present, be fixed at 20.52". The corresponding value of the solar parallax is 8.782". In addition to the doubt thrown on this result by the discrepancy between various determinations of the constant of aberration, it is sometimes doubted whether the latter constant necessarily expresses with entire precision the ratio of the velocity of the earth to the velocity of light. While the theory that it does seems highly probable, it cannot be regarded as absolutely certain. tions of the transits of Venus. So exact is the latter determination that, were there no weak point in the subsequent parts of the See also:process, this method would give far the most certain result for the solar parallax. Its weak point is that the apparent motion of the See also:node depends partly upon the motion of the See also:ecliptic, which cannot be determined with equal precision. The derivation of the distance of the sun by it is of such See also:interest from its simplicity that we shall show the computation. From the observed motion of the node of Venus, as shown by the four transits of 1761, 1769, 1874 and 1882, is found Mass of (earth+moon) =Mass of sun 332600 In gravitational See also:units of mass, based on the See also:metre and second as units of length and time, See also:Log. earth's mass =14.60052 „ moon's ,, =12.6895. The sum of the corresponding See also:numbers multiplied by 332600 gives Log. sun's mass = 20.12773. Putting a for the mean distance of the earth from the sun, and n for its mean motion in one second, we use the fundamental See also:equation a'n2 = Mo-J- M', Mo being the sun's mass, and M' the combined masses of the earth and moon, which are, however, too small to affect the result. For the mean motion of the earth in one second in circular measure, we have 2r _ n 3155814-9--; log. n =7'29907 the denominator of the fraction being the number of seconds in the sidereal See also:year. Then, from the See also:formula co Mo 120.127731 = n2= . -15'59814 Log. a in metres = 11.17653 Log. equat. rad. 6.80470 Sine p 's eq. See also:hor. par. 5.62817 Sun's eq. hor. par. 8.762”. IV. The determination of the solar parallax through the parallactic inequality of the moon's motion also involves two Motion of elements—one of observation, the other of purely Moon. mathematical theory. The inequality in question has its greatest negative value near the time of the moon's first See also:quarter, and the greatest See also:positive value near the third quarter. Meridian observations of the moon have been heretofore made by observing the transit of its illuminated limb. At first quarter its first limb is illuminated; at third quarter, its second limb. In each case the results of the observations may be systematically in error, not only from the uncertain See also:diameter of the moon, but in a still greater degree from the varying effect of irradiation and the See also:personal equation of the observers. The theoretical element is the ratio of the parallactic inequality to the solar parallax. The determination of this ratio is one of the most difficult problems in the lunar theory. Accepting the definitive result of the researches of E. W. See also: . 8.773" lunar equation . . 8.818" The question of the possible or probable error of these results is one on which there is a marked divergence of opinion among investigators. Probably no general agreement could now be reached on a statement more definite than this; the last result may be left out of consideration, and the value of the solar parallax is probably contained between the limits 8.77" and 8.80." The most likely distance of the sun may be stated in round numbers as 93,000,000 See also:miles. (S. Additional information and CommentsThere are no comments yet for this article.
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