Parallax

Parallax (Gr. irapaXX6, alternately), in astronomy, the apparent change in the direction of a heavenly body when viewed from two different points. Geocentric parallax is the angle between the direction of the body as seen from the surface of the earth and the direction in which it appears from the centre of the earth. Annual parallax is the angle between the direction in which a star appears from the earth and the direction in which it appears from the centre of the sun. For stellar parallaxes see Star; the solar parallax is discussed below.

Solar PARALLAx. - The problem of the distance of the sun has always been regarded as the fundamental one of celestial measurement. The difficulties in the way of solving it are very great, and up to the present time the best authorities are not agreed as to the result, the effect of half a century of research having been merely to reduce the uncertainty within continually narrower limits. The mutations of opinion on the subject during the last fifty years have been remarkable. Up to about the middle of the 19th century it was supposed that transits of Venus across the disk of the sun afforded the most trustworthy method of making the determination in question; and when 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 light would be thrown on the subject. The parallax 8.5776" found by Encke was therefore accepted without question, and was employed in the Nautical Almanac from 1834 to 1869. Doubt was first thrown on the accuracy of this number by an announcement from Hansen in 1862 that the observed parallactic inequality of the 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 evidence appear to be that the value 8.95" was used in the Nautical Almanac from 1870 to 1881. ' The most remarkable feature of the discussion since 1862 is that the successive 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. Newcomb, was used from 1882 to 'goo; and since then the value 8.80" has been employed, having been adopted at a Paris conference in 1896.1 Five fundamentally different methods of determining the distance of the sun have been worked out and applied. They are as follows: I. That of direct measurement. - From the measures of the parallax of either Venus or Mars the parallax of the sun can 1 R.Š. Ball, Spherical Astronomy, p. 303.

be immediately derived, because the ratios of distances in the solar system are known with the last degree of of  precision. Transits of Venus and observations of Deterrnina-  various kinds on Mars are all to be included in this tion.

class.

II. The second method is in principle extremely simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the sun to the earth. The velocity is now well determined; the difficulty is to determine the time of passage.

III. The third method is through the determination of the mass of the earth relative to that of the sun. In astronomical practice the masses of the planets are commonly expressed as fractions of the mass of the sun, the latter being taken as unity. When we know the mass of the earth in gravitational measure, its product by the denominator of the fraction just mentioned gives the mass of the sun in gravitational measure. From this the distance of the sun can be at once determined by a fundamental equation of planetary motion.

IV. The fourth method is through the parallactic inequality in the moon's 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 planets, due to the motion of the earth round the common centre of gravity of the earth and moon. It requires a precise knowledge of the moon's 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 place in the discussion. No purely astronomical enterprise was ever carried out on so Transits of P large a scale or at so great an expenditure of 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 left the 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 heliometer to measure the distance between the limbs of Venus and the sun during the whole time that the planet was seen projected on the solar disk, and the other was to take photographs of the sun during the period of the transit and subsequently measure the negatives. The Germans laid the greatest stress on measures with the heliometer; the Americans, English, and French on the photographic 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 modern times, contact observations could be made with much greater precision than in 1761 and 1769, yet, for some reason which it is not easy to explain completely, the modern observations were but little better than the older ones. Discrepancies difficult to 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 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 American photographs the distances between the centres of Venus and the sun, and the angles between the line adjoining the centres and the meridian, could be separately measured and a separate result for the  parallax derived from each. The results were: Transit of 1874: Distances; par. =8-888". Pos. angles; „ =8-873".

Distances; =8-873".

Transit of 1882: °?

Pos. angles; „ =8-772".

The German measures with the heliometer gave apparently concordant results, as follows: Transit of 1874: par. =8.876".

Transit of 1882: „ =8.879".

From Victoria,

| 

7 = 8.801 " 0o06".  
  
---|---  
  
„ Sappho,

„ Iris,

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8.79 8 " 0.011 ".

it 8.812" Eo.009".  
  
The combined result from both these methods is 8.857", while the combination of all the contact observations made by all the parties gave the much smaller result, 8.794". Had the internal contacts alone been used, which many astronomers would have considered the proper course, the result would have been 8.776" In 1877 Sir David Gill organized an expedition to the island of Ascension to observe the parallax of Mars with the heliometer. By measurements giving the position of Mars among Planetary the neighbouring stars in the morning and evening, Parallaxes.  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 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 international effort carried out on the initiative of Sir David Gill to measure the parallax by observations on those minor planets which approach nearest the earth. The scheme of observations was organized on an extended scale. The three bodies chosen for observation were: Victoria (June 10 to Aug. 26, 1889); Iris (Oct. 12 to Dec. 10, 1888); and Sappho (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 drawback of greater distance was, however, in Gill's opinion, more than compensated by the accuracy with which the observations could be made. The 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 telescope as star-like objects which could be compared with the stars with much greater accuracy than a planetary disk like that of Mars, the apparent form of which was changed by its varying phase, due to the different directions of the sun's illumination. These observations.were worked up and discussed by Gill with great elaboration in the Annals of the Cape Observatory, vols. vi. and vii. The results were for the solar parallax 7r: The general mean result was 8.802". From the meridian observations of the same planets made for the purpose of controlling the elements of motion of the planets Auwers found it = 8.806".

In 1898 the remarkable minor planet Eros was discovered, which, on those rare occasions when in opposition near perihelion, would approach the earth to a distance of 0 . 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 case of any other planet. Such an approach had occurred in 1892, but the planet was not then discovered. At the opposition of1900-1901the minimum distance was 0.3 2, much less than that of any other planet. 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 object among a great number of stars, the results have taken about ten years to work out. The photographic right ascensions gave the values 8.80" -}- 0.007" -}- 0.0027" (Hinks) and 8.80" + 0 . 0067" 40.0025" (Perrine); the micrometric observations gave the value 8.806"±0004 (Hinks).1 II. The velocity of light (q.v.) has been measured with all the precision necessary for the purpose. The latest result is 299,860 kilometres per second, with a probable error of perhaps 30 kilometres - that is, about the ten-thousandth 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 hope to reach in the solar parallax. The other element which enters into 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 orbit. The only phenomena of the sort available are eclipses of 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 minute. As the entire time required for light to pass over the 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 constant of aberration or the displacement of the stars due to the earth's motion. The minuteness of this displacement, about 20.50", makes its precise determination an extremely difficult matter. The most careful determinations are affected by systematic errors arising from those diurnal and 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. Struve's classic number, universally accepted during the second half of the 19th century, was 20.445". Serious doubt was first cast upon its accuracy by the observations of Nyren with the same instrument during the years 1880-1882, 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 20.492". In 1895 Chandler, from a general discussion of all the observations, derived the value of 20.50". Since then, two elaborate series of observations made with the zenith telescope for the purpose of determining the variation of latitude and the constant of aberration have been carried on by Professor C. L. Doolittle at the Flower Observatory near Philadelphia, and Professor J. K. Rees and his assistants at the observatory of Columbia University, New York. Each of these 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 20.47"; Doolittle's, from 20.46" to 20.56". This last value agrees very closely with a determination made by Gill at the Cape of Good Hope, and most other recent determinations give values exceeding 20.50". On the whole it is probable that the value exceeds 20.50"; 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.

III. The combined mass of the earth and moon admits of being determined by its effect in changing the position of the plane of the orbit of Venus. The motion of the node of this plane is found with great exactness from observaMass, of the g tions of the transits of Venus. So exact is the latter determination that, were there no weak point in the subsequent parts of the process, this method would give far the most certain result for the solar parallax. Its weak point is that the apparent motion of the node depends partly upon the motion of the ecliptic, which cannot be determined with equal precision. The derivation of the distance of the sun by it is of such 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 units of mass, based on the metre and second as units of length and time, Log. earth's mass =14.60052 „ moon's „ =12.6895.

The sum of the corresponding 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 equation a3 n2 = Mo-1-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 n 8149' l o g. n=7.29907  3155 the denominator of the fraction being the number of seconds in the sidereal year. Then, from the formula a 3 - Mo - [20.12773] - n - 15.59814  Log. a in metres = 1 117653 Log. equat. rad. ® 6.80470 Sine 0 's eq. 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 elements - one of observation, the other of purely mathematical theory. The inequality in question has its greatest negative value near the time of the moon's first quarter, and the greatest 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 diameter of the moon, but in a still greater degree from the varying effect of irradiation and the 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. Brown the value of the solar parallax derived by this method is about 8.773".

V. The fifth method is, as we have said, the most uncertain of all; it will therefore suffice to quote the result, which is 7r= 8.818".

The following may be taken as the most probable values of the solar parallax, as derived independently by the five methods we have described: From measures of parallax. 8.802" „ velocity of light.. 8.781" „ mass of the earth.. 8.762" par. ineq. of moon.. 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 9 3,000,000 miles. (S. N.)