PARALLAX may be defined, generally, as the change produced in the apparent place of an object when it is viewed from a point other than that of reference. In astronomy, the places of the moon and planets are referred to the centre of the earth, those of the fixed stars to the centre of the sun. It is shown in Astronomy (vol. it p. 775) that, the maximum or horizontal parallax of a celestial object being known, its parallax from any point of observation can be calculated. The present article will be restricted to an account of the methods employed for determining the solar and lunar parallaxes and those of the fixed stars.
Solar Parallax. —The sun’s mean equatorial horizontal parallax (termed briefly the “solar parallax ”) is the angle which the equatorial radius of the earth would subtend to an observer at the sun when the earth is at mean distance from the sun. For its determination it would appear only necessary to observe the sun’s apparent position simultaneously from two widely different points on the earth’s surface; the difference of the apparent positions will be due to displacement by parallax, from which displacement the mean equatorial horizontal parallax can be readily deduced.
The requirements of modern astronomy demand that the solar parallax shall be determined with an accuracy of -∏⅛τj∙ part of its amount—that is, within less than τ ⅛ π part of a second of arc. But measures in the neighbourhood of the sun cannot be made with any approach to this accuracy, not only on account of the effect of the sun’s heat on the various parts of the instruments employed, but also of the atmospheric currents created by heat, which tend to destroy steady atmospheric definition and to render the solar image incapable of exact observation. It is thus hopeless to look for any solution of the problem by the most direct method. Two courses remain—either to seek some method which affords a larger angle to measure, or one which permits a mode of observation affording a higher precision. There are many relations to the solar parallax which are well established.
(1) The parallax of the moon is known with very considerable precision by direct determination. The proportion of this parallax to that of the sun is an important term in the lunar theory, and the constant of this term (the ftarallactic inequality ) is a known function of the solar parallax. Hence, if the constant of the parallactic inequality is independently determined, the solar parallax becomes known. The elements of the orbits of Venus and Mars undergo secular variations which increase from year to year, from century to century, and at last acquire very large values. These secular variations (on the assumption that all the terms of the theories of the planets are mathematically accurate) have also a well-determined relation to the solar parallax, and therefore afford a means of determining that parallax with an accuracy which increases by the continuance of observation.
(2) It has been shown (Astronomy, vol. ii. p. 779 sq., and Mechanics, vol. xv. p. 708) that the proportions of the interplanetary distances can be very accurately determined and tables be computed from observations of the apparent places of the planets, without any knowledge or assumption as to absolute distances (although an accurate knowledge of the solar parallax is required for giving final perfection to the lunar and planetary tables). In astronomical ephemerides therefore the distances of planets from the earth are accurately expressed in terms of the earth’s mean distance from the sun, the latter being reckoned unity. Hence, to determine the solar parallax, it is only necessary to measure, at some favourable opportunity, the parallax of any planet, and to multiply the parallax so found by the number which expresses the relation of the distance of the planet from the earth to the earth’s mean distance from the sun.
(3) When Jupiter is in opposition he is nearer the earth by the diameter of the earth’s orbit than when in conjunction; hence, since light occupies a very sensible time to travel, eclipses of Jupiter’s satellites will seem to occur too soon in the first case and too late in the latter, the difference between the extremes of acceleration and retardation being the time occupied by light in crossing the earth’s orbit. This time is about 16⅜ minutes for the mean diameter of the earth’s orbit; hence, if the velocity of light can be independently determined, the diameter of the earth’s orbit becomes known. The determination, by employing the velocity of light, is also arrived at in another way. The constant of aberration (see Astronomy, p. 757), or the maximum apparent change of a star’s true place due to the motion of the observer, depends on the relative velocity of the motion of the observer in space compared with the velocity of light. The angular velocity of the observer is perfectly known; hence if his linear velocity is known his radius of motion is known. Thus, if the constant of aberration and the velocity of light are independently determined, the radius of motion (that is the sun’s parallax) will be found.
There are thus three distinctive typical methods:—(1) the gravitational method, depending on terms in the lunar and planetary theories, the constants of which are determined by observation; (2) the geometrical, or direct observational, method; and (3) the physical method.
1. The, Gτosυitational Method.— The moon’s parallactic inequality appears, at first sight, to furnish a very accurate method, as its constant is about 125", or fourteen times as great as the solar parallax, and the existing observations are very numerous. Unfortunately its determination is inextricably mixed up with the determination of the moon’s diameter—a diameter increased by irradiation, and therefore different for every telescope, and perhaps for every observer. But this is not all. The maximum and minimum effect of the parallactic inequality occur at first and last quarter, ι.e., when the moon is half full. One half of the observations for parallactic inequality therefore are made when the sun is above the horizon, and a great portion of the other half during twilight; whilst those on which the moon’s diameter depend are made at midnight, when the irradiation is a totally different quantity from what it is in daylight or during twilight. Newcomb has attempted to determine the correction of the diameter by the errors in right ascension, derived by comparing Hansen’s tables of the moon with observations made by daylight and at night; but he confesses that the result is so mixed up with the correction of the coefficient of the variation (and, he might have added, with the observer’s personality and the telescope employed) that it cannot be relied upon.
The following are the most important discussions :— [table]
One cannot look with confidence upon a method which thus permits discordance of more than one per cent, in the discussion of the same observations by different astronomers. The result arrived at must depend on the adopted corrections of the moon’s diameter, and, since that diameter is not capable of determination under the same circumstances of illumination as those in which the observations for parallactic inequality are made, the judgment of the theorist must step in and assign some more or less hypothetical grounds for the adoption of a particular diameter; and upon this assumption will turn the whole of the quantity of which we are in search.
It is, however, not impossible that the method of observing a spot near the centre of the moon, instead of the moon’s limb, may lead to a more reliable result. But it will have to be shown by independent methods that the position of the selected spot is not systematically affected by phase.
Attention was first called to the method which employs the secular variations of the elements of the orbits of Venus and Mars for determining the solar parallax by a most able and comprehensive paper communicated by Leverrier to the Paris Academy of Sciences, and published in their Comptes Rendus for 1872, July 22. The most important of these variations is that of the perihelion of Mars. The earth’s attraction increases the heliocentric position of Mars at perihelion by about 50" in a century, and this change at a favourable opposition subtends an angle of 185" at the earth.
On 1672, October 1, the star ψ Aquarii was occulted by Mars. The appulse was observed by Richer at Cayenne, by Picard near Beaufort, and by Romer at Paris. The separate comparisons differ only 0" , 5, 0" ’ 8, and 0" - 3 respectively; and the star ψ Aquarii was very frequently observed by Bradley. The increase in two centuries of the geocentric longitude, corresponding to the distance of the planet Mars from the earth on 1672, October 1, is 294". Hence Μ. Leverrier concludes that (attributing an error not greater than 1" to the determination of the observed variation) the time has arrived when the solar parallax can be determined with a probable error not exceeding y⅜ 75 ∙ of its amount, or the concluded parallax will be exact to nearly ±0" 01. The value of the parallax so deduced Μ. Leverrier finds to be
8"∙866.
Similarly he finds from the latitude of Venus, determined by the transits of Venus in 1761 and 1769, combined with the latitude determined by meridian observations of the present day
8' ’ ∙853.
From the discussion of the meridional observations of Venus in an interval of one hundred and six years, he finds
8"∙859.
These values from the theories of Venus and Mars accord in a wonderful manner, and would appear at first sight to justify considerable confidence in the result. But it is impossible to forget the extraordinary intricacy of the processes through which these results have been evolved, and the liability to some systematic source of error, such, for example, as some neglected term producing a long inequality which may become mixed up with the secular variation.
In 1874 the tabular errors of Venus, as determined by the planet’s transit across the sun’s disk, amounted to more than 5" of arc both in R.A. and declination, and the tabular errors of Mars amounted to more than 8" in R.A. and to about 3" in declination at the opposition of 1877, equivalent to an error of 2'' , 45 in heliocentric longitude (Jfem. R. A. S., vol. xlvi. p. 172). Leverrier’s planetary tables do not, therefore, possess the accuracy attributed to them by their distinguished author, and the conclusions at which he arrived probably require some further modification. Tisserand (Comptes Rendus, 1881, March 21) has continued the researches of Leverrier, and finds that they require modification, and are also subject to very considerable probable error. The later researches of Tisserand appear to point to a value of the solar parallax smaller than that found by Leverrier, but his work has not yet been brought to final conclusion.
2. The Geometrical Method.— The most favourable opportunities for the application of this method are afforded, in a geometrical sense, by the planets Venus and Mars, when the former is in conjunction and the latter in opposition. Of these Venus approaches the earth within one-fourth of the sun’s mean distance, whilst Mars, in the most favourable circumstances, approaches only within one-third of the same distance.
When Venus is near conjunction she is only visible as a slender crescent in the neighbourhood of the sun, and at conjunction is only visible on the occasion of a transit across the sun’s disk. It generally happens, therefore, that the only means of determining the apparent position of Venus near conjunction is to refer that position to the sun’s limb or sun’s centre. But the sun’s place is also affected by parallax, so that when the position of Venus is referred to the sun the parallactic displacement is only the difference of the parallax of the sun and Venus. Mars, on the other hand, can be referred to stars of which the parallax is absolutely insensible; thus it happens that the advantage of Venus in point of parallactic displacement is diminished till the geometrical conditions are only 5 per cent, in favour of Venus. Transits of Venus across the sun’s disk have been observed for parallax in 1761, 1769, 1874, and 1882.
If an astronomer at each of two widely separated stations observes the absolute instant of apparent internal contact of Venus with the sun’s limb, he is sure that the centres of the sun and Venus are separated by an angular distance equal to the “semidiameter of the sun minus the semidiameter of Venus.” The difference of the absolute times at the two stations is due to parallactic displacement, and, the planet’s tabular motion being accurately known, the amount of displacement becomes known. If instead of one contact only the two observers note the instants of internal contact both at ingress and egress, then they practically find the chords described by the planet as seen from both stations. The difference of length of these chords (in time) being known, as well as the approximate diameters of the sun and Venus, and their tabular motion, we have the data for computing the difference of least distance of centres of the sun and Venus at the two stations, and this distance being due to parallax, we have the means of computing the parallax of Venus and thence the solar parallax. This latter method (originally proposed by Halley in 1716) has the advantage of not requiring a rigid determination of the absolute instant of each contact, but merely of the duration of the transit; in other words, it involves no very rigid determination of the longitude or clock error, but only an exact knowledge of the clock rate.
It was Halley’s opinion that the instants of contact could be observed with an accuracy within two or three tenths of a second of time, but experience has gone to show that the actual errors are from ten to forty times this amount, and the causes of those errors can now be assigned with considerable certainty. These causes are— (1) irradiation and diffraction; (2) disturbance of the image by irregular refraction in the earth’s atmosphere; (3) the effect of the atmosphere of Venus in complicating the phenomena at the point of contact.
(1) Irradiation increases the diameter of the sun and diminishes that of Venus. Its extent depends on the aperture of the telescope, the perfection of its optical quality, and the perfection of the focal adjustment. Its amount is also changed by the brilliancy of the sun, i.e., is affected by the transparency of the sky and the density of the sun shade employed. Also, when the space between the limbs of the sun and Venus becomes smaller than the diffraction disk of the object-glass employed, a greyness or shadow is perceived at the point of past or approaching contact; therefore, within a minute angle equal to the separating power (the diameter of the diffraction disk) of the object-glass, the actual instant of contact can only be estimated by changes in the diffraction phenomena. (2) When the images are thrown into rapid vibration by irregular refraction in the earth’s atmosphere, it becomes impossible to distinguish between the vibration of the image of the dark body of Venus across the sun’s limb near the point of contact and the regular phenomena of irradiation, provided that the atmospheric vibrations are sufficiently rapid to produce a persistent image on the retina of the observer’s eye. Thus at the transit of Venus in 1882 observers were instructed to note at ingress the time when there was “a well-marked and persistent discontinuity in the illumination of the apparent limb of the sun.” Now it so happened that at the Royal Observatory, Cape of Good Hope, the definition was very bad—a south-easter was blowing, the effect of which was, as is almost invariably the case, to create a rapid minute vibration in the images of celestial objects (see Sir John Herschel’s Results of Observations at the Cape of Good Hope, p. xiv.). Thus “a well-marked and persistent discontinuity in the illumination of the apparent limb of the sun” was seen by all of five observers at the Royal Observatory from 10 to 20 seconds of time longer than at the adjoining stations in the Cape Colony, where the images were seen comparatively steady and well-defined. The instant of occurrence of the above-described phase is therefore a function of the state of the atmospheric definition, and no accurate means exist of estimating such influence. (3) The observation is besides complicated by the illuminated atmosphere of Venus, which forms an arc of light round the planet near the point of contact. In many cases this light has been confounded with the light of the sun, and has thus caused very considerable errors of observation.
From these various causes the apparent phenomena are different at different stations; and probably also the same phenomena are described by different observers in very different language. The real difficulty of the discussion of the results arises when these different and differently described phenomena have to be combined. It is of no consequence whether a real or seeming contact has been observed; it is only necessary to be certain that those observations are combined which represent the same phenomenon. The same phenomenon would correspond with the same apparent angular distance of centres of the sun and Venus, if all the telescopes were alike, if all the telescopes were in perfect focal adjustment, and if the atmospheric conditions of definition at all the stations were perfect or identical. But if these conditions are not realized (and they cannot be realized in practice) the same apparent phenomena will not represent corresponding phases; and, further, the observers at different stations use such different language to express what they saw that it becomes impossible to select even apparent corresponding phases with any certainty.
The value of the solar parallax deduced from a series of observations of the contacts of Venus with the sun’s limb will therefore entirely depend upon the interpretation put .upon the language of the various observers. The result will besides be systematically affected if the state of atmospheric definition is systematically different in the opposite stations.
It is thus not surprising that very different results have been arrived at by different astronomers from different transits, and even from different discussions of the same observations of the same transit. [table]
Besides observing the contacts, another method was employed by the Germans, the Russians, the Dutch, by Lord Lindsay’s expedition at the transit of Venus in 1874, and by the Germans in 1882, viz., the heliometric method. This consists in observing with the heliometer (see Micrometer, vol. xvi. pp. 252-254) the distance of V en us from opposite limbs of the sun along known position-angles nearly in the line of greatest and least distance of Venus from the sun’s limb. The method possesses many apparent advantages, because it affords the opportunity of multiplying the observations and of eliminating many sources of error.
At first sight it seems as if the method is free from the necessity for any accurate determination of the scale-value of the instrument, because, if measures are made from opposite limbs of the sun, the sun’s diameter may be taken as the standard for all observers, and the place of the planet may be interpolated relatively to the opposite limbs. Unfortunately it happens that there is à very marked difference in observing the sun’s diameter due both to instrument and observer. Thus two observers with different instruments, who have compared scale-value by a number of pairs of stars, or zones of stars, will measure sun-diameters with a marked constant difference. If the sun’s diameter is assumed to be constant, it, in fact, determines the scale-value; hence the distance of centres measured by the two observers will differ by the proportionate part × ∆d, where s is the distance of centres, d the true diameter, and ∆d the difference of diameter as measured by the two observers. Thus it is only when s = 0 (that is, when the planetis near the centre of the sun) that this method can be used,—a condition that does not exist in practice.
In the case of the transit of Venus fully one-third of the whole of this personality would enter into the result by this method of reduction. For rigid reduction therefore it is absolutely necessary to have a rigid determination of scale-value in seconds of arc. Unfortunately this value, when determined for any uniform instrumental condition of temperature, is liable to change, because, in observations of the sun, difference of temperature between the tube, the object-glass, and the scale of the instrument is produced, and the focal adjustment is also disturbed. The scalevalue depends on the relation of the focal length of the objectglass to the length of a part of the scale, and is besides affected by abnormal focal adjustment of the eyepiece.
Drs Auwers and Winnecke adopted a very complete scheme for determining the scale-value at any instant.
1. The scale-value was determined for a uniform condition of the temperature of the instrument by measuring zones of stars whose places were rigidly determined by meridian observations; and by the same means the temperature coefficient of the instrument was determined for different temperatures, the various parts of the instrument being assumed of a uniform temperature in observations of stars by night.
2. The effect of a displacement of focus was determined by measuring the sun’s diameter and distances of pairs of stars with different positions of the focal adjustment.
3. The focal point was determined during sun-observations by adjusting the focus on a telescope fixed in a specially prepared chamber, where its temperature would change very slowly, and the temperature of the scale (and hence its length) were measured by a metallic thermometer; hence the change of the proportion of the scale-length to the focal length became known.
In Lord Lindsay’s expedition similar precautions were employed, excepting that in the last case an attempt was made to determine the temperature of the tube by thermometers and that of the objectglass by a thermo-pile, and the position of the focal point was calculated from these data.
The uncertainties of all these operations are considerable, and, though from the extraordinary labour and care bestowed upon the determination of the necessary corrections by the German astronomers a fairly reliable result may be arrived at, it is certain that the method of determining the solar parallax from heliometric observation of transits of Venus can now be surpassed by methods more direct, more reliable, and at the same time less laborious and costly.
If photographs can be obtained during a transit in which the limbs of the sun and Venus are sufficiently well-defined, the distance of the centres of the sun and planet can be determined (as in the heliometer method) provided only that the pictures are affected by no systematic errors. That this latter condition may be fulfilled the following are the essential conditions.
1. The picture must be formed on the photographic plate without distortion, or, if it is affected by distortion, that distortion must be ascertained and allowed for.
2. No change must take place in the process of developing and fixing the picture, or, if such change is possible, means must be provided for its detection and elimination.
3. The angular value of one inch on the plate must be accurately known, so as to convert measured distances into arc—for the same reasons as in the heliometer method.
It is necessary to employ an image of considerable size, because otherwise the particles of collodion, if magnified so much as to permit measurement of the requisite accuracy, give an irregularity to the limbs that is fatal to accurate estimation. Thus it becomes necessary either to employ a lens of very considerable focal length (40 feet was generally adopted), or to introduce a secondary lens to magnify the image formed in the primary focus. The first of these methods was employed by the Americans, by the French, and in Lord Lindsay’s expedition at the transit of 1874, the second by the British, German, Russian, and Dutch expeditions.
The use of an object-glass of long focus renders mounting of the lens in the usual manner, though not a practical impossibility, yet at least a matter of extreme inconvenience. Accordingly, where lenses of long focus were employed, the telescope was mounted in a horizontal position, and the sun’s rays were reflected by a plane mirror in the direction of the tube’s length. It is not easy to conceive that any sensible distortion in the image can be produced by a lens of such long focus even if only of mediocre quality of figure; indeed the method may be assumed free from any such error; but it is undoubtedly exposed to all the errors of distortion which may be produced by the plane. From the perfection now attained in the construction of optical planes, and the means which exist for testing them, the errors due to this cause may also probably be safely neglected, except in so far as the figure of the plane is distorted by the heat of the sun, and it is not impossible that some sources of systematic error may be thus created.
To determine the angular value of one inch (or other unit of length) on the photographic plate, it is only necessary to measure the distance of the plate from the posterior surface of the objectglass, and then to determine the distance of the optical centre of the lens from that surface; the sum of these two distances is the radius of which lines on the surface of the plate (reckoned from the centre of the plate) are tangents.
The French adopted the daguerrotype method of photography, in which it is impossible to imagine any errors due to contraction of the photographic film, as in the collodion process, because the picture is virtually a portion of the silver plate on which it is taken. But in adopting this process the advantage of measuring the photographs by transmitted light was lost; and it is a practical question, which experience has not yet decided, whether the loss or the gain is the greater.
The Americans, and Lord Lindsay in 1874, using the collodion process, took the precaution to provide means for the detection of possible contraction of the film during development of the picture or drying of the film. This was done by placing the sensitive plate near to or in contact with a reticule ruled on glass near the primary focus; this reticule was thus photographed simultaneously with each photograph of the sun; hence any change produced during the development would cause a similar change in the relative positions of the images of the ruled lines on the developed plate. As a matter of fact the American astronomers have found fairly reliable results from their photographic operations, but the accuracy arrived at is by no means very considerable, the probable error of the complete measurements of an average plate amounting to ± 0"∙5.
But the difficulties of dealing with systematic errors are enormously increased when a secondary magnifier is employed, because it is theoretically impossible with the present optical glass (employing spherical curves) to construct a perfect secondary magnifier in which the scale value should be absolutely the same in every part of the field; still less is it possible, when the attempt is made, to combine the visual and photographic rays in the same focus; hence every photoheliograph of this construction must be separately studied for distortion of the image. The results of actual trial prove that the distortion is even greater than was expected, and is besides not the same in each radius, and the latter error may be produced by a very small error of centring in the lenses which compose the secondary magnifier. The investigation of such errors with the required accuracy would be a laborious and at best an unsatisfactory operation, and is rendered practically impossible by the fact that, whenever the instrument is turned upon the sun, the object-glass becomes heated, its focal length changed, and the optical relation of the secondary magnifier to the image in the principal focus of the object-glass changed also.
For these reasons the photographic observations in which secondary magnifiers were employed might be expected to prove a failure, and this expectation has been confirmed by the result of experience.
The observation of the transit of Venus on a large scale of national expenditure was certainly justified in 1761 and 1769. In those days there were no refined means of measuring angles with high accuracy, and the employment of the motion of Venus and a time-scale of measurement was the best available method of determining the solar parallax. But since 1820 the art of measurement has so advanced, and such refined instruments and methods have been thus introduced, that it may be a matter of some surprise and question to future generations of astronomers why so much labour and money were expended upon so imperfect a method in 1874 and 1882. The justification of these expeditions must be found, not in the reliability of the value of the solar parallax determined by them, but in the impulse given to the construction of instruments, the awakening of a widespread interest in astronomy, the stimulus to invention of new methods of research, and the accurate determination of the latitudes and longitudes of a large number of important and previously undetermined stations on the earth’s surface.
If an opposition of Mars occurs when that planet is near perihelion and the earth near aphelion the planet is then about one-third of the sun’s distance from the earth. When these conditions are nearly realized the opportunity is a favourable one for determining the solar parallax.
On 1672, October 1, the star ψ Aquarii was occulted by Mars. Estimations of the distance of the planet from the star were made at well-observed instants of time by Richer, Picard, and Romer, as already noticed, and from these observations the first approximate determination of the solar parallax was made by Cassini, viz., 9" ’ 5.
The method of observing Mars that has been most largely employed consists in observing the apparent declination of the planet by means of the transit circle—at observatories both in the northern and in the southern hemisphere. To increase the accuracy of the result, the same stars near the planet are observed at the various observatories, so that the method is reduced to measuring the difference of declination between the planet and neighbouring stars. The effects of periodic error in the gradua tion of the circles, of flexure of the instruments, and of abnormal refraction are thus nearly eliminated, and there remain only the systematic errors which may be supposed to arise from the difference of the habit of the observers in bisecting a star and a planet. To some extent these errors could be eliminated by the use of a reversing prism applied in the place of a sun-shade between the eyepiece and the observer’s eye. By the use of such a prism the motion of the spider-web and the limbs of the planet can be reversed with respect to the vertical, and such errors as depend on a different habit of bisecting a similar apparent upper and lower limb would be thus eliminated. But on account of the chromatic dispersion of the atmosphere the lower limb of the planet is coloured red and the upper limb violet; and in the illuminated field of the telescope it is probable that the observer has a tendency to cut with his spider-web more deeply into the feeble violet limb than into the more glaring red limb. The effect of his so doing would be to increase the value of the resulting parallax, and it seems not improbable that from this cause a larger value of the parallax has been obtained by this than by other methods.
The following are the most important series of observations, and their discussion by this method :— [table]
In 1872 (Ast. Nach., No. 1897) Dr Galle of Breslau proposed a method of determining the solar parallax which appears to be the foundation of the method of the future, viz., to measure, by means of the equatorial, the difference of declination between selected stars and a minor planet, or rather to interpolate the declination of a minor planet relative to two stars of comparison. A minor planet presents precisely the appearance of a star, and it is impossible to conceive any personality which can affect the observation of such a planet and a star. The interpolation of the planet’s declination relative to two including stardeclinations (i.e., measurement from stars nearly equally north and south of the planet) entirely eliminates errors due to error of the adopted arc-value of the micrometer screw. It is true that in the case of minor planets the parallax factor can hardly exceed 1∣, whilst in the case of Mars that factor may be 3; but their disks present objects which are capable of being observed with quite two and a half times the accuracy of Mars. Hence the conditions of accidental accuracy are equalized for a single opposition, whilst the advantages of systematic accuracy are entirely in favour of the minor planets. Moreover, the opportunities offered by favourable oppositions of minor planets are much more frequent than in the case of Mars. The opposition of the minor planet Flora in 1874 was observed, at the request of Dr Galle, by a considerable number of observers in both northern and southern hemispheres, but unfortunately only in very few cases with the precaution, care, and perfection of instrumental equipment necessary. In 1882 the minor planets Victoria and Sappho were similarly observed at the request of Gill. The work was taken up by a number of astronomers in both hemispheres, in a much more complete and systematic manner, with better instrumental means, and with the benefit of former experience. The results have riot yet been reduced, but it is believed they will afford a valuable contribution to the problem in question. The results of Dr Galle’s discussion of the observations of Flora in 1874 give for the solar parallax
8 n ∙87±0"∙042; but the same results when the relative weights are assigned in a more legitimate manner lead to the value
8"∙82±0"O6.
But in any plan requiring numerous and widely spread observers it is very difficult to secure that entire sympathy with the end in view, that scrupulous care in minute detail, which is essential in the highest class of observation, and it becomes impossible to alter the previously prepared programme in such a case, should circumstances render it desirable to do so; nor does it always happen that distant observatories can be supplied with the necessary instrumental details in sufficient time. In the case of the Victoria and Sappho observations of 1882 the requisite sympathy and care were accorded in a very remarkable degree, but on account of the errors of the planetary tables (discoverable only when the observations were begun) the selected stars of comparison were not by any means the most favourable that could have been chosen, and were consequently not the stars that a single observer would have selected at the time. Hence arises the desirability of a method not requiring co-operation, in which success depends upon a single observer, who may obtain independently by his own observations a complete series of results.
In 1857 Airy, in an address to the Royal Astronomical Society on the methods available for determining the solar parallax during the next twenty-five years, called attention to the favourable opposition of Mars in 1877, and declared his opinion that the best method of finding the solar parallax was to determine at an equatorial station the difference of right ascension of that planet and neighbouring stars in the evening and early morning, by observing transits of stars and planet across the webs of a well-adjusted rigidly mounted equatorial. The motion of the earth’s rotation would transport the observer 6000 or 7000 miles between the evening and morning observations, and the requisite displacement would thus be obtained. In other w words, the observer would avail himself of the diurnal displacement to determine the parallax of the planet. Of course a very large number of observations would be required, because tne observation of a transit over the webs of a telescope is not so exact as the micrometric comparison of two points. Only one observer availed himself of Airy’s suggestion, but a very good series of observations by this method was obtained by Maxwell Hall at the island of Jamaica. The detailed observations are printed in Mem. R. A. S., vol. xliv. p. 121; the resulting value of the solar parallax is
8"∙79±0"∙06.
In 1874 (in connexion with Lord Lindsay’s expedition to Mauritius) Gill, combining the suggestion of Galle as to the employment of a minor planet and Airy’s suggestion as to the employment of the diurnal displacement, observed the minor planet Juno, which was at that time favourably situated for the purpose. But instead of employing the method of transits of the planet and stars across spiderwebs he used a heliometer, and measured with that instrument the distance of the planet from the same star in the evening and morning. In order to eliminate the effects of changes in the scale-value, Gill selected stars on opposite sides of the planet, and so arranged his observations as to measure simultaneously the angle between the planet and both comparison stars. That is to say, if the two distances in question are called a and b, the measures were arranged in the order «,6,6,α or b,a,a,b. Thus any abnormal scale-value of the instrument applicable to the measurement a would be equally applicable to the measurement 6. If the places of the comparison stars are thus determined by meridian observations, the scale-value may be derived from the observations themselves with all desirable accuracy, and the effect of change in the scale-value (which alone is all-important) be absolutely eliminated. The observations so made at Mauritius showed that the position of the planet Juno relative to two stars of comparison could be so interpolated with a probable error less than - τ 1 τ τ th of a second of arc. A full account of these observations, together with a description of the heliometer, is given in the Dunecht Publications, vol ii. Lord Lindsay’s yacht, which conveyed the heliometer to Mauritius, unfortunately did not reach her destination till the most favourable time for making the observations was past; but sufficient observations were obtained to test the method thoroughly and to prove its capabilities. The value of the solar parallax resulting from the observations of Juno at Mauritius was
8"∙77±0∙"04 ·
In 1877, instead of observing the favourable opposition of Mars of that year by Airy’s method, Gill proposed to the Royal Astronomical Society to employ a heliometer (kindly lent by Lord Lindsay) to observe the planet in a similar manner to that in which he had observed the minor planet J uno at Mauritius in 1874. The offer was accepted. Gill selected the island of Ascension, and there carried out the necessary observations. The stars of comparison, by the kind and hearty co-operation of astronomers, were observed at thirteen of the principal observatories with meridian instruments, a combination of their results affording standard places of high accuracy. In general the angular distance of the planet was measured both in the evening and morning from each of three surrounding stars. The observed readings of the heliometer were corrected for the effects of refraction and phase, for the errors of division of the scales and of the micrometer screw, and were then converted into arc on an assumed value of one revolution of the micrometer screw (or rather of half an interval of the scale divisions).
The tabular apparent distance of the centre of Mars from each star for the instant of each observation was then computed with an assumed approximate value of the solar parallax (8" ’ 80). The calculation of the solar parallax and the elimination of errors of scale-value were then easily effected as follows :·—
Let ∆α, ∆δ = the corrections in seconds of arc to be applied to the tabular right ascension and declination respectively to obtain the true right ascension and declination of Mars at the epoch τ 0 .
p = the position angle of the planet referred to the star of comparison.
δ 0 =the approximate mean declination of the star and planet.
κ = the daily rate of increase of ∆α for the epoch τ 0 . κ' = the daily rate of increase of ∆δ for the epoch τ 0 , τ = the Greenwich mean time of observation.
π = the number of τ ⅞- δ ∙ parts (or the percentage) that the assumed solar parallax must be increased.
z = the correction required to be applied to an observed arc of 10000" reduced on the assumed scale-value.
the observed distance in seconds of arc v = 10000
O =the observed angular distance, computed with the assumed scale-value.
C =the calculated or tabular distance computed with the assumed value of the solar parallax.
Then each observation furnishes an equation of condition of the following form—
∕'∆α +∕"∆δ +f'"n - v z≈ (O - C) -∕'(τ - τ 0 )κ -∕"(τ - τ 0 )√; where f' = sinpcosδ 0 f"≈cosp
/parallax in R. A. ∖ z , , (parallax in declination , ∖ f ,, j “V 100 ) j ∖ 100 ) j ’
the parallaxes in f'" being in seconds of arc.
The equations resulting from each group of observations are then combined, care being taken to combine together in one group such observations only as have been made nearly simultaneously and where the value of z may therefore be assumed to be the same.
The combination of a group of evening with a group of morning observations (in which the term representing the error of scalevalue must then be represented by z and s') thus affords six equations involving five unknown quantities, from which the most probable value of n can be eliminated with its weight by the method of least squares, in terms of κ and κ'.
Care, however, must be taken to confine the combination to such groups as depend on measures from the same stars, if it is desired to eliminate the effects of errors in the adopted star places. Also, since it is assumed that κ and κ vary proportionally with the time, it is necessary that only such observations should be combined. as have been made at epochs sufficiently near together to render this a safe assumption.
Finally the absolute values of κ and κ' for the various combinations are deduced by developing the values of ∆α and ∆δ from each combination in terms of the time, and thus the definitive values of n are obtained.
The combination of these values of n, having regard to the weight of each, led to the result n= -0 ’ 209.
Whence the value of the solar parallax was 8"∙78±O"O12.
It should be remarked that in these observations a reversing prism was so employed as to eliminate any systematic error on the part of the observer which might be due to astigmatism of his eye, or a habit of placing the image of the star otherwise than truly central on the image of Mars. The probable error of one observation of distance having weight unity was found to be ± 0" , 24. Twelve such observations were generally made (and often more) on each night, and complete combinations of observations were secured on twenty-five nights.
This probable error does not exceed that of a single observation of contact on the occasion of a transit of Venus, and yet one hundred and ninety-six such observations were secured, as compared with two which is the utmost that can be secured as the result of any single observer’s expedition to observe a transit of Venus.
It is impossible, however, to say with certainty that the above result is entirely free from systematic error. There is one possible source of such error to be suspected, viz., the possible effect of the chromatic dispersion of the atmosphere which colours the limbs of Mars in the manner already described. In the case of heliometer observations the effect is certainly πμnimized from the fact that the star disk which is compared with the limb of Mars is coloured precisely in the same way as the limb—but whether all error is so eliminated it is impossible to say. A detailed account of these observations and their reductions is given in Mem. R. A. S., vol. xlvi. pp. 1-172.
If a minor planet, however, is observed in the above described manner, no suspicion of the error in question can attach to the final result; and, so far as is known, that method affords the only geometrical means of arriving at an absolutely definitive value of the solar parallax.
The following table represents the oppositions of minor planets that will be available for determining the solar parallax till the end of the present century. [table]
The results of many hundreds of observations for stellar parallax by Gill and Elkin (Mem. R. A. S., vol. xlviii. part 1) prove that the difference of two opposite angular distances each not greater than 2° can be measured by a small heliometer with a probable error not exceeding ±0" ’ 15 when the objects measured are points of light such as two stars (or a star and a minor planet). Hence it is easy to show, that a single observer at an equatorial station (furnished with a suitable heliometer) can determine the solar parallax by the carefid observation of two or three of the more favourable of the above oppositions with a probable error not exceeding ÷O"O1, and with absolute freedom from systematic error. Such a result is not possible by any other known method.
3. The Physical Method.— The determination of the velocity of light has recently been the subject of very refined and accurate measurement by the methods both of Fizeau and of Foucault (see Light, vol. xiv. p. 585). The results of the most recent and best determinations of the velocity of light, expressed in kilometres per second, are the following (Sidereal Messenger, vol. ii. No. 6) :— [table]
If we denote by k the interval required by light to cross the mean radius of the earth’s orbit, any independent determination of k will obviously afford, when combined with the velocity of light, a determination of the sun’s distance, i.e., of the solar parallax (see Light, vol. xiv. p. 584). Such a determination of k is afforded by a discussion of the eclipses of Jupiter’s satellites. Only two such discussions that have any claim to acceptance exist :—the first by Delambre in the early part of the present century, from a discussion of an immense mass of eclipses of the satellites of Jupiter comprising observations from 1662 to 1802; the second by Glasenapp, in a Russian thesis, in which there are discussed the observations of the first satellite of Jupiter from 1848 to 1873. [table]
Todd, in calling attention to Glasenapp’s results (Am. Journal of Science, vol. xix. p. 62), remarks on these two values as follows :—
“The former determination rests on a much greater number of observations than the latter; but it is difficult to form a just estimate of the work of an average last-century observation of an eclipse of a satellite of Jupiter. And, moreover, astronomers have no means of knowing the process which led the distinguished French astronomer to his result—which was adopted in his own tables of the satellites, and which was adopted by Damoiseau in his Tables Ecliptiques, published in 1836. The latter determination rests upon a mass of observations of definite excellence, which have been discussed after the modern fashion.”
Astronomers, however, whilst generally endorsing these remarks, will not be inclined to follow Todd in combining Dalambre’s value with Glasenapp’s by giving double weight to the latter. Having regard to those portions of Todd’s remarks which we have printed in italics, astronomers would generally be of opinion that only Glasenapp’s value of k can be seriously considered at the present day. This value, combined with the above mean value of the velocity of light, leads to 8 "∙76±o"∙O2 as the value of the solar parallax.
The photometric observation of the eclipses of Jupiter’s satellites as now being carried out at Cambridge, U. S., under Prof. Pickering, will probably ere long furnish the data for a much more accurate determination of k, and it is not impossible that very refined heliometric observations of the distance of the first satellite (when apparently near the planet) from the other satellites may likewise yield a reliable value of i.
On the relations between the constant of aberration, the solar parallax, and the velocity of light, see Light, vol. xiv. pp. 584, 585.
The mean of the nine best modern determinations of the constant of aberration {i.e., from 1830 to 1855) gives 20"∙496.
The most recent and valuable paper on this constant is that of Nyrén {Mém. de I’ Acad, de St Pétersbourg, 7th ser., vol. xxxi. No. 9), in which the constant is derived from independent researches extending over many years, with each of the three great fixed instruments of the Pulkowa observatory. The independent mean results are— [table]
This result, combined with the above quoted values of the velocity of light, gives the following values of the solar parallax :— [table]
There still remain some little theoretical difficulties with regard to the theory of aberration. That theory is perfectly obvious on the emission theory of light, but is a priori by no means so obvious on the undulatory theory. Is it certain that the velocity of light in the celestial spaces is identical with (or bears an exactly known relation to) the velocity of light which, having travelled a certain space in air, undergoes reflexion and returns? This is a question for the physicist, and a question that probably demands a practical as well as a theoretical answer.
Also Villarceau {Comptes Rendus, 1872, October 14) points out that in the ordinary theory of aberration no account is taken of the sun’s motion of translation through space, and shows that, if the normal constant of aberration is A, the constant for any particular star is A + A × α, where a depends on the angle which the star’s direction makes with the direction of the sun’s translation in space. In the observations of Nyrén, above referred to, there is a well-marked periodic variation in the values of the constant of aberration derived for twenty-seven stars, which seems to be a function of the right ascension of the stars. This variation may be due to some cause (such as lateral refraction in the north-and-south direction) depending on the seasons, or it may have a real physical significance on the theory of Villarceau. If further observation (especially in the southern hemisphere, where the seasons are reversed) should confirm the latter hypothesis, two important conclusions result:—
(a) We obtain some idea of the direction and amount of motion of the milky way, combined with that of the solar system in space; and
(δ) We may conclude that our theory of light is correct, which supposes that a ray of light is transmitted through space with uniform velocity, independently of the velocity of the source of light, and that ether is fixed and infinite—that is, nowhere limited in extent.
On the other hand a negative result would go far to show that our conception of ether is not correct, at least would force us to adopt one of two conclusions,—either that the milky way is stationary in space (within limits of our power of measurement), or that the ether accom panies the milky way and is not fixed in space and not infinite.
It is, however, a priori improbable that from any of these causes the deduced value of the solar parallax will be affected by τ √< nr of its deduced amount.
The tendency of the best modern determinations is to fix the solar parallax at
8"∙78 or 8"∙79, and hence the mean distance of the earth from the sun at 93 millions of miles, a result which is almost certainly exact within 200,000 miles.
Lunar Parallax. —The constant of the lunar parallax may be determined by a method precisely similar to that followed in the meridian declination observations of Mars. Our knowledge of the parallax of the moon depends at present entirely on such observations made nearly simultaneously at the Royal Observatories of Greenwich and the Cape of Good Hope. The resulting values of the parallax, found directly from these observations, are then multiplied by a factor which expresses the relation between the constant of the lunar parallax (Astronomy, vol. ii. p. 798) to the moon’s tabular parallax at the time; thus each nearly simultaneous observation at the two observatories gives an independent determination of the constant of the lunar parallax.
A better method, however, will be found when the results of numerous occultations of stars have been employed to determine the constants of a new and more accurate lunar theory—a work about to be undertaken by Prof. Simon Newcomb.
The best determination of the constant of the lunar parallax is that of Mr Stone, viz., 3422" ’ 71 {Mem. R.A.S., vol. xxxiv. pp. 11-16), derived from meridian observations at Greenwich and the Cape of Good Hope.
Stellar Parallax. —The constant of parallax of a fixed star is the maximum angle which a line equal to the earth’s mean distance from the sun would subtend if viewed at the star.
The distances of the fixed stars are so remote that till very recent times their parallaxes have been found to be insensible; that is to say, the earth’s orbit viewed from the nearest fixed star presents a disk (or ellipse) too small for measurement.
The limits of this article do not permit a detailed history of the early attempts of astronomers to determine the parallaxes of the fixed stars. The reader is referred on this point to Peters’s Précis historique des travaux sur la parallaxe des étoiles fixes, forming the first section of his celebrated work .Recherches sur la Parallaxe des étoiles fixes {Mém. de l ’ Acad. Imp. de St Pétersbourg, sec. Math, et Physiques, vol. v.). The most notable incident in that history was the discovery of aberration by Bradley, in 1728, when engaged in an unsuccessful attempt to determine the parallax of the star γ Draconis.
The first determination of the parallax of a fixed star iâ due to Henderson, His Majesty’s astronomer at the Cape of Good Hope in 1832 and 1833. It was followed by the nearly simultaneous discoveries of the parallax of 61 Cygni by Bessel and that of α Lyrae by Struve from observations made in the years preceding 1840. Since that time similar researches have been prosecuted with gradually increasing success.
The methods of observation may be divided into two classes,—the absolute and the differential.
The absolute method depends on observation of the zenith distance of a star about the epochs of maximum parallactic displacement in declination—in practice, however, generally throughout the whole year. The differences of declination so observed, after allowing for the effects of refraction, precession, aberration, nutation, and proper motion, afford the means of deducing the parallax of the star. The most notable series of observations of this character are those of Maclear at the Cape of Good Hope, by which he confirmed the results of his predecessor Henderson and those of Peters at Pulkowa in the second section of his work above mentioned. The latter is the most classic work in existence on refined observations of absolute declination, and it is by no means certain that, in more modern meridian observations, the work and methods of that distinguished observer have been equalled—except perhaps at Pulkowa. The minute precautions necessary in such work will be found in Peters’s paper above mentioned (see also Transit Circle). But not with all the skill of Peters, nor with every refinement of equipment and observation, can the difficulties caused by refraction and minute change of instrumental flexure, <fcc,, be completely overcome; the method of absolute altitudes does not, in fact, respond in accuracy to the demands of the problem.
The differential method depends on measuring the difference of declination, of distance, or of position angle between the star whose parallax is to be determined and one or more stars of comparison. It is assumed that the stars most likely to have sensible parallax are those which are remarkable for brilliancy or proper motion, and that the parallaxes of the stars of comparison (having little or no sensible proper motion and · faint magnitude) are so small as to be insignificant. So far as our knowledge goes these assumptions are justified.
Researches on stellar parallax by these methods have been followed of late years with considerable success. The instruments employed have been the heliometer and the filar micrometer (see Micrometer, vol. xvi. pp. 243-248), the latter instrument being used in conjunction with an ordinary equatorial (see Telescope). The precautions required to determine and eliminate systematic error, and to secure the necessary refinement of accuracy, demand more space for their description than the limits of this article admit. The reader is referred for these particulars to the undermentioned papers on the subject.
The heliometer method seems to present the greatest facilities for extensive researches on stellar parallax, not only because measures with this instrument seem, on the whole, to possess the highest accuracy, but because (on account of the large angles that can be measured) a much wider selection of suitable stars of comparison is available. Gyldén of Stockholm has applied the method of observing the differences of right ascension between the star whose parallax is to be determined and each of two comparison stars, and the same method has also been applied by Auwers {Math. Abhand. Berliner Acad., 1867); but the results obtained in this way do not compare at all favourably with the accuracy of properly conducted heliometer measu
The diagram (fig. 1) represents observations made by Gill to determine the parallax of a Centauri, with a heliometer at the Cape of Good Hope. The ordinates of the curve are the time reckoned from 1882Ό, the abscissae the changes in the place of a Centauri due to the parallax computed from the observations. Each dot represents the observations of each single night, and the reader will be able to judge of the accuracy of the observations from the agreement of the dots with the curve. Fig. 2 in like manner represents a series of observations of Sirius.
These and many other results show that, with similar means, it is now possible to detect any differential parallax amounting to 0"O5 with certainty, by a series consisting of a reasonable number of like observations—thus opening up a wide and important field for future research.
The following table contains a list of those stars of which the parallax is known with considerable accuracy,— Nos. 1 to 13 being in the northern and Nos. 14 to 22 in the southern hemisphere. [table]
A glance at the table is sufficient to show that neither apparent magnitude nor apparent motion affords a criterion of the parallax of any fixed star. Similar researches must, in fact, be carried out on a much more extended scale before any definite conclusions can be drawn. At present we can only conclude that different stars really differ· greatly in absolute brightness and absolute motion.
The following are the formulae which will be found most useful in computing the corrections for parallax :— [table]
1. To find the parallax ot the moon in zenith distance and azimuth, from the observed (or apparent) zenith distance and azimuth. Put 7=(≠ ~ 0 , ) cos A'.
Then sin(C-C) = psinτrsin(< ’ '-γ)5
sin (A'-A) = psill,rsin( ^~/ )sinA ·.
sin <
The corresponding quantities are found with all desirable precision for the sun and planets by the formulae—
ζ' - ^=pπ∙sin (ζ' -y); or approximately = τr sin £';
A' - A=pτr sin (φ - φ') sin A'cos ζ'; the latter quantity may generally be neglected.
2. To find the parallax of the moon in right ascension and declination from the true (or geocentric) right ascension and declination „ , . „ p sin π cos φ' cos t
Put sin θ =? ; cos δ
then tan (a-a)= tan θ tan (45° + ⅛0) tan t.
„ . , tan φ' cos ⅜(α - α )
cos [i + ⅛(α -α ’)]
s in θ ' = P sia *∙ s i n ≠'c0s(7-δ).
sin 7 ’
then tan (δ - δ') = tan θ' tan (45° + ⅛0') tan (7 - δ).
3. To find the parallax of the moon in right ascension and declination from the observed (or apparent) right ascension and declination.
. , ,. p sin ιr cos φ' sin t! sιn(α-α = i - ⅛-;
COS δ ’
tan<∕>' cos⅜(g -α ’).
7 cos[i ’ — ∣(α — α ’)] ’
sin (δ - δ')- p ≡* *i n ≠>in( 7 -δ').
sin 7
4. To find the parallax of the sun, planets, or comets in right ascension or declination.
, pπ cos φ' sin t! a- a = -— -. s
COS δ
tan φ' tan 7; cos t
δ - δ' = P 7r sin ≠' 3in (7 ~ 8 ') sin 7
When the distance of the object from the earth (∆) is given (the earth’s mean distance from the sun being reckoned unity), as is usually the case in ephemerides of minor planets and comets, we have mean solar parallax 8"*78
∆ δ - ‘
The reader will find the proof of these formulae in Chauvenet’s Spherical and Practical Astronomy, vol. L pp. 104-127.
For the Parallax of the Fixed Stars.
Put 7) =the maximum angle subtended by the mean distance of the earth from the sun at the distance of the star, = the star’s annual parallax; e =the obliquity of the ecliptic;
Øand r ≈ the sun’s longitude and radius vector; a and a! ) = the star’s true and apparent right ascensions and δ and δ' ) declinations respectively.
1. To find the heliocentric parallax of a star in right ascension and declination, its annual parallax (p) being known.
a - a = -pr sec δ(cos 0 sin a - sin 0 cos e cos a);
δ' - δ = -pr sin 0(cos e sin δ sin a - sin e cos δ) -pr cos 0 cos «.
2. To find the effect of parallax on the distance (s) and position angle (P) of two stars, one of which has sensible annual parallax. ∆s≈prm cos (0- Μ); ∆V=prm' cos (0 - Μ'); where m sin Μ = ( - cos a sin P + sin δ sin a cos P) cos e - cos δ cos P sin e; m cos M=sin <*sin P + sin δ cos a cos P; m' sin Μ' = — - (cos a cos P + sin δ sin a sin P) cos e + cos δ sin P sin;
∞'cosM' = - sinαcos P-sinδcosαsinPl. ,
«L J
