Parallax: Wikis

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A simplified illustration of the parallax of an object against a distant background due to a perspective shift. When viewed from "Viewpoint A", the object appears to be in front of the blue square. When the viewpoint is changed to "Viewpoint B", the object appears to have moved in front of the red square.
This animation is an example of parallax. As the viewpoint moves side to side, the objects in the distance appear to move more slowly than the objects close to the camera.

Parallax is an apparent displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.[1][2] The term is derived from the Greek παράλλαξις (parallaxis), meaning "alteration". Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances. Astronomers use the principle of parallax to measure distances to objects (typically stars) beyond the Solar System. The Hipparcos satellite has taken these measurements for over 100,000 nearby stars. This provides the basis for all other distance measurements in astronomy, the cosmic distance ladder. Here, the term "parallax" is the angle or semi-angle of inclination between two sightlines to the star.

Parallax also affects optical instruments such as binoculars, microscopes, and twin-lens reflex cameras that view objects from slightly different angles. Many animals, including humans, have two eyes with overlapping visual fields to use parallax to gain depth perception; this process is known as stereopsis.

A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use a "needle" type speedometer gauge (when the needle is mounted in front of its dial scale in a way that leaves a noticeable spacing between them). When viewed from directly in front, the speed may show 60 (i.e. the needle appears against the '60' mark on the dial behind); but when viewed from the passenger seat (i.e. from an oblique angle) the needle can appear against a slightly lower or higher mark (depending on whether it is viewed from the left or from the right), because of the combined effect of the spacing and the angle of view.

Contents

Distance measurement in astronomy

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Stellar parallax

On an interstellar scale, parallax created by the different orbital positions of the Earth causes nearby stars to appear to move relative to more distant stars. By observing parallax, measuring angles and using geometry, one can determine the distance to various objects. When the object in question is a star, the effect is known as stellar parallax.

Stellar parallax is most often measured using annual parallax, defined as the difference in position of a star as seen from the Earth and Sun, i. e. the angle subtended at a star by the mean radius of the Earth's orbit around the Sun. The parsec (3.26 light-years) is defined as the distance for which the annual parallax is 1 arcsecond. Annual parallax is normally measured by observing the position of a star at different times of the year as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars. The first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer.[3] Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on radar reflection off the surfaces of planets.[4]

This image demonstrates parallax. The Sun is visible above the streetlight. The reflection in the water shows a virtual image of the Sun and the streetlight. The location of the virtual image is below the surface of the water and thus simultaneously offers a different vantage point of the streetlight, which appears to be shifted relative to the stationary, background Sun.

The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and thus the star with the largest parallax), Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec.[5] This angle is approximately that subtended by an object 2 centimeters in diameter located 5.3 kilometers away.

The fact that stellar parallax was so small that it was unobservable at the time was used as the main scientific argument against heliocentrism during the early modern age. It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed entirely implausible: it was one of Tycho's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere (the fixed stars).[6]

In 1989, the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motions of nearby stars, increasing the reach of the method tenfold. Even so, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of our galaxy. The European Space Agency's Gaia mission, due to launch in 2011 and come online in 2012, will be able to measure parallax angles to an accuracy of 10 microarcseconds, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from earth.[7]

Computation

Stellar parallax motion

Distance measurement by parallax is a special case of the principle of triangulation, which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 arcsecond,[3] leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Assuming the angle is small (see derivation below), the distance to an object (measured in parsecs) is the reciprocal of the parallax (measured in arcseconds): d(pc) = 1 / p(arcsec). For example, the distance to Proxima Centauri is 1/0.7687=1.3009 parsecs (4.243 ly).[5]

Diurnal parallax

Diurnal parallax is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the terrestrial planets or asteroids seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.[8][9]

Lunar parallax

Lunar parallax (often short for lunar horizontal parallax or lunar equatorial horizontal parallax), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body, has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.[10]

The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified. Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface. Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth:- one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram); and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an Earth-surface position corresponding roughly to one of the blue dots on the modified diagram).

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth[11] -- equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth. The Earth-Moon linear distance varies continuously as the Moon follows its perturbed and approximately elliptical orbit around the Earth. The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.[10] The Astronomical Almanac and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of lunar theory.

Diagram of daily lunar parallax

Parallax can also be used to determine the distance to the Moon.

One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000 km. This procedure was first used by Aristarchus of Samos[12] and Hipparchus, and later found its way into the work of Ptolemy.[citation needed] The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed.

Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated:

\mathrm{distance}_{\textrm{moon}} = \frac {\mathrm{distance}_{\mathrm{observerbase}}} {\tan (\mathrm{angle})}
Example of lunar parallax: Occultation of Pleiades by the Moon

This is the method referred to by Jules Verne in From the Earth to the Moon:

Until then, many people had no idea how one could calculate the distance separating the Moon from the Earth. The circumstance was exploited to teach them that this distance was obtained by measuring the parallax of the Moon. If the word parallax appeared to amaze them, they were told that it was the angle subtended by two straight lines running from both ends of the Earth's radius to the Moon. If they had doubts on the perfection of this method, they were immediately shown that not only did this mean distance amount to a whole two hundred thirty-four thousand three hundred and forty-seven miles (94,330 leagues), but also that the astronomers were not in error by more than seventy miles (≈ 30 leagues).

Solar parallax

After Copernicus proposed his heliocentric system, with the Earth in revolution around the Sun, it was possible to build a model of the whole solar system without scale. To ascertain the scale, it is necessary only to measure one distance within the solar system, e.g., the mean distance from the Earth to the Sun (now called an astronomical unit, or AU). When found by triangulation, this is referred to as the solar parallax, the difference in position of the Sun as seen from the Earth's centre and a point one Earth radius away, i. e., the angle subtended at the Sun by the Earth's mean radius. Knowing the solar parallax and the mean Earth radius allows one to calculate the AU, the first, small step on the long road of establishing the size and expansion age[13] of the visible Universe.

A primitive way to determine the distance to the Sun in terms of the distance to the Moon was already proposed by Aristarchus of Samos in his book On the Sizes and Distances of the Sun and Moon. He noted that the Sun, Moon, and Earth form a right triangle (right angle at the Moon) at the moment of first or last quarter moon. He then estimated that the Moon, Earth, Sun angle was 87°. Using correct geometry but inaccurate observational data, Aristarchus concluded that the Sun was slightly less than 20 times farther away than the Moon. The true value of this angle is close to 89° 50', and the Sun is actually about 390 times farther away.[12] He pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the Sun was around 20 times larger than the Moon; this conclusion, although incorrect, follows logically from his incorrect data. It does suggest that the Sun is clearly larger than the Earth, which could be taken to support the heliocentric model.

Measuring Venus transit times to determine solar parallax

Although Aristarchus' results were incorrect due to observational errors, they were based on correct geometric principles of parallax, and became the basis for estimates of the size of the solar system for almost 2000 years, until the transit of Venus was correctly observed in 1761 and 1769.[12] This method was proposed by Edmond Halley in 1716, although he did not live to see the results. The use of Venus transits was less successful than had been hoped due to the black drop effect, but the resulting estimate, 153 million kilometers, is just 2% above the currently accepted value, 149.6 million kilometers.

Much later, the Solar System was 'scaled' using the parallax of asteroids, some of which, like Eros, pass much closer to Earth than Venus. In a favourable opposition, Eros can approach the Earth to within 22 million kilometres.[14] Both the opposition of 1901 and that of 1930/1931 were used for this purpose, the calculations of the latter determination being completed by Astronomer Royal Sir Harold Spencer Jones.[15]

Also radar reflections, both off Venus (1958) and off asteroids, like Icarus, have been used for solar parallax determination. Today, use of spacecraft telemetry links has solved this old problem. The currently accepted value of solar parallax is 8".794 143.[16]

Dynamic or moving-cluster parallax

The open stellar cluster Hyades in Taurus extends over such a large part of the sky, 20 degrees, that the proper motions as derived from astrometry appear to converge with some precision to a perspective point north of Orion. Combining the observed apparent (angular) proper motion in seconds of arc with the also observed true (absolute) receding motion as witnessed by the Doppler redshift of the stellar spectral lines, allows estimation of the distance to the cluster (151 light-years) and its member stars in much the same way as using annual parallax.[17]

Dynamic parallax has sometimes also been used to determine the distance to a supernova, when the optical wave front of the outburst is seen to propagate through the surrounding dust clouds at an apparent angular velocity, while its true propagation velocity is known to be the speed of light.[18]

Derivation

For a right triangle,

\sin p = \frac {1 AU} {d} ,

where p is the parallax, 1 AU (149,600,000 km) is approximately the average distance from the Sun to Earth, and d is the distance to the star. Using small-angle approximations (valid when the angle is small compared to 1 radian),

\sin x \approx x\textrm{\ radians} = x \cdot \frac {180} {\pi} \textrm{\ degrees} = x \cdot 180 \cdot \frac {3600} {\pi} \textrm{\ arcseconds} ,

so the parallax, measured in arcseconds, is

p'' \approx \frac {1 \textrm{\ AU}} {d} \cdot 180 \cdot \frac{3600} {\pi} .

If the parallax is 1", then the distance is

d = 1 \textrm{\ AU} \cdot 180 \cdot \frac {3600} {\pi} = 206,265 \textrm{\ AU} = 3.2616 \textrm{\ ly} \equiv 1 \textrm{\ parsec} .

This defines the parsec, a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply d = 1 / p, when the parallax is given in arcseconds.[19]

Parallax error

Precise parallax measurements of distance have an associated error. However this error in the measured parallax angle does not translate directly into an error for the distance, except for relatively small errors. The reason for this is that an error toward a smaller angle results in a greater error in distance than an error toward a larger angle.

However, an approximation of the distance error can be computed by

\delta d = \delta \left( {1 \over p} \right) =\left| {\partial \over \partial p} \left( {1 \over p} \right) \right| \delta p ={\delta p \over p^2}

where d is the distance and p is the parallax. The approximation is far more accurate for parallax errors that are small relative to the parallax than for relatively large errors.

Visual perception

Because the eyes of humans and other highly evolved animals are in different positions on the head, they present different views simultaneously. This is the basis of stereopsis, the process by which the brain exploits the parallax due to the different views from the eye to gain depth perception and estimate distances to objects.[20] Animals also use motion parallax, in which the animal (or just the head) moves to gain different viewpoints. For example, pigeons (whose eyes do not have overlapping fields of view and thus cannot use stereopsis) bob their heads up and down to see depth.[21]

Parallax and measurement instruments

If an optical instrument — e.g., a telescope, microscope, or theodolite — is imprecisely focused, its cross-hairs will appear to move with respect to the object focused on if one moves one's head horizontally in front of the eyepiece. This is why it is important, especially when performing measurements, to focus carefully in order to eliminate the parallax, and to check by moving one's head.

Also, in non-optical measurements the thickness of a ruler can create parallax in fine measurements. To avoid parallax error, one should take measurements with one's eye on a line directly perpendicular to the ruler so that the thickness of the ruler does not create error in positioning for fine measurements. A similar error can occur when reading the position of a pointer against a scale in an instrument such as a galvanometer (for example, in an analog-display multimeter.) To help the user avoid this problem, the scale is sometimes printed above a narrow strip of mirror, and the user positions his eye so that the pointer obscures its own reflection. This guarantees that the user's line of sight is perpendicular to the mirror and therefore to the scale.

Parallax can cause a speedometer reading to appear different to a car's passenger than to the driver.

Photogrammetric parallax

Aerial picture pairs, when viewed through a stereo viewer, offer a pronounced stereo effect of landscape and buildings. High buildings appear to 'keel over' in the direction away from the centre of the photograph. Measurements of this parallax are used to deduce the height of the buildings, provided that flying height and baseline distances are known. This is a key component to the process of photogrammetry.

Parallax error in photography

Parallax error can be seen when taking photos with many types of cameras, such as twin-lens reflex cameras and those including viewfinders (such as rangefinder cameras). In such cameras, the eye sees the subject through different optics (the viewfinder, or a second lens) than the one through which the photo is taken. As the viewfinder is often found above the lens of the camera, photos with parallax error are often slightly lower than intended, the classic example being the image of person with his or her head cropped off. This problem is addressed in single-lens reflex cameras, in which the viewfinder sees through the same lens through which the photo is taken (with the aid of a movable mirror), thus avoiding parallax error.

In computer graphics

In many early graphical applications, such as video games, the scene was constructed of independent layers that were scrolled at different speeds when the player/cursor moved. Some hardware had explicit support for such layers, such as the Super Nintendo Entertainment System. This gave some layers the appearance of being farther away than others and was useful for creating an illusion of depth, but only worked when the player was moving. Now, most games are based on much more comprehensive three-dimensional graphic models, although portable game systems (such as Nintendo DS) still often use parallax. Parallax based graphics remain to be used for many online applications where the bandwidth required by three-dimensional graphics is excessive.

In gunfire

Owing to the positioning of gun turrets on a warship or in the field, each one has a slightly different perspective of the target relative to the location of the fire control system itself. Therefore, when aiming its guns at the target, the fire control system must compensate for parallax in order to assure that fire from each turret converges on the target.

This is also true of small arms, as the distance between the sighting mechanism and the weapon's bore can introduce significant errors when firing at close range, particularly when firing at small targets.

As a metaphor

In a philosophic/geometric sense: An apparent change in the direction of an object, caused by a change in observational position that provides a new line of sight. The apparent displacement, or difference of position, of an object, as seen from two different stations, or points of view. In contemporary writing parallax can also be the same story, or a similar story from approximately the same time line, from one book told from a different perspective in another book. The word and concept feature prominently in James Joyce's 1922 novel, Ulysses. Orson Scott Card also used the term when referring to Ender's Shadow as compared to Ender's Game.

The metaphor is invoked by Slovenian philosopher Slavoj Žižek in his work The Parallax View. Žižek borrowed the concept of "parallax view" from the Japanese philosopher and literary critic Kojin Karatani. "The philosophical twist to be added (to parallax), of course, is that the observed distance is not simply subjective, since the same object that exists 'out there' is seen from two different stances, or points of view. It is rather that, as Hegel would have put it, subject and object are inherently mediated so that an 'epistemological' shift in the subject's point of view always reflects an ontological shift in the object itself. Or—to put it in Lacanese—the subject's gaze is always-already inscribed into the perceived object itself, in the guise of its 'blind spot,' that which is 'in the object more than object itself', the point from which the object itself returns the gaze. Sure the picture is in my eye, but I am also in the picture."[22]

The word is used in the title of Alan J. Pakula's 1974 movie The Parallax View, in which a reporter (Warren Beatty) investigates an assassination. The word in this case refers to a fictional corporation portrayed in the film.

Notes

  1. ^ Shorter Oxford English Dictionary. 1968. "Mutual inclination of two lines meeting in an angle". 
  2. ^ "Parallax". Oxford English Dictionary (Second Edition ed.). 1989. http://dictionary.oed.com/cgi/entry/50171114?single=1&query_type=word&queryword=parallax&first=1&max_to_show=10. "Astron. Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view, and constituting a measure of the distance of the object.". 
  3. ^ a b Zeilik & Gregory 1998, p. 44.
  4. ^ Zeilik & Gregory 1998, § 22-3.
  5. ^ a b Benedict, G. Fritz et al. (1999). "Interferometric Astrometry of Proxima Centauri and Barnard's Star Using HUBBLE SPACE TELESCOPE Fine Guidance Sensor 3: Detection Limits for Substellar Companions". The Astronomical Journal 118 (2): 1086–1100. doi:10.1086/300975. http://adsabs.harvard.edu/abs/1999astro.ph..5318B. Retrieved 2010-02-17. 
  6. ^ See p.51 in The reception of Copernicus' heliocentric theory: proceedings of a symposium organized by the Nicolas Copernicus Committee of the International Union of the History and Philosophy of Science, Torun, Poland, 1973, ed. Jerzy Dobrzycki, International Union of the History and Philosophy of Science. Nicolas Copernicus Committee; ISBN 9027703116, ISBN 9789027703118
  7. ^ Henney, Paul J.. "ESA's Gaia Mission to study stars". Astronomy Today. http://www.astronomytoday.com/exploration/gaia.html. Retrieved 2008-03-08. 
  8. ^ Seidelmann, P. Kenneth (2005). Explanatory Supplement to the Astronomical Almanac. University Science Books. pp. 123–125. ISBN 1891389459. 
  9. ^ Barbieri, Cesare (2007). Fundamentals of astronomy. CRC Press. pp. 132–135. ISBN 0750308869. 
  10. ^ a b Astronomical Almanac e.g. for 1981, section D
  11. ^ Astronomical Almanac, e.g. for 1981: see Glossary; for formulae see Explanatory Supplement to the Astronomical Almanac, 1992, p.400
  12. ^ a b c Gutzwiller, Martin C. (1998). "Moon-Earth-Sun: The oldest three-body problem". Reviews of Modern Physics 70: 589. doi:10.1103/RevModPhys.70.589. 
  13. ^ Freedman, W.L. (2000). "The Hubble constant and the expansion age of the Universe". Physics Reports 333: 13. doi:10.1016/S0370-1573(00)00013-2. arΧiv:astro-ph/9909076. http://adsabs.harvard.edu/abs/2000PhR...333...13F. 
  14. ^ Whipple 2007, p. 47.
  15. ^ Whipple 2007, p. 117.
  16. ^ US Naval Observatory, Astronomical Constants
  17. ^ Vijay K. Narayanan; Andrew Gould (1999). "A Precision Test of Hipparcos Systematics toward the Hyades". The Astrophysical Journal 515: 256. doi:10.1086/307021. arΧiv:astro-ph/9808284. 
  18. ^ Panagia, N. (1991). "Properties of the SN 1987A circumstellar ring and the distance to the Large Magellanic Cloud". The Astrophysical Journal 380: L23. doi:10.1086/186164. 
  19. ^ Similar derivations are in most astronomy textbooks. See, e. g., Zeilik & Gregory 1998, § 11-1.
  20. ^ Steinman, Scott B.; Garzia, Ralph Philip (2000), Foundations of Binocular Vision: A Clinical perspective, McGraw-Hill Professional, pp. 2–5, ISBN 0-8385-2670-5 
  21. ^ Steinman & Garzia 2000, p. 180.
  22. ^ Žižek, Slavoj (2006). The Parallax View. The MIT Press. pp. 17. ISBN 0262240513. 

References

  • Hirshfeld, Alan w. (2001), Parallax: The Race to Measure the Cosmos, New York: W. H. Freeman, ISBN 0716737116 
  • Whipple, Fred L. (2007), Earth Moon and Planets, Read Books, ISBN 1406764132 .
  • Zeilik, Michael A.; Gregory, Stephan A. (1998), Introductory Astronomy & Astrophysics (4th ed.), Saunders College Publishing, ISBN 0030062284 .

See also

External links


1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

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 " 0 o06".

„ Sappho,

„ Iris,

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"±0 004 (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.)


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Simple English


Parallax is the perceived change in location of an object seen from two different places. In astronomy, parallax is the only direct way to measure distance to stars outside of the solar system. In essence, parallax is the phenomenon which occurs when an object is viewed from different positions and its' position is changed.

Many animals, including humans, have two eyes to use parallax to have depth perception; this is called stereopsis.


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