Parallax

Parallax

Overview


Parallax is a 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. 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
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 use the principle of parallax to measure distances to celestial objects including to the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

, the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, and to star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s beyond the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

.
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Encyclopedia


Parallax is a 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. 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
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 use the principle of parallax to measure distances to celestial objects including to the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

, the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, and to star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s beyond the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

. For example, the Hipparcos
Hipparcos
Hipparcos was a scientific mission of the European Space Agency , launched in 1989 and operated between 1989 and 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial objects on the sky...

 satellite took measurements for over 100,000 nearby stars. This provides a basis for other distance measurements in astronomy, the cosmic distance ladder
Cosmic distance ladder
The cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only for those objects that are "close enough" to Earth...

. Here, the term "parallax" is the angle or semi-angle of inclination between two sight-lines to the star.

Parallax also affects optical instruments such as binoculars
Binoculars
Binoculars, field glasses or binocular telescopes are a pair of identical or mirror-symmetrical telescopes mounted side-by-side and aligned to point accurately in the same direction, allowing the viewer to use both eyes when viewing distant objects...

, microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

s, and twin-lens reflex camera
Twin-lens reflex camera
A twin-lens reflex camera is a type of camera with two objective lenses of the same focal length. One of the lenses is the photographic objective or "taking lens" , while the other is used for the viewfinder system, which is usually viewed from above at waist level...

s that view objects from slightly different angles. Many animals, including humans, have two eye
Human eye
The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

s with overlapping visual fields
Visual perception
Visual perception is the ability to interpret information and surroundings from the effects of visible light reaching the eye. The resulting perception is also known as eyesight, sight, or vision...

 that use parallax to gain depth perception
Depth perception
Depth perception is the visual ability to perceive the world in three dimensions and the distance of an object. Depth sensation is the ability to move accurately, or to respond consistently, based on the distances of objects in an environment....

; this process is known as stereopsis
Stereopsis
Stereopsis refers to impression of depth that is perceived when a scene is viewed with both eyes by someone with normal binocular vision. Binocular viewing of a scene creates two slightly different images of the scene in the two eyes due the the eyes' different positions on the head...

. In computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...

 the effect is used for computer stereo vision
Computer stereo vision
Computer stereo vision is the extraction of 3D information from digital images, such as obtained by a CCD camera. By comparing information about a scene from two vantage points, 3D information can be extracted by examination of the relative positions of objects in the two panels...

, and there is a device called a parallax rangefinder
Coincidence rangefinder
A coincidence rangefinder is a type of rangefinder that uses mechanical and optical principles to allow an operator to determine the distance to a visible object....

 that uses it to find range, and in some variations also altitude to a target.

A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use an older-style "needle" speedometer gauge. When viewed from directly in front, the speed may show exactly 60; but when viewed from the passenger seat the needle may appear to show a slightly different speed, due to the angle of viewing.

Visual perception




As the eyes of humans and other animals are in different positions on the head, they present different views simultaneously. This is the basis of stereopsis
Stereopsis
Stereopsis refers to impression of depth that is perceived when a scene is viewed with both eyes by someone with normal binocular vision. Binocular viewing of a scene creates two slightly different images of the scene in the two eyes due the the eyes' different positions on the head...

, 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. Animals also use motion parallax, in which the animals (or just the head) move 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.

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
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

 angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s and using geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, one can determine the distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

 to various objects. When the object in question is a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

, 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
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....

 (3.26 light-year
Light-year
A light-year, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres...

s) 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
Year
A year is the orbital period of the Earth moving around the Sun. For an observer on Earth, this corresponds to the period it takes the Sun to complete one course throughout the zodiac along the ecliptic....

 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
Friedrich Bessel
-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...

 in 1838 for the star 61 Cygni
61 Cygni
61 Cygni,Not to be confused with 16 Cygni, a more distant system containing two G-type stars harboring the gas giant planet 16 Cygni Bb. sometimes called Bessel's Star or Piazzi's Flying Star, is a binary star system in the constellation Cygnus...

 using a heliometer
Heliometer
Heliometer is an instrument originally designed for measuring the variation of the sun's diameter at different seasons of the year, but applied now to the modern form of the instrument which is capable of much wider use....

. 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
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

 reflection off the surfaces of planets.

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
Proxima Centauri
Proxima Centauri is a red dwarf star about 4.2 light-years distant in the constellation of Centaurus. It was discovered in 1915 by Robert Innes, the Director of the Union Observatory in South Africa, and is the nearest known star to the Sun, although it is too faint to be seen with the naked eye...

, has a parallax of 0.7687 ± 0.0003 arcsec. This angle is approximately that subtended
Subtended
In geometry, an angle subtended by an arc, line or other curve is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context. For example, one may speak of the angle subtended by an arc of a circumference when the angle's vertex is a point on the...

 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
Heliocentrism
Heliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...

 during the early modern age. It is clear from Euclid's
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

 geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 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
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...

's principal objections to Copernican heliocentrism
Copernican heliocentrism
Copernican heliocentrism is the name given to the astronomical model developed by Nicolaus Copernicus and published in 1543. It positioned the Sun near the center of the Universe, motionless, with Earth and the other planets rotating around it in circular paths modified by epicycles and at uniform...

 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).

In 1989, the satellite Hipparcos
Hipparcos
Hipparcos was a scientific mission of the European Space Agency , launched in 1989 and operated between 1989 and 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial objects on the sky...

 was launched primarily for obtaining parallaxes and proper motion
Proper motion
The proper motion of a star is its angular change in position over time as seen from the center of mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600 arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of change in...

s 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-year
Light-year
A light-year, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres...

s away, a little more than one percent of the diameter of the Milky Way Galaxy. The European Space Agency
European Space Agency
The European Space Agency , established in 1975, is an intergovernmental organisation dedicated to the exploration of space, currently with 18 member states...

's Gaia mission, due to launch in 2012 and come online in 2013, 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.

Computation


Distance measurement by parallax is a special case of the principle of triangulation
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...

, 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, 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 parsec
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....

s) is the reciprocal of the parallax (measured in arcseconds): For example, the distance to Proxima Centauri
Proxima Centauri
Proxima Centauri is a red dwarf star about 4.2 light-years distant in the constellation of Centaurus. It was discovered in 1915 by Robert Innes, the Director of the Union Observatory in South Africa, and is the nearest known star to the Sun, although it is too faint to be seen with the naked eye...

 is 1/0.7687=1.3009 pc.

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 planet
Terrestrial planet
A terrestrial planet, telluric planet or rocky planet is a planet that is composed primarily of silicate rocks or metals. Within the Solar System, the terrestrial planets are the inner planets closest to the Sun...

s or asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

s seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.

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.

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 -- 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'. The Astronomical Almanac
Astronomical Almanac
The Astronomical Almanac is an almanac published by the United States Naval Observatory and Her Majesty's Nautical Almanac Office, containing solar system ephemeris and catalogs of selected stellar and extragalactic objects....

 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
Lunar theory
Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

.

Parallax can also be used to determine the distance to the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

.

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
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...

 and Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

, and later found its way into the work of Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

. 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:

This is the method referred to by Jules Verne
Jules Verne
Jules Gabriel Verne was a French author who pioneered the science fiction genre. He is best known for his novels Twenty Thousand Leagues Under the Sea , A Journey to the Center of the Earth , and Around the World in Eighty Days...

 in From the Earth to the Moon
From the Earth to the Moon
From the Earth to the Moon is a humorous science fantasy novel by Jules Verne and is one of the earliest entries in that genre. It tells the story of the president of a post-American Civil War gun club in Baltimore, his rival, a Philadelphia maker of armor, and a Frenchman, who build an enormous...

:
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
Nicolaus Copernicus
Nicolaus Copernicus was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology which displaced the Earth from the center of the universe....

 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
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

, or AU). When found by triangulation
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...

, 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
Age of the universe
The age of the universe is the time elapsed since the Big Bang posited by the most widely accepted scientific model of cosmology. The best current estimate of the age of the universe is 13.75 ± 0.13 billion years within the Lambda-CDM concordance model...

 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
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...

 in his book On the Sizes and Distances of the Sun and Moon
Aristarchus On the Sizes and Distances
On the Sizes and Distances is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who flourished circa 280–240 BC...

. 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
Lunar phase
A lunar phase or phase of the moon is the appearance of the illuminated portion of the Moon as seen by an observer, usually on Earth. The lunar phases change cyclically as the Moon orbits the Earth, according to the changing relative positions of the Earth, Moon, and Sun...

. He then estimated that the Moon, Earth, Sun angle was 87°. Using correct geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 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. He pointed out that the Moon and Sun have nearly equal apparent angular sizes
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 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.

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
Transit of Venus
A transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and Earth, becoming visible against the solar disk. During a transit, Venus can be seen from Earth as a small black disk moving across the face of the Sun...

 was correctly observed in 1761 and 1769. This method was proposed by Edmond Halley
Edmond Halley
Edmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.-Biography and career: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
Black drop effect
The black drop effect is an optical phenomenon visible during a transit of Venus and, to a lesser extent, a transit of Mercury.Just after second contact, and again just before third contact during the transit, a small black "teardrop" appears to connect Venus' disk to the limb of the Sun, making it...

, 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 asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

s, some of which, like Eros
433 Eros
433 Eros is a near-Earth asteroid discovered in 1898, and the first asteroid to be orbited by a probe . It is an S-type asteroid approximately 34.4×11.2×11.2 km in size, the second-largest NEA after 1036 Ganymed, and belongs to the Amor group.Eros is a Mars-crosser asteroid, the first known...

, pass much closer to Earth than Venus. In a favourable opposition, Eros can approach the Earth to within 22 million kilometres. 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
Astronomer Royal
Astronomer Royal is a senior post in the Royal Household of the Sovereign of the United Kingdom. There are two officers, the senior being the Astronomer Royal dating from 22 June 1675; the second is the Astronomer Royal for Scotland dating from 1834....

 Sir Harold Spencer Jones
Harold Spencer Jones
Sir Harold Spencer Jones KBE FRS was an English astronomer. Although born "Jones", his surname became "Spencer Jones"....

.

Also radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

 reflections, both off Venus (1958) and off asteroids, like Icarus
1566 Icarus
1566 Icarus is an Apollo asteroid that at perihelion comes closer to the Sun than Mercury, i.e. it is a Mercury-crossing asteroid. It is also a Venus and Mars-crosser. It is named after Icarus of Greek mythology, who flew too close to the Sun...

, have been used for solar parallax determination. Today, use of spacecraft
Spacecraft
A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....

 telemetry
Telemetry
Telemetry is a technology that allows measurements to be made at a distance, usually via radio wave transmission and reception of the information. The word is derived from Greek roots: tele = remote, and metron = measure...

 links has solved this old problem. The currently accepted value of solar parallax is 8".794 143.

Dynamic or moving-cluster parallax


The open stellar cluster Hyades
Hyades (star cluster)
The Hyades is the nearest open cluster to the Solar System and one of the best-studied of all star clusters. The Hipparcos satellite, the Hubble Space Telescope, and infrared color-magnitude diagram fitting have been used to establish a distance to the cluster's center of ~153 ly...

 in Taurus
Taurus (constellation)
Taurus is one of the constellations of the zodiac. Its name is a Latin word meaning 'bull', and its astrological symbol is a stylized bull's head:...

 extends over such a large part of the sky, 20 degrees, that the proper motions as derived from astrometry
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...

 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
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

 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.

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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

.

Derivation


For a right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

,

where is the parallax, 1 AU is approximately the average distance from the Sun to Earth, and is the distance to the star.
Using small-angle approximations (valid when the angle is small compared to 1 radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

),

so the parallax, measured in arcseconds, is
If the parallax is 1", then the distance is
This defines the parsec
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....

, a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply , when the parallax is given in arcseconds.

Parallax error


Precise parallax measurements of distance have an associated error
Error
The word error entails different meanings and usages relative to how it is conceptually applied. The concrete meaning of the Latin word "error" is "wandering" or "straying". Unlike an illusion, an error or a mistake can sometimes be dispelled through knowledge...

. 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
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.

Parallax and measurement instruments


The thickness of a ruler can create parallax in fine measurements. This is avoided by taking measurements with the 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 occurs when reading the position of a pointer against a scale in an instrument such as a galvanometer
Galvanometer
A galvanometer is a type of ammeter: an instrument for detecting and measuring electric current. It is an analog electromechanical transducer that produces a rotary deflection of some type of pointer in response to electric current flowing through its coil in a magnetic field. .Galvanometers were...

 (for example, in an analog-display multimeter
Multimeter
A multimeter or a multitester, also known as a VOM , is an electronic measuring instrument that combines several measurement functions in one unit. A typical multimeter may include features such as the ability to measure voltage, current and resistance...

.) To help the user avoid this problem, the scale is sometimes printed above a narrow strip of mirror
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...

, and the user positions his eye
Human eye
The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

 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. Similar error also happens when a cars speedometer speed reading appears different to car's passenger from what it does 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
Photogrammetry
Photogrammetry is the practice of determining the geometric properties of objects from photographic images. Photogrammetry is as old as modern photography and can be dated to the mid-nineteenth century....

.

Parallax error in photography



Parallax error can be seen when taking photos with many types of cameras, such as twin-lens reflex camera
Twin-lens reflex camera
A twin-lens reflex camera is a type of camera with two objective lenses of the same focal length. One of the lenses is the photographic objective or "taking lens" , while the other is used for the viewfinder system, which is usually viewed from above at waist level...

s and those including viewfinder
Viewfinder
In photography, a viewfinder is what the photographer looks through to compose, and in many cases to focus, the picture. Most viewfinders are separate, and suffer parallax, while the single-lens reflex camera lets the viewfinder use the main optical system. Viewfinders are used in many cameras of...

s (such as rangefinder camera
Rangefinder camera
A rangefinder camera is a camera fitted with a rangefinder: a range-finding focusing mechanism allowing the photographer to measure the subject distance and take photographs that are in sharp focus...

s). 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 camera
Single-lens reflex camera
A single-lens reflex camera is a camera that typically uses a semi-automatic moving mirror system that permits the photographer to see exactly what will be captured by the film or digital imaging system, as opposed to pre-SLR cameras where the view through the viewfinder could be significantly...

s, 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.

Parallax is also an issue in image stitching
Image stitching
Image stitching or photo stitching is the process of combining multiple photographic images with overlapping fields of view to produce a segmented panorama or high-resolution image. Commonly performed through the use of computer software, most approaches to image stitching require nearly exact...

, such as for panoramas.

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
Super Nintendo Entertainment System
The Super Nintendo Entertainment System is a 16-bit video game console that was released by Nintendo in North America, Europe, Australasia , and South America between 1990 and 1993. In Japan and Southeast Asia, the system is called the , or SFC for short...

. 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 continue to be used for many online applications where the bandwidth required by three-dimensional graphics is excessive.

Parallax in sights


Parallax affects sights
Sight (device)
A sight is a device used to assist aligning or aim weapons, surveying instruments, or other items by eye. Sights can be a simple set or system of markers that have to be aligned together as well as aligned with the target...

 in many ways. On sights fitted to small arms, bows in archery, etc the distance between the sighting mechanism and the weapon's bore or axis can introduce significant errors when firing at close range, particularly when firing at small targets. This difference is generally referred to as "sight height" and is compensated for (when needed) via calculations that also take in other variables such as bullet drop, windage
Windage
Windage is a force created on an object by friction when there is relative movement between air and the object.There are two causes of windage:# the object is moving and being slowed by resistance from the air...

, and the distance at which the target is expected to be.

Parallax in optical sights



In optical sights parallax refers to the apparent movement of the reticle in relationship to the target when the user moves his/her head laterally behind the sight (up/down or left/right), i.e. it is an error where the reticle does not stay aligned with the sight's own optical axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

.

In optical instruments such as telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

s, microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

s, or in telescopic sight
Telescopic sight
A telescopic sight, commonly called a scope, is a sighting device that is based on an optical refracting telescope. They are equipped with some form of graphic image pattern mounted in an optically appropriate position in their optical system to give an accurate aiming point...

s used on small arms and theodolite
Theodolite
A theodolite is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are mainly used for surveying applications, and have been adapted for specialized purposes in fields like metrology and rocket launch technology...

s, the error occurs when the optics are not precisely focused: the reticle will appear to move with respect to the object focused on if one moves one's head sideways in front of the eyepiece. Some firearm telescopic sights are equipped with a parallax compensation mechanism which basically consists of a movable optical element that enables the optical system to project the picture of objects at varying distances and the reticle crosshairs pictures together in exactly the same optical plane. Telescopic sights may have no parallax compensation because they can perform very acceptably without refinement for parallax with the sight being permanently adjusted for the distance that best suits their intended usage. Typical standard factory parallax adjustment distances for hunting telescopic sights are 100 yd or 100 m to make them suited for hunting shots that rarely exceed 300 yd/m. Some target and military style telescopic sights without parallax compensation may be adjusted to be parallax free at ranges up to 300 yd/m to make them better suited for aiming at longer ranges. Scopes for rimfires, shotgun
Shotgun
A shotgun is a firearm that is usually designed to be fired from the shoulder, which uses the energy of a fixed shell to fire a number of small spherical pellets called shot, or a solid projectile called a slug...

s, and muzzleloader
Muzzleloader
A muzzleloader is any firearm into which the projectile and usually the propellant charge is loaded from the muzzle of the gun . This is distinct from the more popular modern designs of breech-loading firearms...

s will have shorter parallax settings, commonly 50 yd/m for rimfire scopes and 100 yd/m for shotguns and muzzleloaders. Scopes for airguns are very often found with adjustable parallax, usually in the form of an adjustable objective, or AO. These may adjust down as far as 3 yards (2.74 m).

Non-magnifying reflector or "reflex" sights
Reflector sight
A reflector or reflex sight is a generally non-magnifying optical device that allows the user to look through a partially reflecting glass element and see an illuminated projection of an aiming point or some other image superimposed on the field of view...

 have the ability to be theoretically "parallax free". But since these sights use parallel collimated light
Collimated light
Collimated light is light whose rays are parallel, and therefore will spread slowly as it propagates. The word is related to "collinear" and implies light that does not disperse with distance , or that will disperse minimally...

 this is only true when the target is at infinity. At finite distances eye movement perpendicular to the device will cause parallax movement in the reticle image in exact relationship to eye position in the cylindrical column of light created by the collimating optics. Firearm sights, such as some red dot sights, try to correct for this via not focusing the reticle at infinity, but instead at some finite distance, a designed target range where the reticle will show very little movement due to parallax. Some manufactures market reflector sight models they call "parallax free", but this refers to an optical system that compensates for off axis spherical aberration
Spherical aberration
thumb|right|Spherical aberration. A perfect lens focuses all incoming rays to a point on the [[Optical axis|optic axis]]. A real lens with spherical surfaces suffers from spherical aberration: it focuses rays more tightly if they enter it far from the optic axis than if they enter closer to the...

, an error that can cause the reticle position to diverge off the sight's optical axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

 with change in eye position.

Artillery gunfire


Because of the positioning of field
Field artillery
Field artillery is a category of mobile artillery used to support armies in the field. These weapons are specialized for mobility, tactical proficiency, long range, short range and extremely long range target engagement....

 or naval artillery
Naval artillery
Naval artillery, or naval riflery, is artillery mounted on a warship for use in naval warfare. Naval artillery has historically been used to engage either other ships, or targets on land; in the latter role it is currently termed naval gunfire fire support...

 guns, each one has a slightly different perspective of the target relative to the location of the fire-control system
Fire-control system
A fire-control system is a number of components working together, usually a gun data computer, a director, and radar, which is designed to assist a weapon system in hitting its target. It performs the same task as a human gunner firing a weapon, but attempts to do so faster and more...

 itself. Therefore, when aiming its guns at the target, the fire control system must compensate for parallax in order to assure that fire
Gunfire
Gunfire can refer to:* Gunshot, the discharge of a firearm* Gunfire , a DC comic book superhero* Gunfire , a 1934 Western starring Rex Bell* Gunfire , a 1950 Western starring Don "Red" Barry...

 from each gun converges on the target.

Parallax rangefinders


A coincidence rangefinder
Coincidence rangefinder
A coincidence rangefinder is a type of rangefinder that uses mechanical and optical principles to allow an operator to determine the distance to a visible object....

 or parallax rangefinder can be used to find distance to a target.

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
James Joyce
James Augustine Aloysius Joyce was an Irish novelist and poet, considered to be one of the most influential writers in the modernist avant-garde of the early 20th century...

's 1922 novel, Ulysses
Ulysses (novel)
Ulysses is a novel by the Irish author James Joyce. It was first serialised in parts in the American journal The Little Review from March 1918 to December 1920, and then published in its entirety by Sylvia Beach on 2 February 1922, in Paris. One of the most important works of Modernist literature,...

. Orson Scott Card
Orson Scott Card
Orson Scott Card is an American author, critic, public speaker, essayist, columnist, and political activist. He writes in several genres, but is primarily known for his science fiction. His novel Ender's Game and its sequel Speaker for the Dead both won Hugo and Nebula Awards, making Card the...

 also used the term when referring to Ender's Shadow
Ender's Shadow
Ender's Shadow is a parallel science fiction novel by the American author Orson Scott Card, taking place at the same time as the novel Ender's Game and depicting the same events from the point of view of Bean, a supporting character in the original novel. It was originally to be titled Urchin, but...

 as compared to Ender's Game
Ender's Game
Ender's Game is a science fiction novel by American author Orson Scott Card. The book originated as the short story "Ender's Game", published in the August 1977 issue of Analog Science Fiction and Fact. Elaborating on characters and plot lines depicted in the novel, Card later wrote additional...

. The artist Sarah Morris
Sarah Morris
Sarah Morris , is a British-born American artist.-Education and exhibitions:Morris double majored in Semiotics and Political Philosophy at Brown University, graduating magna cum laude...

 named her studio Parallax, in reference to her parallel production of paintings and films.

The metaphor is invoked by Slovenian philosopher Slavoj Žižek
Slavoj Žižek
Slavoj Žižek is a Slovenian philosopher, critical theorist working in the traditions of Hegelianism, Marxism and Lacanian psychoanalysis. He has made contributions to political theory, film theory, and theoretical psychoanalysis....

 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 Lacan
Lacan
Lacan is surname of:* Jacques Lacan , French psychoanalyst and psychiatrist** The Seminars of Jacques Lacan** From Bakunin to Lacan: Anti-Authoritarianism and the Dislocation of Power, a book on political philosophy by Saul Newman** Lacan at the Scene* Judith Miller, née Lacan...

ese—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."

See also

  • Triangulation
    Triangulation
    In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...

    , wherein a point is calculated given its angles from other known points
  • Trilateration
    Trilateration
    In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

    , wherein a point is calculated given its distances from other known points
  • Disparity
    Binocular disparity
    Binocular disparity refers to the difference in image location of an object seen by the left and right eyes, resulting from the eyes' horizontal separation. The brain uses binocular disparity to extract depth information from the two-dimensional retinal images in stereopsis...

  • Spectroscopic parallax
    Spectroscopic parallax
    Spectroscopic parallax is an astronomical method for measuring the distances to stars. Despite its name, it does not rely on the apparent change in the position of the star . This technique can be applied to any main sequence star for which a spectrum can be recorded...

  • Trigonometry
    Trigonometry
    Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

  • Xallarap
    Xallarap
    Xallarap is a variation in a gravitational lensing observation caused by the orbital motion of the source. A more traditional and similar effect, parallax, is the variation caused by motion of the earth around the sun. Since the two effects are converses of each other, this led to the name...


External links