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Encyclopedia > Parallax

Parallax, or more accurately motion parallax (Greek: παραλλαγή (parallagé) = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, caused by the motion of an observer. Simply put, it is the shift of an object against a background caused by a change in observer position. If there is no parallax between two objects then they are side by side at the exact same height. Parallax is the change of angular position of two stationary points relative to each other as seen by an observer, due to the observers motion. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... Stationary points (red pluses) and inflection points (green circles). ...

Figure 1: A simplified example of parallax

This parallax is often thought of as the 'apparent motion' of an object against a distant background because of a perspective shift, as seen in Figure 1. When viewed from Viewpoint A, the object appears to be closer to the blue square. When the viewpoint is changed to Viewpoint B, the object appears to have moved in front of the red square. It is most commonly used in astronomy. Image File history File links Parallax_Example. ... Image File history File links Parallax_Example. ...

## Use in distance measurement

By observing parallax, measuring angles and using geometry, one can determine the distance to various objects. When this is in reference to stars, the effect is known as stellar parallax. The first successful measurements of a stellar parallax were made by Friedrich Bessel in 1838 , for the star 61 Cygni. Observation basically means watching something and taking note of anything it does. ... Various meters Measurement is an observation that reduces an uncertainty expressed as a quantity. ... This article is about angles in geometry. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... Friedrich Wilhelm Bessel (July 22, 1784 â€“ March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... 61 Cygni is a star in the constellation Cygnus. ...

Distance measurement by parallax is a special case of the principle of triangulation, where 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 only one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of a triangulation network covering the whole nation. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side and the small top angle (the other two being close to 90 degrees), the long sides (in practice equal) can be determined. Triangulation can be used to find the distance from the shore to the ship. ...

### Parallax error

Precise parallax measurements of distance usually have an associated error. Thus a parallax may be described as some angle ± some angle-error. However this "± angle-error" will not translate directly into a ± error for the range, 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. The word error has different meanings in different domains. ...

However an approximation of the distance error can be computed by means of the following:

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

where d is the distance and p is the parallax. The approximation is far more accurate for relatively small values of the parallax error when compared to the parallax.

### 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 these 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 usually found above the lens of the camera, photos with parallax error are characterized as being slightly lower than intended, the classic example being the image of person with his/her head cropped off. This problem is addressed in single-lens reflex cameras, where the viewfinder sees through the same lens through which the photo is taken, thus avoiding parallax error. The front of a Kinaflex twin-lens reflex camera Sketch of an early 20th century twin-lens reflex camera 1957 Kodak Duaflex IV, an inexpensive fixed-focus TLR A twin-lens reflex camera (TLR) is a type of camera with two objective lenses of the same focal length. ... In photography a viewfinder is what the photographer looks through to compose, and in many cases to focus, the picture. ... A Foca camera of 1947 at the MusÃ©e des Arts et MÃ©tiers in Paris. ... Cross-section view of SLR system: 1) Lens 2) Mirror 3) Shutter 4) Film or sensor 5) Focusing screen 6) Condensing lens 7) Pentaprism 8) Eyepiece The single-lens reflex (SLR) is a type of camera that uses a movable mirror placed between the lens and the film to project...

## Parallax and measurement instruments

If an optical instrument — telescope, microscope, theodolite — is imprecisely focused, the 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 carefully focus in order to 'eliminate the parallax', and to check by moving one's head. This article does not cite any references or sources. ... Robert Hookes microscope (1665) - an engineered device used to study living systems. ... An optical theodolite, manufactured in the Soviet Union in 1958 and used for topographic surveying. ...

Also in non-optical measurements the thickness of a ruler can create parallax in fine measurements. One is always cautioned in science classes to "avoid parallax." By this it is meant that one should always 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. To help the user to 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. It has been suggested that Tangent galvanometer be merged into this article or section. ... A mirror, reflecting a vase. ... For other uses, see Eye (disambiguation). ...

In photography, one also talks about the parallax of a camera viewfinder: for nearby objects, a viewfinder mounted on top of the camera will show something different from what the lens 'sees', and people's heads may be cut off. The problem does not exist for the single lens reflex camera, where the viewfinder looks (with the aid of a movable mirror) through the same lens as is used for taking the photograph. The single-lens reflex camera, more commonly known by the abbreviation SLR, uses a mirror placed between the lens and the film to project the image seen through the lens to a matte focusing screen. ...

## Photogrammetric parallax

Aerial pic 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 is a remote sensing technology in which geometric properties about objects are determined from photographic images. ...

## Lunar parallax

Definition: Lunar parallax involves the arithmetic of parallel stars between two inverted axis which rotate in angles that are only visible by the moon.

Example of lunar parallax: Occultation of Pleiades by the Moon

Jules Verne, From the Earth to the Moon (1865). "Up till 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)." Example of lunar parallax from 4 points on earth This is a simulated image, combining of 4 views of the sky and the moons location relative to the background stars at a single point in time. ... Example of lunar parallax from 4 points on earth This is a simulated image, combining of 4 views of the sky and the moons location relative to the background stars at a single point in time. ... This article is about the French author. ... The projectile, as pictured in an engraving from the 1872 Illustrated Edition. ... 1865 (MDCCCLXV) is a common year starting on Sunday. ...

Another way to use parallax to determine the distance to the Moon would be to take two pictures of the Moon at exactly the same time from two locations on Earth, and compare the position of the Moon relative to the visible stars. Using the orientation of the Earth, and those two points, and a perpendicular displacement, a distance to the Moon can be triangulated.

$textit{distance}_{textit{moon}} = frac {textit{distance}_{textit{observerbase}}} {tan (mathit{angle})}$

## Solar parallax

After Copernicus proposed his heliocentric system, with an Earth in revolution around the Sun, it was possible to build a scale model of the whole solar system, but without the 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 or astronomical unit (AU). When done 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 thus, according to Big Bang theory, the minimum age — of the visible Universe. Nicolaus Copernicus (in Latin; Polish Miko&#322;aj Kopernik, German Nikolaus Kopernikus - February 19, 1473 &#8211; May 24, 1543) was a Polish astronomer, mathematician and economist who developed a heliocentric (Sun-centered) theory of the solar system in a form detailed enough to make it scientifically useful. ... Sol redirects here. ... The astronomical unit (AU or au or a. ... Triangulation can be used to find the distance from the shore to the ship. ... For other uses, see Big Bang (disambiguation). ...

A primitive way of determining 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 argued that the Sun, Moon, and Earth form a right triangle 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. 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; which, although wrong, follows logically from his incorrect data. It does suggest that the Sun is clearly larger than the Earth, which can be taken to support the heliocentric model. Although his 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. For other uses of this name, including the grammarian Aristarchus of Samothrace, see Aristarchus Statue of Aristarchus at Aristotle University in Thessalonica, Greece Aristarchus (Greek: á¼ˆÏÎ¯ÏƒÏ„Î±ÏÏ‡Î¿Ï‚; 310 BC - ca. ... Lunar phase refers to the appearance of the illuminated portion of the Moon as seen by an observer, usually on Earth. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... This article is about angles in geometry. ... The 2004 transit of Venus A transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and Earth, obscuring a small portion of the Suns disk. ...

This method was proposed by Edmond Halley in 1716 , although he did not live to see the results. Download high resolution version (898x441, 26 KB)Self-drawn. ... // Portrait of Edmond Halley painted around 1687 by Thomas Murray (Royal Society, London) Portrait of Edmond Halley Bust of Edmond Halley in the Museum of the Royal Greenwich Observatory Edmond Halley FRS (sometimes Edmund; IPA: ) (November 8, 1656 â€“ January 14, 1742) was an English astronomer, geophysicist, mathematician, meteorologist, and physicist. ...

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% over the currently accepted value, 149.6 million kilometers. The black drop effect is an optical phenomenon visible during a transit of Venus. ...

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. 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. 253 Mathilde, a C-type asteroid. ... The asteroid 433 Eros (eer-os) was named after the Greek god of love Eros. ... Sir Harold Spencer Jones (March 29, 1890 â€“ November 3, 1960) was a British astronomer. ...

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 completely. For other uses, see Radar (disambiguation). ... 1566 Icarus is an Apollo asteroid (a sub-class of near-Earth asteroid) whose unusual characteristic is that at perihelion it is closer to the Sun than Mercury; it is said to be a Mercury-crosser asteroid. ... The Space Shuttle Discovery as seen from the International Space Station. ... Telemetry is a technology that allows the remote measurement and reporting of information of interest to the system designer or operator. ...

## Stellar parallax

Stellar parallax motion

On an interstellar scale, parallax created by the different orbital positions of the Earth causes nearby stars to appear to move relative to the more distant stars. However, this effect is so small it is undetectable without extremely precise measurements. Image File history File links Stellarparallax2. ... Image File history File links Stellarparallax2. ...

The annual parallax is 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. Given two points on opposite ends of the orbit, the parallax is half the maximum parallactic shift evident from the star viewed from the two points. The parsec is the distance for which the annual parallax is 1 arcsecond. A parsec equals 3.26 light years. A parsec is the distance from the Earth to an astronomical object which has a parallax angle of one arcsecond. ... A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 â‰ˆ 7. ...

The distance of an object (in parsecs) can be computed as the reciprocal of the parallax. For instance, the Hipparcos satellite measured the parallax of the nearest star, Proxima Centauri, as .77233 seconds of arc (±.00242"). Therefore, the distance is 1/0.772=1.29 parsecs or about 4.22 light years (±.01 ly). The reciprocal function: y = 1/x. ... Hipparcos (for High Precision Parallax Collecting Satellite) was an astrometry mission of the European Space Agency (ESA) dedicated to the measurement of stellar parallax and the proper motions of stars. ... Proxima Centauri (Latin proximus, -a, -um: meaning next to or nearest to)[4] is a red dwarf star that is likely a part of the Alpha Centauri star system and is the nearest star to the Sun at a distance of 4. ... A parsec is the distance from the Earth to an astronomical object which has a parallax angle of one arcsecond. ... A light-year or lightyear (symbol: ly) is a unit of measurement of length, specifically the distance light travels in vacuum in one year. ...

The angles involved in these calculations are very small. For example, .772 arcseconds is roughly the angle subtended by an object about 2 centimeters in diameter (roughly the size of a U.S. Quarter Dollar) located about 5.3 kilometers away. In mathematics the term subtended usually refers to the direct relationship between an angle and its arc length. ... A quarter is a coin worth one-quarter of a United States dollar, or 25 cents. ...

### Computation

The parallax $p'' = frac {1 AU} {d} cdot 180 cdot frac {3600} {pi}$ in arc seconds

where

1 AU = 1 astronomical unit = Average distance from the Sun to earth = 1.4959 · 1011 m
d = distance to the star

right triangle
$sin p = frac {1 AU} {d}$
small angle approximation
$sin x ~= xtextrm{ radians} = x cdot frac {180} {pi} textrm{ degrees} = x cdot 180 cdot frac {3600} {pi}$ arcseconds
parallax $p'' ~= frac {1 AU} {d} cdot 180 cdot frac{3600} {pi}$
If the parallax is 1", then the distance is $d = 1 AU cdot 180 cdot frac {3600} {pi}$ = 206,265 AU = 3.2616 lyr = 1 parsec (This defines the parsec)
The parallax $p = frac {1} {d}$ arcseconds, when the distance is given in parsecs

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 a gigantic size seemed entirely implausible. Heliocentric Solar System Heliocentrism (lower panel) in comparison to the geocentric model (upper panel) In astronomy, heliocentrism is the theory that the sun is at the centre of the Universe and/or the Solar System. ... For other uses, see Euclid (disambiguation). ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...

Measurements of the annual parallax as the earth goes through its orbit was the first reliable way to determine the distances to the closest stars. This method was first successfully used by Friedrich Wilhelm Bessel in 1838 when he measured the distance to 61 Cygni with a heliometer, and it remains the standard for calibrating other measurement methods (after the size of the orbit of the earth is measured by radar reflection on other planets). STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... Friedrich Wilhelm Bessel (July 22, 1784 â€“ March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... 61 Cygni is a star in the constellation Cygnus. ... Heliometer (from Greek sun and measure) is an instrument originally designed for measuring the variation of the suns diameter at different seasons of the year, but applied now to the modern form of the instrument which is capable of much wider use. ... For other uses, see Radar (disambiguation). ...

In 1989 , a satellite called "Hipparcos" was launched with the main purpose of 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 bit more than one percent of the diameter of our galaxy. Hipparcos (for High Precision Parallax Collecting Satellite) was an astrometry mission of the European Space Agency (ESA) dedicated to the measurement of stellar parallax and the proper motions of stars. ... The proper motion of a star is the motion of the position of the star in the sky (the change in direction in which we see it, as opposed to the radial velocity) after eliminating the improper motions of the stars, which affect their measured coordinates but are not real... For other uses, see Milky Way (disambiguation). ...

## 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 us to estimate the distance of the cluster (151 light years) and its member stars in much the same way as using annual parallax. The Hyades (Ã†Î¥Î¬Î´ÎµÏ‚ also known as Melotte 25 or Collinder 50 or Caldwell 41) is an open star cluster located in the constellation Taurus. ... Taurus (IPA: , Latin: , symbol , ) is one of the constellations of the zodiac. ... Illustration of the use of optical wavelength interferometry to determine precise positions of stars. ... This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ...

Dynamic parallax has sometimes also been used to determine the distance to a supernova, when the optical wave front of the outburst was seen to propagate through the surrounding dust clouds at an apparent angular velocity, when we know its true propagation velocity to be the speed of light. â€œLightspeedâ€ redirects here. ...

## The scale of the Universe

All these various astronomical parallax methods allow us to establish the first rungs on the cosmic scale ladder, out to a few hundred light years. Beyond that, other methods must be taken into use: e.g., "spectroscopic parallaxes" — not really parallaxes at all. It is a prototype of a "standard candle" method, where we observe the apparent brightness of an object we know, based on some physical theory, the true brightness of. For groups of stars we have the Hertzsprung-Russell diagram which allows us to derive a star's absolute brightness or magnitude M from its spectral type. The observed (apparent) brightness or magnitude being m, we can then derive its parallax p by A light-year or lightyear (symbol: ly) is a unit of measurement of length, specifically the distance light travels in vacuum in one year. ... Spectroscopy is the study of spectra, ie. ... The Hertzsprung-Russell diagram (usually referred to by the abbreviation H-R diagram or HRD, also known as a Colour-Magnitude diagram, or CMD) shows the relationship between absolute magnitude, luminosity, classification, and effective temperature of stars. ... In astronomy, absolute magnitude is the apparent magnitude, m, an object would have if it were at a standard luminosity distance away from us, in the absence of interstellar extinction. ... The apparent magnitude (m) of a star, planet or other celestial body is a measure of its apparent brightness as seen by an observer on Earth. ...

$M - m = 5 log (p) + 5 ,$

called "spectroscopic parallax".

Further methods, mostly of the standard candle variety, are the variable stars called Cepheids — the absolute brightness of which depends on their observed period of variation —, supernova brightnesses, globular cluster sizes and brightnesses, complete galaxy brightnesses etc. These are all much more uncertain as they are not based on simple geometry. Yet, parallaxes are the basis of everything, as they allow the calibration of these more uncertain methods in the Solar neighbourhood. A standard candle is an astronomical object that has a known luminosity. ... A Cepheid variable is a member of a particular class of variable stars, notable for a fairly tight correlation between their period of variability and absolute stellar luminosity. ... Multiwavelength X-ray image of the remnant of Keplers Supernova, SN 1604. ... The Globular Cluster M80 in the constellation Scorpius is located about 28,000 light years from the Sun and contains hundreds of thousands of stars. ... For other uses, see Galaxy (disambiguation). ...

A very modern method which is not a traditional parallax method but also geometric in nature, is "gravitational lensing parallax". It depends on observing the differential time delay of brightness variations from a remote quasar reaching us by two different paths through the gravitational field or "lens" of a foreground galaxy. If the redshifts of both the quasar and the foreground galaxy are known, one can show that the absolute distances of both are proportional to the differential delay, and can in fact be calculated given also the geometry of the gravitational lens on the celestial sphere. This article or section is in need of attention from an expert on the subject. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

All these independent techniques aim at determining Hubble's constant, the constant describing how the redshift of galaxies, due to the Universe's expansion, depends on these galaxies' distance from us. Knowing Hubble's constant again allows us to determine, by simply running the film of the cosmic expansion backwards, how long ago it was when all these galaxies were collected in a single point -- the Big Bang. Current knowledge puts this at some 13.7 billion years ago, but with considerable uncertainty and dependence on various model assumptions. Hubbles law is the statement in astronomy that the redshift in light coming from distant galaxies is proportional to their distance. ... Redshift of spectral lines in the optical spectrum of a supercluster of distant galaxies (right), as compared with that of the Sun (left). ... For other uses, see Big Bang (disambiguation). ...

## Sources

Results from FactBites:

 Parallax - Wikipedia, the free encyclopedia (2765 words) Parallax, or more accurately motion parallax (Greek: παραλλαγή (parallagé) = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, due to the motion of an observer. Distance measurement by parallax is a special case of the principle of triangulation, where 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 only one side has been measured. Dynamic parallax has sometimes also been used to determine the distance to a supernova, when the optical wave front of the outburst was seen to propagate through the surrounding dust clouds at an apparent angular velocity, when we know its true propagation velocity to be the speed of light.
More results at FactBites »

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