A planet is an astronomical body orbiting a star or stellar remnant, massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, has cleared its neighbouring region of planetesimals. The term planet is ancient, with ties to history, science and religion. Five planets in the Solar System are visible to the naked eye; these were regarded by many early cultures as emissaries of deities. As scientific knowledge advanced, human perception of the planets changed, incorporating a number of disparate objects. In 2006, the International Astronomical Union adopted a resolution defining planets within the Solar System; this definition is controversial because it excludes many objects of planetary mass based on where or what they orbit. Although eight of the planetary bodies discovered before 1950 remain "planets" under the modern definition, some celestial bodies, such as Ceres, Pallas and Vesta, Pluto, that were once considered planets by the scientific community, are no longer viewed as such.
The planets were thought by Ptolemy to orbit Earth in epicycle motions. Although the idea that the planets orbited the Sun had been suggested many times, it was not until the 17th century that this view was supported by evidence from the first telescopic astronomical observations, performed by Galileo Galilei. About the same time, by careful analysis of pre-telescopic observational data collected by Tycho Brahe, Johannes Kepler found the planets' orbits were elliptical rather than circular; as observational tools improved, astronomers saw that, like Earth, each of the planets rotated around an axis tilted with respect to its orbital pole, some shared such features as ice caps and seasons. Since the dawn of the Space Age, close observation by space probes has found that Earth and the other planets share characteristics such as volcanism, hurricanes and hydrology. Planets are divided into two main types: large low-density giant planets, smaller rocky terrestrials. There are eight planets in the Solar System.
In order of increasing distance from the Sun, they are the four terrestrials, Venus and Mars the four giant planets, Saturn and Neptune. Six of the planets are orbited by one or more natural satellites. Several thousands of planets around other stars have been discovered in the Milky Way; as of 1 April 2019, 4,023 known extrasolar planets in 3,005 planetary systems, ranging in size from just above the size of the Moon to gas giants about twice as large as Jupiter have been discovered, out of which more than 100 planets are the same size as Earth, nine of which are at the same relative distance from their star as Earth from the Sun, i.e. in the circumstellar habitable zone. On December 20, 2011, the Kepler Space Telescope team reported the discovery of the first Earth-sized extrasolar planets, Kepler-20e and Kepler-20f, orbiting a Sun-like star, Kepler-20. A 2012 study, analyzing gravitational microlensing data, estimates an average of at least 1.6 bound planets for every star in the Milky Way.
Around one in five Sun-like stars is thought to have an Earth-sized planet in its habitable zone. The idea of planets has evolved over its history, from the divine lights of antiquity to the earthly objects of the scientific age; the concept has expanded to include worlds not only in the Solar System, but in hundreds of other extrasolar systems. The ambiguities inherent in defining planets have led to much scientific controversy; the five classical planets, being visible to the naked eye, have been known since ancient times and have had a significant impact on mythology, religious cosmology, ancient astronomy. In ancient times, astronomers noted how certain lights moved across the sky, as opposed to the "fixed stars", which maintained a constant relative position in the sky. Ancient Greeks called these lights πλάνητες ἀστέρες or πλανῆται, from which today's word "planet" was derived. In ancient Greece, China and indeed all pre-modern civilizations, it was universally believed that Earth was the center of the Universe and that all the "planets" circled Earth.
The reasons for this perception were that stars and planets appeared to revolve around Earth each day and the common-sense perceptions that Earth was solid and stable and that it was not moving but at rest. The first civilization known to have a functional theory of the planets were the Babylonians, who lived in Mesopotamia in the first and second millennia BC; the oldest surviving planetary astronomical text is the Babylonian Venus tablet of Ammisaduqa, a 7th-century BC copy of a list of observations of the motions of the planet Venus, that dates as early as the second millennium BC. The MUL. APIN is a pair of cuneiform tablets dating from the 7th century BC that lays out the motions of the Sun and planets over the course of the year; the Babylonian astrologers laid the foundations of what would become Western astrology. The Enuma anu enlil, written during the Neo-Assyrian period in the 7th century BC, comprises a list of omens and their relationships with various celestial phenomena including the motions of the planets.
Venus and the outer planets Mars and Saturn were all identified by Babylonian astronomers. These would remain the only known planets until the invention of the telescope in early modern times; the ancient Greeks did not attach as much significance to the planets as the Babylonians. The Pythagoreans, in the 6th and 5t
The parsec is a unit of length used to measure large distances to astronomical objects outside the Solar System. A parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond, which corresponds to 648000/π astronomical units. One parsec is equal to 31 trillion kilometres or 19 trillion miles; the nearest star, Proxima Centauri, is about 1.3 parsecs from the Sun. Most of the stars visible to the unaided eye in the night sky are within 500 parsecs of the Sun; the parsec unit was first suggested in 1913 by the British astronomer Herbert Hall Turner. Named as a portmanteau of the parallax of one arcsecond, it was defined to make calculations of astronomical distances from only their raw observational data quick and easy for astronomers. For this reason, it is the unit preferred in astronomy and astrophysics, though the light-year remains prominent in popular science texts and common usage. Although parsecs are used for the shorter distances within the Milky Way, multiples of parsecs are required for the larger scales in the universe, including kiloparsecs for the more distant objects within and around the Milky Way, megaparsecs for mid-distance galaxies, gigaparsecs for many quasars and the most distant galaxies.
In August 2015, the IAU passed Resolution B2, which, as part of the definition of a standardized absolute and apparent bolometric magnitude scale, mentioned an existing explicit definition of the parsec as 648000/π astronomical units, or 3.08567758149137×1016 metres. This corresponds to the small-angle definition of the parsec found in many contemporary astronomical references; the parsec is defined as being equal to the length of the longer leg of an elongated imaginary right triangle in space. The two dimensions on which this triangle is based are its shorter leg, of length one astronomical unit, the subtended angle of the vertex opposite that leg, measuring one arc second. Applying the rules of trigonometry to these two values, the unit length of the other leg of the triangle can be derived. One of the oldest methods used by astronomers to calculate the distance to a star is to record the difference in angle between two measurements of the position of the star in the sky; the first measurement is taken from the Earth on one side of the Sun, the second is taken half a year when the Earth is on the opposite side of the Sun.
The distance between the two positions of the Earth when the two measurements were taken is twice the distance between the Earth and the Sun. The difference in angle between the two measurements is twice the parallax angle, formed by lines from the Sun and Earth to the star at the distant vertex; the distance to the star could be calculated using trigonometry. The first successful published direct measurements of an object at interstellar distances were undertaken by German astronomer Friedrich Wilhelm Bessel in 1838, who used this approach to calculate the 3.5-parsec distance of 61 Cygni. The parallax of a star is defined as half of the angular distance that a star appears to move relative to the celestial sphere as Earth orbits the Sun. Equivalently, it is the subtended angle, from that star's perspective, of the semimajor axis of the Earth's orbit; the star, the Sun and the Earth form the corners of an imaginary right triangle in space: the right angle is the corner at the Sun, the corner at the star is the parallax angle.
The length of the opposite side to the parallax angle is the distance from the Earth to the Sun (defined as one astronomical unit, the length of the adjacent side gives the distance from the sun to the star. Therefore, given a measurement of the parallax angle, along with the rules of trigonometry, the distance from the Sun to the star can be found. A parsec is defined as the length of the side adjacent to the vertex occupied by a star whose parallax angle is one arcsecond; the use of the parsec as a unit of distance follows from Bessel's method, because the distance in parsecs can be computed as the reciprocal of the parallax angle in arcseconds. No trigonometric functions are required in this relationship because the small angles involved mean that the approximate solution of the skinny triangle can be applied. Though it may have been used before, the term parsec was first mentioned in an astronomical publication in 1913. Astronomer Royal Frank Watson Dyson expressed his concern for the need of a name for that unit of distance.
He proposed the name astron, but mentioned that Carl Charlier had suggested siriometer and Herbert Hall Turner had proposed parsec. It was Turner's proposal. In the diagram above, S represents the Sun, E the Earth at one point in its orbit, thus the distance ES is one astronomical unit. The angle SDE is one arcsecond so by definition D is a point in space at a distance of one parsec from the Sun. Through trigonometry, the distance SD is calculated as follows: S D = E S tan 1 ″ S D ≈ E S 1 ″ = 1 au 1 60 × 60 × π
Hanyu Pinyin abbreviated to pinyin, is the official romanization system for Standard Chinese in mainland China and to some extent in Taiwan. It is used to teach Standard Mandarin Chinese, written using Chinese characters; the system includes four diacritics denoting tones. Pinyin without tone marks is used to spell Chinese names and words in languages written with the Latin alphabet, in certain computer input methods to enter Chinese characters; the pinyin system was developed in the 1950s by many linguists, including Zhou Youguang, based on earlier forms of romanizations of Chinese. It was published by revised several times; the International Organization for Standardization adopted pinyin as an international standard in 1982, was followed by the United Nations in 1986. The system was adopted as the official standard in Taiwan in 2009, where it is used for international events rather than for educational or computer-input purposes, but "some cities and organizations, notably in the south of Taiwan, did not accept this", so it remains one of several rival romanization systems in use.
The word Hànyǔ means'the spoken language of the Han people', while Pīnyīn means'spelled sounds'. In 1605, the Jesuit missionary Matteo Ricci published Xizi Qiji in Beijing; this was the first book to use the Roman alphabet to write the Chinese language. Twenty years another Jesuit in China, Nicolas Trigault, issued his Xi Ru Ermu Zi at Hangzhou. Neither book had much immediate impact on the way in which Chinese thought about their writing system, the romanizations they described were intended more for Westerners than for the Chinese. One of the earliest Chinese thinkers to relate Western alphabets to Chinese was late Ming to early Qing dynasty scholar-official, Fang Yizhi; the first late Qing reformer to propose that China adopt a system of spelling was Song Shu. A student of the great scholars Yu Yue and Zhang Taiyan, Song had been to Japan and observed the stunning effect of the kana syllabaries and Western learning there; this galvanized him into activity on a number of fronts, one of the most important being reform of the script.
While Song did not himself create a system for spelling Sinitic languages, his discussion proved fertile and led to a proliferation of schemes for phonetic scripts. The Wade–Giles system was produced by Thomas Wade in 1859, further improved by Herbert Giles in the Chinese–English Dictionary of 1892, it was popular and used in English-language publications outside China until 1979. In the early 1930s, Communist Party of China leaders trained in Moscow introduced a phonetic alphabet using Roman letters, developed in the Soviet Oriental Institute of Leningrad and was intended to improve literacy in the Russian Far East; this Sin Wenz or "New Writing" was much more linguistically sophisticated than earlier alphabets, but with the major exception that it did not indicate tones of Chinese. In 1940, several thousand members attended a Border Region Sin Wenz Society convention. Mao Zedong and Zhu De, head of the army, both contributed their calligraphy for the masthead of the Sin Wenz Society's new journal.
Outside the CCP, other prominent supporters included Sun Fo. Over thirty journals soon appeared written in Sin Wenz, plus large numbers of translations, some contemporary Chinese literature, a spectrum of textbooks. In 1940, the movement reached an apex when Mao's Border Region Government declared that the Sin Wenz had the same legal status as traditional characters in government and public documents. Many educators and political leaders looked forward to the day when they would be universally accepted and replace Chinese characters. Opposition arose, because the system was less well adapted to writing regional languages, therefore would require learning Mandarin. Sin Wenz fell into relative disuse during the following years. In 1943, the U. S. military engaged Yale University to develop a romanization of Mandarin Chinese for its pilots flying over China. The resulting system is close to pinyin, but does not use English letters in unfamiliar ways. Medial semivowels are written with y and w, apical vowels with r or z.
Accent marks are used to indicate tone. Pinyin was created by Chinese linguists, including Zhou Youguang, as part of a Chinese government project in the 1950s. Zhou is called "the father of pinyin," Zhou worked as a banker in New York when he decided to return to China to help rebuild the country after the establishment of the People's Republic of China in 1949, he became an economics professor in Shanghai, in 1955, when China's Ministry of Education created a Committee for the Reform of the Chinese Written Language, Premier Zhou Enlai assigned Zhou Youguang the task of developing a new romanization system, despite the fact that he was not a professional linguist. Hanyu Pinyin was based on several existing systems: Gwoyeu Romatzyh of 1928, Latinxua Sin Wenz of 1931, the diacritic markings from zhuyin. "I'm not the father of pinyin," Zhou said years later. It's a lo
In observational astronomy, an asterism is a popularly-known pattern or group of stars that can be seen in the night sky. This colloquial definition makes it appear quite similar to a constellation, but they differ in that a constellation is an recognized area of the sky, while an asterism is a visually obvious collection of stars and the lines used to mentally connect them; this distinction between terms remains somewhat inconsistent. An asterism may be understood as an informal group of stars within the area of an official or defunct former constellation; some include stars from more than one constellation. Asterisms range from simple shapes of just few stars to more complex collections of many bright stars, they are useful for people. For example, the asterisms known as The Plough comprises the seven brightest stars in the International Astronomical Union recognised constellation Ursa Major. Another is the asterism of the Southern Cross. In many early civilizations, it was common to associate groups of stars in connect-the-dots stick-figure patterns.
This process was arbitrary, different cultures have identified different constellations, although a few of the more obvious patterns tend to appear in the constellations of multiple cultures, such as those of Orion and Scorpius. As anyone could arrange and name a grouping of stars there was no distinct difference between a constellation and an asterism. E.g. Pliny the Elder in his book Naturalis Historia mentions 72 asterisms. A general list containing 48 constellations began to develop with the astronomer Hipparchus, was accepted as standard in Europe for 1,800 years; as constellations were considered to be composed only of the stars that constituted the figure, it was always possible to use any leftover stars to create and squeeze in a new grouping among the established constellations. Furthermore, exploration by Europeans to other parts of the globe exposed them to stars unknown to them. Two astronomers known for expanding the number of southern constellations were Johann Bayer and Nicolas Louis de Lacaille.
Bayer had listed twelve figures made out of stars. Many of these proposed constellations have been formally accepted, but the rest have remained as asterisms. In 1928, the International Astronomical Union divided the sky into 88 official constellations following geometric boundaries encompassing all of the stars within them. Any additional new selected groupings of stars or former constellations are considered as asterisms. However, depending on the particular literature source, any technical distinctions between the terms'constellation' and'asterism' remain somewhat ambiguous. E.g. Both the open clusters The Pleiades or Seven Sisters and The Hyades in Taurus are sometimes considered as an asterisms, but this depends on the source. Component stars of asterisms mark out simple geometric shapes; the Great Diamond consisting of Arcturus, Spica and Cor Caroli. An East-West line from Arcturus to Denebola forms an equilateral triangle with Cor Caroli to the North, another with Spica to the South; the Arcturus, Spica triangle is sometimes given the name Spring Triangle.
Together these two triangles form the Diamond. Formally, the stars of the Diamond are in the constellations Boötes, Virgo and Canes Venatici; the Summer Triangle of Deneb and Vega — α Cygni, α Aquilae, α Lyrae — is recognized in the northern hemisphere summer skies, as its three stars are all of the 1st magnitude. The stars of the Triangle are in the band of the Milky Way which marks the galactic equator, are in the direction of the galactic center; the Great Square of Pegasus is the quadrilateral formed by the stars Markab, Scheat and Alpheratz, representing the body of the winged horse. The asterism was recognized as the constellation ASH. IKU "The Field" on the MUL. APIN cuneiform tablets from about 1100 to 700 BC. One-third of the 1st-magnitude stars visible in the sky are in the so-called Winter Hexagon with Capella, Rigel, Sirius and Pollux with 2nd-magnitude Castor, on the periphery, Betelgeuse off-center. Although somewhat flattened, thus more elliptical than circular, the figure is so large that it cannot be taken in all at once, thus making the lack of true circularity less noticeable.
Some prefer to regard it as a Heavenly'G'. The Winter Triangle visible in the northern sky's winter and comprise the first magnitude stars Procyon and Sirius. A familiar asterism is the Big Dipper, Plough or Charles's Wain, composed of the seven brightest stars in Ursa Major; these stars delineate the Bear's hindquarters and exaggerated tail, or alternatively, the "handle" forming the upper outline of the bear's head and neck. With its longer tail, Ursa Minor hardly appears bearlike at all, is known by its pseudonym, the Little Dipper; the Northern Cross in Cygnus. The upright runs from Deneb in the Swan's tail to Albireo in the beak; the transverse runs from Aljanah i
Minute and second of arc
A minute of arc, arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn, one minute of arc is 1/21600 of a turn – it is for this reason that the Earth's circumference is exactly 21,600 nautical miles. A minute of arc is π/10800 of a radian. A second of arc, arcsecond, or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, π/648000 of a radian; these units originated in Babylonian astronomy as sexagesimal subdivisions of the degree. To express smaller angles, standard SI prefixes can be employed; the number of square arcminutes in a complete sphere is 4 π 2 = 466 560 000 π ≈ 148510660 square arcminutes. The names "minute" and "second" have nothing to do with the identically named units of time "minute" or "second"; the identical names reflect the ancient Babylonian number system, based on the number 60. The standard symbol for marking the arcminute is the prime, though a single quote is used where only ASCII characters are permitted.
One arcminute is thus written 1′. It is abbreviated as arcmin or amin or, less the prime with a circumflex over it; the standard symbol for the arcsecond is the double prime, though a double quote is used where only ASCII characters are permitted. One arcsecond is thus written 1″, it is abbreviated as arcsec or asec. In celestial navigation, seconds of arc are used in calculations, the preference being for degrees and decimals of a minute, for example, written as 42° 25.32′ or 42° 25.322′. This notation has been carried over into marine GPS receivers, which display latitude and longitude in the latter format by default; the full moon's average apparent size is about 31 arcminutes. An arcminute is the resolution of the human eye. An arcsecond is the angle subtended by a U. S. dime coin at a distance of 4 kilometres. An arcsecond is the angle subtended by an object of diameter 725.27 km at a distance of one astronomical unit, an object of diameter 45866916 km at one light-year, an object of diameter one astronomical unit at a distance of one parsec, by definition.
A milliarcsecond is about the size of a dime atop the Eiffel Tower. A microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth. A nanoarcsecond is about the size of a penny on Neptune's moon Triton as observed from Earth. Notable examples of size in arcseconds are: Hubble Space Telescope has calculational resolution of 0.05 arcseconds and actual resolution of 0.1 arcseconds, close to the diffraction limit. Crescent Venus measures between 66 seconds of arc. Since antiquity the arcminute and arcsecond have been used in astronomy. In the ecliptic coordinate system and longitude; the principal exception is right ascension in equatorial coordinates, measured in time units of hours and seconds. The arcsecond is often used to describe small astronomical angles such as the angular diameters of planets, the proper motion of stars, the separation of components of binary star systems, parallax, the small change of position of a star in the course of a year or of a solar system body as the Earth rotates.
These small angles may be written in milliarcseconds, or thousandths of an arcsecond. The unit of distance, the parsec, named from the parallax of one arc second, was developed for such parallax measurements, it is the distance at which the mean radius of the Earth's orbit would subtend an angle of one arcsecond. The ESA astrometric space probe Gaia, launched in 2013, can approximate star positions to 7 microarcseconds. Apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red giant with a diameter of 0.05 arcsecond. Because of the effects of atmospheric seeing, ground-based telescopes will smear the image of a star to an angular diameter of about 0.5 arcsecond. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1 arcsecond. Space telescopes are diffraction limited. For example, the Hubble Space Telescope can reach an angular size of stars down to about 0.1″. Techniques exist for improving seeing on the ground. Adaptive optics, for example, can produce images around 0.05 arcsecond on a 10 m class telescope.
Minutes and seconds of arc are used in cartography and navigation. At sea level one minute of arc
The zodiac is an area of the sky that extends 8° north or south of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The paths of the Moon and visible planets are within the belt of the zodiac. In Western astrology, astronomy, the zodiac is divided into twelve signs, each occupying 30° of celestial longitude and corresponding to the constellations Aries, Gemini, Leo, Libra, Sagittarius, Capricorn and Pisces; the twelve astrological signs form a celestial coordinate system, or more an ecliptic coordinate system, which takes the ecliptic as the origin of latitude and the Sun's position at vernal equinox as the origin of longitude. The English word zodiac derives from zōdiacus, the Latinized form of the Ancient Greek zōidiakòs kýklos, meaning "cycle or circle of little animals". Zōidion is the diminutive of zōion; the name reflects the prominence of animals among the twelve signs. The zodiac was in use by the Roman era, based on concepts inherited by Hellenistic astronomy from Babylonian astronomy of the Chaldean period, which, in turn, derived from an earlier system of lists of stars along the ecliptic.
The construction of the zodiac is described in the Almagest. Although the zodiac remains the basis of the ecliptic coordinate system in use in astronomy besides the equatorial one, the term and the names of the twelve signs are today associated with horoscopic astrology; the term "zodiac" may refer to the region of the celestial sphere encompassing the paths of the planets corresponding to the band of about eight arc degrees above and below the ecliptic. The zodiac of a given planet is the band. By extension, the "zodiac of the comets" may refer to the band encompassing most short-period comets; the division of the ecliptic into the zodiacal signs originates in Babylonian astronomy during the first half of the 1st millennium BC. The zodiac draws on stars in earlier Babylonian star catalogues, such as the MUL. APIN catalogue, compiled around 1000 BC; some of the constellations can be traced further back, to Bronze Age sources, including Gemini "The Twins", from MAŠ. TAB. BA. GAL. GAL "The Great Twins", Cancer "The Crab", from AL.
LUL "The Crayfish", among others. Around the end of the 5th century BC, Babylonian astronomers divided the ecliptic into twelve equal "signs", by analogy to twelve schematic months of thirty days each; each sign contained thirty degrees of celestial longitude, thus creating the first known celestial coordinate system. According to calculations by modern astrophysics, the zodiac was introduced between 409 and 398 BC and within a few years of 401 BC Unlike modern astronomers, who place the beginning of the sign of Aries at the place of the Sun at the vernal equinox; the divisions do not correspond to where the constellations started and ended in the sky. The Sun in fact passed through at least 13, not 12 Babylonian constellations. In order to align with the number of months in a year, designers of the system omitted the major constellation Ophiuchus. Including smaller figures, astronomers have counted up to 21 eligible zodiac constellations. Changes in the orientation of the Earth's axis of rotation means that the time of year the sun is in a given constellation has changed since Babylonian times.
Because the division was made into equal arcs, 30° each, they constituted an ideal system of reference for making predictions about a planet's longitude. However, Babylonian techniques of observational measurements were in a rudimentary stage of evolution and they measured the position of a planet in reference to a set of "normal stars" close to the ecliptic as observational reference points to help positioning a planet within this ecliptic coordinate system. In Babylonian astronomical diaries, a planet position was given with respect to a zodiacal sign alone, less in specific degrees within a sign; when the degrees of longitude were given, they were expressed with reference to the 30° of the zodiacal sign, i.e. not with a reference to the continuous 360° ecliptic. In astronomical ephemerides, the positions of significant astronomical phenomena were computed in sexagesimal fractions of a degree. For daily ephemerides, the daily positions of a planet were not as important as the astrologically significant dates when the planet crossed from one zodiacal sign to the next.
Knowledge of the Babylonian zodiac is reflected in the Hebrew Bible. Some authors have linked the twelve tribes of Israel with the twelve signs and/or the lunar Hebrew calendar having 12 lunar months in a lunar year. Martin and others have argued that the arrangement of the tribes around the Tabernacle corresponded to the order of the Zodiac, with Judah, Reuben and Dan representing the middle signs of Leo, Aquarius and Scorpio, respectively; such connectio
Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distant stars. The components for proper motion in the equatorial coordinate system are given in the direction of right ascension and of declination, their combined value is computed as the total proper motion. It has dimensions of angle per time arcseconds per year or milliarcseconds per year. Knowledge of the proper motion and radial velocity allows calculations of true stellar motion or velocity in space in respect to the Sun, by coordinate transformation, the motion in respect to the Milky Way. Proper motion is not "proper", because it includes a component due to the motion of the Solar System itself. Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same constellations over historical time.
Ursa Major or Crux, for example, looks nearly the same now. However, precise long-term observations show that the constellations change shape, albeit slowly, that each star has an independent motion; this motion is caused by the movement of the stars relative to the Solar System. The Sun travels in a nearly circular orbit about the center of the Milky Way at a speed of about 220 km/s at a radius of 8 kPc from the center, which can be taken as the rate of rotation of the Milky Way itself at this radius; the proper motion is a two-dimensional vector and is thus defined by two quantities: its position angle and its magnitude. The first quantity indicates the direction of the proper motion on the celestial sphere, the second quantity is the motion's magnitude expressed in arcseconds per year or milliarcsecond per year. Proper motion may alternatively be defined by the angular changes per year in the star's right ascension and declination, using a constant epoch in defining these; the components of proper motion by convention are arrived at.
Suppose an object moves from coordinates to coordinates in a time Δt. The proper motions are given by: μ α = α 2 − α 1 Δ t, μ δ = δ 2 − δ 1 Δ t; the magnitude of the proper motion μ is given by the Pythagorean theorem: μ 2 = μ δ 2 + μ α 2 ⋅ cos 2 δ, μ 2 = μ δ 2 + μ α ∗ 2, where δ is the declination. The factor in cos2δ accounts for the fact that the radius from the axis of the sphere to its surface varies as cosδ, for example, zero at the pole. Thus, the component of velocity parallel to the equator corresponding to a given angular change in α is smaller the further north the object's location; the change μα, which must be multiplied by cosδ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", μδ the "proper motion in declination". If the proper motion in right ascension has been converted by cosδ, the result is designated μα*. For example, the proper motion results in right ascension in the Hipparcos Catalogue have been converted. Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.
The position angle θ is related to these components by: μ sin θ = μ α cos δ = μ α ∗, μ cos θ = μ δ. Motions in equatorial coordinates can be converted to motions in galactic coordinates. For the majority of stars seen in the sky, the observed proper motions are small and unremarkable; such stars are either faint or are distant, have changes of below 10 milliarcseconds per year, do not appear to move appreciably over many millennia. A few do have significant motions, are called high-proper motion stars. Motions can be in seemingly random directions. Two or more stars, double stars or open star clusters, which are moving in similar directions, exhibit so-called shared or common proper motion, suggesting they may be gravitationally attached or share similar motion in space. Barnard's Star has the largest proper motion of all stars, moving at 10.3 seconds of arc per year. L