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Equatorial coordinate system

The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the celestial equator), a primary direction towards the March equinox, and a right-handed convention.

The origin at the centre of Earth means the coordinates are geocentric, that is, as seen from the centre of Earth as if it were transparent. The fundamental plane and the primary direction mean that the coordinate system, while aligned with Earth's equator and pole, does not rotate with the Earth, but remains relatively fixed against the background stars. A right-handed convention means that coordinates increase northward from and eastward around the fundamental plane.

Primary direction
This description of the orientation of the reference frame is somewhat simplified; the orientation is not quite fixed. A slow motion of Earth's axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the ecliptic, and a small oscillation of the Earth's axis, nutation.

In order to fix the exact primary direction, these motions necessitate the specification of the equinox of a particular date, known as an epoch, when giving a position. The three most commonly used are:
 * Mean equinox of a standard epoch (usually J2000.0, but may include B1950.0, B1900.0, etc.): is a fixed standard direction, allowing positions established at various dates to be compared directly.
 * Mean equinox of date: is the intersection of the ecliptic of "date" (that is, the ecliptic in its position at "date") with the mean equator (that is, the equator rotated by precession to its position at "date", but free from the small periodic oscillations of nutation). Commonly used in planetary orbit calculation.
 * True equinox of date: is the intersection of the ecliptic of "date" with the true equator (that is, the mean equator plus nutation). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.

A position in the equatorial coordinate system is thus typically specified true equinox and equator of date, mean equinox and equator of J2000.0, or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.

Use in astronomy
A star's spherical coordinates are often expressed as a pair, right ascension and declination, without a distance coordinate. The direction of sufficiently distant objects is the same for all observers, and it is convenient to specify this direction with the same coordinates for all. In contrast, in the horizontal coordinate system, a star's position differs from observer to observer based on their positions on the Earth's surface, and is continuously changing with the Earth's rotation.

Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects. Setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on the celestial sphere.

Declination
The declination symbol $δ$, (lower case "delta", abbreviated DEC) measures the angular distance of an object perpendicular to the celestial equator, positive to the north, negative to the south. For example, the north celestial pole has a declination of +90°. The origin for declination is the celestial equator, which is the projection of the Earth's equator onto the celestial sphere. Declination is analogous to terrestrial latitude.

Right ascension

The right ascension symbol $α$, (lower case "alpha", abbreviated RA) measures the angular distance of an object eastward along the celestial equator from the March equinox to the hour circle passing through the object. The March equinox point is one of the two points where the ecliptic intersects the celestial equator. Right ascension is usually measured in sidereal hours, minutes and seconds instead of degrees, a result of the method of measuring right ascensions by timing the passage of objects across the meridian as the Earth rotates. There are $360°⁄24^{h}$ = 15° in one hour of right ascension, and 24h of right ascension around the entire celestial equator.

When used together, right ascension and declination are usually abbreviated RA/Dec.

Hour angle
Alternatively to right ascension, hour angle (abbreviated HA or LHA, local hour angle), a left-handed system, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the hour circle passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of Earth. Hour angle may be considered a means of measuring the time since upper culmination, the moment when an object contacts the meridian overhead.

A culminating star on the observer's meridian is said to have a zero hour angle (0h). One sidereal hour (approximately 0.9973 solar hours) later, Earth's rotation will carry the star to the west of the meridian, and its hour angle will be 1h. When calculating topocentric phenomena, right ascension may be converted into hour angle as an intermediate step.

Geocentric equatorial coordinates

There are a number of rectangular variants of equatorial coordinates. All have:
 * The origin at the centre of the Earth.
 * The fundamental plane in the plane of the Earth's equator.
 * The primary direction (the $x$ axis) toward the March equinox, that is, the place where the Sun crosses the celestial equator in a northward direction in its annual apparent circuit around the ecliptic.
 * A right-handed convention, specifying a $y$ axis 90° to the east in the fundamental plane and a $z$ axis along the north polar axis.

The reference frames do not rotate with the Earth (in contrast to Earth-centred, Earth-fixed frames), remaining always directed toward the equinox, and drifting over time with the motions of precession and nutation. \frac{X}{R} = \frac{\xi}{\mathit{\Delta}} &= \cos \delta \cos \alpha \\ \frac{Y}{R} = \frac{\eta}{\mathit{\Delta}} &= \cos \delta \sin \alpha \\ \frac{Z}{R} = \frac{\zeta}{\mathit{\Delta}} &= \sin \delta \end{align}$$
 * In astronomy:
 * The position of the Sun is often specified in the geocentric equatorial rectangular coordinates $z$, $x$, $y$ and a fourth distance coordinate, $z$ $X$, in units of the astronomical unit.
 * The positions of the planets and other Solar System bodies are often specified in the geocentric equatorial rectangular coordinates $Y$, $Z$, $R$ and a fourth distance coordinate, $(= \sqrt{X + Y + Z})$ (equal to $ξ$), in units of the astronomical unit.These rectangular coordinates are related to the corresponding spherical coordinates by $$\begin{align}
 * In astrodynamics:
 * The positions of artificial Earth satellites are specified in geocentric equatorial coordinates, also known as geocentric equatorial inertial (GEI), Earth-centred inertial (ECI), and conventional inertial system (CIS), all of which are equivalent in definition to the astronomical geocentric equatorial rectangular frames, above. In the geocentric equatorial frame, the $η$, $ζ$ and $Δ$ axes are often designated $\sqrt{ξ + η + ζ}$, $x$ and $y$, respectively, or the frame's basis is specified by the unit vectors $z$, $I$ and $J$.
 * The Geocentric Celestial Reference Frame (GCRF) is the geocentric equivalent of the International Celestial Reference Frame (ICRF). Its primary direction is the equinox of J2000.0, and does not move with precession and nutation, but it is otherwise equivalent to the above systems.

Heliocentric equatorial coordinates
In astronomy, there is also a heliocentric rectangular variant of equatorial coordinates, designated $K$, $Î$, $Ĵ$, which has:

 * The origin at the centre of the Sun.
 * The fundamental plane in the plane of the Earth's equator.
 * The primary direction (the $K̂$ axis) toward the March equinox.
 * A right-handed convention, specifying a $x$ axis 90° to the east in the fundamental plane and a $y$ axis along Earth's north polar axis.

This frame is in every way equivalent to the $z$, $x$, $y$ frame, above, except that the origin is removed to the centre of the Sun. It is commonly used in planetary orbit calculation. The three astronomical rectangular coordinate systems are related by $$\begin{align} \xi &= x + X \\ \eta &= y + Y \\ \zeta &= z + Z \end{align}$$

title: Equatorial coordinate system


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Horizontal coordinate system

The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles of a spherical coordinate system: altitude and azimuth. Therefore, the horizontal coordinate system is sometimes called the az/el system, the alt/az system, or the alt-azimuth system, among others. In an altazimuth mount of a telescope, the instrument's two axes follow altitude and azimuth.

Definition
This celestial coordinate system divides the sky into two hemispheres: The upper hemisphere, where objects are above the horizon and are visible, and the lower hemisphere, where objects are below the horizon and cannot be seen, since the Earth obstructs views of them. The great circle separating the hemispheres is called the celestial horizon, which is defined as the great circle on the celestial sphere whose plane is normal to the local gravity vector. In practice, the horizon can be defined as the plane tangent to a quiet, liquid surface, such as a pool of mercury. The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir.

The following are two independent horizontal angular coordinates:
 * Altitude (alt.), sometimes referred to as elevation (el.) or, is the angle between the object and the observer's local horizon. For visible objects, it is an angle between 0° and 90°.
 * Azimuth (az.) is the angle of the object around the horizon, usually measured from true north and increasing eastward. Exceptions are, for example, ESO's FITS convention where it is measured from the south and increasing westward, or the FITS convention of the Sloan Digital Sky Survey where it is measured from the south and increasing eastward.

A horizontal coordinate system should not be confused with a topocentric coordinate system. Horizontal coordinates define the observer's orientation, but not location of the origin, while topocentric coordinates define the origin location, on the Earth's surface, in contrast to a geocentric celestial system.

General features
The horizontal coordinate system is fixed to a location on Earth, not the stars. Therefore, the altitude and azimuth of an object in the sky changes with time, as the object appears to drift across the sky with Earth's rotation. In addition, since the horizontal system is defined by the observer's local horizon, the same object viewed from different locations on Earth at the same time will have different values of altitude and azimuth.

The cardinal points on the horizon have specific values of azimuth that are helpful references.

Horizontal coordinates are very useful for determining the rise and set times of an object in the sky. When an object's altitude is 0°, it is on the horizon. If at that moment its altitude is increasing, it is rising, but if its altitude is decreasing, it is setting. However, all objects on the celestial sphere are subject to diurnal motion, which always appears to be westward.

A northern observer can determine whether altitude is increasing or decreasing by instead considering the azimuth of the celestial object:
 * If the azimuth is between 0° and 180° (north–east–south), the object is rising.
 * If the azimuth is between 180° and 360° (south–west–north), the object is setting.

There are the following special cases:

 * All directions are south when viewed from the North Pole, and all directions are north when viewed from the South Pole, so the azimuth is undefined in both locations. When viewed from either pole, a star (or any object with fixed equatorial coordinates) has constant altitude and thus never rises or sets. The Sun, Moon, and planets can rise or set over the span of a year when viewed from the poles because their declinations are constantly changing.
 * When viewed from the equator, objects on the celestial poles stay at fixed points, perched on the horizon.

title: Horizontal coordinate system


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Ecliptic coordinate system

In astronomy, the ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bodies have orbits with only slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the March equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.

Ecliptic grid globe.png ecliptic coordinates as seen from outside the celestial sphere. {{legend|red|Ecliptic longitude; measured along the ecliptic from the March equinox}} {{legend|yellow|Ecliptic latitude; measured perpendicular to the ecliptic}}

A full globe is shown here, although high-latitude coordinates are seldom seen except for certain comets and asteroids.]]

Primary direction

The celestial equator and the ecliptic are slowly moving due to perturbing forces on the Earth, therefore the orientation of the primary direction, their intersection at the March equinox, is not quite fixed. A slow motion of Earth's axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the ecliptic, and a small oscillation of the Earth's axis, nutation.

In order to reference a coordinate system which can be considered as fixed in space, these motions require specification of the equinox of a particular date, known as an epoch, when giving a position in ecliptic coordinates. The three most commonly used are: A position in the ecliptic coordinate system is thus typically specified true equinox and ecliptic of date, mean equinox and ecliptic of J2000.0, or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.
 * Mean equinox of a standard epoch: (usually the J2000.0 epoch, but may include B1950.0, B1900.0, etc.) is a fixed standard direction, allowing positions established at various dates to be compared directly.
 * Mean equinox of date: is the intersection of the ecliptic of "date" (that is, the ecliptic in its position at "date") with the mean equator (that is, the equator rotated by precession to its position at "date", but free from the small periodic oscillations of nutation). Commonly used in planetary orbit calculation.
 * True equinox of date: is the intersection of the ecliptic of "date" with the true equator (that is, the mean equator plus nutation). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.

Spherical coordinates

 * Ecliptic longitude
 * Ecliptic longitude or celestial longitude (symbols: heliocentric $l$, geocentric $λ$) measures the angular distance of an object along the ecliptic from the primary direction. Like right ascension in the equatorial coordinate system, the primary direction (0° ecliptic longitude) points from the Earth towards the Sun at the March equinox. Because it is a right-handed system, ecliptic longitude is measured positive eastwards in the fundamental plane (the ecliptic) from 0° to 360°. Because of axial precession, the ecliptic longitude of most "fixed stars" (referred to the equinox of date) increases by about 50.3 arcseconds per year, or 83.8 arcminutes per century, the speed of general precession. However, for stars near the ecliptic poles, the rate of change of ecliptic longitude is dominated by the slight movement of the ecliptic (that is, of the plane of the Earth's orbit), so the rate of change may be anything from minus infinity to plus infinity depending on the exact position of the star.

 * Ecliptic latitude
 * Ecliptic latitude or celestial latitude (symbols: heliocentric $b$, geocentric $β$), measures the angular distance of an object from the ecliptic towards the north (positive) or south (negative) ecliptic pole. For example, the north ecliptic pole has a celestial latitude of +90°. Ecliptic latitude for "fixed stars" is not affected by precession.

 * Distance
 * Distance is also necessary for a complete spherical position (symbols: heliocentric $r$, geocentric $Δ$). Different distance units are used for different objects. Within the Solar System, astronomical units are used, and for objects near the Earth, Earth radii or kilometers are used.

Historical use
From antiquity through the 18th century, ecliptic longitude was commonly measured using twelve zodiacal signs, each of 30° longitude, a practice that continues in modern astrology. The signs approximately corresponded to the constellations crossed by the ecliptic. Longitudes were specified in signs, degrees, minutes, and seconds. For example, a longitude of ♌ 19° 55′ 58″ is 19.933° east of the start of the sign Leo. Since Leo begins 120° from the March equinox, the longitude in modern form is 139° 55′ 58″.

In China, ecliptic longitude is measured using 24 Solar terms, each of 15° longitude, and are used by Chinese lunisolar calendars to stay synchronized with the seasons, which is crucial for agrarian societies.

Rectangular coordinates

A rectangular variant of ecliptic coordinates is often used in orbital calculations and simulations. It has its origin at the center of the Sun (or at the barycenter of the Solar System), its fundamental plane on the ecliptic plane, and the $x$-axis toward the March equinox. The coordinates have a right-handed convention, that is, if one extends their right thumb upward, it simulates the $y$-axis, their extended index finger the $z$-axis, and the curl of the other fingers points generally in the direction of the $x$-axis.

These rectangular coordinates are related to the corresponding spherical coordinates by

 * $$\begin{align}

x &= r \cos b \cos l \\ y &= r \cos b \sin l \\ z &= r \sin b \end{align}$$

Conversion from ecliptic coordinates to equatorial coordinates
$$ \begin{bmatrix} x_\text{equatorial} \\ y_\text{equatorial} \\ z_\text{equatorial} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \varepsilon & -\sin \varepsilon \\ 0 & \sin \varepsilon & \cos \varepsilon \\ \end{bmatrix} \begin{bmatrix} x_\text{ecliptic} \\ y_\text{ecliptic} \\ z_\text{ecliptic} \\ \end{bmatrix} $$

Conversion from equatorial coordinates to ecliptic coordinates
$$ \begin{bmatrix} x_\text{ecliptic} \\ y_\text{ecliptic} \\ z_\text{ecliptic} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \varepsilon & \sin \varepsilon \\ 0 & -\sin \varepsilon & \cos \varepsilon \\ \end{bmatrix} \begin{bmatrix} x_\text{equatorial} \\ y_\text{equatorial} \\ z_\text{equatorial} \\ \end{bmatrix} $$ where $z$ is the obliquity of the ecliptic.

title: Ecliptic coordinate system


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Eudoxus of Cnidus

Eudoxus of Cnidus (Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios; c. 390) was an ancient Greek astronomer, mathematician, doctor, and lawmaker. He was a student of Archytas and Plato. All of his original works are lost, though some fragments are preserved in Hipparchus' Commentaries on the Phenomena of Aratus and Eudoxus. Spherics by Theodosius of Bithynia may be based on a work by Eudoxus.

Life
Eudoxus, son of Aeschines, was born and died in Cnidus (also transliterated Knidos), a city on the southwest coast of Anatolia. The years of Eudoxus' birth and death are not fully known but Diogenes Laërtius gave several biographical details, mentioned that Apollodorus said he reached his acme in the 103rd Olympiad (368–365 BC), and claimed he died in his 53rd year. From this 19th century mathematical historians reconstructed dates of 408–355 BC, but 20th century scholars found their choices contradictory and prefer a birth year of c. 390 BC. His name Eudoxus means "honored" or "of good repute" (εὔδοξος, from eu "good" and doxa "opinion, belief, fame", analogous to the Latin Benedictus).

According to Diogenes Laërtius, crediting Callimachus' Pinakes, Eudoxus studied mathematics with Archytas (of Tarentum, Magna Graecia) and studied medicine with Philiston the Sicilian. At the age of 23, he traveled with the physician Theomedon—who was his patron and possibly his lover —to Athens to study with the followers of Socrates. He spent two months there—living in Piraeus and walking 7 mi each way every day to attend the Sophists' lectures—then returned home to Cnidus. His friends then paid to send him to Heliopolis, Egypt for 16 months, to pursue his study of astronomy and mathematics. From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis. He traveled south to the court of Mausolus. During his travels he gathered many students of his own.

Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, c. 367 he assumed headship (scholarch) of the Academy during Plato's period in Syracuse, and taught Aristotle. He eventually returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on theology, astronomy, and meteorology. He had one son, Aristagoras, and three daughters, Actis, Philtis, and Delphis.

In mathematical astronomy, his fame is due to the introduction of the concentric spheres, and his early contributions to understanding the movement of the planets.

His work on proportions shows insight into irrational numbers and the linear continuum: it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science, and inspired Richard Dedekind's work on the real numbers.

Craters on Mars and the Moon are named in his honor. An algebraic curve (the Kampyle of Eudoxus) is also named after him.

Mathematics
Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all Antiquity second only to Archimedes. Eudoxus was probably the source for most of book V of Euclid's. He rigorously developed Antiphon's method of exhaustion, a precursor to the integral calculus which was also used in a masterly way by Archimedes in the following century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one-third the volume of a prism with the same base and altitude, and the volume of a cone is one-third that of the corresponding cylinder.

Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of irrational numbers. In doing so, he reversed a Pythagorean emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics. Some Pythagoreans, such as Eudoxus's teacher Archytas, had believed that only arithmetic could provide a basis for proofs. Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms. The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra.

The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem (Elements I.47), by using addition of areas and only much later (Elements VI.31) a simpler proof from similar triangles, which relies on ratios of line segments.

Ancient Greek mathematicians calculated not with quantities and equations as we do today; instead, a proportionality expressed a relationship between geometric magnitudes. The ratio of two magnitudes was not a numerical value, as we think of it today; the ratio of two magnitudes was a primitive relationship between them.

Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V.

In Definition 5 of Euclid's Book V we read:

"Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order."

By using modern-day notation, this is clarified as follows. If we take four quantities: a, b, c, and d, then the first and second have a ratio $$a/b$$; similarly the third and fourth have a ratio $$c/d$$ Now to say that $$a/b = c/d.$$ we do the following: for any two arbitrary integers, m and n, form the equimultiples m·a and m·c of the first and third; likewise form the equimultiples n·b and n·d of the second and fourth. If it happens that m·a > n·b, then we must also have m·c > n·d. If it happens that m·a = n·b, then we must also have m·c = n·d. Finally, if it happens that m·a < n·b, then we must also have m·c < n·d.

This means that $$a/b = c/d$$ if and only if the rational numbers $$n/m$$ that are larger than $$a/b$$ are the same as the ones that are larger than $$c/d,$$ and similarly for "equal" and "smaller". This is to be compared with Dedekind cuts that define a real number by the set of rational numbers that are larger, equal or smaller than the number to be defined.

Eudoxus' definition depends on comparing the similar quantities m·a and n·b, and the similar quantities m·c and n·d, and does not depend on the existence of a common unit for measuring these quantities.

The complexity of the definition reflects the deep conceptual and methodological innovation involved. It brings to mind the famous fifth postulate of Euclid concerning parallels, which is more extensive and complicated in its wording than the other postulates.

The Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, just as do the modern epsilon-delta definitions of limit and continuity.

Additionally, the Archimedean property stated as definition 4 of Euclid's book V is originally due not to Archimedes but to Eudoxus.

Astronomy
In ancient Greece, astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions. Identifying the astronomical work of Eudoxus as a separate category is therefore a modern convenience. Some of Eudoxus's astronomical texts whose names have survived include:

 * Disappearances of the Sun, possibly on eclipses
 * Oktaeteris (Ὀκταετηρίς), on an eight-year lunisolar-Venus cycle of the calendar
 * Phaenomena (Φαινόμενα) and Enoptron (Ἔνοπτρον), on spherical astronomy, probably based on observations made by Eudoxus in Egypt and Cnidus
 * On Speeds, on planetary motions

We are fairly well informed about the contents of Phaenomena, for Eudoxus's prose text was the basis for a poem of the same name by Aratus. Hipparchus quoted from the text of Eudoxus in his commentary on Aratus.

Eudoxan planetary models
A general idea of the content of On Speeds can be gleaned from Aristotle's Metaphysics XII, 8, and a commentary by Simplicius of Cilicia (6th century AD) on De caelo, another work by Aristotle. According to a story reported by Simplicius, Plato posed a question for Greek astronomers: "By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?" Plato proposed that the seemingly chaotic wandering motions of the planets could be explained by combinations of uniform circular motions centered on a spherical Earth, apparently a novel idea in the 4th century BC.

In most modern reconstructions of the Eudoxan model, the Moon is assigned three spheres:

 * The outermost rotates westward once in 24 hours, explaining rising and setting.
 * The second rotates eastward once in a month, explaining the monthly motion of the Moon through the zodiac.
 * The third also completes its revolution in a month, but its axis is tilted at a slightly different angle, explaining motion in latitude (deviation from the ecliptic), and the motion of the lunar nodes.

The Sun is also assigned three spheres. The second completes its motion in a year instead of a month. The inclusion of a third sphere implies that Eudoxus mistakenly believed that the Sun had motion in latitude.

The five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn) are assigned four spheres each:

 * The outermost explains the daily motion.
 * The second explains the planet's motion through the zodiac.
 * The third and fourth together explain retrogradation, when a planet appears to slow down, then briefly reverse its motion through the zodiac. By inclining the axes of the two spheres with respect to each other, and rotating them in opposite directions but with equal periods, Eudoxus could make a point on the inner sphere trace out a figure-eight shape, or hippopede.

Importance of Eudoxan system
Callippus, a Greek astronomer of the 4th century, added seven spheres to Eudoxus's original 27 (in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars). Aristotle described both systems, but insisted on adding "unrolling" spheres between each set of spheres to cancel the motions of the outer set. Aristotle was concerned about the physical nature of the system; without unrollers, the outer motions would be transferred to the inner planets.

A major flaw in the Eudoxian system is its inability to explain changes in the brightness of planets as seen from Earth. Because the spheres are concentric, planets will always remain at the same distance from Earth. This problem was pointed out in Antiquity by Autolycus of Pitane. Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance. However, Eudoxus's importance to astronomy and in particular to Greek astronomy is considerable.

Ethics
Aristotle, in the Nicomachean Ethics, attributes to Eudoxus an argument in favor of hedonism—that is, that pleasure is the ultimate good that activity strives for. According to Aristotle, Eudoxus put forward the following arguments for this position:
 * 1) All things, rational and irrational, aim at pleasure; things aim at what they believe to be good; a good indication of what the chief good is would be the thing that most things aim at.
 * 2) Similarly, pleasure's opposite—pain—is universally avoided, which provides additional support for the idea that pleasure is universally considered good.
 * 3) People don't seek pleasure as a means to something else, but as an end in its own right.
 * 4) Any other good that you can think of would be better if pleasure were added to it, and it is only by good that good can be increased.
 * 5) Of all of the things that are good, happiness is peculiar for not being praised, which may show that it is the crowning good.

title: Eudoxus of Cnidus


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Galactic coordinate system

The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way Galaxy, and the fundamental plane parallel to an approximation of the galactic plane but offset to its north. It uses the right-handed convention, meaning that coordinates are positive toward the north and toward the east in the fundamental plane.

Galactic longitude

Longitude (symbol $l$) measures the angular distance of an object eastward along the galactic equator from the Galactic Center. Analogous to terrestrial longitude, galactic longitude is usually measured in degrees (°).

Galactic latitude
Latitude (symbol $b$) measures the angle of an object northward of the galactic equator (or midplane) as viewed from Earth. Analogous to terrestrial latitude, galactic latitude is usually measured in degrees (°).

Definition
The first galactic coordinate system was used by William Herschel in 1785. A number of different coordinate systems, each differing by a few degrees, were used until 1932, when Lund Observatory assembled a set of conversion tables that defined a standard galactic coordinate system based on a galactic north pole at RA 12h 40m, dec +28° (in the B1900.0 epoch convention) and a 0° longitude at the point where the galactic plane and equatorial plane intersected.

In 1958, the International Astronomical Union (IAU) defined the galactic coordinate system in reference to radio observations of galactic neutral hydrogen through the hydrogen line, changing the definition of the Galactic longitude by 32° and the latitude by 1.5°. In the equatorial coordinate system, for equinox and equator of 1950.0, the north galactic pole is defined at right ascension 12h 49m, declination +27.4°, in the constellation Coma Berenices, with a probable error of ±0.1°. Longitude 0° is the great semicircle that originates from this point along the line in position angle 123° with respect to the equatorial pole. The galactic longitude increases in the same direction as right ascension. Galactic latitude is positive towards the north galactic pole, with a plane passing through the Sun and parallel to the galactic equator being 0°, whilst the poles are ±90°. Based on this definition, the galactic poles and equator can be found from spherical trigonometry and can be precessed to other epochs; see the table.

The IAU recommended that during the transition period from the old, pre-1958 system to the new, the old longitude and latitude should be designated $l$ and $b$ while the new should be designated $l$ and $b$. This convention is occasionally seen.

Radio source Sagittarius A*, which is the best physical marker of the true Galactic Center, is located at 17h 45m 40.0409s, −29° 00′ 28.118″ (J2000). Rounded to the same number of digits as the table, 17h 45.7m, −29.01° (J2000), there is an offset of about 0.07° from the defined coordinate center, well within the 1958 error estimate of ±0.1°. Due to the Sun's position, which currently lies $56.75 ly$ north of the midplane, and the heliocentric definition adopted by the IAU, the galactic coordinates of Sgr A* are latitude +0° 07′ 12″ south, longitude 0° 04′ 06″. Since as defined the galactic coordinate system does not rotate with time, Sgr A* is actually decreasing in longitude at the rate of galactic rotation at the sun, $Ω$, approximately 5.7 milliarcseconds per year (see Oort constants).

Conversion between equatorial and galactic coordinates
An object's location expressed in the equatorial coordinate system can be transformed into the galactic coordinate system. In these equations, $l$ is right ascension, $b$ is declination. NGP refers to the coordinate values of the north galactic pole and NCP to those of the north celestial pole.

\begin{align} \sin(b) &= \sin(\delta_\text{NGP})\sin(\delta)+\cos(\delta_\text{NGP})\cos(\delta)\cos(\alpha-\alpha_\text{NGP}) \\ \cos(b) \sin(l_\text{NCP}-l) &= \cos(\delta)\sin(\alpha-\alpha_\text{NGP}) \\ \cos(b) \cos(l_\text{NCP}-l) &= \cos(\delta_\text{NGP}) \sin(\delta) - \sin(\delta_\text{NGP}) \cos(\delta) \cos(\alpha - \alpha_\text{NGP}) \end{align} $$

The reverse (galactic to equatorial) can also be accomplished with the following conversion formulas.

\begin{align} \sin (\delta) &= \sin (\delta_\text{NGP}) \sin(b) + \cos (\delta_\text{NGP}) \cos(b) \cos(l_\text{NCP} - l) \\ \cos (\delta) \sin(\alpha - \alpha_\text{NGP}) &= \cos(b) \sin(l_\text{NCP} - l) \\ \cos (\delta) \cos(\alpha - \alpha_\text{NGP}) &= \cos (\delta_\text{NGP}) \sin(b) - \sin(\delta_\text{NGP}) \cos(b) \cos(l_\text{NCP} - l) \end{align} $$

Where:

 * $$\alpha_\text{NGP}=12^h 51.4^m \qquad \delta_\text{NGP}=27.13^{\circ} \qquad l_\text{NCP}=122.93314^{\circ}$$

Rectangular coordinates
In some applications use is made of rectangular coordinates based on galactic longitude and latitude and distance. In some work regarding the distant past or future the galactic coordinate system is taken as rotating so that the $l$-axis always goes to the centre of the galaxy.

There are two major rectangular variations of galactic coordinates, commonly used for computing space velocities of galactic objects. In these systems the $b$-axes are designated $α$, but the definitions vary by author. In one system, the $δ$ axis is directed toward the Galactic Center ($x$ = 0°), and it is a right-handed system (positive towards the east and towards the north galactic pole); in the other, the $xyz$ axis is directed toward the galactic anticenter ($UVW$ = 180°), and it is a left-handed system (positive towards the east and towards the north galactic pole).

In the constellations
The galactic equator runs through the following constellations:

 * Sagittarius
 * Serpens
 * Scutum
 * Aquila
 * Sagitta
 * Vulpecula
 * Cygnus
 * Cepheus
 * Cassiopeia
 * Camelopardalis
 * Perseus
 * Auriga
 * Taurus
 * Gemini
 * Orion
 * Monoceros
 * Canis Major
 * Puppis
 * Vela
 * Carina
 * Crux
 * Centaurus
 * Circinus
 * Norma
 * Ara
 * Scorpius
 * Ophiuchus

title: Galactic coordinate system


text:
Phenytoin

Phenytoin (PHT), sold under the brand name Dilantin among others, is an anti-seizure medication. It is useful for the prevention of tonic-clonic seizures (also known as grand mal seizures) and focal seizures, but not absence seizures. The intravenous form, fosphenytoin, is used for status epilepticus that does not improve with benzodiazepines. It may also be used for certain heart arrhythmias or neuropathic pain. It can be taken intravenously or by mouth. The intravenous form generally begins working within 30 minutes and is effective for roughly 24 hours. Blood levels can be measured to determine the proper dose.

Common side effects include nausea, stomach pain, loss of appetite, poor coordination, increased hair growth, and enlargement of the gums. Potentially serious side effects include sleepiness, self harm, liver problems, bone marrow suppression, low blood pressure, and toxic epidermal necrolysis. There is evidence that use during pregnancy results in abnormalities in the baby. It appears to be safe to use when breastfeeding. Alcohol may interfere with the medication's effects.

Phenytoin was first made in 1908 by the German chemist Heinrich Biltz and found useful for seizures in 1936. It is on the World Health Organization's List of Essential Medicines. Phenytoin is available as a generic medication. In 2020, it was the 260th most commonly prescribed medication in the United States, with more than 1million prescriptions.

Seizures
Though phenytoin has been used to treat seizures in infants, as of 2023, its effectiveness in this age group has been evaluated in only one study. Due to the lack of a comparison group, the evidence is inconclusive.
 * Tonic-clonic seizures: Mainly used in the prophylactic management of tonic-clonic seizures with complex symptomatology (psychomotor seizures). A period of 5–10 days of dosing may be required to achieve anticonvulsant effects.
 * Focal seizures: Mainly used to protect against the development of focal seizures with complex symptomatology (psychomotor and temporal lobe seizures). Also effective in controlling focal seizures with autonomic symptoms.
 * Absence seizures: Not used in treatment of pure absence seizures due to risk for increasing frequency of seizures. However, can be used in combination with other anticonvulsants during combined absence and tonic-clonic seizures.
 * Seizures during surgery: A 2018 meta-analysis found that early antiepileptic treatment with either phenytoin or phenobarbital reduced the risk of seizure in the first week after neurosurgery for brain tumors.
 * Status epilepticus: Considered after failed treatment using a benzodiazepine due to slow onset of action.

Other

 * Abnormal heart rhythms: may be used in the treatment of ventricular tachycardia and sudden episodes of atrial tachycardia after other antiarrhythmic medications or cardioversion has failed. It is a class Ib antiarrhythmic.
 * Digoxin toxicity: Intravenous phenytoin formulation is a medication of choice for arrhythmias caused by cardiac glycoside toxicity.
 * Trigeminal neuralgia: Second choice drug to carbamazepine.

Special considerations

 * Phenytoin has a narrow therapeutic index. Its therapeutic range for both anticonvulsant and antiarrhythmic effect is 10–20 μg/mL.
 * Avoid giving intramuscular formulation unless necessary due to skin cell death and local tissue destruction.
 * Elderly patients may show earlier signs of toxicity.
 * In the obese, ideal body weight should be used for dosing calculations.
 * Pregnancy: Pregnancy category D due to risk of fetal hydantoin syndrome and fetal bleeding. However, optimal seizure control is very important during pregnancy so drug may be continued if benefits outweigh the risks. Due to decreased drug concentrations as a result of plasma volume expansion during pregnancy, dose of phenytoin may need to be increased if only option for seizure control.
 * Breastfeeding: The manufacturer does not recommend breastfeeding since low concentrations of phenytoin are excreted in breast milk.
 * Liver disease: Do not use oral loading dose. Consider using decreased maintenance dose.
 * Kidney disease: Do not use oral loading dose. Can begin with standard maintenance dose and adjust as needed.
 * Intravenous use is contraindicated in patients with sinus bradycardia, sinoatrial block, second- or third-degree atrioventricular block, Stokes-Adams syndrome, or hypersensitivity to phenytoin, other hydantoins or any ingredient in the respective formulation.

Side effects
Common side effects include nausea, stomach pain, loss of appetite, poor coordination, increased hair growth, and enlargement of the gums. Potentially serious side effects include sleepiness, self harm, liver problems, bone marrow suppression, low blood pressure, and toxic epidermal necrolysis. There is evidence that use during pregnancy results in abnormalities in the baby. Its use appears to be safe during breastfeeding. Alcohol may interfere with the medication's effects.

Heart and blood vessels
Severe low blood pressure and abnormal heart rhythms can be seen with rapid infusion of IV phenytoin. IV infusion should not exceed 50 mg/min in adults or 1–3 mg/kg/min (or 50 mg/min, whichever is slower) in children. Heart monitoring should occur during and after IV infusion. Due to these risks, oral phenytoin should be used if possible.

Neurological
At therapeutic doses, phenytoin may produce nystagmus on lateral gaze. At toxic doses, patients experience vertical nystagmus, double vision, sedation, slurred speech, cerebellar ataxia, and tremor. If phenytoin is stopped abruptly, this may result in increased seizure frequency, including status epilepticus.

Phenytoin may accumulate in the cerebral cortex over long periods of time which can cause atrophy of the cerebellum. The degree of atrophy is related to the duration of phenytoin treatment and is not related to dosage of the medication.

Phenytoin is known to be a causal factor in the development of peripheral neuropathy.

Blood
Folate is present in food in a polyglutamate form, which is then converted into monoglutamates by intestinal conjugase to be absorbed by the jejunum. Phenytoin acts by inhibiting this enzyme, thereby causing folate deficiency, and thus megaloblastic anemia. Other side effects may include: agranulocytosis, aplastic anemia, decreased white blood cell count, and a low platelet count.

Pregnancy
Phenytoin is a known teratogen, since children exposed to phenytoin are at a higher risk of birth defects than children born to women without epilepsy and to women with untreated epilepsy. The birth defects, which occur in approximately 6% of exposed children, include neural tube defects, heart defects and craniofacial abnormalities, including broad nasal bridge, cleft lip and palate, and smaller than normal head. The effect on IQ cannot be determined as no study involves phenytoin as monotherapy, however poorer language abilities and delayed motor development may have been associated with maternal use of phenytoin during pregnancy. This syndrome resembles the well-described fetal alcohol syndrome. and has been referred to as "fetal hydantoin syndrome". Some recommend avoiding polytherapy and maintaining the minimal dose possible during pregnancy, but acknowledge that current data fails to demonstrate a dose effect on the risk of birth defects. Data now being collected by the Epilepsy and Antiepileptic Drug Pregnancy Registry may one day answer this question definitively.

Cancer
There is no good evidence to suggest that phenytoin is a human carcinogen. However, lymph node abnormalities have been observed, including malignancies.

Mouth
Phenytoin has been associated with drug-induced gingival enlargement (overgrowth of the gums), probably due to above-mentioned folate deficiency; indeed, evidence from a randomized controlled trial suggests that folic acid supplementation can prevent gingival enlargement in children who take phenytoin. Plasma concentrations needed to induce gingival lesions have not been clearly defined. Effects consist of the following: bleeding upon probing, increased gingival exudate, pronounced gingival inflammatory response to plaque levels, associated in some instances with bone loss but without tooth detachment.

Skin
Hypertrichosis, Stevens–Johnson syndrome, purple glove syndrome, rash, exfoliative dermatitis, itching, excessive hairiness, and coarsening of facial features can be seen in those taking phenytoin.

Phenytoin therapy has been linked to the life-threatening skin reactions Stevens–Johnson syndrome (SJS) and toxic epidermal necrolysis (TEN). These conditions are significantly more common in patients with a particular HLA-B allele, HLA-B*1502. This allele occurs almost exclusively in patients with ancestry across broad areas of Asia, including South Asian Indians.

Phenytoin is primarily metabolized to its inactive form by the enzyme CYP2C9. Variations within the CYP2C9 gene that result in decreased enzymatic activity have been associated with increased phenytoin concentrations, as well as reports of drug toxicities due to these increased concentrations. The U.S. Food and Drug Administration (FDA) notes on the phenytoin drug label that since strong evidence exists linking HLA-B*1502 with the risk of developing SJS or TEN in patients taking carbamazepine, consideration should be given to avoiding phenytoin as an alternative to carbamazepine in patients carrying this allele.

Immune system
Phenytoin has been known to cause drug-induced lupus.

Phenytoin is also associated with induction of reversible IgA deficiency.

Psychological
Phenytoin may increase risk of suicidal thoughts or behavior. People on phenytoin should be monitored for any changes in mood, the development or worsening depression, and/or any thoughts or behavior of suicide.

Bones
Chronic phenytoin use has been associated with decreased bone density and increased bone fractures. Phenytoin induces metabolizing enzymes in the liver. This leads to increased metabolism of vitamin D, thus decreased vitamin D levels. Vitamin D deficiency, as well as low calcium and phosphate in the blood cause decreased bone mineral density.

Interactions
Phenytoin is an inducer of the CYP3A4 and CYP2C9 families of the P450 enzyme responsible for the liver's degradation of various drugs.

A 1981 study by the National Institutes of Health showed that antacids administered concomitantly with phenytoin "altered not only the extent of absorption but also appeared to alter the rate of absorption. Antacids administered in a peptic ulcer regimen may decrease the AUC of a single dose of phenytoin. Patients should be cautioned against concomitant use of antacids and phenytoin."

Warfarin and trimethoprim increase serum phenytoin levels and prolong the serum half-life of phenytoin by inhibiting its metabolism. Consider using other options if possible.

In general, phenytoin can interact with the following drugs:

 * Antidepressants drugs
 * Antifungal drugs such as fluconazole, ketoconazole
 * antibiotics such as metronidazole, chloramphenicol, clarithromycin, azithromycin
 * Cortones (such as betamethasone, dexamethasone, hydrocortisone and prednisolone
 * L-DOPA (phenytoin can cause the beneficial effect of levodopa to disappear.)

Mechanism of action
Phenytoin is believed to protect against seizures by causing voltage-dependent block of voltage gated sodium channels. This blocks sustained high frequency repetitive firing of action potentials. This is accomplished by reducing the amplitude of sodium-dependent action potentials through enhancing steady-state inactivation. Sodium channels exist in three main conformations: the resting state, the open state, and the inactive state.

Phenytoin binds preferentially to the inactive form of the sodium channel. Because it takes time for the bound drug to dissassociate from the inactive channel, there is a time-dependent block of the channel. Since the fraction of inactive channels is increased by membrane depolarization as well as by repetitive firing, the binding to the inactive state by phenytoin sodium can produce voltage-dependent, use-dependent and time-dependent block of sodium-dependent action potentials.

The primary site of action appears to be the motor cortex where spread of seizure activity is inhibited. Possibly by promoting sodium efflux from neurons, phenytoin tends to stabilize the threshold against hyperexcitability caused by excessive stimulation or environmental changes capable of reducing membrane sodium gradient. This includes the reduction of post-tetanic potentiation at synapses which prevents cortical seizure foci from detonating adjacent cortical areas. Phenytoin reduces the maximal activity of brain stem centers responsible for the tonic phase of generalized tonic-clonic seizures.

Pharmacokinetics
Phenytoin elimination kinetics show mixed-order, non-linear elimination behaviour at therapeutic concentrations. Where phenytoin is at low concentration it is cleared by first order kinetics, and at high concentrations by zero order kinetics. A small increase in dose may lead to a large increase in drug concentration as elimination becomes saturated. The time to reach steady state is often longer than 2 weeks.

History
Phenytoin (diphenylhydantoin) was first synthesized by German chemist Heinrich Biltz in 1908. Biltz sold his discovery to Parke-Davis, which did not find an immediate use for it. In 1938, other physicians, including H. Houston Merritt and Tracy Putnam, discovered phenytoin's usefulness for controlling seizures, without the sedative effects associated with phenobarbital.

According to Goodman and Gilman's Pharmacological Basis of Therapeutics:

"In contrast to the earlier accidental discovery of the antiseizure properties of potassium bromide and phenobarbital, phenytoin was the product of a search among nonsedative structural relatives of phenobarbital for agents capable of suppressing electroshock convulsions in laboratory animals."

It was approved by the FDA in 1953 for use in seizures.

Jack Dreyfus, founder of the Dreyfus Fund, became a major proponent of phenytoin as a means to control nervousness and depression when he received a prescription for Dilantin in 1966. He has claimed to have supplied large amounts of the drug to Richard Nixon throughout the late 1960s and early 1970s, although this is disputed by former White House aides and Presidential historians. Dreyfus' experience with phenytoin is outlined in his book, A Remarkable Medicine Has Been Overlooked. Despite more than $70 million in personal financing, his push to see phenytoin evaluated for alternative uses has had little lasting effect on the medical community. This was partially because Parke-Davis was reluctant to invest in a drug nearing the end of its patent life, and partially due to mixed results from various studies.

In 2008, the drug was put on the FDA's Potential Signals of Serious Risks List to be further evaluated for approval. The list identifies medications with which the FDA has identified potential safety issues, but has not yet identified a causal relationship between the drug and the listed risk. To address this concern, the Warnings and Precautions section of the labeling for Dilantin injection was updated to include additional information about Purple glove syndrome in November 2011.

Economics
Phenytoin is available as a generic medication.

Since September 2012, the marketing licence in the UK has been held by Flynn Pharma Ltd, of Dublin, Ireland, and the product, although identical, has been called Phenytoin Sodium xxmg Flynn Hard Capsules. (The xxmg in the name refers to the strength—for example "Phenytoin sodium 25 mg Flynn Hard Capsules"). The capsules are still made by Pfizer's Goedecke subsidiary's plant in Freiburg, Germany, and they still have Epanutin printed on them. After Pfizer's sale of the UK marketing licence to Flynn Pharma, the price of a 28-pack of 25 mg phenytoin sodium capsules marked Epanutin rose from 66p (about $0.88) to GB£15.74 (about $25.06). Capsules of other strengths also went up in price by the same factor—2,384%, costing the UK's National Health Service an extra GB£43 million (about $68.44 million) a year. The companies were referred to the Competition and Markets Authority (CMA) who found that they had exploited their dominant position in the market to charge "excessive and unfair" prices.

The CMA imposed a record GB£84.2 million fine on the manufacturer Pfizer, and a GB£5.2 million fine on the distributor Flynn Pharma and ordered the companies to reduce their prices.

Brand names
Phenytoin is marketed under many brand names worldwide.

In the US, Dilantin is marketed by Viatris after Upjohn was spun off from Pfizer.

Research
Tentative evidence suggests that topical phenytoin is useful in wound healing in people with chronic skin wounds. A meta-analysis also supported the use of phenytoin in managing various ulcers. Phenytoin is incorporated into compounded medications to optimize wound treatment, often in combination with misoprostol.

Some clinical trials have explored whether phenytoin can be used as neuroprotector in multiple sclerosis.

title: Phenytoin