1.
Elliptic coordinate system
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In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F1 and F2 are generally taken to be fixed at − a and + a, respectively, on the x -axis of the Cartesian coordinate system. The most common definition of coordinates is x = a cosh μ cos ν y = a sinh μ sin ν where μ is a nonnegative real number. On the complex plane, an equivalent relationship is x + i y = a cosh These definitions correspond to ellipses, in an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the coordinates are equal to h μ = h ν = a sinh 2 μ + sin 2 ν = a cosh 2 μ − cos 2 ν. Using the double argument identities for hyperbolic functions and trigonometric functions, other differential operators such as ∇ ⋅ F and ∇ × F can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates. An alternative and geometrically intuitive set of coordinates are sometimes used. Hence, the curves of constant σ are ellipses, whereas the curves of constant τ are hyperbolae, the coordinate τ must belong to the interval, whereas the σ coordinate must be greater than or equal to one. The coordinates have a relation to the distances to the foci F1 and F2. For any point in the plane, the sum d 1 + d 2 of its distances to the foci equals 2 a σ, thus, the distance to F1 is a, whereas the distance to F2 is a. A drawback of these coordinates is that the points with Cartesian coordinates and have the coordinates, so the conversion to Cartesian coordinates is not a function. The scale factors for the elliptic coordinates are h σ = a σ2 − τ2 σ2 −1 h τ = a σ2 − τ21 − τ2. Hence, the area element becomes d A = a 2 σ2 − τ2 d σ d τ. Other differential operators such as ∇ ⋅ F and ∇ × F can be expressed in the coordinates by substituting the scale factors into the general found in orthogonal coordinates. Elliptic coordinates form the basis for several sets of orthogonal coordinates. The elliptic cylindrical coordinates are produced by projecting in the z -direction, some traditional examples are solving systems such as electrons orbiting a molecule or planetary orbits that have an elliptical shape. The geometric properties of elliptic coordinates can also be useful, for concreteness, r, p and q could represent the momenta of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products. Curvilinear coordinates Generalized coordinates Hazewinkel, Michiel, ed. Elliptic coordinates, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Korn GA, mathematical Handbook for Scientists and Engineers, McGraw-Hill
2.
Celestial coordinate system
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In astronomy, a celestial coordinate system is a system for specifying positions of celestial objects, satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, the coordinate systems are implemented in either spherical coordinates or rectangular coordinates. Spherical coordinates, projected on the sphere, are analogous to the geographic coordinate system used on the surface of the Earth. These differ in their choice of fundamental plane, which divides the sphere into two equal hemispheres along a great circle. Rectangular coordinates, in units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental plane. Each coordinate system is named after its choice of fundamental plane, the following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane, the primary direction is the starting point of the longitudinal coordinates. The origin is the distance point, the center of the celestial sphere. The horizontal, or altitude-azimuth, system is based on the position of the observer on Earth, the positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of relative to an observers ideal horizon. The equatorial coordinate system is centered at Earths center, but fixed relative to the celestial poles, the coordinates are based on the location of stars relative to Earths equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the solar system, the equatorial system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescopes or other instruments scales so that they match the equatorial coordinates of the object to observe. There are also subdivisions into mean of date coordinates, which average out or ignore nutation, and true of date, the fundamental plane is the plane of the Earths orbit, called the ecliptic plane. The geocentric ecliptic system was the coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon. The heliocentric ecliptic system describes the orbital movement around the Sun. The system is used for computing the positions of planets and other solar system bodies
3.
Orbit
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In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet about a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating path around a body, to a close approximation, planets and satellites follow elliptical orbits, with the central mass being orbited at a focal point of the ellipse, as described by Keplers laws of planetary motion. For ease of calculation, in most situations orbital motion is adequately approximated by Newtonian Mechanics, historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and it assumed the heavens were fixed apart from the motion of the spheres, and was developed without any understanding of gravity. After the planets motions were accurately measured, theoretical mechanisms such as deferent. Originally geocentric it was modified by Copernicus to place the sun at the centre to help simplify the model, the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Second, he found that the speed of each planet is not constant, as had previously been thought. Third, Kepler found a relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter,5. 23/11.862, is equal to that for Venus,0. 7233/0.6152. Idealised orbits meeting these rules are known as Kepler orbits, isaac Newton demonstrated that Keplers laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the sizes are in inverse proportion to their masses. Where one body is more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, in a dramatic vindication of classical mechanics, in 1846 le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits, in relativity theory, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions but the differences are measurable. Essentially all the evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy
4.
Solar System
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The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets, of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury. The Solar System formed 4.6 billion years ago from the collapse of a giant interstellar molecular cloud. The vast majority of the mass is in the Sun. The four smaller inner planets, Mercury, Venus, Earth and Mars, are terrestrial planets, being composed of rock. The four outer planets are giant planets, being more massive than the terrestrials. All planets have almost circular orbits that lie within a flat disc called the ecliptic. The Solar System also contains smaller objects, the asteroid belt, which lies between the orbits of Mars and Jupiter, mostly contains objects composed, like the terrestrial planets, of rock and metal. Beyond Neptunes orbit lie the Kuiper belt and scattered disc, which are populations of trans-Neptunian objects composed mostly of ices, within these populations are several dozen to possibly tens of thousands of objects large enough that they have been rounded by their own gravity. Such objects are categorized as dwarf planets, identified dwarf planets include the asteroid Ceres and the trans-Neptunian objects Pluto and Eris. In addition to two regions, various other small-body populations, including comets, centaurs and interplanetary dust clouds. Six of the planets, at least four of the dwarf planets, each of the outer planets is encircled by planetary rings of dust and other small objects. The solar wind, a stream of charged particles flowing outwards from the Sun, the heliopause is the point at which pressure from the solar wind is equal to the opposing pressure of the interstellar medium, it extends out to the edge of the scattered disc. The Oort cloud, which is thought to be the source for long-period comets, the Solar System is located in the Orion Arm,26,000 light-years from the center of the Milky Way. For most of history, humanity did not recognize or understand the concept of the Solar System, the invention of the telescope led to the discovery of further planets and moons. The principal component of the Solar System is the Sun, a G2 main-sequence star that contains 99. 86% of the known mass. The Suns four largest orbiting bodies, the giant planets, account for 99% of the mass, with Jupiter. The remaining objects of the Solar System together comprise less than 0. 002% of the Solar Systems total mass, most large objects in orbit around the Sun lie near the plane of Earths orbit, known as the ecliptic
5.
Planet
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The term planet is ancient, with ties to history, astrology, science, mythology, and religion. Several planets in the Solar System can be seen with the naked eye and these were regarded by many early cultures as divine, or as emissaries of deities. As scientific knowledge advanced, human perception of the planets changed, in 2006, the International Astronomical Union officially adopted a resolution defining planets within the Solar System. This definition is controversial because it excludes many objects of mass based on where or what they orbit. The planets were thought by Ptolemy to orbit Earth in deferent, at about the same time, by careful analysis of pre-telescopic observation data collected by Tycho Brahe, Johannes Kepler found the planets orbits were not circular but elliptical. As observational tools improved, astronomers saw that, like Earth, the planets rotated around tilted axes, and 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 planets share characteristics such as volcanism, hurricanes, tectonics. Planets are generally divided into two types, large low-density giant planets, and smaller rocky terrestrials. Under IAU definitions, there are eight planets in the Solar System, in order of increasing distance from the Sun, they are the four terrestrials, Mercury, Venus, Earth, and Mars, then the four giant planets, Jupiter, Saturn, Uranus, 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. e. in the 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. A2012 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 lights of antiquity to the earthly objects of the scientific age. The concept has expanded to include not only in the Solar System. 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, and ancient astronomy. In ancient times, astronomers noted how certain lights moved across the sky, as opposed to the fixed stars, ancient Greeks called these lights πλάνητες ἀστέρες or simply πλανῆται, from which todays word planet was derived. In ancient Greece, China, Babylon, and indeed all pre-modern civilizations, it was almost universally believed that Earth was the center of the Universe and that all the planets circled Earth. The first civilization known to have a theory of the planets were the Babylonians
6.
Mercury (planet)
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Mercury is the smallest and innermost planet in the Solar System. Its orbital period around the Sun of 88 days is the shortest of all the planets in the Solar System and it is named after the Roman deity Mercury, the messenger to the gods. Like Venus, Mercury orbits the Sun within Earths orbit as a planet, so it can only be seen visually in the morning or the evening sky. Also, like Venus and the Moon, the displays the complete range of phases as it moves around its orbit relative to Earth. Seen from Earth, this cycle of phases reoccurs approximately every 116 days, although Mercury can appear as a bright star-like object when viewed from Earth, its proximity to the Sun often makes it more difficult to see than Venus. Mercury is tidally or gravitationally locked with the Sun in a 3,2 resonance, as seen relative to the fixed stars, it rotates on its axis exactly three times for every two revolutions it makes around the Sun. As seen from the Sun, in a frame of reference that rotates with the orbital motion, an observer on Mercury would therefore see only one day every two years. Mercurys axis has the smallest tilt of any of the Solar Systems planets, at aphelion, Mercury is about 1.5 times as far from the Sun as it is at perihelion. Mercurys surface appears heavily cratered and is similar in appearance to the Moons, the polar regions are constantly below 180 K. The planet has no natural satellites. Mercury is one of four planets in the Solar System. It is the smallest planet in the Solar System, with a radius of 2,439.7 kilometres. Mercury is also smaller—albeit more massive—than the largest natural satellites in the Solar System, Ganymede, Mercury consists of approximately 70% metallic and 30% silicate material. Mercurys density is the second highest in the Solar System at 5.427 g/cm3, Mercurys density can be used to infer details of its inner structure. Although Earths high density results appreciably from gravitational compression, particularly at the core, Mercury is much smaller, therefore, for it to have such a high density, its core must be large and rich in iron. Geologists estimate that Mercurys core occupies about 55% of its volume, Research published in 2007 suggests that Mercury has a molten core. Surrounding the core is a 500–700 km mantle consisting of silicates, based on data from the Mariner 10 mission and Earth-based observation, Mercurys crust is estimated to be 35 km thick. One distinctive feature of Mercurys surface is the presence of narrow ridges
7.
Small Solar System body
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A Small Solar System Body is an object in the Solar System that is neither a planet, nor a dwarf planet, nor a natural satellite. The term was first defined in 2006 by the International Astronomical Union, all other objects, except satellites, orbiting the Sun shall be referred to collectively as Small Solar System Bodies. These currently include most of the Solar System asteroids, most Trans-Neptunian Objects, comets and this encompasses all comets and all minor planets other than those that are dwarf planets. Except for the largest, which are in equilibrium, natural satellites differ from small Solar System bodies not in size. The orbits of satellites are not centered on the Sun, but around other Solar System objects such as planets, dwarf planets. Some of the larger small Solar System bodies may be reclassified in future as dwarf planets, the orbits of the vast majority of small Solar System bodies are located in two distinct areas, namely the asteroid belt and the Kuiper belt. These two belts possess some internal structure related to perturbations by the planets, and have fairly loosely defined boundaries. Other areas of the Solar System also encompass small bodies in smaller concentrations and these include the near-Earth asteroids, centaurs, comets, and scattered disc objects
8.
Orbital inclination
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Orbital inclination measures the tilt of an objects orbit around a celestial body. It is expressed as the angle between a plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Earth directly above the equator, the plane of the orbit is the same as the Earths equatorial plane. The general case is that the orbit is tilted, it spends half an orbit over the northern hemisphere. If the orbit swung between 20° north latitude and 20° south latitude, then its orbital inclination would be 20°, the inclination is one of the six orbital elements describing the shape and orientation of a celestial orbit. It is the angle between the plane and the plane of reference, normally stated in degrees. For a satellite orbiting a planet, the plane of reference is usually the plane containing the planets equator, for planets in the Solar System, the plane of reference is usually the ecliptic, the plane in which the Earth orbits the Sun. This reference plane is most practical for Earth-based observers, therefore, Earths inclination is, by definition, zero. Inclination could instead be measured with respect to another plane, such as the Suns equator or the invariable plane, the inclination of orbits of natural or artificial satellites is measured relative to the equatorial plane of the body they orbit, if they orbit sufficiently closely. The equatorial plane is the perpendicular to the axis of rotation of the central body. An inclination of 30° could also be described using an angle of 150°, the convention is that the normal orbit is prograde, an orbit in the same direction as the planet rotates. Inclinations greater than 90° describe retrograde orbits, thus, An inclination of 0° means the orbiting body has a prograde orbit in the planets equatorial plane. An inclination greater than 0° and less than 90° also describe prograde orbits, an inclination of 63. 4° is often called a critical inclination, when describing artificial satellites orbiting the Earth, because they have zero apogee drift. An inclination of exactly 90° is an orbit, in which the spacecraft passes over the north and south poles of the planet. An inclination greater than 90° and less than 180° is a retrograde orbit, an inclination of exactly 180° is a retrograde equatorial orbit. For gas giants, the orbits of moons tend to be aligned with the giant planets equator, the inclination of exoplanets or members of multiple stars is the angle of the plane of the orbit relative to the plane perpendicular to the line-of-sight from Earth to the object. An inclination of 0° is an orbit, meaning the plane of its orbit is parallel to the sky. An inclination of 90° is an orbit, meaning the plane of its orbit is perpendicular to the sky
9.
Ecliptic
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The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. The ecliptic is actually the apparent path of the Sun throughout the course of a year, because Earth takes one year to orbit the Sun, the apparent position of the Sun also takes the same length of time to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day, again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, for example, the Sun is north of the celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time, if the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes. The Sun, in its apparent motion along the ecliptic, crosses the equator at these points, one from south to north. The crossing from south to north is known as the equinox, also known as the first point of Aries. The crossing from north to south is the equinox or descending node. Likewise, the ecliptic itself is not fixed, the gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earths orbit, and hence of the ecliptic, known as planetary precession. The combined action of two motions is called general precession, and changes the position of the equinoxes by about 50 arc seconds per year. Once again, this is a simplification, periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earths axis, and hence the celestial equator, known as nutation. Obliquity of the ecliptic is the used by astronomers for the inclination of Earths equator with respect to the ecliptic. It is about 23. 4° and is currently decreasing 0.013 degrees per hundred years due to planetary perturbations, the angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. From 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated, jPLs fundamental ephemerides have been continually updated. J. Laskar computed an expression to order T10 good to 0″. 04/1000 years over 10,000 years, all of these expressions are for the mean obliquity, that is, without the nutation of the equator included