Apsidal Precession
In celestial mechanics, apsidal precession (or apsidal advance) is the precession (gradual rotation) of the line connecting the apsis, apsides (line of apsides) of an orbiting body, astronomical body's orbit. The apsides are the orbital points farthest (apoapsis) and closest (periapsis) from its primary (astronomy), primary body. The apsidal precession is the first time derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°, which takes the Earth about 112,000 years and the Moon about 8.85 years. History The ancient Greek astronomer Hipparchus noted the apsidal precession of the Moon's orbit (as the revolution of the Moon's apogee with a period of approximately 8.85 years); it is corrected for in the Antikythera Mechanism (circa 80 BC ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Precessing Kepler Orbit 280frames E0
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called ''nutation''. In physics, there are two types of precession: torque-free and torque-induced. In astronomy, ''precession'' refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the Earth, known as the precession of the equinoxes. Torque-free or torque neglected Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the angular momentum is a constant, but ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity. Mathematical definition The quadrupole moment tensor ''Q'' is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. Q_ + Q_ + Q_ = 0). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and zero-trace property, in this form only five of these are independent. For a discrete system of \ell point charges or masses in the case of a gravitational quadrupole, each with charge q_\ell, or mass m_\ell, and position \mathbf_\ell = \left(r_, r_, r_\right) relative to the coordinate system origin, the components of the ''Q'' matrix are defined by: Q_ = \sum_\ell q_\ell\left(3r_ r_ - \left\, \mathbf_\el ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Urbain Le Verrier
Urbain Jean Joseph Le Verrier (; 11 March 1811 – 23 September 1877) was a French astronomer and mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune using only mathematics. The calculations were made to explain discrepancies with Uranus's orbit and the laws of Kepler and Newton. Le Verrier sent the coordinates to Johann Gottfried Galle in Berlin, asking him to verify. Galle found Neptune the same night he received Le Verrier's letter, within 1° of the predicted position. The discovery of Neptune is widely regarded as a dramatic validation of celestial mechanics, and is one of the most remarkable moments of 19th-century science. Life Early years Urbain Le Verrier was born at Saint-Lô, Manche, France, to a modest bourgeois family, his parents being Louis-Baptiste Le Verrier and Marie-Jeanne-Josephine-Pauline de Baudre. He studied at the École Polytechnique – briefly chemistry, under Gay-Lussac, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Mercury (planet)
Mercury is the first planet from the Sun. It is a rocky planet with a trace atmosphere. While it is the List of Solar System objects by size, smallest and least massive planet of the Solar System, its surface gravity is slightly higher than that of Mars. The surface of Mercury is similar to Earth's Moon, heavily Impact crater, cratered, with expansive rupes system, generated from thrust faults, and bright ray systems, formed by ejecta. Its largest crater, Caloris Planitia, has a diameter of , which is about one-third the diameter of the planet (). Being the most inferior planet, inferior orbiting planet it appears in Earth's sky, always close to the Sun, either as a "morning star" or an "evening star". It stays most of the time the closest to all other planets and is the planet with the highest delta-v needed to travel to from all other planets of the Solar System. Mercury's sidereal year (88.0 Earth days) and sidereal day (58.65 Earth days) are in a 3:2 ratio. This relation ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Perihelion Precession
In celestial mechanics, apsidal precession (or apsidal advance) is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points farthest (apoapsis) and closest (periapsis) from its primary body. The apsidal precession is the first time derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°, which takes the Earth about 112,000 years and the Moon about 8.85 years. History The ancient Greek astronomer Hipparchus noted the apsidal precession of the Moon's orbit (as the revolution of the Moon's apogee with a period of approximately 8.85 years); it is corrected for in the Antikythera Mechanism (circa 80 BCE) (with the supposed value of 8.88 years ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. T ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Newton's Law Of Universal Gravitation
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product of their masses and Proportionality (mathematics)#Inverse proportionality, inversely proportional to the square of the distance between their centers of mass. Separated objects attract and are attracted Shell theorem, as if all their mass were concentrated at their centers. The publication of the law has become known as the "Unification (physics)#Unification of gravity and astronomy, first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors. This is a general physical law derived from empirical observations by what Isaac Newton called ''inductive reasoning''. It is a part of classical mechanics and was formulated in Newton's work ''Philosophiæ Natura ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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WASP-12b
WASP-12b is a hot Jupiter (a class of extrasolar planets) orbiting the star WASP-12, discovered in April of 2008, by the SuperWASP planetary transit survey. The planet takes only a little over one Earth day to orbit its star, in contrast to about 365.25 days for the Earth to orbit the Sun. Its distance from the star (approximately ) is only \textstyle\frac the Earth's distance from the Sun, with an eccentricity the same as Jupiter's. Consequently, it has one of the lowest densities for exoplanets ("inflated" by the flux of energy from the star). On December 3, 2013, scientists working with the Hubble Space Telescope (HST) reported detecting water in the atmosphere of the exoplanet. In July 2014, NASA announced finding very dry atmospheres on three exoplanets ( HD 189733b, HD 209458b, WASP-12b) orbiting sun-like stars. In September 2017, researchers working on the HST announced that WASP-12b reflects just 6% of the light that shines on its surface. As a result, the exoplanet h ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hot Jupiter
Hot Jupiters (sometimes called hot Saturns) are a class of gas giant exoplanets that are inferred to be physically similar to Jupiter (i.e. Jupiter analogue, Jupiter analogues) but that have very short orbital periods (). The close proximity to their stars and high surface-atmosphere temperatures resulted in their informal name "hot Jupiters". Hot Jupiters are the easiest extrasolar planets to detect via the radial velocity, radial-velocity method, because the oscillations they induce in their parent stars' motion are relatively large and rapid compared to those of other known types of planets. One of the best-known hot Jupiters is . Discovered in 1995, it was the first extrasolar planet found orbiting a Sun-like star. has an orbital period of about four days. General characteristics Though there is diversity among hot Jupiters, they do share some common properties. * Their defining characteristics are their large masses and short orbital periods, spanning 0.36–11.8 ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Bertrand's Theorem
In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits. The first such potential is an inverse-square central force such as the gravitational or electrostatic potential: V(r) = -\frac with force f(r) = -\frac = -\frac. The second is the radial harmonic oscillator potential: V(r) = \frac k r^2 with force f(r) = -\frac = -k r. The theorem is named after its discoverer, Joseph Bertrand. Derivation All attractive central forces can produce circular orbits, which are naturally closed orbits. The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circu ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |