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Jupiter Trojan
The Jupiter trojans, commonly called trojan asteroids or simply trojans, are a large group of asteroids that share the planet Jupiter's orbit around the Sun. Relative to Jupiter, each trojan librates around one of Jupiter's stable Lagrange points: either ', existing 60° ahead of the planet in its orbit, or ', 60° behind. Jupiter trojans are distributed in two elongated, curved regions around these Lagrangian points with an average semi-major axis of about 5.2  AU. The first Jupiter trojan discovered, 588 Achilles, was spotted in 1906 by German astronomer Max Wolf. More than 9,800 Jupiter trojans have been found . By convention, they are each named from Greek mythology after a figure of the Trojan War, hence the name "trojan". The total number of Jupiter trojans larger than 1 km in diameter is believed to be about , approximately equal to the number of asteroids larger than 1 km in the asteroid belt. Like main-belt asteroids, Jupiter trojans form families. ...
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Formation And Evolution Of The Solar System
The formation of the Solar System began about 4.6 billion years ago with the gravitational collapse of a small part of a giant molecular cloud. Most of the collapsing mass collected in the center, forming the Sun, while the rest flattened into a protoplanetary disk out of which the planets, moons, asteroids, and other small Solar System bodies formed. This model, known as the nebular hypothesis, was first developed in the 18th century by Emanuel Swedenborg, Immanuel Kant, and Pierre-Simon Laplace. Its subsequent development has interwoven a variety of scientific disciplines including astronomy, chemistry, geology, physics, and planetary science. Since the dawn of the space age in the 1950s and the discovery of extrasolar planets in the 1990s, the model has been both challenged and refined to account for new observations. The Solar System has evolved considerably since its initial formation. Many moons have formed from circling discs of gas and dust around their pa ...
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Lagrange Point
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem in which two bodies are far more massive than the third. Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit. Small objects placed in orbit at Lagrange points are in equilibrium in at least two directions relative to the center of mass of the large bodies. For any combination of two orbital bodies there are five Lagrange points, L1 to L5, all in the orbital plane of the two l ...
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Horseshoe Orbit
In celestial mechanics, a horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The osculating (instantaneous) orbital period of the smaller body remains very near that of the larger body, and if its orbit is a little more eccentric than that of the larger body, during every period it appears to trace an ellipse around a point on the larger object's orbit. However, the loop is not closed but drifts forward or backward so that the point it circles will appear to move smoothly along the larger body's orbit over a long period of time. When the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, with the larger body between the 'horns'. Asteroids in horseshoe orbits with respect to Earth include 54509 YORP, , , and possibly . A broader definition includes 3753 Cruithne, which can be said to be in a compound ...
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Tadpole Orbit
In celestial mechanics, a horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The osculating (instantaneous) orbital period of the smaller body remains very near that of the larger body, and if its orbit is a little more eccentric than that of the larger body, during every period it appears to trace an ellipse around a point on the larger object's orbit. However, the loop is not closed but drifts forward or backward so that the point it circles will appear to move smoothly along the larger body's orbit over a long period of time. When the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, with the larger body between the 'horns'. Asteroids in horseshoe orbits with respect to Earth include 54509 YORP, , , and possibly . A broader definition includes 3753 Cruithne, which can be said to be in a compound ...
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Restricted Three-body Problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the -body problem. Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, Earth, and the Sun. In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles. Mathematical description The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions \mathbf = (x_i, y_i, z_i ...
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Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJoseph-Louis Lagrange, comte de l’Empire
''Encyclopædia Britannica''
or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian and , later naturalized

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Maximilian Franz Joseph Cornelius Wolf
Maximilian Franz Joseph Cornelius Wolf (21 June 1863 – 3 October 1932) was a German astronomer and a pioneer in the field of astrophotography. He was the chairman of astronomy at the University of Heidelberg and director of the Heidelberg-Königstuhl State Observatory from 1902 until his death in 1932. Early life Max Wolf was born in Heidelberg, Germany on 21 June 1863, the son of medical doctor Franz Wolf. His father encouraged an interest in science and built an observatory for his son in the garden of the family home. It is from here that Wolf was credited with his first astronomical discovery, comet 14P/Wolf, in 1884. Life at the university Wolf attended his local university and, in 1888, at the age of 25, was awarded a Ph.D. by the University of Heidelberg. He spent one year of post-graduate study in Stockholm, the only significant time he would spend outside of Heidelberg in his life. He returned to the University of Heidelberg and accepted the position of ''p ...
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Earth Trojan
An Earth trojan is an asteroid that orbits the Sun in the vicinity of the Earth–Sun Lagrangian points (leading 60°) or (trailing 60°), thus having an orbit similar to Earth's. Only two Earth trojans have so far been discovered. The name "trojan" was first used in 1906 for the Jupiter trojans, the asteroids that were observed near the Lagrangian points of Jupiter's orbit. Members (leading) * : A 300-metre diameter asteroid, discovered using the Wide-field Infrared Survey Explorer (WISE) satellite in January 2010. * : Discovered by the Pan-STARRS survey in December 2020 and later recognized as an Earth trojan in January 2021. (trailing) * No known objects are currently thought to be trojans of Earth. Searches An Earth-based search for objects was conducted in 1994, covering 0.35 square degrees of sky, under poor observing conditions. Received 24 November 1997; revised 13 April 1998. That search failed to detect any objects ...
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Uranus Trojan
There are two known Uranus trojans, or minor planets orbiting in the Lagrangian points of Uranus Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of .... Both are in the region: * * See also * Trojan (astronomy) {{set index, astronomical objects 7 ...
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Neptune Trojan
Neptune trojans are bodies that orbit the Sun near one of the stable Lagrangian points of Neptune, similar to the trojans of other planets. They therefore have approximately the same orbital period as Neptune and follow roughly the same orbital path. Twenty-eight Neptune trojans are currently known, of which 24 orbit near the Sun–Neptune Lagrangian point 60° ahead of Neptune and four orbit near Neptune's region 60° behind Neptune. The Neptune trojans are termed 'trojans' by analogy with the Jupiter trojans. The discovery of in a high-inclination (>25°) orbit was significant, because it suggested a "thick" cloud of trojans (Jupiter trojans have inclinations up to 40°), which is indicative of freeze-in capture instead of in situ or collisional formation. It is suspected that large (radius ≈ 100 km) Neptune trojans could outnumber Jupiter trojans by an order of magnitude. E. I. Chiang and Y. Lithwick ''Neptune Trojans as a Testbed for Planet Formation'', The Astroph ...
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Mars Trojan
The Mars trojans are a group of trojan objects that share the orbit of the planet Mars around the Sun. They can be found around the two Lagrangian points 60° ahead of and behind Mars. The origin of the Mars trojans is not well understood. One theory suggests that they were primordial objects left over from the formation of Mars that were captured in its Lagrangian points as the Solar System was forming. However, spectral studies of the Mars trojans indicate this may not be the case. Another explanation involves asteroids chaotically wandering into the Mars Lagrangian points later in the Solar System's formation. This is also questionable considering the short dynamical lifetimes of these objects. The spectra of Eureka and two other Mars trojans indicates an olivine-rich composition. Since olivine-rich objects are rare in the asteroid belt it has been suggested that some of the Mars trojans are captured debris from a large orbit-altering impact on Mars when it encountered a plane ...
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