1.
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

2.
Apsis
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An apsis is an extreme point in an objects orbit. The word comes via Latin from Greek and is cognate with apse, for elliptic orbits about a larger body, there are two apsides, named with the prefixes peri- and ap-, or apo- added to a reference to the thing being orbited. For a body orbiting the Sun, the point of least distance is the perihelion, the terms become periastron and apastron when discussing orbits around other stars. For any satellite of Earth including the Moon the point of least distance is the perigee, for objects in Lunar orbit, the point of least distance is the pericynthion and the greatest distance the apocynthion. For any orbits around a center of mass, there are the terms pericenter and apocenter, periapsis and apoapsis are equivalent alternatives. A straight line connecting the pericenter and apocenter is the line of apsides and this is the major axis of the ellipse, its greatest diameter. For a two-body system the center of mass of the lies on this line at one of the two foci of the ellipse. When one body is larger than the other it may be taken to be at this focus. Historically, in systems, apsides were measured from the center of the Earth. In orbital mechanics, the apsis technically refers to the distance measured between the centers of mass of the central and orbiting body. However, in the case of spacecraft, the family of terms are used to refer to the orbital altitude of the spacecraft from the surface of the central body. The arithmetic mean of the two limiting distances is the length of the axis a. The geometric mean of the two distances is the length of the semi-minor axis b, the geometric mean of the two limiting speeds is −2 ε = μ a which is the speed of a body in a circular orbit whose radius is a. The words pericenter and apocenter are often seen, although periapsis/apoapsis are preferred in technical usage, various related terms are used for other celestial objects. The -gee, -helion and -astron and -galacticon forms are used in the astronomical literature when referring to the Earth, Sun, stars. The suffix -jove is occasionally used for Jupiter, while -saturnium has very rarely used in the last 50 years for Saturn. The -gee form is used as a generic closest approach to planet term instead of specifically applying to the Earth. During the Apollo program, the terms pericynthion and apocynthion were used when referring to the Moon, regarding black holes, the term peri/apomelasma was used by physicist Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002

3.
Irregular moon
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In astronomy, an irregular moon, irregular satellite or irregular natural satellite is a natural satellite following a distant, inclined, and often eccentric and retrograde orbit. They have been captured by their parent planet, unlike regular satellites,113 irregular satellites have been discovered, orbiting all four of the giant planets. The largest of each planet are Himalia of Jupiter, Phoebe of Saturn, Sycorax of Uranus, in 1997, the first two Uranian irregulars were discovered, Caliban and Sycorax. It is currently thought that the satellites were captured from heliocentric orbits near their current locations. An alternative theory, that they originated further out in the Kuiper belt, is not supported by current observations, there is no widely accepted precise definition of an irregular satellite. Informally, satellites are considered irregular if they are far enough from the planet that the precession of their plane is primarily controlled by the Sun. The radius of the Hill sphere is given in the adjacent table, the orbits of the known irregular satellites are extremely diverse, but there are certain patterns. Retrograde orbits are far more common than prograde orbits, no satellites are known with orbital inclinations higher than 55°. In addition, some groupings can be identified, in one large satellite shares a similar orbit with a few smaller ones. Given their distance from the planet, the orbits of the satellites are highly perturbed by the Sun. The semi-major axis of Pasiphae, for example, changes as much as 1.5 Gm in two years, the inclination around 10°, and the eccentricity as much as 0.4 in 24 years. Consequently, mean orbital elements are used to identify the groupings rather than osculating elements at the given date, irregular satellites have been captured from heliocentric orbits. This could involve, a collision of a body and a satellite, resulting in the incoming body losing energy. A close encounter between an incoming binary object and the planet, resulting in one component of the binary being captured, such a route has been suggested as most likely for Triton. After the capture, some of the satellites could break up leading to groupings of smaller moons following similar orbits, resonances could further modify the orbits making these groupings less recognizable. The current orbits of the moons are stable, in spite of substantial perturbations near the apocenter. The cause of stability in a number of irregulars is the fact that they orbit with a secular or Kozai resonance. The satellites enter the zone of the moons and are lost or ejected via collision

4.
N-body problem
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In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, in the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is more difficult to solve. Having done so, he and others soon discovered over the course of a few years, Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the problem in the early 17th century. Ironically, this conformity led to the wrong approach, after Newtons time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of gravitational interactive forces. Newton said in his Principia, paragraph 21, And hence it is that the force is found in both bodies. The Sun attracts Jupiter and the planets, Jupiter attracts its satellites. Two bodies can be drawn to other by the contraction of rope between them. Newton concluded via his third law of motion according to this Law all bodies must attract each other. This last statement, which implies the existence of gravitational forces, is key. The problem of finding the solution of the n-body problem was considered very important. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, in case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem, the version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n =3. The n-body problem considers n point masses mi, i =1,2, …, n in a reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi, Newtons second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass

5.
Orbital eccentricity
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The orbital eccentricity of an astronomical object is a parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is an orbit, values between 0 and 1 form an elliptical orbit,1 is a parabolic escape orbit. The term derives its name from the parameters of conic sections and it is normally used for the isolated two-body problem, but extensions exist for objects following a rosette orbit through the galaxy. In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit, the eccentricity of this Kepler orbit is a non-negative number that defines its shape. The limit case between an ellipse and a hyperbola, when e equals 1, is parabola, radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits have zero angular momentum and hence eccentricity equal to one, keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1. For a repulsive force only the trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that arcsin yields the projection angle of a circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury, next, tilt any circular object by that angle and the apparent ellipse projected to your eye will be of that same eccentricity. From Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros out of the center, from ἐκ- ek-, eccentric first appeared in English in 1551, with the definition a circle in which the earth, sun. Five years later, in 1556, a form of the word was added. The eccentricity of an orbit can be calculated from the state vectors as the magnitude of the eccentricity vector, e = | e | where. For elliptical orbits it can also be calculated from the periapsis and apoapsis since rp = a and ra = a, where a is the semimajor axis. E = r a − r p r a + r p =1 −2 r a r p +1 where, rp is the radius at periapsis. For Earths annual orbit path, ra/rp ratio = longest_radius / shortest_radius ≈1.034 relative to center point of path, the eccentricity of the Earths orbit is currently about 0.0167, the Earths orbit is nearly circular. Venus and Neptune have even lower eccentricity, over hundreds of thousands of years, the eccentricity of the Earths orbit varies from nearly 0.0034 to almost 0.058 as a result of gravitational attractions among the planets. The table lists the values for all planets and dwarf planets, Mercury has the greatest orbital eccentricity of any planet in the Solar System. Such eccentricity is sufficient for Mercury to receive twice as much solar irradiation at perihelion compared to aphelion, before its demotion from planet status in 2006, Pluto was considered to be the planet with the most eccentric orbit

6.
Escape velocity
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The escape velocity from Earth is about 11.186 km/s at the surface. More generally, escape velocity is the speed at which the sum of a kinetic energy. With escape velocity in a direction pointing away from the ground of a massive body, once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. When given a speed V greater than the speed v e. In these equations atmospheric friction is not taken into account, escape velocity is only required to send a ballistic object on a trajectory that will allow the object to escape the gravity well of the mass M. The existence of escape velocity is a consequence of conservation of energy, by adding speed to the object it expands the possible places that can be reached until with enough energy they become infinite. For a given gravitational potential energy at a position, the escape velocity is the minimum speed an object without propulsion needs to be able to escape from the gravity. Escape velocity is actually a speed because it does not specify a direction, no matter what the direction of travel is, the simplest way of deriving the formula for escape velocity is to use conservation of energy. Imagine that a spaceship of mass m is at a distance r from the center of mass of the planet and its initial speed is equal to its escape velocity, v e. At its final state, it will be a distance away from the planet. The same result is obtained by a calculation, in which case the variable r represents the radial coordinate or reduced circumference of the Schwarzschild metric. All speeds and velocities measured with respect to the field, additionally, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point. In common usage, the point is on the surface of a planet or moon. On the surface of the Earth, the velocity is about 11.2 km/s. However, at 9,000 km altitude in space, it is less than 7.1 km/s. The escape velocity is independent of the mass of the escaping object and it does not matter if the mass is 1 kg or 1,000 kg, what differs is the amount of energy required. For an object of mass m the energy required to escape the Earths gravitational field is GMm / r, a related quantity is the specific orbital energy which is essentially the sum of the kinetic and potential energy divided by the mass. An object has reached escape velocity when the orbital energy is greater or equal to zero

7.
Interplanetary Transport Network
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The Interplanetary Transport Network is a collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little or no energy and these points have the peculiar property of allowing objects to orbit around them, despite lacking an object to orbit. While they use energy, the transport can take a very long time. The key to discovering the Interplanetary Transport Network was the investigation of the nature of the winding paths near the Earth-Sun. They were first investigated by Jules-Henri Poincaré in the 1890s and he noticed that the paths leading to and from any of those points would almost always settle, for a time, on an orbit about that point. There are in fact a number of paths taking one to the point and away from it. When plotted, they form a tube with the orbit about the Lagrange point at one end, the derivation of these paths traces back to mathematicians Charles C. Hiten, Japans first lunar probe, was moved into orbit using similar insight into the nature of paths between the Earth and the Moon. They referred to it as an Interplanetary Superhighway As it turns out and this makes sense, since the orbit is unstable, which implies one will eventually end up on one of the outbound paths after spending no energy at all. However, with careful calculation, one can pick which outbound path one wants and this turned out to be useful, as many of these paths lead to some interesting points in space, such as the Earths Moon or between the Galilean moons of Jupiter. As a result, for the cost of reaching the Earth–Sun L2 point, but the trip from earth to Mars or other distant location would likely take thousands of years. The transfers are so low-energy that they travel to almost any point in the Solar System possible. On the downside, these transfers are very slow, for trips from earth to other planets, they are not useful for manned or unmanned probes, as the trip would take many generations. The network is also relevant to understanding Solar System dynamics, Comet Shoemaker–Levy 9 followed such a trajectory on its path with Jupiter. In a more recent example, the Chinese spacecraft Change 2 used the ITN to travel from lunar orbit to the Earth-Sun L2 point, then on to fly by the asteroid 4179 Toutatis. In addition to orbits around Lagrange points, the dynamics that arise from the gravitational pull of more than one mass yield interesting trajectories. For example, the gravity environment of the Sun–Earth–Moon system allows spacecraft to travel distances on very little fuel. Launched in 1978, the ISEE-3 spacecraft was sent on a mission to orbit around one of the Lagrange points, the spacecraft was able to maneuver around the Earths neighborhood using little fuel by taking advantage of the unique gravity environment

8.
Lissajous orbit
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Lyapunov orbits around a Lagrangian point are curved paths that lie entirely in the plane of the two primary bodies. In contrast, Lissajous orbits include components in this plane and perpendicular to it, halo orbits also include components perpendicular to the plane, but they are periodic, while Lissajous orbits are not. In practice, any orbits around Lagrangian points L1, L2, or L3 are dynamically unstable, as a result, spacecraft in these Lagrangian point orbits must use their propulsion systems to perform orbital station-keeping. These orbits can however be destabilized by other nearby massive objects, several missions have used Lissajous orbits, ACE at Sun–Earth L1, SOHO at Sun-Earth L1, DSCOVR at Sun–Earth L1, WMAP at Sun–Earth L2, and also the Genesis mission collecting solar particles at L1. On 14 May 2009, the European Space Agency launched into space the Herschel and Planck observatories, eSAs current Gaia mission also uses a Lissajous orbit at Sun–Earth L2. In 2011, NASA transferred two of its THEMIS spacecraft from Earth orbit to Lunar orbit by way of Earth-Moon L1 and L2 Lissajous orbits. In the 2005 science fiction novel Sunstorm by Arthur C. Clarke and Stephen Baxter, the shield is described to have been in a Lissajous orbit at L1. In the story a group of wealthy and powerful people shelter opposite the shield at L2 so as to be protected from the storm by the shield, the Earth. Koon, W. S. M. W. Lo, J. E. Marsden, dynamical Systems, the Three-Body Problem, and Space Mission Design

9.
Lambert's problem
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In celestial mechanics Lamberts problem is concerned with the determination of an orbit from two position vectors and the time of flight, solved by Johann Heinrich Lambert. It has important applications in the areas of rendezvous, targeting, guidance, suppose a body under the influence of a central gravitational force is observed to travel from point P1 on its conic trajectory, to a point P2 in a time T. Stated another way, Lamberts problem is the value problem for the differential equation r ¯ ¨ = − μ ⋅ r ^ r 2 of the two-body problem for which the Kepler orbit is the general solution. The precise formulation of Lamberts problem is as follows, Two different times t 1, t 2, find the solution r ¯ satisfying the differential equation above for which r ¯ = r ¯1 r ¯ = r ¯2. The value A is positive or negative depending on which of the points P1 and P2 that is furthest away from the point F1, the point F1 is either on the left or on the right branch of the hyperbola depending on the sign of A. The semi-major axis of this hyperbola is | A | and the eccentricity E is d | A | and this hyperbola is illustrated in figure 2. e. First one separates the cases of having the orbital pole in the direction r ¯1 × r ¯2 or in the direction − r ¯1 × r ¯2, then r ¯ will continue to pass through r ¯2 every orbital revolution. If t 2 − t 1 is in the range that can be obtained with an elliptic Kepler orbit corresponding y value can then be using an iterative algorithm. Selecting the parameter y as 30000 km one gets a time of 3072 seconds assuming the gravitational constant to be μ =398603 km3/s2. Corresponding orbital elements are semi-major axis =23001 km eccentricity =0.566613 true anomaly at time t1 = −7. 577° true anomaly at time t2 =92. 423° This y-value corresponds to Figure 3. The radial and tangential velocity components can then be computed with the formulas V r = μ p ⋅ e ⋅ sin θ V t = μ p ⋅, the transfer times from P1 to P2 for other values of y are displayed in Figure 4. The most typical use of this algorithm to solve Lamberts problem is certainly for the design of interplanetary missions and this approach is often used in conjunction with the patched conic approximation. This is also a method for orbit determination, from MATLAB central PyKEP a Python library for space flight mechanics and astrodynamics

10.
Jacobi integral
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In celestial mechanics, Jacobis integral is the only known conserved quantity for the circular restricted three-body problem. Unlike in the problem, the energy and momentum of the system are not conserved separately. The integral has been used to derive numerous solutions in special cases, as the system co-rotates with the two masses, they remain stationary and positioned at and 1. In this system of reference, the forces act on the particle are the two gravitational attractions, the centrifugal force and the Coriolis force. For a direct proof, see below. In the inertial, sidereal co-ordinate system, the masses are orbiting the barycentre, in these co-ordinates the Jacobi constant is expressed by, C J =2 +2 n −. The left side represents the square of the velocity v of the test particle in the co-rotating system, 1This co-ordinate system is non-inertial, which explains the appearance of terms related to centrifugal and Coriolis accelerations. Rotating reference frame Tisserands Criterion Carl D. Murray and Stanley F. Dermot Solar System Dynamics, pages 68–71