1.
Orbital elements
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Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in classical two-body systems. There are many different ways to describe the same orbit. A real orbit changes over time due to perturbations by other objects. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time, the traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion. When viewed from a frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the center of mass. When viewed from a non-inertial frame centred on one of the bodies, only the trajectory of the body is apparent. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference, the reference body is called the primary, the other body is called the secondary. The primary does not necessarily possess more mass than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary. The main two elements that define the shape and size of the ellipse, Eccentricity —shape of the ellipse, semimajor axis —the sum of the periapsis and apoapsis distances divided by two. For circular orbits, the axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass. For paraboles or hyperboles, this is infinite, tilt angle is measured perpendicular to line of intersection between orbital plane and reference plane. Any three points on an ellipse will define the ellipse orbital plane, the plane and the ellipse are both two-dimensional objects defined in three-dimensional space. Longitude of the ascending node —horizontally orients the ascending node of the ellipse with respect to the reference frames vernal point, and finally, Argument of periapsis defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis. True anomaly at epoch defines the position of the body along the ellipse at a specific time. The mean anomaly is a mathematically convenient angle which varies linearly with time and it can be converted into the true anomaly ν, which does represent the real geometric angle in the plane of the ellipse, between periapsis and the position of the orbiting object at any given time. Thus, the anomaly is shown as the red angle ν in the diagram

2.
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

3.
Orbital node
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An orbital node is one of the two points where an orbit crosses a plane of reference to which it is inclined. An orbit that is contained in the plane of reference has no nodes, common planes of reference include, For a geocentric orbit, the Earths equatorial plane. In this case, non-inclined orbits are called equatorial, for a heliocentric orbit, the ecliptic. In this case, non-inclined orbits are called ecliptic, for an orbit outside the Solar System, the plane through the primary perpendicular to a line through the observer and the primary. If a reference direction from one side of the plane of reference to the other is defined, the two nodes can be distinguished. For geocentric and heliocentric orbits, the node is where the orbiting object moves north through the plane of reference. The position of the node may be used as one of a set of parameters, called orbital elements and this is done by specifying the longitude of the ascending node The line of nodes is the intersection of the objects orbital plane with the plane of reference. It passes through the two nodes, the symbol of the ascending node is, and the symbol of the descending node is. In medieval and early times the ascending and descending nodes were called the dragons head and dragons tail. These terms originally referred to the times when the Moon crossed the apparent path of the sun in the sky, also, corruptions of the Arabic term such as ganzaar, genzahar, geuzaar and zeuzahar were used in the medieval West to denote either of the nodes. Pp. 196–197, p.65, pp. 95–96, the Greek terms αναβιβάζων and καταβιβάζων were also used for the ascending and descending nodes, giving rise to the English words anabibazon and catabibazon. For the orbit of the Moon around the Earth, the plane is taken to be the ecliptic. The gravitational pull of the Sun upon the Moon causes its nodes, called the nodes, to precess gradually westward. Eclipse Euler angles Longitude of the ascending node

4.
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

5.
Heliocentric orbit
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A heliocentric orbit is an orbit around the barycenter of the Solar System, which is usually located within or very near the surface of the Sun. All planets, comets, and asteroids in the Solar System are in such orbits, the moons of planets in the Solar System, by contrast, are not in heliocentric orbits as they orbit their respective planet. A similar phenomenon allows the detection of exoplanets by way of the radial velocity method, the helio- prefix is derived from the Greek word helios, meaning sun, and also Helios, the personification of the Sun in Greek mythology. The first spacecraft to be put in an orbit is Luna 1. A trans-Mars injection is an orbit in which a propulsive maneuver is used to set a spacecraft on a trajectory, also known as Mars transfer orbit. Every two years, low-energy transfer windows open up which allow movement between planets with the lowest possible delta-v requirements, transfer injections can place spacecraft into either a Hohmann transfer orbit or bi-elliptic transfer orbit. Trans-Mars injections can be either a single maneuver burn, such as used by the NASA MAVEN orbiter, or a series of perigee kicks. Earths orbit Geocentric orbit Heliocentrism Astrodynamics Low-energy transfer List of artificial objects in heliocentric orbit List of orbits

6.
Geocentric orbit
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A geocentric orbit or Earth orbit involves any object orbiting the Earth, such as the Moon or artificial satellites. In 1997 NASA estimated there were approximately 2,465 artificial satellite orbiting the Earth and 6,216 pieces of space debris as tracked by the Goddard Space Flight Center. Over 16,291 previously launched objects have decayed into the Earths atmosphere, altitude as used here, the height of an object above the average surface of the Earths oceans. Analemma a term in astronomy used to describe the plot of the positions of the Sun on the celestial sphere throughout one year, apogee is the farthest point that a satellite or celestial body can go from Earth, at which the orbital velocity will be at its minimum. Eccentricity a measure of how much an orbit deviates from a perfect circle, eccentricity is strictly defined for all circular and elliptical orbits, and parabolic and hyperbolic trajectories. Equatorial plane as used here, an imaginary plane extending from the equator on the Earth to the celestial sphere, escape velocity as used here, the minimum velocity an object without propulsion needs to have to move away indefinitely from the Earth. An object at this velocity will enter a parabolic trajectory, above this velocity it will enter a hyperbolic trajectory, impulse the integral of a force over the time during which it acts. Inclination the angle between a plane and another plane or axis. In the sense discussed here the reference plane is the Earths equatorial plane, orbital characteristics the six parameters of the Keplerian elements needed to specify that orbit uniquely. Orbital period as defined here, time it takes a satellite to make one orbit around the Earth. Perigee is the nearest approach point of a satellite or celestial body from Earth, sidereal day the time it takes for a celestial object to rotate 360°. For the Earth this is,23 hours,56 minutes,4.091 seconds, solar time as used here, the local time as measured by a sundial. Velocity an objects speed in a particular direction, since velocity is defined as a vector, both speed and direction are required to define it. The following is a list of different geocentric orbit classifications, Low Earth orbit - Geocentric orbits ranging in altitude from 160 kilometers to 2,000 kilometres above mean sea level. At 160 km, one revolution takes approximately 90 minutes, medium Earth orbit - Geocentric orbits with altitudes at apogee ranging between 2,000 kilometres and that of the geosynchronous orbit at 35,786 kilometres. Geosynchronous orbit - Geocentric circular orbit with an altitude of 35,786 kilometres, the period of the orbit equals one sidereal day, coinciding with the rotation period of the Earth. The speed is approximately 3,000 metres per second, high Earth orbit - Geocentric orbits with altitudes at apogee higher than that of the geosynchronous orbit. A special case of high Earth orbit is the elliptical orbit

7.
Longitude of the ascending node
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The longitude of the ascending node is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a direction, called the origin of longitude, to the direction of the ascending node. The ascending node is the point where the orbit of the passes through the plane of reference. Commonly used reference planes and origins of longitude include, For a geocentric orbit, Earths equatorial plane as the plane. In this case, the longitude is called the right ascension of the ascending node. The angle is measured eastwards from the First Point of Aries to the node, for a heliocentric orbit, the ecliptic as the reference plane, and the First Point of Aries as the origin of longitude. The angle is measured counterclockwise from the First Point of Aries to the node, the angle is measured eastwards from north to the node. pp.40,72,137, chap. In the case of a star known only from visual observations, it is not possible to tell which node is ascending. In this case the orbital parameter which is recorded is the longitude of the node, Ω, here, n=<nx, ny, nz> is a vector pointing towards the ascending node. The reference plane is assumed to be the xy-plane, and the origin of longitude is taken to be the positive x-axis, K is the unit vector, which is the normal vector to the xy reference plane. For non-inclined orbits, Ω is undefined, for computation it is then, by convention, set equal to zero, that is, the ascending node is placed in the reference direction, which is equivalent to letting n point towards the positive x-axis. Kepler orbits Equinox Orbital node perturbation of the plane can cause revolution of the ascending node

8.
Astrodynamics
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Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of objects is usually calculated from Newtons laws of motion. It is a discipline within space mission design and control. General relativity is an exact theory than Newtons laws for calculating orbits. Until the rise of space travel in the century, there was little distinction between orbital and celestial mechanics. At the time of Sputnik, the field was termed space dynamics, the fundamental techniques, such as those used to solve the Keplerian problem, are therefore the same in both fields. Furthermore, the history of the fields is almost entirely shared, johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1605. Isaac Newton published more general laws of motion in his 1687 book. The following rules of thumb are useful for situations approximated by classical mechanics under the assumptions of astrodynamics outlined below the rules. The specific example discussed is of a satellite orbiting a planet, Keplers laws of planetary motion, Orbits are elliptical, with the heavier body at one focus of the ellipse. Special case of this is an orbit with the planet at the center. A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured, the square of a satellites orbital period is proportional to the cube of its average distance from the planet. Without applying force, the period and shape of the satellites orbit wont change, a satellite in a low orbit moves more quickly with respect to the surface of the planet than a satellite in a higher orbit, due to the stronger gravitational attraction closer to the planet. If thrust is applied at one point in the satellites orbit, it will return to that same point on each subsequent orbit. Thus one cannot move from one orbit to another with only one brief application of thrust. Thrust applied in the direction of the satellites motion creates an elliptical orbit with an apoapse 180 degrees away from the firing point, the consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the circular orbit and wish to dock, unless they are very close. This will change the shape of its orbit, causing it to gain altitude and actually slow down relative to the leading craft, the space rendezvous before docking normally takes multiple precisely calculated engine firings in multiple orbital periods requiring hours or even days to complete

9.
Kepler orbit
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In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. It is thus said to be a solution of a case of the two-body problem. As a theory in classical mechanics, it also does not take account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways, in most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of mass can be described as Kepler orbits around their common center of mass. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path, as measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543, Nicolaus Copernicus published a model of the solar system. In 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe, Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion, the first law states, The orbit of every planet is an ellipse with the sun at a focus. More generally, the path of an object undergoing Keplerian motion may also follow a parabola or a hyperbola, alternately, the equation can be expressed as, r = p 1 + e cos Where p is called the semi-latus rectum of the curve. This form of the equation is useful when dealing with parabolic trajectories. Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions, between 1665 and 1666, Isaac Newton developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the Principia, in which he outlined his laws of motion, newtons law of gravitation states, Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The laws of Kepler and Newton formed the basis of modern celestial mechanics until Albert Einstein introduced the concepts of special and general relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in astronomy and astrodynamics. See also Orbit Analysis To solve for the motion of an object in a two body system, two simplifying assumptions can be made,1, the bodies are spherically symmetric and can be treated as point masses. There are no external or internal forces acting upon the other than their mutual gravitation. The shapes of large bodies are close to spheres

10.
Orbital mechanics
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Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of objects is usually calculated from Newtons laws of motion. It is a discipline within space mission design and control. General relativity is an exact theory than Newtons laws for calculating orbits. Until the rise of space travel in the century, there was little distinction between orbital and celestial mechanics. At the time of Sputnik, the field was termed space dynamics, the fundamental techniques, such as those used to solve the Keplerian problem, are therefore the same in both fields. Furthermore, the history of the fields is almost entirely shared, johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1605. Isaac Newton published more general laws of motion in his 1687 book. The following rules of thumb are useful for situations approximated by classical mechanics under the assumptions of astrodynamics outlined below the rules. The specific example discussed is of a satellite orbiting a planet, Keplers laws of planetary motion, Orbits are elliptical, with the heavier body at one focus of the ellipse. Special case of this is an orbit with the planet at the center. A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured, the square of a satellites orbital period is proportional to the cube of its average distance from the planet. Without applying force, the period and shape of the satellites orbit wont change, a satellite in a low orbit moves more quickly with respect to the surface of the planet than a satellite in a higher orbit, due to the stronger gravitational attraction closer to the planet. If thrust is applied at one point in the satellites orbit, it will return to that same point on each subsequent orbit. Thus one cannot move from one orbit to another with only one brief application of thrust. Thrust applied in the direction of the satellites motion creates an elliptical orbit with an apoapse 180 degrees away from the firing point, the consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the circular orbit and wish to dock, unless they are very close. This will change the shape of its orbit, causing it to gain altitude and actually slow down relative to the leading craft, the space rendezvous before docking normally takes multiple precisely calculated engine firings in multiple orbital periods requiring hours or even days to complete

11.
Box orbit
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In stellar dynamics a box orbit refers to a particular type of orbit that can be seen in triaxial systems, i. e. systems that do not possess a symmetry around any of its axes. They contrast with the orbits that are observed in spherically symmetric or axisymmetric systems. In a box orbit, the star oscillates independently along the three different axes as it moves through the system, as a result of this motion, it fills in a box-shaped region of space. Unlike loop orbits, the stars on box orbits can come close to the center of the system. As a special case, if the frequencies of oscillation in different directions are commensurate, such orbits are sometimes called boxlets. Horseshoe orbit Lissajous curve List of orbits

12.
Parabolic trajectory
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In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit and it is also sometimes referred to as a C3 =0 orbit. Parabolic trajectories are escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits. At any position the body has the escape velocity for that position. This is entirely equivalent to the energy being 0, C3 =0 Barkers equation relates the time of flight to the true anomaly of a parabolic trajectory. There are two cases, the move away from each other or towards each other. At any time the speed from t =0 is 1.5 times the current speed

13.
Circular orbit
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A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis. Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions, here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion. In this case, not only the distance, but also the speed, angular speed, potential, there is no periapsis or apoapsis. This orbit has no radial version, transverse acceleration causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have a = v 2 r = ω2 r where, v is velocity of orbiting body. The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of a is measured in meters per second per second, then the values for v will be in meters per second, r in meters. μ = G M is the standard gravitational parameter, the orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to, r = h 2 μ where, h = r v is specific angular momentum of the orbiting body. Maneuvering into a circular orbit, e. g. It is also a matter of maneuvering into the orbit, for the sake of convenience, the derivation will be written in units in which c = G =1. The four-velocity of a body on an orbit is given by. The dot above a variable denotes derivation with respect to proper time τ

14.
Elliptic orbit
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In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1, this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0, in a wider sense it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1, in a gravitational two-body problem with negative energy both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit, examples of elliptic orbits include, Hohmann transfer orbit, Molniya orbit and tundra orbit. A is the length of the semi-major axis, the velocity equation for a hyperbolic trajectory has either +1 a, or it is the same with the convention that in that case a is negative. Conclusions, For a given semi-major axis the orbital energy is independent of the eccentricity. ν is the true anomaly. The angular momentum is related to the cross product of position and velocity. Here ϕ is defined as the angle which differs by 90 degrees from this and this set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit, the two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit, another set of six parameters that are commonly used are the orbital elements. In the Solar System, planets, asteroids, most comets, the following chart of the perihelion and aphelion of the planets, dwarf planets and Halleys Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity, note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halleys Comet and Eris. A radial trajectory can be a line segment, which is a degenerate ellipse with semi-minor axis =0. Although the eccentricity is 1, this is not a parabolic orbit, most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed and it is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, the velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity. The radial elliptic trajectory is the solution of a problem with at some instant zero speed

15.
Horseshoe orbit
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A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The orbital period of the body is very nearly the same as for the larger body. The loop is not closed but will drift forward or backward slightly each 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, asteroids in horseshoe orbits with respect to Earth include 54509 YORP,2002 AA29,2010 SO16,2015 SO2 and possibly 2001 GO2. A broader definition includes 3753 Cruithne, which can be said to be in a compound and/or transition orbit, saturns moons Epimetheus and Janus occupy horseshoe orbits with respect to each other. The following explanation relates to an asteroid which is in such an orbit around the Sun, the asteroid is in almost the same solar orbit as Earth. Both take approximately one year to orbit the Sun and it is also necessary to grasp two rules of orbit dynamics, A body closer to the Sun completes an orbit more quickly than a body further away. If a body accelerates along its orbit, its orbit moves outwards from the Sun, if it decelerates, the orbital radius decreases. The horseshoe orbit arises because the attraction of the Earth changes the shape of the elliptical orbit of the asteroid. The shape changes are small but result in significant changes relative to the Earth. The horseshoe becomes apparent only when mapping the movement of the relative to both the Sun and the Earth. The asteroid always orbits the Sun in the same direction, however, it goes through a cycle of catching up with the Earth and falling behind, so that its movement relative to both the Sun and the Earth traces a shape like the outline of a horseshoe. Starting at point A, on the ring between L5 and Earth, the satellite is orbiting faster than the Earth and is on its way toward passing between the Earth and the Sun. But Earths gravity exerts an outward accelerating force, pulling the satellite into an orbit which decreases its angular speed. When the satellite gets to point B, it is traveling at the speed as Earth. Earths gravity is still accelerating the satellite along the orbital path, eventually, at Point C, the satellite reaches a high and slow enough orbit such that it starts to lag behind Earth. It then spends the next century or more appearing to drift backwards around the orbit when viewed relative to the Earth and its orbit around the Sun still takes only slightly more than one Earth year. Given enough time, the Earth and the satellite will be on opposite sides of the Sun, eventually the satellite comes around to point D where Earths gravity is now reducing the satellites orbital velocity

16.
Hyperbolic trajectory
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In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central objects gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola, in more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one. Under standard assumptions a body traveling along this trajectory will coast to infinity, similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories. The specific energy of a hyperbolic trajectory orbit is positive, planetary flybys, used for gravitational slingshots, can be described within the planets sphere of influence using hyperbolic trajectories. Like an elliptical orbit, a trajectory for a given system can be defined by its semi major axis. However, with a hyperbolic orbit other parameters may more useful in understanding a bodys motion, the following table lists the main parameters describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them. The semi major axis is not immediately visible with an hyperbolic trajectory, usually, by convention, it is negative, to keep various equations are consistent with elliptical orbits. With a hyperbolic trajectory the orbital eccentricity is greater than 1, the eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a v shape. At e =2 the asymptotes are at right angles, with e >2 the asymptotes are more that 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line, with bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola. In the situation of a spacecraft or comet approaching a planet, if the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planets radius an impact should be expected, a body approaching Jupiter from the outer solar system with a speed of 5.5 km/h, will need the impact parameter to be at least 770, 000km or 11 times Jupiter radius to avoid collision. As, typically, all variables can be determined accurately. μ = b v ∞2 tan δ /2 where δ =2 θ ∞ − π is the angle the body is deflected from a straight line in its course. Where μ is a parameter w and a is the semi-major axis of the orbit. The flight path angle is the angle between the direction of velocity and the perpendicular to the direction, so it is zero at periapsis. For example, at a place where escape speed is 11.2 km/s,11.62 −11.22 =3.02 This is an example of the Oberth effect

17.
Parking orbit
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A parking orbit is a temporary orbit used during the launch of a satellite or other space probe. A launch vehicle boosts into the orbit, then coasts for a while. The alternative to an orbit is direct injection, where the rocket fires continuously until its fuel is exhausted. There are several reasons why a parking orbit may be used, geostationary spacecraft require an orbit in the plane of the equator. Getting there requires a transfer orbit with an apogee directly above the equator. Unless the launch site itself is close to the equator. Instead, the craft is placed with a stage in an inclined parking orbit. When the craft crosses the equator, the stage is fired to raise the spacecrafts apogee to geostationary altitude. Finally, a burn is required to raise the perigee to the same altitude. In order to reach the Moon or a planet at a desired time, using a preliminary parking orbit before final injection can widen this window from seconds or minutes, to several hours. For the Apollo programs manned lunar missions, a parking orbit allowed time for spacecraft checkout while still close to home, use of a parking orbit requires a rocket upper stage to perform the injection burn while under zero g conditions. Often, the upper stage which performs the parking orbit injection is used for the final injection burn. During the parking orbit coast, the propellants will drift away from the bottom of the tank and this must be dealt with through the use of tank diaphragms, or ullage rockets to settle the propellant back to the bottom of the tank. A reaction control system is needed to orient the stage properly for the final burn, cryogenic propellants must be stored in well-insulated tanks, to prevent excessive boiloff during coast. Battery life and other consumables must be sufficient for the duration of the parking coast, the Centaur and Agena families of upper stages were designed for such restarts and have often been used in this manner. The last Agena flew in 1987, but Centaur is still in production, the Briz-M stage often performs the same role for Russian rockets. The Apollo program used parking orbits, for all the mentioned above except those that pertain to geostationary orbits. When the Space Shuttle orbiter launched interplanetary probes such as Galileo, the Ariane 5 does not use parking orbits

18.
Hohmann transfer orbit
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In orbital mechanics, the Hohmann transfer orbit /ˈhoʊ. mʌn/ is an elliptical orbit used to transfer between two circular orbits of different radii in the same plane. The orbital maneuver to perform the Hohmann transfer uses two engine impulses, one to move a spacecraft onto the orbit and a second to move off it. This maneuver was named after Walter Hohmann, the German scientist who published a description of it in his 1925 book Die Erreichbarkeit der Himmelskörper, Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets. The diagram shows a Hohmann transfer orbit to bring a spacecraft from a circular orbit into a higher one. It is one half of an orbit that touches both the lower circular orbit the spacecraft wishes to leave and the higher circular orbit that it wishes to reach. The transfer is initiated by firing the engine in order to accelerate it so that it will follow the elliptical orbit. When the spacecraft has reached its orbit, its orbital speed must be increased again in order to change the elliptic orbit to the larger circular one. The engine is fired again at the lower distance to slow the spacecraft into the lower circular orbit. The Hohmann transfer orbit is based on two instantaneous velocity changes, extra fuel is required to compensate for the fact that the bursts take time, this is minimized by using high thrust engines to minimize the duration of the bursts. Low thrust engines can perform an approximation of a Hohmann transfer orbit and this requires a change in velocity that is up to 141% greater than the two impulse transfer orbit, and takes longer to complete. Typically μ is given in units of m3/s2, as such be sure to use meters not kilometers for r 1 and r 2, the total Δ v is then, Δ v t o t a l = Δ v 1 + Δ v 2. In application to traveling from one body to another it is crucial to start maneuver at the time when the two bodies are properly aligned. In the smaller circular orbit the speed is 7.73 km/s, in the elliptical orbit in between the speed varies from 10.15 km/s at the perigee to 1.61 km/s at the apogee. The Δv for the two burns are thus 10.15 −7.73 =2.42 and 3.07 −1.61 =1.46 km/s, together 3.88 km/s. It is interesting to note that this is greater than the Δv required for an orbit,10.93 −7.73 =3.20 km/s. Applying a Δv at the LEO of only 0.78 km/s more would give the rocket the escape speed, as the example above demonstrates, the Δv required to perform a Hohmann transfer between two circular orbits is not the greatest when the destination radius is infinite. The Δv required is greatest when the radius of the orbit is 15.58 times that of the smaller orbit. This number is the root of x3 − 15x2 − 9x −1 =0

19.
Geosynchronous orbit
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A geosynchronous orbit is an orbit about the Earth of a satellite with an orbital period that matches the rotation of the Earth on its axis of approximately 23 hours 56 minutes and 4 seconds. Over the course of a day, the position in the sky traces out a path, typically in a figure-8 form, whose precise characteristics depend on the orbits inclination. Satellites are typically launched in an eastward direction, a special case of geosynchronous orbit is the geostationary orbit, which is a circular geosynchronous orbit at zero inclination. A satellite in a geostationary orbit appears stationary, always at the point in the sky. Popularly or loosely, the term geosynchronous may be used to mean geostationary, specifically, geosynchronous Earth orbit may be a synonym for geosynchronous equatorial orbit, or geostationary Earth orbit. A semi-synchronous orbit has a period of ½ sidereal day. Relative to the Earths surface it has twice this period, examples include the Molniya orbit and the orbits of the satellites in the Global Positioning System. Circular Earth geosynchronous orbits have a radius of 42,164 km, all Earth geosynchronous orbits, whether circular or elliptical, have the same semi-major axis.4418 km3/s2. In the special case of an orbit, the ground track of a satellite is a single point on the equator. A geostationary equatorial orbit is a geosynchronous orbit in the plane of the Earths equator with a radius of approximately 42,164 km. A satellite in such an orbit is at an altitude of approximately 35,786 km above sea level. It maintains the position relative to the Earths surface. The theoretical basis for this phenomenon of the sky goes back to Newtons theory of motion. In that theory, the existence of a satellite is made possible because the Earth rotates. Such orbits are useful for telecommunications satellites, a perfectly stable geostationary orbit is an ideal that can only be approximated. Elliptical geosynchronous orbits can be and are designed for satellites in order to keep the satellite within view of its assigned ground stations or receivers. A satellite in a geosynchronous orbit appears to oscillate in the sky from the viewpoint of a ground station. Satellites in highly elliptical orbits must be tracked by ground stations

20.
Geostationary orbit
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A geostationary orbit, geostationary Earth orbit or geosynchronous equatorial orbit is a circular orbit 35,786 kilometres above the Earths equator and following the direction of the Earths rotation. An object in such an orbit has a period equal to the Earths rotational period and thus appears motionless, at a fixed position in the sky. Using this characteristic, ocean color satellites with visible and near-infrared light sensors can also be operated in geostationary orbit in order to monitor sensitive changes of ocean environments, the notion of a geostationary space station equipped with radio communication was published in 1928 by Herman Potočnik. The first appearance of an orbit in popular literature was in the first Venus Equilateral story by George O. Smith. Clarke acknowledged the connection in his introduction to The Complete Venus Equilateral, the orbit, which Clarke first described as useful for broadcast and relay communications satellites, is sometimes called the Clarke Orbit. Similarly, the Clarke Belt is the part of space about 35,786 km above sea level, in the plane of the equator, the Clarke Orbit is about 265,000 km in circumference. Most commercial communications satellites, broadcast satellites and SBAS satellites operate in geostationary orbits, a geostationary transfer orbit is used to move a satellite from low Earth orbit into a geostationary orbit. The first satellite placed into an orbit was the Syncom-3. A worldwide network of operational geostationary meteorological satellites is used to provide visible and infrared images of Earths surface, a geostationary orbit can only be achieved at an altitude very close to 35,786 km and directly above the equator. This equates to a velocity of 3.07 km/s and an orbital period of 1,436 minutes. This ensures that the satellite will match the Earths rotational period and has a footprint on the ground. All geostationary satellites have to be located on this ring. 85° per year, to correct for this orbital perturbation, regular orbital stationkeeping manoeuvres are necessary, amounting to a delta-v of approximately 50 m/s per year. A second effect to be taken into account is the longitude drift, there are two stable and two unstable equilibrium points. Any geostationary object placed between the points would be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation. The correction of this effect requires station-keeping maneuvers with a maximal delta-v of about 2 m/s per year, solar wind and radiation pressure also exert small forces on satellites, over time, these cause them to slowly drift away from their prescribed orbits. In the absence of servicing missions from the Earth or a renewable propulsion method, hall-effect thrusters, which are currently in use, have the potential to prolong the service life of a satellite by providing high-efficiency electric propulsion. This delay presents problems for latency-sensitive applications such as voice communication, geostationary satellites are directly overhead at the equator and become lower in the sky the further north or south one travels. At latitudes above about 81°, geostationary satellites are below the horizon, because of this, some Russian communication satellites have used elliptical Molniya and Tundra orbits, which have excellent visibility at high latitudes

21.
Sun-synchronous orbit
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A Sun-synchronous orbit is a geocentric orbit that combines altitude and inclination in such a way that the satellite passes over any given point of the planets surface at the same local solar time. Such an orbit can place a satellite in constant sunlight and is useful for imaging, spy, more technically, it is an orbit arranged in such a way that it precesses once a year. The surface illumination angle will be nearly the same time that the satellite is overhead. This consistent lighting is a characteristic for satellites that image the Earths surface in visible or infrared wavelengths. For example, a satellite in sun-synchronous orbit might ascend across the twelve times a day each time at approximately 15,00 mean local time. This is achieved by having the orbital plane precess approximately one degree each day with respect to the celestial sphere, eastward. Typical sun-synchronous orbits are about 600–800 km in altitude, with periods in the 96–100 minute range, riding the terminator is useful for active radar satellites as the satellites solar panels can always see the Sun, without being shadowed by the Earth. The dawn/dusk orbit has been used for solar observing scientific satellites such as Yohkoh, TRACE, Hinode and PROBA2, Sun-synchronous orbits can happen around other oblate planets, such as Mars. A satellite around the almost spherical Venus, for example, will need an outside push to be in a sun-synchronous orbit.696 deg. Note that according to this approximation cos i equals −1 when the semi-major axis equals 12352 km, the period can be in the range from 88 minutes for a very low orbit to 3.8 hours. If one wants a satellite to fly over some given spot on Earth every day at the same hour, it can do between 7 and 16 orbits per day, as shown in the following table. When one says that a Sun-synchronous orbit goes over a spot on the earth at the local time each time. The Sun will not be in exactly the same position in the sky during the course of the year, very often a frozen orbit is therefore selected that is slightly higher over the Southern hemisphere than over the Northern hemisphere. ERS-1, ERS-2 and Envisat of European Space Agency as well as the MetOp spacecraft of EUMETSAT are all operated in Sun-synchronous, orbital perturbation analysis Analemma Geosynchronous orbit Geostationary orbit List of orbits Polar orbit World Geodetic System Sandwell, David T. The Gravity Field of the Earth - Part 1 Sun-Synchronous Orbit dictionary entry, centennial of Flight Commission NASA Q&A Boain, Ronald J. The A-B-Cs of Sun Synchronous Orbit Design, List of satellites in Sun-synchronous orbit

22.
Low Earth orbit
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A low Earth orbit is an orbit around Earth with an altitude between 160 kilometers, and 2,000 kilometers. Objects below approximately 160 kilometers will experience very rapid orbital decay, with the exception of the 24 human beings who flew lunar flights in the Apollo program during the four-year period spanning 1968 through 1972, all human spaceflights have taken place in LEO or below. The International Space Station conducts operations in LEO, the altitude record for a human spaceflight in LEO was Gemini 11 with an apogee of 1,374.1 kilometers. All crewed space stations to date, as well as the majority of satellites, have been in LEO, objects in LEO encounter atmospheric drag from gases in the thermosphere or exosphere, depending on orbit height. Due to atmospheric drag, satellites do not usually orbit below 300 km, objects in LEO orbit Earth between the denser part of the atmosphere and below the inner Van Allen radiation belt. The mean orbital velocity needed to maintain a stable low Earth orbit is about 7.8 km/s, calculated for circular orbit of 200 km it is 7.79 km/s and for 1500 km it is 7.12 km/s. The delta-v needed to achieve low Earth orbit starts around 9.4 km/s, atmospheric and gravity drag associated with launch typically adds 1. 3–1.8 km/s to the launch vehicle delta-v required to reach normal LEO orbital velocity of around 7.8 km/s. Equatorial low Earth orbits are a subset of LEO and these orbits, with low inclination to the Equator, allow rapid revisit times and have the lowest delta-v requirement of any orbit. Orbits with an inclination angle to the equator are usually called polar orbits. Higher orbits include medium Earth orbit, sometimes called intermediate circular orbit, orbits higher than low orbit can lead to early failure of electronic components due to intense radiation and charge accumulation. Although the Earths pull due to gravity in LEO is not much less than on the surface of the Earth, people, a low Earth orbit is simplest and cheapest for satellite placement. It provides high bandwidth and low communication time lag, but satellites in LEO will not be visible from any point on the Earth at all times. Earth observation satellites and spy satellites use LEO as they are able to see the surface of the Earth more clearly as they are not so far away and they are also able to traverse the surface of the Earth. A majority of satellites are placed in LEO, making one complete revolution around the Earth in about 90 minutes. The International Space Station is in a LEO about 400 km above the Earths surface, since it requires less energy to place a satellite into a LEO and the LEO satellite needs less powerful amplifiers for successful transmission, LEO is used for many communication applications. Because these LEO orbits are not geostationary, a network of satellites is required to provide continuous coverage, lower orbits also aid remote sensing satellites because of the added detail that can be gained. Remote sensing satellites can also take advantage of sun-synchronous LEO orbits at an altitude of about 800 km, envisat is one example of an Earth observation satellite that makes use of this particular type of LEO. The LEO environment is becoming congested with space debris due to the frequency of object launches and this has caused growing concern in recent years, since collisions at orbital velocities can easily be dangerous, and even deadly

23.
Medium Earth orbit
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Medium Earth orbit, sometimes called intermediate circular orbit, is the region of space around the Earth above low Earth orbit and below geostationary orbit. The most common use for satellites in this region is for navigation, communication, the most common altitude is approximately 20,200 kilometres ), which yields an orbital period of 12 hours, as used, for example, by the Global Positioning System. Other satellites in medium Earth orbit include Glonass and Galileo constellations, communications satellites that cover the North and South Pole are also put in MEO. The orbital periods of MEO satellites range from about 2 to nearly 24 hours, telstar 1, an experimental satellite launched in 1962, orbits in MEO. The orbit is home to a number of artificial satellites

24.
Molniya orbit
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A Molniya orbit is a type of highly elliptical orbit with an inclination of 63.4 degrees, an argument of perigee of −90 degrees and an orbital period of one half of a sidereal day. Molniya orbits are named after a series of Soviet/Russian Molniya communications satellites which have been using this type of orbit since the mid-1960s.4 degrees north, to get a continuous high elevation coverage of the Northern Hemisphere, at least three Molniya spacecraft are needed. The reason that the inclination should have the value 63. 4° is that then the argument of perigee is not perturbed by the J2 term of the field of the Earth. Much of the area of the former Soviet Union, and Russia in particular, is located at high latitudes, to broadcast to these latitudes from a geostationary orbit would require considerable power due to the low elevation angles. A satellite in a Molniya orbit is better suited to communications in these regions because it looks directly down on them, an additional advantage is that considerably less launch energy is needed to place a spacecraft into a Molniya orbit than into a geostationary orbit. It is necessary to have at least three spacecraft if permanent high elevation coverage is needed for an area like the whole of Russia where some parts are as far south as 45° N. If three spacecraft are used, each spacecraft is active for periods of eight hours per orbit centered at apogee as illustrated in figure 9. The Earth completes half a rotation in 12 hours, so the apogees of successive Molniya orbits will alternate between one half of the hemisphere and the other half. For example if the apogee longitudes are 90° E and 90° W, said next spacecraft has the visibility displayed in figure 3 and the switch-over can take place. Note that the two spacecraft at the time of switch-over are separated about 1500 km, so that the stations only have to move the antennas a few degrees to acquire the new spacecraft. To avoid this expenditure of fuel, the Molniya orbit uses an inclination of 63. 4° and that this is the case follows from equation of the article Orbital perturbation analysis as the factor then is zero. The reason why the orbital period shall be half a day is that the geometry relative to the ground stations should repeat every 24 hours. In fact, the precise ideal orbital period resulting in a ground track repeating every 24 hours is not precisely half a sidereal day, but rather half a synodic day. For a Molniya orbit, the inclination is selected such that Δ ω as given by the formula above is zero but Δ Ω, as given by the other equation, will be −0. 0742° per orbit. The rotational period of the Earth relative to the node will therefore be only 86,129 seconds,35 seconds less than the day which is 86,164 seconds. The primary use of the Molniya orbit was for the satellite series of the same name. After two launch failures in 1964, the first successful satellite to use this orbit was Molniya 1-01 launched on April 23,1965. The early Molniya-1 satellites were used for military communications starting in 1968

25.
Orbit of the Moon
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Not to be confused with Lunar orbit. The Moon orbits Earth in the direction and completes one revolution relative to the stars in approximately 27.323 days. Earth and the Moon orbit about their barycentre, which lies about 4,600 km from Earths center, on average, the Moon is at a distance of about 385,000 km from Earths centre, which corresponds to about 60 Earth radii. With a mean velocity of 1.022 km/s, the Moon appears to move relative to the stars each hour by an amount roughly equal to its angular diameter. The Moon differs from most satellites of planets in that its orbit is close to the plane of the ecliptic. The plane of the orbit is inclined to the ecliptic by about 5°. The properties of the orbit described in this section are approximations, the Moons orbit around Earth has many irregularities, whose study has a long history. The orbit of the Moon is distinctly elliptical, with an eccentricity of 0.0549. The non-circular form of the lunar orbit causes variations in the Moons angular speed and apparent size as it moves towards, the mean angular movement relative to an imaginary observer at the barycentre is 13. 176° per day to the east. The Moons elongation is its angular distance east of the Sun at any time, at new moon, it is zero and the Moon is said to be in conjunction. At full moon, the elongation is 180° and it is said to be in opposition, in both cases, the Moon is in syzygy, that is, the Sun, Moon and Earth are nearly aligned. When elongation is either 90° or 270°, the Moon is said to be in quadrature, the orientation of the orbit is not fixed in space, but rotates over time. This orbital precession is also called apsidal precession and is the rotation of the Moons orbit within the orbital plane, the Moons apsidal precession is distinct from, and should not be confused with its axial precession. The mean inclination of the orbit to the ecliptic plane is 5. 145°. The rotational axis of the Moon is also not perpendicular to its plane, so the lunar equator is not in the plane of its orbit. Therefore, the angle between the ecliptic and the equator is always 1. 543°, even though the rotational axis of the Moon is not fixed with respect to the stars. The period from moonrise to moonrise at the poles is quite close to the sidereal period, when the sun is the furthest below the horizon, the moon will be full when it is at its highest point. The nodes are points at which the Moons orbit crosses the ecliptic, the Moon crosses the same node every 27.2122 days, an interval called the draconic or draconitic month

26.
Polar orbit
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A polar orbit is one in which a satellite passes above or nearly above both poles of the body being orbited on each revolution. It therefore has an inclination of 90 degrees to the equator, a satellite in a polar orbit will pass over the equator at a different longitude on each of its orbits. Polar orbits are used for earth-mapping, earth observation, capturing the earth as time passes from one point, reconnaissance satellites. The Iridium satellite constellation also uses a polar orbit to provide telecommunications services, the disadvantage to this orbit is that no one spot on the Earths surface can be sensed continuously from a satellite in a polar orbit. Near-polar orbiting satellites commonly choose a Sun-synchronous orbit, meaning each successive orbital pass occurs at the same local time of day. To keep the local time on a given pass, the time period of the orbit must be kept as short as possible. However, very low orbits of a few hundred kilometers rapidly decay due to drag from the atmosphere, commonly used altitudes are between 700 km and 800 km, producing an orbital period of about 100 minutes. The half-orbit on the Sun side then takes only 50 minutes, list of orbits Molniya orbit Vandenberg Air Force Base, a major United States launch location for polar orbits Orbital Mechanics

27.
Tundra orbit
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A Tundra orbit is a highly elliptical geosynchronous orbit with a high inclination and an orbital period of one sidereal day. A satellite placed in this orbit spends most of its time over an area of the Earth. The ground track of a satellite in an orbit is a closed figure eight. These orbits are similar to Molniya orbits, which have the same inclination. The only current known user of Tundra orbits is the EKS satellite, until 2016, Sirius Satellite Radio, now part of Sirius XM Holdings operated a constellation of three satellites in Tundra orbits for satellite radio. The RAAN and mean anomaly of each satellite was offset by 120° so that when one satellite moved out of position, the three satellites were launched in 2000 and moved into circular disposal orbits in 2016, Sirius XM now broadcasts only from geostationary satellites. Tundra and Molniya orbits are used to high latitude users with higher elevation angles than a geostationary orbit. An argument of perigee of 270° places apogee at the northernmost point of the orbit, an argument of perigee of 90° would likewise serve the high southern latitudes. An argument of perigee of 0° or 180° would cause the satellite to dwell over the equator, the Tundra and Molniya orbits use a sin−1 √4/5 ≈63. 4° inclination to null the secular perturbation of the argument of perigee caused by the Earths equatorial bulge. With any inclination other than 63. 4° or its supplement,116. 6°, the argument of perigee would change steadily over time, and apogee would occur either before or after the highest latitude is reached

28.
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

29.
Lunar orbit
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In astronomy, lunar orbit refers to the orbit of an object around the Moon. As used in the program, this refers not to the orbit of the Moon about the Earth. The Soviet Union sent the first spacecraft to the vicinity of the Moon and it passed within 6,000 kilometres of the Moons surface, but did not achieve lunar orbit. This craft provided the first pictures of the far side of the Lunar surface, the Soviet Luna 10 became the first spacecraft to actually orbit the Moon in April 1966. It studied micrometeoroid flux, and lunar environment until May 30,1966, the first United States spacecraft to orbit the Moon was Lunar Orbiter 1 on August 14,1966. The first orbit was an elliptical orbit, with an apolune of 1,008 nautical miles, then the orbit was circularized at around 170 nautical miles to obtain suitable imagery. Five such spacecraft were launched over a period of thirteen months, all of which successfully mapped the Moon, the most recent was the Lunar Atmosphere and Dust Environment Explorer, which became a ballistic impact experiment in 2014. The Apollo programs Command/Service Module remained in a parking orbit while the Lunar Module landed. The combined CSM/LM would first enter an orbit, nominally 170 nautical miles by 60 nautical miles. Orbital periods vary according to the sum of apoapsis and periapsis, the LM began its landing sequence with a Descent Orbit Insertion burn to lower their periapsis to about 50,000 feet, chosen to avoid hitting lunar mountains reaching heights of 20,000 feet. These anomalies are significant enough to cause an orbit to change significantly over the course of several days. The Apollo 11 first manned landing mission employed the first attempt to correct for the perturbation effect. The parking orbit was circularized at 66 nautical miles by 54 nautical miles, but the effect was overestimated by a factor of two, at rendezvous the orbit was calculated to be 63.2 nautical miles by 56.8 nautical miles. The Apollo 15 subsatellite PFS-1 and the Apollo 16 subsatellite PFS-2, PFS-1 ended up in a long-lasting orbit, at 28 degrees inclination, and successfully completed its mission after one and a half years. PFS-2 was placed in a particularly unstable orbital inclination of 11 degrees, list of orbits Mass concentration Orbital mechanics

30.
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

31.
Semi-major and semi-minor axes
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In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction

32.
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

33.
Mean anomaly
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In celestial mechanics, the mean anomaly is an angle used in calculating the position of a body in an elliptical orbit in the classical two-body problem. Define T as the time required for a body to complete one orbit. In time T, the radius vector sweeps out 2π radians or 360°. The average rate of sweep, n, is then n =2 π T or n =360 ∘ T, define τ as the time at which the body is at the pericenter. From the above definitions, a new quantity, M, the mean anomaly can be defined M = n, because the rate of increase, n, is a constant average, the mean anomaly increases uniformly from 0 to 2π radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter, π radians at the apocenter, if the mean anomaly is known at any given instant, it can be calculated at any later instant by simply adding n δt where δt represents the time difference. Mean anomaly does not measure an angle between any physical objects and it is simply a convenient uniform measure of how far around its orbit a body has progressed since pericenter. The mean anomaly is one of three parameters that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly. Define l as the longitude, the angular distance of the body from the same reference direction. Thus mean anomaly is also M = l − ϖ, mean angular motion can also be expressed, n = μ a 3, where μ is a gravitational parameter which varies with the masses of the objects, and a is the semi-major axis of the orbit. Mean anomaly can then be expanded, M = μ a 3, and here mean anomaly represents uniform angular motion on a circle of radius a

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True anomaly
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In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the focus of the ellipse. The true anomaly is usually denoted by the Greek letters ν or θ, as shown in the image, the true anomaly f is one of three angular parameters that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly. Note that the satellite P orbits around the planet which is at position F, for circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely-determined periapsis. Instead the argument of u is used, u = arccos n ⋅ r | n | | r | where. For circular orbits with zero inclination the argument of latitude is also undefined, the radius is related to the true anomaly by the formula r = a ⋅1 − e 21 + e cos ν where a is the orbits semi-major axis. Keplers laws of planetary motion Eccentric anomaly Mean anomaly Ellipse Hyperbola Murray, C. D. & Dermott, S. F.1999, Solar System Dynamics, Cambridge University Press,1960, An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York