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.
Apsidal precession
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In celestial mechanics, perihelion precession, apsidal precession or orbital precession is the precession of the orbit of a celestial body. More precisely, it is the rotation of the line joining the apsides of an orbit. Perihelion is the closest point to the Sun, the apsidal precession is the first derivative of the argument of periapsis, one of the six primary orbital elements of an orbit. The precession of the apsides was discovered in the eleventh century by al-Zarqālī. ωtotal = ωGeneral Relativity + ωquadrupole + ωtide + ωperturbations For Mercury, from classical mechanics, if stars and planets are considered to be purely spherical masses, then they will obey a simple 1/r2 force law and hence execute closed elliptical orbits. The good resulting approximation of the bulge is useful for understanding the interiors of such planets. For the shortest-period planets, the interior induces precession of a few degrees per year. Newton derived an early theorem which attempted to explain apsidal precession and this theorem is historically notable, but it was never widely used and it proposed forces which have been found not to exist, making the theorem invalid. This theorem of revolving orbits remained largely unknown and undeveloped for over three centuries until 1995, however, his theorem did not account for the apsidal precession of the Moon without giving up the inverse-square law of Newtons law of universal gravitation. Additionally, the rate of apsidal precession calculated via Newtons theorem of revolving orbits is not as accurate as it is for newer methods such as by perturbation theory. An apsidal precession of the planet Mercury was noted by Urbain Le Verrier in the mid-19th century and accounted for by Einsteins general theory of relativity. In the case of Mercury, half of the axis is circa 5. 79×1010 m, the eccentricity of its orbit is 0.206. From these and the speed of light, it can be calculated that the apsidial precession during one period of revolution is ϵ =5. 028×10−7 radians. Because of apsidal precession the Earths argument of periapsis slowly increases, the Earths polar axis, and hence the solstices and equinoxes, precess with a period of about 26000 years in relation to the fixed stars. This interaction between the anomalistic and tropical cycle is important in the climate variations on Earth, called the Milankovitch cycles. An equivalent is known on Mars. The figure illustrates the effects of precession on the northern hemisphere seasons, notice that the areas swept during a specific season changes through time

3.
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

4.
Argument of periapsis
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The argument of periapsis, symbolized as ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the ascending node to its periapsis. For specific types of orbits, words such as perihelion, perigee, periastron, an argument of periapsis of 0° means that the orbiting body will be at its closest approach to the central body at the same moment that it crosses the plane of reference from South to North. An argument of periapsis of 90° means that the body will reach periapsis at its northmost distance from the plane of reference. Adding the argument of periapsis to the longitude of the ascending node gives the longitude of the periapsis, however, especially in discussions of binary stars and exoplanets, the terms longitude of periapsis or longitude of periastron are often used synonymously with argument of periapsis. In the case of equatorial orbits, the argument is strictly undefined, where, ex and ey are the x- and y-components of the eccentricity vector e. In the case of circular orbits it is assumed that the periapsis is placed at the ascending node. Kepler orbit Orbital mechanics Orbital node

5.
Attitude control
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Attitude control is controlling the orientation of an object with respect to an inertial frame of reference or another entity. The integrated field that studies the combination of sensors, actuators and algorithms is called Guidance, Navigation, a spacecrafts attitude must typically be stabilized and controlled for a variety of reasons. Propulsion system thrusters are fired only occasionally to make desired changes in spin rate, if desired, the spinning may be stopped through the use of thrusters or by yo-yo de-spin. The Pioneer 10 and Pioneer 11 probes in the solar system are examples of spin-stabilized spacecraft. Three-axis stabilization is a method of spacecraft attitude control in which the spacecraft is held fixed in the desired orientation without any rotation. One method is to use thrusters to continually nudge the spacecraft back. Thrusters may also be referred to as control systems, or reaction control systems. The space probes Voyager 1 and Voyager 2 employ this method, another method for achieving three-axis stabilization is to use electrically powered reaction wheels, also called momentum wheels, which are mounted on three orthogonal axes aboard the spacecraft. They provide a means to trade angular momentum back and forth between spacecraft and wheels, to rotate the vehicle on a given axis, the reaction wheel on that axis is accelerated in the opposite direction. To rotate the back, the wheel is slowed. This is done during maneuvers called momentum desaturation or momentum unload maneuvers, most spacecraft use a system of thrusters to apply the torque for desaturation maneuvers. A different approach was used by the Hubble Space Telescope, which had sensitive optics that could be contaminated by thruster exhaust, there are advantages and disadvantages to both spin stabilization and three-axis stabilization. Many spacecraft have components that require articulation, Voyager and Galileo, for example, were designed with scan platforms for pointing optical instruments at their targets largely independently of spacecraft orientation. Many spacecraft, such as Mars orbiters, have solar panels that must track the Sun so they can provide power to the spacecraft. Cassinis main engine nozzles are steerable, knowing where to point a solar panel, or scan platform, or a nozzle — that is, how to articulate it—requires knowledge of the spacecrafts attitude. Because AACS keeps track of the attitude, the Suns location. It logically falls to one subsystem, then, to manage both attitude and articulation, the name AACS may even be carried over to a spacecraft even if it has no appendages to articulate. Many sensors generate outputs that reflect the rate of change in attitude and these require a known initial attitude, or external information to use them to determine attitude

6.
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

7.
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

8.
Eccentric anomaly
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In orbital mechanics, eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three parameters that define a position along an orbit, the other two being the true anomaly and the mean anomaly. Consider the ellipse with equation given by, x 2 a 2 + y 2 b 2 =1, where a is the semi-major axis and b is the semi-minor axis. For a point on the ellipse, P = P, representing the position of a body in an elliptical orbit. The eccentric anomaly, E, is observed by drawing a right triangle with one vertex at the center of the ellipse, having hypotenuse a, the eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure as f. The equation sin E = −y/b is immediately able to be ruled out since it traverses the ellipse in the wrong direction. It can also be noted that the equation can be viewed as being the similar triangle with adjacent side through P and the minor auxiliary circle, hypotenuse b. The eccentricity e is defined as, e =1 −2, with this result the eccentric anomaly can be determined from the true anomaly as shown next. The true anomaly is the angle labeled f in the figure, located at the focus of the ellipse, the true anomaly and the eccentric anomaly are related as follows. Hence, tan E = sin E cos E =1 − e 2 sin θ e + cos θ. Angle E is therefore the adjacent angle of a triangle with hypotenuse 1 + e cos θ, adjacent side e + cos θ. The eccentric anomaly E is related to the mean anomaly M by Keplers equation and it is usually solved by numerical methods, e. g. the Newton–Raphson method. Murray, Carl D. & Dermott, Stanley F, solar System Dynamics, Cambridge University Press, Cambridge, GB Plummer, Henry C. K. An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York, NY Eccentricity vector Orbital eccentricity

9.
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

10.
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

11.
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

12.
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

13.
Free-return trajectory
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The first spacecraft traveled using free-return trajectory on October 4,1959 was Russian «Луна-3». Then, free return trajectories were introduced by Arthur Schwaniger of NASA in 1963 with reference to the Earth-Moon system, the spacecraft passes behind the Moon. It moves there in a direction opposite to that of the Moon, if the crafts orbit begins in a normal direction near Earth, then it makes a figure 8 around the Earth and Moon. Flight time for a cislunar free-return trajectory decreases with increasing periselenum radius and it has a period of about 650 hours. Considering the trajectory in a frame of reference, the perigee occurs directly under the moon when the moon is on one side of the earth. Speed at perigee is about 10.91 km/s, after three days it reaches the moons orbit, but now more or less on the opposite side of the earth from the moon. After a few days, the craft reaches its apogee and begins to fall back toward the Earth. The craft passes on the side of the moon at a radius of 2150 km and is thrown back outwards where it reaches a second apogee. It then falls back toward the earth, goes around to the first side, there will of course be similar trajectories with periods of about two sidereal months, three sidereal months, and so on. In each case, the two apogees will be further and further away from Earth and these were not considered by Schwaniger. This kind of trajectory can occur of course for similar three-body problems, while in a true free-return trajectory no propulsion is applied, in practice there may be small mid-course corrections or other maneuvers. A free-return trajectory may be the trajectory to allow a safe return in the event of a systems failure, this was applied in the Apollo 8, Apollo 10. In such a case a free return to a suitable reentry situation is more useful than returning to near the Earth, since all went well these Apollo missions did not have to take advantage of the free return, and inserted into orbit upon arrival at the Moon. They then performed a maneuver to change to a trans-Lunar trajectory that was not a free return. In fact, within hours after the accident, Apollo 13 used the module to maneuver from its planned trajectory to a free-return trajectory. Apollo 13 was the only Apollo mission to turn around the Moon in a free-return trajectory. A free-return transfer orbit to Mars is also possible, as with the moon, this option is mostly considered for manned missions. Robert Zubrin, in his book The Case for Mars, discusses various trajectories to Mars for his mission design Mars Direct, the naive Hohmann transfer orbit can be made free-return

14.
Frozen orbit
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In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which natural drifting due to the central bodys shape has been minimized by careful selection of the orbital parameters. This results in a stable orbit that minimizes the use of stationkeeping propellant. For many spacecraft, changes to orbits are caused by the oblateness of the Earth, gravitational attraction from the Sun and Moon, solar radiation pressure and they must be counteracted by maneuvers to keep the spacecraft in the desired orbit. For a geostationary spacecraft, correction maneuvers on the order of 40–50 m/s per year are required to counteract these forces, for Sun-synchronous spacecraft, intentional shifting of the orbit plane can be used for the benefit of the mission. For these missions, an orbit with an altitude of 600–900 km is used. An appropriate inclination is selected so that the precession of the plane is equal to the rate of movement of the Earth around the Sun - or about 1 degree per day. As a result, the spacecraft will pass over points on the Earth that have the time of day during every orbit. For instance, if the orbit is square to the Sun, the vehicle will always pass over points at which it is 6 a. m. on the north-bound portion and this is called a Dawn-Dusk orbit. Alternatively, if the plane is perpendicular to the Sun, the vehicle will always pass over Earth noon on the north-bound leg. Such orbits are desirable for many Earth observation missions such as weather, imagery, perturbing forces caused by the oblateness of the Earth will also change the shape of the orbit. To compensate, there are near-circular orbits where there are no secular/long periodic perturbations of the eccentricity, such an orbit is called a frozen orbit. These orbits are often the choice for Earth observation missions where repeated observations under constant conditions are desirable. The Earth observation satellites ERS-1, ERS-2 and Envisat are operated in Sun-synchronous frozen orbits, similarly the quadratic terms of the eccentricity vector components in can be ignored for almost circular orbits, i. e. The latter figure means that the eccentricity vector will have described a circle in 1569 orbits. In this way the secular perturbation of the eccentricity vector caused by the J2 term is used to counteract all secular perturbations, applying this algorithm for the case discussed above, i. e. e. Including also the forces due to the higher zonal terms the value changes to. With the modern theory this explicit closed form expression is not directly used and these directions r ^ and λ ^ are illustrated in Figure 1. e. Towards the blue point of Figure 2

15.
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

16.
Hill sphere
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An astronomical bodys Hill sphere is the region in which it dominates the attraction of satellites. The outer shell of that constitutes a zero-velocity surface. To be retained by a planet, a moon must have an orbit that lies within the planets Hill sphere and that moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, one simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus. In more precise terms, the Hill sphere approximates the gravitational sphere of influence of a body in the face of perturbations from a more massive body. It was defined by the American astronomer George William Hill, based on the work of the French astronomer Édouard Roche, for this reason, it is also known as the Roche sphere. In the example to the right, the Hill sphere extends between the Lagrangian points L1 and L2, which lie along the line of centers of the two bodies. The region of influence of the body is shortest in that direction. When eccentricity is negligible, this becomes r ≈ a m 3 M3, in the Earth example, the Earth orbits the Sun at a distance of 149.6 million km. The Hill sphere for Earth thus extends out to about 1.5 million km, all stable satellites of the Earth must have an orbital period shorter than seven months. The previous formula can be re-stated as follows,3 r 3 a 3 ≈ m M and this is also convenient, because many planetary astronomers work in and remember distances in units of planetary radii. The Hill sphere is only an approximation, and other forces can eventually perturb an object out of the sphere and this third object should also be of small enough mass that it introduces no additional complications through its own gravity. The region of stability for retrograde orbits at a distance from the primary, is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter, however, a sphere of this size and mass would be denser than lead. In fact, in any low Earth orbit, a body must be more dense than lead in order to fit inside its own Hill sphere. A spherical geostationary satellite, however, would only need to be more than 6% of the density of water to support satellites of its own. Within the Solar System, the planet with the largest Hill radius is Neptune, with 116 million km, or 0.775 au, an asteroid from the asteroid belt will have a Hill sphere that can reach 220000 km, diminishing rapidly with decreasing mass. The Hill sphere of 1999 KW4, a Mercury-crosser asteroid that has a moon, a typical extrasolar hot Jupiter, HD209458 b, has a Hill sphere radius of 593,000 km, about 8 times its physical radius of approx 71,000 km

17.
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

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Inertial frame of reference
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In classical physics and special relativity, an inertial frame of reference is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. The physics of a system in an inertial frame have no causes external to the system, all inertial frames are in a state of constant, rectilinear motion with respect to one another, an accelerometer moving with any of them would detect zero acceleration. Measurements in one frame can be converted to measurements in another by a simple transformation. In general relativity, in any region small enough for the curvature of spacetime and tidal forces to be negligible, systems in non-inertial frames in general relativity dont have external causes because of the principle of geodesic motion. Physical laws take the form in all inertial frames. For example, a ball dropped towards the ground does not go straight down because the Earth is rotating. Someone rotating with the Earth must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion, another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force. The motion of a body can only be described relative to something else—other bodies, observers and these are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary, for example, suppose a free body that has no external forces on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, however, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, indeed, an intuitive summary of inertial frames can be given as, In an inertial reference frame, the laws of mechanics take their simplest form. In an inertial frame, Newtons first law, the law of inertia, is satisfied, Any free motion has a constant magnitude, the force F is the vector sum of all real forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. The extra terms in the force F′ are the forces for this frame. The first extra term is the Coriolis force, the second the centrifugal force, also, fictitious forces do not drop off with distance. For example, the force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. All observers agree on the forces, F, only non-inertial observers need fictitious forces. The laws of physics in the frame are simpler because unnecessary forces are not present. In Newtons time the stars were invoked as a reference frame

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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

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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

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Jacobi coordinates
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In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, an algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. In words, the algorithm is described as follows, Let mj, the position coordinates xj and xk are replaced by their relative position rjk = xj − xk and by the vector to their center of mass Rjk = /. The node in the tree corresponding to the virtual body has mj as its right child. The order of children indicates the relative coordinate points from xk to xj, repeat the above step for N −1 bodies, that is, the N −2 original bodies plus the new virtual body

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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

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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

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Kepler's equation
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In orbital mechanics, Keplers equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, the equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics. Keplers equation is where M is the anomaly, E is the eccentric anomaly. The eccentric anomaly E is useful to compute the position of a point moving in a Keplerian orbit, numerical analysis and series expansions are generally required to evaluate E. There are several forms of Keplers equation, each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits, the hyperbolic Kepler equation is used for hyperbolic orbits. The radial Kepler equation is used for linear orbits, Barkers equation is used for parabolic orbits. When e =1, Keplers equation is not associated with an orbit, when e =0, the orbit is circular. Increasing e causes the circle to flatten into an ellipse, when e =1, the orbit is completely flat, and it appears to be either a segment if the orbit is closed, or a ray if the orbit is open. An infinitesimal increase to e results in an orbit with a turning angle of 180 degrees. Further increases reduce the angle, and as e goes to infinity. The Hyperbolic Kepler equation is, where H is the eccentric anomaly. This equation is derived by multiplying Keplers equation by 1/2 making the replacement E =2 sin −1 , calculating M for a given value of E is straightforward. However, solving for E when M is given can be more challenging. Keplers equation can be solved for E analytically by Lagrange inversion, the solution of Keplers equation given by two Taylor series below. Confusion over the solvability of Keplers equation has persisted in the literature for four centuries, Kepler himself expressed doubt at the possibility of ﬁnding a general solution. +13 M99. + ⋯, e ≠1 These series can be reproduced in Mathematica with the InverseSeries operation, InverseSeries InverseSeries These functions are simple Taylor series. Taylor series representations of functions are considered to be definitions of those functions

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Kepler's laws of planetary motion
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In astronomy, Keplers laws of planetary motion are three scientific laws describing the motion of planets around the Sun. The orbit of a planet is an ellipse with the Sun at one of the two foci, a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. The square of the period of a planet is proportional to the cube of the semi-major axis of its orbit. Most planetary orbits are circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars, whose published values are somewhat suspect, from this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. Keplers work improved the theory of Nicolaus Copernicus, explaining how the planets speeds varied. Isaac Newton showed in 1687 that relationships like Keplers would apply in the Solar System to a approximation, as a consequence of his own laws of motion. Keplers laws are part of the foundation of modern astronomy and physics, Keplers laws improve the model of Copernicus. Keplers corrections are not at all obvious, The planetary orbit is not a circle, the Sun is not at the center but at a focal point of the elliptical orbit. Neither the linear speed nor the speed of the planet in the orbit is constant, but the area speed is constant.015. The calculation is correct when perihelion, the date the Earth is closest to the Sun, the current perihelion, near January 4, is fairly close to the solstice of December 21 or 22. It took nearly two centuries for the current formulation of Keplers work to take on its settled form, voltaires Eléments de la philosophie de Newton of 1738 was the first publication to use the terminology of laws. The Biographical Encyclopedia of Astronomers in its article on Kepler states that the terminology of laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in An account of the discoveries of Kepler that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, further, the current usage of Keplers Second Law is something of a misnomer. Kepler had two versions, related in a sense, the distance law and the area law. The area law is what became the Second Law in the set of three, but Kepler did himself not privilege it in that way, Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Keplers third law was published in 1619 and his first law reflected this discovery

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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

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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

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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