1.
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

2.
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

3.
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

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