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.
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