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This problem contains three degrees of freedom (the three Cartesian coordinates of the orbiting body). Therefore, each particular Keplerian ( = unperturbed) orbit is fully defined by six quantities - the initial values of the Cartesian components of the body's position and velocity. For this reason, all sets of orbital elements contain exactly six parameters. For a mathematically accurate explanation of this fact see the Discussion and references therein. (''See also'': Orbital State Vectors ). KEPLERIAN ELEMENTS The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his Kepler's Laws . The Keplerian elements are six:
We see that the first three orbital elements are simply the Eulerian angles defining the orientation of the orbit relative to some fiducial coordinate system. The above six elements parameterise a conic orbit emerging in an unperturbed two-body problem - an ellipse, a parabola, or a hyperbola. A realistic perturbed trajectory is represented as a sequence of such instantaneous conics that share one of their foci. In case the orbital elements are postulated to parameterise a sequence of conics that are always tangent to the trajectory, these orbital elements are called osculating. Instead of the Mean Anomaly At Epoch , , they often employ the Mean Anomaly . Sometimes the Mean Longitude , or the True Anomaly or, rarely, the Eccentric Anomaly are used instead of the mean anomaly at epoch. Sometimes the epoch itself is used as the sixth orbital element, instead of the mean anomaly at epoch. Keplerian elements can be obtained from Orbital State Vectors using VEC2TLE Software or by some Direct Computations . Other orbital parameters, such as the Period , can then be calculated from the Keplerian elements. In some cases, the period is used as an orbital element instead of semi-major axis. The elements can be seen as defining the orbit by degrees:
Because the simple Newtonian model of orbital motion of idealised points in free space is not exact, the orbital elements of real objects tend to change over time. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, due to the nonsphericity of the primary, due to the atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, etc... This evolution is described by the so-called planetary equations, which come in the form of Lagrange, or in the form of Gauss, or in the form of Delaunay, or in the form of Poincare, or in the form of Hill. (The latter is a very exotic option, emerging in the case when the true anomaly enters the set of six orbital elements. Hill considered this kind of orbit parameterisation back in 1913.) For more information, see the Discussion. TWO LINE ELEMENTS Keplerian elements parameters can be encoded as text in a number of formats. The most common of them is the NASA / NORAD "two-line elements"(TLE) format {Link without Title} , originally designed for use with 80-column punched cards, but still in use because it is the most common format, and works as well as any other. TLEs older than 30 days become considerably inaccurate. Orbital positions and heights can be calculated from TLEs through the SGP/ SGP4 / SDP4 /SGP8/SDP8 algorithms.
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