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( Created by: Zsolt Sandor & Peter Klagyivik , Eötvös Lorand University) . Secular Perturbations. Eccentric and Mean anomalies Kepler’s equation f,g functions Universal variables for hyperbolic and eccentric orbits Disturbing function - PowerPoint PPT Presentation

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Secular Perturbations

Secular PerturbationsEccentric and Mean anomaliesKeplers equationf,g functions Universal variables for hyperbolic and eccentric orbitsDisturbing functionLow eccentricity expansions for Disturbing functionSecular terms at low eccentricityPrecession of angle of perihelionApsidal resonancePericenter glow models for eccentric holes in circumstellar disks(Created by: Zsolt Sandor & Peter Klagyivik, Etvs Lorand University) raf = true anomalyfEllipsebcenter of ellipseSun is focal pointb=semi-minor axisa=semi-major axis

EraE = Eccentric anomalyf = true anomalyfOrbit from center of ellipseEllipsebaeRelationship between Eccentric, True and Mean anomaliesUsing expressions for x,y in terms of true and Eccentric anomalies we find thatSo if you know E you know f and can find position in orbitWrite dr/dt in terms of n, r, a, e Then replace dr/dt with function depending on E, dE/dtMean Anomaly and Keplers equationNew angle M defined such that or Integrate dE/dt finding

Keplers equationMust be solved to integrate orbit in time.The mean anomaly is not an angle defined on the orbital planeIt is an angle that advances steadily in time It is related to the azimuthal angle in the orbital plane, for a circular orbit, the two are the same and f=MChange t, increase M. Compute E numerically using Keplers equationCompute f using relation between E and fRotate to take into account argument of perihelionCalculate x,y in plane of orbitRotate two more times for inclination and longitude of ascending node to final Cartesian position

Keplers equationMust be solved to integrate orbit in time.Procedure for integrating orbit or for converting orbital elements to a Cartesian position:

Inclination and longitude of ascending nodeSign of terms depends on sign of hz.I inclination.retrograde orbits have /2