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Orbital Mechanics

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Orbital Mechanics Orbital Mechanics Part 1 Orbital Forces Why a Sat. remains in orbit ? Bcs the centrifugal force caused by the Sat. rotation around earth is counter- balanced by the Earth's Pull. Kepler’s Laws The Satellite (Spacecraft) which orbits the earth follows the same laws that govern the motion of the planets around the sun. J. Kepler (1571-1630) was able to derive empirically three laws describing planetary motion I. Newton was able to derive Keplers laws from his own laws of mechanics [gravitation theory] Kepler’s 1st Law (Law of Orbits) The path followed by a Sat. (secondary body) orbiting around the primary body will be an ellipse. The center of mass (barycenter) of a two-body system is always centered on one of the foci (earth center). Kepler’s 1st Law (Law of Orbits) The eccentricity (abnormality) e: a 2 b2 e a b- semiminor axis , a- semimajor axis VIN: e=0 circular orbit 0<e<1 ellip. orbit Orbit Calculations Ellipse is the curve traced by a point moving in a plane such that the sum of its distances from the foci is constant. Kepler’s 2nd Law (Law of Areas) For equal time intervals, a Sat. will sweep out equal areas in its orbital plane, focused at the barycenter VIN: S1>S2 at t1=t2 V1>V2 Max(V) at Perigee & Min(V) at Apogee Kepler’s 3rd Law (Harmonic Law) The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies. a 3 n 2 n- mean motion of Sat. (radian/sec) , - earth geocentric gravitational constant = 3.986005x1014 The mean distance is equal to the semimajor axis a VIN: With n in rad/sec, the orbital period in sec. : P=2/n Kepler’s 3rd Law (Harmonic Law) In English: Orbits with the same semi-major axis will have the same period Examples Examples Orbital Elements Definition A set of mathematical parameters that enables us to accurately describe satellite motion Purpose Discriminate one satellite from other satellites Predict where a satellite will be in the future or has been in the past Determine amount and direction of maneuver or perturbation Orbital Elements Semi-Major Axis (Size) Eccentricity (Shape) Inclination Right Ascension (Orientation) Argument of Perigee Epoch Time (Location within orbit) True Anomaly – Mean Anomaly Equatorial Plane Inclination ( i) Orbital Plane Inclination Prograde: 0 i < 90 Equatorial: i= 0 or 180 Retrograde: Polar: i = 90 90 i < 180 i Argument of Perigee () Right ascension() Direction of True satellite motion Anomaly () Locate with Respect to Space Orbital Elements Definition A set of mathematical parameters that enables us to accurately describe satellite motion Purpose Discriminate one satellite from other satellites Predict where a satellite will be in the future or has been in the past Determine amount and direction of maneuver or perturbation Orbital Elements Semi-Major Axis (Size) Eccentricity (Shape) Inclination Right Ascension (Orientation) Argument of Perigee Epoch Time (Location within orbit) True Anomaly – Mean Anomaly Example Example Orbit Perturbations.
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  • Appendix a Orbits
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  • Multi-Body Trajectory Design Strategies Based on Periapsis Poincaré Maps
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