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General Motion

General Motion

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General Motion

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  1. General Motion • Rest: Quasars • Linear: Stars • Keplerian: Binary • Perturbed Keplerian:Asteroids, Satellites • Complex: Planets, Space Vehicles • Rotational: Earth, Moon, Satellites, …

  2. Linear Motion • Radial Motion • Proper Motion = Angular Motion

  3. Quasar/Star Catalog • Epoch • Mean Place (at Epoch) • Parallax (at Epoch) • Proper Motion • Radial Velocity • Astrophys. Quantities:Magnitude, Color, … • ICRFnn, HIPPARCOS, FKn, PPM, AGKn

  4. Keplerian • Two Body Motion under Newtonian Mech. • Gravitational Constant • Elements = 6 Constants of Motion • Shape • Orientation • Timing

  5. Units of Mass • SI: kilogram kg • Astronomical: Solar Mass • Newtonian Gravitational Constant • Measurable Quantity = GM = Body-centric Gravitational Constant • Heliocentric Geocentric

  6. Keplerian Elements • Semi-Major Axis: a • Eccentricity: e • Longitude of Ascending Node: W • Inclination: I • Argument of Pericenter: w • Epoch of Pericenter Passage: T

  7. Ellipse • Semi-major axis: a • Semi-minor axis: b

  8. Eccentricity • Eccentricity: e, Co-Eccentricity: e’ F ae

  9. Orbital Orientation • Euler (3-1-3) Angles of Orbital Plane RF • 3 Important Directions • Departure Point: X-axis • Ascending Node: N • Pericenter: P

  10. Orbital Plane Z w P I g W N

  11. Keplerian Orbits • Elliptic: e < 1 • Planets, Satellites, Binary • Parabolic: e = 1 • Good Approximation for Comets • Nearly Parabolic: e ~ 1 • Comets, some peculiar Asteroids • Hyperbolic: e > 1 • Space Vehicles, Virtual (Change of Origin)

  12. Elements to Position, Velocity • Solve Kepler’s Equation • Time Derivative of E • PV in Orbital RF

  13. Elements to PV (contd.) • Backward Euler Rotation

  14. Kepler’s Equation • First Nonlinear Equation in History • Elliptic • Parabolic • Hyperbolic

  15. Elliptic Kepler’s Equation • Eccentric Anomaly: E • Mean Anomaly: M = n ( t – T ) • Kepler’s 3rd Law • True Anomaly: f

  16. Solution of Kepler’s Equation • Reduction of Variable Domain • Newton Method

  17. Initial Guess for Newton Method • Stability Theory • Initial Guess = Upper Bound • Efficient Choice

  18. Perturbed Keplerian Orbits • Elements as Functions of Time • Perturbation Theory • Polynomial + Fourier Series

  19. Complex Motion • Equation of Motion • Numerical/Analytical Solution • Parameter Fitting to Observational Data • Results = Ephemeris

  20. Planetary/Lunar Ephemerides • Numerical: DE series (NASA/JPL), DE405 • Analytical: VSOP/ELP (BdL) • DE: available at NAO/CC • Fortran/C callable routines + Binary file(s) • DE405: 1600-2200, UNIX/Win/Mac • P/V of Sun+Moon+9planets • Base: PN Eq.Motion + Precision Data + Least Square Fitting (Mass, Init. Cond., etc.) • Other Solar System Bodies: HORIZONS • Details: http://ssd.jpl.nasa.gov/