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## General Motion

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**General Motion**• Rest: Quasars • Linear: Stars • Keplerian: Binary • Perturbed Keplerian:Asteroids, Satellites • Complex: Planets, Space Vehicles • Rotational: Earth, Moon, Satellites, …**Linear Motion**• Radial Motion • Proper Motion = Angular Motion**Quasar/Star Catalog**• Epoch • Mean Place (at Epoch) • Parallax (at Epoch) • Proper Motion • Radial Velocity • Astrophys. Quantities:Magnitude, Color, … • ICRFnn, HIPPARCOS, FKn, PPM, AGKn**Keplerian**• Two Body Motion under Newtonian Mech. • Gravitational Constant • Elements = 6 Constants of Motion • Shape • Orientation • Timing**Units of Mass**• SI: kilogram kg • Astronomical: Solar Mass • Newtonian Gravitational Constant • Measurable Quantity = GM = Body-centric Gravitational Constant • Heliocentric Geocentric**Keplerian Elements**• Semi-Major Axis: a • Eccentricity: e • Longitude of Ascending Node: W • Inclination: I • Argument of Pericenter: w • Epoch of Pericenter Passage: T**Ellipse**• Semi-major axis: a • Semi-minor axis: b**Eccentricity**• Eccentricity: e, Co-Eccentricity: e’ F ae**Orbital Orientation**• Euler (3-1-3) Angles of Orbital Plane RF • 3 Important Directions • Departure Point: X-axis • Ascending Node: N • Pericenter: P**Orbital Plane**Z w P I g W N**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)**Elements to Position, Velocity**• Solve Kepler’s Equation • Time Derivative of E • PV in Orbital RF**Elements to PV (contd.)**• Backward Euler Rotation**Kepler’s Equation**• First Nonlinear Equation in History • Elliptic • Parabolic • Hyperbolic**Elliptic Kepler’s Equation**• Eccentric Anomaly: E • Mean Anomaly: M = n ( t – T ) • Kepler’s 3rd Law • True Anomaly: f**Solution of Kepler’s Equation**• Reduction of Variable Domain • Newton Method**Initial Guess for Newton Method**• Stability Theory • Initial Guess = Upper Bound • Efficient Choice**Perturbed Keplerian Orbits**• Elements as Functions of Time • Perturbation Theory • Polynomial + Fourier Series**Complex Motion**• Equation of Motion • Numerical/Analytical Solution • Parameter Fitting to Observational Data • Results = Ephemeris**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/