General Motion Rest: Quasars Linear: Stars Keplerian: Binary Perturbed Keplerian: Asteroids, Satellites Complex: Planets, Space Vehicles Rotational: Earth,

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Presentation transcript:

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’ ae F

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

Z P  N W I w Orbital Plane

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 3 rd 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: , 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: