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

000 ttt vxx

sin

sincos

coscos

rx 0

0

0

0

tt

Vrr R

Quasar/Star Catalog

Epoch

Mean Place (at Epoch)

Parallax (at Epoch)

Proper Motion

Radial Velocity

Astrophys. Quantities:Magnitude, Color, …

ICRFnn, HIPPARCOS, FKn, PPM, AGKn

0t 00 ,

0 ,

RV

Keplerian

Two Body Motion under Newtonian Mech.

Gravitational Constant

Elements = 6 Constants of MotionShapeOrientationTiming

xx

32

2

rdt

d

mMG

ea,,, IΩ

T

Units of Mass

SI: kilogram kg

Astronomical: Solar Mass

Newtonian Gravitational Constant

Measurable Quantity = GM

= Body-centric Gravitational ConstantHeliocentric GeocentricSGM EGM

SM

G

Keplerian Elements

Semi-Major Axis: a

Eccentricity: e

Longitude of Ascending Node: Inclination: I

Argument of Pericenter: Epoch of Pericenter Passage: T

Ellipse

Semi-major axis: a

Semi-minor axis: b 12

2

2

2

b

y

a

x

a

b

Eccentricity

Eccentricity: e, Co-Eccentricity: e’

22

22

1' , ea

be

a

bae

ae

F

Orbital Orientation

Euler (3-1-3) Angles of Orbital Plane RF

3 Important DirectionsDeparture Point: X-axis

Ascending Node: N

Pericenter: P

313313 ,, RRRR II

Z

P

N

I

Orbital Plane

Keplerian Orbits

Elliptic: e < 1Planets, Satellites, Binary

Parabolic: e = 1Good Approximation for Comets

Nearly Parabolic: e ~ 1Comets, some peculiar Asteroids

Hyperbolic: e > 1Space Vehicles, Virtual (Change of Origin)

Elements to Position, Velocity

Solve Kepler’s Equation

Time Derivative of E

PV in Orbital RF

Eb

eEa

sin

cos

TtnEeE sin

Ee

nE

cos1

EEb

EEa

cos

sin

Elements to PV (contd.)

Backward Euler Rotation

00

,,

ΩI, 313Rvx

Kepler’s Equation

First Nonlinear Equation in History

Elliptic

Parabolic

Hyperbolic

MEeE sin

PM3

3

HMFFe sinh

Elliptic Kepler’s Equation

Eccentric Anomaly: E

Mean Anomaly: M = n ( t – T )

Kepler’s 3rd Law

True Anomaly: f

MEeE sin

frEb

freEa

sinsin

coscos

32an

Solution of Kepler’s Equation

Reduction of Variable Domain

Newton Method

Ee

EEEeMEf

Ef

EfEEfE

cos1

sincos

'

*

*

0sin MEeEEf

EM 0 0

Initial Guess for Newton Method

Stability Theory

Initial Guess = Upper Bound

Efficient Choice

0'',0'

,00

EfEf

ff

e

eMeM

e

M

fffE

1

,,

1min

,2

,0min ***0

0* Ef

Perturbed Keplerian Orbits

Elements as Functions of Time

Perturbation Theory

Polynomial + Fourier Series

tΛTΩIeaΛ ,,,,,

kkkkk tStC

tΛtΛΛΛ

sincos

2210

Complex Motion

Equation of Motion

Numerical/Analytical Solution

Parameter Fitting to Observational Data

Results = Ephemeris

xx

32

2

rdt

d

Planetary/Lunar EphemeridesNumerical: DE series (NASA/JPL), DE405Analytical: VSOP/ELP (BdL)DE: available at NAO/CC

Fortran/C callable routines + Binary file(s)DE405: 1600-2200, UNIX/Win/MacP/V of Sun+Moon+9planetsBase: PN Eq.Motion + Precision Data + Least Square Fitting (Mass, Init. Cond., etc.)

Other Solar System Bodies: HORIZONSDetails: http://ssd.jpl.nasa.gov/