General Motion Rest: Quasars Linear: Stars Keplerian: Binary Perturbed Keplerian: Asteroids,...
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Transcript of General Motion Rest: Quasars Linear: Stars Keplerian: Binary Perturbed Keplerian: Asteroids,...
![Page 1: General Motion Rest: Quasars Linear: Stars Keplerian: Binary Perturbed Keplerian: Asteroids, Satellites Complex: Planets, Space Vehicles Rotational: Earth,](https://reader036.fdocuments.us/reader036/viewer/2022082819/56649f1d5503460f94c34e85/html5/thumbnails/1.jpg)
General Motion
Rest: Quasars
Linear: Stars
Keplerian: Binary
Perturbed Keplerian: Asteroids, Satellites
Complex: Planets, Space Vehicles
Rotational: Earth, Moon, Satellites, …
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Linear Motion
Radial Motion
Proper Motion = Angular Motion
000 ttt vxx
sin
sincos
coscos
rx 0
0
0
0
tt
Vrr R
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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
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Keplerian
Two Body Motion under Newtonian Mech.
Gravitational Constant
Elements = 6 Constants of MotionShapeOrientationTiming
xx
32
2
rdt
d
mMG
ea,,, IΩ
T
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Units of Mass
SI: kilogram kg
Astronomical: Solar Mass
Newtonian Gravitational Constant
Measurable Quantity = GM
= Body-centric Gravitational ConstantHeliocentric GeocentricSGM EGM
SM
G
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Keplerian Elements
Semi-Major Axis: a
Eccentricity: e
Longitude of Ascending Node: Inclination: I
Argument of Pericenter: Epoch of Pericenter Passage: T
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Ellipse
Semi-major axis: a
Semi-minor axis: b 12
2
2
2
b
y
a
x
a
b
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Eccentricity
Eccentricity: e, Co-Eccentricity: e’
22
22
1' , ea
be
a
bae
ae
F
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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
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Z
P
N
I
Orbital Plane
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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)
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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
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Elements to PV (contd.)
Backward Euler Rotation
00
,,
ΩI, 313Rvx
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Kepler’s Equation
First Nonlinear Equation in History
Elliptic
Parabolic
Hyperbolic
MEeE sin
PM3
3
HMFFe sinh
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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
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Solution of Kepler’s Equation
Reduction of Variable Domain
Newton Method
Ee
EEEeMEf
Ef
EfEEfE
cos1
sincos
'
*
*
0sin MEeEEf
EM 0 0
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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
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Perturbed Keplerian Orbits
Elements as Functions of Time
Perturbation Theory
Polynomial + Fourier Series
tΛTΩIeaΛ ,,,,,
kkkkk tStC
tΛtΛΛΛ
sincos
2210
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Complex Motion
Equation of Motion
Numerical/Analytical Solution
Parameter Fitting to Observational Data
Results = Ephemeris
xx
32
2
rdt
d
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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/