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3
Inertial system. Newtons laws valid.
Center in
Gravity center
Fixt in relation to
The Fix-stars.
Connection to CT
Through siderial
Time.
Satellite geophysics, 2013-11-10
4
Satellite movent around ideal Earth.
Spherical, homogeous, no athmosphere
Newtons law of attraction: Force= F =G(Mm)/r2
M=Earth mass, m = satellitte mass
G= gravitational constant, r distance from C./
Satellite geophysics, 2013-11-10
5
Orbit is curve in 3D-space.
Orbital curve:
Acceleration Force
2. order differental equation
If in ONE point we know:
Velocity-vector (3 numbers)
Position (3 numbers)
Determines orbit ! (6 numbers)
)(
)(
)(
)(
3
2
1
tc
tc
tc
tc
32
2
/)(
rrGMmdt
tcd
State-vector
Satellite geophysics, 2013-11-10
6
The Kepler laws as consequences of the law of attraction
1. Law: Orbit is elliptic, with 1 focus in the gravity center of the Earth. Orbital plane fix in inertial coordinate-system – tree constants fixed.
With a, e 5 constants fixed !
/ b
aC
f
2
222
a
bae
Satellite geophysics, 2013-11-10
7
Kelper’s 2. law.
Areas covered by the position-vector is proportional with time, t.
Velocity of Satellite is NOT constant.
Minumum: Apogee
Maximum: Perigee
Satellite geophysics, 2013-11-10
8
Kepler’s 3. law.
ant),(
)T time,Revolution(3
2
constaaxismajorsemi
anomalymeanTtnM
velocityangularmeanaGMn
GMaT
)(
/
,4/
3
232
Satellite geophysics, 2013-11-10
9
3. law:
Consequence: 2 satellites with same semi-major axis will have same revolution time, T, independent of the excentricity.
/
Satellite geophysics, 2013-11-10
10
6 Kepler-elements
Position given by
statevector or
6 Kepler- elements
= Ascending nodes
rectancention,
i: orbit inclination,
= perigee
argument
a= semi major axis,
e: excentricity, f=latitude,
Satellite geophysics, 2013-11-10
11
Computation of state-vector from Kepler-elementer
Coordinat system in Orbital plane, center in C. Polar coordinates f, r.
E: excentric
anomaly
Satellite geophysics, 2013-11-10
12
Velocity and angular velocity
Linar in time !
Orbit is straight line expressed in Kepler-elementes in the 6-dimensional space
EeEM
TtnMMedeE
Eef
Eear
sin
)(cos
sin1)tan()tan(
)cos1(
2
Satellite geophysics, 2013-11-10
13
To Inertial system by Rotations:
Position = Rxqq, Velocity = Rxqq’
Composed of 3 rotations
/)()()( 313 RiRRRxq
Satellite geophysics, 2013-11-10
15
Forces acting on the satellite.
• Fc= Ideal spherical Earth,
• Fnc= deviation from ideal
• Fn,Fs from Sun and Moon
• Fr , solar pressure
• Fa=atmosphere,
• Tides,• Magnetic Field
/
Satellite geophysics, 2013-11-10
17
Satellit orbits, solar pressure, atmosphere
Forces depend on shade/non shade of sun.
Relationship masse/surface area. Variations of 2 m.
Depends on density of atmosphere, satellite diameter, mass and velocity.
v=7500 m/s, force 0.000001 m/s2
Neglicible for GPS.
/
Satellite geophysics, 2013-11-10
18
Satellit-orbits – other bodies and mass changes.
• Moon most important, Planets small effect• Earth deformation, tides/loading• Seasonal masse-changes.
Satellite geophysics, 2013-11-10
19
Satellit orbits – description of changes.
16 parametres,
Update
Every hour.
Satellite geophysics, 2013-11-10
20
Satellite orbital parameters for GPS
• Mean anomaly• Mean movement difference• Excentricity
• Square-roor of a• Right acension• Inclination at t0e
• Perigee argument• Time derivative of rectac.• Time derivative of i• Correction to f • Correction to r• Corrections to i• Reference-time
e
rcrs
isic
usuc
t
CC
CC
CC
i
i
a
e
n
M
0
0
0
,
,
,
Satellite geophysics, 2013-11-10
21
Computation of position, Torge p. 132.
GM=3.98608x1014 m3/s2, =7.292115147x10-5 rad/s2
True anomaly fk from time-difference tk=t-t0e
Mean-anomali:
Solution iterativly wrt Ek,
e
kk tnaGMMM )/( 30
)sin( kkk EeEM
Tkkkkkeekek
kiskickk
krskrckk
kuskuckk
k
kk
ruRiRRXthentt
Longitude
fCfCtiii
fCfCEear
fCfCfu
eE
Eeaf
)0,0,)(()()()(
:node ascending of
)(2sin)(2cos
)(2sin)(2cos)cos1(
)(2sin)(2cos
cos
sin1tan
1300
0
2
Satellite geophysics, 2013-11-10