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Satellite orbits . Where is the satellite ? May we see it ?. CTS Referencesystem. Fixed with respect to the Earth. /. Inertial system. Newtons laws valid. Center in Gravity center Fixt in relation to The Fix-stars. Connection to CT Through siderial Time. - PowerPoint PPT Presentation
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1
Satellite orbits.
• Where is the satellite ?
• May we see it ?
Satellite geophysics, 2013-11-10
2
CTS Referencesystem. Fixed with respect to the Earth.
/
Satellite geophysics, 2013-11-10
3
Inertial system. Newtons laws valid.
Center inGravity center
Fixt in relation toThe Fix-stars.
Connection to CTThrough siderial Time.
Satellite geophysics, 2013-11-10
4
Satellite movent around ideal Earth.
Spherical, homogeous, no athmosphereNewtons law of attraction: Force= F =G(Mm)/r2 M=Earth mass, m = satellitte massG= 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
tctctc
tc
32
2
/)( rrGMmdttcd
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 !
/ ba
C
f
2
222
abae
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: ApogeeMaximum: Perigee
Satellite geophysics, 2013-11-10
8
Kepler’s 3. law.
ant),(
)T time,Revolution(3
2
constaaxismajorsemi
anomalymeanTtnMvelocityangularmeanaGMn
GMaT
)(/
,4/3
232
Satellite geophysics, 2013-11-10
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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
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6 Kepler-elements
Position given by statevector or
6 Kepler- elements
= Ascending nodesrectancention, i: orbit inclination, = perigee argumenta= 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: excentricanomaly
Satellite geophysics, 2013-11-10
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Velocity and angular velocity
Linar in time !Orbit is straight line expressed in Kepler-elementes in the
6-dimensional space
EeEMTtnMMed
eEEef
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
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Satellitorbits GPS, i= 55 - Torge 5.2.
Satellite geophysics, 2013-11-10
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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
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Satellite orbits – influence of non-central force.
•
/
Satellite geophysics, 2013-11-10
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Satellit orbits, solar pressure, atmosphereForces 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
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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
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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
CCCCCC
i
i
a
enM
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
LongitudefCfCtiii
fCfCEearfCfCfu
eEEeaf
)0,0,)(()()()(
:node ascending of )(2sin)(2cos
)(2sin)(2cos)cos1()(2sin)(2cos
cossin1tan
1300
0
2
Satellite geophysics, 2013-11-10
22
Satellit orbits .
• LEO: Low Earth Orbit h < 2000 km• MEO: Medium Earth Orbit 5000-20000 km• GEO: Geostationary, h=36000 km
• IGSO: Inclined Geo-syncronous Orbit• HEO: Highly Elliptic Orbit
Satellite geophysics, 2013-11-10