satellite orbits

1

Satellite orbits.

• Where is the satellite ?

• May we see it ?

Satellite geophysics, 2013-11-10

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CTS Referencesystem. Fixed with respect to the Earth.

/


3

Inertial system. Newtons laws valid.

Center inGravity center

Fixt in relation toThe Fix-stars.

Connection to CTThrough siderial Time.


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./


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


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


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


8

Kepler’s 3. law.

ant),(

)T time,Revolution(3

2

constaaxismajorsemi

anomalymeanTtnMvelocityangularmeanaGMn

GMaT

)(/

,4/3

232


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3. law:

Consequence: 2 satellites with same semi-major axis will have same revolution time, T, independent of the excentricity.

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10

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,


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Computation of state-vector from Kepler-elementer

Coordinat system in Orbital plane, center in C. Polar coordinates f, r.

E: excentricanomaly


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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


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To Inertial system by Rotations:

Position = Rxqq, Velocity = Rxqq’

Composed of 3 rotations

/)()()( 313 RiRRRxq


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Satellitorbits GPS, i= 55 - Torge 5.2.


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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

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16

Satellite orbits – influence of non-central force.

•

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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.

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18

Satellit-orbits – other bodies and mass changes.

• Moon most important, Planets small effect• Earth deformation, tides/loading• Seasonal masse-changes.


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Satellit orbits – description of changes.

16 parametres,Update Every hour.


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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

,,,


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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


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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 orbits

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