1 Satellite orbits. Where is the satellite ? May we see it ? Satellite geophysics,

2 CTS Referencesystem. Fixed with respect to the Earth. / Satellite geophysics,

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,

4 Satellite movent around ideal Earth. Spherical, homogeous, no athmosphere Newtons law of attraction: Force= F =G(Mm)/r 2 M=Earth mass, m = satellitte mass G= gravitational constant, r distance from C. / Satellite geophysics,

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) State-vector Satellite geophysics,

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 a C f Satellite geophysics,

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,

8 Kepler’s 3. law. Satellite geophysics,

9 3. law: Consequence: 2 satellites with same semi-major axis will have same revolution time, T, independent of the excentricity. / Satellite geophysics,

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,

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,

12 Velocity and angular velocity Linar in time ! Orbit is straight line expressed in Kepler-elementes in the 6-dimensional space Satellite geophysics,

13 To Inertial system by Rotations: Position = R xq q, Velocity = R xq q’ Composed of 3 rotations / Satellite geophysics,

14 Satellitorbits GPS, i= 55 - Torge 5.2. Satellite geophysics,

15 Forces acting on the satellite. F c = Ideal spherical Earth, F nc = deviation from ideal F n,F s from Sun and Moon F r, solar pressure F a =atmosphere, Tides, Magnetic Field / Satellite geophysics,

16 Satellite orbits – influence of non-central force. / Satellite geophysics,

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 m/s 2 Neglicible for GPS. / Satellite geophysics,

18 Satellit-orbits – other bodies and mass changes. Moon most important, Planets small effect Earth deformation, tides/loading Seasonal masse-changes. Satellite geophysics,

19 Satellit orbits – description of changes. 16 parametres, Update Every hour. Satellite geophysics,

20 Satellite orbital parameters for GPS Mean anomaly Mean movement difference Excentricity Square-roor of a Right acension Inclination at t 0e Perigee argument Time derivative of rectac. Time derivative of i Correction to f Correction to r Corrections to i Reference-time Satellite geophysics,

21 Computation of position, Torge p GM= x10 14 m 3 /s 2, = x10 -5 rad/s 2 True anomaly f k from time-difference t k =t-t 0e Mean-anomali: Solution iterativly wrt E k, Satellite geophysics,

22 Satellit orbits. LEO: Low Earth Orbit h < 2000 km MEO: Medium Earth Orbit km GEO: Geostationary, h=36000 km IGSO: Inclined Geo-syncronous Orbit HEO: Highly Elliptic Orbit Satellite geophysics,