1. Satellite geodesy: Kepler orbits, Kaula Ch. 2+3.I1.2a Basic equation: Acceleration Connects potential, V, and geometry. (We disregard disturbing forces – friction). C.C.Tscherning, 2005-11-03.

2. Velocity: Integration along orbit. Position: one integration more C.C.Tscherning, 2005-11-03.

3. We must know (approximatively) the orbit to make the integration. The equation connects the position and velocity with parameters expressing V. Parameters: kMC ij Orbit integration and parameter determination C.C.Tscherning, 2005-11-03.

4. Directions and distances from Earth using Cameras, lasers, radar-tracking, time- differences Distances from satellites to ”point” on Earth surface (also ”cross-overs”) Range rates: Doppler effect, contineous tracking. Measurements in or between satellites: gradiometry, GPS-positions, ranging Observations C.C.Tscherning, 2005-11-03.

5. Spherical harmonic coefficients, kMC ij Positions of ground tracking stations Changes to Earth Rotation and pole-position Tides (both oceanic and solid earth) Drag-coefficients, air-density Contributions from Sun and Moon. Parameters C.C.Tscherning, 2005-11-03.

6. Ordinary differential equations Change from 3 second order equations to 6 first- order equations: C.C.Tscherning, 2005-11-03.

7. Coordinate transformation in 6D-space New coordinates q i and p i. C.C.Tscherning, 2005-11-03.

8. (q,p) selected so orbits straight lines If Possible also so that kmC ij ”amplified”. C.C.Tscherning, 2005-11-03.

9. Kepler orbit If potential V=km/r: Orbit in plane through origin (0). Is an ellipse with one focus in origin C.C.Tscherning, 2005-11-03.

10. Geometry E and f C.C.Tscherning, 2005-11-03.

11. Kepler elements i=inclination, Ω=longitude of ascending node (DK: knude) e=excentricity, a=semi-major axis, ω=argument of perigaeum, f+ ω=”latitude”. M=E-esinE=Mean anomaly (linear in time !) C.C.Tscherning, 2005-11-03.

12. From CIS to CTS We must transform from Conventional Inertial System to Conventional Terrestrial System using siderial time, θ: Rotation Matrix C.C.Tscherning, 2005-11-03.

13. From q-system to CIS 3 rotations. R i with integer i subscript is rotation about i-axis. R xu is rotation from u to x. C.C.Tscherning, 2005-11-03.

14. Elliptic orbit We use spherical coordinates r,λ in (q 1,q 2 )-plane C.C.Tscherning, 2005-11-03.

15. Angular momentum λ is arbitrary := 0 ! C.C.Tscherning, 2005-11-03.

16. Integration With u=1/r C.C.Tscherning, 2005-11-03.

17. Integration C.C.Tscherning, 2005-11-03.

18. Ellipse as solution If ellipse with center in (0,0) C.C.Tscherning, 2005-11-03.

19. Expressed in orbital plane C.C.Tscherning, 2005-11-03.

20. Parameter change C.C.Tscherning, 2005-11-03.

21. Further substitution C.C.Tscherning, 2005-11-03.

22. Transformation to CIS C.C.Tscherning, 2005-11-03.

23. Velocity C.C.Tscherning, 2005-11-03.

24. From orbital plane to CIS. C.C.Tscherning, 2005-11-03.

25. Determination of f. C.C.Tscherning, 2005-11-03.