1 / 25

Satellite geodesy: Kepler orbits, Kaula Ch. 2+3.I1.2a

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, 2007-10-25. Velocity: Integration along orbit. Position: one integration more. C.C.Tscherning, 2007-10-25.

calvin
Download Presentation

Satellite geodesy: Kepler orbits, Kaula Ch. 2+3.I1.2a

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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, 2007-10-25.

  2. Velocity: Integration along orbit. • Position: one integration more C.C.Tscherning, 2007-10-25.

  3. Orbit integration and parameter determination • We must know (approximatively) the orbit to make the integration. • The equation connects the position and velocity with parameters expressing V. • Parameters: kMCij C.C.Tscherning, 2007-10-25.

  4. Observations • 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 C.C.Tscherning, 2007-10-25.

  5. Parameters • Spherical harmonic coefficients, kMCij • 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. C.C.Tscherning, 2007-10-25.

  6. Ordinary differential equations • Change from 3 second order equations to 6 first-order equations: C.C.Tscherning, 2007-10-25.

  7. Coordinate transformation in 6D-space • New coordinates qi and pi . C.C.Tscherning, 2007-10-25.

  8. (q,p) selected so orbits straight lines • If • Possible also so that kmCij ”amplified”. C.C.Tscherning, 2007-10-25.

  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, 2007-10-25.

  10. Geometry • E and f C.C.Tscherning, 2007-10-25.

  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, 2007-10-25.

  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, 2007-10-25.

  13. From q-system to CIS • 3 rotations. Ri with integer i subscript is rotation about i-axis. Rxu is rotation from u to x. C.C.Tscherning, 2007-10-25.

  14. Elliptic orbit • We use spherical coordinates r,λ in (q1,q2)-plane C.C.Tscherning, 2007-10-25.

  15. Angular momentum • λ is arbitrary := 0 ! C.C.Tscherning, 2007-10-25.

  16. Integration • With u=1/r C.C.Tscherning, 2007-10-25.

  17. Integration C.C.Tscherning, 2007-10-25.

  18. Ellipse as solution • If ellipse with center in (0,0) C.C.Tscherning, 2007-10-25.

  19. Expressed in orbital plane C.C.Tscherning, 2007-10-25.

  20. Parameter change C.C.Tscherning, 2007-10-25.

  21. Further substitution C.C.Tscherning, 2007-10-25.

  22. Transformation to CIS C.C.Tscherning, 2007-10-25.

  23. Velocity C.C.Tscherning, 2007-10-25.

  24. From orbital plane to CIS • . C.C.Tscherning, 2007-10-25.

  25. Determination of f • . C.C.Tscherning, 2007-10-25.

More Related