Satellite geodesy: Kepler orbits, Kaula Ch. 2+3.I1.2a Basic equation: Acceleration Connects potential, V, and geometry. (We disregard disturbing forces.

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Presentation transcript:

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,

Velocity: Integration along orbit. Position: one integration more C.C.Tscherning,

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,

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,

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,

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

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

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

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

Geometry E and f C.C.Tscherning,

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,

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

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,

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

Angular momentum λ is arbitrary := 0 ! C.C.Tscherning,

Integration With u=1/r C.C.Tscherning,

Integration C.C.Tscherning,

Ellipse as solution If ellipse with center in (0,0) C.C.Tscherning,

Expressed in orbital plane C.C.Tscherning,

Parameter change C.C.Tscherning,

Further substitution C.C.Tscherning,

Transformation to CIS C.C.Tscherning,

Velocity C.C.Tscherning,

From orbital plane to CIS. C.C.Tscherning,

Determination of f. C.C.Tscherning,