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

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Satellite geodesy Kepler orbits, Kaula Ch.
23.I1.2a

• Basic equation Acceleration
• Connects potential, V, and geometry. (We
  disregard disturbing forces friction).

C.C.Tscherning, 2007-10-25.
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Velocity Integration along orbit.

• Position one integration more

C.C.Tscherning, 2007-10-25.
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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.
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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.
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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.
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Ordinary differential equations

• Change from 3 second order equations to 6
  first-order equations

C.C.Tscherning, 2007-10-25.
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Coordinate transformation in 6D-space

• New coordinates qi and pi .

C.C.Tscherning, 2007-10-25.
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(q,p) selected so orbits straight lines

• If
• Possible also so that kmCij amplified.

C.C.Tscherning, 2007-10-25.
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Kepler orbit

• If potential Vkm/r
• Orbit in plane through origin (0).
• Is an ellipse with one focus in origin

C.C.Tscherning, 2007-10-25.
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Geometry

• E and f

C.C.Tscherning, 2007-10-25.
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Kepler elements

• iinclination, Olongitude of ascending node (DK
  knude)
• eexcentricity, asemi-major axis,
• ?argument of perigaeum, f ?latitude.
• ME-esinEMean anomaly (linear in time !)

C.C.Tscherning, 2007-10-25.
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From CIS to CTS

• We must transform from Conventional Inertial
  System to Conventional Terrestrial System using
  siderial time, ?
• Rotation Matrix

C.C.Tscherning, 2007-10-25.
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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.
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Elliptic orbit

• We use spherical coordinates r,? in (q1,q2)-plane

C.C.Tscherning, 2007-10-25.
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Angular momentum

• ? is arbitrary 0 !

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

• With u1/r

C.C.Tscherning, 2007-10-25.
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Integration
C.C.Tscherning, 2007-10-25.
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Ellipse as solution

• If ellipse with center in (0,0)

C.C.Tscherning, 2007-10-25.
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Expressed in orbital plane
C.C.Tscherning, 2007-10-25.
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Parameter change
C.C.Tscherning, 2007-10-25.
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Further substitution
C.C.Tscherning, 2007-10-25.
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Transformation to CIS
C.C.Tscherning, 2007-10-25.
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Velocity

• 
  

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

• .

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

• .

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