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

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Presentation transcript:

From CIS to CTS We must transform from Conventional Inertial System to Conventional Terrestrial System using siderial time, θ: Rotation Matrix C.C.Tscherning, 2011-10-25.

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, 2011-10-25.

Elliptic orbit We use spherical coordinates r,λ in (q1,q2)-plane C.C.Tscherning, 2011-10-25.

Angular momentum λ is arbitrary := 0 ! C.C.Tscherning, 2011-10-25.

Integration With u=1/r C.C.Tscherning, 2011-10-25.

Integration C.C.Tscherning, 2011-10-25.

If ellipse with center in (0,0) Ellipse as solution If ellipse with center in (0,0) C.C.Tscherning, 2011-10-25.

Expressed in orbital plane C.C.Tscherning, 2011-10-25.

Parameter change C.C.Tscherning, 2011-10-25.

Further substitution C.C.Tscherning, 2011-10-25.

Transformation to CIS C.C.Tscherning, 2011-10-25.

Velocity C.C.Tscherning, 2011-10-25.

From orbital plane to CIS . C.C.Tscherning, 2011-10-25.

Determination of f . C.C.Tscherning, 2011-10-25.

General equations of motion (Kaula 3.2)I2.1a C.C.Tscherning, 2009-10-25.

Change of variables . C.C.Tscherning, 2009-10-25.

Kaula (3.38) . C.C.Tscherning, 2009-10-25.

We take the zero term out: Force Function We take the zero term out: C.C.Tscherning, 2009-10-25.

Conversion of spherical harmonics (Kaula, 3.3)I2.2a We want to express the terms in the expansion in Kepler variables: . C.C.Tscherning, 2011-10-25.

Kaula 3.72, 3.73. C.C.Tscherning, 2011-10-25.

Kaula 3.74. C.C.Tscherning, 2011-10-25.

Kaula 3.75. With C20=-0.00010827, e=0.001, a=1.2ae C.C.Tscherning, 2011-10-25.

Orbit with repeating ground track Applications Orbit with repeating ground track Orbit which gives resonance with specific term(s) Orbit which is sun-syncroneous Orbit which enables close ”encounter” with an object, such as the poles. C.C.Tscherning, 2011-10-25.

Sol-synkron bane Så må vi have: C.C.Tscherning, 2011-10-25.

Geostationær C.C.Tscherning, 2011-10-25.