Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Orbit ComputationInter Satellite Linksx⎡cosΩcosω⎢⎢− sin Ωsinωcosi⎢sin Ω cosω=⎢+cosΩsinωcosi⎢sin ωsin i⎢⎢⎣− cosΩsinω− sin Ωcosωcosi− sin ΩsinωIx OP+ cosΩcosωcosicosωsin i⎤sin Ωsin i ⎥⎥− cos Ωsin i⎥⋅⎥cosi⎥⎥⎥⎦Eq. 4.1-16Likewise, the transformation from the inertial to an earth centred earth fixed frame (e.g.WGS-84) is achieved by a similar operation.withx⎡ cosΘsin Θ=⎢⎢− sin Θ cosΘ⎢⎣0 00⎤0⎥⎥⋅1⎥⎦ECEFx IΘ hour angleEq. 4.1-174.1.2 Accounting for Secular PerturbationsA satellite trajectory computed using the Keplerian equations would diverge very soon fromthe actual one. Most of the acting forces cause periodically varying perturbations, althoughwith increasing amplitude. The main secular perturbations are caused by the oblate shape ofthe earth's gravity field. The major deviation is due to the nodal regression caused by theoblateness. The following equation gives the derivative of the right ascension with respect totime.d Ω 23 RE= −n⋅ ⋅ J2⋅ ⋅cos iEq. 4.1-18dt 2 22a ⋅ 1− εwith J 2 being the oblateness coefficient. The oblate gravity field has also an impact on the lineof apsis2( 4 − 5sin i)2d ω 3 RE= ⋅ n ⋅ JEq. 4.1-192dt 42 2a 1− εand a minor impact on the mean motionM = M + nJ2nJ 20⋅t2( 3cos i −1)( ) ⎥ ⎥ ⎤⋅ cos2 3⋅ 1− ε ⎦⎡2⎢3 RE= n01+⋅ J2⋅i⎢ 4 2⎣ awithM mean anomaly at time tEq. 4.1-20Page 22R. Wolf