Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Orbit ComputationInter Satellite LinksU =U =withGMrGMr+ GM+ GMN∑∑n= 2 m=0Nnn∑∑n= 2 m=0arnen+1arcosnen+m+1Pm(m)nϕ⋅P(m)n(sin ϕ)(C(sin ϕ)(Cnmrmnmcoscosmλ + Smnmϕ⋅cosmλ + Ssin mλ)nmrmcosmϕ⋅sin mλ)Eq. 4.2-26x = r cosϕcosλEq. 4.2-27y = r cosϕsinλz = r sin ϕIntroducing the following abbreviationsm mξ=r cos ϕcosmλmmη = r cos ϕsinmλmmyields for the gradient of the gravity potentialU U Ugrad Ugrad GM N nn∂ ∂ ∂ae( m)( ) = ( , , ) = ( ) + GM∑ ∑ grad{ P C Sn m n(sin ϕ)( nmξm +nmηm)}+ + 1∂x∂y∂zrrn=2 m=0ξ , η can be computed recursively using the following simple expressionsmmEq. 4.2-28Eq. 4.2-29ξ = 1Eq. 4.2-30ηξη00mm= 0= ξ= ξm−1m−1x − ηy + ηm−1m−1yxIn the following the partials of the above expression are givenn∂ aea{ } =− ( n+ m+1 )n mn∂x rrne+ + 1 + m+3n∂ aea{ } =− ( n+ m+1 )n mn∂y rrne+ + 1 + m+3n∂ aea{ } =− ( n+ m+1 )n mn∂z rrne+ + 1 + m+3xyzEq. 4.2-31Eq. 4.2-32Eq. 4.2-33∂ϕϕ∂x P ( m) zxr P ( m+1) Eq. 4.2-34[n(sin )] =−n(sin )3∂ϕϕ∂y P ( m) zyr P ( m+1)Eq. 4.2-35[n(sin )] =−n(sin )3Page 30R. Wolf