Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksOrbit Computation2∂ϕϕ∂z P ( m) 1 zr r PEq. 4.2-36( m+1)[n(sin )] = −n(sin )3∂ξm∂x∂ξm∂y∂ξm∂z∂ηm∂x∂ηm∂ym xsin λ= [ ξm + tanϕcosλsinϕξm + ηm = mξm−r rcosϕ]1m ycosλ= [ ξm + tanϕsinλsinϕξηm−m= −mηm−r rcosϕm z= [ ξm− tanϕcos ϕξm] = 0r r]1m xsin λ= [ ηm + tanϕcosλsinϕηm − ξm = mηm−r rcosϕ]1m ycosλ= [ ηm + tanϕsinλsinϕηm + ξm = mξm−r rcosϕ]1Eq. 4.2-37Eq. 4.2-38Eq. 4.2-39Eq. 4.2-40Eq. 4.2-41∂ηm∂zm z= [ ηm− tanϕcos ϕηm] = 0r rEq. 4.2-42Especially the computation of the η and ξ is subject to numerical problems because they arein the order of magnitude of r m . For a high order spherical expansion it is advantageous tocomputeηη =mrξξ =mrEq. 4.2-43which is dimensionless and restricted to the range between 0 and 1. The remaining factor r mcan be multiplied with the termsranen+m+1andarnen+m+3This has the additional advantage of bringing them into a numerical stable formEq. 4.2-44R. Wolf Page 31