Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksOrbit ComputationU Gravity potentialGM Earth's gravity constantr magnitude of radius vector (of an arbitrary point)a Earth's equatorial radiusn,m Degree and order of spherical harmonicsP nm Legendre functionsC nm ,S nm Coefficients of spherical harmonicsϕ Latitudeλ LongitudeThe Legendre polynomials P n and associated functions P nm are defined asandPn1 d 2 nn (x) = (x −1)n n2 n! dxEq. 4.2-12Pnm(x) = (1 − x2 m / 2)dmPnmdx(x)Eq. 4.2-13The force acting on a point in the gravitation field is obtained by computing the gradient ofthe potential. Eq. 4.2-14∂U∂U∂Ug = grad(U) = ( , , )∂x∂y∂zThis analytical expression is not very well suited for implementation. Soop (1994) indicates arecursive method for computing the Legendre polynomials and functions, as well as thepartial derivatives required to compute the gravity force.4.2.1.1 Computation of Legendre Polynomials and FunctionsThe Legendre polynomials can be computed recursively using starting values for the first twoterms:P (x) = 1;0P (x) = x2n −1pn(x) = ⋅ xPnn1−1n −1(x) − ⋅ Pnn−2(x) if n ≥ 2Eq. 4.2-15The associated Legendre functions are obtained in two steps. First, the m-fold derivative ofeach polynomial P n (x) is computedmPnm(m) d (x)P n (x) =dxEq. 4.2-16R. Wolf Page 27