Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksOrbit ComputationP (x) = 1;02n −1Pn(x) = ⋅ x ⋅ Pnnn −1− ⋅ Pnn−2(x) ⋅P (x) = x ⋅1− 1(x) ⋅2n + 12n − 332n + 12n −1if n ≥ 2Eq. 4.2-21The normalisation factors of the associated Legendre functions contains faculties, whichshould not be computed explicitly.P(m)nmd Pn(x)(x) = ⋅mdx2( ) ( n −2n + 1)m !( n + m)!Eq. 4.2-22Fortunately, they can be reduced in the resulting recursive normalisation factors. Therecursive algorithm for fully normalised Legendre functions is given asPPP(m)n(m)n(m)n(x) = 0(x) = 1⋅3⋅...⋅ (2m −1)⋅2n −1(x) = ⋅ x ⋅ Pn − mn + m −1− ⋅ Pn − mif n < m(m)n−2(x) ⋅2n + 1( 2n −1)( n + m)( n + m −1)( 2n + 1)( n − m)1(x) ⋅( 2n −1)( n + m)( 2n + 1)( n − m)( n − m −1)if( 2n − 3)( n + m)( n + m −1)(m)n−with the normalized starting values:P (1)0(x) = 0P (1)1(x) =3ifn > mn = mEq. 4.2-23Eq. 4.2-24P (2)1(x) = 0The method described above is numerically very stable and has been successfully used tocompute Legendre functions up to degree and order 700. A drawback of this method is thatthe computational burden is about twice as high as for non-normalised Legendre functions.Thus, for a spherical harmonics expansion up to degree and order of say 15 –20 the denormalisationof the coefficients would be favourable.4.2.1.3 Computation of GravityThe expression of the gravity potential in terms of a spherical harmonics expansionU =GMr+ GMNn∑∑n= 2 m=0raneP ϕ λ + λn+ 1 nm(sin )(Cnmcos m Snmsin mcan be rearranged the following way (Colombo 1981))Eq. 4.2-25R. Wolf Page 29