Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksOrbit ComputationM 0 mean anomaly at time T 0Applying these equations, the Kepler orbits can be computed with Kepler parameterscorrected for the influence of the oblate earth, thus leading to a somewhat more accurate orbitcomputation. Note, that the transformation into the earth fixed frame has to be conductedusing the corrected values for Ω and ω.4.2 Numerical Integration of the Equations of MotionThe equations of motion of a satellite are described by the following system of six ordinarylinear differential equations, which has to be solved to obtain the satellites position andvelocity vector in time.dx= xDdtdxD= DD x =dtdyD= DD y =dtdzD= DD z =dt,dy= yDdt,Fx,k= ∑ax,kk m k=Fy,k= ∑ay,kk m k=Fz,k= ∑az,km=∑∑∑kkdz= zDdtf (x, y,z)f (x, y,z)f (x, y, z)Eq. 4.2-1The integration of such a system of 1 st order linear differential equations can not be doneanalytically, but is a well known problem to numerical mathematics. There are severalstandard procedures to solve it, e.g. Runge-Kutta or Adams-Bashford-Moulton. These twoshall be briefly outlined in this section.One of the most versatile numerical integration algorithms is the Runge-Kutta procedure. It isa one-step algorithm, requiring only the preceding state vector to compute the actual one. Itsolves differential equations of the typexD (t) = f (x, t)Eq. 4.2-2xii(t0i) = c0applying the following difference equationxkn+1i= xn+= h ⋅f (tnv∑i=1w k+ c h, xiijn+i−1∑j=1aijkj)Eq. 4.2-3with h step width (in time)ci, ai coefficients, determined by the order and stage number of the algorithmR. Wolf Page 23