High-order analytic continuation and numerical stability analysis for the classical two-body problem
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Several methods exist for integrating the Keplerian Motion of two gravitationally interacting bodies. Lagrange introduced three vector-valued invariants that allowed the position and velocity vectors to be expanded. Alternatively, the classical F & G Lagrangian coefficients provide a mapping of the initial position and velocity into current time values. Nevertheless, classically, it has proven difficult to develop high-order Taylor series expansions, even though the governing equations are well-defined. This work presents two scalar Lagrange-like invariants (f = r r and g = f-n/2) that enable all of the higherorder vector-valued derivatives to be recursively evaluated using Leibniz product rule. The calculations for the higher-order trajectory derivatives are developed by deriving a differential equation for g that eliminates fractional powers. The basic methodology is extended to handle perturbed acceleration force models for the J2 zonal gravity harmonic term. The approach is both efficient and accurate. The two-body problem is handled by computing r = -rg, where r = [x,y,z] denotes the inertial relative coordinate vector that locates an object relative to the Earth and = 398601.2 km2/sec2 is the gravitational constant. Given r, r, r, Leibniz product rule is used to recursively develop {f, g, r}, {f, g, r}., Series convergence issues are addressed by invoking a vector version of a Pad series approximation for each component of the vector- valued Taylor series r(t + h) = r(t) + r(t)h + r(t)h2 / 2!+r(t)h3/3!+. The behavior of the Pad roots is studied as a function of the number of derivative terms retained in the series approximation and solution accuracy for the series approximation. Numerical results are presented that compare the solution accuracy and integration time required by ODE45 and RKN1210, and the analytic power series methods developed in this work for two-body plus J 2 perturbation terms.
