Abstract
In this work, the Taylor series based technique, Analytic Continuation is implemented to develop a method for the computation of the gravity and drag perturbed State Transition Matrix (STM) incorporating adaptive time steps and expansion order. Analytic Continuation has been developed for the two-body problem based on two scalar variables f and gp and their higher order time derivatives using Leibniz rule. The method has been proven to be very precise and efficient in trajectory propagation. The method is expanded to include the computation of the STM for the perturbed two-body problem. Leibniz product rule is used to compute the partials for the recursive formulas and an arbitrary order Taylor series is used to compute the STM. Four types of orbits, LEO, MEO, GTO and HEO, are presented and the simulations are run for 10 orbit periods. The accuracy of the STM is evaluated via RMS error for the unperturbed cases, symplectic check for the gravity perturbed cases and error propagation for the gravity and drag perturbed orbits. The results are compared against analytical and high order numerical solvers (ODE45, ODE113 and ODE87) in terms of accuracy. The results show that the method maintains double-precision accuracy for all test cases and 1-2 orders of magnitude improvement in linear prediction results compared to ODE87. The present approach is simple, adaptive and can readily be expanded to compute the full spherical harmonics gravity perturbations as well as the higher order state transition tensors.
Highlights
In Astrodynamics, the State Transition Matrix (STM) of the two-body problem works as a sensitivity of the current states to the initial conditions
In this work, the Taylor series based technique, Analytic Continuation is implemented to develop a method for the computation of the gravity and drag perturbed State Transition Matrix (STM) incorporating adaptive time steps and expansion order
Simulation results are presented for four different types of orbits: low earth orbit (LEO), MEO, GTO and HEO, for up to 10,000 orbit periods and compared with ODE45, ODE113 and ODE87
Summary
In Astrodynamics, the State Transition Matrix (STM) of the two-body problem works as a sensitivity of the current states to the initial conditions. Algorithms 1 – 3 in the Appendix show the steps for the full implementation of the analytic continuation method to compute the J2 − J6 and drag perturbed STM for the two-body problem. For the J2 − J6 perturbed cases, the results of the 10 orbit periods of the elements of the [Esym.] matrix of every step using Analytic Continuation method are compared with the results using ODE87, ODE113 and ODE45 In this case, the initial and final time is provided to the solvers and the flexibility is given to the methods to select the time steps for the calculation. The error propagation compares J2 − J6 gravity and drag perturbed Analytic Continuation results for the four orbit types in Table 1 against ODE87 for 10 orbit periods. The results of the error propagation of the position and velocity are shown in Figs. 14, 15, 16, 17, 18, 19, 20, and 21
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