A Computation Process for the Higher Order State Transition Tensors of the Gravity and Drag Perturbed Two-Body Problem Using Adaptive Analytic Continuation Technique

Abstract

Abstract In this work, the Taylor series based integration approach, Analytic Continuation, has been implemented to compute Higher Order State Transition Tensors for the \(J_2-J_6\) gravity and drag perturbed Two-body problem. Analytic Continuation is an integration method applied to solve different fundamental problems of astrodynamics. In this method, two scalar quantities f and \(g_p\) are defined and differentiated to arbitrary order using the Leibniz product rule to obtain higher order time derivatives of the variables which are implemented in the Taylor series expansion of the solution. Previously, this method has been proved to be highly precise and computationally efficient in propagating Two-body trajectories with full spherical harmonics gravity and atmospheric drag perturbation. More recently, this method has been implemented in propagating gravity and drag perturbed State Transition Matrix (STM) with machine precision level of accuracy. An expansion of the procedure to compute \(J_2\) gravity perturbed Higher Order State Transition Tensors has also been presented in the subsequent work. In this paper, the method is further expanded to incorporate up to \(J_6\) gravity and drag perturbation in the computation of Higher Order State Transition Tensors. Four types of orbits are considered for numerical simulations: LEO, MEO, GTO, and HEO. First, RMS error of the unperturbed STMs comparing to the closed form solution of Battin are presented. Then, the error in the symplectic nature of the gravity perturbed STM and Higher Order State Transition Tensors are checked, showing double precision accuracy of the STMs and tensors. Finally, initial error of the states of the \(J_2-J_6\) gravity and drag-perturbed orbits are propagated using the computed perturbed Higher Order State Transition Tensors and compared against the results obtained using the perturbed STM, showing at least 3 to 4 digits of accuracy improvement while using Higher Order Tensors.