STATE TRANSITION TENSOR MODELS FOR THE UNCERTAINTY PROPAGATION OF THE TWO-BODY PROBLEM

Several methods exist for integrating the Keplerian Motion of two gravitationally interacting bodies, even when gravitational perturbation terms are included. The challenge is that the equations of motion become very stiff when the perturbation terms are included, which forces the use of small time steps, higher-order methods, or extended precision calculations. Recently, Turner and Elgohary have shown that by introducing two scalar Lagrange-like invariants that it is possible to integrate the two-body and two-body plus J2 perturbation term using a recursive formulation for developing an analytic continuation-based power series that overcomes the limitations of standard integration methods. Numerical comparisons with RK12(10), and other state of the art integration methods indicate performance improvements of 70X, while maintaining mm accuracy for the orbit predictions. Extensions for J3 through J6 are currently under development. With accurate trajectories available, the next important theoretical development becomes extending the series-based solution for the state transition matrices (STM) for both the two-body and two-body plus J2 perturbation. STMs are useful for many celestial mechanics optimization calculations. Second and third order STM models are developed to support uncertainty propagation investigations. The application of scalar Lagrange-like invariants generates highly efficient state trajectory, STM, and higher-order STMs models. The proposed mathematical models are expected to be broadly useful for celestial mechanic applications for optimization, uncertainty propagation, and nonlinear estimation theory.