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<p>This work presents a semi-analytical time-series propagation framework for the circular restricted three-body problem (CR3BP) in which the solution flow is represented as an explicit truncated Taylor polynomial in time. Higher-order time-derivative coefficients are derived symbolically from the CR3BP equations of motion and evaluated recursively, enabling closed-form short-interval propagation of both the state and the associated state transition matrix (STM) obtained by differentiating the same truncated series. Long-arc propagation is achieved by recursively reinitializing the series with an adaptive step size selected from a remainder bound based on Taylor’s theorem, so accuracy is governed by the chosen truncation order together with the error-controlled interval length rather than by numerical integration of the equations of motion and variational equations. Orbit-propagation verification over planar, spatial, and non-periodic Earth–Moon trajectories is performed by monitoring the Jacobi constant under recursive propagation with the remainder-based controller. STM verification is performed using segment-wise preservation of canonical Hamiltonian structure through determinant and symplecticity diagnostics and using long-arc periodic accuracy through composition of segment STMs to form the monodromy matrix and comparison of stability-index quantities against JPL catalog values. Finally, the framework is demonstrated in two standard applications, impulsive targeting using STM subblocks to map terminal-state corrections to initial-epoch maneuvers and relative orbit determination using range, range-rate, and optical-angle measurements within an extended Kalman filter (EKF) and an unscented Kalman filter (UKF) that propagate the state using the same time-series model. Results show that the method provides accurate state and sensitivity propagation for nonlinear cislunar dynamics while maintaining a consistent closed-form treatment of the CR3BP flow map.</p>

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Semi-Analytical Time Series Solutions to the Circular Restricted Three-Body Problem

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  • 发表日期：

    2026年7月30日

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  • Ryan J. Menges,
  • Daniel J. Scheeres

摘要

This work presents a semi-analytical time-series propagation framework for the circular restricted three-body problem (CR3BP) in which the solution flow is represented as an explicit truncated Taylor polynomial in time. Higher-order time-derivative coefficients are derived symbolically from the CR3BP equations of motion and evaluated recursively, enabling closed-form short-interval propagation of both the state and the associated state transition matrix (STM) obtained by differentiating the same truncated series. Long-arc propagation is achieved by recursively reinitializing the series with an adaptive step size selected from a remainder bound based on Taylor’s theorem, so accuracy is governed by the chosen truncation order together with the error-controlled interval length rather than by numerical integration of the equations of motion and variational equations. Orbit-propagation verification over planar, spatial, and non-periodic Earth–Moon trajectories is performed by monitoring the Jacobi constant under recursive propagation with the remainder-based controller. STM verification is performed using segment-wise preservation of canonical Hamiltonian structure through determinant and symplecticity diagnostics and using long-arc periodic accuracy through composition of segment STMs to form the monodromy matrix and comparison of stability-index quantities against JPL catalog values. Finally, the framework is demonstrated in two standard applications, impulsive targeting using STM subblocks to map terminal-state corrections to initial-epoch maneuvers and relative orbit determination using range, range-rate, and optical-angle measurements within an extended Kalman filter (EKF) and an unscented Kalman filter (UKF) that propagate the state using the same time-series model. Results show that the method provides accurate state and sensitivity propagation for nonlinear cislunar dynamics while maintaining a consistent closed-form treatment of the CR3BP flow map.