The exponential growth of space objects in near-Earth and geostationary orbits has posed severe threats to space environment safety, with debris clouds from spacecraft breakup events being a critical concern. Debris cloud tracing, as a key technology for locating breakup points, faces dual challenges of insufficient precision in analytical methods and excessive computational load in numerical methods. To balance traceability accuracy with computational efficiency, this paper proposes a breakup time determination method integrating a clustering algorithm and the minimization of average relative distance. The method first calculates the average relative distance between fragment pairs and preliminarily estimates the breakup epoch using a golden section step-size optimization strategy. Subsequently, the density peak clustering (DPC) algorithm is introduced to eliminate abnormal fragments. The breakup epoch is then refined based on the cleansed fragment dataset, achieving high-precision localization. Validation through simulations of real breakup events demonstrates that this method significantly improves localization accuracy. It establishes a highly reliable temporal benchmark for space collision tracing, debris diffusion prediction, and orbital safety management.

Modern space missions such as spacecraft formation flying and on-orbit servicing impose higher requirements on the accuracy of relative motion dynamics models. The traditional Clohessy-Wiltshire equations and Tschauner-Hempel equations are based on the two-body problem assumption and ignore the J2 term in Earth's non-spherical gravitational perturbation, leading to significant errors in long-term relative motion prediction. This paper systematically studies the modeling and analytical solution of spacecraft relative motion under J2 perturbation. Firstly, linearized relative motion equations considering the J2 perturbation difference are established in the Local Vertical Local Horizontal (LVLH) coordinate system. To address the challenge of solving this set of variable-coefficient differential equations, this paper proposes a key variable substitution method that transforms the system into a set of constant-coefficient differential equations, and then derives analytical solutions for three-axis relative motion expressed as functions of the initial state. This analytical solution clearly reveals the coupling mechanism between long-term drift and periodic motion induced by J2 perturbation, providing a new theoretical tool for high-precision relative orbit design, drift suppression, and control strategies.