AI 中文总结
本文提出一种数值方法，通过连续对齐构造平面牛顿三体问题的单参数相对周期轨道族，得到庞加莱、希尔及双星型解族，其结果可作为受限四体模型的规定轨迹。
AI 中文摘要
相对周期轨道（Relative Periodic Orbits, RPOs）是三体问题的解，在均匀旋转参考系中呈周期性，而在惯性坐标系中通常为准周期性。本文提出一种数值方法，用于计算和延拓平面牛顿三体问题的单参数RPO族。该方法利用连续朔望点（此处指三体共线且速度满足对应对称条件的构型），通过匹配两次连续对齐时的位置和动量，将RPO的计算简化为低维非线性问题。随后对该问题的解进行数值延拓，并通过旋转单值矩阵的非平凡特征值确定线性稳定性，计算时需剔除与守恒量和连续对称性相关的中性方向。将该方法应用于多种质量分布和初始构型，得到庞加莱型、希尔型及双星型解族。这些族呈现出从近圆到高偏心率运动的转变、共振附近及转折点处的稳定性变化，以及当旋转角为2π的有理倍数时的绝对周期解。在希尔型族中，随着最小天体失去对中等质量天体的引力束缚，延拓过程将卫星构型与绕恒星运动的构型连接起来；在双星 regime 中还得到了绕双星和绕恒星运动的构型。研究结果阐明了RPO的动力学多样性，并提供了连贯的三体运动，可作为受限四体模型中的规定轨迹。
英文摘要
Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.
