太阳系和致密星系统后牛顿N体问题动力学

In general relativity，this proposal will be devoted to studying the dynamics of post-Newtonian (PN) N-body problems in the Solar system and systems of compact stars. Explicit symplectic integrators for inseparable Hamiltonian systems will be designed. Some differences that may exist in the two approaches to a strong gravitational system of compact objects at the same PN order will be given analytically and numerically. It may be at different PN orders that a Lagrangian is equivalent to a Hamiltonian. The integrability or nonintegrability of this Hamiltonian can show that of the Lagrangian. In this way, a comparable mass compact binary system with one object spinning can be proved to be integrable and nonchaotic. Thus, this conflicting result on the presence of and the absence of chaos in the system will.be no longer existent. In addition to this, differences between the PN Lagrangian and Hamiltonian approaches to a three-body problem of spinning compact stars with/without gravitational radiation reaction will be discussed in detail. The related results on chaos in this problem that were given by the fractal basin boundary method will be rechecked by means of appropriate integrators and invariant chaos indicators. On the other hand, a high order symplectic integrator will be used to obtain high-precision long-term solutions of the Earth and of the Earth’s spin axis in the system of the Sun and the 8 main planets. Therefore, the long-term evolution of the Earth’s eccentricity and the Earth’s spin direction can be understood well. As an application of the PN N-body dynamics in the Solar system, high-precision astrometric models on our deep space exploration will be given. As a result, the present proposal may have a breakthrough on research of chaos in spinning compact binaries, and will be an important reference for the study of PN approximation theory and our deep space exploration. It is interesting to investigate the stability of the Solar system. It will be helpful to develop the interdiscipline of nonlinear celestial mechanics and general relativity, and to achieve one of the goals in relativistic basic astronomy in our country.

本项目研究致密星和太阳系后牛顿多体问题动力学。构建不可分系统的显辛算法；揭示同阶后牛顿拉格朗日和哈密顿之间的可能差异,找到与拉格朗日等价的哈密顿量，并借助后者来探究前者的可积与否，论证一体自转的相当质量比致密双星可积，解决该系统混沌与不混沌的冲突难题；比较引力辐射有无的自转致密星后牛顿三体问题的拉格朗日与哈密顿的动力学差异，再用相对论不变混沌指标检验分形盆方法所得结果的真伪；构建太阳系1阶后牛顿系统的高阶显辛方法，获得地球平动和转动的高精度长期历表，定量给出地球轨道偏心率和自转轴指向的长期演化规律；在后牛顿N体问题内建立并完善服务于深空导航与测控的高精度测量模型。这项研究可望取得自转致密双星混沌研究的突破进展，对后牛顿理论研究有重要价值，为深空探测提供理论参考，也对太阳系稳定性研究具有重要意义，促进非线性天体力学和相对论等交叉领域发展，实现我国相对论基本天文学的发展目标之一。

本项目围绕相对论后牛顿理论，开展太阳系和黑洞双星、三星等致密天体的动力学基础性前沿研究。构建几类高效的后牛顿系统几何数值积分算法，以便为这些系统的长期定性演化研究提供可靠数值结果，丰富和发展了天体力学数值方法。在后牛顿理论方面，与国际广义相对论著名学者Damour观点不同，我们指出同阶后牛顿拉格朗日与哈密顿系统一般存在差异、不等价以及一体旋转黑洞双星是有序非混沌的；还给出了自洽无截断的后牛顿拉格朗日运动方程，理论上自动严格保持所有运动常数。正确揭示同阶后牛顿拉格朗日自洽方程、截断方程与哈密顿三种表述之间的可能差异，纠正有关文献错误，彻底澄清有关旋转致密双星系统混沌的争议，为后牛顿拉格朗日系统的可积性提供了判据,丰富致密星二、三体问题动力学内涵，为未来的后牛顿系统动力学研究和引力波探测提供参考。给出了太阳系主要基频长期演化的数值模型与地球轨道长期历表的分析表达，将对太阳系天体轨道长期演化乃至太阳系稳定性等经典天体力学课题的研究有意义。建立深空探测的高精度测量模型，可以应用于我国未来的火星探测任务。项目组1人获得国务院政府特殊津贴，共发表学术论文57篇，其中SCI收录53篇。项目对研究队伍形成和研究生培养发挥了重要作用。