力梯度辛算法在外限制性系外行星三体问题中的应用

Application of Force Gradient Symplectic Algorithm to External-restricted Three-body Problem with two Stars and an Extroplanet

旋转质心坐标系下的椭圆型外限制性系外行星三体问题的哈密顿方程含有坐标和动量的交叉项,并且显含时间变量,系统不再守恒,显式力梯度辛算法无法直接应用。对此,通过扩大相空间将非保守哈密顿系统变换为自治的哈密顿系统,并重新构造力梯度辛算法,实现力梯度辛算法在椭圆型外限制性三体问题中的应用。结果表明,构造的力梯度辛算法的精度优于非力梯度辛算法,并且优化后的力梯度辛算法的精度优于未优化的力梯度辛算法。此外,采用优化的力梯度算法,以及快速Lyapunov指数对椭圆型外限制性系外行星三体系统进行相空间扫描,获得了各参数对行星轨道动力学稳定性的影响。

The three-body system is not integrable,and can be chaotic in some circumstances.The description of chaotic motion depends on reliable numerical methods and chaos indicators.Symplectic algorithms can keep Hamiltonian phase flow and overcome the problem that traditional algorithms have secular drifts in energy errors.Because of this,they are the best integrators for studying the long-term dynamic evolution of the Hamiltonian system.The Hamiltonian of an elliptic external-restricted three-body problem with two stars and an extroplanet in the rotating centroid coordinate system contains the cross term of coordinates and momentum,and explicitly depends on time variable.The Hamiltonian system is no longer conserved,and the explicit force gradient symplectic algorithm cannot be directly applied.Extending the phase space to transform the non-conservative Hamiltonian system into a conservative Hamiltonian system,we can establish force gradient symplectic algorithms.The standard fourth-order symplectic algorithm,the optimized fourth-order symplectic algorithm,the optimized fourth-order force gradient symplectic algorithm and the optimized fourth-order force gradient symplectic algorithm are independently used to solve the elliptic external-restricted three-body problem.It is found that the accuracy of the force gradient symplectic algorithm constructed in this paper is better than that of the non-force gradient symplectic algorithm,and the accuracy of the optimized force gradient symplectic algorithm is better than that of the symplectic algorithms without optimization.The fourth-order optimized force gradient algorithm and fast Lyapunov index are used to scan the phase space of the elliptic external-restricted three-body system,and the effects of each parameter on the planetary orbit dynamic stability are obtained.