Ruling out chaos in comparable mass compact binary systems with one body spinning
Ruling out chaos in comparable mass compact binary systems with one body spinning
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DOI：
10.1093/mnras/stv1485
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发表时间：
2015-09
影响因子：
4.8
通讯作者：
Xin Wu;Guo-Qing Huang
中科院分区：
文献类型：
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作者：
Xin Wu;Guo-Qing Huang
Levin（，Phys. Rev. D，74，124027）对只有一个质量相当的双星自旋和自旋效应仅限于首阶自旋-轨道耦合的系统的混沌行为给出了两个相反的主张。利用分形盆边界方法，研究了一类二阶后牛顿（2 PN）调和坐标拉格朗日运动方程的混沌现象。然而，在另一组2 PN Arnowitt-Deser-Misner（ADM）的哈密顿运动方程没有混沌被证实与参数的解决方案的帮助。当只有一个物体旋转时，保守的PN拉格朗日和哈密顿方法对可比质量双星的动力学有混沌吗？这仍然是一个悬而未决的问题。一篇关于正则共轭自旋变量的论文（Wu和Xie，，Phys. Rev. D，81，084045）直接表明，这些哈密顿方法是可积的和非混沌的，与PN阶和自旋效应无关。在这个意义上，我们需要回答的只是拉格朗日方法是否允许混沌的问题。正如Wu et al.（，Phys. Rev. D，91，024042），在ADM坐标中，在某一阶的这些拉格朗日方法中的任何一个通常具有从分析观点来看在无限阶或从数值观点来看在某一足够高的有限阶的分析数学等效哈密顿量。哈密顿量是完全正则的，并且在八维相空间中有四个总能量和总角动量的积分，因此它通常是可积的。我们用它来证明拉格朗日量中没有混沌。另一方面，我们使用快速李雅普诺夫指数的方法，重新审视了2 PN调和坐标拉格朗日动力学的领先阶自旋轨道耦合的一个身体自旋。研究发现，分形方法是不足以支持混沌不稳定的并合双星，即使辐射反应被关闭。总之，无论是PN保守拉格朗日公式还是PN保守哈密顿公式，在ADM/调和坐标系下的情况下，一个机构的自旋都不会是混沌的。有了这个结果，就有可能结束在相关文献中关于具有可比质量的具有一体旋转的双星的混沌行为的两种不同主张的争论。
Levin (, Phys. Rev. D, 74, 124027) has given two contrary claims on the chaotic behaviour of a system in which only one body of comparable mass binaries spins and spin effects are restricted to the leading order spin–orbit couplings. Chaos in one set of second post-Newtonian (2PN) harmonic coordinate Lagrangian equations of motion was allowed via the fractal basin boundary method. However, in another set of 2PN Arnowitt–Deser–Misner (ADM) Hamiltonian equations of motion no chaos was confirmed with the aid of parametric solutions. Is there chaos for conservative PN Lagrangian and Hamiltonian approaches to the dynamics of comparable mass binaries when only one object spins? This is still an open question. A paper on canonical, conjugate spin variables (Wu and Xie, , Phys. Rev. D, 81, 084045) has directly shown that these Hamiltonian approaches are integrable and non-chaotic regardless of PN orders and spin effects. In this sense, what we are required to answer is only the question of whether the Lagrangian approaches allow chaos. As recently confirmed by Wu et al. (, Phys. Rev. D, 91, 024042), in ADM coordinates, any one of these Lagrangian approaches at a certain order generally has an analytical mathematical equivalent Hamiltonian at an infinite order from an analytical point of view or at a certain high enough finite order from a numerical point of view. The Hamiltonian is completely canonical and has four integrals of the total energy and total angular momentum in an eight-dimensional phase space, and therefore it is typically integrable. We use this to show the absence of chaos in the Lagrangian. On the other hand, we use the method of fast Lyapunov exponents to revisit the 2PN harmonic coordinate Lagrangian dynamics with the leading-order spin–orbit coupling of one body spinning. It is found that the fractal method is not sufficient to support chaos in unstable merging binaries, even if the radiation reaction is turned off. In summary, neither the PN conservative Lagrangian formulations nor the PN conservative Hamiltonian formulations can be chaotic in ADM/harmonic coordinates for the case of one body spinning. With this result, it is possible to end the dispute in the related literature regarding the two different claims on the chaotic behaviour of comparable mass binaries with one body spinning.