相对论性毕达哥拉斯三体问题
The relativistic Pythagorean three-body problem
摘要：
研究了相对论对不稳定三体系统的混沌性质和动力学结果的影响;勾股三体问题。为此，我们扩展的布鲁特斯N-体代码，包括后牛顿成对条款高达2.5阶，和一阶泰勒展开的爱因斯坦-因菲尔德-霍夫曼运动方程。我们的系统是相对论性的程度取决于总质量的标度（单位大小为1秒差距）。使用布鲁图斯收敛方法，我们测试的时间可逆性在保守制度，并证明，我们能够获得明确的解决方案的相对论性三体问题。它也被证实，一个成功的时间可逆性测试所需的最小数值精度与初始扰动的放大因子相关。当我们考虑通过引力波发射的耗散效应时，我们发现共振的持续时间和小扰动的指数增长量取决于质量标度。对于单位质量<= 10 MSun，系统的行为与牛顿的情况没有区别，共振总是以一个二元体和一个逃逸体结束。当质量增加到1 e7 MSun时，相对论逐渐变得更加突出，但大多数系统仍然溶解。第一次合并开始出现在质量约为1 e5 MSun的情况下，在1 e7 MSun和1 e9 MSun之间，所有系统都过早地结束合并。这些合并之前，由一个引力波驱动的螺旋。对于质量标度>= 1 e9 MSun，所有系统在第一次近距离接触时都会导致引力波合并。相对论性三体碰撞为解决最终的秒差距问题提供了一条有效的途径。在1 e7 MSun的特征质量尺度上合并的开始可能会在超大质量黑洞的质量函数中留下印记。
英文摘要：
We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the Brutus N-body code to include Post-Newtonian pairwise terms up to 2.5 order, and the first order Taylor expansion to the Einstein-Infeld-Hoffmann equations of motion. The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec). Using the Brutus method of convergence, we test for time-reversibility in the conservative regime, and demonstrate that we are able to obtain definitive solutions to the relativistic three-body problem. It is also confirmed that the minimal required numerical accuracy for a successful time-reversibility test correlates with the amplification factor of an initial perturbation. When we take into account dissipative effects through gravitational wave emission, we find that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling. For a unit mass <= 10 MSun, the system behavior is indistinguishable from the Newtonian case, and the resonance always ends in a binary and one escaping body. For a mass scaling up to 1e7 MSun, relativity gradually becomes more prominent, but the majority of the systems still dissolve. The first mergers start to appear for a mass of ~1e5 MSun, and between 1e7 MSun and 1e9 MSun all systems end prematurely in a merger. These mergers are preceded by a gravitational wave driven in-spiral. For a mass scaling >= 1e9 MSun, all systems result in a gravitational wave merger upon the first close encounter. Relativistic three-body encounters thus provide an efficient pathway for resolving the final parsec problem. The onset of mergers at the characteristic mass scale of 1e7 MSun potentially leaves an imprint in the mass function of supermassive black holes.
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