论文标题
共振连锁的不稳定性发作
The onset of instability in resonant chains
论文作者
论文摘要
有证据表明，一旦从系统中删除了来自天然气的耗散动作，尤其是对于$ n $（行星的数量）和$ k $（表明链条的紧凑），大多数$ k $ type $ k $：$ k-1 $的平均运动谐振将变得不稳定。我们提出了一种新型的动力学机制，可以解释这些不稳定性的起源，从而解释系群样品中共鸣系统的缺乏。它依赖于副频率$2π（1/p_1-1/p_2）$的一小部分二次共振与平均运动共振中的库频率之间的出现。这些次级共振激发了平均运动共振的库振幅，从而导致不稳定性。我们通过执行明确的扰动方案来详细介绍这些次要共鸣的出现，以在行星质量中二阶并隔离与之相关的谐波项。在3：2-3：2中的三个行星的情况下，平均运动共振作为一个例子，获得了其中一个共振之一的简单但一般的分析模型，描述了一个次要共振的激活的初始阶段。还简要描述了激发系统的动力学。该方案显示了如何获得对这些共振的出现以及它们触发的动力学的分析见解。最后，对于任意$ n $和$ k $，获得了这种动态机制的概括。这导致了对以前的数值实验的解释，该实验对谐振链的稳定性进行了解释，这表明了为什么随着$ n $和$ k $的增加，关键的行星质量允许稳定性降低。
There is evidence that most chains of mean motion resonances of type $k$:$k-1$ among exoplanets become unstable once the dissipative action from the gas is removed from the system, particularly for large $N$ (the number of planets) and $k$ (indicating how compact the chain is). We present a novel dynamical mechanism that can explain the origin of these instabilities and thus the dearth of resonant systems in the exoplanet sample. It relies on the emergence of secondary resonances between a fraction of the synodic frequency $2 π(1/P_1-1/P_2)$ and the libration frequencies in the mean motion resonance. These secondary resonances excite the amplitudes of libration of the mean motion resonances thus leading to an instability. We detail the emergence of these secondary resonances by carrying out an explicit perturbative scheme to second order in the planetary masses and isolating the harmonic terms that are associated with them. Focusing on the case of three planets in the 3:2 -- 3:2 mean motion resonance as an example, a simple but general analytical model of one of these resonances is obtained which describes the initial phase of the activation of one such secondary resonance. The dynamics of the excited system is also briefly described. This scheme shows how one can obtain analytical insight into the emergence of these resonances, and into the dynamics that they trigger. Finally, a generalisation of this dynamical mechanism is obtained for arbitrary $N$ and $k$. This leads to an explanation of previous numerical experiments on the stability of resonant chains, showing why the critical planetary mass allowed for stability decreases with increasing $N$ and $k$.
