共振链中不稳定性的开始
The onset of instability in resonant chains
摘要：
有证据表明，一旦气体的耗散作用从系外行星中移除，大多数$k$：$k-1$型平均运动共振链就会变得不稳定，特别是对于大的$N$（行星数量）和$k$（表明链有多紧凑）。我们提出了一种新的动力学机制，可以解释这些不稳定性的起源，从而在系外行星样本的共振系统的缺乏。它依赖于一小部分会合频率$2 π(1/P_1-1/P_2)$和平均运动共振中的天平动频率之间的次级共振的出现。这些次级共振激发平均运动共振的振动振幅，从而导致不稳定性。我们详细介绍了这些二次共振的出现进行了明确的微扰计划，第二阶的行星质量和孤立的谐波与他们相关的条款。以三颗行星在3：2 - 3：2平均运动共振中的情况为例，得到了其中一个共振的简单而一般的解析模型，它描述了一个这样的次级共振激活的初始阶段。文中还简要介绍了激发系统的动力学特性。这个方案展示了人们如何能够获得对这些共振的出现以及它们触发的动态的分析洞察。最后，这个动力学机制的推广得到任意$N$和$k$。这导致以前的数值实验的共振链的稳定性的解释，显示为什么允许稳定的临界行星质量随着增加$N$和$k$减少。
英文摘要：
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 $k$.
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