Closely-packed multi-planet systems are known to experience dynamical instability if the spacings between the planets are too small. Such instability can be tempered by the frictional forces acting on the planets from gaseous discs. A similar situation applies to stellar-mass black holes embedded in AGN discs around supermassive black holes. In this paper, we use $N$-body integrations to evaluate how the frictional damping of orbital eccentricity affects the growth of dynamical instability for a wide range of planetary spacing and planet-to-star mass ratios. We find that the stability of a system depends on the damping timescale $\tau$ relative to the zero-friction instability growth timescale $t_{\rm inst}$. In a two-planet system, the frictional damping can stabilise the dynamical evolution if $t_{\rm inst}\gtrsim\tau$. With three planets, $t_{\rm inst} \gtrsim 10\tau - 100\tau$ is needed for stabilisation. When the separations between the planetary orbits are sufficiently small, $t_{\rm inst}$ can be less than the synodic period between the planets, which makes frictional stabilisation unlikely to occur. As the orbital spacing increases, the instability timescale tends to grow exponentially on average, but it can vary by a few orders of magnitude depending on the initial orbital phases of the planets. In general, the stable region (at large orbital spacings) and unstable region (at small orbital spacings) are separated by a transition zone, in which the (in)stability of the system is not guaranteed. We also devise a linear map to analyse the dynamical instability of the "planet + test-mass" system, and we find qualitatively similar results to the $N$-body simulations.

中文翻译：

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具有气体摩擦的多轨道系统的动态不稳定性

众所周知，如果行星之间的间距太小，密集的多行星系统会经历动态不稳定。这种不稳定性可以通过气态盘作用在行星上的摩擦力来缓和。类似的情况也适用于嵌入在超大质量黑洞周围的活动星系核盘中的恒星质量黑洞。在本文中，我们使用 $N$-body 积分来评估轨道偏心率的摩擦阻尼如何影响各种行星间距和行星与恒星质量比的动态不稳定性的增长。我们发现系统的稳定性取决于阻尼时间尺度 $\tau$ 相对于零摩擦不稳定性增长时间尺度 $t_{\rm inst}$。在双行星系统中，如果$t_{\rm inst}\gtrsim\tau$，摩擦阻尼可以稳定动力学演化。拥有三颗行星，$t_{\rm inst} \gtrsim 10\tau - 100\tau$ 是稳定所必需的。当行星轨道之间的间隔足够小时，$t_{\rm inst}$ 可以小于行星之间的会合周期，这使得摩擦稳定不太可能发生。随着轨道间距的增加，不稳定性时间尺度趋于平均呈指数增长，但根据行星的初始轨道相位，它可以变化几个数量级。一般来说，稳定区（大轨道间距）和不稳定区（小轨道间距）被一个过渡区隔开，其中系统的（非）稳定性得不到保证。我们还设计了一个线性图来分析“行星+测试质量”系统的动态不稳定性，我们发现与$N$-body 模拟在性质上相似的结果。