Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets
Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets
复制标题
DOI：
10.1086/338504
复制
发表时间：
2001-08
期刊：
影响因子：
--
通讯作者：
M. Lee;S. Peale
中科院分区：
文献类型：
--
作者：
M. Lee;S. Peale
Marcy和他的同事们发现两颗行星在围绕gj876恒星的2:1轨道共振中，这一发现得到了劳克林和钱伯斯的数据的动力学拟合的补充，劳克林和钱伯斯的数据将这两颗行星放置在共面轨道上，在2:1平均运动可通约性的三个共振中深入。通过动态拟合选择这种几乎奇异的状态意味着谐振几乎肯定是真实的，并且谐振变量的振动幅度很小，无限稳定。gj876系统揭示了2:1共振的几个不寻常的性质。最低阶平均运动共振变量和长期共振变量，θ1 = λ1 - 2λ2 + ϖ1, θ2 = λ1 - 2λ2 + ϖ2，和θ3 = ϖ1 - ϖ2的摆动约为0°（其中λ1,2是内外行星的平均经度，ϖ1,2是periperid的经度）与我们熟悉的木卫二对的几何结构不同，其中θ2和θ3摆动约为180°。通过考虑θ1和θ2稳定同步振动时1 = 2的条件，我们证明了gj876的几何形状是由大的轨道偏心率ei造成的，而在Io-Europa系统中，极小的轨道偏心率导致后者的几何形状。令人惊讶的是，具有θ1， θ2和θ3全部振动的gj876结构在e1达到0.86时保持稳定，并且在θ1的振幅接近45°时保持稳定，这进一步支持了现有系统的不确定稳定性。任何驱动原本广泛分离的轨道相互靠近的过程都可能导致捕获到2:1可通约性的观测共振。我们发现gj876系统外行星的强迫向内迁移导致了一定程度的捕获，当初始e1 > 0.06和e2 > 0.03时，迁移速率为| > 2/a2| > 3 × 10-2(a2/AU)-3/2 year -1。较大的偏心率可能导致在达到2:1可通约性之前捕获到更高阶的共振。这些行星的质量足以打开围绕年轻的gj876的星云盘的间隙，并清除它们之间的盘状物质，由此产生的行星-星云相互作用通常迫使外行星在盘粘性时间尺度上向内迁移，其倒数比|2/a2|的上界小3个数量级。如果没有偏心阻尼，则偏心增长迅速，在共振内继续迁移，在半长轴ai进一步降低仅7%后，ei超过观测值。当偏心率阻尼i/ei = -K|i/ai|时，偏心率达到平衡值，在共振内任意长时间的迁移中保持恒定。当K≈100时，仅存在外行星的迁移和阻尼时，平衡偏心率接近观测到的偏心率，而当K≈10时，也存在内行星的迁移和阻尼时，平衡偏心率接近观测到的偏心率。只要i/ei = -K|i/ai|，该结果与迁移速率i的大小或函数形式无关。尽管现有的对行星-星云相互作用影响的分析估计与这种形式的偏心阻尼在某些圆盘参数值上是一致的，但尚不清楚这种相互作用是否能产生获得观测到的偏心所需的大K值。恒星或行星内部潮汐耗散的偏心阻尼是完全可以忽略不计的，所以观测到的gj876系统的动力学特性可能需要一个不太可能的共振捕获时间的微调，以接近星云生命周期的结束。
The discovery by Marcy and coworkers of two planets in 2 : 1 orbital resonance about the star GJ 876 has been supplemented by a dynamical fit to the data by Laughlin & Chambers, which places the planets in coplanar orbits deep in three resonances at the 2 : 1 mean-motion commensurability. The selection of this almost singular state by the dynamical fit means that the resonances are almost certainly real, and with the small amplitudes of libration of the resonance variables, indefinitely stable. Several unusual properties of the 2 : 1 resonances are revealed by the GJ 876 system. The libration of both lowest order mean-motion resonance variables and the secular resonance variable, θ1 = λ1 - 2λ2 + ϖ1, θ2 = λ1 - 2λ2 + ϖ2, and θ3 = ϖ1 - ϖ2, about 0° (where λ1,2 are the mean longitudes of the inner and outer planet and ϖ1,2 are the longitudes of periapse) differs from the familiar geometry of the Io-Europa pair, where θ2 and θ3 librate about 180°. By considering the condition that 1 = 2 for stable simultaneous librations of θ1 and θ2, we show that the GJ 876 geometry results from the large orbital eccentricities ei, whereas the very small eccentricities in the Io-Europa system lead to the latter's geometry. Surprisingly, the GJ 876 configuration, with θ1, θ2, and θ3 all librating, remains stable for e1 up to 0.86 and for amplitude of libration of θ1 approaching 45° with the current eccentricities—further supporting the indefinite stability of the existing system. Any process that drives originally widely separated orbits toward each other could result in capture into the observed resonances at the 2 : 1 commensurability. We find that forced inward migration of the outer planet of the GJ 876 system results in certain capture into the observed resonances if initially e1 ≲ 0.06 and e2 ≲ 0.03 and the migration rate |2/a2| ≲ 3 × 10-2(a2/AU)-3/2 yr-1. Larger eccentricities lead to likely capture into higher order resonances before the 2 : 1 commensurability is reached. The planets are sufficiently massive to open gaps in the nebular disk surrounding the young GJ 876 and to clear the disk material between them, and the resulting planet-nebular interaction typically forces the outer planet to migrate inward on the disk viscous timescale, whose inverse is about 3 orders of magnitude less than the above upper bound on |2/a2| for certain capture. If there is no eccentricity damping, eccentricity growth is rapid with continued migration within the resonance, with ei exceeding the observed values after a further reduction in the semimajor axes ai of only 7%. With eccentricity damping i/ei = -K|i/ai|, the eccentricities reach equilibrium values that remain constant for arbitrarily long migration within the resonances. The equilibrium eccentricities are close to the observed eccentricities for K ≈ 100 if there is migration and damping of the outer planet only, but for K ≈ 10 if there is also migration and damping of the inner planet. This result is independent of the magnitude or functional form of the migration rate i as long as i/ei = -K|i/ai|. Although existing analytic estimates of the effects of planet-nebula interaction are consistent with this form of eccentricity damping for certain disk parameter values, it is as yet unclear that such interaction can produce the large value of K required to obtain the observed eccentricities. The alternative eccentricity damping by tidal dissipation within the star or the planets is completely negligible, so the observed dynamical properties of the GJ 876 system may require an unlikely fine-tuning of the time of resonance capture to be near the end of the nebula lifetime.