In this work, we investigate configuration formation of two inner terrestrial planets near mean motion resonance (MMRs) induced by the perturbation of a distant gas-giant for the Kepler-68 system, by conducting thousands of numerical simulations. The results show that the formation of terrestrial planets is relevant to the speed of Type I migration, the mass of planets, and the existence of giant planet. The mass and eccentricity of the giant planet may play a crucial role in shaping the final configuration of the system. The inner planet pair can be trapped in 5:3 or 7:4 MMRs if the giant planet revolves the central star with an eccentric orbit, which is similar to the observed configuration of Kepler-68. Moreover, we find that the eccentricity of the middle planet can be excited to roughly 0.2 if the giant planet is more massive than 5 $M_J$,otherwise the terrestrial planets are inclined to remain near-circular orbits. Our study may provide a likely formation scenario for the planetary systems that harbor several terrestrial planets near MMRs inside and one gas-giant exterior to them.

中文翻译：

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巨行星引起的类地行星对近平均运动共振：在Kepler-68系统中的应用

在这项工作中，我们通过进行数千次数值模拟，研究了由 Kepler-68 系统的遥远气体巨星的扰动引起的两颗靠近平均运动共振 (MMR) 的内部类地行星的构型形成。结果表明，类地行星的形成与I型迁移的速度、行星的质量以及巨行星的存在有关。这颗巨行星的质量和离心率可能在塑造系统的最终配置方面发挥着至关重要的作用。如果这颗巨行星以偏心轨道围绕中心恒星旋转，这对内行星可能会被困在 5:3 或 7:4 的 MMR 中，这与观测到的 Kepler-68 配置相似。此外，我们发现如果这颗巨行星的质量大于 5 $M_J$，中间行星的离心率可以被激发到大约 0.2，否则类地行星倾向于保持接近圆形的轨道。我们的研究可能为行星系统提供一个可能的形成场景，这些行星系统在 MMR 附近有几颗类地行星，在它们的外部有一颗气态行星。