We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. A reasonable situation, where two planets with known orbits are more massive than the third one, is discussed. The boundaries of possible motions are controlled by the parameter c2E . If the actual value of c2E is less than or equal to a critical value (cE)cr , the regions of possible motions are bounded and therefore the system is stable. The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems, and then the results are checked using N-body integrator.

中文翻译：

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共面行星四体系统的稳定性

我们考虑共面行星四体问题，其中三颗行星围绕一颗大恒星运行，但它们的轨道没有交叉。如果没有交换或轨道交叉，系统是稳定的。从Sundman不等式出发，推导出运动学边界方程。讨论了一种合理的情况，即已知轨道的两颗行星的质量比第三颗大。可能运动的边界由参数 c2E 控制。如果 c2E 的实际值小于或等于临界值 (cE)cr ，则可能运动的区域是有界的，因此系统是稳定的。在特殊情况下获得的标准应用于太阳系和目前已知的太阳系外行星系统，然后使用 N 体积分器检查结果。