Hill stability of the satellite in the elliptic restricted four-body problem

  • Chao Liu
  • Shengping GongEmail author
Original Article


We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant \(C\) determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant \(C_{\mathrm{ac}}\ ( {>} 0 ) \) is bigger than or equal to the critical energy constant \(C_{\mathrm{cr}}\), the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth–Moon system.


Four-body problem Binary subsystem Restricted Hill stability Asteroid capture 



This work was supported by the National Natural Science Foundation of China (Grant No. 11772167).


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Aerospace EngineeringTsinghua UniversityBeijingChina

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