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On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

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Abstract

In the present paper, using the first-order approximation of the \(n\)-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem in the Solar system. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass-distance and velocity-speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to good numerical accuracy in the conservation of the Jacobi constant and almost allow for an equivalence between the Lagrangian and Hamiltonian approaches at the same post-Newtonian order. Accordingly, the dynamics of the system is analyzed in terms of the Poincaré sections method and Lyapunov exponents, finding that for specific values of the Jacobi constant the dynamics can be either chaotic or regular. Our results suggest that the chaoticity of the post-Newtonian system is slightly increased in comparison with its Newtonian counterpart.

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Notes

  1. In the Newtonian limit \(1/c^{2}\rightarrow 0\), \(J=\omega L-E\), with \(E\) the total energy and \(L\) the angular momentum.

  2. For the case \(c=1\) in Huang and Wu (2014), the Jacobi constant still has higher accuracy when \(a=1\) and the velocity at any time is smaller than \(10^{-4}\), or when the distance \(a\) takes a larger value (see Su et al. 2016 and Chen and Wu 2016).

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Acknowledgements

We thank the anonymous referee for constructive criticisms and suggestions that helped us improve this paper. FLD acknowledges financial support from Universidad de los Llanos, under Grants Commission: Postdoctoral Fellowship Scheme. FDLC and GAG gratefully acknowledges the financial support provided by VIE-UIS, under grants numbers 1822, 1785 and 1838, and COLCIENCIAS, Colombia, under Grant No. 8840.

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Authors and Affiliations

  1. Facultad de Ciencias Humanas y de la Educación, Universidad de los Llanos, Villavicencio, Colombia

    F. L. Dubeibe

  2. Grupo de Investigación en Relatividad y Gravitación, Escuela de Física, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia

    F. L. Dubeibe, F. D. Lora-Clavijo & Guillermo A. González

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  1. F. L. Dubeibe
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  2. F. D. Lora-Clavijo
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  3. Guillermo A. González
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Correspondence to F. L. Dubeibe.

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Dubeibe, F.L., Lora-Clavijo, F.D. & González, G.A. On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem. Astrophys Space Sci 362, 97 (2017). https://doi.org/10.1007/s10509-017-3076-1

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