Abstract
We present a planar four-body model, the Circular Planar (2+2)-Body Problem, for the motion of two asteroids (having small but positive masses) moving under the gravitational attraction of each other and under the gravitational attraction of two primaries (with masses much larger than the two smaller mass bodies) moving in uniform circular motion about their center of mass. We show the Circular Planar (2+2)-Body Problem has (at least) 6 relative equilibria and (at least) 10 one-parameter families of periodic orbits, two of which are of Hill-type. The existence of six relative equilibria and eight one-parameter families of periodic orbits is obtained by a reduction of the Circular Planar (2+2)-Body Problem in which the primaries have equal mass, the asteroids have equal mass, and the positions of the asteroids are symmetric with respect to the origin. The remaining two one-parameter families of periodic orbits, which are of comet-type, are obtained directly in the Circular Planar (2+2)-Body Problem.
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References
Alvarez-Ramírez, M., Llibre, J.: The symmetric central configuration of the \(4\)-body problem with masses \(m_1=m_2\ne m_3=m_4\). Appl. Math. Comput. 219, 5996–6001 (2013). https://doi.org/10.1016/j.amc.2012.12.036
Alvarez-Ramírez, M., Skea, J.E.F., Stuchi, T.J.: Nonlinear stability analysis in a equilateral restricted four-body problem. Astrophys. Space Sci. (2015). https://doi.org/10.1007/s10509-015-2333-4
Bakker, L.F., Murri, J., Simmons, S.: Collinear relative equilibria in a circular (2+2)-body problem. In preparation, 2023
Corbera, M., Llibre, J.: Central configurations of the \(4\)-body problem with masses \(m_1=m_2>m_3=m_4>0\) and small. Appl. Math. Comput. 246, 121–147 (2014). https://doi.org/10.1016/j.amc.2014.07.109
Llibre, J., Stoica, C.: Comet- and hill-type periodic orbits in restricted \((n+1)\)-body problems. J. Differ. Equ. 250, 1747–1766 (2011). https://doi.org/10.1016/j.jde.2010.08.005
Llibre, J., Paşca, D., Valls, C.: The circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. Celest. Mech. Dyn. Astron. (2021). https://doi.org/10.1007/s10569-021-10052-6
Long, Y., Sun, S.: Four-body central configurations with some equal masses. Arch. Rational Mech. Anal. 162, 25–44 (2002). https://doi.org/10.1007/s002050100183
Meyer, K.R., Offin, D.C.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, volume 90 of Applied Mathematical Sciences. Springer International Publishing, 3rd edition (2017)
Roy, A.E., Steves, B.A.: Some special restricted four-body problems-ii: from caledonia to copenhagen. Planet Space Sci. 46(11–12), 1475–1486 (1998)
Shoaib, M., Faye, I.: Collinear equilibrium solutions of four-body problems. J. Astrophys. Astr. 32, 411–423 (2011)
Simó, C.: Relative equilibrium solutions in the four body problem. Celest. Mech. 18, 165–184 (1978)
Szebehely, V.: The Theory of Orbits. Academic Press, New York (1967)
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The authors thank the anonymous peer-reviewers for their thoughtful feedback that helped improve the clarity and concision of the paper.
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Bakker, L.F., Freeman, N.J. Relative equilibria and periodic orbits in a Circular Planar (2+2)-Body Problem. Celest Mech Dyn Astron 135, 58 (2023). https://doi.org/10.1007/s10569-023-10173-0
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DOI: https://doi.org/10.1007/s10569-023-10173-0
Keywords
- Planar four-body problem
- Relative equilibria
- Periodic orbits
- Convex kite configuration
- Hill orbits
- Comet orbits