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Exchange orbits: an interesting case of co-orbital motion

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Abstract

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.

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Notes

  1. Note that for the semi-major axis of P1 we have taken \(1~\text{ AU }-\delta a_{P1}\) and for P2 1 AU \(+\) \(\delta a_{P1}.\)

  2. \(\mu =M_{Planet}/(M_{Planet}+M_{Star})\), where \(M_{Planet} = M_{Planet1} = M_{Planet2}.\)

  3. In this study we were interested whether two terrestrial planets can perform a stable xch-e motion when a perturbing body is present.

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Acknowledgments

BF and RS wants to acknowledge the support by the Austrian FWF project P23810-N16. RD wants to acknowledge the support by the Austrian FWF NFN project S 11603-N16.

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  1. Türkenschanzstarsse 17, 1180 , Vienna, Austria

    Barbara Funk, Rudolf Dvorak & Richard Schwarz

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  1. Barbara Funk
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  2. Rudolf Dvorak
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  3. Richard Schwarz
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Correspondence to Barbara Funk.

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Funk, B., Dvorak, R. & Schwarz, R. Exchange orbits: an interesting case of co-orbital motion. Celest Mech Dyn Astr 117, 41–58 (2013). https://doi.org/10.1007/s10569-013-9497-4

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  • DOI: https://doi.org/10.1007/s10569-013-9497-4

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