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Escape dynamics and fractal basins boundaries in the three-dimensional Earth-Moon system

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Abstract

The orbital dynamics of a spacecraft, or a comet, or an asteroid in the Earth-Moon system in a scattering region around the Moon using the three dimensional version of the circular restricted three-body problem is numerically investigated. The test particle can move in bounded orbits around the Moon or escape through the openings around the Lagrange points \(L_{1}\) and \(L_{2}\) or even collide with the surface of the Moon. We explore in detail the first four of the five possible Hill’s regions configurations depending on the value of the Jacobi constant which is of course related with the total orbital energy. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits in several two-dimensional types of planes and distinguishing between four types of motion: (i) ordered bounded, (ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we locate the different basins and we relate them with the corresponding spatial distributions of the escape and collision times. Our outcomes reveal the high complexity of this planetary system. Furthermore, the numerical analysis suggests a strong dependence of the properties of the considered basins with both the total orbital energy and the initial value of the \(z\) coordinate, with a remarkable presence of fractal basin boundaries along all the regimes. Our results are compared with earlier ones regarding the planar version of the Earth-Moon system.

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Notes

  1. For the value of the mass ratio \(\mu\) we adopted the same precision (number of significant decimal digits) as in Paper I.

  2. In Paper I the authors classified two dimensional orbits with \(\dot{x_{0}} = 0\) and \(\dot {y_{0}} > 0\). In our work we expand into three dimensions this choice of initial conditions thus considering orbits with \(\dot{x_{0}} = \dot{z_{0}} = 0\) and \(\dot{y_{0}} > 0\). The case with non-zero values in \(x\) and \(z\) components of the velocity of the test particle is another choice which however is not considered here, obviously for saving space.

  3. The set of initial conditions of orbits which lead to a certain final state (escape, collision or bounded motion) is defined as a basin.

  4. Obviously, if we numerically integrate initial conditions inside this region we will see that they lead to immediate collision to the Moon.

  5. When we state that an area is fractal we simply mean that it has a fractal-like geometry without conducting any specific calculations as in Aguirre et al. (2009).

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Acknowledgements

I would like to express my warmest thanks to the anonymous referee for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

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  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Zotos, E.E. Escape dynamics and fractal basins boundaries in the three-dimensional Earth-Moon system. Astrophys Space Sci 361, 94 (2016). https://doi.org/10.1007/s10509-016-2683-6

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