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08-10-2016

Investigating the planar circular restricted three-body problem with strong gravitational field

Author: Euaggelos E. Zotos

Published in: Meccanica | Issue 9/2017

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Abstract

The case of the planar circular restricted three-body problem where one of the two primaries has a stronger gravitational field with respect to the classical Newtonian field is investigated. We consider the case where two primaries have the same mass, so as the the only difference between them to be the strength of the gravitational field which is controlled by the power p of the potential. A thorough numerical analysis takes place in several types of two dimensional planes in which we classify initial conditions of orbits into three main categories: (1) bounded, (2) escaping and (3) collision. Our results reveal that the power of the gravitational potential has a huge impact on the nature of orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collision basins and we managed to correlate them with the corresponding escape and collision time of the orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.

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In the “Appendix” we shall demonstrate that Keplerian circular orbits are also possible in the case where we have gravity stronger than the classical Newtonian one.

The most safe and efficient way to determine if an orbit escapes or not is the value of the total orbital energy of the particle measured by an observer in the inertial frame of reference. In particular, if the total orbital energy in the inertial frame is negative, the test particle might return back to the scattering region. On the contrary, if the total orbital energy becomes positive the test particle escapes, beyond any doubt, and it will never come back [6]. Our previous numerical experience (e.g., [53‐55]) strongly suggests that the total orbital energy of the test-particle in the inertial frame becomes positive much sooner than it takes for the massless particle to cross the disk with radius \(R_d = 10\). Thus we may claim that our escape criterion used in the previous series of papers, and also in the present one, is both correct and safe. In the following Section we will present numerical evidence proving the validity of our escape criterion.

We choose the \({\dot{\phi }} < 0\) instead of the \({\dot{\phi }} > 0\) part simply because in [53] we seen that it contains more interesting orbital content.

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 [2].

An infinite number of regions of (stable) quasi-periodic (or small scale chaotic) motion is expected from classical chaos theory.

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Title: Investigating the planar circular restricted three-body problem with strong gravitational field
Author: Euaggelos E. Zotos
Publication date: 08-10-2016
Publisher: Springer Netherlands
Published in: Meccanica / Issue 9/2017
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-016-0548-2

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