The rectilinear three-body problem |
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Authors: | Victor Vladimirovich Orlov Anna V Petrova Kiyotaka Tanikawa Masaya M Saito Alija I Martynova |
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Institution: | (1) Sobolev Astronomical Institute, St. Petersburg State University, Universitetskij pr. 28, Staryj Peterhof , St. Petersburg, 198504, Russia;(2) National Astronomical Observatory, Mitaka, Tokyo 181, Japan;(3) Department of Astronomical Science, SOKENDAI, Mitaka, Tokyo, Japan;(4) State Forest Technical Academy, Institutskij per. 5, St. Petersburg, 194021, Russia |
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Abstract: | The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that
covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities,
escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our
numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios
is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in
the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their
dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple
approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches
or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds
to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this
periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics,
we study the fine structure of the region of initial conditions, in particular the chaotic scattering region. |
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Keywords: | Three-body problem Rectilinear three-body problem Triple approaches Schubart periodic orbit Escapes Ejections |
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