The rectilinear three-body problem

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.Keywords Three-body problem · Rectilinear three-body problem · Triple approaches · Schubart periodic orbit · Escapes · Ejections A. I. Martynova State Forest Technical Academy, Institutskij per. 5, St. Petersburg 194021, Russia 123 94 V. V. Orlov et al.

Power-law tails in triple system decay statistics

“…It is also known that the "figure-eight" orbit becomes unstable when relative differenses in masses reach approximately 10 −2 [11], so this very interesting orbit is hardly important for any real astrophysical problem. The Schubart orbit [12] remaines stable in a much broader interval of masses [13] On the other hand, famous triangle Lagrange points become stable in the situation of suffitiently different masses, and we can imagine that set of stable orbits have rather nontrivial dependence on mass differences.…”

Non-schubart periodic orbits in the rectilinear three-body problem

“…This problem, called the rectilinear three-body problem, has been extensively studied (Mikkola and Hietarinta 1989;Hietarinta and Mikkola 1993;Tanikawa and Mikkola 2000a;Tanikawa and Mikkola 2000b) and recent progress is reviewed in Orlov et al (2009). The present paper is the third of our works on this system (Saito and Tanikawa 2007;Saito and Tanikawa 2009) referred to as Papers I and II).…”

The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

“…It should be noted that there is a narrow escape region between any two neighbouring strata in an arch. According to numerical results of Orlov et al (2008), the interplay time is almost the same inside each of these escape regions, whereas it is longer as an escape region is located nearer to the Schubart region.…”