http://swrc.ontoware.org/ontology#Thesis
The Free-Fall Three-Body Problem : Escape and Collision
en
梅原 広明
ウメハラ ヒロアキ
UMEHARA Hiroaki
総研大甲第316号
The escape phenomena in the three-body problem with zero initial velocities and equal masses are studied both numerically and analytically. In particular, the effects of triple and binary collisions are considered in detail. Here, escape means that two particles form a binary and the third particle recedes from the binary to infinity. Collision is defined as the event when the distance between particles vanishes. First, escape orbits are searched by a numerical survey of the initial-value space and compared with collision orbits obtained by Tanikawa et al. (Cele. Mech. Dyna. Astr., 62 (1995) 335-362). Most escape phenomena occur after the triple encounter which is of the slingshot type. A particle passes through between the other particles which are receding from each other. It is found that escape orbits due to slingshot distribute around a particular family of binary-collision orbits which maintains nearly isosceles configuration. The configuration and the velocity vectors are almost symmetric. Moreover, if orbits approach sufficiently close to triple collision, all escape orbits distribute around the binary-collision orbits. Furthermore, orbits without escape during the first triple encounter are also found sufficiently close to the triple-collision orbit. Therefore, it becomes clear that explaining the distribution of escape orbits only by triple-collision orbits is impossible. The particular family of binary collisions has a central role in escape phenomena as well as triple collision does. Discovery of escape orbits due to exchange encounter is also one of the results. Two close approaches between two particles successively occur. The dynamical features of the slingshot and the exchange are compared with each other. Escape probabilities and increments of binding energies are evaluated statistically for the respective encounter-types. It is shown that some of slingshot encounters result in more energetic evolution than all of exchange encounters. So the restrictive and the favorable conditions of slingshot configurations leading to escape are searched for. Using the slingshot conditions, it is answered why slingshot-escape orbits distribute around the particular binary-collision orbits showing nearly symmetrical motion. Finally, it is proved analytically that both escape and non-escape orbits after the first triple encounter exist arbitrarily close to the particular triple-collision orbit. The homothetic-equilateral triple-collision orbit is considered. This orbit maintains the equilateral-triangle configuration. It is proved that in the initial-value space escape orbits distribute around three kinds of isosceles orbits where different particles escape and non-escape orbits are distributed in between. In order to show this, it is proved that the homothetic-equilateral orbit is isolated from other triple-collision orbits so far as the collision during the first triple encounter concerns. Moreover, the escape criterion is formulated in the planar isosceles problem and translated into the words of regularizing variables. The results explain the orbital structure numerically obtained in the beginning of the present thesis. With the aid of numerical integrations, it is shown that the distribution of escape orbits around another triple-collision orbit are topologically similar to the one around the homothetic-equilateral orbit. Here, it is found that the isosceles motion has an important role in determining the dynamical evolution.
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