A search for collision orbits in the free-fall three-body problem I. Numerical procedure

Celestial Mech Dyn Astr

2007

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The dynamical structure of phase space of gravitational Newtonian three bodies which lie on a line (rectilinear three-body system) is studied. We take an initial value plane and classify the points on the plane according to the fate of the orbits starting from the points, using symbol sequences. The structure appearing on the initial value plane with this classification was well studied for the equal-mass case (Tanikawa and Mikkola 2000, Chaos 10, 649-657). In this paper, we follow and clarify the changes of this structure with the mass ratio of three particles.

Section: Fig 11mentioning

confidence: 76%

The rectilinear three-body problem using symbol sequence I. Role of triple collision

Saito

Celestial Mech Dyn Astr

2007

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“…[1][2][3][15][16][17] Conversely, any triangle is similar to one of the triangles formed by three mass points m 1 , m 2 , and m 3 .…”

Section: B Free-fallmentioning

confidence: 99%

Numerical experiments of the planar equal-mass three-body problem: The effects of rotation

Kuwabara

Chaos: An Interdisciplinary Journal of Nonlinear Science

2007

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The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k

“…A region formed with n-th-escape points will be called the n-th-escape region. Tanikawa et al (1995) found initial points T i , i ∈ N on C for a triple collision. Such triple collisions occur during the first triple-encounter, since m 2 approaches the center of gravity of m 1 and m 3 monotonically before a triple collision.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…As a model problem, we used the free-fall three-body problem, since we knew the structure of the phase space (Tanikawa et al 1995;Tanikawa, Umehara 1998). We integrated orbits up to the third triple-encounter using the respective criteria, saw the different results, and looked for the best criterion.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…There are various ways of discretization in the three-body problem. One counts the number of binary collisions (Tanikawa et al 1995;Zare, Chesley 1998;Tanikawa, Mikkola 2000) or counts the number of triple encounters (Boyd, McMillan 1992, 1993Tanikawa, Umehara 1998). Thus, there is an increasing demand to accurately define the state of a triple encounter.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improvement of the Triple-Encounter Criterion

Umehara¹,

Publications of the Astronomical Society of Japan

2001

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