The stability of vertical motion in the N-body circular Sitnikov problem

T. Bountis, K. E. Papadakis

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N - 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z 0 values, where z 0 lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, "islands" of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N - 1 primary masses.

Original languageEnglish
Pages (from-to)205-225
Number of pages21
JournalCelestial Mechanics and Dynamical Astronomy
Volume104
Issue number1-2
DOIs
Publication statusPublished - Jun 2009
Externally publishedYes

Fingerprint

vertical motion
Vertical
intervals
Orbits
Interval
Motion
orbits
N-body Problem
many body problem
infinity
Periodic Orbits
Orbit
Infinity
Ring
bifurcation
rings
Equations of motion
escape
Equations of Motion
Periodic Solution

Keywords

  • 3-dimensional periodic orbits
  • Escape regions
  • N-body problem
  • Ordered motion
  • Sitnikov motions
  • Stability
  • Uniform rotating ring

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science

Cite this

The stability of vertical motion in the N-body circular Sitnikov problem. / Bountis, T.; Papadakis, K. E.

In: Celestial Mechanics and Dynamical Astronomy, Vol. 104, No. 1-2, 06.2009, p. 205-225.

Research output: Contribution to journalArticle

Bountis, T. ; Papadakis, K. E. / The stability of vertical motion in the N-body circular Sitnikov problem. In: Celestial Mechanics and Dynamical Astronomy. 2009 ; Vol. 104, No. 1-2. pp. 205-225.
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