scholarly journals A Complete Family of Periodic Solutions of the Planar Three-Body Problem and their Stability

1977 ◽  
Vol 33 ◽  
pp. 159-159
Author(s):  
M. Hénon
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Three Body Problem ◽  
Complete Family ◽  
First Order ◽  
Hierarchical Arrangement ◽  
The Family ◽  
Three Body

AbstractWe give a complete description of a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with a rectilinear orbit, computed by Schubart in 1956. It ends in retrograde revolution, i.e., a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the “interplay” type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.

Networks of periodic orbits in the circular restricted three-body problem with first order post-Newtonian terms

Meccanica ◽  
2019 ◽  
Vol 54 (15) ◽  
pp. 2339-2365 ◽  
Author(s):  
Euaggelos E. Zotos ◽  
K. E. Papadakis ◽  
Md Sanam Suraj ◽  
Amit Mittal ◽  
Rajiv Aggarwal
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
First Order ◽  
Three Body

scholarly journals New Phenomena in the Spatial Isosceles Three-Body Problem

2015 ◽  
Vol 25 (09) ◽  
pp. 1550116 ◽  
Author(s):  
Duokui Yan ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Initial Condition ◽  
Collision Orbit ◽  
Three Body Problem ◽  
Straight Line ◽  
Three Body ◽  
New Phenomena ◽  
Two Bodies

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions.

scholarly journals The family of Quasi-satellite periodic orbits in the circular co-planar RTBP

2014 ◽  
Vol 9 (S310) ◽  
pp. 172-173
Author(s):  
Alexandre Pousse ◽  
Philippe Robutel ◽  
Alain Vienne
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Real Problem ◽  
Three Body Problem ◽  
The Family ◽  
Circular Case ◽  
Validity Limit ◽  
Stable Family ◽  
Three Body ◽  
Very High

AbstractIn the circular case of the coplanar Restricted Three-body Problem, we studied how the family of quasi-satellite (QS) periodic orbits allows to define an associated libration center. Using the averaged problem, we highlighted a validity limit of this one: for QS orbits with low eccentricities, the averaged problem does not correspond to the real problem. We do the same procedure to L3, L4 and L5 emerging periodic orbits families and remarked that for very high eccentricities ${\cal F}_{L_4}$ and ${\cal F}_{L_5}$ merge with ${\cal F}_{L_3}$ which bifurcates to a stable family.

scholarly journals Asymmetric Periodic Orbits in the Three-Body Problem and Their Stability

1983 ◽  
Vol 74 ◽  
pp. 249-256
Author(s):  
A. Tsouroplis ◽  
C.G. Zagouras
Keyword(s):  
Periodic Solutions ◽  
Numerical Integration ◽  
Periodic Orbits ◽  
Body Problem ◽  
Numerical Determination ◽  
Three Body Problem ◽  
Stability Parameters ◽  
Variational Equations ◽  
Three Body

AbstractAn algorithm for the numerical determination of asymmetric periodic solutions of the planar general three body problem is described. The elements of the “variational” matrix which are used in this algorithm are computed by numerical integration of the corresponding “variational” equations. These elements are also used in the study of the linear isoenergetic stability. A number of asymmetric periodic orbits are presented and their stability parameters are given.

GLOBAL BEHAVIOR OF THE CHARGED ISOSCELES THREE-BODY PROBLEM

1994 ◽  
Vol 04 (04) ◽  
pp. 865-884 ◽  
Author(s):  
PAU ATELA ◽  
ROBERT I. McLACHLAN
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Body Problem ◽  
Kepler Problem ◽  
Global Bifurcation ◽  
Three Body Problem ◽  
Anisotropic Kepler Problem ◽  
The One ◽  
Three Body ◽  
Limiting Value

We study the global bifurcation diagram of the two-parameter family of ODE’s that govern the charged isosceles three-body problem. (The classic isosceles three-body problem and the anisotropic Kepler problem (two bodies) are included in the same family.) There are two major sources of periodic orbits. On the one hand the “Kepler” orbit, a stable orbit exhibiting the generic bifurcations as the multiplier crosses rational values. This orbit turns out to be the continuation of the classical circular Kepler orbit. On the other extreme we have the collision-ejection orbit which exhibits an “infinite-furcation.” Up to a limiting value of the parameter we have finitely many periodic orbits (for each fixed numerator in the rotation number), passed this value there is a sudden birth of an infinite number of them. We find that these two bifurcations are remarkably connected forming the main “skeleton” of the global bifurcation diagram. We conjecture that this type of global connection must be present in related problems such as the classic isosceles three-body problem and the anisotropic Kepler problem.

Sign in / Sign up

Export Citation Format

Share Document