Numerical exploration of the restricted three-body problem

1966 ◽  
Vol 25 ◽  
pp. 157-169 ◽  
Author(s):  
M. Hénon
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
General Picture ◽  
Numerical Exploration ◽  
The One ◽  
Three Body ◽  
Area Preserving

A number of orbits have been computed in the plane restricted three-body problem; the two main bodies have the same mass and move on a circular orbit. By consideration of the successive intersections of the orbit with thexaxis, the problem can be reduced to the study of a plane area-preserving mapping. A second integral, distinct from Jacobi's integral, seems to exist inside given ranges of initial conditions, but not outside. The general picture is quite similar to the one found in the problem of the third integral of galactic motion. Extension of this work to other values of the mass ratio is under way.

scholarly journals Progression of bifurcated family f type periodic orbits in the circular restricted three-body problem

2017 ◽  
Vol 5 (2) ◽  
pp. 69
Author(s):  
Nishanth Pushparaj ◽  
Ram Krishan Sharma
Keyword(s):  
Solar Radiation ◽  
Periodic Orbits ◽  
Radiation Pressure ◽  
Body Problem ◽  
Solar Radiation Pressure ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Semi Major Axis ◽  
Three Body

Progression of f-type family of periodic orbits, their nature, stability and location nearer the smaller primary for different mass ratios in the framework of circular restricted three-body problem is studied using Poincaré surfaces of section. The orbits around the smaller primary are found to decrease in size with increase in Jacobian Constant C, and move very close towards the smaller primary. The orbit bifurcates into two orbits with the increase in C to 4.2. The two orbits that appear for this value of C belong to two adjacent separate families: one as direct orbit belonging to family g of periodic orbits and other one as retrograde orbit belonging to family f of periodic orbits. This bifurcation is interesting. These orbits increase in size with increase in mass ratio. The elliptic orbits found within the mass ratio 0 < µ ≤ 0.1 have eccentricity less than 0.2 and the orbits found above the mass ratio µ > 0.1 are elliptical orbits with eccentricity above 0.2. Deviations in the parameters: eccentricity, semi-major axis and time period of these orbits with solar radiation pressure q are computed in the frame work of photogravitational restricted Three-body problem in addition to the restricted three-body problem. These parameters are found to decrease with increase in the solar radiation pressure.

scholarly journals Retrograde Satellites in the Circular Plane Restricted Three-Body Problem

1974 ◽  
Vol 62 ◽  
pp. 129-129
Author(s):  
D. Benest
Keyword(s):  
Body Problem ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Three Body ◽  
Circular Plane

Characteristics and stability of simple-periodic retrograde satellites of the lighter body are presented for Hill's case and for all values of the mass ratio m2/(m1+m2) between 0 and 0.5.

scholarly journals Robe's Restricted Three-Body Problem with Variable Masses and Perturbing Forces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jagadish Singh ◽  
Oni Leke
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Second Primary ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Small Perturbations ◽  
Triangular Points ◽  
Numerical Exploration ◽  
Three Body ◽  
Infinitesimal Mass

The linear stability of equilibrium points of a test particle of infinitesimal mass in the framework of Robe's circular restricted three-body problem, as in Hallan and Rana, together with effect of variation in masses of the primaries with time according to the combined Meshcherskii law, is investigated. It is seen that, due to a small perturbation in the centrifugal force and an arbitrary constant of a particular integral of the Gylden-Meshcherskii problem, every point on the line joining the centers of the primaries is an equilibrium point provided they lie within the shell. Further, a number of pairs of equilibrium points lying on the -plane and forming triangles with the centers of the shell and the second primary exist, for some values of . The points collinear with the center of the shell are found to be stable under some conditions and the range of stability depends on the small perturbations and , while the triangular points are unstable. Illustrative numerical exploration is given to indicate significant improvement of the problem in Hallan and Rana.

Perturbed Location of L1 point in the photogravitational relativistic restricted three-body problem (R3BP) with oblate primaries

2015 ◽  
Vol 93 (3) ◽  
pp. 300-311 ◽  
Author(s):  
S.E. Abd El-Bar ◽  
F.A. Abd El-Salam ◽  
M. Rassem
Keyword(s):  
Body Problem ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Series Form ◽  
Analytical Result ◽  
Three Body

The restricted three-body problem is studied in the post-Newtonian framework. The primaries are assumed oblate radiant sources. The perturbed location of the L1 point is computed, and a series form of the location of this point is obtained as a new analytical result. To introduce a semianalytical view, a Mathematica 9 program is constructed so as to draw the location of L1 versus the mass ratio μ ∈ (0, 0.5) taking into account one or more of the considered perturbations. All the obtained illustrations are analyzed.

scholarly journals Libration of Retrograde Satellite Orbits in the Circular Plane Restricted Three-Body Problem

1979 ◽  
Vol 81 ◽  
pp. 41-44
Author(s):  
Daniel Benest
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
The Body ◽  
Satellite Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Limited Region ◽  
Lagrange Points ◽  
Three Body ◽  
Circular Plane

In the circular plane restricted three-body problem, we study the stable large retrograde non-periodic satellite orbits. We use rotating axes with the origin in the body around which turns the satellite, called its primary. We choose the initial conditions such as Yo=0 and Uo=0, so that an orbit can be represented by a point in the (Xo,Vo) plane. In this plane, the set of stable orbits is represented by a limited region, which we call the stability zone. This zone is composed in general by a large continental region, approximately limited by Lagrange points, and a peninsula more or less elongated.

scholarly journals Stability of the Triangular Lagrangian Solutions of the Photo Gravitational Restricted Three-Body Problem in the Three-Dimensional Case

1993 ◽  
Vol 132 ◽  
pp. 291-308
Author(s):  
Md. Ghulam Murtuza ◽  
Vijay Kumar ◽  
R.K. Choudhry
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
The Stability ◽  
Three Body

AbstractThe stability of the triangular Lagrangian solutions for the photo-gravitational restricted three-body problem in the three-dimensional case is investigated for the case when the resonances are absent and also when the resonances are present. Stability is proved for most (in the sense of Lebesgue) initial conditions for all μ < μ0 except for the resonance cases.

scholarly journals Perturbed locations and linear stability of collinear Lagrangian points in the elliptic restricted three body problem with triaxial primaries

2019 ◽  
Vol 28 (1) ◽  
pp. 145-153
Author(s):  
Walid Ali Rahoma ◽  
Akram Masoud ◽  
Fawzy Ahmed Abd El-Salam ◽  
Elamira Hend Khattab
Keyword(s):  
Linear Stability ◽  
Body Problem ◽  
Equilibrium Points ◽  
Classical Case ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
The Difference ◽  
The Stability ◽  
Three Body

Abstract This paper aims to study the effect of the triaxiality and the oblateness as a special case of primaries on the locations and stability of the collinear equilibrium points of the elliptic restricted three body problem (in brief ERTBP). The locations of the perturbed collinear equilibrium points are first determined in terms of mass ratio of the problem (the smallest mass divided by the total mass of the system) and different concerned perturbing factors. The difference between the locations of collinear points in the classical case of circular restricted three body problem and those in the perturbed case is represented versus mass ratio over its range. The linear stability of the collinear points is discussed. It is observed that the stability regions for our model depend mainly on the eccentricity of the orbits in addition to the considered perturbations.

scholarly journals J+-Like invariants of periodic orbits of the second kind in the restricted three-body problem

2018 ◽  
Vol 12 (03) ◽  
pp. 675-712 ◽  
Author(s):  
Joontae Kim ◽  
Seongchan Kim
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Three Body

We determine three invariants: Arnold’s [Formula: see text]-invariant as well as [Formula: see text] and [Formula: see text] invariants, which were introduced by Cieliebak–Frauenfelder–van Koert, of periodic orbits of the second kind near the heavier primary in the restricted three-body problem, provided that the mass ratio is sufficiently small.

scholarly journals On the Stability ofL4,5in the Relativistic R3BP with Oblate Secondary and Radiating Primary

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Nakone Bello ◽  
Jagadish Singh
Keyword(s):  
Stability Region ◽  
Body Problem ◽  
Critical Mass ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Triangular Points ◽  
Numerical Exploration ◽  
The Stability ◽  
Oblate Secondary ◽  
Three Body

We consider a version of the relativistic restricted three-body problem (R3BP) which includes the effects of oblateness of the secondary and radiation of the primary. We determine the positions and analyze the stability of the triangular points. We find that these positions are affected by relativistic, oblateness, and radiation factors. It is also seen that both oblateness of the secondary and radiation of the primary reduce the size of stability region. Further, a numerical exploration computing the positions of the triangular points and the critical mass ratio of some binaries systems consisting of the Sun and its planets is given in the tables.

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