scholarly journals Motion near The Unit Circle in The Three-Body Problem

1999 ◽  
Vol 172 ◽  
pp. 281-290
Author(s):  
Roger A. Broucke
Keyword(s):  
Linear System ◽  
Unit Circle ◽  
Restricted Problem ◽  
Body Problem ◽  
Equations Of Motion ◽  
Original System ◽  
Polar Coordinates ◽  
Three Body Problem ◽  
Simplified Models ◽  
Three Body

Many of the important applications of the circular planar restricted problem of three bodies involve motion in the vicinity of the unit circle, (as defined in canonical units). It is then of interest to develop simplified models which are valid in this region. These models preserve the gross characteristics of the original system but they possess simpler equations of motion.We will also show that several simplified models can be seen as a perturbation of a very well known simple linear system: the Clohessy-Wiltshire equations used by NASA in all their rendezvous operations. These are actually very close to the well-known Hill problem. We will thus consider the Restricted problem as a perturbed Hill or Clohessy-Wiltshire problem. We also introduce the Clohessy-Wiltshire Lagrangian in polar coordinates.

scholarly journals On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

2021 ◽  
pp. 56-73
Author(s):  
S. E. Abd El-Bar
Keyword(s):  
Characteristic Equation ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Three Body Problem ◽  
Total Potential ◽  
The Mean ◽  
The Stability ◽  
Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

Orbits of Trojan Asteroids

1974 ◽  
Vol 62 ◽  
pp. 63-69 ◽  
Author(s):  
G. A. Chebotarev ◽  
N. A. Belyaev ◽  
R. P. Eremenko
Keyword(s):  
Solar System ◽  
Body Problem ◽  
Equations Of Motion ◽  
Orbital Evolution ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Trojan Asteroids ◽  
Actual Motion ◽  
Three Body

In this paper the orbital evolution of Trojan asteroids are studied by integrating numerically the equations of motion over the interval 1660–2060, perturbations from Venus to Pluto being taken into account. The comparison of the actual motion of Trojans in the solar system with the theory based on the restricted three-body problem are given.

scholarly journals Nekhoroshev-Stability ofL4andL5in the Spatial Restricted Problem

1999 ◽  
Vol 172 ◽  
pp. 445-446 ◽  
Author(s):  
Giancarlo Benettin ◽  
Francesco Fassò ◽  
Massimiliano Guzzo
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Critical Mass ◽  
Kam Theory ◽  
Birkhoff Normal Form ◽  
Three Body Problem ◽  
Arnold Diffusion ◽  
Convexity Assumption ◽  
Three Body ◽  
Nekhoroshev Stability

The Lagrangian equilateral pointsL4andL5of the restricted circular three-body problem are elliptic for all values of the reduced massμbelow Routh’s critical massμR≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting byda convenient distance from the equilibrium point, one asks whetherfor any small єe > 0, with positiveaandb. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus onμ(see e.g .Giorgilli, 1989). Our aim was instead considering all values ofμup toμR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassòet al, 1998; Guzzoet al, 1998; Niedermann, 1998).

Displacement of the Lagrange Equilibrium Points in the Restricted Three Body Problem With Rigid Body Satellite

1978 ◽  
Vol 41 ◽  
pp. 305-314
Author(s):  
W.J. Robinson
Keyword(s):  
Rigid Body ◽  
Restricted Problem ◽  
Body Problem ◽  
Equilibrium Points ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

AbstractIn the restricted problem of three point masses, the positions of the equilibrium points are well known and are tabulated. When the satellite is a rigid body, these values no longer correspond to the equilibrium points. This paper seeks to determine the magnitudes of the discrepancies.

scholarly journals Orbital stability of high inclination asteroids

1996 ◽  
Vol 172 ◽  
pp. 187-192
Author(s):  
N. A. Solovaya ◽  
E. M. Pittich
Keyword(s):  
Solar System ◽  
Body Problem ◽  
Equations Of Motion ◽  
Orbital Stability ◽  
Dynamical Model ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

The orbital evolutions of fictitious asteroids with high inclinations have been investigated. The selected initial orbits represent asteroids with movement, which corresponds to the conditions of the Tisserand invariant for C = C (L1) in the restricted three body problem. Initial eccentricities of the orbits cover the interval 0.0–0.4, inclinations the interval 40–80°, and arguments of perihelion the interval 0–360°. The equations of motion of the asteroids were numerically integrated from the epoch March 25, 1991 forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all planets. The orbits of the model asteroids are stable at least during the investigated period.

scholarly journals Quasi-Lntegrals of the Plane Restricted Three-Body Problem

1993 ◽  
Vol 132 ◽  
pp. 309-319
Author(s):  
E.M. Nezhinskij
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Test Body ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Jacobi Integral ◽  
Three Body ◽  
Two Bodies

AbstractThe paper is concerned with studying the domain of possible motion and a field of the test body velocities in the plane restricted problem of three bodies. The study is based on existence of a quasi-integral of areas (similar to an integral of areas in the problem of two bodies) as well as on the Jacobi integral. The method of constructing the quasi-integrals is a standard one (see, for example, [1],[2].

scholarly journals Stationary solutions, critical mass, Tadpole orbits in the circular restricted three-body problem with the more massive primary as an oblate spheroid

2020 ◽  
Vol 13 (39) ◽  
pp. 4168-4188
Author(s):  
A Arantza Jency
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Critical Mass ◽  
Equations Of Motion ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Mean Motion ◽  
The Mean ◽  
Three Body

Background: The location and stability of the equilibrium points are studied for the Planar Circular Restricted Three-Body Problem where the more massive primary is an oblate spheroid. Methods: The mean motion of the equations of motion is formulated from the secular perturbations as derived by(1) and used in(2–4). The singularities of the equations of motion are found for locating the equilibrium points. Their stability is analysed using the linearized variational equations of motion at the equilibrium points. Findings: As the effect of oblateness in the mean motion expression increases, the location and stability of the equilibrium points are affected by the oblateness of the more massive primary. It is interesting to note that all the three collinear points move towards the more massive primary with oblateness. It is a new result. Among the shifts in the locations of the five equilibrium points, the y–location of the triangular equilibrium points relocate the most. It is very interesting to note that the eccentricities (e) of the orbits around L1 and L3 increase, while it decreases around L2 with the addition of oblateness with the new mean motion. The decrease in e is significant in Saturn-Mimas system from 0.95036 to 0.87558. Similarly, the value of the critical mass ratio mc, which sets the limit for the linear stability of the triangular points, further reduces significantly from 0:285: : :A1 to 0:365: : :A1 with the new mean motion. The mean motion sz in the z-direction increases significantly with the new mean motion from 9A1/4 to 9A1/2.

scholarly journals Minimal velocity surface in the restricted circular Three-Body-Problem

2020 ◽  
Vol 65 (4) ◽  
pp. 734-742
Author(s):  
Konstantin V. Kholshevnikov ◽  
◽  
Vladimir B. Titov ◽  
◽  
Keyword(s):  
Body Problem ◽  
Original System ◽  
Three Body Problem ◽  
Zero Mass ◽  
Jacobi Integral ◽  
Velocity Surface ◽  
Minimal Velocity ◽  
Zero Velocity Surface ◽  
Three Body ◽  
Minimum Velocity

In the framework of the restricted circular Three-Body-Problem, the concept of the minimum velocity surface S is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of Hill surface requires occurrence of the Jacobi integral. The minimum velocity surface, other than the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of the main bodies motion. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system by longitude of the main bodies. Properties of S are investigated. Here is the most significant. The set of possible motions of the zero-mass body bounded by the surface S is compact. As an example the surfaces S for four small moons of Pluto are considered in the framework of the averaged problem Pluto — Charon — small satellite. In all four cases, S represents a topological torus with small cross section, having a circumference in the plane of motion of the main bodies as the center line.

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