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.

scholarly journals Approximate Analytical Periodic Solutions to the Restricted Three-Body Problem with Perturbation, Oblateness, Radiation and Varying Mass

Universe ◽  
2020 ◽  
Vol 6 (8) ◽  
pp. 110
Author(s):  
Fabao Gao ◽  
Yongqing Wang
Keyword(s):  
Periodic Solutions ◽  
Body Problem ◽  
Three Dimensional ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
The Third ◽  
Velocity Surface ◽  
Solution Point ◽  
Zero Velocity Surface ◽  
Three Body

Against the background of a restricted three-body problem consisting of a supergiant eclipsing binary system, the two primaries are composed of a pair of bright oblate stars whose mass changes with time. The zero-velocity surface and curve of the problem are numerically studied to describe the third body’s motion area, and the corresponding five libration points are obtained. Moreover, the effect of small perturbations, Coriolis and centrifugal forces, radiative pressure, and the oblateness and mass parameters of the two primaries on the third body’s dynamic behavior is discussed through the bifurcation diagram. Furthermore, the second- and third-order approximate analytical periodic solutions around the collinear solution point L3 in two-dimensional plane and three-dimensional spaces are presented by using the Lindstedt-Poincaré perturbation method.

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 Is Jacobi theorem valid in the singly averaged restricted circular Three-Body-Problem?

2021 ◽  
Vol 8 (1) ◽  
pp. 179-184
Author(s):  
Konstantin V. Kholshevnikov ◽  
◽  
Keyword(s):  
Total Energy ◽  
Body Problem ◽  
Negative Energy ◽  
Mass Point ◽  
Restricted Three Body Problem ◽  
Energy Integral ◽  
Three Body Problem ◽  
Zero Mass ◽  
N Body Problem ◽  
Three Body

C. Jacobi found that in the General N-Body-Problem (including N = 3) for the Lagrangian stability of any solution necessary is the negativity of the total energy of the system. For the restricted three-body-problem, this statement is trivial, since a zero-mass body introduces zero contribution to the energy of the system. If we consider only the equations describing the movement of the zero mass point, then the energy integral disappears. However, if we average the equations over the longitudes of the main bodies, the energy integral appears again. Is the Jacobi theorem valid in this case? It turned out not. For arbutrary large values of total energy, there exist bounded periodic orbits. At the same time the negative energy is sufficient for the boundedness of an orbit in the configuration space.

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.

Rigid-Rotator and Fixed-Shape Solutions to the Coulomb Three-Body Problem

1997 ◽  
Vol 22 (1) ◽  
pp. 37-60 ◽  
Author(s):  
A. Santander ◽  
J. Mahecha ◽  
F. Pérez
Keyword(s):  
Body Problem ◽  
Three Body Problem ◽  
Rigid Rotator ◽  
Three Body
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