Escape of a Subsystem in a N-Body Problem. Application to the Three Body Problem

In terms of stability around the primary, it is widely known that the semimajor axis of the retrograde satellites is much larger than the corresponding semimajor axis of the prograde satellites. Usually this conclusion is obtained numerically, since precise analytical derivation is far from being easy, especially, in the case of two or more disturbers. Following the seminal idea that what is unstable in the restricted three-body problem is also unstable in the general N-body problem, we present a simplified model which allows us to derive interesting resonant configurations. These configurations are responsible for cumulative perturbations which can give birth to strong instability that may cause the ejection of the satellite. Then we obtain, analytically, approximate bounds of the stability of prograde and retrograde satellites. Although we recover quite well previous results of other authors, we comment very briefly some weakness of these bounds.

Download Full-text

New Periodic Orbits for the n-Body Problem

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.2338323 ◽

2006 ◽

Vol 1 (4) ◽

pp. 307-311 ◽

Cited By ~ 8

Author(s):

Cristopher Moore ◽

Michael Nauenberg

Keyword(s):

Periodic Orbits ◽

Body Problem ◽

Fourier Coefficients ◽

Three Dimensional ◽

Inertia Tensor ◽

Three Body Problem ◽

Cubic Symmetry ◽

Numerical Evidence ◽

N Body Problem ◽

Three Body

Since the discovery of the figure-eight orbit for the three-body problem [Moore, C., 1993, Phys. Rev. Lett., 70, pp. 3675–3679] a large number of periodic orbits of the n-body problem with equal masses and beautiful symmetries have been discovered. However, most of those that have appeared in the literature are either planar or are obtained from perturbations of planar orbits. Here we exhibit a number of new three-dimensional periodic n-body orbits with equal masses and cubic symmetry, including some whose moment of inertia tensor is a scalar. We found these orbits numerically, by minimizing the action as a function of the trajectories’ Fourier coefficients. We also give numerical evidence that a planar three-body orbit first found in [Hénon, M., 1976, Celest. Mech., 13, pp. 267–285], rediscovered by [Moore, 1993], and found to exist for different masses by [Nauenberg, M., 2001, Phys. Lett., 292, pp. 93–99], is dynamically stable.

Download Full-text

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

Vestnik of Saint Petersburg University Mathematics Mechanics Astronomy ◽

10.21638/spbu01.2021.116 ◽

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.

Download Full-text