26 research outputs found
Symmetric Periodic Solutions of the Anisotropic Manev Problem
We consider the Manev Potential in an anisotropic space, i.e., such that the
force acts differently in each direction. Using a generalization of the
Poincare' continuation method we study the existence of periodic solutions
for weak anisotropy. In particular we find that the symmetric periodic orbits
of the Manev system are perturbed to periodic orbits in the anisotropic
problem.Comment: Late
Stable manifolds and homoclinic points near resonances in the restricted three-body problem
The restricted three-body problem describes the motion of a massless particle
under the influence of two primaries of masses and that circle
each other with period equal to . For small , a resonant periodic
motion of the massless particle in the rotating frame can be described by
relatively prime integers and , if its period around the heavier primary
is approximately , and by its approximate eccentricity . We give a
method for the formal development of the stable and unstable manifolds
associated with these resonant motions. We prove the validity of this formal
development and the existence of homoclinic points in the resonant region.
In the study of the Kirkwood gaps in the asteroid belt, the separatrices of
the averaged equations of the restricted three-body problem are commonly used
to derive analytical approximations to the boundaries of the resonances. We use
the unaveraged equations to find values of asteroid eccentricity below which
these approximations will not hold for the Kirkwood gaps with equal to
2/1, 7/3, 5/2, 3/1, and 4/1.
Another application is to the existence of asymmetric librations in the
exterior resonances. We give values of asteroid eccentricity below which
asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2
resonances for any however small. But if the eccentricity exceeds these
thresholds, asymmetric librations will exist for small enough in the
unaveraged restricted three-body problem