Antonio Elipe

Frozen Orbits About the Moon

Frozen orbits are of special interest to mission designers of artificial satellites. On average the eccentricity and argument of the perigee of such orbits remain stationary. Frozen orbits correspond to equilibria in an averaged form of the zonal problem and are almost periodic solutions of the full (nonaveraged) problem. In the zonal problem of a satellite around the moon, we numerically continue natural families of periodic orbits with the polar component of the angular momentum as the parameter. Three families of frozen orbits are discovered.

Bifurcations and equilibria in the extended N-body ring problem

Abstract We consider the motion of an infinitesimal particle under the gravitational field of (n+1) bodies in ring configuration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m0=βm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter β). This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the topology of the problem is modified.

On the equilibrium solutions in the circular planar restricted three rigid bodies problem

In this paper, the restricted problem of three rigid bodies under central forces is considered, and the collinear and triangular equilibrium solutions are obtained. Finally, an application to the case of axisymmetric ellipsoids is made.

Linearization: Laplace vs. Stiefel

The method for processing perturbed Keplerian systems known today as the linearization was already known in the XVIIIth century; Laplace seems to be the first to have codified it. We reorganize the classical material around the Theorem of the Moving Frame. Concerning Stiefels own contribution to the question, on the one hand, we abandon the formalism of Matrix Theory to proceed exclusively in the context of quaternion algebra; on the other hand, we explain how, in the hierarchy of hypercomplex systems, both the KS-transformation and the classical projective decomposition emanate by doubling from the Levi-Civita transformation. We propose three ways of stretching out the projective factoring into four-dimensional coordinate transformations, and offer for each of them a canonical extension into the moment space. One of them is due to Ferrándiz; we prove it to be none other than the extension of Burdets focal transformation by Liouvilles technique. In the course of constructing the other two, we examine the complementarity between two classical methods for transforming Hamiltonian systems, on the one hand, Stiefels method for raising the dimensions of a system by means of weakly canonical extensions, on the other, Liouvilles technique of lowering dimensions through a Reduction induced by ignoration of variables.

Analytical Model to Find Frozen Orbits for a Lunar Orbiter

Analytical theories based on Lie-Deprit transforms are used to obtain families of periodic orbits for the problem of an orbiter around the moon. Low and moderately high orbit models are analyzed. Equilibria of the normalized equations of motion provide the representation of a global portrait of families of frozen orbits depending on values of the inclination, eccentricity, and semimajor axis. By means of the inverse transformation it is possible to refine the initial conditions for frozen orbits of a simplified model, and these initial conditions may be used as starters of numerical continuation methods when more complex models are considered.

The Serret Andoyer Formalism in Rigid-Body Dynamics: I. Symmetries and Perturbations

This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics, and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, that stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret-Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into action-angle ones, using the method of Sadov.

Periodic Orbits Around a Massive Straight Segment

In this paper, we consider the motion of a particle under the gravitational field of a massive straight segment. This model is used as an approximation to the gravitational field of irregular shaped bodies, such as asteroids, comet nuclei and planetss moons. For this potential, we find several families of periodic orbits and bifurcations.

Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory

In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations.

Automatic programming of recurrent power series

The integration of differential equations by recurrent power series is a classical method in ODE. This method is valid on very long spans of integration and unusually large step-sizes. However, this method is rarely used, mainly since each problem requires a specific formulation.

Non-linear stability of the equilibria in the gravity field of a finite straight segment

We study the non-linear stability of the equilibria corresponding to the motion of a particle orbiting around a finite straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by elongated celestial bodies. By means of the Arnolds theorem for non-definite quadratic forms we determine the orbital stability of the equilibria, for all values of the parameter k of the problem, resonant cases included.