Martin Lara

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.

Long-lifetime lunar repeat ground track orbits

A high degree and order lunar gravitational field is superimposed on the Earth-moon restricted three-body model to capture the dominating forces on a spacecraft in the vicinity of the moon. For the synchronously rotating moon, periodic orbits in this unaveraged model map repeat ground tracks and represent higher-order solutions to the frozen orbit problem. The stable or near-stable solutions are found over a wide range of defining characteristics, making them suitable for long-lifetime parking applications such as science orbits, crew exploration vehicle parking orbits, and global coverage constellation orbits. A full ephemeris is considered for selected orbits to evaluate the validity of the time-invariant, simplified model. Of the most promising results are the low-altitude families of near-circular, inclined orbits that maintain long-term stability despite the highly nonspherical lunar gravity. The method is systematic and enables rapid design and analysis of long-life orbits around any tidally locked celestial body with an arbitrarily high degree and order spherical harmonic gravity field.

Dynamic Behavior of an Orbiter Around Europa

The dynamics of an orbiter close to a planetary satellite are known to be unstable from a wide range of inclinations encompassing polar orbits. Taking the Jupiter-Europa system as our model, we use numerically determined periodic orbits to investigate the stability of motion over three-dimensional space for this problem. We have found that the change in stability is produced by a bifurcation in phase space. At a certain critical inclination, almost circular periodic orbits change their stability character to instability and new families of (stable) elliptic orbits appear.

Computation of a Science Orbit About Europa

T HE instrument requirements for a scientific mission about Europa constrain the orbit design to a subset of limited values of the orbital elements. Near-circular, low-altitude, high-inclination orbits are normally required for mapping missions, and spacemission designers try to minimize the altitude variation of the satellite over the surface of the body by searching for orbits with small eccentricity and with a fixed argument of periapsis. These orbits are usually called frozen orbits [1–3]. However, due to third-body perturbations, high-inclination orbits around planetary satellites are known to be unstable [4–6], and thus emerges the problem of maximizing the orbital lifetime. One proposed approach to maximize orbit lifetime resorts to dynamical systems theory. This approach has been shown to be useful in orbit maintenance routines, in which the stable manifold associated with unstable nominal orbits provides a efficient way of maximizing time between maneuvers [7]. In the same fashion, paths in the plane of argument of periapsis and eccentricity that yield long lifetime near polar orbits around Europa have been recently identified [8]. In this paper we take a different approach. Periodic orbits around Europa are known to exist and have been previously used in the investigation of stability regions around Europa [9–11]. Periodic orbits in the rotating frame are ideal, nominal, repeat ground-track orbits that, for long enough repetition cycles, are suitable for mapping missions. We compute low-altitude, near-circular, highly inclined, repeat ground-track, unstable periodic orbits, and find that these kinds of solutions enjoy long lifetimes. Our procedure is based on fast numerical algorithms that are easily automated. The numerical search for initial conditions of repeat ground-track orbits is very simple and feasible even for higher-order gravity fields [12,13]. Safe recurrences for computing the gradient and Hessian of the gravitational potential can be found in the reference list (see, for instance, [14] and references therein). For our search, we use a simplified dynamical model that considers the mean gravitational field of a synchronously rotating and orbiting moon, and take into account the perturbations of the third body in the Hill problem approximation [5,15,16]. Tests on the validity of the solutions aremade in an ephemerismodel that includes perturbations of the sun, the other Galileans, the nonsphericity of Jupiter, the other gas giants, and a Europa gravity model that is consistent with synchronousmoon theory andNASA’sGalileo close encounters [17]. In passing from the simplified to ephemeris model we introduce a one-dimensional parameter scaling of the initial conditions that proved efficient in the past [10]. On one side it provides a simple and feasible optimization for a given epoch. But it also shows how isolated the optimized solution is in the ephemerismodel, thus giving a reasonable estimation of the robustness of the solution in the presence of realistic perturbing forces. With respect to theEuropa gravityfield, it turns out that theGalileo flyby data cannot detect valid signatures for any gravitational terms for Europa beyond , J2, and C2;2 [17]. However, based on observations of other celestial bodies, it seems reasonable to speculate that Europa could be top or bottom heavy [18], and previous studies have shown that J3 can play an important role [8,18,19]. Therefore, we study the influence of J3 in the proposed orbits, and find repeat ground-track orbits with higher eccentricities than the second-order gravity field solutions, but with similar lifetimes.

Classification of the distant stability regions at europa

Long-term stable trajectories around Europa, one of the Galilean moons of Jupiter, are analyzed for their potential applications in spacecraft trajectory design, such as end-of-mission disposal options, backup orbits, or intermediary targets for transfer trajectories. The phase space is analyzed via the computation of families of periodic orbits and the estimation of their stability character. Although the core analysis of the paper uses the circular restricted three-body problem, a selected set of long-term stable solutions is checked by integrating the corresponding initial conditions in an ephemeris model over several years. The current model and methods can be readily applied to other planetary satellites including the other Galileans at Jupiter, the many satellites at Saturn, and the Earths moon.

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.

Repeat Ground Track Orbits of the Earth Tesseral Problem as Bifurcations of the Equatorial Family of Periodic Orbits

Orbits repeating their ground track on the surface of the earth are found to be members of periodic-orbit families (in a synodic frame) of the tesseral problem of the Earth artificial satellite. Families of repeat ground track orbits appear as vertical bifurcations of the equatorial family of periodic orbits, and they evolve from retrograde to direct motion throughout the 180 degrees of inclination. These bifurcations are always close to the resonances of the Earths rotation rate and the mean motion of the orbiter.

Simplified Equations for Computing Science Orbits Around Planetary Satellites

An analytical theory for spacecraft motion close to synchronously orbiting and rotating planetary satellites is provided. The ratio rotation rate of the satellite mean motion of the orbiter is the small parameter of the theory. A double-averaging over the mean anomaly and the argument of the node reduces the problem to one degree of freedom in the eccentricity and the argument of the periapsis. The theory is based on the Lie-Deprit perturbation method, which permits recovering the short- and long-period terms through explicit transformations. An application to the computation of (unstable) science orbits is presented, a case in which the transformation equations admit dramatic simplifications.

Phase Space Structure Around Oblate Planetary Satellites

The dynamics of an orbiter around planetary satellites are modeled using Hills equations perturbed by the nonsphericity of the satellite. Classically, the long-term behavior of this problem is studied by averaging techniques. The double-averaged problem is integrable. However, up to second order, it presents a symmetry of direct and retrograde inclination orbits that do not exist in the original problem. Lie transforms are used to reduce the problem to an integrable one, with the transformations performed up to third order where the inclination symmetry is broken. Then, by the use of the double reduced space, which is a sphere, a full description of families of frozen orbits and their bifurcations is given. Saddle-center and pitchfork bifurcations related to stable, frozen orbits are identified. Finally, for the specific case of a Europa orbiter, the equilibria of the reduced problem are related to periodic solutions of the nonaveraged problem in a synodic frame.