Juan F. San-Juan

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.

MathATESAT: a symbolic-numeric environment in astrodynamics and celestial mechanics

The increase in the facilities of the general computer algebra system, in particular Mathematica, and hardware evolution have supplied us with the possibility of developing a new environment called MathATESAT. Our new system collects all the tools necessary to carry out high accuracy analytical theories in order to analyze the quantitative and qualitative behaviour of a dynamic system. Real applications for the calculation of several expansions of Disturbing Functions, classical expansions of the Kepler problem and some details involved in the qualitative analysis of dynamic systems, are used to illustrate the capability of MathATESAT.

ATESAT: a symbolic processor for artificial satellite theory

Analytical theories for the artificial satellite motion involve operations with the so called Poisson series. Even if only a second order theory is required, the amount of terms involved is so huge, that it is almost an impossible task to carry out by hand the theory. Thus, algebraic manipulators are essential in this field, and even more, since general purpose manipulators are not completely satisfactory, specific manipulators are necessary. Aware of this fact, we build ATESAT (Automatization of Theories and Ephemeris in the artificial Satellite problem), that provides the automatic generation of programs for computing the ephemeris of the satellite from the analytical theory chosen. ATESAT is built with PSPC, an algebraic manipulator from our own, especially designed for manipulating Poisson series.

Secular motion around synchronously orbiting planetary satellites

We investigate the secular motion of a spacecraft around the natural satellite of a planet. The satellite rotates synchronously with its mean motion around the planet. Our model takes into account the gravitational potential of the satellite up to the second order, and the third-body perturbation in Hills approximation. Close to the satellite, the ratio of rotation rate of the satellite to mean motion of the orbiter is small. When considering this ratio as a small parameter, the Coriolis effect is a first-order perturbation, while the third-body tidal attraction, the ellipticity effect, and the oblateness perturbation remain at higher orders. Then, we apply perturbation theory and find that a third-order approach is enough to show the influence of the satellites ellipticity in the pericenter dynamics. Finally, we discuss the averaged system in the three-dimensional parametric space, and provide a global description of the flow.

Short Term Evolution of Artificial Satellites

When the elimination of the parallax and the elimination of the perigee is applied to the zonal problem of the artificial satellite, a one-degree of freedom Hamiltonian is obtained. The classical way to integrate this Hamiltonian is by applying the Delaunay normalization, however, changing the time to the perturbed true anomaly and the variable to the inverse of the distance, the Hamilton equations become a perturbed harmonic oscillator. In this paper we apply the Krylov—Bogoliubov—Mitropolsky (KBM) method to integrate the perturbed harmonic oscillator as an alternative method to the Delaunay normalization. This method has no problem with small eccentricities and inclinations, and shows good numerical results in the evaluation of ephemeris of satellites.

Delaunay variables approach to the elimination of the perigee in Artificial Satellite Theory

Analytical integration in Artificial Satellite Theory may benefit from different canonical simplification techniques, like the elimination of the parallax, the relegation of the nodes, or the elimination of the perigee. These techniques were originally devised in polar-nodal variables, an approach that requires expressing the geopotential as a Pfaffian function in certain invariants of the Kepler problem. However, it has been recently shown that such sophisticated mathematics are not needed if implementing both the relegation of the nodes and the parallax elimination directly in Delaunay variables. Proceeding analogously, it is shown here how the elimination of the perigee can be carried out also in Delaunay variables. In this way the construction of the simplification algorithm becomes elementary, on one hand, and the computation of the transformation series is achieved with considerable savings, on the other, reducing the total number of terms of the elimination of the perigee to about one third of the number of terms required in the classical approach.

Averaging Tesseral Effects: Closed Form Relegation versus Expansions of Elliptic Motion

Longitude-dependent terms of the geopotential cause nonnegligible short-period effects in orbit propagation of artificial satellites. Hence, accurate analytical and semianalytical theories must cope with tesseral harmonics. Modern algorithms for dealing analytically with them allow for closed form relegation. Nevertheless, current procedures for the relegation of tesseral effects from subsynchronous orbits are unavoidably related to orbit eccentricity, a key fact that is not enough emphasized and constrains application of this technique to small and moderate eccentricities. Comparisons with averaging procedures based on classical expansions of elliptic motion are carried out, and the pros and cons of each approach are discussed.

An Economic Hybrid Analytical Orbit Propagator Program Based on SARIMA Models

We present a new economic hybrid analytical orbit propagator program based on SARIMA models, which approximates to a 4×4 tesseral analytical theory for a Quasi-Spot satellite. The J2 perturbation is described by a first-order closed-form analytical theory, whereas the effects produced by the higher orders of J2 and the perturbation of the rest of zonal and tesseral harmonic coefficients are modelled by SARIMA models. Time series analysis is a useful statistical prediction tool, which allows building a model for making future predictions based on the study of past observations. The combination of the analytical techniques and time series analysis allows an increase in accuracy without significant loss in efficiency of the new propagators, as a consequence of modelling higher-order terms and other perturbations are not taken into account in the analytical theory.

Precise Analytical Computation of Frozen-Eccentricity, Low Earth Orbits in a Tesseral Potential

Classical procedures for designing Earth’s mapping missions rely on a preliminary frozen-eccentricity orbit analysis. This initial exploration is based on the use of zonal gravitational models, which are frequently reduced to a simple analysis. However, the model may not be accurate enough for some applications. Furthermore, lower order truncations of the geopotential are known to fail in describing the behavior of elliptic frozen orbits properly. Inclusion of a higher degree geopotential, which also takes into account the short-period effects of tesseral harmonics, allows for the precise computation of frozen-eccentricity, low Earth orbits that show smaller long-period effects in long-term propagations than those obtained when using the zonal model design.

On Bounded Satellite Motion under Constant Radial Propulsive Acceleration

The Hamiltonian formulation of the constant radial propulsive acceleration problem in nondimensional units reveals that the problem does not depend on any physical parameter. The qualitative description of the integrable flow is given in terms of the energy and the angular momentum, showing that the different regimes are the result of a bifurcation phenomenon. The solution via the Hamilton-Jacobi equation demonstrates that the elliptic integrals of the three kinds are intrinsic to the problem.