José M. Ferrándiz

Improved Bettis Methods for Long-Term Prediction

In this paper we introduce a modification of Bettis’ method in order to improve the long-term numerical integration of perturbed oscillators. We give several examples, involving both single and coupled oscillators, to illustrate the efficiency of this approach with respect to the methods of Bettis and Adams-Bashforth.

Long-Time Predictions of Satellite Orbits by Numerical Integration

In this paper we aim at establishing limits of predictability to find the extension of time for which meaningful analytical and/or numerical predictions can be made in orbital behaviour of artificial satellites. These limits depend, of course, on the accuracy required, on the specific dynamical models formulated, on the sets of variables chosen to describe them, on the numerical or analytical techniques used and, especially, on the specific trajectories to be established. In order to check the reliability of the predictions, first integrals, constraints among redundant variables and backwards integrations from the ending point to the initial conditions have been used.

Exact linearization of non-planar intermediary orbits in the satellite theory

In this paper we consider the reduction of the equations of motion for non-planar perturbed two body problems into linear form. It is seen that this can be easily accomplished for any element of the class of radial intermediaries to the satellite problem proposed by Deprit in 1981, since they have a functional dependence suitable for linearization. The transformation is worked out by using an adequate set of redundant variables. Four harmonic oscillators are obtained, of which two are coupled through gyroscopic terms. Their constant frequencies contain the secular contribution of the main problem of artificial satellite theory up to the order of the considered intermediary. Therefore, this result may well be interesting in relation to the study and prediction of accurate long-term solutions to satellite problems.

General Motion of a Triaxial Rigid Body in a Newtonian Force Field

The study of the general motion of rigid bodies is of increasing importance since highly accurate solutions are required not only for the classical cases of precession and nutation of natural bodies but also for the attitude of artificial satellites. Intrinsically, this problem is of some difficulty. Although the general motion of a rigid body can be split up into the trnslation of its center of mass and the rotation about that point, both are usually coupled and therefore we are faced with a generally non-reducible 6-dimensional problem.

A Note on the Canonical Character of the Stiefel-Scheifele Time Element

In this paper we present an easy way of transforming some standard canonical sets with the special feature that, in the new set, the time element by Stiefel and Scheifele appears as a canonical variable.

New Intermediaries for the Main Problem in Satellite Theory

After reviewing the original approach leading to the introduction of intermediaries in Satellite Theory, a general procedure to define intermediaries for the Main Problem in this Theory is proposed. This procedure is susceptible to an intuitive interpretation analogous to solving a simple puzzle. The application of this method to the Main Problem allows us not only to recover the well known classical intermediaries but also to obtain several completely new ones, all admitting simple solutions.

Application of Spherically Exact Algorithms to Numerical Predictability in Two-Body Problems

Numerical predictions are strongly dependent on the algorithms used in the integration, even in cases as simple as the two-body problem, perturbed or not. In this contribution we show some numerical experiments comparing the results obtained by applying different codes. Among them we include some with special preservation properties, such as being spherically exact.

A family of multistep methods to integrate orbits on spheres

SummaryA trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution, rather than in finding the actual correspondence between the values of the parameter and the points on that curve. This paper introduces a family of multi-stage, multi-step numerical methods to integrate trajectory problems whose solution is on a spherical surface. It has been shown that this kind of algorithms has good numerical properties: consistency, stability, convergence and others that are not standard. The latest ones make them a better choice for certain problems.

Poincaré-Similar Variables Including J 2 -Secular Effects

In a previous paper we had defined a set of eight generalized canonical Delaunay- Similar (CDS) variables incorporating the first-order secular effects present in the Main Problem in the Theory of Earth’s Artificial Satellite. The new CDS set wsls derived by means of a canonical transformation whose generating function is inspired by Deprit’s radial intermediary and can be considered as a generalization of a canonical set introduced by Scheifele. When applied to Deprit’s intermediary, the proposed variables lead to a simple solution, the momenta being constant and the co-ordinates being either a constant or a linear function of the independent variable. As a further step, a set of generalized Poincare-Similar (PS) canonical variables corresponding to the aforesaid DS ones is now constructed; the new GPS set also exhibits the same feature of containing the whole first-order secular contribution of the J2 zonal harmonic of the Earth’s potential and is free from singularities.

A New Transformation of the Two-Body Problem to Oscillator Form

In this paper we propose a change of variables, similar to Bond’s,which reduces the Kepler’s problem to a three dimensional harmonic oscillator, the new time being proportional to the eccentric anomaly. This transformation is extended to include arbitrary anomalies and can be used to linearize some perturbed problems.