Alberto Abad

University of Zaragoza | UNIZAR · Department of Theoretical Physics

In Abad et al., we studied the time evolution of orbital elements in the so-called “Gyldén–Mestchersky cases.” Actually the plots represent the evolution of orbital elements with respect to the eccentric anomaly, related with time through Kepler’s equation. When reproducing figures with respect to time, they are globally similar to the ones we obta...

In this work, we study the evolution of the families of simple symmetric periodic orbits in the restricted three-body problem whatever the value of the mass parameter μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \set...

The dynamics of the two-body problem with general mass-loss functions that depend on both the independent variable and the radial distance are studied. These functions have been considered by Docobo and coworkers to explain the so-called “periastron” effect. By means of some appropriate changes of variables we reduce the integration of this problem...

We consider a relatively simple model to represent some types of asteroids that are very elongated and with two protuberances on their end points. A conspicuous example of this type of body is asteroid 216-Kleopatra. To model such bodies, we consider a dipole segment, which consists of a massive segment with two spherical masses at the end point of...

This Note presents an explicit relation between time and true anomaly for Keple-rian orbits, with no need to use the Kepler's equation which introduces an intermediary angle, the eccentric anomaly for elliptic motions and the hyperbolic anomaly for hyperbolic orbits, and which is the common way in computing the position on the orbit. Because of som...

This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, perturbed Kepler Hamiltonians which perturbation is of the form ∑j=2NAj∕rj with Aj constant. Although there are many applications of these Hamiltonians in Physics, Astronomy and Astrodynamics, we focus our interest on a particular case in the core of Artifi...

The hierarchical three-body problem is one of the classical issues of celestial mechanics, but recently it has regained importance due to its applications to new scenarios, like compact objects and exoplanets. In this paper we realize a computational study of this problem using the TIDES software package, which is applied not only to a set of theor...

We study the dynamics of two-body problems with gravitational parameters μ = μ ( t ) decreasing with the time. In particular we focus our attention on the behavior of the orbital elements in the first two Gyldén–Mestchersky cases: in the first case (GM-I) , where μ 0 is the constant initial value of the parameter and α a small positive constant. In...

In Artificial Satellite Theory, for the so called main problem (the two-body problem is only perturbed by the equatorial planet's bulge) Deprit's intermediary plays an important role because of its simplicity and integrability while maintaining a similar structure to the original Hamiltonian. To integrate it, we prove that there is no need to intro...

This paper shows the precise relative motion of different orbiters located at the geostationary region thanks to high precision astrometric coordinates, which are calculated thanks to different accurate observations taken from the Venezuelan National Observatory. These orbiters are close to each-other and present different relative motions although...

We consider the planar restricted four-body problem proposed by Moulton. One infinitesimal mass moves under the attraction of three mass points in collinear Euler’s configuration, namely \(P_0 (m_0 = \mu \,m)\) is placed at the origin, and other two identical points \(P_1 (m)\) and \(P_2 (m)\) are placed at the same distance from the origin. The pr...

In this paper, a three dimensional case of the restricted four-body problem with radiation pressure is considered. The three primaries are supposed to be in a collinear central configuration where both masses and both radiation forces of peripheral bodies are equal. In addition to the analysis of the equilibria in the planar problem introduced in a...

In this paper, a restricted four-body problem with radiation pressure is considered. The three primaries are supposed in a collinear central configuration where both masses and both radiation forces of peripheral bodies are equal. After an adequate formulation, the problem is reduced to a tri-parametric one. A complete analysis of the position of e...

In this paper, we study the problem of computing periodic orbits of Hamiltonian systems providing large families of such orbits. Periodic orbits constitute one of the most important invariants of a system, and this paper provides a comprehensive analysis of two efficient computational approaches for Hamiltonian systems. First, a new version of the...

In this work we investigate orbits containing Keplerian closed arcs. We characterize these orbits and compute them by using Lambert's problem. By analyzing its orbital elements, we can obtain orbits with a given inclination and repeated ground track such as the Molniya and Tundra orbits.

In this paper we introduce a set of symplectic variables, based on the intrinsic frame or Frenet frame. Applications to numerical propagation of Keplerian motions are made using EFRKG, a symplectic exponentially fitted Runge-Kutta- Gauss numerical method.

In this paper we introduce a set of symplectic variables, based on the intrinsic frame or Frenet frame. Applications to Keplerian motions are made using EFRKG, a symplectic exponentially fitted Runge-Kutta-Gauss numerical method.

An evolution strategy algorithm belonging to the general field of genetic algorithms is developed to detect periodic orbits in dynamical problems. The algorithm is applied to the problem of motion of a particle under the gravitational field of a solid circular wire.

In this paper we analyze the concept of space orbit to extract the parameters and properties that characterize it in the same way that object oriented programing encapsulate the objects. We use these parameters as the base to construct a software, named Orbits, to handle orbits in an educational environement.

A new method, based on automatic differentiation technique, has been proposed in this paper to compute the derivatives of the gravity potential. Using this method we can obtain derivatives up to any order. Instead of explicit expressions of the derivatives we use an iterative scheme to simultaneously compute the value of all the desired derivatives...

This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES provides an easy-to-use interface for standard double precision integrations, but also for quadruple precision and multiple precision integrations. The motivation for the development of this pa...

This paper discusses the use of recently developed techniques and software in the numerical propagation of uncertainties in initial coordinates and/or parameters for initial value problems. We present an approach based on several validated numerical integration techniques but focusing on the propagation of boxes. The procedure uses a multivariable...

This paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision. These very high requirements are useful, for example, in the studies of complex pole location in many physical systems. The algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation (ODE) sol...

This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge–Kutta or similar schemes; indeed, in some cases thes...

In many branches of the science, the numerical solution of differential equations is a natural request. However, the way to solve those equations is not always the same. Sometimes it is needed to solve the equation as fast as possible. In other cases good precision is required, even 100 digits or more to guarantee the final results. Sensitivity ana...

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 i...

Analytical theories based on Lie-Deprit transforms are used to ob-tain families of periodic orbits for the problem of an orbiter around the Moon. Low and high orbit models are analyzed. Equilibria of the normalized equations of motion provides the representation of a global portrait of families of frozen orbits depending on values of the inclinatio...

In the analytical approach to the main problem in satellite theory, the consideration of the physical parameters imposes a lower bound for normalized Hamiltonian. We show that there is no elliptic frozen orbits, at critical inclination, when we consider small values of H, the third component of the angular momentum. The argument used suggests that...

The design of spatial missions to Mars requires the development of analytical theories in order to put artificial satellites in orbit around Mars. In this paper, we present a complete third order analytical model of a satellite perturbed by the zonal J 2, ..., J 6 harmonics of the Mars potential. Two Lie transformations, the elimination of the Par...

Computer Algebra Systems are developing very fast and it is now possible to use new computational power very efficiently to analytically integrate dynamical systems. However, the task of producing an appropriate program is time consuming and requires a considerable amount of skills and practice. Here the merits of numerical versus computer algebrai...

Kepler’s generalized equation is a transcendental nonlinear equation that must be solved in order to compute the position and velocity of an artificial satellite at any instant t. In this paper, we propose a method to solve analytically that equation. The method is based on the properties of non canonical Lie transformations and, under certain cond...

Computer Algebra Systems are developing very fast and it is now possible to use new computational power very efficiently to analytically integrate dynamical systems. However, the task of producing an appropriate program is time consuming and requires a considerable amount of skills and practice. Here the merits of numerical versus computer algebrai...

The importance of the development of analytical theories for the motion of artificial satellites is well known, however the involved algebra makes very difficult each advance in the efficience of such theories. We propose here a method to improve the precision of the theories with less computational effort: a detailed study of the relative value of...

The first-order analytical theory of spacecraft having significant shape ellipticity about second degree and order gravity field was described. The body was assumed to have uniform rotation around its axis of greatest inertia and this model included all main perturbations such as Keplarian attraction, Coriolis force and ellipticity perturbations. A...

Efficiency in handling Poisson series is essential to obtain high-accuracy analytical theories in celestial mechanics and non-linear dynamics in general. A good knowledge of the mathematical structure of these objects is fundamental to create data structures to store and handle efficiently its equivalent computational object. In this paper we analy...

Special efficient Poisson series processors (PSP) have been created. Poison series appear frequently in problems of non-linear dynamics and celestial mechanics and their size makes their manipulation by means of general computer algebra systems (CAS) inefficient, but the characteristics of the problems suggest the use of general CAS with other gene...

Various phenomena of resonances and bifurcations are explained by one-parameter models made of a simple pendulum under perturbation. The model we bring here involves two parameters, and thus covers more generic situations than the simple pendulum. The solutions are exact; they are expressed in elliptic functions. Periodic perturbations of the model...

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...

ith a unit element. We call it series by an abuse of language, because although it may have a finite or infinite number of terms, in practice, in order to define an equivalent computational object, we need to consider only series with a finite number of terms. Usually, high order theories in Celestial Mechanics include series with a very big number...

An abstract is not available.

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 ge...

Poisson series appear frequently in problems of non-linear dynamics and celestial mechanics. The size of such mathematical objects makes their manipulation by means of general symbolic processors (GSP) inefficient. Special processors named Poisson series processors (PSP) have been created to handle them in a more efficient way. We propose here a wa...

In this paper, we deal with the stellar three body problem, that is one star is far away from the other two stars. The outer orbit is assumed to be Keplerian. To analyze the effect of the distant star on the orbit of the close stars, we use the Gauss method; this method consist in replacing the gravitational attraction of the third star by the grav...

We present a new way of handling perturbed non-linear oscillators. The solution of the equations of motion of a non-linear oscillator, generally, involves elliptic functions. The main key of our method consists in keeping these elliptic functions in the unperturbed part, and expanding the perturbation as a Fourier series of the amplitude, in which...

This article discusses the stellar three-body problem using an approximation in which the outer orbit is assumed to be Keplerian. The equations of motion are integrated by the stroboscopic method, i.e., basically at successive periods of a rapidly changing variable (the eccentric anomaly of the inner orbit). The theory is applied to the triple-star...

When perturbation methods are applied to the coupled translational- rotational motion of two rigid bodies, there are several possibilities in defining the different terms which constitute the Hamiltonian form; even the zero order may be different depending on the various hypotheses. Usually in the literature, authors choose the model and in most of...

The equations of the motion of a perturbed keplerian orbit are formulated in a non standard way which represents plainly the dynamical behavior of the solution, and avoids singularities in inclination. Quaternions are used for describing rotations between the different frames with a consequent increase of speed and precision. Several tests are perf...

The new techniques of algebraic manipulation, especially the Poisson Series Processor, permit the analytical integration of the more and more complex problems of celestial mechanics. The authors are developing a new Poisson Series Processor, PSPC, and they use it to solve the Lagrange equation of the orbital motion. They integrate the Lagrange equa...

Previous work by the authors has been used to revise the orbit of the binary 94 Aquarii, using both interferometric and spectroscopic observations.

The rotation of a rigid body which is moving near L(4) is considered. Based upon Euler angles, an extended set of quaternions and its conjugate moments is employed, and the canonical equations of the motion are established. Finally, a numerical example is given.

An extension of Docobo's analytical method for calculating the orbits of visual double stars allows all orbital elements to be obtained from three interferometer observations (θ,ρt) and three spectroscopic observations (v;t).

An algebraic processor plus an interactive graphics system transform a desk computer into a very handy tool for research in non linear mechanics.

This paper deals with the application of the stroboscopic method to a canonical set of variables; namely, Hill variables. The equations of the method, up to the first order, are given and are applied to the zonal artificial satellite. In this way the number of equations to be integrated is reduced.

The behaviour of a subsystem of a many-body problem may be studied using the hierarchical relative coordinates defined by Nugeyre and Bouvier (1981), The present article employs a notation allowing the position vectors of the particles to be expressed explicity as functions of the hierarchical relative coordinates, This machinery is used to general...

The second order stroboscopic method of Roth (1979) is modified and applied to artificial satellite theory in order to solve the equations of motion of an intermediary radial formulated in Hill variables. The strobosopic method allows easy inclusion of all types of perturbations and increases the speed and accuracy of computations, even after many...

Two different analytical techniques for solving visual double star orbits with inclination equal to 90° are given. The first is based on Docobo's method, and the second is derived by using the Fourier transform for angular distances. Both techniques are applied to the pairs 19192S2442 and 19338S2339.

A new method of improving a preliminary orbit is found by means of the corrections of the Fourier coefficients. An application of the method is made for the systems HD 149240 and 105 Herculis.

The rotational motion of a rigid body satellite whose center of mass is moving near the libration point L4 is studied using the Euler parameters. The Hamiltonian form of the equations of motion is obtained, and inherent difficulties of the problem (singularities, triaxial complexity, etc) do not appear.

An algorithm to compute periodic orbits of dynamical systems up to an arbitrary number of precision digits is presented. The algorithm is based on an optimized Newton-Raphson method combined with a new numerical ODE solver, TIDES, that uses a Taylor series method. Finally, we present some numerical tests for the Lorenz model and the Hénon-Heiles Ha...

Languages like C++, based on the object oriented programming, add new posibilities to the creation of more modern and efficient symbolic software. In this paper we present the Lie trasformations as a new computational object that complement to the Poisson Series. We characterize, from a computational ponit of view, not only the canonical and non-ca...