A. Elipe - Academia.edu (original) (raw)

Papers by A. Elipe

Acta Astronautica, 2021

This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, pert... more This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, perturbed Kepler Hamiltonians which perturbation is of the form N j=2 A j /r j with A j 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 Artificial Satellite Theory, the Cid's radial intermediary. For this problem, we integrate the equations of motion in two different ways, by means of the elliptic P-Weierstrass function and by using the Krilov-Bogoliubov averaging method to integrate a perturbed harmonic oscillator. In this case, the resulting solution is given in terms of the classical Kepler's equation, with no need of introducing more complex generalized Kepler's equation.

Impact of Modern Dynamics in Astronomy, 1999

In this paper, we consider the motion of a particle under the gravitational field of a massive st... more 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 planets's moons. For this potential, we find several families of periodic orbits and bifurcations.

Chaos, Solitons & Fractals, 2007

Many dynamical systems may be represented in a set of non-canonical coordinates that generate an ... more Many dynamical systems may be represented in a set of non-canonical coordinates that generate an su(2) algebraic structure. The topology of the phase space is the one of the S 2 sphere, the Poisson structure is the one of the rigid body, and the Hamiltonian is a parametric quadratic form in these "spherical " coordinates. However, there are other problems in which the Poisson structure losses its symmetry. In this paper we analyze this case and, we show how the loss of the spherical symmetry affects the phase flow and parametric bifurcations for the bi-parametric cases.

Advances in Space Research, 2012

The dynamics of orbits around planetary satellites, taking into account the gravitational attract... more The dynamics of orbits around planetary satellites, taking into account the gravitational attraction of a third-body and the non-uniform distribution of mass of the planetary satellite, is studied. The Hamiltonian considered is explicitly time-dependent. Conditions for frozen orbits are presented. Low-altitude, near-polar orbits, very desirable for scientific missions to study planetary satellites such as the Jupiter's moon Europa, are analyzed. Lifetimes for these orbits are computed through the single and double averaged method. Comparison between the results obtained by the single and double averaged method is presented. The single-averaged model is more realistic, since it does not eliminate the term due to the equatorial ellipticity of the planetary satellite as done by the double-averaged problem. Considering the single-averaged method, we found unstable frozen orbits where the satellite does not impact with the surface of Europa for at least 200 days. We present an approach using the unaveraged disturbing potential to analyze the effects of these terms in the amplitude of the eccentricity.

International Journal of Remote Sensing, 2003

Page 1. “garcia de galdeano” garcía degaldeano seminario matemático n. 23 PRE-PUBLICACIONES del .... more Page 1. “garcia de galdeano” garcía degaldeano seminario matemático n. 23 PRE-PUBLICACIONES del ... 11 Page 13. References [1] Born, GH, Mitchel, JL and Heyler, GA, 1987, GEOSAT ERM—Mission Design. Journal of the Astronautical Sciences, 35, 119–134. ...

Applied Mathematics and Computation, 2013

AbstractThis paper reviews the SerretAndoyer (SA) canonical formalism in rigid-body dynam-ics, an... more AbstractThis paper reviews the SerretAndoyer (SA) canonical formalism in rigid-body dynam-ics, 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 oered 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 SerretAndoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specically, 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...

In the software market, a number of applications designed for symbolic computations and running o... more In the software market, a number of applications designed for symbolic computations and running on desk computers are continously appearing. Among them the authors use Mathematica to handle the very discouraging expressions appearing in the attitude dynamics problem. The graphic capability of Mathematica is used in the qualitative analysis of the attitude of an artificial satellite, in order to locate where the possible bifurcations take place.

In this paper, stationary motions and their stabilities are investigated for a system of two rigi... more In this paper, stationary motions and their stabilities are investigated for a system of two rigid bodies, one spherical and the other with a nonspherical distribution of density, but whose ellipsoid of inertia is a sphere. Under the assumption of a planar orbit, stationary solutions and stability conditions are found.

“garcia de galdeano” garcía de galdeano seminario matemáticon. 22 PRE-PUBLICACIONES del seminario... more “garcia de galdeano” garcía de galdeano seminario matemáticon. 22 PRE-PUBLICACIONES del seminario matematico 2002

Nonlinear Science and Complexity, 2011

Astrophysics and Space Science Library, 1986

Two sufficient conditions for the existence of the Jacobian constant in the restricted circular t... more Two sufficient conditions for the existence of the Jacobian constant in the restricted circular three body problem, when the satellite body is a gyrostat, are given. In both conditions, the analytical expression of Jacobian constant is found. The rigid body and the classical cases are obtained as particular solutions.

Astronomical & Astrophysical Transactions, 2001

Central forces are very interesting in Mechanics, since they are simple and model quite well seve... more Central forces are very interesting in Mechanics, since they are simple and model quite well several physical problems. Here we consider a potential function of the type U (r) = −A/r − B/(2r 2). For this potential by means of the regulazing Sundman's transformation, and making use of Stumpff's functions, we obtain a solution that is regular and valid for all possible types of motion, bounded or not.

In the software market, a number of applications designed for symbolic computations and running o... more In the software market, a number of applications designed for symbolic computations and running on desk computers are continously appearing. Among them the authors use Mathematica to handle the very discouraging expressions appearing in the attitude dynamics problem. The graphic capability of Mathematica is used in the qualitative analysis of the attitude of an artificial satellite, in order to locate where the possible bifurcations take place.

The authors consider a dynamical system consisting of a couple of elliptic oscillators in 1:1 res... more The authors consider a dynamical system consisting of a couple of elliptic oscillators in 1:1 resonance, perturbed by a quartic potential. The perturbed Hamiltonian is expressed in terms of Lissajous variables, and after that, it is normalized up to the first order. The averaged Hamiltonian is analyzed, and the phase flow evolution with the parameters is shown. In this way,

Acta Astronautica, 2021

This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, pert... more This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, perturbed Kepler Hamiltonians which perturbation is of the form N j=2 A j /r j with A j 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 Artificial Satellite Theory, the Cid's radial intermediary. For this problem, we integrate the equations of motion in two different ways, by means of the elliptic P-Weierstrass function and by using the Krilov-Bogoliubov averaging method to integrate a perturbed harmonic oscillator. In this case, the resulting solution is given in terms of the classical Kepler's equation, with no need of introducing more complex generalized Kepler's equation.

Impact of Modern Dynamics in Astronomy, 1999

In this paper, we consider the motion of a particle under the gravitational field of a massive st... more 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 planets's moons. For this potential, we find several families of periodic orbits and bifurcations.

Chaos, Solitons & Fractals, 2007

Many dynamical systems may be represented in a set of non-canonical coordinates that generate an ... more Many dynamical systems may be represented in a set of non-canonical coordinates that generate an su(2) algebraic structure. The topology of the phase space is the one of the S 2 sphere, the Poisson structure is the one of the rigid body, and the Hamiltonian is a parametric quadratic form in these "spherical " coordinates. However, there are other problems in which the Poisson structure losses its symmetry. In this paper we analyze this case and, we show how the loss of the spherical symmetry affects the phase flow and parametric bifurcations for the bi-parametric cases.

Advances in Space Research, 2012

The dynamics of orbits around planetary satellites, taking into account the gravitational attract... more The dynamics of orbits around planetary satellites, taking into account the gravitational attraction of a third-body and the non-uniform distribution of mass of the planetary satellite, is studied. The Hamiltonian considered is explicitly time-dependent. Conditions for frozen orbits are presented. Low-altitude, near-polar orbits, very desirable for scientific missions to study planetary satellites such as the Jupiter's moon Europa, are analyzed. Lifetimes for these orbits are computed through the single and double averaged method. Comparison between the results obtained by the single and double averaged method is presented. The single-averaged model is more realistic, since it does not eliminate the term due to the equatorial ellipticity of the planetary satellite as done by the double-averaged problem. Considering the single-averaged method, we found unstable frozen orbits where the satellite does not impact with the surface of Europa for at least 200 days. We present an approach using the unaveraged disturbing potential to analyze the effects of these terms in the amplitude of the eccentricity.

International Journal of Remote Sensing, 2003

Page 1. “garcia de galdeano” garcía degaldeano seminario matemático n. 23 PRE-PUBLICACIONES del .... more Page 1. “garcia de galdeano” garcía degaldeano seminario matemático n. 23 PRE-PUBLICACIONES del ... 11 Page 13. References [1] Born, GH, Mitchel, JL and Heyler, GA, 1987, GEOSAT ERM—Mission Design. Journal of the Astronautical Sciences, 35, 119–134. ...

Applied Mathematics and Computation, 2013

AbstractThis paper reviews the SerretAndoyer (SA) canonical formalism in rigid-body dynam-ics, an... more AbstractThis paper reviews the SerretAndoyer (SA) canonical formalism in rigid-body dynam-ics, 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 oered 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 SerretAndoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specically, 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...

In the software market, a number of applications designed for symbolic computations and running o... more In the software market, a number of applications designed for symbolic computations and running on desk computers are continously appearing. Among them the authors use Mathematica to handle the very discouraging expressions appearing in the attitude dynamics problem. The graphic capability of Mathematica is used in the qualitative analysis of the attitude of an artificial satellite, in order to locate where the possible bifurcations take place.

In this paper, stationary motions and their stabilities are investigated for a system of two rigi... more In this paper, stationary motions and their stabilities are investigated for a system of two rigid bodies, one spherical and the other with a nonspherical distribution of density, but whose ellipsoid of inertia is a sphere. Under the assumption of a planar orbit, stationary solutions and stability conditions are found.

“garcia de galdeano” garcía de galdeano seminario matemáticon. 22 PRE-PUBLICACIONES del seminario... more “garcia de galdeano” garcía de galdeano seminario matemáticon. 22 PRE-PUBLICACIONES del seminario matematico 2002

Nonlinear Science and Complexity, 2011

Astrophysics and Space Science Library, 1986

Two sufficient conditions for the existence of the Jacobian constant in the restricted circular t... more Two sufficient conditions for the existence of the Jacobian constant in the restricted circular three body problem, when the satellite body is a gyrostat, are given. In both conditions, the analytical expression of Jacobian constant is found. The rigid body and the classical cases are obtained as particular solutions.

Astronomical & Astrophysical Transactions, 2001

Central forces are very interesting in Mechanics, since they are simple and model quite well seve... more Central forces are very interesting in Mechanics, since they are simple and model quite well several physical problems. Here we consider a potential function of the type U (r) = −A/r − B/(2r 2). For this potential by means of the regulazing Sundman's transformation, and making use of Stumpff's functions, we obtain a solution that is regular and valid for all possible types of motion, bounded or not.

In the software market, a number of applications designed for symbolic computations and running o... more In the software market, a number of applications designed for symbolic computations and running on desk computers are continously appearing. Among them the authors use Mathematica to handle the very discouraging expressions appearing in the attitude dynamics problem. The graphic capability of Mathematica is used in the qualitative analysis of the attitude of an artificial satellite, in order to locate where the possible bifurcations take place.

The authors consider a dynamical system consisting of a couple of elliptic oscillators in 1:1 res... more The authors consider a dynamical system consisting of a couple of elliptic oscillators in 1:1 resonance, perturbed by a quartic potential. The perturbed Hamiltonian is expressed in terms of Lissajous variables, and after that, it is normalized up to the first order. The averaged Hamiltonian is analyzed, and the phase flow evolution with the parameters is shown. In this way,