Jacques Henrard
Université de Namur
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Featured researches published by Jacques Henrard.
Celestial Mechanics and Dynamical Astronomy | 1990
Jacques Henrard
A detailed account is given of a semi-numerical perturbation method which has been proposed and improved upon in a succession of previous papers. The method analyses the first order effect of a small perturbation applied to a non-trivial two-degreeof-freedom separable Hamiltonian system (including the description of resonances) and construct approximate surfaces of section of the perturbed system. When the separable Hamiltonian system is already the description of a resonance, as it is the case in the problems we have investigated in the previous papers, these resonances are actually secondary resonances. The key role of the method is the numerical description of the angle-action variables of the separable system. The method is thus able to describe the perturbations of non-trivial separable systems and is not confined to the analysis of a small neigborhood of their periodic orbits.
The Astronomical Journal | 1967
André Deprit; Jacques Henrard
Abstract : In reference to any solution of a conservative dynamical system with two degrees of freedom, Hills equation is generalized to encompass non- necessarily isoenergetic displacements as well as the isoenergetic displacements caused by a variation of a parameter. This new variational equation is made the foundation of a methodical procedure for continuing numerically natural families of periodic orbits. The method consists of two steps-- an isoenergetic corrector and a tangential predictor. Although the algorithm makes no assumption of symmetry on the periodic orbits to be continued, special attention is paid to the symmetric orbits, but only to show how in these cases the method can be simplified substantially.
Celestial Mechanics and Dynamical Astronomy | 1987
Jacques Henrard; Charles Murigande
We analyse in details a simple dynamical system proposed by G. Colombo for the description of the rotational state of planets and satellites. We show that the derivatives of the critical areas are simple analytical functions of the parameters of the problem. These quantities are instrumental in computing the probabilities of capture of the precession of the spin axis in resonance with the precession of the orbit.
Celestial Mechanics and Dynamical Astronomy | 1990
Jacques Henrard; N.D. Caranicolas
The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdoms perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdoms uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance ωp = 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode.
Celestial Mechanics and Dynamical Astronomy | 1991
Alessandro Morbmelli; Jacques Henrard
In this paper a theoretical perturbation approach to the problem of the dynamics in secular resonance is exposed. This approach avoids any expansion of the main term of the Hamiltonian (linear term in the masses) with respect to the eccentricity or the inclination of the asteroid, in order to achieve results valid for any value of these variables. Moreover suitable action-angle variables are introduced to take properly into account the dynamics related to the motion of the argument of perihelion of the asteroid, which is relevant at high inclination. A class of secular resonances wider than that usually considered is found. An explicit computation of the location of the main secular resonances, estimating also the contribution of the quadratic term in the masses by means of classical series expansion, is reported in the last sections. The accuracy of computations obtained by series expansion is discussed in the paper.
Icarus | 1987
Jacques Henrard; Anne Lemaitre
Following the ideas of J. Wisdom (1985, Icarus 63, 272–289), an analytical perturbation theory for the 21 Jovian resonance in the planar elliptic-restricted problem is presented. The predictions of the theory are in good agreement with the features found numerically by C.D. Murray (1986, Icarus 65, 70–82) for the problem truncated at the second order in the eccentricities. On the whole it is found that large parts of the phase space are preserved from chaotic motions or large perturbations in eccentricity. Wisdoms effect (spreading by Jupiter eccentricity of the chaotic motion generated close to the critical curve) is present but confined to a small volume of the phase space. Murrays “central region” where the eccentricity starting from e = 0.15 can reach values larger than 0.5 is due to truncation effects.
Icarus | 1983
Jacques Henrard; Anne Lemaitre
Abstract In this paper an analytical model describing the effect of a displacement of the Jovian resonances in the asteroid belt is analyzed. It is found that small displacement can transform a truncated uniform density distribution of asteroids into a gap. As a possible explanation for the displacement, the effect of the removal of an accretion disk in the early stage of the solar system is investigated. It is found that removal of a disk containing a few percent of the solar mass between the orbit of the asteroids and the orbit of Jupiter is sufficient to account for the observed Hecuba gap.
Journal of Differential Equations | 1973
Jacques Henrard
Abstract Lyapunovs center theorem relative to the existence of families of periodic orbits emanating from an equilibrium is generalized to cases where a resonance occurs between two basic frequencies. Analytical Hamiltonian systems are considered and the theorems depend on the nonannulation of an invariant of the system. The proof is performed in two steps. In a first step the theorems are shown to be valid for some approximation of the Hamiltonian system. These results are described in a previous paper ( Henrard, 1970 ) and are only summarized here. In a second step Poincares perturbation theorem is generalized in order to transfer to the original system the conclusions relatives to its approximations. In the conclusion, our results are compared with similar results published recently.
Celestial Mechanics and Dynamical Astronomy | 1991
Alessandro Morbidelli; Jacques Henrard
In this paper an analytical model, suitable for a global description of the dynamics in a secular resonance of order 1, is derived from the general perturbation study developed in a previous paper (Morbidelli and Henrard (1991)). Such a model is then used to study the secular resonances ν6, ν5 and ν16, and pictures illustrating the secular motion are obtained. The peculiarities of the ν5 resonances are discussed in detail. The results are compared with those obtained by the theories of Yoshikawa and Nakai-Kinoshita. Some numerical simulations performed by Ch. Froeschlé and H. Scholl are discussed in the light of the new theoretical results. New numerical experiments on the ν6 resonance are also presented.
Celestial Mechanics and Dynamical Astronomy | 1986
Jacques Henrard
Resonance in the restricted three body problem usually lead by averaging to one-degree of freedom Hamiltonian systems described by the Hamiltonian Ho (p,P).