Anne Lemaitre

A perturbative treatment of the 21 Jovian resonance

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.

A mechanism of formation for the Kirkwood gaps

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.

The reducing transformation and Apocentric Librators

We propose a canonical transformation reducing the averaged planar planetary problem near resonance to a one degree of freedom problem when the perturbation is truncated at the first order in the eccentricities.This reducing transformation leads to a very simple explanation of the puzzling behaviour of the Apocentric Librators, a class of asteroids identified by Franklinet al. (1975).An exploration of the phase space of the average problem with the use of the mapping technique shows that the alternation of two libration mechanism is a common feature for initial conditions near, but not inside, the deep resonance region.

THE UNTANGLING TRANSFORMATION

It is always possible to find canonical forms for quadratic Hamiltonians. In cases in which the eigenvalues of the associated linear system are simple, either real or pure imaginary, the canonical form introduces action-angle coordinates that are most useful for the application of perturbation theory; this is the case in which the quadratic Hamiltonian is the first term in the expansion of a nonlinear Hamiltonian around an equilibrium. The general theory is rather involved, and it may be worthwhile to find shortcuts in simple situations. We present here such a shortcut for a quadratic Hamiltonian of 2 degrees of freedom with a symmetry that is often present in celestial mechanics. The canonical transformation proposed has the added advantage of having a clear geometric interpretation and of being a generalization of the so-called reducing transformation that has been useful in several problems.

Secular resonances in the primitive solar nebula

We present here a very simple model that could explain the relatively high eccentricities and inclinations observed in the minor planet belt. This model is based upon the sweeping of the secular resonances ν6 and ν16 through the belt due to the gravitational effect of the dissipation of a primitive solar nebula. The sweeping of the ν16 secular resonance (responsible for the high inclinations) is very sensitive to the density profile of the nebula. For the model to work we need a density profile proportional to ϱ−k with κ between 1.0 and 1.5.

Coupled rotational motion of Mercury

We present a simple dynamical model of the rotation of Mercury in which the Hermean rotation is composed of two commensurabilities: (i) a 3:2 spin-orbit resonance between fast variables and (ii) a 1:1 synchronous precession of both orbital and rotational nodes. We investigate the coupling between these two degrees of freedom. First, we study the global phase space of Mercury and quantify the libration areas. Second, we concentrate on the present location of Mercury. The impact of the slow degree of freedom on the fast one can be modeled through the adiabatic invariant, whereas the impact of the fast degree of freedom on the slow one is clearly represented by Poincare sections. In addition, the adiabatic invariant theory leads to a simple analytical model of the rotation of Mercury where the two coupled degrees of freedom are taken into account. This model can be used in different applications that require a first-order rotational motion such as the one describing the influence of the precession and rotation of the planet on the orbit of an artificial satellite around Mercury.

Dissipative forces and external resonances

We analyze the process of resonance trapping due to Poynting–Robertson drag and Stokes drag in the frame of the restricted 3-body problem and in the case of external mean motion resonances. The numerical simulations presented are computed by using the 3-dimensional extended Schubart averaging (ESA) integrator developed by Moons (1994) for all mean motion resonances. We complete it by adding the contributions of the dissipative forces. To follow the philosophy of the initial integrator, we average the drag terms, but we do not make any expansion in series of eccentricity or inclination. We show our results, especially capture around asymmetric equilibria, and compare them to those found by Beaue and Ferraz-Mello (1993, 1994) and Liou et al. (1979).

Asteroid Motion Near the 2:1 Resonance: A Symplectic Mapping Approach

We present a symplectic mapping model that is valid close to the 2:1 resonance in a Sun-Jupiter-asteroid system, for planar motion. The mapping is based on the averaged Hamiltonian close to this resonance, where an additional correction term has been introduced in order to restore the correct position and stability of the fixed points. The topology of the mapping is similar to that of the Poincare map of the real system. The evolution of the eccentricity of the asteroid is presented and it is shown that for a large region of phase space there is a diffusion to very large values, leading to escape of the asteroid from the 2:1 resonance. There are however regions in phase space where the eccentricity remains bounded, in the framework of this model.

Proper Elements: What are They ?

The general ideas about the calculation of proper elements are given here, followed by a comparison of the hypotheses and simplifications made in three different theories: YKM theory, composition of the contributions of Yuasa, Kneževic and Milani, W theory corresponding to Williams’s work and HLM theory, paper of Lemaitre and Morbidelli, based on Henrard’s semi numerical method. Some short numerical comparisons conclude the paper.

Second Fundamental Model of Resonance with Asymmetric Equilibria

We study the position and the stability of the equilibria for a generic Hamiltonian function developed up to the second harmonic and depending on two parameters; we describe the topology of the phase space for fixed values of these parameters. We show that for some values of the parameters asymmetric equilibria (unstable or stable) may appear. We deduce the conditions of capture into first order resonances for slowly drifting systems. We apply this model to the restricted three-body problem.