Asteroidal Resonances Research Articles

A Global Study of the 3[rcolon]1 Resonance Neighborhood: A Search for Unstable Asteroids

The recent discovery of two asteroids with unstable orbits in the neighborhood of the 5 : 2 resonance raises the question whether there exist unstable asteroids in the neighborhood of the more important 3 : 1 asteroidal resonance. The two asteroids have short dynamical lifetimes of about 10 and 50 Myr. Semimajor-axis drift due to the Yarkovsky effect can explain their existence in such a short-lived orbit. Since this effect is not specific to the 5 : 2 resonance, a natural question is whether there exist short-lived asteroids near other resonance borders. This question is interesting because by studying such objects we can better understand the production of near-Earth asteroids (NEAs) via the collision and Yarkovsky processes. We search for short-lived asteroids in the neighborhood of the 3 : 1 resonance. The borders of this resonance and the positions of the asteroids with respect to them are determined by using the mean elliptic planar restricted three-body problem and a method that invokes the concept of a representative A semianalytical global study of the orbits for this nonlinear dynamical system is made through this convenient plane. The limit between the circulation and libration or chaotic orbits in this simple problem is taken as the edge of the resonance. We then choose 41 numbered asteroids that are closest to the resonance borders and have inclinations smaller than 10°. The orbits of these asteroids are numerically integrated using the full equations of motion in the gravitational field of the Sun and the planets Jupiter, Saturn, Mars, Earth, and Venus. We find that about 40% of the studied asteroids enter into the 3 : 1 resonance within 100 Myr and are subsequently dynamical eliminated. The existence of a nonnegligible number of asteroids in the 3 : 1 resonance neighborhood implies that a process must be continually acting to replenish this region. We suggest that these asteroids may have wandered to the resonance borders through small semimajor-axis displacements due to the Yarkovsky effect. This result also confirms that asteroids are continually feeding the chaotic 3 : 1 resonance region and consequently the Mars crossers and NEA population.

A symplectic mapping approach of the dynamics of the Hecuba gap

In this work, we develop a symplectic mapping allowing the study of the dynamical behavior of asteroidal resonances. To obtain such a mapping, we combine the symplectic scheme proposed by Hadjidemetriou together with an asymmetric expansion of the disturbing function which takes into account the inclinations of both the perturber and the disturbed bodies. This mapping is applied to the 2/1 and 3/2 mean motion resonances in the asteroidal belt. We explore a wide range of initial conditions in the phase space in order to get a large number of results to allow some statistical conclusions about the diffusion mechanisms acting in these resonances.

The depletion of the Hecuba gap vs the long-lasting Hilda group

This paper presents a comparative analysis of the 2 1 and 3 2 asteroidal resonances based on several analytical and numerical tools. The frequency map analysis was used to obtain a refined estimation of the chaotic transport. Fourier and wavelet analyses were used to construct the web of inner resonances and showed that they are the seat of the strongly unstable motion observed in the numerical simulations. The most regular regions in both resonances were classified. A fast symplectic mapping allowed a number of direct runs over 10 8 years of the orbits initially in these regions. The stability of orbits over the age of the solar system was discussed and compared to the distribution of the observed asteroids in both resonances.

The Determinant Role of Jupiter's Great Inequality in the Depletion of the Hecuba Gap

This paper deals with the influence of Jupiter's Great Inequality (GI) on the orbital evolution of 2:1 resonant asteroids. This perturbation of Jupiter's motion enhances the chaotic diffusion of orbits and the depletion of asteroids in the Hecuba gap. The failure of models that adopt for Jupiter an elliptic motion with only secular perturbations is explained. We also show the dependence of the diffusion rates on the GI period, and the maximum diffusion rates that would take place if the GI period were closer to the libration period. Significant similar effects are absent in the 3:2 asteroidal resonance.

Escape of asteroids from the Hecuba gap

The dynamics of the 2 1 mean-motion asteroidal resonance with Jupiter is studied by numerical integration of the equations of motion of the Sun-Jupiter-Saturn-asteroid system. The measurement of the fundamental asteroidal frequencies by means of Fourier and wavelet analyses allows us to construct the web of the secular, secondary and Kozai resonances inside the 2 1 - resonance boundaries. The structure of the phase space of the 2 1 resonance is discussed with emphasis on the acting depletion mechanisms due to presence of these inner resonances. Special attention is paid to the study of the middle-eccentricity depleted region. The importance of the great inequality of the Jupiter-Saturn system in the acceleration of the diffusion processes in this region is pointed out. The existence of a group of asteroids like (3789) Zhongguo, inside the 2 1 resonance, is also discussed.

A symplectic mapping approach to the study of the stochasticity in asteroidal resonances

In the case of the 2:1 and 3:2 resonances with Jupiter, it has not been yet possible to have a complete identification of all chaotic diffusion processes at work, mainly because the time scale of some of them are of an order still out of the reach of precise integrations. A planar Hadjidemetriou's mapping, using expansions valid for high eccentricities and scaled in order to accelerate the diffusion processes, was derived. The solutions obtained with the mapping show huge eccentricity variations in all orbits starting in the middle of the 2:I resonance, when the main short-period perturbations of Jupiter's orbit are considered. The solutions starting in the middle of the 3:2 resonance do not show any important diffusion.

Kirkwood Gaps and Resonant Groups

This paper is a short review of the dynamics of the asteroidal resonances as currently determined from maps and simulations over 106 – 107 years. The main recent results concern the extensive exploration of the phase space to determine domains of initial conditions leading to close approaches to the inner planets, the topological dynamics of the planar Sun-Jupiter-asteroid problem at very high eccentricities and the differences amongst 2/1 and 3/2 resonances able to explain the existence of a gap in the asteroidal belt at the 2/1 resonance and of a group of asteroids in the 3/2 resonance. Current results point to a confirmation of Wisdom's theory for the formation of the gaps by gravitational evolution and scattering by the inner planets.

Chaotic behaviour of trajectories for the asteroidal resonances

Chaotic behaviour of trajectories for the asteroidal resonances

Mappings for the First Order Asteroidal Resonance

We construct a two step algebraic mapping from Sessin's simplified model for the first order resonance. The orbits obtained with this mapping are compared to the ones calculated with the exact solution. We also derive a reduced Hamiltonian. A plane Poincaré mapping, using delta periodic function, is constructed and compared to the reduced Hamiltonian contour curves showing the splitting of the separatrix due to delta perturbation technique.

ASTEROIDAL SOURCE OF ORDINARY CHONDRITES*

The orbital evolution of asteroidal fragments with diameters ranging from 10 cm to 20 km, injected into the 3:1 Kirkwood gap at 2.50 A.U., has been investigated using Monte Carlo techniques. It is assumed that this material can become Earth‐crossing on a time scale of 106 years, as a result of a chaotic zone discovered by Wisdom, associated with the 3:1 resonance. This phenomenon, as well as close encounter planetary perturbations, the v6 secular resonance, and the ablative effects of the Earth's atmosphere are included in the determination of the orbital characteristics of meteorites impacting the Earth derived by fragmentation of this asteroidal material. It is found that the predicted meteorite orbits closely match those found for observed ordinary chondrites, and the total flux is in approximate agreement with the observed fall rate of ordinary chondrites. About 10% of the predicted impacting bodies are meteorite‐size bodies originating directly from the asteroid belt. The remainder are obtained by subsequent fragmentation of larger (∼1 m to 20 km diameter) Earth‐crossing asteroidal fragments. The largest of these fragments are observable as Apollo‐Amor objects. Thus the apparent paradox between the orbital characteristics of observed ordinary chondrites and those predicted from Apollo object sources is reconciled. Both appear to be complementary aspects of the same phenomena. No other asteroidal resonance is found to be satisfactory as a source of ordinary chondrites. These meteorites are therefore most likely to be derived from S asteroids in this limited region of the asteroidal belt, the largest of which are 11 Parthenope, 17 Thetis, and 29 Amphitrite.