Asteroid motion near the 3 : 1 resonance

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.     

Numerical investigation of motions of resonant asteroids in the three-dimensional space

Motions of asteroids in mean motion resonances with Jupiter are studied in three-dimensional space. Orbital changes of fictitious asteroids in the Kirkwood gaps are calculated by numerical integrations for 10⁵ – 10⁶ years. The main results are as follows: (1) There are various motions of resonant asteroids, and some of them are very complicated and chaotic and others are regular. (2) The eccentricity of some asteroids becomes very large, and the variation of the inclination is large while the eccentricity is large. (3) In the 3:1 resonance, there is a long periodic change in the variation of the inclination, when (7 : ) is a simple ratio (7: longitude of perihelion, : longitude of node). (4) In the 7:3 resonance, the variation of the inclination of some resonant asteroids is so large that prograde motion becomes retrograde. Some asteroids in the 7:3 resonance can collide with the Sun as well as with the inner planets.     

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist.     

Meteorites from the asteroid 6 Hebe

We have numerically integrated the orbits of 18 fictitious fragments ejected from the asteroid 6 Hebe, an S-type object about 200km across which is located very close to theg=g ₆ (orv ₆) secular resonance at a semimajor axis of 2.425AU and a (proper) inclination of 15° .0. A realistic ejection velocity distribution, with most fragments escaping at relative speeds of a few hundredsm/s, has been assumed. In four cases we have found that the resonance pumps up the orbital eccentricity of the fragments to values >0.6, which result into Earth-crossing, within a time span of 1Myr; subsequent close encounters with the Earth cause strongly chaotic orbital evolution. The closest Earth and Mars encounters recorded in our integration occur at miss distances of a few thousandths ofAU, implying collision lifetimes <10⁹ yr. Some other fragments affected by the secular resonance become Mars-crossers but not Earth-crossers over the integration time span. Two bodies are injected into the 3 : 1 mean motion resonance with Jupiter, and also display macroscopically chaotic behaviour leading to Earth-crossing. 6 Hebe is the first asteroid for which a realistic collisional/dynamical evolutionroute to generate meteorites has been fully demonstrated. It may be the parent body of one of the ordinary chondrite classes.     

Stability of planar oscillations of a satellite in an elliptic orbit

A problem of stability of odd 2-periodic oscillations of a satellite in the plane of an elliptic orbit of arbitrary eccentricity is considered. The motion is supposed to be only under the influence of gravitational torques.Stability of plane oscillations was investigated earlier (Zlatoustovet al., 1964) in linear approximation. In the present paper a problem of stability is solved in the non-linear mode. Terms up to the forth order inclusive are taken into consideration in expansion of Hamiltonian in a series.It is shown that necessary conditions of stability obtained in linear approximation coincide with sufficient conditions for almost all values of parameters ande (inertial characteristics of the satellite and eccentricity of the orbit). Exceptions represent either values of the parameters ,e when a problem of stability cannot be solved in a strict manner by non-linear approximation under consideration, or values of the parameters which correspond to resonances of the third and fourth orders. At the resonance of the third order oscillations are unstable, but at the resonance of the fourth order both unstability and stability of the satellite's oscillations take place depending on the values of the parameters ,e.     

Resonant motion in the restricted three body problem

The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.     

Ideal Resonance Problem: the Post-Post-Pendulum Approximation

Explicit construction of the solutions of the Hamiltonian system given by H = H ₀(J) – A(J) cos (ideal resonance problem), two orders of approximation beyond the well-known pendulum approximation. The given solutions are valid for libration amplitudes of order . The procedure used is extended to allow the construction of the solutions of Hamiltonians with perturbations involving two degrees of freedom; the post-pendulum solution of an example of this kind is constructed.     

A perturbation method for problems with two critical arguments

We develop a semi-numerical perturbation method for problems with two critical arguments. We apply it to a truncated model of the restricted, elliptic three body problem in case of a resonance 2/1 We identify regions of the phase space where chaotic motion is expected because of the presence of homoclinic orbits. One of these regions, the largest one, sits at the entrance to the resonance zone and is associated with a 2/1 resonance between the two critical arguments. The results are compared with numerical results due to Murray (1986)     

Irregular behaviour and eccentricity increase in the trajectories of a dynamical system modelling asteroidal motions

Many trajectories of the third body are integrated numerically in a modified elliptical restricted three body problem (ERTBP), in which the eccentricity, e, of the orbit of the second primary varies sinusoidally with time. It is found that, in the case of the 2:1 resonance, the introduction of the time variability of e modifies significantly the behaviour of the trajectories of the third body. In particular their osculating eccentricity e, present the following two notable features: (a) In all cases it shows a definite chaotic variation, which appears at significantly shorter time-scales than the one found by Wisdom in the e = constant case. (b) In many cases it shows a significant increase, up and beyond the (critical) value e _crit = 0.52. As a result the third body approaches the first primary at distances smaller than 0.29 (where by we denote the semi-major axis of the trajectory of the second primary around the first), which in the actual Sun-Jupiter-asteroid problem corresponds to the semi-major axis of Mars. Our result might be of interest in the context of explaining the Kirkwood gaps at the resonances where the osculating eccentricity of asteroid trajectories calculated in the classical (e = constant) ERTBP does not reach Mars crosser values.     

One to One Resonance at High Inclination

We report results from long term numerical integrations and analytical studies of particular orbits in the circular restricted three-body problem. These are mostly high-inclination trajectories in 1 : 1 resonance starting at or near the triangular Lagrangian L₅ point. In some intervals of inclination these orbits have short Lyapunov times, from 100 to a few hundred periods, yet the osculating semi-major axis shows only relatively small fluctuations and there are no escapes from the 1 : 1 resonance. The eccentricity of these chaotic orbits varies in an erratic manner, some of the orbits becoming temporarily almost rectilinear. Similarly the inclination experiences large variations due to the conservation of the Jacobi constant. We studied such orbits for up to 10⁹ periods in two cases and for 10⁶ periods in all others, for inclinations varying from 0° to 180°. Thus our integrations extend from thousands to 10 million Lyapunov times without escapes of the massless particle. Since there are no zero-velocity curves restricting the motion this opens questions concerning the reason for the persistence of the 1 : 1 resonant motion. In the theory sections we consider the mechanism responsible for the observed behavior. We derive the averaged 1 : 1 resonance disturbing function, to second order in eccentricity, to calculate a critical inclination found in the numerical experiment, and examine motion close to this inclination. The cause of the chaos observed in the numerical experiments appears to be the emergence of saddle points in the averaged disturbing potential. We determine the location of several such saddle points in the (, ) plane, with being the mean longitude difference and the argument of pericentre. Some of the saddle points are illustrated with the aid of contour plots of the disturbing function. Motion close to these saddles is sensitive to initial conditions, thus causing chaos.This revised version was published online in October 2005 with corrections to the Cover Date.     

The main mean motion commensurabilities in the planar circular and elliptic problem

We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances ₅ and ₆ are located.     

Resonances and close approaches. I. The Titan-Hyperion case

The orbits of Titan and Hyperion represent an interesting case of orbital resonance of order one (ratio of periods 3/4), which can be studied within a reasonable accuracy by means of the planar restricted three-body problem. The behaviour of this resonance has been investigated by numerical integrations, of which we show the results in terms of the Poincaré mapping in the plane of the coordinates = [(2L – 2G)] cos ( _H – t)and = –,[(2L – 2G)] sin ( _H –t)keeping a constant value of the Jacobi integral throughout all integrations. We find the numerical invariant curves corresponding to low and high eccentricity resonance locking (which seem stable, at least during the limited time span of our experiments) and show that the observed libration of Hyperion's pericenter about the conjunction lies inside the stable high eccentricity region. If initial conditions are chosen outside the stable zones, we have no more stable librations, but a chaotic behaviour causing successive close approaches to Titan.We discuss these results both from the point of view of the mathematical theory of invariant curves, and with the aim of understanding the origin of the resonance locking in this case. The tidal evolution theory cannot be rigorously tested by such experiments (because of the dissipative terms which change the Jacobi constant); however, we note that the time scale of chaotic evolution is by many orders of magnitude smaller than the tidal dissipation time scale, so that the chaotic regions of the phase space cannot be crossed by a slow and smooth evolution. Therefore, our results seem to favour the hypothesis that Hyperion was formed via accumulation of the planetesimals originally inside a stable island of libration, while Titan was depleting by collisions or ejections the zones where the bodies could not escape the chaotic behaviour.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

A simple analytical model for the secular resonance ν6 in the asteroidal belt

When asteroids are in the secular resonance ₆, the variation of the eccentricity becomes very large. In this paper, the dynamics of this secular resonance ₆ is investigated by a simple analytical model, in which the third degree terms of the eccentricity and inclination are taken into account. The eccentricity variations of asteroids located near this resonance are represented clearly by the diagrams of equi-Hamiltonian curves on the plane of versuse ( the longitude of perihelion of asteroids and Saturn,e: the eccentricity of asteroids). These diagrams predict that the eccentricity of these asteroids suffers a large increase or decrease, and that the secular resonance argument librates about 0° and 180°. In order to confirm these predictions, numerical integrations are carried out over one million years. By these integrations, it is found that the eccentricity of secular resonant asteroids becomes more than 0.8, and that the libration about 0° also exists, as well as the libration about 180°. The strongly depopulated region in the asteroidal belt, which corresponds to the position of the secular resonance ₆, is also explained well by this analytical model.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

Exploring the 7:4 mean motion resonance—I: Dynamical evolution of classical transneptunian objects

In the transneptunian classical region (), an unexpected orbital excitation in eccentricity and inclination, dynamically distinct populations and the presence of chaotic regions are observed. For instance, the 7:4 mean motion resonance () appears to have been causing unique dynamical excitation according to observational evidences, namely, an apparent shallow gap in number density and anomalies in the colour distribution, both features enhanced near the 7:4 mean motion resonance location. In order to investigate the resonance dynamics, we present extensive computer simulation results totalizing almost 10,000 test particles under the effect of the four giant planets for the age of the solar system. A chaotic diffusion experiment was also performed to follow tracks in phase space over 4-5 Gyr. The 7:4 mean motion resonance is weakly chaotic causing irregular eccentricity and inclination evolution for billions of years. Most 7:4 resonant particles suffered significant eccentricities and/or inclinations excitation, an outcome shared even by those located in the vicinity of the resonance. Particles in stable resonance locking are rare and usually had 0.25<e<0.3. For other regions, 7:4 resonants had quite large mobility in phase space typically leaving the resonance (and being scattered) after reaching a critical e∼0.2. The escape happened in 10⁸-10⁹ yr time scales. Concerning the inclination dependence for 7:4 resonants, we found strong instability islands for approximately i>10°. Taking into account those particles still locked in the resonance at the end of the simulations, we determined a retainability of 12-15% for real 7:4 resonant transneptunian objects (TNOs). Lastly, our results demonstrate that classical TNOs associated with the 7:4 mean motion resonance have been evolving continuously until present with non-negligible mixing of populations.     

Numerical results to the Sitnikov-problem

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.     

The three principal secular resonances ν5, ν6, and ν16 in the asteroidal belt

We review theoretical and numerical results obtained for secular resonant motion in the asteroidal belt. William's theory (1969) yields the locations of the principal secular resonances ₅, ₆, and ₁₆ in the asteroidal belt. Theories by Nakai and Kinoshita (1985) and by Yoshikawa (1987) allow us to model the basic features of orbital evolution at the secular resonances ₁₆ and ₆, respectively. No theory is available for the secular resonance v₅. Numerical experiments by Froeschlé and Scholl yield quantitative and new qualitative results for orbital evolutions at the three principal secular resonances ₅, ₆, and ₁₆. These experiments indicate possible chaotic motion due to overlapping resonances. A secular resonance may overlap with another secular resonance or with a mean motion resonance. The role of the secular resonances as possible sources of meteorites is discussed.     

Delaunay normalisations

Too many terms are generated by a Delaunay normalisation when the perturbation is developed in powers of the eccentricity. Ways of bypassing the expansion are discussed. There are: (i) Brouwer's method of implicit variables; (ii) the preparation by canonical transformations; and (iii) the application of representation theory for Lie algebras. Illustrations of the techniques are drawn from the main problem of satellite theory and from the (1–1) resonance at the triangular equilibrium in the restricted problem of three bodies.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.