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 Beaué and Ferraz-Mello (1993, 1994) and Liou et al. (1979).     

Ringlet–satellite interactions

We study the interaction of a satellite and a nearby ringlet on eccentric and inclined orbits. Secular torques originate from mean motion resonances and the secular interaction potential which represents the m  = 1 global modes of the ring. The torques act on the relative eccentricity and inclination. The resonances damp the relative eccentricity. The inclination instability owing to the resonances is turned off by a finite differential eccentricity of the order of 0.27 for nearly coplanar systems. The secular potential torque damps the eccentricity and inclination and does not affect the relative semi-major axis; also, it suppresses the inclination instability that persists at small differential eccentricities. The damping of the relative eccentricity and inclination forces an initially circular and planar small mass ringlet to reach the eccentricity and inclination of the satellite. When the planet is oblate, the interaction of the satellite damps the proper precession of a small mass ringlet so that it precesses at the satellite's rate independently of their relative distance. The oblateness of the primary modifies the long-term eccentricity and inclination magnitudes and introduces a constant shift in the apsidal and nodal lines of the ringlet with respect to those of the satellite. These results are applied to Saturn's F-ring, which orbits between the moons Prometheus and Pandora.     

Tridimensional Dissipative Semi-Numerical Model

Our research combines mean motion resonances and dissipative forces in the averaged elliptic spatial restricted three-body problem. The models presented can be applied in many contexts mixing resonances and dissipations,e.g., asteroid belt, transneptunian region, exoplanets, systems of planetary rings, etc. We propose a semi-numerical model that simulates the behaviour of test particles under the effect of generic forces, functions of powers of the position and/or of the velocity. This model is valid for any orbital eccentricities or inclinations, even at high values. Captures around symmetric and asymmetric equilibria are reproduced and the apparitions of a plateau of inclination for long periods of time are dectected.     

Atlas of the mean motion resonances in the Solar System

The aim of this work is to present a systematic survey of the strength of the mean motion resonances (MMRs) in the Solar System. We know by applying simple formulas where the resonances with the planets are located but there is no indication of the strength that these resonances have. We propose a numerical method for the calculation of this strength and we present an atlas of the MMRs constructed with this method. We found there exist several resonances unexpectedly strong and we look and find in the small bodies population several bodies captured in these resonances. In particular in the inner Solar System we find one asteroid in the resonance 6:5 with Venus, five asteroids in resonance 1:2 with Venus, three asteroids in resonance 1:2 with Earth and six asteroids in resonance 2:5 with Earth. We find some new possible co-orbitals of Earth, Mars, Saturn, Uranus and Neptune. We also present a discussion about the behavior of the resonant disturbing function and where the stable equilibrium points can be found at low and high inclination resonant orbits.     

Relative Effect of Inclinations for Moonlets in the Triple Asteroidal Systems

We present the analysis and computational results for the inclination relative effect of moonlets of triple asteroidal systems. Perturbations on moonlets due to the primary’s non-sphericity gravity, the solar gravity, and moonlets’ relative gravity are discussed. The inclination vector for each moonlet follows a periodic elliptical motion; the motion period depends on the moonlet’s semi-major axis and the primary’s J2 perturbations. Perturbation on moonlets from the Solar gravity and moonlet’s relative gravity makes the motion of the x component of the inclination vector of moonlet 1 and the y component of the inclination vector of moonlet 2 to be periodic. The mean motion of x component and the y component of the inclination vector of each moonlet forms an ellipse. However, the instantaneous motion of x component and the y component of the inclination vector may be an elliptical disc due to the coupling effect of perturbation forces. Furthermore, the x component of the inclination vector of moonlet 1 and the y component of the inclination vector of moonlet 2 form a quasi-periodic motion. Numerical calculation of dynamical configurations of two triple asteroidal systems (216) Kleopatra and (153591) 2001 SN263 validates the conclusion.     

Review of the dynamics in the Kirkwood gaps

The study of mean motion resonance dynamics was motivated by the search for an explanation for the puzzling problem of the Kirkwood gaps. The most important contributions in this field within the last 32 years are reviewed here. At the beginning of that period, which coincides with the first long-term numerical investigations of resonant motion, different hypotheses (collisional, gravitational, statistical and cosmological) to explain the origin of the gaps were still competing with each other. At present, a general theory, based on gravitational mechanisms only, is capable of explaining in a uniform way all the Kirkwood gaps except the 2/1 one. Indeed, in the 4/1, 3/1, 5/2 and 7/3 mean motion commensurabilities, the overlap of secular resonances leads to almost overall chaos where asteroids undergo large and wild variations in their orbital elements. Such asteroids, if not thrown directly into the Sun, are sooner or later subject to strong close encounters with the largest inner planets, the typical time scale of the whole process being of the order of a few million years. Unfortunately, this mechanism is not capable of explaining the 2/1 gap where the strong chaos produced by the overlapping secular resonances does not attain orbits with moderate eccentricity, of low inclination and with low to moderate amplitude of libration. In the light of the most recent studies, it appears that the 2/1 gap is the global consequence of slow diffusive processes. At present, the origin of these processes remains under study.     

Possible effects of secular resonances in Phobos and Triton

Due to the tides, the orbits of Phobos and Triton are contracting. While their semi major axes are decreasing, several possibilities of secular resonances involving node, argument of the pericenter and mean motion of the Sun will take place. In the case of Mars, if the obliquity (ε), during the passage through some resonances, is not so small, very significant variations of the inclination will appear. In one case, capture is almost certain provided that ε?20°. For Triton there are also similar situations, but capture seems to be not possible, mainly because in S₁ state, Triton's orbit is sufficiently inclined (far) with respect to the Neptune's equator. Following Chyba et al. (Astron. Astrophys. 219 (1989) 123), a simplified equation that gives the evolution of the inclination versus the semi major axis, is derived. The time needed for Triton crash onto Neptune is longer than that one obtained by these authors, but the main difference is due to the new data used here. In general, even in the case of non-capture passages, some significant jumps in inclination and in eccentricities are possible.     

Repeat Ground Track Orbits of the Earth Tesseral Problem as Bifurcations of the Equatorial Family of Periodic Orbits

Orbits repeating their ground track on the surface of the earth are found to be members of periodic-orbit families (in a synodic frame) of the tesseral problem of the Earth artificial satellite. Families of repeat ground track orbits appear as vertical bifurcations of the equatorial family of periodic orbits, and they evolve from retrograde to direct motion throughout the 180 degrees of inclination. These bifurcations are always close to the resonances of the Earth's rotation rate and the mean motion of the orbiter.     

A Symplectic Mapping Model as a Tool to Understand the Dynamics of 2/1 Resonant Asteroid Motion

We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear. The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics of planets in retrograde mean motion resonance

In a previous paper (Gayon and Bois 2008a), we have shown the general efficiency of retrograde resonances for stabilizing compact planetary systems. Such retrograde resonances can be found when two-planets of a three-body planetary system are both in mean motion resonance and revolve in opposite directions. For a particular two-planet system, we have also obtained a new orbital fit involving such a counter-revolving configuration and consistent with the observational data. In the present paper, we analytically investigate the three-body problem in this particular case of retrograde resonances. We therefore define a new set of canonical variables allowing to express correctly the resonance angles and obtain the Hamiltonian of a system harboring planets revolving in opposite directions. The acquiring of an analytical “rail” may notably contribute to a deeper understanding of our numerical investigation and provides the major structures related to the stability properties. A comparison between our analytical and numerical results is also carried out.     

The Flora Family: A Case of the Dynamically Dispersed Collisional Swarm?

Asteroid families are believed to originate by catastrophic disruptions of large asteroids. They are nowadays identified as clusters in the proper orbital elements space. The proper elements are analytically defined as constants of motion of a suitably simplified dynamical system. Indeed, they are generally nearly constant on a 10⁷-10⁸-year time scale. Over longer time intervals, however, they may significantly change, reflecting the accumulation of the tiny nonperiodic evolutions provided by chaos and nonconservative forces. The most important effects leading to a change of the proper orbital elements are (i) the chaotic diffusion in narrow mean motion resonances, (ii) the Yarkovsky nongravitational force, and (iii) the gravitational impulses received at close approaches with large asteroids. A natural question then arises: How are the size and shape of an asteroid family modified due to evolution of the proper orbital elements of its members over the family age? In this paper, we concentrate on the dynamical dispersion of the proper eccentricity and inclination, which occurs due to (i), but with the help of (ii) and (iii). We choose the Flora family as a model case because it is unusually dispersed in eccentricity and inclination and, being located in the inner main belt, is intersected by a large number of effective mean motion resonances with Mars and Jupiter. Our results suggest that the Flora family dynamically disperses on a few 10⁸-year time scale and that its age may be significantly less than 10⁹ years. We discuss the possibility that the parent bodies of the Flora family and of the ordinary L chondrite meteorites are the same object. In a broader sense, this work suggests that the common belief that the present asteroid families are simple images of their primordial dynamical structure should be revised.     

Construction of a Nekhoroshev like result for the asteroid belt dynamical system

We investigate the possibility of obtaining a Nekhoroshev like result for the dynamical system describing the motion of an asteroid in the main belt, From the mathematical point of view this is a new result since the problem is degenerate and we want to control also the motion of degenerate actions, We find that there are regions, such as the resonances of low order among the fast angles (mean motion resonances), where a Nekhoroshev like result cannot be proved a priori, Conversely, we are able to confine the motions in the mean motion resonances of logarithmically large order in the perturbation parameters, as well as in the non-resonant region, We discuss also the connection with the existence of invariant tori.     

Long term dynamical evolution and classification of classical TNOs

Classical trans-Neptunian objects (TNOs) are believed to represent the most dynamically pristine population in the trans-Neptunian belt (TNB) offering unprecedented clues about the formation of our Solar System. The long term dynamical evolution of classical TNOs was investigated using extensive simulations. We followed the evolution of more than 17000 particles with a wide range of initial conditions taking into account the perturbations from the four giant planets for 4 Gyr. The evolution of objects in the classical region is dependent on both their inclination and semimajor axes, with the inner (a<45 AU) and outer regions (a>45 AU) evolving differently. The reason is the influence of overlapping secular resonances with Uranus and Neptune (40–42 AU) and the 5:3 (a∼ ∼42.3 AU), 7:4 (a∼ ∼43.7 AU), 9:5 (a∼ ∼44.5 AU) and 11:6 (a∼ ∼ 45.0 AU) mean motion resonances strongly sculpting the inner region, while in the outer region only the 2:1 mean motion resonance (a∼ ∼47.7 AU) causes important perturbations. In particular, we found: (a) A substantial erosion of low-i bodies (i<10°) in the inner region caused by the secular resonances, except those objects that remained protected inside mean motion resonances which survived for billion of years; (b) An optimal stable region located at 45 AU<a<47 AU, q>40 AU and i>5° free of major perturbations; (c) Better defined boundaries for the classical region: 42–47.5 AU (q>38 AU) for cold classical TNOs and 40–47.5 AU (q>35 AU) for hot ones, with i=4.5° as the best threshold to distinguish between both populations; (d) The high inclination TNOs seen in the 40–42 AU region reflect their initial conditions. Therefore they should be classified as hot classical TNOs. Lastly, we report a good match between our results and observations, indicating that the former can provide explanations and predictions for the orbital structure in the classical region.     

Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location

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.     

Resonances in Multiple Planetary Systems

Since the first extrasolar planet was discovered about 10 years ago, a major point of dynamical investigations was the determination of stable regions in extrasolar planetary systems where additional planets may exist. Using numerical methods, we investigate the dynamical stability in known multiple planetary systems (HD74156, HD38529, HD168443, HD169830) with special interest on the region between the two known planets and on the mean motion resonances inside this region. As a dynamical model we take the restricted 4-body problem containing the host star, the two planets and massless test-planets. For our numerical integrations, we used the Lie-integrator and additionally the Fast Lyapunov Indicators as a tool for detecting chaotic motion. We also investigated the inner resonances with the outer planet and the outer resonances with the inner planet with test-planets located inside the resonances.     

Motion of Pasiphae’s perijove and node

We investigated the motion of the perijove and ascending node of the 8th satellite of Jupiter, Pasiphae. The main perturbations by the Sun on the satellite permitted to use an intermediate orbit obtained by approximated solutions of differential equations previously transformed by the Von Zeipel method. The orbit is a non-Keplerian ellipse. The secular motion of the ascending node, argument of perijove, and essential periodic perturbations were taken into account. Using our theory we showed that the inclination and eccentricity of Pasiphae can acquire values by which the orbit becomes a librating one; but, within Pasiphae’s observation period, the motion of its perijove is circulating. Taking into account the results of our previous works on Pasiphae motion, we can conclude that the mean motion of the ascending node is similar for different values of the satellite inclination and eccentricity. But the mean motion of the perijove strongly depends on the orbit inclination and eccentricity, according to the Lidov–Kozai mechanism.     

Stable Chaos versus Kirkwood Gaps in the Asteroid Belt: A Comparative Study of Mean Motion Resonances

We have shown, in previous publications, that stable chaos is associated with medium/high-order mean motion resonances with Jupiter, for which there exist no resonant periodic orbits in the framework of the elliptic restricted three-body problem. This topological “defect” results in the absence of the most efficient mechanism of eccentricity transport (i.e., large-amplitude modulation on a short time scale) in three-body models. Thus, chaotic diffusion of the orbital elements can be quite slow, while there can also exist a nonnegligible set of chaotic orbits which are semiconfined (stable chaos) by “quasi-barriers” in the phase space. In the present paper we extend our study to all mean motion resonances of order q≤9 in the inner main belt (1.9-3.3 AU) and q≤7 in the outer belt (3.3-3.9 AU). We find that, out of the 34 resonances studied, only 8 possess resonant periodic orbits that are continued from the circular to the elliptic three-body problem (regular families), namely, the 2/1, 3/1, 4/1, and 5/2 in the inner belt and the 7/4, 5/3, 11/7, and 3/2 in the outer belt. Numerical results indicate that the 7/3 resonance also carries periodic orbits but, unlike the aforementioned resonances, 7/3-periodic orbits belong to an irregular family. Note that the five inner-belt resonances that carry periodic orbits correspond to the location of the main Kirkwood gaps, while the three outer-belt resonances correspond to gaps in the distribution of outer-belt asteroids noted by Holman and Murray (1996, Astron. J.112, 1278-1293), except for the 3/2 case where the Hildas reside. Fast, intermittent eccentricity increase is found in resonances possessing periodic orbits. In the remaining resonances the time-averaged elements of chaotic orbits are, in general, quite stable, at least for times t∼250 Myr. This slow diffusion picture does not change qualitatively, even if more perturbing planets are included in the model.     

The occurrence of high-order resonances and Kozai mechanism in the scattered disk

By means of numerical methods we explore the relevance of the high-order exterior mean motion resonances (MMR) with Neptune that a scattered disk object (SDO) can experience in its diffusion to the Oort cloud. Using a numerical method for estimate the strength of these resonances we show that high-eccentricity or high-inclination resonant orbits should have evident dynamical effects. We investigate the properties of the Kozai mechanism (KM) for non-resonant SDO's and the conditions that generate the KM inside a MMR associated with substantial changes in eccentricity and inclination. We found that the KM inside a MMR is typical for SDO's with Pluto-like or greater inclinations and is generated by the oscillation of ω inside the mixed (e,i) resonant terms of the disturbing function. A SDO diffusing to the Oort cloud should experience temporary captures in MMR, preferably of the type 1:N, and when evolving inside a MMR and experiencing the KM it can reach regions where the strength of the resonance drops and consequently there is a possibility of being decoupled from the resonance generating by this way a long-lived high-perihelion scattered disk object (HPSDO).     

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60^° of mutual planetary inclination, but in most families, the stability does not exceed 20^°–30^°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.     

Collective resonant phenomena on small bodies in the solar system

For both asteroids and meteor streams, and also for comets, resonances play a major role for their orbital evolutions but on different time scales. For asteroids both mean motion resonances and secular resonances not only structure the phase space of regular orbits but are mainly at the origin for the inherent chaos of planet crosser objects.For comets and their chaotic routes temporary trapping into orbital resonances is a well known phenomenon. In addition for slow diffusion through the Kuiper belt resonances are the only candidates for originating a slow chaos.Like for asteroids, resonances with Jupiter play a major role for the orbital evolution of meteor streams. Crossing of separatrix like zones appears to be crucial for the formation of arcs and for the dissolution of streams. In particular the orbital inclination of a meteor stream appears to be a critical parameter for arc formation. Numerical results obtained in an other context show that the competition between the Poynting-Robertson drag and the gravitational interaction of grains near the 2/1 resonance might be very important in the long run for the structure of meteor streams.