The nekhoroshev theorem and the asteroid belt dynamical system

The present paper reviews the Nekhoroshev theorem from the point of view of physicists and astronomers. We point out that Nekhoroshev result is strictly connected with the existence of a specific structure of the phase space, the existence of which can be checked with several numerical tools. This is true also for a degenerate system such as the one describing the motion of an asteroid in the so called main belt. The main difference is that in some parts of the belt, the Nekhoroshev result cannot apply a priori. Mean motion resonances of order smaller than the logarithm of the mass of Jupiter and first order secular resonances must be excluded. In the remaining parts, conversely, the Nekhoroshev theorem can be proved, provided someparameters, such as the masses, the eccentricities and the inclinations of the planets are small enough. At the light of this result, a massive campaign of numerical integrations of real and fictitious asteroids should allow to understand which is the real dynamical structure of the asteroid belt.     

Local And Global Diffusion Along Resonant Lines in Discrete Quasi-integrable Dynamical Systems

We detect and measure diffusion along resonances in a quasi-integrable symplectic map for different values of the perturbation parameter. As in a previously studied Hamiltonian case (Lega et al., 2003) results agree with the prediction of the Nekhoroshev theorem. Moreover, for values of the perturbation parameter slightly below the critical value of the transition between Nekhoroshev and Chirikov regime we have also found a diffusion of some orbits along macroscopic portions of the phase space. Such a diffusion follows in a spectacular way the peculiar structure of resonant lines.     

Probing the Nekhoroshev Stability of Asteroids

We apply the spectral formulation of the Nekhoroshev theorem to investigate the long-term stability of real main belt asteroids. We find numerical indication that some asteroids are in the so-called Nekhoroshev stability regime, that is they are on chaotic orbits but their motion is stable over very long times. We have analyzed the motion of bodies in different regions of the belt, to assess the sensitivity of our method. We found that it allows us to clearly discriminate between different dynamical regimes, such as the one described by the Nekhoroshev stability, the one well described by the KAM theory, and the unstable chaotic regime in which diffusion in phase space can be detected over time spans much shorter than the age of the solar system.     

The Role of the Inclination in the Captures in External Resonances in the Three Body Problem

Interested in the role of the inclination in our results (Jancart and Lemaitre, 2001), we analyze the process of resonance trapping due to general dissipation forces in the frame of the spatial restricted three body problem and in the case of external mean motion resonances. We compute our simulations by using the three-dimensional Extended Schubart Averaging integrator developed by Moons (1994) for all mean motion resonances. We complete it by adding the averaged contributions of general dissipative forces like Murray has proposed in the article (1994). The behavior of the inclination is especially pointed out.     

Long-Term Stability Analysis of Quasi Integrable Degenerate Systems Through the Spectral Formulation of the Nekhoroshev Theorem

We describe a numerical application of the Nekhoroshev theorem to investigate the long-term stability of quasi-integrable systems. We extend the results of a previous paper to a class of degenerate systems, which are typical in celestial mechanics.     

A numerical study of the size of the homoclinic tangle of hyperbolic tori and its correlation with Arnold diffusion in Hamiltonian systems

Using a three degrees of freedom quasi-integrable Hamiltonian as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to single resonances. We measure an exponential dependence of the splitting of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. We also detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. The variation of the size of the homoclinic tangle as well as the topological transitions turn out to be correlated to the speed of Arnold diffusion.     

Measure of the exponential splitting of the homoclinic tangle in four-dimensional symplectic mappings

Using four-dimensional symplectic maps as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to the single resonances. We measure an exponential dependence of the size of the lobes of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. The variation of the size of the lobes turns out to be correlated to the diffusion coefficient.     

The Dynamical Evolution of the Atens

In this investigation the orbits of 21 Atens (semimajor axes smaller than the Earth) are studied with the aid of numerical integrations over the time interval of one million years. The dynamical model was a 6-body Solar System, where the perturbations of Uranus and Neptune were ignored, and where Mercury's mass was added to the Sun's mass. Thus mean motion resonances, secular resonances and the Kozai resonance were fully taken into account. The evolution of the semimajor axes shows the typical step function like pattern which we know also from comets although some Atens have a very fuzzy development of the orbital elements, and some of them stay in a mean motion resonance for very long time. The evolution from Atens to Apollos (with semimajor axes larger than the Earth) and vice versa is also a phenomenon which we could observe. The main goal was the study of encounters of the Atens with the Earth and Venus. We found out that Venus encounters occur somewhat more often than Earth encounters (approximately one within the distance Earth - Moon every 40000 years with Venus, one every 50000 years with the Earth). This revised version was published online in July 2006 with corrections to the Cover Date.     

Reducing the probability of capture into resonance

A migrating planet can capture planetesimals into mean motion resonances. However, resonant trapping can be prevented when the drift or migration rate is sufficiently high. Using a simple Hamiltonian system for first- and second-order resonances, we explore how the capture probability depends on the order of the resonance, drift rate and initial particle eccentricity. We present scaling factors as a function of the planet mass and resonance strength to estimate the planetary migration rate above which the capture probability drops to less than half. Applying our framework to multiple extrasolar planetary systems that have two planets locked in resonance, we estimate lower limits for the outer planet's migration rate, allowing resonance capture of the inner planet.
Mean motion resonances are comprised of multiple resonant subterms. We find that the corotation subterm can reduce the probability of capture when the planet eccentricity is above a critical value. We present factors that can be used to estimate this critical planet eccentricity. Applying our framework to the migration of Neptune, we find that Neptune's eccentricity is near the critical value that would make its 2 : 1 resonance fail to capture twotinos. The capture probability is affected by the separation between resonant subterms and so is also a function of the precession rates of the longitudes of periapse of both planet and particle near resonance.     

Massive identification of asteroids in three-body resonances

An essential role in the asteroidal dynamics is played by the mean motion resonances. Two-body planet–asteroid resonances are widely known, due to the Kirkwood gaps. Besides, so-called three-body mean motion resonances exist, in which an asteroid and two planets participate. Identification of asteroids in three-body (namely, Jupiter–Saturn–asteroid) resonances was initially accomplished by Nesvorný and Morbidelli (Nesvorný D., Morbidelli, A. [1998]. Astron. J. 116, 3029–3037), who, by means of visual analysis of the time behaviour of resonant arguments, found 255 asteroids to reside in such resonances. We develop specialized algorithms and software for massive automatic identification of asteroids in the three-body, as well as two-body, resonances of arbitrary order, by means of automatic analysis of the time behaviour of resonant arguments. In the computation of orbits, all essential perturbations are taken into account. We integrate the asteroidal orbits on the time interval of 100,000 yr and identify main-belt asteroids in the three-body Jupiter–Saturn–asteroid resonances up to the 6th order inclusive, and in the two-body Jupiter–asteroid resonances up to the 9th order inclusive, in the set of ～250,000 objects from the “Asteroids – Dynamic Site” (AstDyS) database. The percentages of resonant objects, including extrapolations for higher-order resonances, are determined. In particular, the observed fraction of pure-resonant asteroids (those exhibiting resonant libration on the whole interval of integration) in the three-body resonances up to the 6th order inclusive is ≈0.9% of the whole set; and, using a higher-order extrapolation, the actual total fraction of pure-resonant asteroids in the three-body resonances of all orders is estimated as ≈1.1% of the whole set.     

Confined chaotic motion in three-body resonances: trapping of trans-Neptunian material induced by gas-drag

Jiang & Yeh proposed gas-drag-induced resonant capture as a mechanism able to explain the dominant 3:2 resonance observed in the trans-Neptunian belt. Using a model of a disc–star–planet system they concluded that gaseous drag in a protoplanetary disc can trap trans-Neptunian object (TNO) embryos into the 3:2 resonance rather easily although it could not trap objects into the 2:1 resonance. Here we further investigate this scenario using numerical simulations within the context of the planar restricted four-body problem by including both present-day Uranus and Neptune. Our results show that mean motion and corotation resonances are possible and trapping into both the 3:2 and 2:1 resonances as well as other resonances is observed. The associated corotation centres may easily form larger planetesimals from smaller ones. Corotation resonances evolve into pure Lindblad resonances in a time-scale of 0.5 Myr. The non-linear corotation and mean motion resonances produced are very size selective. The 3:2 resonance is dominant for submetric particles but for larger particles the 2:1 resonance is stronger. In summary, our calculations show that confined chaotic motion around the resonances not only increases trapping efficiency but also the orbital eccentricities of the trapped material, modifying the relative abundance of trapped particles in different resonances. If we assume a more compact planetary system, instead of using the present-day values of the orbital elements of Uranus and Neptune, our results remain largely unchanged.     

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.     

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.     

Stability of higher order resonances in the restricted three-body problem

Third and fourth order mean motion resonances are studied in the model of the restricted three-body problem by numerical methods for mass parameters corresponding approximately to the Sun?CJupiter and Sun?CNeptune systems. In the case of inner resonances, it is shown that there are two regions of libration in the 8:5 and 7:4 resonances, one at low, the other at high eccentricities. In the 9:5 and 7:3 resonances libration can exist only in one region at high eccentricities. The 5:2 and 4:1 resonances are very regular, with one librational zone existing for all eccentricities. There is no visible region of libration at any eccentricities in the 5:1 resonance, the transition between the regions of direct and retrograde circulation is very sharp. In the case of outer resonances, the 8:5 and 7:4 resonances have also two regions of libration, but the 9:5 resonance has three, the 7:3 resonance two librational zones. The 5:2 resonance is again very regular, but it is parted for two regions of libration at high eccentricities. Libration is possible in the 4:1 resonance only at high eccentricities. The 5:1 resonance is very symmetric. In the case of outer resonances, a comparison is made with trans-Neptunian objects (TNO) in higher order mean motion resonances. Several new librating TNOs are identified.     

Dynamics of Asteroid 3200 Phaethon Under Overlap of Different Resonances

The paper is focused on studying the motion of asteroid 3200 Phaethon which approached the Earth in December 2017. We consider the dynamics of asteroid 3200 Phaethon, reveal its encounters with planets, mean motion and secular resonances, and estimate the predictability time and the causes of chaoticity. A peculiar feature in the dynamics of the object is that it passes through the unstable orbital resonance 3/7 with Venus and exhibits a gamut of apsidal-nodal resonances with Mercury, Venus, Earth, Mars, and Jupiter, as well as a large number of close encounters with terrestrial planets. These properties result in a chaotic character of the motion beyond a time interval between the years 1780 and 2350.

     

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.     

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).     

The resonant structure of Jupiter's Trojan asteroids – I. Long-term stability and diffusion

We study the global dynamics of the jovian Trojan asteroids by means of the frequency map analysis. We find and classify the main resonant structures that serve as skeleton of the phase space near the Lagrangian points. These resonances organize and control the long-term dynamics of the Trojans. Besides the secondary and secular resonances, that have already been found in other asteroid sets in mean motion resonance (e.g. main belt, Kuiper belt), we identify a new type of resonance that involves secular frequencies and the frequency of the great inequality, but not the libration frequency. Moreover, this new family of resonances plays an important role in the slow transport mechanism that drives Trojans from the inner stable region to eventual ejections. Finally, we relate this global view of the dynamics with the observed Trojans, identify the asteroids that are close to these resonances and study their long-term behaviour.     

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.     

Capture of grains into resonances through Poynting-Robertson drag

We review here some relevant problems connected to the evolution of circumstellar dust grains, subjected to Poynting-Robertson (PR) drag, and perturbed by first-order resonances with a planet on a circular orbit. We show that only outer mean motion resonances are able to counteract the damping effect of PR drag. However, the high orbital eccentricities reached by the particle lead to orbit crossings with the planet. This is a serious difficulty for a permanent trapping to be achieved. In any case, we show that the time spent in the resonance is long enough for statistical effects (accumulation at the resonant radius) to be significant. We underline some difficulties associated with this problem, namely, the non-adiabaticity of motion in the resonance phase space and the existence of close encounters with the planet at high eccentricities.