Are higher order resonances really interesting?

It is shown that higher order resonances of equilibria of conservative Hamiltonian systems do not give rise to an exchange of energies between the different degrees of freedom for most of the initial values on a certain long time-scale. The topology of these resonances is analysed, using Bott-Morse theory.     

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.     

Nonlinear stability of the libration points in the photogravitational restricted three body problem

The stability of the triangular libration points in the case when the first and the second order resonances appear was investigated. It was proved that the first order resonances do not cause instability. The second order resonances may lead to instability. Domains of the instability in the two-dimensional parameter space were determined.     

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.     

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.     

Stability of the Triangular Libration Points in the Unrestricted Planar Problem of a Symmetric Rigid Body and a Point Mass

In papers (Godziewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.In this paper, we investigate the stability of triangular equilibria permissible by the dynamics of the model, under the assumption of low-order resonances. We analyze all resonances of order smaller than four, and we examine the stability with application of theorems by Markeev and Sokolsky. These are the possible following cases: the non-diagonal resonance of the first order with two null characteristic frequencies (unstable); resonances of the first order with one nonzero frequency (diagonal and non-diagonal, stable and unstable); the second-order resonance, which is non-diagonal and stable, and the third-order resonance which is generically unstable, except for three points in the parameters' space, corresponding to stable equilibria.We discuss a perturbed version of Kokoriev and Kirpichnikov model, and we find that if the perturbation is small and depends on the coordinates only, the triangular equilibria persist, except if for the unperturbed equilibria the first-order resonance occurs. We show that the resonances of the order higher than two are also preserved if the perturbation acts.     

Resonances and bifurcations in axisymmetric scale-free potentials

We investigate an analytical treatment of bifurcations of families of resonant 'thin' tubes in axisymmetric galactic potentials. We verify that the most relevant bifurcations are due to the (1:1) resonance producing the 'inclined' orbits through two different mechanisms: from the disc orbit and from the 'thin' tube associated with the vertical oscillation. The closest resonances occurring after these are the (4:3) resonance in the oblate case and the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a straightforward way using a second-order truncated normal form. The higher order resonances are instead cumbersome to investigate, because the normal form has to be truncated to a high degree and the number of terms grows very rapidly. We therefore adopt a further simplification giving analytical formulae for the values of the parameters at which bifurcations ensue and compare them with selected numerical results. Thanks to the asymptotic nature of the series involved, the predictions are reliable well beyond the convergence radius of the original series.     

Superhumps, magnetic fields and the mass ratio in AM Canum Venaticorum

We show that the observed K velocities and periodicities of AM CVn can be reconciled given a mass ratio   q ≈ 0.22  and a secondary star with a modest magnetic field of surface strength   B ∼ 1 T  . We see that the new mass ratio implies that the secondary is most likely semidegenerate. The effect of the field on the accretion disc structure is examined. The theory of precessing discs and resonant orbits is generalized to encompass higher order resonances than 3 : 2 and shown to retain consistency with the new mass ratio.     

Secular resonances between bodies on close orbits II: prograde and retrograde orbits for irregular satellites

In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133–160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one-to-one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits \( b_1 \Delta \varpi + b_2 \Delta \varOmega \), forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that \({>}20\) and \({>}40\%\) of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with \(b_1+|b_2|\le 3\) persist, while even higher-order resonances, up to \(b_1+|b_2|\ge 7\), survive. Depending on the mass, between 10 and 60% of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model.     

The web of three-planet resonances in the outer Solar System: II. A source of orbital instability for Uranus and Neptune

The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they overlap in a region of 10⁻³ AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10⁻⁴ AU/Gyr.     

Asteroids in three-body mean-motion resonances with Jupiter and Mars

We identify the asteroids in three-body mean-motion resonances with Jupiter and Mars on the set of all known on April 2016 numbered asteroids (467308 objects). The resonant objects are identified by the direct analysis of the behavior (libration/circulation) of the resonant arguments on 100000 yrs. All essential perturbations during the integration of the equations of the motion are taken into account. The number of the asteroids in different resonances has been calculated for all possible resonances with the order less or equal 6.     

Secular perturbation theory and computation of asteroid proper elements

A new theory for the calculation of proper elements, taking into account terms of degree four in the eccentricities and inclinations, and also terms of order two in the mass of Jupiter, has been derived and programmed in a self contained code. It has many advantages with respect to the previous ones. Being fully analytical, it defines an explicit algorithm applicable to any chosen set of orbits. Unlike first order theories, it takes into account the effect of shallow resonances upon the secular frequencies; this effect is quite substantial, e.g. for Themis. Short periodic effects are corrected for by a rigorous procedure. Unlike linear theories, it accounts for the effects of higher degree terms and can thus be applied to asteroids with low to moderate eccentricity and inclination; secular resonances resulting from the combination of up to four secular frequencies can be accounted for. The new theory is self checking : the proper elements being computed with an iterative algorithm, the behaviour of the iteration can be used to define a quality code. The amount of computation required for a single set of osculating elements, although not negligible, is such that the method can be systematically applied on long lists of osculating orbital elements, taken either from catalogues of observed objects or from the output of orbit computations. As a result, this theory has been used to derive proper elements for 4100 numbered asteroids, and to test the accuracy by means of numerical integrations. These results are discussed both from a quantitative point of view, to derive an a posteriori accuracy of the proper elements sets, and from a qualitative one, by comparison with the higher degree secular resonance theory.     

On the Instability of Jupiter's Trojans

We have numerically explored the mechanisms that destabilize Jupiter's Trojan orbits outside the stability region defined by Levison et al. (1997, Nature385, 42-44). Different models have been exploited to test various possible sources of instability on timescales on the order of ∼10⁸ years.In the restricted three-body model, only a few Trojan orbits become unstable within 10⁸ years. This intrinsic instability contributes only marginally to the overall instability found by Levison et al.In a model where the orbital parameters of both Jupiter and Saturn are fixed, we have investigated the role of Saturn and its gravitational influence. We find that a large fraction of Trojan orbits become unstable because of the direct nonresonant perturbations by Saturn. By shifting its semimajor axis at constant intervals around its present value we find that the near 5:2 mean motion resonance between the two giant planets (the Great Inequality) is not responsible for the gross instability of Jupiter's Trojans since short-term perturbations by Saturn destabilize Trojans, even when the two planets are far out of the resonance.Secular resonances are an additional source of instability. In the full six-body model with the four major planets included in the numerical integration, we have analyzed the effects of secular resonances with the node of the planets. Trojan asteroids have relevant inclinations, and nodal secular resonances play an important role. When a Trojan orbit becomes unstable, in most cases the libration amplitude of the critical argument of the 1:1 mean motion resonance grows until the asteroid encounters the planet. Libration amplitude, eccentricity, and nodal rate are linked for Trojan orbits by an algebraic relation so that when one of the three parameters is perturbed, the other two are affected as well. There are numerous secular resonances with the nodal rate of Jupiter that fall inside the region of instability and contribute to destabilize Trojans, in particular the ν₁₆. Indeed, in the full model the escape rate over 50 Myr is higher compared to the fixed model.Some secular resonances even cross the stability region delimited by Levison et al. and cause instability. This is the case of the 3:2 and 1:2 nodal resonances with Jupiter. In particular the 1:2 is responsible for the instability of some clones of the L4 Trojan (3540) Protesilaos.     

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.     

Normal Form Invariants Around Spin-orbit Periodic Orbits

We consider a model of spin-orbit interaction, describing the motion of an oblate satellite rotating about an internal spin-axis and orbiting about a central planet. The resulting second order differential equation depends upon the parameters provided by the equatorial oblateness of the satellite and its orbital eccentricity. Normal form transformations around the main spin-orbit resonances are carried out explicitly. As an outcome, one can compute some invariants; the fact that these quantities are not identically zero is a necessary condition to prove the existence of nearby periodic orbits (Birkhoff fixed point theorem). Moreover, the nonvanishing of the invariants provides also the stability of the spin-orbit resonances, since it guarantees the existence of invariant curves surrounding the periodic orbit.     

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.     

Geometric interpretation for the spectral stability in the charged three-body problem

We present a geometric interpretation of the spectral stability of the triangular libration points in the charged three-body problem. We obtain that the spectral stability varies with the position of the center of mass of the three charges with respect to the circumcenter of the triangle configuration, which does not depend directly of the charges. If the center of mass is outside or on the circumference of a well defined radius ??, then spectral stability occurs. In addition, we analyze the existence of resonances within the spectral region of stability under this geometric interpretation, determining resonance curves of order 2, 3, 4, . . ., some of them with multiple resonances.     

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.     

Chaotic behaviour of trajectories for the asteroidal resonances

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3).