Exterior Mean Motion Resonances Research Articles

Non-perturbative investigation of low-eccentricity exterior mean motion resonances

ABSTRACT Mean motion resonances are important in the analysis and understanding of the dynamics of planetary systems. While perturbative approaches have been dominant in many previous studies, recent non-perturbative approaches have revealed novel properties in the low-eccentricity regime for interior mean motion resonances of Jupiter in the fundamental model of the circular planar restricted three-body model. Here, we extend the non-perturbative investigation to exterior mean motion resonances in the low-eccentricity regime (up to about 0.1) and for perturber mass in the range of ∼5 × 10−5 to 1 × 10−3 (in units of the central mass). Our results demonstrate that first-order exterior resonances have two branches at low eccentricity as well as low-eccentricity bridges connecting neighbouring first-order resonances. With increasing perturber mass, higher order resonances dissolve into chaos, whereas low-order resonances persist with larger widths in their radial extent but smaller azimuthal widths. For low-order resonances, we also detect secondary resonances arising from small-integer commensurabilities between resonant librations and the synodic frequency. These secondary resonances contribute significantly to generating the chaotic sea that typically occurs near mean motion resonances of higher mass perturbers.

The phase space structure of retrograde mean motion resonances with Neptune: the 4/5, 7/9, 5/8 and 8/13 cases

We compute planar and three-dimensional retrograde periodic orbits in the vicinity of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and we concentrate on the dynamics of higher-order exterior mean motion resonances with Neptune. By using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 4/5, 7/9, 5/8 and 8/13 resonances. We determine the bifurcation points from the planar circular to the planar elliptic problem and we find all the corresponding families. In order to obtain a global view of the families of periodic orbits, the eccentricity of the primaries takes values in the whole interval 0<e'<1. Then, we find all the possible vertical critical orbits (v.c.o) of the planar circular problem and we proceed to the three-dimensional circular restricted 3-body problem. In this model, retrograde periodic orbits are generated mainly from the retrograde v.c.o. Also, if we continue families of direct orbits for i>90^circ , then we can obtain families of 3D symmetric retrograde periodic orbits. The linear stability is examined too. Stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. In order to study the phase space structure of the above resonances, we construct dynamical stability maps for the whole inclination interval (0<i<180^circ ) by using the well-known “MEGNO Chaos Indicator”. Finally, we discuss about TNOs which are currently located at these resonances.

Dynamical classification of the asteroids in the Hungaria group II: The objects affected by the exterior mean-motion resonances 3:4, 5:7, 7:10 and 8:11 with Mars

In this work, we present a dynamical study of the Hungaria asteroids close to the exterior 3:4, 5:7, 7:10 and 8:11 mean-motion resonances with Mars. We aim to extend the classification of Hungaria asteroids affected by mean-motion resonances initiated in our previous work with the analysis of the 2:3 resonance. To do it, we use the semi-analytical model developed in our previous study to identify the region of candidates possibly affected for each particular resonance. Then, we numerically integrate the temporal evolution of each candidate for 2 Myrs, considering the N-body problem where the asteroids are assumed to be massless bodies affected by the gravitational perturbation of all the planets in the Solar System. We find 38 asteroids captured inside the considered mean-motion resonances and 582 whose dynamical evolution is affected by these structures. Finally, we provide mean values for the orbital elements that may be taken as first approximations of proper elements, for those bodies that do not have these quantities previously determined.

Dynamical classification of the asteroids in the Hungaria group: The objects affected by the exterior mean-motion resonance 2:3 with Mars

In this paper, we perform a dynamical study of the population of Hungaria asteroids, close to the exterior 2:3 mean-motion resonance (MMR) with Mars. The aim of this work is to initiate a catalog of asteroids captured in mean-motion resonances (MMRs) with Mars in the Hungaria region, by first identifying the minor bodies affected by the 2:3 MMR. To achieve our goal, we follow the temporal evolution of massless test particles by 2 Myr in the proximity of the 2:3 resonance considering the gravitational influence of all planets of the Solar System. We present a semi-analytical model to calculate the extension of the resonant region for a population with the dynamical characteristics of the Hungaria asteroids. We show that the resonance can disturb the asteroids in two ways: by capture or by indirect influence. We find 22 asteroids captured inside the 2:3 MMR and 89 asteroids which exhibit temporary librations around this resonance. The predictions of our model are in good agreement with the numerical results, which make it useful for the study of other MMRs in this region. For asteroids without determined synthetic proper elements, we use the results of our numerical simulations to calculate the mean orbital elements, which can be used as a first approximation of the proper elements.

Retrograde periodic orbits in 1/2, 2/3 and 3/4 mean motion resonances with Neptune

We study planar and three-dimensional retrograde periodic orbits, using the model of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and focusing on the dynamics of resonant trans-Neptunian objects (TNOs). The position and the stability character of the periodic orbits can provide important piece of information on the stability and long-term evolution of small TNOs in retrograde motion. Using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 1/2, 2/3 and 3/4 exterior mean motion resonances with Neptune. The bifurcations for planar families of the elliptic model and families of the circular spatial model are determined and the bifurcated families are computed. In our study of the planar elliptic model, we consider the eccentricity of the primaries in the whole interval $$0<e'<1$$ for dynamical completeness. In the spatial circular model, retrograde periodic orbits are obtained mainly from bifurcations of the retrograde planar orbits. Also, we obtain retrograde periodic motion from continuing direct orbits for inclination values larger than $$90^\circ $$ . The linear stability of orbits is of major importance. Generally, stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. TNOs of retrograde motion are not common, but new discoveries cannot be excluded.

Neptune’s resonances in the scattered disk

The Scattered Disk Objects (SDOs) are thought to be a small fraction of the ancient population of leftover planetesimals in the outer solar system that were gravitationally scattered by the giant planets and have managed to survive primarily by capture and sticking in Neptune's exterior mean motion resonances (MMRs). In order to advance understanding of the role of MMRs in the dynamics of the SDOs, we investigate the phase space structure of a large number of Neptune's MMRs in the semi-major axis range 33--140~au by use of Poincar\'e sections of the circular planar restricted three body model for the full range of particle eccentricity pertinent to SDOs. We find that, for eccentricities corresponding to perihelion distances near Neptune's orbit, distant MMRs have stable regions with widths that are surprisingly large and of similar size to those of the closer-in MMRs. We identify a phase-shifted second resonance zone that exists in the phase space at planet-crossing eccentricities but not at lower eccentricities; this second resonance zone plays an important role in the dynamics of SDOs in lengthening their dynamical lifetimes. Our non-perturbative measurements of the sizes of the stable resonance zones confirm previous results and provide an additional explanation for the prominence of the $N$:1 sequence of MMRs over the $N$:2, $N$:3 sequences and other MMRs in the population statistics of SDOs; our results also provide a tool to more easily identify resonant objects.

Inclined asymmetric librations in exterior resonances

Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $\pi$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.

Dust particles in mean motion resonances influenced by an interstellar gas flow

The orbital evolution of a dust particle captured in a mean motion resonance with a planet in circular orbit under the action of the Poynting-Robertson effect, radial stellar wind and an interstellar gas flow of is investigated. The secular time derivative of Tisserand parameter is analytically derived for arbitrary orbit orientation. From the secular time derivative of Tisserand parameter a general relation between the secular time derivatives of eccentricity and inclination is obtained. In the planar case (the case when the initial dust particle position vector, initial dust particle velocity vector and interstellar gas velocity vector lie in the planet orbital plane) is possible to calculate directly the secular time derivative of eccentricity. Using numerical integration of equation of motion we confirmed our analytical results in the three-dimensional case and also in the planar case. Evolutions of eccentricity of the dust particle captured in an exterior mean motion resonance under the action of the Poynting-Robertson effect, radial stellar wind for the cases with and without the interstellar gas flow are compared. Qualitative properties of the orbital evolution in the planar case are determined. Two main groups of the secular orbital evolutions exist. In the first group the eccentricity and argument of perihelion approach to some values. In the second group the eccentricity oscillates and argument of perihelion rapidly shifts.

Kepler constraints on planets near hot Jupiters

We present the results of a search for planetary companions orbiting near hot Jupiter planet candidates (Jupiter-size candidates with orbital periods near 3d) identified in the Kepler data through its sixth quarter of science operations. Special emphasis is given to companions between the 21 interior and exterior mean-motion resonances. A photometric transit search excludes companions with sizes ranging from roughly two-thirds to five times the size of the Earth, depending upon the noise properties of the target star. A search for dynamically induced deviations from a constant period (transit timing variations) also shows no significant signals. In contrast, comparison studies of warm Jupiters (with slightly larger orbits) and hot Neptune-size candidates do exhibit signatures of additional companions with these same tests. These differences between hot Jupiters and other planetary systems denote a distinctly different formation or dynamical history.

Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem

We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries.

Inner Solar System dynamical analogs of plutinos

By studying orbits of asteroids potentially in 3:2 exterior mean motion resonance with Earth, Venus, and Mars, we have found plutino analogs. We identify at least 27 objects in the inner Solar System dynamically protected from encounter through this resonance. These are four objects associated with Venus, six with Earth, and seventeen with Mars. Bodies in the 3:2 exterior resonance (including those in the plutino resonance associated with Neptune) orbit the Sun twice for every three orbits of the associated planet, in such a way that with sufficiently low libration amplitude close approaches to the planet are impossible. As many as 15% of Kuiper Belt objects share the 3:2 resonance, but are poorly observed. One of several resonance sweeping mechanisms during planetary migration is likely needed to explain the origin and properties of 3:2 resonant Kuiper Belt objects. Such a mechanism likely did not operate in the inner Solar System. We suggest that scattering by the next planet out allows entry to, and exit from, 3:2 resonance for objects associated with Venus or Earth. 3:2 resonators of Mars, on the other hand, do not cross the paths of other planets, and have a long lifetime. There may exist some objects trapped in the 3:2 Mars resonance which are primordial, with our tests on the most promising objects known to date indicating lifetimes of at least tens of millions of years. Identifying 3:2 resonant systems in the inner Solar System permits this resonance to be studied on shorter timescales and with better determined orbits than has been possible to date, and introduces new mechanisms for entry into the resonant configuration.

Embedded star clusters and the formation of the Oort cloud: II. The effect of the primordial solar nebula

This paper deals with Oort cloud formation while the Sun was in an embedded cluster and surrounded by its primordial nebula. This work is a continuation of Brasser et al. [Brasser, R., Duncan, M., Levison, H., 2006. Icarus 184, 59–82], building on the model presented therein, and adding the aerodynamic drag and gravitational potential of the primordial solar nebula. Results are presented of numerical simulations of comets subject to the gravitational influence of the Sun, Jupiter, Saturn, star cluster and primordial solar nebula; some of the simulations included the gravitational influence of Uranus and Neptune as well. The primordial solar nebula was approximated by the minimum-mass Hayashi model [Hayashi, C., Nakozawa, K., Nakagawa, Y., 1985. In: Black, D.C., Matthews, M.S. (Eds.). Protostars and Planets II. Univ. of Arizona Press, Tucson, AZ] whose inner and outer radii have been truncated at various distances from the Sun. A comet size of 1.7 km was used for most of our simulations. In all of our simulations, the density of the primordial solar nebula decayed exponentially with an e-folding time of 2 Myr. It turns out that when the primordial solar nebula extends much beyond Saturn or Neptune, virtually no material will end up in the Oort cloud (OC) during this phase. Instead, the majority of the material will be on circular orbits inside of Jupiter if the inner edge of the disk is well inside Jupiter's orbit. If the disk's inner edge is beyond Jupiter's orbit, most comets end up on orbits in exterior mean-motion resonances with Saturn when Uranus and Neptune are not present. In those cases where the outer edge of the disk is close to Saturn or Neptune, the fraction of material that ends up in the subsequently formed OC is much less than that found in Brasser et al. [Brasser, R., Duncan, M., Levison, H., 2006. Icarus 184, 59–82] for the same cluster densities. This implies that for comets of roughly 2 km in size, the presence of the primordial solar nebula hinders OC formation. A byproduct of some of our simulations are endresults with a substantial fraction of the comets in the Uranus–Neptune scattered disk. A subsequent followup of this material is planned for the near future. In order to determine the effect of the size of the comets on OC formation efficiency, a set of runs with the same initial conditions but different cometary radii have been performed as well, from which it is determined that the threshold comet size to begin producing significant Oort clouds is roughly 20 km. This implies that the presence of the primordial solar nebula acts as a size-sorting mechanism, with large bodies unaffected by the gas drag and ending up in the OC while small bodies remain trapped in the planetary region, in the models studied.

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).

Three dimensional periodic orbits in exterior mean motion resonances with Neptune

In the framework of the three dimensional restricted three-body problem we study the dynamics of the exterior mean motion resonances with Neptune. The basis of our study is the computation of periodic orbits and their linear stability. The position and stability of periodic orbits critically determine the phase space topology. Stable periodic orbits are centers of regular librations where resonant trapping of minor bodies can take place. The periodic orbits are continued analytically from the planar model to the three-dimensional one forming families, which, in most cases, extend up to high inclination values and pass to the regime of retrograde motion. A classification of the families, according to their continuation and termination, is given. The computations show that resonant librating motion is present up to high inclination values and for particular eccentricity domains. When the ellipticity of the Neptune's orbit is taken into account, periodic orbits are found only for very high inclination values and all of them are unstable.

A Safety Net for Fast Migrators: Interactions between Gap‐opening and Sub–Gap‐opening Bodies in a Protoplanetary Disk

Young planets interact with their parent gas disks through tidal torques. An imbalance between inner and outer torques causes bodies of mass $\ga 0.1$ Earth masses to lose angular momentum and migrate inward rapidly relative to the disk; this is known as ``Type I'' migration. However, protoplanets that grow to gas giant mass, O($10^2) M_\oplus$, open a gap in the disk and are subsequently constrained to migrate more slowly, locked into the disk's viscous evolution in what is called migration. In a young planetary system, both Type I and Type II bodies likely coexist; if so, differential migration ought to result in close encounters when the former originate on orbits exterior to the latter. We investigate the resulting dynamics, using two different numerical approaches: an N-body code with dissipative forces added to simulate the effect of the gas disk, and a hybrid code which combines an N-body component with a 1-dimensional viscous disk model, treating planet-disk interactions in a more self-consistent manner. In both cases, we find that sub-gap-opening bodies have a high likelihood of being resonantly captured when they encounter a gap-opening body. A giant planet thus tends to act as a barrier in a protoplanetary disk, collecting smaller protoplanets outside of its orbit. Such behavior has two important implications for giant planet formation: First, for captured protoplanets it mitigates the problem of the migration timescale becoming shorter than the growth timescale. Secondly, it suggests one path to forming systems with multiple giant planets: Once the first has formed, it traps/accretes the future solid core of the second in an exterior mean-motion resonance, and so on. The most critical step in giant planet formation may thus be the formation of the very first one.

Symmetric and Nonsymmetric Periodic Orbits in the Exterior Mean Motion Resonances with Neptune

We present families of periodic orbits and their stability for the exterior mean motion resonances 1:2, 1:3 and 1:4 with Neptune in the framework of the planar circular restricted three-body problem. We found that in each resonance there exist two branches of symmetric elliptic periodic orbits with stable and unstable segments. Asymmetric periodic orbits bifurcate from the corresponding symmetric ones. Asymmetric periodic orbits are stable and the motion in their neighbourhood is a libration with respect to the resonant angle variable. In all the families of asymmetric periodic orbits the eccentricity extends to high values. Poincare sections reveal the changes of the topology in phase space.

Bifurcations of periodic orbits and potential stability regions in Kuiper belt dynamics

In the framework of the restricted three body problem, the resonant periodic orbits associated with the Kuiper belt dynamics are studied. Particularly, all the first, second and third order exterior mean motion resonances with Neptune located up to 50A.U. and the asymmetric resonances (beyond the 48 A.U.) are considered. We present the bifurcation points of families of periodic orbits of the planar circular problem from which families of periodic orbits are generated in the planar elliptic and in the 3D circular problem. Similarities and differences between the various resonant cases are noticed. The relation between the distribution of the bifurcation points and the population of small bodies at the particular resonances is discussed.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincare maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.

Resonant and Secular Families of the Kuiper Belt

We review ongoing efforts to identify occupants of mean-motion resonances (MMRs) and collisional families in the Edgeworth—Kuiper belt. Direct integrations of trajectories of Kuiper belt objects (KBOs) reveal the 1:1 (Trojan), 5:4, 4:3, 3:2 (Plutino), 5:3, 7:4, 9:5, 2:1 (Twotino), and 5:2 MMRs to be inhabited. Apart from the Trojan, resonant KBOs typically have large orbital eccentricities and inclinations. The observed pattern of resonance occupation is consistent with resonant capture and adiabatic excitation by a migratory Neptune; however, the dynamically cold initial conditions prior to resonance sweeping that are typically assumed by migration simulations are probably inadequate. Given the dynamically hot residents of the 5:2 MMR and the substantial inclinations observed in all exterior MMRs, a fraction of the primordial belt was likely dynamically pre-heated prior to resonance sweeping. A pre-heated population may have arisen as Neptune gravitationally scattered objects into trans-Neptunian space. The spatial distribution of Twotinos offers a unique diagnostic of Neptune’s migration history. The Neptunian Trojan population may rival the Jovian Trojan population, and the former’s existence is argued to rule out violent orbital histories for Neptune. Finally, lowest-order secular theory is applied to several hundred non-resonant KBOs with well-measured orbits to update proposals of collisional families. No convincing family is detected.

The Resonance Region of Plutinos under the Perturbation of Outer Planets

Three resonances, the 3:2 exterior mean motion resonance with Neptune, Kozai resonance and 1:1 super resonance, are known to govern concurrently the stability of the motion of Pluto. We explore numerically the resonance zones in which the three resonance coexist. There might exist plutinos with relatively large masses in these zones. Considering that Pluto's perturbation is important to the long-term evolution of plutinos, the resonance zone is mainly explored in the mirror region of Pluto, which is a mirror image of the region around Pluto with respect to the invariant plane of the solar system. We find six resonance zones in the mirror region. The orbit elements at the centers of the six zones and the corresponding heliocentric distances, longitudes and latitudes on July 1, 2003 have been computed and listed for the reference of observation.