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

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

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.     

The origin of the Kuiper Belt high-inclination population

I simulate the orbital evolution of the four major planets and a massive primordial planetesimal disk composed of 10⁴ objects, which perturb the planets but not themselves. As Neptune migrates by energy and angular momentum exchange with the planetesimals, a large number of primordial Neptune-scattered objects are formed. These objects may experience secular, Kozai, and mean motion resonances that induce temporary decrease of their eccentricities. Because planets are migrating, some planetesimals can escape those resonances while in a low-eccentricity incursion, thus avoiding the return path to Neptune close encounter dynamics. In the end, this mechanism produces stable orbits with high inclination and moderate eccentricities. The population so formed together with the objects coming from the classical resonance sweeping process, originates a bimodal distribution for the Kuiper Belt orbits. The inclinations obtained by the simulations can attain values above 30° and their distribution resembles a debiased distribution for the high-inclination population coming from the real classical Kuiper Belt.     

The Dynamics of Objects in the Inner Edgeworth–Kuiper Belt

Objects in 3:2 mean motion resonance with Neptune are protected from close encounters with Neptune by the resonance. Bodies in orbits with semi-major axis between 39.5 and about 42 AU are not protected by the resonance; indeed due to overlapping secular resonances, the eccentricities of orbits in this region are driven up so that a close encounter with Neptune becomes inevitable. It is thus expected that such orbits are unstable. The list of known Trans-Neptunian objects shows a deficiency in the number of objects in this gap compared to the 43–50 AU region, but the gap is not empty. We numerically integrate models for the initial population in the gap, and also all known objects over the age of the Solar System to determine what fraction can survive. We find that this fraction is significantly less than the ratio of the population in the gap to that in the main belt, suggesting that some mechanism must exist to introduce new members into the gap. By looking at the evolution of the test body orbits, we also determine the manner in which they are lost. Though all have close encounters with Neptune, in most cases this does not lead to ejection from the Solar System, but rather to a reduced perihelion distance causing close encounters with some or all of the other giant planets before being eventually lost from the system, with Saturn appearing to be the cause of the ejection of most of the objects.     

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 Poincaré 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.     

Kuiper带天体的轨道分布特性

1992年9月,夏威夷大学的D.Jewitt和加利福尼亚大学的J.Lun发现了海王星外绕太阳运行的第一个小天体1992QB1[1],开创了人类对于海王星外天体的实际观测的研究.近10年的接连不断发现,已经证实了海王星轨道外面存在着一个由大量的环绕太阳运动的小天体组成的环带[2].由于G.P.Kuiper曾在1951年的文章中提出过在冥王星的外边可能存在小天体的问题,因此人们一般把这个环带称为Kuiper带,你这些天体为“KuiperBelt　Objects”(KBOs),或从逻辑上称它们为“Trans-NeptunianObjects”(TNOs)[3]     

Structure of the 3:1 Jovian resonance: A comparison of numerical methods

The motion of asteroids near the 3:1 Jovian resonance in the restricted planar case is studied using three numerical methods: (a) integrating the full equations of motion, (b) integrating the averaged equations of motion, and (c) using an algebraic mapping recently developed by Wisdom (1982, Astron. J.87, 577–593). The relative merits of each method are investigated. It is concluded that in the regular regions of the phase space, methods b and c give excellent agreement with each other and that provided the maximum eccentricity e_max < 0.4 differences with the exact solution (method a) are <6% in e_max and <27% in the period of the oscillations. The additions of higher order terms in the expansion of the averaged Hamiltonian provides marginally better agreement with the full integration. This is probably due to the slow convergence of the expansion of the disturbing function at large eccentricities (e > 0.3). In chaotic regions of the phase there is little agreement between the orbital elements at any given time calculated by each method. However, all methods reflect the qualitative behavior of the chaotic trajectories and give good agreement on the bounds of the motion. Since the map is at least 200 times faster than solving the full equations of motion it is an efficient method of rapidly exploring accessible regions of the chaotic phase space.     

The origin of TNO 2004 XR190 as a primordial scattered object

Numerical integrations of the equations of motion of the giant planets and scattering particles show that there is a possible orbital itinerary that a particle may follow from a scattering mode up to a stable position near the orbit of 2004 XR₁₉₀. This orbital evolution requires that the particle gets trapped in a mean motion resonance with Neptune coupled with the Kozai resonance. Imposing migration on Neptune while a particle is experiencing both resonances can entail an escape from resonance at a low particle’s eccentricity. This eccentricity and the associated inclination are always similar to those of 2004 XR₁₉₀. I conclude that 2004 XR₁₉₀ was most likely a scattered object that went through those resonance processes and was eventually deposited at its current position. By the same argument, it is expected that there must exist several other objects with similar semimajor axis, eccentricity and inclination as those of 2004 XR₁₉₀.     

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.     

Asteroid motion near the 3 : 1 resonance

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

Dynamic portrait of the planetary 2/1 mean-motion resonance – I. Systems with a more massive outer planet

We analyse the global structure of the phase space of the planar planetary 2/1 mean-motion resonance in cases where the outer planet is more massive than its inner companion. Inside the resonant domain, we show the existence of two families of periodic orbits, one associated to the librational motion of resonant angle (σ-family) and the other related to the circulatory motion of the difference in longitudes of pericentre (  Δϖ  -family). The well-known apsidal corotation resonances (ACR) appear as intersections between both families. A complex web of secondary resonances is also detected for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system.
The construction of dynamical maps for various values of the total angular momentum shows the evolution of the families of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems. For low–moderate eccentricities, several different stable modes exist outside the ACR. For larger eccentricities, however, all stable solutions are associated to oscillations around the stationary solutions.
Finally, we present a possible link between these stable families and the process of resonance capture, identifying the most probable routes from the secular region to the resonant domain, and discussing how the final resonant configuration may be affected by the extension of the chaotic layer around the resonance region.     

Exploring the 7:4 mean motion resonance—II: Scattering evolutionary paths and resonance sticking

In our preliminary study, we have investigated basic properties and dynamical evolution of classical TNOs around the 7:4 mean motion resonance with Neptune (a∼43.7 AU), motivated by observational evidences that apparently present irregular features near this resonance (see [Lykawka and Mukai, 2005a. Exploring the 7:4 mean motion resonance—I. Dynamical evolution of classical trans-Neptunian objects (TNOs). Space Planet. Sci. 53, 1175-1187]; hereafter “Paper I”). In this paper, we aim to explore the dynamical long-term evolution in the scattered disk (but not its early formation) based on the computer simulations performed in Paper I together with extra computations. Specifically, we integrated the orbital motion of test particles (totalizing a bit more than 10,000) placed around the 7:4 mean motion resonance under the effect of the four giant planets for the age of the Solar System. In order to investigate chaotic diffusion, we also conducted a special simulation with on-line computation of proper elements following tracks in phase space over 4-5 Gyr. We found that: (1) A few percent (1-2%) of the test particles survived in the scattered disk with direct influence of other Neptunian mean motion resonances, indicating that resonance sticking is an extremely common phenomenon and that it helps to enhance scattered objects longevity. (2) In the same region, the so-called extended scattered TNOs are able to form via very long resonance trapping under certain conditions. Namely, if the body spends more than about 80% of its dynamical lifetime trapped in mean motion resonance(s) and there is the action of a k+1 or (k+2)/2 mean motion resonance (e.g., external mean motion resonances with Neptune described as (j+k)/j with j=1 and 2, respectively). According to this hypothetical mechanism, 5-15% of current scattered TNOs would possess thus probably constituting a significant part of the extended scattered disk. (3) Moreover, considering hot orbital initial conditions, it is likely that the trans-Neptunian belt (or Edgeworth-Kuiper belt) has been providing members to the scattered disk, so that scattered TNOs observed today would consist of primordial scattered bodies mixed with TNOs that came from unstable regions of the trans-Neptunian belt in the past.Considering the three points together, our results demonstrated that the scattered disk has been evolving continuously since early times until present.     

Symplectic Mapping for Trojan-Type Motion in the Elliptic Restricted Three-Body Problem

A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn.     

Symmetric and asymmetric 3:1 resonant periodic orbits with an application to the 55Cnc extra-solar system

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.     

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.     

Numerical studies of chaotic Hilda-type orbits

Earlier work indicates a comparatively rapid chaotic evolution of the orbits of some Hilda asteroids that move at the border of the domain occupied by the characteristic parameters of the objects at the 3/2 mean motion resonance. A simple Jupiter–Saturn model of the forces leads to numerical results on some of these cases and allows a search for additional resonances that can contribute to the chaotic evolution. In this context the importance of the secondary resonances that depend on the period of revolution of the argument of perihelion is pointed out. Among the studied additional resonances there are three-body resonances with arguments that depend on the mean longitudes of Jupiter, Saturn, and asteroid, but on slowly circulating angular elements of the asteroid as well, and the frequency of these arguments is close to a rational ratio with respect to the frequency of the libration due to the basic resonance.     

Origin of the structure of the Kuiper belt during a dynamical instability in the orbits of Uranus and Neptune

We explore the origin and orbital evolution of the Kuiper belt in the framework of a recent model of the dynamical evolution of the giant planets, sometimes known as the Nice model. This model is characterized by a short, but violent, instability phase, during which the planets were on large eccentricity orbits. It successfully explains, for the first time, the current orbital architecture of the giant planets [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461], the existence of the Trojans populations of Jupiter and Neptune [Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465], and the origin of the late heavy bombardment of the terrestrial planets [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469]. One characteristic of this model is that the proto-planetary disk must have been truncated at roughly 30 to 35 AU so that Neptune would stop migrating at its currently observed location. As a result, the Kuiper belt would have initially been empty. In this paper we present a new dynamical mechanism which can deliver objects from the region interior to ∼35 AU to the Kuiper belt without excessive inclination excitation. In particular, we show that during the phase when Neptune's eccentricity is large, the region interior to its 1:2 mean motion resonance becomes unstable and disk particles can diffuse into this area. In addition, we perform numerical simulations where the planets are forced to evolve using fictitious analytic forces, in a way consistent with the direct N-body simulations of the Nice model. Assuming that the last encounter with Uranus delivered Neptune onto a low-inclination orbit with a semi-major axis of ∼27 AU and an eccentricity of ∼0.3, and that subsequently Neptune's eccentricity damped in ∼1 My, our simulations reproduce the main observed properties of the Kuiper belt at an unprecedented level. In particular, our results explain, at least qualitatively: (1) the co-existence of resonant and non-resonant populations, (2) the eccentricity-inclination distribution of the Plutinos, (3) the peculiar semi-major axis—eccentricity distribution in the classical belt, (4) the outer edge at the 1:2 mean motion resonance with Neptune, (5) the bi-modal inclination distribution of the classical population, (6) the correlations between inclination and physical properties in the classical Kuiper belt, and (7) the existence of the so-called extended scattered disk. Nevertheless, we observe in the simulations a deficit of nearly-circular objects in the classical Kuiper belt.     

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

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

Neptune migration model with one extra planet

We explore conventional Neptune migration model with one additional planet of mass at 0.1-2.0M_⊕. This planet inhabited in the 3:2 mean motion resonance with Neptune during planet migration epoch, and then escaped from the Kuiper belt when jovian planets parked near the present orbits. Adding this extra planet and assuming the primordial disk truncated at about 45 AU in the conventional Neptune migration model, it is able to explain the complex structure of the observed Kuiper belt better than the usual Neptune migration model did in several respects, which are the following. (1) High-inclination Plutinos with i?15-35° are produced. (2) Generating the excitation of the classical Kuiper belt objects, which have moderate eccentricities and inclinations. (3) Producing the larger ratio of Neptune’s 3:2 to 2:1 resonant particles, and the lower ratio of particles in the 3:2 resonance to those in the classical belt, which may be more consistent with observations. (4) Finally, several Neptune’s 5:2 resonant particles are obtained. However, numerical experiments imply that this model is a low-probability event. In addition to the low probability, two features produced by this model may be inconsistent with the observations. They are small number of low-inclination particles in the classical belt, and the production of a remnant population with near-circular and low-inclination orbit within . According to our present study, including one extra planet in the conventional Neptune migration model as the scenario we explored here may be unsuitable because of the low probability, and the two drawbacks mentioned above, although this model can explain better several features which is hard to produce by the conventional Neptune migration model. The issues of low-probability event and the lack of low-inclination KBOs in the classical belt are interesting and may be studied further under a more realistic consideration.     

The scattered trans-Neptunian object 1998 XY95

On 1998 December 12 a new trans-Neptunian object, 1998 XY95, was discovered as part of a deep search. Recent observations of this object have placed it amongst the class of objects known as the scattered trans-Neptunian objects (TNOs). A total of 39 CCD images of 1998 XY95 were taken over two nights, and these were used to search for a light curve, but no significant periodicity was found. An examination of the possible orbital evolution gives no indication of how it may have arrived on its present orbit. The current best-fitting orbit is unstable, but remains within a band of semi-major axis approximately 2 au wide. The bottom of this band is due to 3:1 mean motion resonance with Neptune, while the reason for the top of the band remains unclear.