High-order Mean Motion Resonances Research Articles

A study of the high-inclination population in the Kuiper belt – IV. High-order mean-motion resonances in the classical region

ABSTRACT In our previous study of Neptune’s 4:7 mean-motion resonance (MMR), we discovered that its resonant angle can only librate within a specific eccentricity (e) versus inclination (i) region, determined by a theoretical limiting curve. This ‘permissible region’ is independent of time and encompasses the entire possible stable region. We now generalize this theory to investigate all high-order MMRs embedded in the main classical Kuiper belt (MCKB). We first consider the second-order 3:5 MMR in the framework of planet migration and resonance capture, and have further validated our limiting curve theory for both captured and observed 3:5 resonators. It suggests that only the (e, i) pairs inside the individual permissible regions should be chosen as initial conditions for studying the in situ evolution of high-order resonators. With such a new setting, we proceed to explore the long-term stability (for 4 Gyr) of different resonant populations, and our simulations predict that (1) the 3:5 and 4:7 resonators are comparable in number, and they could have inclinations up to 40°, (2) the populations of objects in the higher order 5:9, 6:11, 7:12, and 7:13 resonances are about 1/10 of the 3:5 (or 4:7) resonator population, and nearly all of them are found on the less inclined orbits with i < 10°, and (3) for these high-order resonances, almost all resonators reside in their individual permissible regions. In summary, our results make predictions for the number and orbital distributions of potential resonant objects that will be discovered in the future throughout the MCKB.

A Dynamical Survey of the Trans-Neptunian Region. I. Mean-motion Resonances with Neptune

In this paper, we present a large-scale dynamical survey of the trans-Neptunian region, with particular attention to mean-motion resonances (MMRs). We study a set of 4121 trans-Neptunian objects (TNOs), a sample far larger than in previous works. We perform direct long-term numerical integrations that enable us to examine the overall dynamics of the individual TNOs as well as to identify all MMRs. For the latter purpose, we apply an own-developed FAIR method that allows the semi-automatic identification of even very high-order MMRs. Apart from searching for the more frequent eccentricity-type resonances that previous studies concentrated on, we set our method to allow the identification of inclination-type MMRs, too. Furthermore, we distinguish between TNOs that are locked in a given MMR throughout the whole integration time span (108 yr) and those that are only temporarily captured in resonances. For a more detailed dynamical analysis of the trans-Neptunian space, we also construct dynamical maps using test particles. Observing the fine structure of the 34–80 au region underlines the stabilizing role of the MMRs, with the regular regions coinciding with the positions of real TNOs.

Bridges and gaps at low-eccentricity first-order resonances

ABSTRACT Previous works on the divergence of first-order mean-motion resonances (MMRs) have studied in detail the extent of the pericentric and apocentric libration zones of adjacent first-order MMRs, highlighting possible bridges between them in the low eccentricity circular restricted three-body problem. Here, we describe the previous results in the context of periodic orbits and show that the so-called circular family of periodic orbits is the path that can drive the passage between neighbouring resonances under dissipative effects. We illustrate that the circular family can bridge first-order and higher order resonances, while its gaps at first-order MMRs can serve as boundaries that stop transitions between resonances. In particular, for the Sun–asteroid–Jupiter problem, we show that, during the migration of Jupiter in the protoplanetary disc, a system initially evolving below the apocentric branch of a first-order MMR follows the circular family and can either be captured into the pericentric branch of an adjacent first-order MMR if the orbital migration is rapid or in a higher order MMR in case of slow migration. Radial transport via the circular family can be extended to many small body and planetary system configurations undergoing dissipative effects (e.g. tidal dissipation, solar mass-loss, and gas drag).

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–Jupiter and Sun–Neptune 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.

INCLINATION MIXING IN THE CLASSICAL KUIPER BELT

We investigate the long-term evolution of the inclinations of the known classical and resonant Kuiper belt objects (KBOs). This is partially motivated by the observed bimodal inclination distribution and by the putative physical differences between the low- and high-inclination populations. We find that some classical KBOs undergo large changes in inclination over gigayear timescales, which means that a current member of the low-inclination population may have been in the high-inclination population in the past, and vice versa. The dynamical mechanisms responsible for the time-variability of inclinations are predominantly distant encounters with Neptune and chaotic diffusion near the boundaries of mean motion resonances. We reassess the correlations between inclination and physical properties including inclination time-variability. We find that the size-inclination and color-inclination correlations are less statistically significant than previously reported (mostly due to the increased size of the data set since previous works with some contribution from inclination variability). The time-variability of inclinations does not change the previous finding that binary classical KBOs have lower inclinations than non-binary objects. Our study of resonant objects in the classical Kuiper belt region includes objects in the 3:2, 7:4, 2:1, and eight higher-order mean motion resonances. We find that these objects (some of which were previously classified as non-resonant) undergo larger changes in inclination compared to the non-resonant population, indicating that their current inclinations are not generally representative of their original inclinations. They are also less stable on gigayear timescales.

TRANSIT TIMING VARIATIONS FOR INCLINED AND RETROGRADE EXOPLANETARY SYSTEMS

We perform numerical calculations of the expected transit timing variations (TTVs) induced on a Hot-Jupiter by an Earth-mass perturber. Motivated by the recent discoveries of retrograde transiting planets, we concentrate on an investigation of the effect of varying relative planetary inclinations, up to and including completely retrograde systems. We find that planets in low order (E.g. 2:1) mean-motion resonances (MMRs) retain approximately constant TTV amplitudes for $0<\,^{\circ}i<170\,^{\circ}$, only reducing in amplitude for $i > 170\,^{\circ}$. Systems in higher order MMRs (E.g. 5:1) increase in TTV amplitude as inclinations increase towards $45\,^{\circ}$, becoming approximately constant for $45 < i < 135$, and then declining for $i > 135\,^{\circ}$. Planets away from resonance slowly decrease in TTV amplitude as inclinations increase from 0 to 180, where-as planets adjacent to resonances can exhibit a huge range of variability in TTV amplitude as a function of both eccentricity and inclination. For highly retrograde systems ($135\,^{\circ} < i \leq 180\,^{\circ}$), TTV signals will be undetectable across almost the entirety of parameter space, with the exceptions occurring when the perturber has high eccentricity or is very close to a MMR. This high inclination decrease in TTV amplitude (on and away from resonance) is important for the analysis of the known retrograde and multi-planet transiting systems, as inclination effects need to be considered if TTVs are to be used to exclude the presence of any putative planetary companions: absence of evidence is not evidence of absence.

A giant impact origin for Pluto's small moons and satellite multiplicity in the Kuiper belt

The two newly discovered satellites of Pluto (P1 and P2) have masses that are small compared to both Pluto and Charon-that is, between 5 x 10(-4) and 1 x 10(-5) of Pluto's mass, and between 5 x 10(-3) and 1 x 10(-4) of Charon's mass. This discovery, combined with the constraints on the absence of more distant satellites of Pluto, reveal that Pluto and its moons comprise an unusual, highly compact, quadruple system. These facts naturally raise the question of how this puzzling satellite system came to be. Here we show that P1 and P2's proximity to Pluto and Charon, the fact that P1 and P2 are on near-circular orbits in the same plane as Pluto's large satellite Charon, along with their apparent locations in or near high-order mean-motion resonances, all probably result from their being constructed from collisional ejecta that originated from the Pluto-Charon formation event. We also argue that dust-ice rings of variable optical depths form sporadically in the Pluto system, and that rich satellite systems may be found--perhaps frequently--around other large Kuiper belt objects.

Neptune's Migration into a Stirred-Up Kuiper Belt: A Detailed Comparison of Simulations to Observations

We use N-body simulations to examine the consequences of Neptune's outward migration into the Kuiper Belt, with the simulated end states being compared rigorously and quantitatively to the observations. These simulations confirm the 2003 findings of Chiang and coworkers, who showed that Neptune's migration into a previously stirred-up Kuiper Belt can account for the Kuiper Belt objects (KBOs) known to librate at Neptune's 5 : 2 resonance. We also find that capture is possible at many other weak, high-order mean-motion resonances, such as 11 : 6, 13 : 7, 13 : 6, 9 : 4, 7 : 3, 12 : 5, 8 : 3, 3 : 1, 7 : 2, and 4 : 1. The more distant of these resonances, such as the 9 : 4, 7 : 3, 5 : 2, and 3 : 1, can also capture particles in stable, eccentric orbits beyond 50 AU, in the region of phase space conventionally known as the "Scattered Disk." Indeed, 90% of the simulated particles that persist over the age of the solar system in the Scattered-Disk zone never had a close encounter with Neptune but instead were promoted into these eccentric orbits by Neptune's resonances during the migration epoch. This indicates that the observed Scattered Disk might not be so scattered. This model also produced only a handful of Centaurs, all of which originated at Neptune's mean-motion resonances in the Kuiper Belt. However, a noteworthy deficiency of the migration model considered here is that it does not account for the observed abundance of Main Belt KBOs having inclinations higher than 15°. In order to rigorously compare the model end state with the observed Kuiper Belt in a manner that accounts for telescopic selection effects, Monte Carlo methods are used to assign sizes and magnitudes to the simulated particles that survive over the age of the solar system. If the model considered here is indeed representative of the outer solar system's early history, then the following conclusions are obtained: (1) The observed 3 : 2 and 2 : 1 resonant populations are both depleted by a factor of ∼20 relative to model expectations; this depletion is likely due to unmodeled effects, possibly perturbations by other large planetesimals. (2) The size distribution of those KBOs inhabiting the 3 : 2 resonance is significantly shallower than the Main Belt's size distribution. (3) The total number of KBOs having radii R > 50 km and orbiting interior to Neptune's 2 : 1 resonance is N ∼ 1.7 × 105; these bodies have a total mass of M ∼ 0.08(ρ/1 g cm-3)(p/0.04)-3/2 M⊕, assuming they have a material density ρ and an albedo p. We also report estimates of the abundances and masses of the Belt's various subpopulations (e.g., the resonant KBOs, the Main Belt, and the so-called Scattered Disk) and provide upper limits on the abundance of Centaurs and Neptune's Trojans, as well as upper limits on the sizes and abundances of hypothetical KBOs that might inhabit the a > 50 AU zone.

The dynamical structure of the habitable zone in the HD 38529, HD 168443 and HD 169830 systems

The dynamical structure of the habitable zone in the multiple exoplanetary systems HD 38529, HD 168443 and HD 169830 is investigated. By using long-time numerical integration and fast chaos-detection methods, it is shown that the habitable zone of all three systems is mostly chaotic. There is a stable region between the two known planets only in the system HD 38529, near the inner edge of the habitable zone, where a third planet could exist. This region is stable for a wide range of orbital eccentricity and mass of the larger outer planet. The stability region is divided by higher-order mean motion resonances, which are studied in detail. It seems that higher-order resonances are important in exoplanetary systems due to the large planetary masses and large orbital eccentricities.

Stable Chaos in the Asteroid Belt

Most asteroids are characterized by positive Lyapounov Characteristic Exponents (LCE), which means that their orbits are chaotic. Indeed, the striking fact is that for many of them the maximum LCE is very large, with a corresponding Lyapounov time shorter (often much shorter) than 100,000 years, as we report in this work. This finding is in apparent contradiction with the overall orbital stability observed in numerical integrations covering timespans orders of magnitude longer than the Lyapounov times. More importantly, it is in contradiction with the estimated astronomical age of these small planets. The phenomenon, of which other examples are already known, is referred to as stable chaos. We have studied in particular a region, that of the Veritas asteroid family, where other causes of chaotic motion are not relevant and stable chaos can be investigated as such in its various appearances. Stable chaos is found to be due to high order mean motion resonances with Jupiter in combination with secular perturbations on the perihelia of the asteroids. These perturbations cause a large number of critical arguments to get in and out of the resonance and can move the orbit from one high resonance to another, in a typical irregular behavior. Since chaos is driven by secular perturbations, it is no surprise that the Lyapounov times are very close to the timescales of these perturbations. The quasi-integrals of motion of asteroids with very short Lyapounov times are less stable and a slow diffusion is observed in eccentricity and inclination. There is essentially no diffusion in semimajor axis, although variations are large (e.g., between two high order resonances). These asteroids can therefore be regarded as cases of stable chaos.

On the past orbital history of comet P/Halley

In recent papers (Carusi et al.; 1986, 1987) the authors reported the discovery of temporary librating behaviours about high order mean motion resonances with Jupiter of a number of short-period comets of Halley type, taking place in the period 1585 – 2406 AD. These comets are: P/Brorsen-Metcalf, P/Pons-Brooks and P/O1bers, all librating about the 6∶1 resonance, P/Dubiago, about the 5∶1, and P/Vaisala 2, about the 7∶1.