Resonant Kuiper Belt Research Articles

A new multiple-arc model of the resonant Kuiper belt objects – plutinos

Abstract To incorporate the gravitational influence of Kuiper belt objects (KBOs) in planetary ephemerides, uniform-ring models are commonly employed. In this paper, for representing the KBO population residing in Neptune’s 2:3 mean motion resonance (MMR), known as the Plutinos, we introduce a three-arc model by considering their resonant characteristics. Each ‘arc’ refers to a segment of the uniform ring and comprises an appropriate number of point masses. Then the total perturbation of Plutinos is numerically measured by the change in the Sun-Neptune distance (ΔdSN). We conduct a comprehensive investigation to take into account various azimuthal and radial distributions associated with the resonant amplitudes (A) and eccentricities (e) of Plutinos, respectively. The results show that over a 100-year period: (1) at the smallest e = 0.05, the Sun-Neptune distance change ΔdSN caused by Plutinos decreases significantly as A reduces. It can deviate from the value of ΔdSN obtained in the ring model by approximately 100 km; (2) as e increases in the medium range of 0.1-0.2, the difference in ΔdSN between the arc and ring models becomes increasingly significant; (3) at the largest e ≳ 0.25, ΔdSN can approach zero regardless of A, and the arc and ring models exhibit a substantial difference in ΔdSN, reaching up to 170 km. Then the applicability of our three-arc model is further verified by comparing it to the perturbations induced by observed Plutinos on the positions of both Neptune and Saturn. Moreover, the concept of the multiple-arc model, designed for Plutinos, can be easily extended to other MMRs densely populated by small bodies.

Large-step neural network for learning the symplectic evolution from partitioned data

ABSTRACT In this study, we focus on learning Hamiltonian systems, which involves predicting the coordinate ($\boldsymbol q$) and momentum ($\boldsymbol p$) variables generated by a symplectic mapping. Based on Chen & Tao (2021), the symplectic mapping is represented by a generating function. To extend the prediction time period, we develop a new learning scheme by splitting the time series ($\boldsymbol q_i$, $\boldsymbol p_i$) into several partitions. We then train a large-step neural network (LSNN) to approximate the generating function between the first partition (i.e. the initial condition) and each one of the remaining partitions. This partition approach makes our LSNN effectively suppress the accumulative error when predicting the system evolution. Then we train the LSNN to learn the motions of the 2:3 resonant Kuiper belt objects for a long time period of 25 000 yr. The results show that there are two significant improvements over the neural network constructed in our previous work: (1) the conservation of the Jacobi integral and (2) the highly accurate predictions of the orbital evolution. Overall, we propose that the designed LSNN has the potential to considerably improve predictions of the long-term evolution of more general Hamiltonian systems.

How the formation of Neptune shapes the Kuiper belt

Hydrodynamical simulations predict the inward migration of giant planets during the gas phase of the protoplanetary disc. This phenomenon is also invoked to explain resonant and near-resonant exoplanetary system structures. The early inward migration may also have affected our Solar System and sculpted its different minor planet reservoirs. In this study we explore how the early inward migration of the giant planets shapes the Kuiper belt. We test different scenarios with only Neptune and Uranus and with all the four giant planets, also including some models with the subsequent outward planetesimal-driven migration of Neptune after the gas dispersal. We find objects populating mean motion resonances even when Neptune and Uranus do not migrate at all or only migrate inwards. When the planets are fixed, planetesimals stick only temporarily to the mean motion resonances, while inwards migration yields a new channel to populate the resonances without invoking convergent migration. However, in these cases, it is hard to populate mean motion resonances that do not cross the planetesimal disc (such as 2:1 and 5:2) and there is a lack of resonant Kuiper belt objects (KBOs) crossing Neptune’s orbit. These Neptune crossers are an unambiguous signature of the outward migration of Neptune. The starting position and the growth rate of Neptune have consequences for the contamination of the classical Kuiper belt region from neighbouring regions. The eccentricity and inclination space of the hot classical Kuiper belt objects and the scattered disc region become much more populated when all the giant planets are included. The 5:2 resonance with Neptune becomes increasingly populated with deeper inward migrations of Neptune. However, the overall inclination distribution is still narrower than suggested by observations, as is generally the case for Kuiper belt population models.

An analytically treatable model of long-term dynamics in a mean motion resonance with coexisting resonant modes

We examine a 2-DOF Hamiltonian system modeling some previously unexplored dynamical effects in first-order mean motion resonances in the spatial circular restricted three-body problem “star-planet-asteroid.” In distinction from the well-known integrable model of Sessin and Ferraz-Mello (Celest Mech Dyn Astron 32:307–332, 1984) and Wisdom (Celest Mech Dyn Astron 38:175–180, 1986), in our analysis, we kept more terms in the expansion for the disturbing function. This allowed us to study the phenomena caused by coexisting resonant modes, that were lost in the integrable model. In addition, we identified the areas of dynamical chaos in the phase space of the considered system. Despite the non-integrability of our model, we were able to compute analytically many quantities characterizing its dynamical behavior. Finally, we illustrate some of our results through the examples of the known resonant Kuiper belt objects.

Phase Curves of Nix and Hydra from the New Horizons Imaging Cameras

Abstract NASA’s New Horizons spacecraft’s voyage through the Pluto system centered on 2015 July 14 provided images of Pluto’s small satellites Nix and Hydra at viewing angles unattainable from Earth. Here, we present solar phase curves of the two largest of Pluto’s small moons, Nix and Hydra, observed by the New Horizons LOng Range Reconnaissance Imager and Multi-spectral Visible Imaging Camera, which reveal the scattering properties of their icy surfaces in visible light. Construction of these solar phase curves enables comparisons between the photometric properties of Pluto’s small moons and those of other icy satellites in the outer solar system. Nix and Hydra have higher visible albedos than those of other resonant Kuiper Belt objects and irregular satellites of the giant planets, but not as high as small satellites of Saturn interior to Titan. Both Nix and Hydra appear to scatter visible light preferentially in the forward direction, unlike most icy satellites in the outer solar system, which are typically backscattering.

THE FIRST HIGH-PHASE OBSERVATIONS OF A KBO: NEW HORIZONS IMAGING OF (15810) 1994 JR1 FROM THE KUIPER BELT

ABSTRACT NASA’s New Horizons spacecraft observed (15810) 1994 JR1, a 3:2 resonant Kuiper belt object (KBO), using the LOng Range Reconnaissance Imager (LORRI) on 2015 November 2 from a distance of 1.85 au and again on 2016 April 7 from a distance of 0.71 au. These were the first close observations of any KBO other than Pluto. Combining ground-based and Hubble Space Telescope (HST) observations at small phase angles and the LORRI observations at higher phase angles, we produced the first disk-integrated solar phase curve of a typical KBO from α = 0.°6–58°. Observations at these geometries, attainable only from a spacecraft in the outer solar system, constrain surface properties such as macroscopic roughness and the single particle phase function. 1994 JR1 has a rough surface with a 37° ± 5° mean topographic slope angle and has a relatively rapid rotation period of 5.47 ± 0.33 hr. 1994 JR1 is currently 2.7 au from Pluto; our astrometric points enable high-precision orbit determination and integrations that show that it comes this close to Pluto every 2.4 million years (104 heliocentric orbits), causing Pluto to perturb 1994 JR1. During the November spacecraft observation, the KBO was simultaneously observed using HST in two colors, confirming its very red spectral slope. These observations have laid the groundwork for numerous potential future distant KBO observations in the New Horizons–Kuiper belt extended mission.

Impact and cratering rates onto Pluto

The New Horizons spacecraft fly-through of the Pluto system in July 2015 will provide humanity’s first data for the crater populations on Pluto and its binary companion, Charon. In principle, these surfaces could be dated in an absolute sense, using the observed surface crater density (# craters/km2 larger than some threshold crater diameter D). Success, however, requires an understanding of both the cratering physics and absolute impactor flux. The Canada-France Ecliptic Plane Survey (CFEPS) L7 synthetic model of classical and resonant Kuiper belt populations (Petit, J.M. et al. [2011]. Astron. J. 142, 131–155; Gladman, B. et al. [2012]. Astron. J. 144, 23–47) and the scattering object model of Kaib et al. (Kaib, N., Roškar, R., Quinn, T. [2011]. Icarus 215, 491–507) calibrated by Shankman et al. (Shankman, C. et al. [2013]. Astrophys. J. 764, L2–L5) provide such impact fluxes and thus current primary cratering rates for each dynamical sub-population. We find that four sub-populations (the q<42AU hot and stirred main classicals, the classical outers, and the plutinos) dominate Pluto’s impact flux, each providing ≈15–25% of the total rate. Due to the uncertainty in how the well-characterized size distribution for Kuiper belt objects (with impactor diameter d>100km) connects to smaller projectiles, we compute cratering rates using five model impactor size distributions: a single power-law, a power-law with a knee, a power-law with a divot, as well as the “wavy” size distributions described in Minton et al. (Minton, D.A. et al. [2012]. Asteroids Comets Meteors Conf. 1667, 6348) and Schlichting et al. (Schlichting, H.E., Fuentes, C.I., Trilling, D.E. [2013]. Astron. J. 146, 36–42). We find that there is only a small chance that Pluto has been hit in the past 4Gyr by even one impactor with a diameter larger than the known break in the projectile size distribution (d≈100km) which would create a basin on Pluto (D⩾400km in diameter). We show that due to present uncertainties in the impactor size distribution between d=1–100km, computing absolute ages for the surface of Pluto is entirely dependent on the extrapolation to small sizes and thus fraught with uncertainty. We show, however, what the ages would be for several cases and illustrate the relative importance of each Kuiper belt sub-population to the cratering rate, both now and integrated into the past. In addition, we compute the largest “fresh” crater expected to have formed in 1Gyr on the surface of Pluto and in 3Gyr on Charon (to 95% confidence) and use the “wavy” size distribution models to predict whether these largest “fresh” craters will provide surfaces for which portions of the crater production function can be measured should most of the target’s surface appear saturated. The fly-through results coupled with telescopic surveys that bridge current uncertainties in the d=10–100km regime should eventually result in the population estimate uncertainties for the Kuiper belt sub-populations, and thus the impact fluxes onto Pluto and Charon, dipping to <30%. We also compute “disruption timescales” (to a factor of three accuracy) for Pluto’s smaller satellites: Styx, Nix, Kerberos, and Hydra. We find that none of the four satellites have likely undergone a catastrophic disruption and reassembly event in the past ≈4Gyr. In addition, we find that for a knee size distribution with αfaint⩽0.4 (down to sub-km diameters), satellites of all sizes are able to survive catastrophic disruption over the past 4Gyr.

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.

CHAOTIC DIFFUSION OF RESONANT KUIPER BELT OBJECTS

We carried out extensive numerical orbit integrations to probe the long-term chaotic dynamics of the two strongest mean motion resonances of Neptune in the Kuiper belt, the 3:2 (Plutinos) and 2:1 (Twotinos). Our primary results include a computation of the relative volumes of phase space characterized by large- and small-resonance libration amplitudes, and maps of resonance stability measured by mean chaotic diffusion rate. We find that Neptune's 2:1 resonance has weaker overall long-term stability than the 3:2 -- only ~15% of Twotinos are projected to survive for 4 Gyr, compared to ~27% of Plutinos, based on an extrapolation from our 1-Gyr integrations. We find that Pluto has only a modest effect, causing a ~4% decrease in the Plutino population that survives to 4 Gyr. Given current observational estimates, and assuming an initial distribution of particles proportional to the local phase space volume in the resonance, we conclude that the primordial populations of Plutinos and Twotinos formerly made up more than half the population of the classical and resonant Kuiper Belt. We also conclude that Twotinos were originally nearly as numerous as Plutinos; this is consistent with predictions from early models of smooth giant planet migration and resonance sweeping of the Kuiper Belt, and provides a useful constraint for more detailed models.

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.

Collisional velocities and rates in resonant planetesimal belts

We consider a belt of small bodies around a star, captured in one of the external or 1:1 mean-motion resonances with a massive perturber. The objects in the belt collide with each other. Combining methods of celestial mechanics and statistical physics, we calculate mean collisional velocities and collisional rates, averaged over the belt. The results are compared to collisional velocities and rates in a similar, but non-resonant belt, as predicted by the particle-in-a-box method. It is found that the effect of the resonant lock on the velocities is rather small, while on the rates more substantial. The collisional rates between objects in an external resonance are by about a factor of two higher than those in a similar belt of objects not locked in a resonance. For Trojans under the same conditions, the collisional rates may be enhanced by up to an order of magnitude. Our results imply, in particular, shorter collisional lifetimes of resonant Kuiper belt objects in the solar system and higher efficiency of dust production by resonant planetesimals in debris disks around other stars.

Dynamical determination of the mass of the Kuiper Belt from motions of the inner planets of the Solar system

In this paper we determine dynamically the mass of the Kuiper Belt Objects by exploiting the latest least-squares determinations of the extra rates of perihelia of the inner planets of the Solar system. By modelling classical Kuiper Belt Objects as an ecliptic ring of finite thickness, we obtain 0.033 ± 0.115 in units of terrestrial masses. For resonant Kuiper Belt Objects, a two-ring model yields 0.018 ± 0.063. These values are consistent with recent determinations obtained using ground- and space-based optical techniques. Some implications for precise tests of Einsteinian and post-Einsteinian gravity are briefly discussed.

Brownian Motion in Planetary Migration

A residual planetesimal disk of mass 10-100 Earth masses remained in the outer solar system following the birth of the giant planets, as implied by the existence of the Oort cloud, coagulation requirements for Pluto, and inefficiencies in planet formation. Upon gravitationally scattering planetesimal debris, planets migrate. Orbital migration can lead to resonance capture, as evidenced here in the Kuiper and asteroid belts, and abroad in extra-solar systems. Finite sizes of planetesimals render migration stochastic (noisy). At fixed disk mass, larger (fewer) planetesimals generate more noise. Extreme noise defeats resonance capture. We employ order-of-magnitude physics to construct an analytic theory for how a planet's orbital semi-major axis fluctuates in response to random planetesimal scatterings. To retain a body in resonance, the planet's semi-major axis must not random walk a distance greater than the resonant libration width. We translate this criterion into an analytic formula for the retention efficiency of the resonance as a function of system parameters, including planetesimal size. We verify our results with tailored numerical simulations. Application of our theory reveals that capture of Resonant Kuiper belt objects by a migrating Neptune remains effective if the bulk of the primordial disk was locked in bodies having sizes 1000 km was less than a few percent. Coagulation simulations produce a size distribution of primordial planetesimals that easily satisfies these constraints. We conclude that stochasticity did not interfere with, nor modify in any substantive way, Neptune's ability to capture and retain Resonant Kuiper belt objects during its migration.

A Non-Planar Circular Model for the 4/7 Resonance

An adiabatic approximation for the non-planar, circular, restricted 3BP is presented for the external resonance 4/7. It can be used as a model for resonant Kuiper belt objects. The Hamiltonian is truncated at the fourth order in eccentricities and inclinations. After averaging, we have a system of two degrees of freedom with two frequencies. Numerical calculations show that the ratio of these frequencies is ~102. Having introduced suitable canonical variables, we used the adiabatic approach introduced by Wisdom in a different context. We left slow variables frozen and after solving the pendulum problem for fast variables, we used the averaged effect of fast variables on slow variables. In this way we obtained the guiding trajectories for slow variables as contour lines of adiabatic invariant. We discuss the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds to the separatrix of fast pendulum motion, where the assumption of sharp division between fast and slow frequencies is not correct and the adiabatic theory fails. The model works well for e ~ 0.1 and can be used for finding the chaotic regions, but for e~ 0.17 it becomes unsatisfactory due to truncation and bad convergence of the Laplace expansion. Qualitatively it can, however, help us to understand how the protective mechanism works as the interplay of mean motion and Kozai–Lidov resonance.

Resonance Occupation in the Kuiper Belt: Case Examples of the 52 and Trojan Resonances

As part of our ongoing Deep Ecliptic Survey (DES) of the Kuiper belt, we report on the occupation of the 1 : 1 (Trojan), 4 : 3, 3 : 2, 7 : 4, 2 : 1, and 5 : 2 Neptunian mean motion resonances (MMRs). The previously unrecognized occupation of the 1 : 1 and 5 : 2 MMRs is not easily understood within the standard model of resonance sweeping by a migratory Neptune over an initially dynamically cold belt. Among all resonant Kuiper belt objects (KBOs), the three observed members of the 5 : 2 MMR discovered by DES possess the largest semimajor axes (a ≈ 55.4 AU), the highest eccentricities (e ≈ 0.4), and substantial orbital inclinations (i ≈ 10°). Objects (38084) 1999HB12 and possibly 2001KC77 can librate with modest amplitudes of ~90° within the 5 : 2 MMR for at least 1 Gyr. Their trajectories cannot be explained by close encounters with Neptune alone, given the latter's current orbit. The dynamically hot orbits of such 5 : 2 resonant KBOs, unlike hot orbits of previously known resonant KBOs, may imply that these objects were preheated to large inclination and large eccentricity prior to resonance capture by a migratory Neptune. Our first discovered Neptunian Trojan, 2001QR322, may not owe its existence to Neptune's migration at all. The trajectory of 2001QR322 is remarkably stable; the object can undergo tadpole-type libration about Neptune's leading Lagrange (L4) point for at least 1 Gyr with a libration amplitude of 24°. Trojan capture probably occurred while Neptune accreted the bulk of its mass. For an assumed albedo of 12%–4%, our Trojan is ~130–230 km in diameter. Model-dependent estimates place the total number of Neptune Trojans resembling 2001QR322 at ~20–60. Their existence helps to rule out violent orbital histories for Neptune.

Present status of chestnut blight

The New Horizons spacecraft fly-through of the Pluto system in July 2015 will provide humanity’s first data for the crater populations on Pluto and its binary companion, Charon. In principle, these surfaces could be dated in an absolute sense, using the observed surface crater density (# craters/km2 larger than some threshold crater diameter D). Success, however, requires an understanding of both the cratering physics and absolute impactor flux. The Canada-France Ecliptic Plane Survey (CFEPS) L7 synthetic model of classical and resonant Kuiper belt populations (Petit, J.M. et al. [2011]. Astron. J. 142, 131–155; Gladman, B. et al. [2012]. Astron. J. 144, 23–47) and the scattering object model of Kaib et al. (Kaib, N., Roškar, R., Quinn, T. [2011]. Icarus 215, 491–507) calibrated by Shankman et al. (Shankman, C. et al. [2013]. Astrophys. J. 764, L2–L5) provide such impact fluxes and thus current primary cratering rates for each dynamical sub-population. We find that four sub-populations (the q<42AU hot and stirred main classicals, the classical outers, and the plutinos) dominate Pluto’s impact flux, each providing ≈15–25% of the total rate. Due to the uncertainty in how the well-characterized size distribution for Kuiper belt objects (with impactor diameter d>100km) connects to smaller projectiles, we compute cratering rates using five model impactor size distributions: a single power-law, a power-law with a knee, a power-law with a divot, as well as the “wavy” size distributions described in Minton et al. (Minton, D.A. et al. [2012]. Asteroids Comets Meteors Conf. 1667, 6348) and Schlichting et al. (Schlichting, H.E., Fuentes, C.I., Trilling, D.E. [2013]. Astron. J. 146, 36–42). We find that there is only a small chance that Pluto has been hit in the past 4 Gyr by even one impactor with a diameter larger than the known break in the projectile size distribution (d≈100km) which would create a basin on Pluto (D⩾400km in diameter). We show that due to present uncertainties in the impactor size distribution between d=1–100km, computing absolute ages for the surface of Pluto is entirely dependent on the extrapolation to small sizes and thus fraught with uncertainty. We show, however, what the ages would be for several cases and illustrate the relative importance of each Kuiper belt sub-population to the cratering rate, both now and integrated into the past. In addition, we compute the largest “fresh” crater expected to have formed in 1 Gyr on the surface of Pluto and in 3 Gyr on Charon (to 95% confidence) and use the “wavy” size distribution models to predict whether these largest “fresh” craters will provide surfaces for which portions of the crater production function can be measured should most of the target’s surface appear saturated. The fly-through results coupled with telescopic surveys that bridge current uncertainties in the d=10–100km regime should eventually result in the population estimate uncertainties for the Kuiper belt sub-populations, and thus the impact fluxes onto Pluto and Charon, dipping to <30%. We also compute “disruption timescales” (to a factor of three accuracy) for Pluto’s smaller satellites: Styx, Nix, Kerberos, and Hydra. We find that none of the four satellites have likely undergone a catastrophic disruption and reassembly event in the past ≈4Gyr. In addition, we find that for a knee size distribution with αfaint⩽0.4 (down to sub-km diameters), satellites of all sizes are able to survive catastrophic disruption over the past 4 Gyr.