Mean Motion Resonances at High Eccentricities: The 2:1 and the 3:2 Interior Resonances

Abstract

Mean motion resonances [MMRs] play an important role in the formation and evolution of planetary systems and have significantly influenced the orbital properties and distribution of planets and minor planets in the solar system as well as exo-planetary systems. Most previous theoretical analyses have focused on the low-to-moderate eccentricity regime, but with new discoveries of high eccentricity resonant minor planets and even exoplanets, there is increasing motivation to examine MMRs in the high eccentricity regime. Here we report on a study of the high eccentricity regime of MMRs in the circular planar restricted three-body problem. Non-perturbative numerical analyses of the 2:1 and the 3:2 interior resonances are carried out for a wide range of secondary-to-primary mass ratio, and for a wide range of eccentricity of the test particle. The surface-of-section technique is used to study the phase space structure near resonances. We identify transitions in phase space at certain critical eccentricities related to the geometry of resonant orbits; new stable libration zones appear at high eccentricity at libration centers shifted from those at low eccentricities. We present novel results on the mass and eccentricity dependence of the resonance libration centers and their widths in semi-major axis. Our results show that MMRs have sizable libration zones at high eccentricities, comparable to those at lower eccentricities.

Dynamic Portrait of the Retrograde 1:1 Mean Motion Resonance

“…Unlike the case of prograde resonance, where the libration width decreases with increasing eccentricity above the planet-crossing curve (Wang & Malhotra 2017), we notice that the width of retrograde 1:1 resonance keeps growing with increasing eccentricity. However, our evaluation of libration width based on averaged perturbation is just a preliminary estimation.…”

“…While considering other mean mo-tion commensurabilities like 2:1 or 3:1, the series expansion is only valid for small eccentricity. When the eccentricity of the asteroid is above the planet-crossing limit (e > 1/α − 1 for outer perturber and e > 1 − 1/α for inner perturber), a new libration zone arises (Moons & Morbidelli 1993;Wang & Malhotra 2017), which is now explained as the phase protection mechanism from planetary collisions provided by mean motion resonance (Morbidelli 2002). To sum up, the series expansion can only be used when orbits under consideration do not cross (Batygin & Morbidelli 2017;Mardling 2013) and the ratio of semi-major axes is never unity.…”

Trans-Neptunian Objects Transiently Stuck in Neptune’s Mean-motion Resonances: Numerical Simulations of the Current Population

“…We examined the libration behavior for many known objects and determined that these slips through f=0 are rare and typically so short that our averaging procedure outlined above would be insensitive to them; in other cases, our procedure would merely result in the single stick being split into two separate sticks (divided in time at the point where the crossing occurs). We note that at very high eccentricities, stable libration in resonance is theoretically possible around f=0 (see Wang & Malhotra 2017;Malhotra et al 2018); however, for Neptune's resonances, this stable zone occurs at very high, deeply planet-crossing eccentricities, which should result in only very short sticks. Malhotra et al (2018), for example, noted an observed TNO that is experiencing a temporary stick around f=0 in Neptune's 5:2 resonance.…”

Neptune’s resonances in the scattered disk

“…As illustrated in Figures 2 and 9, the phase space structure changes with eccentricity, with at least one particularly significant transition when new stable islands appear when eccentricity exceeds the Neptune-crossing value. We show in Sections 4 and 5 that these phase space transitions are related to the shape of the resonant orbit in the rotating frame and its relationship to Neptune's orbit (see also Wang & Malhotra (2017) and Malhotra et al (2018)).…”

“…A disadvantage is that non-co-planarity, the effects of a non-circular Neptune orbit, and the perturbations of other planets are not included; still, the map of stable libration zones of moderate-to-high-eccentricity MMRs in the simplified model yields new insights, as we will show. In the context of the outer solar system, this non-perturbative method has been previously employed by Pan & Sari (2004) for their study of exterior resonant orbits of Jacobi constant close to 3 (pericenter distance close to the planet's orbit); Wang & Malhotra (2017) employed it for a general study of the 2:1 and 3:2 interior resonances in the high eccentricity regime. Malhotra (1996) and Malhotra et al (2018) made use of this method for a small number of Neptune's exterior MMRs.…”