We apply the analytical disturbing function for arbitrary inclination derived in our previous work to characterize resonant width and libration of mean motion resonances at arbitrary inclination obtained from direct numerical simulations of the three-body problem. We examine the 2:1 and 3:1 inner Jupiter and 1:2 and 1:3 outer Neptune resonances and their possible asymmetric librations using a new analytical pendulum model of resonance that includes the simultaneous libration of multiple arguments and their second harmonics. The numerically-derived resonance separatrices are obtained using the Mean Exponential Growth factor of Nearby Orbits (MEGNO chaos indicator). We find that the analytical and numerical estimates are in agreement and that resonance width is determined by the first few fundamental resonance modes that librate simultaneously on the resonant timescale. Our results demonstrate that the new pendulum model may be used to ascertain resonance width analytically, and more generally, that the disturbing function for arbitrary inclination is a powerful analytical tool that describes resonance dynamics of low as well as high inclination asteroids in the solar system.
Aim. The eccentric von Zeipel-Lidov-Kozai (ZLK) effect is widely used to explain dynamical phenomena in varieties of astrophysical systems. The purpose of this work is to make clear the dynamical essence of the eccentric ZLK effect by constructing an inherent connection between such an effect and dynamics of secular resonance in restricted hierarchical planetary systems. Methods. Dynamical structures of apsidal resonance are analytically studied by means of perturbative treatments. The resonant model is formulated by averaging the Hamiltonian (up to octupole order) over rotating ZLK cycles, pr…
Orbital resonances play an important role in the dynamics of planetary systems. Classical theoretical analyses found in textbooks report that libration widths of first order mean motion resonances diverge for nearly circular orbits. Here we examine the nature of this divergence with a non-perturbative analysis of a few first order resonances interior to a Jupiter-mass planet. We show that a first order resonance has two branches, the pericentric and the apocentric resonance zone. As the eccentricity approaches zero, the centers of these zones diverge away from the nominal resonance location bu…
In spite of planetary resonances being a common dynamical mechanism acting on planetary systems, no general model exists for describing their properties, particularly for commensurabilities of any order and arbitrary values of the eccentricities and inclinations. The present work presents a semi-analytical model that describes the resonance strength, width, location and stability of fixed points, as well as periods of small-amplitude librations. The model is valid to any two gravitationally interacting massive bodies, and thus applicable to planets around single or binary stars. Using a theore…
In this study, dynamics of secular resonances for inner test particles are investigated under the octupole-level approximation by taking non-perturbative approaches. In practice, webs of the major secular resonances are produced by identifying families of stable periodic orbits and the associated stable libration zones are obtained by analysing Poincaré surfaces of section. By taking different values of the factor $ε$ ($ε$ measures the contribution of octupole terms), the influences of the octupole-order terms upon the dynamical structures are evaluated. Under the condition of $ε= 0$ (no octup…
In this work, retrograde mean motion resonances (MMRs) are investigated by means of analytical and numerical approaches. Initially, we define a new resonant angle to describe the retrograde MMRs and then perform a series of canonical transformations to formulate the resonant model, in which the phase portrait, resonant centre and resonant width can be analytically determined. To validate the analytical developments, the non-perturbative analysis is made by taking advantage of Poincaré surfaces of section. Some modifications are introduced in the production of Poincaré sections and, in particul…
Shell-wise orbital slotting in Low Earth Orbit (LEO) can improve space safety, simplify space traffic coordination and management, and optimize orbital capacity. This paper describes two methods to generate 2D Lattice Flower Constellations (2D-LFCs) that are defined with respect to either an arbitrary degree or an arbitrary degree and order Earth geopotential. By generating shells that are quasi-periodic and frozen with respect to the Earth geopotential, it is possible to safely stack shells with vertical separation distances smaller than the osculating variation in semi-major axis of each she…
A study of the high-inclination population in the Kuiper belt -- V. Mean motion resonances beyond 50 AU
Nanjing University（南京大学）；School of Astronomy and Space Science, Nanjing University（南京大学天文与空间科学学院）；Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University（南京大学现代天文与天体物理教育部重点实验室）
Perturbative analyses of planetary resonances commonly predict singularities and/or divergences of resonance widths at very low and very high eccentricities. We have recently re-examined the nature of these divergences using non-perturbative numerical analyses, making use of Poincaré sections but from a different perspective relative to previous implementations of this method. This perspective reveals fine structure of resonances which otherwise remains hidden in conventional approaches, including analytical, semi-analytical and numerical-averaging approaches based on the critical resonant ang…
Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the $2g+h$ lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity space for fixed semi-major axis. This allows to compute the change in the width of the $2g+h$ resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, $i_{L}$. Since $i_L$ is comparabl…
