The process of capture in the coorbital region of a solar system planet is studied. Absolute capture likelihood in the 1:1 resonance is determined by randomly constructed statistical ensembles numbering $7.24\times 10^5$ of massless asteroids that are set to migrate radially from the outer to the inner boundaries of the coorbital region of a Jupiter-mass planet. Orbital states include coorbital capture, ejection, collisions with the Sun and the planet and free-crossing of the coorbital region. The relative efficiency of retrograde capture with respect to prograde capture is confirmed as an intrinsic property of the coorbital resonance. Half the asteroids cross the coorbital region regardless of eccentricity and for any inclination less than $120^\circ$. We also find that the recently discovered retrograde coorbital of Jupiter, asteroid 2015 BZ509, lies almost exactly at the capture efficiency peak associated with its orbital parameters.
Motivated by the dynamics of resonance capture, we study numerically the coorbital resonance for inclination180 >=I>=0 in the circular restricted three-body problem. We examine the similarities and differences between planar and three dimensional coorbital resonance capture and seek their origin in the stability of coorbital motion at arbitrary inclination. After we present stability maps of the planar prograde and retrograde coorbital resonances, we characterize the new coorbital modes in three dimensions. We see that retrograde mode I (R1) and mode II (R2) persist as we change the relative i…
In this work, we construct and test an analytical and a semianalytical secular models for two planets locked in a coorbital non-coplanar motion, comparing some results with the case of restricted three body problem. The analytical average model replicates the numerical N-body integrations, even for moderate eccentricities ($\lesssim$ 0.3) and inclinations ($\lesssim10^\circ$), except for the regions corresponding to quasi-satellite and Lidov-Kozai configurations. Furthermore, this model is also useful in the restricted three body problem, assuming very low mass ratio between the planets. We al…
对任意倾角的三体问题中的共振捕获进行了数值研究。无质量的粒子被设置为从1：5共振之外与一颗质量为1000的行星漂移，从而遇到行星的不同平均运动共振的网络。随机构造的样本探索0至180度倾角的参数空间，增量为5度，总计近6x10^5个数值模拟。30个共振内部和外部的行星的位置进行监测。我们发现，逆行共振是出乎意料的更有效的捕获比逆行共振，共振顺序不一定是一个很好的指标，在任意倾角的捕获效率。捕获概率显着下降，在中等样本偏心度的初始倾角范围内[10度，110度]。轨道反演是可能的初始圆形轨道与倾角范围[60度，130度]。在1：1共轨共振中的捕获在大的逆行倾角时很可能发生。如果行星的轨道偏心率大于0.1，则通过偏心Kozai-Lidov机制的作用降低捕获概率。捕获的不对称性出现在内部和外部共振之间，因为被捕获的轨道优先被捕获在内部共振中。逆行共振的相对俘获效率表明，逆行轨道上的达摩克洛和半人马座小行星的动力学寿命必须明显大于逆行轨道上的动力学寿命，这意味着最近发现的逆行共振小行星2006 BZ 8，2008 SO 218，2009 QY 6和1999 LE 31（Morais和Nynui，2013，MNRAS 436，L30）可能是在外部巨行星之间徘徊的最古老的小天体之一。
The effect of radial drift rate on mean motion resonance capture is studied for prograde, polar and retrograde orbits. We employ the numerical framework of our earlier exploration of resonance capture at arbitrary inclination. Randomly constructed samples of massless particles are set to migrate radially from outside the orbit of a Jupiter-mass planet at different drift rates totalling more than $1.6\times 10^6$ numerical simulations. Slower drift rates reduce overall capture probability especially for prograde orbits and enhance capture at specific initial inclinations of high order resonance…
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 …
The classical disturbing function of the three-body problem widely used in planetary dynamics studies is an expansion of the gravitational interaction of the three-body problem with respect to zero eccentricity and zero inclination. This restricts its validity to nearly coplanar orbits. Motivated by the dynamical study of asteroids, Centaurs and transneptunian objects with arbitrary inclinations, we derive a series expansion of the gravitational interaction with respect to an arbitrary reference inclination that generalises our work on the polar and retrograde disturbing functions. The new dis…
A fundamental aspect of the three-body problem is its stability. Most stability studies have focused on the co-planar three-body problem, deriving analytic criteria for the dynamical stability of such pro/retrograde systems. Numerical studies of inclined systems phenomenologically mapped their stability regions, but neither complement it by theoretical framework, nor provided satisfactory fit for their dependence on mutual inclinations. Here we present a novel approach to study the stability of hierarchical three-body systems at arbitrary inclinations, which accounts not only for the instantan…
Applying the technique of dynamical maps we study the orbital stability of test particles in the Solar System in the space (a,e,i) defined by 0.1<a<38 au, 0<e<0.9 and 0<i<180 identifying the unstable and stable regions. We find stable niches where small bodies can survive even for very high eccentricities. Mean motion resonances play a fundamental role providing stability against the planetary perturbations specially for high inclination orbits. A stability stripe around i=150 is present all along the Solar System. We found that the population of objects with semimajor axes between 10 and 30 a…
Pebble accretion is an efficient mechanism able to build up the core of the giant planets within the lifetime of the protoplanetary disc gas-phase. The core grows via this process until the protoplanet reaches its pebble isolation mass and starts to accrete gas. During the growth, the protoplanet undergoes a rapid, large-scale, inward migration due to the interactions with the gaseous protoplanetary disc. In our work, we investigate how this early migration would have affected the minor body populations in our solar system. In particular, we focus on the Jupiter Trojans and the Hildas asteroid…
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…
In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination and the harmonic arguments are l…
We propose a semianalytical method for the calculation of widths, libration centers and small amplitude libration periods of the mean motion resonances k_p:k in the framework of the circular restricted three body problem valid for arbitrary eccentricities and inclinations. Applying the model to the trans Neptunian region (TNR) we obtain several atlas of resonances between 30 and 100 au showing their domain in the plane (a,e) for different orbital inclinations. The resonance width may change substantially when varying the argument of the perihelion of the resonant object and in order to take in…
Mean motion resonances play a fundamental role in the dynamics of the small bodies of the Solar System. The last decades of the 20th century gave us a detailed description of the dynamics as well as the process of capture of small bodies in coplanar or small inclination resonant orbits. More recently, semianalytical or numerical methods allowed us to explore the behavior of resonant motions for arbitrary inclination orbits. The emerging dynamics is very rich, including large orbital changes due to secular effects inside mean motion resonances. The process of capture in highly inclined or retro…
We revisit the problem of secular resonance sweeping during the dissipation of a protoplanetary disk and its possible role in exciting the orbits of primordial asteroids, in the light of recent models of solar system evolution. We develop an integrator that incorporates the gravitational effect of a uniformly (or not) depleting, axisymmetric disk with arbitrary surface density profile; its performance is verified by analytical calculations. The secular response of fictitious asteroids, under perturbations from Jupiter, Saturn and a decaying disk, is thoroughly studied. Note that the existence …
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…
We have added to the Chroma+ suite of stellar atmosphere and spectrum modelling codes the ability to synthesize the exo-planet transit lightcurve for planets of arbitrary size up to 10% of the host stellar radius, and arbitrary planetary and stellar mass and orbital radius (thus determining orbital velocity) and arbitrary orbital inclination. The lightcurves are computed in situ, integrated with the radiative transfer solution for the radiation field emerging from the stellar surface, and there is no limb-darkening parameterization. The lightcurves are computed for the Johnson-Bessel photometr…
In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism take…
The existence of a giant planet beyond Neptune -- referred to as Planet Nine (P9) -- has been inferred from the clustering of longitude of perihelion and pole position of distant eccentric Kuiper belt objects (KBOs). After updating calculations of observational biases, we find that the clustering remains significant at the 99.6\% confidence level. We thus use these observations to determine orbital elements of P9. A suite of numerical simulations shows that the orbital distribution of the distant KBOs is strongly influenced by the mass and orbital elements of P9 and thus can be used to infer t…
Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. Perturbations on the time-scale of the orbital period of the perturber
