The integrable cases of the planetary three-body problem at first-order resonance

This paper investigates the stability of the motion in the averaged planar general three-body problem in the case of first-order resonance. The equations of the averaged motion of bodies near the resonance surface is obtained and is analytically integrated by quadratures. The stability of the averaged motion is analytically investigated in relation to the semi-major axes, the eccentricities and the resonance phases. An autonomous second-order equation is obtained for the deviation of semiaxes from the resonance surface. This equation has an energy integral and is analytically integrated by quadratures. The quasi-periodic dependence on time with two-frequency basis of the averaged motion of bodies is found. The basic frequencies are analytically calculated. With the help of the mean functionals calculated along integral curves of the averaged problem the new analytic first integrals are constructed with coefficients periodic in time. The analytic conditions of librations of resonance phases are obtained.     

Mapping models for near-conservative systems with applications

A systematic method is presented to construct a mapping model for a near-conservative system, based on that of Hadjidemetriou for a Hamiltonian system [1]. The mapping we constructed preserves the basic features of the actual system. We apply this method to the forced oscillating system and to the 3 : 1 resonant grain motion in the presence of Poynting-Robertson drag.Supported by the National Natural Science Foundation of China.     

Forced secondary resonances at the 3 : 1 commensurability

For the 3 : 1 Jovian resonance problem, the time scales of the two degrees of freedom of the resonant Hamiltonian are well-separated [5]. With the adiabatic approximation, the solution for the fast oscillations can be found in terms of the slowly varying variables. Thus the rapidly oscillating terms in the slow oscillation equations can be treated as forced terms. We refer to the resonance between the forcing and intrinsic frequencies as a forced secondary one in this paper. We discuss the forced secondary resonances in asteroidal motion at the 3 : 1 commensurability by using Wisdom's method. The results show that the orbits situated originally near the resonance will leave the neighbourhood of resonance and tend to the separatrices and critical points for different energies, respectively. We have not found any stochastic web as expected in this case. Moreover, we study the problem of validity on the approximation of a system.The Project Supported by the National Natural Science Foundation of China.     

Surfaces of section in the Miranda-Umbriel 3:1 inclination problem

The recent numerical simulations of Tittemore and Wisdom (1988, 1989, 1990) and Dermottet al. (1988), Malhotra and Dermott (1990) concerning the tidal evolution through resonances of some pairs of Uranian satellites have revealed interesting dynamical phenomena related to the interactions between close-by resonances. These interactions produce chaotic layers and strong secondary resonances. The slow evolution of the satellite orbits in this dynamical landscape is responsible for temporary capture into resonance, enhancement of eccentricity or inclination and subsequent escape from resonance. The present contribution aims at developing analytical tools for predicting the location and size of chaotic layers and secondary resonances. The problem of the 3:1 inclination resonance between Miranda and Umbriel is analysed.     

Novel Analysis of Tadpole and Horseshoe Orbits

A simple analysis of tadpole and horseshoe orbits is performed using a new perturbation technique. The first-order results are presented explicitely, extension to any perturbation order is delineated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Revisiting natural and laboratory electron spin resonance (ESR) dose response curves of quartz from Chinese loess

Natural and laboratory ESR dose response curves (DRCs) of [AlO₄/h]⁰ and [TiO₄/M⁺]⁰ were investigated for samples of quartz from the Luochuan loess-palaeosol master section, Chinese Loess Plateau. The natural and laboratory DRCs show a clear divergence above ∼1000 Gy, with much lower D₀ values and saturation levels observed for the natural DRCs, which is in agreement with the previous study by Tsukamoto et al. (2018). Young (<15 ka) samples from Luochuan and Jingbian – another site of the Chinese Loess Plateau, together with two modern samples of Chinese loess, were used to investigate the residual signals of [AlO₄/h]⁰ and [TiO₄/M⁺]⁰ centres. Our results are in line with published studies and show that the significant residual signals corresponding to several tens to hundreds of Gy are present in both Al and Ti centres. These need to be taken into account before laboratory DRC construction. ESR pulse annealing experiments performed on samples irradiated with different doses show an apparent dose-dependent thermal instability of [AlO₄/h]⁰ and [TiO₄/M⁺]⁰, with the signals for higher doses decaying faster with increasing temperature. We attribute the change in D₀ with preheat reported in Tsukamoto et al.2018, as well as the difference between laboratory and natural DRCs, to this apparent dose-dependent thermal instability of the signals. The saturation level of the natural DRC, being the result of reaching the equilibrium between filling of the traps and emptying them due to thermal decay, is therefore additionally affected at higher doses, due to the increased thermal instability. The inability to recreate in the laboratory the same response to irradiation as the one observed in nature questions the accuracy of dating samples beyond ∼1000 Gy.     

Resonance trapping and evolution of particles subject to poynting-robertson drag: Adiabatic and non-adiabatic approaches

The problem of resonance trapping for particles subject to Poynting-Robertson drag is approached initially from an adiabatic regime theory. A simplified Hamiltonian system is presented for simple eccentricity-type resonances up to order 3, and expressions related to the trapping process are deduced. The fast dissipation provoked by Poynting-Robertson leads to the employment of a numerical approach for the computation of resonance capture probabilities, for particles in the size range of practical importance. Some aspects of the dynamical evolution of a particle after capture are noticed from results of numerical integrations. Analytical methods are used in order to confirm the numerical results.     

On Motions Near the Lagrange Equilibrium Point L 4 in the Case of Routh's Critical Mass Ratio

We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L ₄ when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L ₄. We also show that in the case of instability of L ₄ the trajectories starting in a vicinity of L ₄ remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Partial Reduction in the N-Body Planetary Problem using the Angular Momentum Integral

We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincaré's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP.     

The integrable cases of the general spatial three-body problem at third-order resonance

For the general three-body problem at third-order resonance an analytical solution is obtained by the use of the Weierstrass functions.