On the relationship between Lyapunov times and macroscopic instability times

On the basis of the general theory of Hamiltonian systems, we consider the relationship between Lyapunov times and macroscopic diffusion times. We find out that there are two regimes: the Nekhoroshev regime and the resonant overlapping regime. In the first case the diffusion time is exponentially long with respect to Lyapunov times. In the second case, the relationship is polynomial although we do not find any theoretical reason for the existence of a universal power law. We show numerical evidences which confirm our theoretical considerations.     

The Effect of the Great Inequality on the Hecuba Gap

The effects of the "great inequality" (the quasi-resonance between Jupiter and Saturn) on the motion in the 2/1 mean motion resonance with Jupiter (the Hecuba gap) is investigated. We confirm the proposition made by Ferraz-Mello and collaborators that the great inequality generates secondary resonances which are likely to produce the slow diffusion observed in numerical investigations. We identify, in the restricted three body problem, the frequencies responsible for these secondary resonances. This revised version was published online in July 2006 with corrections to the Cover Date.     

New Determinations of the Ages of Tide in the North Atlantic Ocean

Ages of tide provide relevant information about the spatial distribution of existing anomalies in the normal modes of the oceans, because a delay may be associated with bottom friction energy dissipation, closely located resonances, bathymetric gradients, or radiational effects. The determination of other parameters, such as the age of diurnal tide or age of parallax, also provide further knowledge about the ocean's hydrodynamical response to acting forces. Following the development of new ocean models and the availability of a greater amount of data, these parameters can be redetermined. We present the spatial distribution of these parameters in the Northeast Atlantic Ocean and the Mediterranean Sea, obtained from 507 stations. The results are discussed in terms of bathymetric models, coastal features, sea surface temperature, wind and other environmental factors.     

Long term dynamical evolution and classification of classical TNOs

Classical trans-Neptunian objects (TNOs) are believed to represent the most dynamically pristine population in the trans-Neptunian belt (TNB) offering unprecedented clues about the formation of our Solar System. The long term dynamical evolution of classical TNOs was investigated using extensive simulations. We followed the evolution of more than 17000 particles with a wide range of initial conditions taking into account the perturbations from the four giant planets for 4 Gyr. The evolution of objects in the classical region is dependent on both their inclination and semimajor axes, with the inner (a<45 AU) and outer regions (a>45 AU) evolving differently. The reason is the influence of overlapping secular resonances with Uranus and Neptune (40–42 AU) and the 5:3 (a∼ ∼42.3 AU), 7:4 (a∼ ∼43.7 AU), 9:5 (a∼ ∼44.5 AU) and 11:6 (a∼ ∼ 45.0 AU) mean motion resonances strongly sculpting the inner region, while in the outer region only the 2:1 mean motion resonance (a∼ ∼47.7 AU) causes important perturbations. In particular, we found: (a) A substantial erosion of low-i bodies (i<10°) in the inner region caused by the secular resonances, except those objects that remained protected inside mean motion resonances which survived for billion of years; (b) An optimal stable region located at 45 AU<a<47 AU, q>40 AU and i>5° free of major perturbations; (c) Better defined boundaries for the classical region: 42–47.5 AU (q>38 AU) for cold classical TNOs and 40–47.5 AU (q>35 AU) for hot ones, with i=4.5° as the best threshold to distinguish between both populations; (d) The high inclination TNOs seen in the 40–42 AU region reflect their initial conditions. Therefore they should be classified as hot classical TNOs. Lastly, we report a good match between our results and observations, indicating that the former can provide explanations and predictions for the orbital structure in the classical region.     

Comparison of global storm activity rate calculated from Schumann resonance background components to electric field intensity E0Z

This work presents the results of a comparison between the global storm activity rate I_RS and electric field intensity E_0Z. The permanent analysis of the I_RS may become an important tool for testing Global Electric Circuit models. I_RS is determined by a new method that uses the background component of the first 7 Schumann resonances (SR). The rate calculations are based on ELF observations carried out in 2005 and 2006 in the observatory station “Hylaty” of the Jagiellonian University in the Eastern Carpathians (Kułak, A., Zięba, S., Micek, S., Nieckarz, Z., 2003. Solar variations in extremely low frequency propagation parameters: I. A two-dimensional telegraph equation (TDTE) model of ELF propagation and fundamental parameters of Schumann resonances, J. Geophys. Res., 108, 1270, doi:10.1029/2002JA009304). Diurnal runs of the I_RS rate were compared with diurnal runs of E_0Z amplitudes registered at the Earth's surface in the Geophysical Observatory of the Polish Academy of Sciences in Świder (Kubicki, M., 2005. Results of Atmospheric Electricity and Meteorological Observations, S. Kalinowski Geophysical Observatory at Świder 2004, Pub. Inst. Geophysics Polish Academy of Sciences, D-68 (383), Warszawa.). The days with the highest values of the correlation coefficient (R) between amplitudes of both observed parameters characterizing atmosphere electric activity are shown. The seasonal changes of R, I_RS and E_0Z are also presented.     

An improved Lorentzian technique for evaluating resonance characteristics of the Earth-ionosphere cavity

The effectiveness of various formulations of the Lorentzian procedure for estimating Schumann resonance (SR) characteristics of the Earth-ionosphere waveguide from transient electromagnetic signals is tested in the limits of a simplified, spherically uniform model of the resonator. It is shown that the major improvement, in comparison with the “classic” Lorentzian formulation, is achieved by consideration of the intra-modal phase interference. The effect of the “limited frequency dispersion” inherent in the “classic” Lorentzian approach – that is of substantial importance at the lowest SR modes – can be effectively neutralized by interpolating the values for the propagation parameter between the adjacent modal frequencies. Several practical aspects of applying the Lorentzian procedure to transient signals are also discussed.     

Asteroid Mean Elements: Higher Order and Iterative Theories

Mean orbital elements are obtained from osculating ones by removing the short periodic perturbations. Large catalogues of asteroid mean elements need to be computed, as a first step in the computation of proper elements, used to study asteroid families. The algorithms for this purpose available so far are only accurate to first order in the masses of the perturbing planet; the mean elements have satisfactory accuracy for most of the asteroid belt, but degraded accuracy in the neighbourhoods of the main mean motion resonances, especially the 2:1. We investigate a number of algorithms capable of improving this approximation; they belong to the two classes of Breiter-type methods and iterative methods. The former are obtained by applying some higher order numerical integration scheme, such as Runge–Kutta, to the differential equation whose solution is a transformation removing the fast angular variables from the equations; they can be used to compute a full second order theory, however, only if the full second order determining function is explicitly computed, and this is computationally too cumbersome for a complicated problem such as the N-body. The latter are fixed point iterative schemes, with the first order theory as an iteration step, used to compute the inverse map from mean to osculating elements; formally the method is first order, but because they implement a fixed frequency perturbation theory, they are more accurate than conventional single iteration methods; a similar method is already in use in our computation of proper from mean elements. Many of these methods are tested on a sample of asteroid orbits taken from the Themis family, up to the edge of the 2:1 resonance, and the dispersion of the values of the computed mean semimajor axis over 100 000 years is used as quality control. The results of these tests indicate that the iterative methods are superior, in this specific application, to the Breiter methods, in accuracy and reliability. This is understood as the result of the cancellations occurring between second order perturbation terms: the incomplete second order theory, resulting from the use of a Breiter method with the first order determining function only, can be less accurate than complete, fixed frequency theories of the first order. We have therefore computed new catalogues of asteroid mean and proper elements, incorporating an iterative algorithm in both steps (osculating to mean and mean to proper elements). This new data set, significantly more reliable even in the previously degraded regions of Themis and Cybele, is in the public domain. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical experiments in the 3/1 andv 6 overlapping resonance region

Numerical experiments of fictitious small bodies with initial eccentricities e=0.1 have been performed in the overlapping region of the 3/1 mean motion resonance and of thev ₆ secular resonance 2.48a2.52AU for different values of the initial inclination 16°i20°. An analysis for thev ₆ secular resonance shows that the topology is different from the one found outside the overlapping region: the critical argument for thev ₆ resonance in the overlapping region rotates in opposite direction as compared to the purev ₆ region. In the 3/1 resonance region the secular resonancev ₅ is dominant, and some secondary secular resonances asv ₆ –v ₁₆ andv ₅ +v ₆ are present.     

Earth impact probability of the Asteroid (25143) Itokawa to be sampled by the spacecraft Hayabusa

The Japanese spacecraft Hayabusa is planed to reach the Asteroid Itokawa in September 2005, and to bring back some samples of its surface to Earth in 2007. We have studied the future possible evolution of this asteroid by integrating numerically over 100 Myr a set of 39 initially indistinguishable orbits (clones), obtained either by small variations of the nominal initial conditions, or by using different computers (introducing different round-off errors). The results indicate that an Earth impact of this 500-m-size asteroid is likely within a million years, which is only a factor of four larger than the average impact frequency of asteroids of this size. The mission Hayabusa may thus sample a good candidate for being among the next 500-m-size Earth impactors.     

On the dynamics in the asteroids belt. Part II: Detailed study of the main resonances

The general theory exposed in the first part of this paper is applied to the following resonances with Jupiter's motion : 3/2, 2/1, 5/2, 3/1, 7/2, 4/1; these are the most relevant resonances for the asteroids. The whole analysis is performed in the framework of the spatial problem of three bodies, both in the circular and in the elliptic case. The results are also compared with the observed distribution of the asteroids.