Integration error over very long time spans

The main limit to the time span of a numerical integration of the planetary orbits is no longer set by the availability of computer resources, but rather by the accumulation of the integration error. By the latter we mean the difference between the computed orbit and the dynamical behaviour of the real physical system, whatever the causes. The analysis of these causes requires an interdisciplinary effort: there are physical model and parameters errors, algorithm and discretisation errors, rounding off errors and reliability problems in the computer hardware and system software, as well as instabilities in the dynamical system. We list all the sources of integration error we are aware of and discuss their relevance in determining the present limit to the time span of a meaningful integration of the orbit of the planets. At present this limit is of the order of 10⁸ years for the outer planets. We discuss in more detail the truncation error of multistep algorithms (when applied to eccentric orbits), the coefficient error, the method of Encke and the associated coordinate change error, the procedures used to test the numerical integration software and their limitations. Many problems remain open, including the one of a realistic statistical model of the rounding off error; at present, the latter can only be described by a semiempirical model based upon the simpleN ² formula (N=number of steps, =machine accuracy), with an unknown numerical coefficient which is determined only a posteriori.     

The shape of the small satellites of Saturn: Gravitational equilibrium vs solid-state strength

In the set of small satellites of Saturn recently imaged by the Voyager probes, we can observe the transition from irregularly-shaped, strength-dominated objects to larger, gravity-dominated bodies with shapes roughly fitting the theoretical equilibrium figures. The transition occurs for a radius of 100±50 km, corresponding to a typical material strength of the order of 10⁷ dynes cm^?2. We discuss briefly the cases of Mimas, Enceladus, Hyperion, Phoebe and the small coorbital and F-ring shepherding moons, showing that an analysis of the shape data can often provide interesting results on the physical properties, origin and collisional history of these objects.     

The shape and structure of Mimas

The Voyager images have shown that Mimas and Enceladus have regular shapes, with topography of the order of 1% of the diameter. Therefore, we can compare the global shapes of these satellites with the corresponding figures of gravitational equilibrium. In the case of Mimas, this comparison rules out a homogeneous interior, but implies the existence of a denser, presumably rocky core within this small icy satellite.     

Resonant structure of the outer asteroid belt

An analysis of ordered and chaotic regions of motion in the outer asteroid belt has shown that once the eccentricity of Jupiter is introduced the chaotic regions of the circular model are quite easily depleted. This suggests that also objects in neighbouring regions must be strongly perturbed. Therefore it is not surprising that many outer belt asteroids have been reported in the literature as resonant or anyway dynamically protected. By using the planar elliptic restricted 3-body model we have investigated the motion of outer belt asteroids which had not been suspected to librate. We find 3 cases of libration and 11 cases of e, coupling that can be explained within the theory of secular resonances. It is thus established that in the outer belt only resonant and dynamically protected asteroids can have lifetimes of the same order as the age of the Solar System.     

New Definition of Discovery for Solar System Objects

We propose a new formal definition of discovery for a Solar System object. It is based on an objective and mathematically rigorous algorithm to assess when a set of observations is enough to constitute a discovery. When this definition is satisfied, in almost all cases the orbit is defined well enough to establish the nature of the object discovered (Main Belt vs. Near Earth Asteroid, Trans-Neptunian vs. long period comet). The frequency of occurrence of exceptions is estimated by a set of numerical experiments. The availability of a non-subjective definition of discovery allows some rules to be adopted for the assignment of discovery credit with a minimum risk of dispute. Such rules should be fair, encourage good practice by the observers and acknowledge the contribution of the orbit computers providing the identifications and the orbits, as well as the one of all the contributing observers.     

Light-time computations for the BepiColombo Radio Science Experiment

The Radio Science Experiment is one of the on board experiments of the Mercury ESA mission BepiColombo that will be launched in 2014. The goals of the experiment are to determine the gravity field of Mercury and its rotation state, to determine the orbit of Mercury, to constrain the possible theories of gravitation (for example by determining the post-Newtonian parameters), to provide the spacecraft position for geodesy experiments and to contribute to planetary ephemerides improvement. This is possible thanks to a new technology which allows to reach great accuracies in the observables range and range rate; it is well known that a similar level of accuracy requires studying a suitable model taking into account numerous relativistic effects. In this paper we deal with the modelling of the space-time coordinate transformations needed for the light-time computations and the numerical methods adopted to avoid rounding-off errors in such computations.     

Photometry of Comet 9P/Tempel 1 during the 2004/2005 approach and the Deep Impact module impact

The results of the 9P/Tempel 1 CARA (Cometary Archive for Amateur Astronomers) observing campaign is presented. The main goal was to perform an extended survey of the comet as a support to the Deep Impact (DI) Mission. CCD R, I and narrowband aperture photometries were used to monitor the Afρ quantity. The observed behavior showed a peak of 310 cm 83 days before perihelion, but we argue that it can be distorted by the phase effect, too. The phase effect is roughly estimated around 0.0275 mag/degree, but we had no chance for direct determination because of the very similar geometry of the observed apparitions. The log-slope of Afρ was around −0.5 between about 180-100 days before the impact but evolved near the steady-state like 0 value by the impact time. The DI module impact caused about a 60% increase in the value of Afρ and a cloud feature in the coma profile which was observed just after the event. The expansion of the ejecta cloud was consistent with a fountain model with initial projected velocity of 0.2 km/s and β=0.73. Referring to a 25,000 km radius area centered on the nucleus, the total cross section of the ejected dust was 0.06 days after the impact, and 1.93 days after the impact (A is the dust albedo). Five days after the event no signs of the impact were detected, nor deviations from the expected activity referring both to the average pre-impact behavior and to the previous apparitions.     

Symmetries and Rank Deficiency in the Orbit Determination Around Another Planet

We study the possible degeneracies for the normal matrix of the observations from the Earth of the motion of a satellite around a planet, and give the possible solutions to the loss of precision in the orbit determination caused by the rank deficiency. Finally we discuss the methods available to control the instability in the orbit determination resulting from the degeneracy.     

Numerical homogenization-based seismic assessment of an English-bond masonry prototype: Structural level application

A multiscale strategy is evaluated at a structural level for the analysis of unreinforced masonry structures. The mechanical characterization of the masonry is deduced from homogenization-based micro-scale finite element (FE) models. The derived data are here employed at a structural level via a discrete FE model. The discrete FE model is composed of quadrilateral rigid plates interconnected through vertical and horizontal interfaces. On the interfaces, between adjoining discrete elements, a model that accounts for the in- and out-of-plane behavior of masonry, with damage and plasticity, is adopted. Such interfaces represent the material pre- and post-peak regimes, its orthotropy, and, depending on the micro-model assumed, account by three-dimensional shear effects that are especially important for multi-leaf walls and complex regular textures. The discrete model has been implemented in an advanced structural analysis software where powerful built-in features as the arc-length method, line-search algorithm, and implicit or explicit solver schemes are available. The multi-scale model is applied for the dynamic study of a small English-bond masonry house prototype subjected to a series of consecutive earthquake records. Detailed comparisons between the experimental and numerical data are presented, including the results obtained through a continuous total strain rotating crack model. Quasi-static and dynamic analyses are conducted. Results demonstrate that when enough experimental information is available on the masonry components under tension, shear, and compression regimes, the approach predicts well the seismic structural response in terms of time-history displacements, seismic capacity, and damage patterns. The required computational cost (CPU time) is very attractive.