## Objective

The effects of resonance sweeping in the asteroid belt in the early ages of the solar system has been explored. One possible explanation of the Kirkwood gaps is a slow motion of the mean motion resonances through the asteroid belt during the early stages of the solar system, due to the effect of drag on primordial dust or to the gravitational effect of the primitive solar nebula.

The basic model for studying such a resonance sweeping is the extension of the adiabatic invariant theory. A refinement of this basic model has been developed which takes into account the effects of secondary resonances and of the thickness of the chaotic layer.

Another puzzling fact about the asteroid belt is the large value of the eccentricities and inclinations of the asteroids compared with those of the planets. A possible method for pumping up these quantities has been proposed which is based upon the possible shift of secular resonances in the asteroid belt. Such a shift could have been caused, at the beginning of the solar system, by the dissipation of a protosolar nebula.

This technique evaluates the effect of a perturbation on an integrable system which can be already complex by itself and highly nonlinear. This is in contrast to the usual perturbation methods which can be applied only on very simple (most of the time linear) unperturbed approximations. The action angle variables of the integrable (but complex) unperturbed problem are introduced numerically and the various steps of the perturbation method (averaging, transformation, etc) are performed numerically. Also the seminumerical purturbation method is based upon the numerical evaluation of the perturbation function and not on its analytical expansion (in power series of the eccentricities and/or inclination). This enables it to deal with high eccentricity and/or high inclination problems.

This method has been applied to studies of the localization of the chaotic layers in the Kirkwood gaps, of the localization and dynamics of the secular resonances (both topics being of importance concerning the transport of material from the main asteroid belt to Earth crossing orbits) and of the computation of proper elements for high eccentricity and high inclination orbits (to be used for the determination of families of orbits).

The identification of asteroid families using wavelet transform has been carried out. A cross comparison has been made between the calculated results and the families obtained from the same data by another method. A good agreement between these 2 different method. A good agreement between these 2 different methods has been established, not only for the well known populous families, but also for the little known ones, the existence of which seemed previously to depend on the method used. It is the first time that such an agreement has been reached in this domain.

An analysis of the dynamics of secular resonances has been carried out and the localization such resonances reported. It has been shown that secular resonances, with characteristic time scales of hundreds of thousands of years, are responsible for slow chaotic diffusion which appears to be responsible for the transport of crossing asteroids and meteorites.

A new theory of proper elements based on Lie series technique plus an ad hoc iterative algorithm has been developed in the form of a software package. These proper elements, derived for many thousands of main belt asteroids (numbered and nonnumbered), were used to identify asteroid families and to assess their reliability through suitable statistical techniques. Extensive numerical tests were carried out to assess the accuracy (ie stability in time) of the proper elements.

The influence of secular resonances on the distribution of asteroids and the quality of proper elements was investigated and their position was accurately mapped in the 3-dimensional proper elements space corresponding to the whole solar system form 2 to 50 astronomical units for the Sun.

Some interesting cases of real main belt asteroids having chaotic orbits, but displaying a long term macroscopically stable behaviour, were studied by monitoring their long term behaviour numerically.

The frequency and outcomes of the close approaches between asteroids and planets have been investigated. The frequency of occurrence of close approaches (from shallow ones down to physical collisions) were reassessed by using the data from the Spaceguard database (400 planet crossing asteroids over a 400000 year time span). Criteria for the reliability of statistical methods for such computations were derived. The stability and the protective role of the resonance lockings in the avoidance of collisions with the planets was investigated. Particular asteroids having anomalously high Earth impact probabilities were identified.

Several aspects of asteroid collision evolution and its influence on their physical properties have been studied. The collision probabilities and average impact velocities throughout the main asteroid belt were calculated using a program based on Wetherill's theory and a bias free sample of about 700 orbits. A complex numerical code to model the effects of collisions (including both cratering and shattering events) on asteroid sizes and spins was developed, tested and used to simulate the evolution of initial asteroid populations over the history of the solar system.

The free parameters involved in the models were derived from comparison with laboratory impact experiments and the plausible ranges were explored systematically. This exploration included the initial size and spin rate distribution of asteroidal planetesimal before the onset of the current regime dominated by disruptive high velocity impacts.

The same physical models were also applied to derive constraints on the collision lifetime of the asteroid 951 Gaspra (met in 1991 by the Galileo probe), to explain the properties of the Saturnian moon Hyperion (observed by the Voyager probes) and to the simulation of the artificial debris cloud orbiting the Earth.

Asteroid physical properties were analyzed using different techniques. The asteroid size distribution was derived by IRAS data and fitted to truncated Pareto functions. Rotation rates and shapes of asteroids measured from lightcurve photometry were analyzed both statistically and for individual objects in order to obtain observational constraints on the origin and collision history of asteroids.

A study on the plausible ages for the main asteroid families was carried out. A new statistical method (hierarchical clustering) to search for families and assess their reliability and robustness was applied on the new proper elements set. About 20 reliable families were identified and analyzed.

The origin of meteorites in asteroid collision and their delive ry from the main belt through mean motion and secular resonances was studied. The basic idea was that of simulating the random ejection of fragments from all the known asteroids and estimating the efficiency of insertion into chaotic (resonant) regions in the phase space, subsequently leading to planet crossing orbits.

This research concentrated on the development of an integrated model for the study of the collision evolution of the asteroid population, the identification and statistical and physical analysis of the asteroid families and on simulations of asteroidal catastrophic disruption by hypervelocity impact experiments.

The basic idea was to integrate experimental results and theoretical modelling to develop an algorithm which is able to describe the simultaneous evolution of asteroid sizes and spin rates over solar history and to take into account the different and numerous constraints, etc. A complex numerical simulation program has been produced.

It has been found that asteroids larger than 300 to 400 km in diameter have had little changes to their spin rates due to collisions and hence they should have preserved their primordial rotation features.

A new set of very accurate proper elements has been computed by means of a second order, fourth degree secular perturbation theory. A multivariate data analysis technique was then applied to identify asteroid clusterings and, at the same time, to assess their reliability by one or more objective methods. In addition to the 3 largest and most significant families, 12 more reliable and robust families were found throughout the belt.

A series of hypervelocity impact experiments have been carried out in the open. These were devoted to the study of the behaviour and properties of fragments produced impacting macroscopic targets by hypervelocity projectiles. Evidence was found of collimated jets which is the ejection of a statistically significant number of fragments all closely aligned about some radial directions. If similar phenomena occur after asteroidal catastrophic collisions they could give rise to rubble pile asteroids, to gravitationally bound small fragments (similar to the binary asteroid 1989 PB) and to asteroid dynamical families with complex nonisotropic morphology.

A new method has been developed to obtain a mapping model for a Hamiltonian system near a resonance for nearly integrable Hamiltonians. The perturbation of the generating function of the unperturbed mapping is introduced by the averaged Hamiltonian of the original system expressed in action angle variables. It has been proved that the fixed points of the mapping thus obtained coincide with the fixed points of the averaged Hamiltonian (periodic orbits of the original system) and have the same stability characteristics. It is essential to use an averaged Hamiltonian which possesses all the fixed points of the original system in order for the mapping to be realistic. A good check to find the range of validity of an averaged Hamiltonian is to compare its fixed points with the corresponding periodic orbits of the actual dynamical system.

A basic dynamical system studied is the restricted 3 body problem with the Sun and Jupiter as primaries. All the families of periodic orbits at this resonance have been computed and compared with the fixed points of the averaged Hamiltonian, valid neat this resonance. The coincidence is good for small values of the eccentricity of the asteroid but the high eccentricities are not present in this model. A simple correction term to the former simple averaged Hamiltonian was found that introduces to the model the missing high eccentricity resonances. This latter corrected averaged Hamiltonian has been used to obtain the mapping model. It is found that the use of this corrected Hamiltonian gives chaotic jumps in the eccentricity of the asteroid. The method of spectral analysis is also used to detect chaos. It gives a sharp distinction between ordered and chaotic motion. The chaotic behaviour of the eccentricity resulting in jumps to high values can be detected by this method by considering small time intervals well before the jump takes place.

Small inetrplentary object (asteroids, comets, meteroids) are a most numeous and diverse population of bodies, and are of outstanding interest for a number of reasons: the understanding of their long-term dynamical evolution is a challenge for celestial mechanics; their physical properties hold important clues to the early formation phase of the solar system; their evolution is due to processes like catastrophic collisions and intense outgassing whose scale is unique in the solar system; they have recently been or will soon be the target of dedicated exploration missions providing a wealth of new data. The joint research effort outlined in this proposal envisages the key to a signifiant progress in this field in the interplay of several different methods and techniques, like celestial mechanics and theory of dynamical systems, numerical integration of orbits, models for the physical structure and evolution of the projects, astronomical observations and laboratory experiments.

### Fields of science

- natural sciencesphysical sciencesastronomyplanetary sciencesasteroids
- natural sciencesmathematicsapplied mathematicsdynamical systems
- natural sciencesphysical sciencesastronomyplanetary sciencesplanets
- natural sciencesphysical sciencesastronomyplanetary sciencescomets
- natural sciencesphysical sciencesastronomyplanetary sciencescelestial mechanics

### Topic(s)

Data not available### Call for proposal

Data not available### Funding Scheme

CSC - Cost-sharing contracts### Coordinator

Namur / Namen

Belgium

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### Participants (4)

54006 Thessaloniki

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06304 Nice

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10025 Pino Torinese

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56127 Pisa

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