A complete list of combinations of the rates of asteroid perihelia and nodes and the corresponding fundamental frequencies of planets, giving rise to secular resonances and involving up to 4 frequencies, is known from the previous work, while for the resonances with 6 frequencies a systematically derived comprehensive list is given here for the first time. There are 28 divisors in the theory of degree up to 4, not all of which can give rise to resonances, while at degree 6 there are (at least) 33 such possibly resonant frequency combinations.

Mapping the secular resonances by plotting the resonant lines in the phase space of proper elements or of secular frequencies, possibly also against the background of known asteroids, enables to straightforwardly identify resonances causing large long periodic variations of asteroid orbital elements, resonances that interact with known families, those that bound the dynamically distinct regions, deplete or disturb asteroids in these regions, etc.

Normal form methods allow one to compute quasi-invariants of a Hamiltonian system, which are referred to as proper elements. The computation of the proper elements turns out to be useful to associate dynamical properties that lead to identify families of space debris, as it was done in the past for families of asteroids. In particular, through proper elements we are able to group fragments generated by the same break-up event and we possibly associate them to a parent body. A qualitative analysis of the results is given by the computation of the Pearson correlation coefficient and the probability of the Kolmogorov-Smirnov statistical test.

We search for possible differences in rotational frequencies, diameters, albedos, and orbital parameters between Trojans belonging to the L4 and L5 swarms using our own observations and literature data. With increasing number of observational data it becomes evident that the L4 and L5 populations have very similar distributions of most parameters with an exception of orbital inclination distribution.

The history of asteroid families, from their discovery back in 1918, until the present time, is briefly reviewed. Two threads have been followed: on the development of the theories of asteroid motion and the computation of proper elements, and on the methods of classification themselves. Three distinct periods can be distinguished: the first one until mid-1930s, devoted to discovery and first attempts towards understanding of the properties of families; the second one, until early 1980s, characterized by a growing understanding of their importance as key evidence of the collisional evolution; the third one, characterized by an explosion of work and results, comprises the contemporary era. An assessment is given of the state-of-the-art and possible directions for the future effort, focusing on the dynamical studies, and on improvements of classification methods to cope with ever increasing data set.

The Large Synoptic Survey Telescope (LSST) is currently by far the most ambitious proposed ground-based optical survey. With initial funding from the National Science Foundation (NSF), Department of Energy (DOE) laboratories, and private sponsors, the design and development efforts are well underway at many institutions, including top universities and national laboratories. Solar System mapping is one of the four key scientific design drivers, with emphasis on efficient Near-Earth Object (NEO) and Potentially Hazardous Asteroid (PHA) detection, orbit determination, and characterization. The LSST system will be sited at Cerro Pachon in northern Chile. In a continuous observing campaign of pairs of 15 s exposures of its 3,200 megapixel camera, LSST will cover the entire available sky every three nights in two photometric bands to a depth of V=25 per visit (two exposures), with exquisitely accurate astrometry and photometry. Over the proposed survey lifetime of 10 years, each sky location would be visited about 1000 times, with the total exposure time of 8 hours distributed over several broad photometric bandpasses. The baseline design satisfies strong constraints on the cadence of observations mandated by PHAs such as closely spaced pairs of observations to link different detections and short exposures to avoid trailing losses. Due to frequent repeat visits LSST will effectively provide its own follow-up to derive orbits for detected moving objects.

Detailed modeling of LSST operations, incorporating real historical weather and seeing data from Cerro Pachon, shows that LSST using its baseline design cadence could find 90% of the PHAs with diameters larger than 250 m, and 75% of those greater than 140 m within ten years. However, by optimizing sky coverage, the ongoing simulations suggest that the LSST system, with its first light in 2013, can reach the Congressional mandate of cataloging 90% of PHAs larger than 140m by 2020. In addition to detecting, tracking, and determining orbits for these PHAs, LSST will also provide valuable data on their physical and chemical characteristics (accurate color and variability measurements), constraining PHA properties relevant for risk mitigation strategies. In order to fulfill the Congressional mandate, a survey with an etendue of at least several hundred m2deg2, and a sophisticated and robust data processing system is required. It is fortunate that the same hardware, software and cadence requirements are driven by science unrelated to NEOs: LSST reaches the threshold where different science drivers and different agencies (NSF, DOE and NASA) can work together to efficiently achieve seemingly disjoint, but deeply connected, goals.

In this work, we review the analytical and semi-analytical tools introduced to deal with resonant proper elements and their applications to the Trojan asteroids, the numerical computation of synthetic proper elements for resonant and non resonant asteroids, and the introduction of proper elements for planet crossing asteroids. We discuss the applications and accuracy of these methods and present some comparisons between them.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The paper presents a review of the recent contributions and open questions concerning the families of asteroids. Due to the availability of very large catalogues (synthetic and analytical proper elements of the asteroids and large observational surveys of their spectra) and to the introduction of non gravitational forces in their determination, the concept of static family has disappeared, to be replaced by this of dynamical families. The proper elements are not constant anymore but are ageing on very long timescales. The size distributions of the populations of asteroids, in and out the families, their ages, the ejection velocities of the fragments after an impact, have been reconsidered by several teams of research, with this new approach. Parallel numerical simulations of collisions and fragmentations of bodies have showed that most of the asteroids are likely rubble piles or agglomerates than monolithic blocks. The methods of classification have been refined and combine, in their newest versions, the dynamics and the observations, working now on 5 dimensional space instead of 3. A series of sub families of the large well-known families have been recently identified, using catalogues with more than 100 000 asteroids (the cluster Karin for example).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The orbits of 13 Trojan asteroids have been calculated numerically in the model of the outer solar system for a time interval of 100 million years. For these asteroids Milani et al. (1997) determined Lyapunov times less than 100 000 years and introduced the notion “asteroids in stable chaotic motion”. We studied the dynamical behavior of these Trojan asteroids (except the asteroid Thersites which escaped after 26 million years) within 11 time intervals - i.e. subintervals of the whole time - by means of: (1) a numerical frequency analysis (2) the root mean square (r.m.s.) of the orbital elements and (3) the proper elements. For each time interval we compared the root mean squares of the orbital elements (a, e and i) with the corresponding proper element. It turned out that the variations of the proper elements ep in the different time intervals are correlated with the corresponding r.m.s.(e); this is not the case for sin Ip with r.m.s.(i).

We propose here the first results of the semi analytical method of Henrard(1990) applied to the calculation of proper elements for the asteroids.