In the study of Dynamics a very important role is played by the integrals of the motion, that is by quantities that are constant during the time evolution of a dynamical system. When the dynamics is non integrable, as it is the case for the N-body problem, such quantities do not exist. Then it is very useful to compute quasi integrals of the motion: they are quantities changing very slowly, and can be considered approximately constant over time intervals not too long. In the context of asteroid dynamics, the quasi integrals are called proper elements.
Proper elements have been computed analytically for the main belt
asteroids [Milani and Knezevic 1992,Milani and Knezevic 1994] and have been successfully used
to determine asteroid collisional families and to understand the
dynamical structure of the asteroid belt, in particular the relevance
of secular resonances. This has been possible because these proper
elements have been proven to be stable over time scales of the order
of 
years. Over an even longer time scale, in many cases slow
diffusion processes change them by significant amounts [Milani and Farinella 1994],
nevertheless the stability of the proper elements is enough to
identify a large number of reliable collisional families [Zappalà et al. 1994].
Another method used to compute proper elements [Williams 1969] is the semianalytic one, that is based upon the averaging, by quadrature, of the equations of motion over the fast angle variables, the ones with periods of a few years. Semianalytic proper elements have been used effectively for asteroids belonging to the main belt but with inclination and eccentricity too high for accurate analytical theories [Lemaitre and Morbidelli 1994].
A small fraction of the known asteroids, and a significant fraction of the comets, are on orbits which can cross the orbits of the terrestrial planets and/or the giant planets. The Earth-crossing population is especially important for the role it plays in the formation of craters on the Earth and on the Moon. It would be desirable to be capable to compute proper elements also for this class of orbits, for the following reasons. First, to detect the possibility and compute the probability of collision of such objects with the Earth over intermediate time scales (thousands to tens of thousands years). Second, to identify the objects whose long term orbital evolution is controlled by one or more of the main secular resonances, to draw a map of the dynamical structure of the Earth crossing region in phase space. Third, to identify meteor streams (that can be observed only when they are crossing the orbit of the Earth) and to give a criterion to be employed in the identification of their parent bodies.
The very fact that the orbits in this class can cross the orbit of some planets results in the presence of the mathematical singularity of collision, either along the orbit or nearby in the phase space. This singularity results in both strong divergence of the perturbative series used in the analytical theories and in the divergence of the quadratures used in the semianalytical averaging. Thus both techniques used to compute proper elements for main belt asteroids are inapplicable to Earth-crossing orbits.
On the other hand we have to realize that the goals for which we would
like to have proper elements are very different in this case, and the
required stability times are much shorter: for most applications, time
spans between 
and 
years are long enough. Over
longer time spans the dynamics is dominated by large changes in
orbital elements, including semimajor axis, resulting from close
approaches and the effects of secular resonances. This kind of
dynamics is strongly chaotic to the point of being impossible to
describe other than in a statistical sense. Thus even a rougher
approximation could be used, provided it is shown to be mathematically
consistent and numerically computable, if it can provide information
stable over time spans comparable to one cycle of the slow, secular
arguments, such as the argument of perihelion 
.
In the paper [Gronchi and Milani 1998] we have developed the mathematical theory of such an approximation, applicable even to orbits which undergo crossings with the orbit of a major planet. In this paper we improve upon this theory, and we derive from it a computationally realistic and numerically stable algorithm to compute the right hand side of the averaged equations of motion (Section 2). We have based a numerical integration method for the averaged equations (Section 3) on this mathematical foundation. In this way we have computed proper elements for all the Near-Earth Asteroids (NEA), and made them available in a catalog (as announced in Section 4). These proper elements allow to discuss qualitative properties of the near-Earth orbits, such as the occurrence and frequency of node crossings with our planet (Section 5), and the occurrence of secular resonances (Section 6). We conclude (in Section 7) by stressing that this is just the first step in the procedure to manufacture reliable and accurate proper element catalogs for NEA (and in the future also for comets and meteors). This needs to be followed by a long series of tests, in which stability problems will be identified, and successive improvements of the computational algorithm will be implemented, as it happened for the analytical proper elements of main belt asteroids [Milani and Knezevic 1988,Milani and Knezevic 1998].