2. Synthetic theory

The idea to develop a theory of motion by means of a set of purely numerical procedures is not new, but so far it has been used in the Solar System dynamics only for subsystems with a few bodies. The numerical approach was used as a complement to the analytical one, to integrate an analytically defined secular system (Laskar [1988]), or even to develop fully numerical theories of motion, but for the systems with fewer bodies. The first long-term direct numerical integrations of the equations of motion of the Solar System were made by Cohen and Hubbard ([1965]), who integrated five outer planets for 120,000 yr. This integration was later extended to 1 Myr by Cohen et al. ([1973]), and to 5 Myr by Kinoshita and Nakai ([1984]). As the computing power increased over the last couple of decades, the time span covered by integrations has also expanded, and Applegate et al. ([1986]), using a dedicated ``Digital Orrery'' computer, integrated orbits of the outer planets for more than 200 Myr, and later for even more. In the framework of the LONGSTOP project (Milani et al. [1987], Carpino et al. [1987], Nobili et al. [1989]) synthetic (as opposed to analytic) theories of motion of the outer planets were derived by means of a set of purely numerical procedures, which consisted in the integration of orbits of the planets for 9.3 Myr, later 100 Myr, with online digital filtering to eliminate short-periodic perturbations, and Fourier analysis of the filtered output to identify the spectral lines. One advantage of such a procedure is that it allows a direct quantitative comparison with the results of analytic theories. This represents the core of the method applied in the present paper, although some adaptations to the asteroid case had to be done, We computed proper elements for numbered and multi-opposition asteroids, with initial conditions taken from the astorb.dat osculating orbital elements data base for the epoch JD 2451100.5 provided by E. Bowell¹. The reason to select only asteroids observed at multiple oppositions was that only for these objects the accuracy of orbital elements is sufficient to match the accuracy of the proper elements we wanted to compute.