1. Introduction

The main motivation to produce ``yet another set of asteroid proper elements'' is that for many asteroids the currently available proper elements, derived by means of an analytical theory by Milani and Knezevic ([1990,1992,1994]), are not (any more) accurate enough, at least when these elements are used to identify and study asteroid dynamical families, and to try to understand the dynamical structure of the asteroid belt. In the densely populated low to moderate eccentricity and inclination region of the asteroid main belt, the proper elements computed analytically are accurate to a level very close to the best result achievable by any analytical theory (Milani and Knezevic [1994]). This because there is an infinite web of resonances (Arnold [1963]), and because of the occurrence of chaotic motions. It is not yet clear whether the asteroid belt is a big chaotic sea (although with diffusion times on the order of billions of years), or it is structured in the sense described by Nekoroshev theorem for quasi-integrable Hamiltonian systems (Morbidelli and Guzzo [1997]). However, it is now established that the motions of asteroids are basically chaotic, and that only the dynamical mechanisms responsible for this chaos, the time scales involved, and the size and form of the resulting macroscopic instabilities are different (Milani et al. [1997]). Moreover, asteroids are not allowed to be less chaotic than the perturbing major planets (Laskar [1989], Nobili et al. [1989]), and this ``induced chaos'' introduces a chaotic noise in the asteroid proper elements of the order of 10^-4 (Milani and Knezevic [1994]). The typical instability of the current analytical proper elements in proper e and sin I is $\le 0.0015$ over 5 Myr (Milani and Knezevic [1994]). This accuracy was enough for Zappalà et al. ([1995]) to reliably identify asteroid families in a sample of 12,487 asteroids and throughout almost the entire belt. However, they also found that there are still some regions of the belt where a reliable identification of families was not possible. This could result from two main problems: either the proper elements in these regions were of degraded accuracy (e.g. near the resonances; see Milani and Knezevic [1999] for an attempt to improve upon the accuracy of mean/proper elements of asteroids in the vicinity of the mean motion resonances), or the background density of asteroids was too high, thus smearing the family borders and forcing the clumping of families which should have been kept separate (this applies especially in the Flora region). In addition, it has been recently recognized that chaotic diffusion, acting over very long time spans, can drive members of the families out of the regions in the phase space occupied by the families. Such a diffusive transport of the bodies seems to be common if a family is cut through by some mean motion resonances, giving rise to chaotic behavior localized to a narrow range of semiamjor axis (Milani and Farinella [1994], Milani et al. [1997], Morbidelli and Nesvorný [1999]). Another efficient mechanism of ``family erosion'' is provided by the Yarkovsky effect, which also appears to transport some family members out of the families by pushing them into the nearby resonances (Vokrouhlichký et al. [1999]). In an attempt to solve the problem, and to produce proper elements more accurate, more stable in time and more reliable, we have tried a different method to compute asteroid proper elements. We adopted an approach similar to the one used for the major outer planets by Carpino et al. ([1987]): by purely numerical techniques, we produced so-called ``synthetic'' proper elements for a large sample of asteroids. The procedure consists of simultaneous integration of the asteroid orbits for a ``long-enough'' time span, online filtering of the short-periodic perturbations and computation of Lyapunov Characteristic Exponents to monitor the chaotic behaviors. The output of the integration is then spectrally resolved under constraints set by d'Alembert rules, and the principal harmonics (proper values) extracted from the time series, together with the associated fundamental frequencies. It is quite obvious that with such an approach we have to handle in different ways the inner and the outer part of the main belt. In the outer part of the belt we are allowed to use a somewhat simplified dynamical model: the direct perturbation are included only for the four outer major planets, while a barycentric correction applied to the initial conditions removes most indirect effects of the inner planets (Milani and Knezevic [1992]). In this way we could speed up the numerical integrations by choosing a longer step size. In the inner part of the main belt, on the other hand, we have to include the effects of the terrestrial planets (at least Mars and Earth), and this makes the model more complex and significantly slows down the integration. In this paper we present the results of the first phase of this work, that is the results of computation of the synthetic proper elements for a total of 10,265 asteroids located in the outer main belt, between 2.5 AU and 4.0 AU. We propagated all the orbits for 2 Myr, but in a number of troublesome cases we extended the integrations to 10 Myr. We have included in this catalog only numbered and multi-opposition objects, to be sure that their osculating orbital elements are accurate enough; there would be no point in producing proper elements with a computational accuracy better than the one of the available osculating elements. We have also excluded the high eccentricity Mars approachers, that is those with initial perihelion q<1.75 AU, because for these the perturbations by Mars should be taken into account. The rest of the paper is organized as follows. In Section 2 we explain what is a synthetic theory, describe the procedures involved, the input data, and other important details and technicalities; in Section 3. we present and discuss the results, with special emphasis on the hyperbolic, resonant and chaotic cases, and we asses the accuracy and stability of the synthetic proper elements we have computed. We also give some examples of the new features, of the dynamical structure of the entire belt and of some families, which are made visible by more accurate and reliable proper elements. Finally, in Section 4 we summarize the results, draw some conclusions, and discuss future work.