SPECTRAL-ANALYSIS OF ASTEROIDAL TRAJECTORIES IN THE 2 1 RESONANCE/

The evolution of the eccentricity of particular asteroidal trajectorie s in the 2:1 resonance is analysed by a spectral scheme. This analysis is an alternative to the well known analysis based on the computation of Liapunov exponents. The method can be used to estimate the long ti me evolution of a trajectory from a segment considerably shorter than the one needed for a Liapunov exponent calculation. The particular tra jectories analysed by our method may be divided into two classes: (a) those appearing to be unstable, in the sense that their eccentricities become overcritical within the time interval of our numerical integra tions, which, in turn, implies close encounters with a planet (Mars, t he Earth or Jupiter) and (b) those appearing, in the above sense, to b e stable. Trajectories are computed in the frame of four models: (i) t he elliptic restricted three body problem (ERTBP) model with a fixed v alue of 0.048 for Jupiter's eccentricity, e(J), (ii) the ERTBP model w ith a fixed e(J) greater than or equal to 0.055, (iii) the ERTBP model with an artificially oscillating e(J) with amplitudes corresponding t o the full problem, including all outer planets, but with a frequency not corresponding to the real problem, and (iv) the six-body problem S un - outer planets - asteroid. The main results are: (a) There are tra jectories which are stable in model (i) and which are chaotic as well as unstable in models (ii)-(iv). This suggests that chaos sets on when Jupiter's eccentricity exceeds a certain threshold value. This is not universal all over the 2:1 resonance region. (b) Trajectories which a re stable in all four models are chaotic in models (ii)-(iv) according to our spectral analysis.