ON THE ORIGINAL DISTRIBUTION OF THE ASTEROIDS .4. NUMERICAL EXPERIMENTS IN THE OUTER ASTEROID BELT

We assumed, in 1973, that the asteroids were the building blocks of the planets and initially were distributed approximately uniformly. We suggested that the present distribution was carved out by the perturbations of Jupiter. To test this hypothesis further, we have now integrated the orbits of 140 asteroids with 14 semimajor axes from 0.63 to 0.76 (in units where aJupiter = 1.0) in increments of 0.01, and at 10 values of the eccentricity from 0.01 to 0.19 in increments of 0.02. The integrations extended for 1 million Jovian years (about 12 million years) unless the asteroid crossed Jupiter's orbit before then. Jupiter was in an elliptic orbit with its eccentricity varying in a 54,000-year period, and its line of apsides rotating with a 300,000-year period induced by the secular perturbations of Saturn. The integrations were confined to a plane. In the main survey, we chose the most "unstable" phases of the asteroid relative to Jupiter; i.e., the apsidal lines of the asteroid and Jupiter were separated by 180°. There are only 8 of the 8032 asteroids with reliable orbits with semimajor axes between 0.68 and 0.74. In our simulation, 64 90 (71%) of the asteroids with semimajor axes equal to or larger than 0.680 were removed. Of the 26 asteroids that remained, 16 had initial eccentricities equal to or less than 0.03. Ten of the remaining asteroids with a≥0.68 were integrated for 10 million Jovian years; 5 became Jupiter crossers and 5 remained. Including these longer integrations, 69 90 (77%) of the asteroids with a≥0.68 escaped. However, the depletion was much reduced in those cases when the initial phases caused conjunctions to occur at the asteroid's perihelion; i.e., the apsidal lines of the asteroid and Jupiter coincided. We found a strong correlation between the Lyapunov exponent and the crossing time for orbits with crossing times less than 1 million Jovian years (and for the few longer integrations). If this result extrapolates to 4.5 billion years, it would indicate that with Jupiter the only perturber, there are stable orbits where there are no asteroids today. © 1992.