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It has been widely accepted that the giant planets would serve as a gravitational shield that effectively prevented planetesimal material originating from the Kuiper Belt and beyond from penetrating the region inside Jupiter’s orbit. Wetherill (1994) employed Öpik’s method in a series of approximate numerical simulations and concluded that the present configuration of the Jovian planets would prevent 99% to 99.9% of planetesimal material from crossing the orbits of the terrestrial planets. Two developments, in the interim, have cast doubt upon this conjecture. First, Öpik’s method has been shown by numerous authors, including ourselves, to be highly inaccurate in situations where planetesimal materials near their perihelion would be situated at a radial distance from the sun commensurate with that of one of the terrestrial planets. Second, more accurate integration methods than the one used by Wetherill revealed that typically 15% to 20% of such material would be injected into the inner solar system. The inference made from these relatively recent publications was that planetesimal materials found in the solar system would therefore present a substantial collision risk to each of the terrestrial planets, despite any confirmatory observational/geologic evidence.
We performed three highly accurate simulations of the planets Earth through to Neptune and a large number of test particles. The initial positions, initial velocities and masses of the planets were the same in all three simulations. The first two simulations had the test particles starting in the Jupiter-Saturn and Saturn-Neptune zones, respectively. The third simulation had the test particles initially on cometary orbits. We refer to these simulations as the JS, SU and Wetherill simulations respectively. We had 500,000 test particles for the JS zone, and 100,000 test particles for each of the SU and Wetherill cometary zones. Five of the 500,000 test particles hit Earth and none of the two groups of 100,000 test particles. We reaffirmed that approximately 15% of the planetesimals were scattered by the giant planets into orbits that crossed those of Earth and Mars. Among other explorations of this problem, only Horner and Jones incorporated Mars and Earth into their simulation, albeit for 107 years in contrast with our 108 years and with one-fifth the number of test particles that we used. Moreover, they performed their calculations with a much less-accurate second-order mixed variable symplectic integrator in contrast with our twelfth order Runge-Kutta-Nyström method. In order to obtain terrestrial impact events in their simulation, and overcome their smaller integration times and planetesimal numbers, they employed an unrealistically large 106 km collision radius for Earth. Their results, suitably scaled to present a realistic Earth radius, are similar to ours. However, we felt it imperative to understand why so few planetesimals entering the inner solar system would collide with Earth.
Generally speaking, the combination of the sun and Jupiter would dominate the evolution of the planetesimal’s orbit, preserving in this three-body problem the Tisserand criterion, an accurate approximation to the Jacobi constant of celestial mechanics. Occasionally, Saturn would act as an external perturber to the planetesimal’s orbit causing the planetesimal to lose energy in the interaction, thereby reducing its semimajor axis a as it becomes more tightly bound. A secondary outcome of the Tisserand criterion is that the planetesimal’s eccentricity e would be substantially increased. As a consequence, while perihelion a ( 1 - e ) of the planetesimal would reside inside the orbit of Jupiter, its aphelion a ( 1 + e ) would reside outside Jupiter's orbit. Moreover, owing to Kepler's second law, the planetesimal would spend most of its orbital period in the outer solar system subject to the gravitational influences of all of the Jovian planets. Therefore, it became clear that the planetesimal material which was introduced into the inner solar system would, nevertheless, be very short-lived owing to their continued exposure to gravitational scattering and ejection from the solar system by the Jovian planets.[BBG(1]
While these conclusions pertain most closely to the present-day configuration of giant planets, they most likely describe earlier epochs of solar system evolution since Jupiter and Saturn, especially, formed as a consequence of hydrodynamic instabilities present in the primitive solar nebula. We offer the conjecture that the recent refutation of the Late Heavy Bombardment scenario can be explained using the mechanisms described above which have demonstrated that the giant planets will effectively shield the inner solar system as Wetherill predicted, albeit by a more complex sequence of events. Using our results, as well as those of Horner and Jones, we estimate that the likelihood of an outer solar system planetesimal striking the Earth is one in 105.