Institute for Advanced Study Informal Astrophysics Seminar

There is now ample evidence of large quantities of planetesimal material present in orbits extending outwards from Jupiter and beyond. Several research groups have shown that as much as 15% of such material currently in orbits residing between Jupiter and Saturn would undergo major change and ultimately cross the orbit of the Earth. Highly accurate integration schemes assure that such orbits are not formally chaotic, but the geologic history of Earth over the last billion years reveals at most five major impact events. We shall show that the likelihood of a collision, in contrast with an Earth orbit crossing, is approximately 10−510−5, an outcome of significance to earlier evolution of our solar system as well as other extra-solar planetary systems. Further, we show why the Earth-orbit crossers have such a low probability of colliding with Earth. While their orbits are governed by the classical three-body problem results of Jacobi and, especially, Tisserand, we show that Saturn will have a significant perturbative effect. In particular, the Earth-crossers have lost substantial energy in their orbits due to their interaction with Saturn. This results in their semi-major axes a being substantially reduced while their orbital eccentricities e become larger as expected (Murray and Dermott). The outcome, then, is that the perihelia of the orbits have decreased but their aphelia continue to reside in the outer solar system where, owing to Kepler’s second law, they spend almost all of their time until consumed by the gas giants or being expelled from our solar system. This result provides a clear description of the mechanism that prevented the speculated “Late Heavy Bombardment” from having taken place.Much earlier in solar system history, planetesimals in the outer solar system had a much more substantive role and possibly possessed some tens of Earth masses of primordial material. Moreover, they were subject not only to the behavior associated with their self-gravitating dynamics but of resonances with the formative gas giants as well as planetesimal-planetesimal collisions. A fundamental issue, then, is whether simulations of such environments is feasible, i.e. lacks sensitivity to initial conditions with e-folding or Lyapunov time scales that are very short. Owing to the complications supplementing the dynamics of the “planetesimal swarm” by gas giant-induced resonances and collision events, we have explored the underlying stability of a swarm of planetesimals moving solely under the influence of their mutual gravitational interactions in their orbits around the sun. In particular, we have explored the dependence of the Lyapunov time scale for growth of uncertainties as a function of the number and mass of individual planetesimals and have observed, for systems similar in size to those associated with the Nice-family of models, that the time scale is only 50 years. During the span of only 1,000 years, the perturbation of a single planetesimal by 1 km results in all planetesimals being displaced approximately 1 AU for their expected positions. Moreover, by adapting analytical results obtained by Goodman, Heggie, and Hut as well as Binney and Tremaine, we have verified our Lyapunov computational results analytically. The outcome of this is that essentially no simulation involving mutually interacting planetesimal swarms remains reliable after a few centuries rendering it computationally impossible simulate the early evolution of our solar system, however realistic the picture that we incorporate into the model. This condition is very reminiscent of Lorenz’s (1963) discovery that the numerical simulation of convective flows, such as that present in our atmosphere, can be pursued extending over periods longer than a few days.