Capability Development for Modeling the Early Galactic Environment
of the Solar System, 15-R9494

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Principal Investigators
David E. Kaufmann
Harold F. Levison

Inclusive Dates:  07/01/04 – 01/01/06

Background - Understanding the formation and evolution of planetary systems like our solar system remains one of the more fundamental problems facing astronomy today. It is also a high scientific priority of NASA. There is growing recognition that the likely early galactic environment of the solar system, a stellar cluster, may have played a significant role in its formation and subsequent dynamical evolution. As yet, it has not been possible to model accurately the late stages of the formation of planets and planetary satellite systems in the environment of a stellar cluster. This inability is because current state-of-the-art computer algorithms either cannot accurately integrate these systems over timescales required for planets to form or cannot accurately integrate these systems through the close stellar passages characteristic of a cluster environment.

Approach - The objectives of this project were (1) to develop a computational algorithm that can efficiently perform long-term integrations of these hierarchical systems while accurately following close encounters between any of the constituent bodies and (2) to demonstrate the effectiveness of the algorithm by using it to test an hypothesis of current scientific interest – can clouds of comets around planetary systems be formed from early close encounters between such systems in their birth clusters? Two important concepts formed the basis of the new algorithm. The first, due to dynamicist Douglas Heggie, is a special formulation of the equations of motion of the gravitational N-body problem. Heggie's formulation removes certain mathematical difficulties in following close encounters between bodies, but at the expense of efficiency. The second is a set of techniques used in the Symplectic Massive Body Algorithm (SyMBA), a sophisticated planetary dynamics code developed under the leadership of Co-Investigator Levison with support from the ACR. These techniques allow computational effort to be focused efficiently where it is needed. This project called for the marriage of these two concepts in a single computational algorithm. To do this we employed the framework of the standard software development cycle: requirements definition, design and prototyping, implementation, and testing and demonstration.

Accomplishments - Unfortunately, we were not successful in our efforts to develop the desired computational algorithm and code. Despite encouraging early results when we applied the new code to rather simple test problems, we quickly encountered numerical instabilities when we applied the code to problems of interest to us (i.e., long-term solar system integrations). We found that the instabilities were quite general in the Heggie algorithm and prevented us from obtaining reliable results. We attempted to stabilize the algorithm by implementing constraints between the integration variables. This approach has been used successfully in the field of molecular dynamics to impose similar constraints (e.g., fixing the bond lengths between constituent atoms of a molecule while simultaneously integrating the equations of motion for each atom separately). In our case, we were unable to include the stabilizing constraints in a feasible way. Some numerical problems remained, and even when we were able to apply the constraints with sufficient accuracy, an inordinately large amount of computational time was required. Given these significant problems with the Heggie algorithm, we feel that future efforts at solving the problem of planetary formation and evolution in the environment of a stellar cluster should be directed elsewhere. A more limited, but perhaps more practical, near-term approach would be to address the dynamics of the stellar cluster and the dynamics of the planetary systems separately using existing codes.

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