SwRI IRD 2004--Capability Development for Modeling the Early Galactic Environment of the Solar System, 15-9494

Capability Development for Modeling the Early Galactic Environment of the Solar System, 15-9494

Background - Understanding the formation and evolution of planetary systems like our solar system remains one of the most 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 the project are: (i) 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 (ii) to demonstrate the effectiveness of the algorithm by using it to test a 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 form 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-I Levison. These techniques allow computational effort to be focused efficiently where it is needed. This project calls for the marriage of these two concepts in a single computational algorithm. To do this, we are employing the framework of the standard software development cycle: requirements definition, design and prototyping, implementation, and testing and demonstration.

Accomplishments - To date we have completed the requirements definition phase and have begun the design and prototyping phase. We have developed a prototype code that implements the Heggie algorithm in a straightforward way. To test the basic functionality of the prototype code, we have run a 100-million-year simulation of the outer solar system (the sun plus Jupiter, Saturn, Uranus, Neptune, and Pluto) and compared the results to those generated by a code based on the Wisdom-Holman mapping, the state-of-the-art algorithm for this type of simulation. Figure 1 shows the total energy as a function of time in the simulation using the prototype Heggie code. Conservation of energy is one measure of the quality of an integration. Ideally, the total energy should be exactly conserved. The prototype Heggie code generally conserved the total energy to 1 part in 10,000. Also of note in Figure 1 is the fact that there is no secular drift in the total energy, an important property of symplectic codes. Figure 2 shows the time evolution of the semimajor axis of Pluto's orbit in the two simulations described above. There is very good agreement between the two integrations in terms of the frequency and amplitude of the variations. In general, there is similarly good agreement between the two simulations as regards Pluto's other orbital elements as well as the orbital elements of the other planets.

Conservation of energy in a simulation using the Heggie algorithm. Energy units are given in a system in which the unit of time is the day, the unit of length is the astronomical unit (AU), and the Newtonian gravitational constant G = 1.

Evolution of the semimajor axis of Pluto's orbit using the Heggie algorithm versus the Wisdom-Holman mapping.