Hope College Department of Physics and Engineering
Research Experiences for Undergraduates
Summer 2001
Project Summary

Project Title: Optimizing the Runge-Kutta Routine for the Integration for the Trajectory of Neutron Stars
Student Name: Katie Sweet
Studentís home institution: Northern Arizona University
Research Advisor(s): Dr Peter Gonthier
Source of Support (NSF-REU, or other): NSF-REU

Everything is in motion. The galaxies are moving as space expands, the galaxies rotate around their core, the stars move through the galaxy, planets orbit a star, and the planets rotate around their own axis. It takes 2.4e8 years for the Sun to travel once around the galaxy. This means that in its lifetime it has done so 54 times. It takes the Earth one year to complete its orbit around the Sun, and in its lifetime has done so 4.5 billion times. Everything is in motion. When a star goes supernova the original orbit is changed. The energy released gives its remnants, a neutron star or black hole, a different trajectory.

In the Runge-Kutta routine we constrained the calculated trajectory so the total energy remains within 1e-8 of the initial energy. For any calculated neutron star that does not conserve its total energy while being evolved is kicked out of the code. The program randomly selects initial conditions for each neutron star such as its age, position and other various characteristics. Therefore, we hoped that by studying the neutron stars that did not make it through the program we could better understand why these events in the code did not reserve their total energy. Also we hoped to use this knowledge to improve the efficiency of the program so to have more successful stars.

By comparing the graphs of the data from both types of neutron stars some noticeable differences appeared. Two examples are the graphs used to compare the characteristics are the time step, dt, taken along its trajectory verses time, and radial acceleration and axial acceleration out of the plane of the galaxy, az, verses time. The successful events tended to have a smoother dt verses time graphs and a more symmetrical wavelike az verses time graphs. On the other hand, the unsuccessful stars had a dt verses time graph that stepped down over time and the az verses time graphs were very rough and lacked symmetry. The steps in the dt verses time graphs were due to the programís attempt to constrain the changes in total energy by at first doubling dt and at the end, as it stepped down, reducing dt by half. Yet, when comparing the trajectory of both types of neutron stars the graphs were both similar. By viewing the graph, for example, in the trajectory in the plane of the galaxy of a successful star and unsuccessful star there was no defining characteristics for both groups. So by just viewing graphs of trajectory there is no way to predict an unsuccessful star. These graphs by themselves do not provide suffice answers as how to predict and to prevent failed stars.

While we were able to decrease the number of stars that did not maintain their total energy within 1e-8 in the program, we did not come to a complete understanding in being able to predict the particular trajectories that need special considerations in order to be able to constrain their total energy. We have been able to decrease the number of these stars by changing some of the initial parameters. Yet we do know that it was due to a large time step, dt, and acceleration that some events no longer conserved energy and therefore were no longer considered by the program. So the changes that we made were to decrease the constraint on the conservation of energy and the controls on change in the time step, dt. One complication of altering the parameters was an increase in execution time in running the program. Optimization of the Runge-Kutta routine will be increased when we are able to predict possible problem trajectories and be able to provide special considerations for those cases.

Slide show of Katie Sweet's work