Astrophysicists who use computers to study the evolution of galaxies have a difficult job. A galaxy, like the weather, is a complex system and notoriously tough to model, but the 7 February print issue of PRL suggests a new way to check the accuracy of many galaxy simulations. The author describes techniques for finding accurate and exact “shadow” solutions that closely follow simulation results. Although shadows have been used in the past, they were limited to only the simplest simulations. The new methods could improve the confidence of researchers looking at complex behaviors such as galaxy collisions.
When astrophysicists want to answer a question–how spiral arms develop, for example, or how a galaxy would evolve with or without a black hole in the middle–they begin with a set of initial positions and velocities of a group of stars. Then, using equations for the gravitational forces stars exert on one another, the researchers can numerically compute that galaxy’s future. Because a galaxy is a chaotic system–a miniscule change in its initial conditions can become a giant change in its shape two billion years later–the computation isn’t easy.
Even the best computers must use approximations to solve problems, so some degree of error is inevitable. “In a chaotic system these errors are magnified exponentially,” says Wayne Hayes of the University of Maryland in College Park. “The real solution of the system could be very different than the numerically computed one.”
A “shadow” is an exact solution to a set of equations that closely follows the path of a numerically computed solution for a long time–perhaps 50 or 100 rotations of a galaxy. A simulation that doesn’t have a shadow isn’t necessarily wrong, but a shadow’s presence is a good indication that a simulation is accurate. Astrophysicists can search for shadows by applying Newton’s method, a mathematical way to refine approximate solutions into exact ones. But doing so requires a lot of computer power. Until Hayes’s new work, it was only possible to search for shadows of systems in which one star was moving in a field of fixed stars–not a very realistic model of a galaxy.
Now Hayes, who performed the research while at the University of Toronto, has found long-lasting shadows of systems with multiple moving stars. In previous work, he had devised algorithms to run Newton’s method faster. In his current paper, he demonstrates that he can approximate the shadow of a system with multiple moving particles by superimposing the shadows of an equivalent number of one-moving-particle systems.
Even using Hayes’s new method, searching for shadows would require too much time and computer power for most astrophysicists to bother. However, Hayes plans to use his work to devise a practical rule of thumb for so-called time-steps. Real stars’ positions and velocities change continuously, but simulations move in discrete steps, re-calculating stellar positions, say, every two million years. The longer the time-step, the greater the possible error. In an upcoming paper, Hayes plans to demonstrate that the maximum time-step that astrophysicists should use is the amount of time it takes a star to move one-third of the average distance between stars. Any longer than that, he says, and the shadows will disappear.
“Hayes used some innovative and elegant computational tricks,” says Timothy Sauer of George Mason University in Fairfax, Virginia. Sauer emphasizes the importance of the paper to astronomical models. “This paper is the first to show that shadowing can be extended to a size that people care about.”