Theory of the N-Body Problem
June 9, 1996
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There are several different types of deviations from the ideal star movement that
show up from the discretization and the round off errors. Most of these errors come from
having one or more of the constants of motion changing over time.
When a star gains or loses energy, instead of the star making an elliptical orbit
around a collapsar, the orbits decay and either spiral in or spiral out. So, instead of a
perfect ellipse, the star movement looks like FIGURE 15.
This type of error tends to make a very uninteresting star system because a lot of
stars will be lost either due to collisions or by stars moving off the screen. Euler's method
tends to gain energy over time, but the fourth order Adam-Bashford method tends to very
slowly lose energy.
Some methods, such as the taylor3 method, both gains and loses energy, depending
on the situation. When a star orbits a collapsar, it keeps it's elliptical orbit, but the orbit is
more elongated than it should be. It is hard to even notice this unless you monitor the total
energy level as the system progresses.
It is hard to predict what this type of error will do to a system. With only two stars
it is hard to tell that this type of error even exists, but with many stars it is clear that the
taylor3 method does not give as accurate of results as other methods.
Some methods maintain the elliptical orbit around a collapsar, but the perihelion
(the spot on the ellipse closest to the collapsar) rotates around the collapsar over time.
Energy is neither gained nor lost over time, but the angular momentum changes.
FIGURE 15. Results of a method that gains/loses energy.
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