Theory of the N-Body Problem
June 9, 1996
The result of this error produces interesting results in XStar. Stars that orbit close
to a collapsar form a solid disk of colors. Should that make this error more acceptable than
The overstep phenomenon is directly caused by the discrete movement of the stars
instead of the continuous movement that exists in the real world. Take the case of a star
that is falling toward a collapsar. As the star gets closer to the collapsar it picks up speed
and with each step the force of gravity gets much stronger. When the star gets to time step
(See FIGURE 17.) it will be so close to the collapsar that the next step will take it well
past the collapsar onto the other side when, in reality, it should have collided with the col-
lapsar. At time t
, the star is now far enough away from the collapsar and moving at such a
high speed that it will just keep on moving. The net result is that the star has gained a great
deal of energy out of nowhere.
It is important to note that this same error can (and usually does) occur when a star
is moving around a collapsar or star on a sharply curved path. Also, the star may not gain
so much energy that it shoots off the screen. Instead, it can end up mixing this added
energy into the rest of the stars, causing all stars to move toward the edges of the screen.
Mathematically this is the result of trying to integrate over the singularity when
the distance to the collapsar is zero, or a pole in the complex plane when the star is on a
curved path. The result is that the interval of convergence for the Taylor series is violated.
There are several ways of fixing this problem. The computer could try and check to
see if a star moved through any other star after each step, but this would be very hard to do
and it would be very expensive to do. The computer could simply say that the stars collide
FIGURE 16. Results of a method that shifts the perihelion
FIGURE 17. The overstep phenomenon
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