Theory of the N-Body Problem
June 9, 1996
at a fairly large distance, say around time step 3 in the above diagram, and this is what
XStar currently does. Another option would be to use a different ODE integration method,
such as the Runge-Kutta method, that handles singularities better thus letting the collision
distance be smaller. Lastly, a "softening" factor can be applied to the system by adding a
small constant to the distances that are calculated. For large distances this has little, if any,
effect, but for cases where the particles might collide, this has the effect of turning the par-
ticles into clouds which can pass through each other.
The sling shot effect isn't an error, it is a real part of physics and it is has actually
been used by the interplanetary space probes. However, the slingshot effect looks so much
like the overstep phenomenon that it is very hard to tell the difference without actually
watching the total energy levels of the star system.
This effect can happen only when there are at least three stars involved. When two
stars interact, the kinetic energy that is gained over the period of time when the stars are
moving closer together will always be exactly matched by the kinetic energy that is lost
over the equivalent time period when the stars are moving away from each other. With
three stars, it is possible for stars to convert some of the binding energy between two stars
into the kinetic energy of a third star. This often happens when stars pass close to each
other, with one star gaining a lot of energy. Since this is the same general situation that the
overstep phenomenon occurs in, it is easy to confuse the two.
The case of using the sling shot effect with an interplanetary probe can make a
good example. In FIGURE 18., a space probe uses Jupiter to gain kinetic energy after
being launched from Earth. At time t
, both the space probe and Jupiter are much more
attracted (bound) to the Sun than they are to each other. The space probe, therefore,
follows a roughly elliptical orbit away from the sun. At time t
, as the space probe has
slowed enough that it is about to start falling back toward the Sun. However, Jupiter is
close by and so the probe starts to "fall" toward Jupiter picking up a great deal of speed.
Jupiter, on the other hand, is slowed down by a very tiny amount, and thus it will orbit the
Sun in a slightly smaller circle. When the space probe passes around the front of Jupiter, it
changes direction and starts to move away from both the Sun and Jupiter. Jupiter is still
more influenced by the Sun than by the probe so it gets pulled away from the probe. Thus,
the space probe loses less kinetic energy while it is moving away from Jupiter than it
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