 Efficient Force Function Evaluation Methods
June 9, 1996
46
4.3 Closely Packed Stars Can Be Moved at the Same Time
The opposite of the situation discussed in Section 4.2 is also an important special
case. In FIGURE 25., we have a binary star system and several distant stars. For a star in
the binary star system, the force function will be dominated by the other star and the effect
of the other stars on the binary star's movement will be negligible. These two stars will
just move in a Kepler (elliptical) orbit, but they will require a very small step size because
they are orbiting so quickly (compare to other movements). Instead of calculate the force
function for these two stars individually, we can replace these two stars with their center of
mass and move the center of mass along. When these two stars get close enough to a third
star to start making a difference, the point along the Kepler orbit can be calculated and the
two stars can be broken apart again.
It isn't just binary stars that we can make a special case of. Even if two stars are
moving too quickly to orbit one another and are just making a close pass, we can calculate
the path along the conic section over a relatively large time step. No method of evaluating
the force function can be faster than these cases where the force function isn't even evalu-
ated.
Even 3 or 4 tightly packed stars can be considered as a special case. Again, the
group of stars can be replaced by their center of mass for the purposes of moving the
group, and the individual paths inside the group can be calculated by the simple Particle-
Particle method. While this is an
O(n
2
)
operation, because
n
is so small, this is actually
more efficient than other methods and keeps the errors from making simplifying assump-
tions from being too large.
Sources: (10:6,8,18:6)
Binary Star
FIGURE 25. Binary star system and several distant stars