Efficient Force Function Evaluation Methods

June 9, 1996

46

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**

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