One of the fundamental limitations of current *N*-body simulations of
stellar systems is that they usually contain several orders of
magnitude fewer particles (``stars'') than real stellar systems
contain. Thus, each simulated particle feels a much more ``lumpy'' or
*granular* gravitational field than a star would feel in a real
system. For collisional systems this may mean that timescales and
statistics are distorted by some approximately computable amount
[38], but for collisionless systems this spurious
collisionality is simply a source of error
[39]. The effect of close encounters can be
moderated via *force softening*, but the granular effect of the
gravitational potential is strong even for particles that are separated
by large distances ([75, section 1.3a,], [68]).
A common force computation method for
collisionless systems is the *particle-mesh* method
(*eg., * [61]), in
which softening is implicit and equal to the distance between mesh
points. For particle-particle or tree methods for force computation,
softening is added explicitly by modifying the gravitational force as

where is the softening parameter. It
is usually chosen to approximate the average inter-particle distance.
However, Sellwood [68] notes that if a disk is being
simulated in two dimensions, then the softening parameter
can instead be used to approximate a disk of
thickness . He also argues that a simple smoothing of
the gravitational potential is probably an oversimplification, since
real spiral galaxies contain giant molecular clouds and star clusters
that make the potential significantly less smooth than the theorist's
ideal. He includes a diagram clearly showing that spurious
spiral structure can develop in a *cold disk* as a result of
discreteness noise. This structure is clearly visible in a disk of
2,000 particles after only 1 revolution, and is visible even in a disk
with 20,000 particles after a few revolutions. Discreteness noise can
also introduce spurious heating of a collisionless *cold disk*, so
that artificial disk-cooling mechanisms must be introduced
[68, 57].

Dyer and Ip [22] note that equation (2) is physically inconsistent: particles act like Plummer distributions when they are contributing to the field, but are points when the field acts upon them, leading to a violation of Newton's Third Law and thus a violation of conservation of momentum. They fix this with a model in which softened particles are treated as uniform density spheres. Non-overlapping spheres interact as points, while a polynomial is used to compute the force on spheres that overlap. (The spheres are not solid in any physical sense. They can pass through each other like ``ghosts''.)

When approximate force computation algorithms are used, they are
generally the next most dominant source of error. Barnes and Hut
[7] analyzed their tree code [6] for
errors and found that it works well for collisionless systems,
but their results were inconclusive about collisional systems.
(Jernigan and Porter [45] introduce a tree method that
is completely different from the Barnes and Hut algorithm, and works
very well for collisional systems.) Barnes and Hut [7]
found that errors in global conservation of energy were dominated by
force errors, and concluded that a modest error tolerance in force of
about one part in per particle was sufficient to conserve global
energy to one part in , and that ``further improvements in
tree methods for a fixed *N* is much less desirable than simply
increasing *N*.'' Hernquist, Hut, and Makino
[39] came to a similar conclusion after doing
many experiments with various accuracies of force computation and
values of *N* from to . They measured individual particle
energies (which are conserved in collisionless systems), and found
that, for a given *N*, the energy deviations were almost identical
regardless of the force error, and that energy errors decrease
uniformly as *N* is increased. In other words, the effects of
discreteness noise dominate those of force errors, so that simply
increasing *N* is the most desirable improvement that can be made to
current *N*-body simulations.
They state that they believe this result is ``*independent of
N*'' (their emphasis). This author believes that they are probably
correct for values of

A bizarre alternative is offered by Bryant [11, 12], who introduces a model in which velocities are always bounded and stellar encounters are interpreted as mutual absorption and emission of the simulated bodies.

Fri Dec 27 17:41:39 EST 1996

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