  A shadow of a numerical solution to a chaotic system is an_exact_ solution to
the equations of motion that remains close to the numerical solution for a long
time. In a collisionless n-body system, we know that particle motion is
governed by the global potential rather than by inter-particle interactions. As
a result, the trajectory of each individual particle in the system is
independently shadowable. It is thus meaningful to measure the number of
particles that have shadowable trajectories as a function of time. We find that
the number of shadowable particles decays exponentially with time as exp(-mu
t), and that for eps in [~0.2,1] (in units of the local mean inter-particle
separation $\bar n$), there is an explicit relationship between the decay
constant mu, the timestep h of the leapfrog integrator, the softening eps, and
the number of particles N in the simulation. Thus, given N and eps, it is
possible to pre-compute the timestep h necessary to acheive a desired fraction
of shadowable particles after a given length of simulation time. We demonstrate
that a large fraction of particles remain shadowable over ~100 crossing times
even if particles travel up to about 1/3 of the softening length per timestep.
However, a sharp decrease in the number of shadowable particles occurs if the
timestep increases to allow particles to travel further than 1/3 the softening
length in one timestep, or if the softening is decreased below ~0.2$\bar n$.
