  Numerical relativity codes now being developed will evolve initial data
representing colliding black holes at a relatively late stage in the collision.
The choice of initial data used for code development has been made on the basis
of mathematical definitiveness and usefulness for computational implementation.
By using the ``particle limit'' (the limit of an extreme ratio of masses of
colliding holes) we recently showed that the standard choice is not a good
representation of astrophysically generated initial data. Here we show that,
for the particle limit, there is a very simple alternative choice that appears
to give excellent results. That choice, ``convective'' initial data is, roughly
speaking, equivalent to the start of a time sequence of parameterized solutions
of the Hamiltonian constraint; for a particle in circular orbit, it is the
initial data of the steady state solution on any hypersurface. The
implementation of related schemes for equal mass holes is discussed.
