  The analysis of gravitational wave data may require greater accuracy than is
afforded by the adiabatic approximation to the trajectory of and field produced
by a particle moving in curved spacetime. Higher accuracy is available with a
two-timescale approach using as an expansion parameter the ratio of orbital
time to radiation reaction time. To avoid apparent divergences at large
distances, the details of the method are important, especially the choice of
the foliation, the spacetime surfaces on which the orbital elements are taken
to be constant. Here we apply the two-timescale approach to a simple linear
model to demonstrate the details of the method. In particular we use it to show
that a null foliation avoids large-distance divergences in the first-order
post-adiabatic approximation, and we argue that this will be true more
generally for a null foliation.
