  We describe a simple approach to the problem of incorporating the response
time of an atom or ion being Doppler-cooled into the theory of the cooling
process. The system being cooled does not in general respond instantly to the
changing laser frequencies it experiences in its rest frame, and this "dynamic
effect" can affect significantly the temperatures attainable. It is
particularly important for trapped ions when there is a slow decay out of the
cooling cycle requiring the use of a repumping beam. We treat the cases of
trapped ions with two and three internal states, then apply the theory to
$^{40}{\rm Ca}^+$. For this ion experimental data exist showing the ion to be
cold under conditions for which heating is predicted if the dynamic effect is
neglected. The present theory accounts for the observed behaviour.
