  We investigate the temporal dynamics of Doppler cooling of an initially hot
single trapped atom in the weak binding regime using a semiclassical approach.
We develop an analytical model for the simplest case of a single vibrational
mode for a harmonic trap, and show how this model allows us to estimate the
initial energy of the trapped particle by observing the fluorescence rate
during the cooling process. The experimental implementation of this temperature
measurement provides a way to measure atom heating rates by observing the
temperature rise in the absence of cooling. This method is technically
relatively simple compared to conventional sideband detection methods, and the
two methods are in reasonable agreement. We also discuss the effects of RF
micromotion, relevant for a trapped atomic ion, and the effect of coupling
between the vibrational modes on the cooling dynamics.
