  Doppler cooling with lasers is essential to ions' trapping and also a
preliminary step towards achievement of ultracold ions. Due to lack of
effective tools, experimentally monitoring the ions' temperature and the
laser-ion coupling is difficult in Doppler cooling. Here we analytically
explore the Doppler cooling process of trapped ions, exemplified by
$^{40}$Ca$^{+}$, by solving the friction coefficient in the Doppler cooling
with respect to a thermal bath, particularly, to a bath with large heating
rate. We show four regions for cooling and heating induced by the three-level
electromagnetically induced transparency and propose a practical method for
measuring the Rabi frequency by the Doppler cooling window. In addition, the
final temperature of the laser-cooled ions can be obtained analytically in the
case of a weak thermal bath, whereas for the strong thermal bath this requires
numerical treatment due to involvement of the Doppler shift and large
Lamb-Dicke parameter. Our analytic results would help for understanding many
experimental observations, such as configuration phase transition, phonon laser
and thermodynamics regarding hot trapped ions.
