  Within density functional theory, we have obtained the structure of $^4$He
droplets doped with neutral calcium atoms. These results have been used, in
conjunction with newly determined {\it ab-initio} $^1\Sigma$ and $^1\Pi$ Ca-He
pair potentials, to address the $4s4p$ $^1$P$_1 \leftarrow 4s^2$ $^1$S$_0$
transition of the attached Ca atom, finding a fairly good agreement with
absorption experimental data. We have studied the drop structure as a function
of the position of the Ca atom with respect of the center of mass of the helium
moiety. The interplay between the density oscillations arising from the helium
intrinsic structure and the density oscillations produced by the impurity in
its neighborhood plays a role in the determination of the equilibrium state,
and hence in the solvation properties of alkaline earth atoms. In a case of
study, the thermal motion of the impurity within the drop surface region has
been analyzed in a semi-quantitative way. We have found that, although the
atomic shift shows a sizeable dependence on the impurity location, the thermal
effect is statistically small, contributing by about a 10% to the line
broadening. The structure of vortices attached to the calcium atom has been
also addressed, and its effect on the calcium absorption spectrum discussed. At
variance with previous theoretical predictions, we conclude that spectroscopic
experiments on Ca atoms attached to $^4$He drops will be likely unable to
detect the presence of quantized vortices in helium nanodrops.
