  Accurate knowledge of interaction potentials among the alkali atoms and
alkaline earth ions is very useful in the studies of cold atom physics. Here we
carry out theoretical studies of the long-range interactions among the Li, Na,
K, and Rb alkali atoms with the Ca$^+$, Ba$^+$, Sr$^+$, and Ra$^+$ alkaline
earth ions systematically which are largely motivated by their importance in a
number of applications. These interactions are expressed as a power series in
the inverse of the internuclear separation $R$. Both the dispersion and
induction components of these interactions are determined accurately from the
algebraic coefficients corresponding to each power combination in the series.
Ultimately, these coefficients are expressed in terms of the electric multipole
polarizabilities of the above mentioned systems which are calculated using the
matrix elements obtained from a relativistic coupled-cluster method and core
contributions to these quantities from the random phase approximation. We also
compare our estimated polarizabilities with the other available theoretical and
experimental results to verify accuracies in our calculations. In addition, we
also evaluate the lifetimes of the first two low-lying states of the ions using
the above matrix elements. Graphical representation of the interaction
potentials versus $R$ are given among all the considered atoms and ions.
