  In this paper, we investigate how the fine structure constant, $\alpha$,
locally varies in the presence of a static and spherically symmetric
gravitational source. The procedure consists in calculating the solution and
the energy eigenvalues of a massive scalar field around that source,
considering the weak-field regimen. From this result, we obtain expressions for
an spatially variable fine structure constant by considering suitable
modifications in the involved parameters admitting some scenarios of
semi-classical and quantum gravities. Constraints on free parameters of the
approached theories are calculated from astrophysical observations of the
emission spectra of a white dwarf. Such constraints are finally compared with
those ones obtained in the literature.
