  Because the distant retrograde orbits dynamics inherently depends on special
functions, approximate analytical solutions in the literature are commonly
constrained to providing rough approximations of the qualitative behavior. We
rely on perturbation methods and succeed in computing a higher order analytical
solution that extends the range of applicability of previous solutions to the
problem of relative, quasi-satellite orbits with large librations. Besides, the
analytical solution provides two design parameters that are effectively used in
the computation of periodic, distant retrograde orbits without constraining to
the typical 1:1 resonance.
