Regular perturbation expansions are used to analyse unsteady, inviscid, slender, incompressible (constant density), axisymmetric, annular liquid jets when the gases enclosed by and surrounding the jet are dynamically passive. Both inertia- and capillarity-dominated annular jets are considered. It is shown that, for inertia-dominated jets, closure of the leading order equations is achieved at second order in the perturbation parameter which is the slenderness ratio, whereas closure is achieved at first order for capillarity-dominated jets. The leading order equations are used to determine the fluid dynamics of steady annular jets.