A simple inviscid model to predict the onset of breakdown in an axisymmetric vortex is proposed. Three problems are considered: the shock-induced breakdown of a compressible vortex, the breakdown of a free compressible vortex, and the breakdown of a free incompressible vortex. The same physical reasoning is used in all three problems to predict the onset of breakdown. It is hypothesized that breakdown is the result of the competing effects of adverse pressure rise and streamwise momentum flux at the vortex centerline. Breakdown is assumed to occur if the pressure rise exceeds the axial momentum flux. A formula with no adjustable constants is derived for the critical swirl number in all three problems. The dependence of the critical swirl number on parameters such as upstream Mach number, excess/deficit in centerline axial velocity, and shock oblique angle is explored. The predictions for the onset of shock-induced breakdown and free incompressible breakdown are compared to experiment and computation, and good agreement is observed. Finally, a new breakdown map is proposed. It is suggested that the adverse pressure rise at the vortex axis be plotted against the axial momentum flux to determine the onset of breakdown. The proposed map allows the simultaneous comparison of data from flows ranging from incompressible breakdown to breakdown induced by a shock wave.