Convexity of intercept time in short-range re-entry vehicle defence: how to optimally intercept a fast re-entry vehicle travelling in a lofted trajectory

Purpose The described anomaly allows an interceptor to intercept a threat that has superior speed. The results account for gravity and would not occur if the trajectories of the threat and the interceptor were modelled as simple straight lines. The algorithm could be tested experimentally. The article also provides a criteria for feasible intercepts in general at short ranges. Design/methodology/approach The derivation of the optimal intercept time with gravity is non-trivial and would be intractable using calculus. The optimization is done in the velocity phase space instead of the time of flight. In addition, the proof of the convexity of the intercept time is complex, requiring intricate arguments. We also make use of a dimensionless scaling technique that simplifies the derivation. Findings There are two main results that arise from the intercept time formula. First, there is a unique optimal delay in launching the interceptor to minimize total intercept time. Second, there is a feasible volume of approach that changes with time, where it is possible to intercept the RV. If the interceptor lies outside of this volume, then an intercept is not possible. Research limitations/implications The derivation is made assuming short-range missile defence scenarios and environmental conditions, such as air friction and tracking, are not modelled. Practical implications The algorithm allows a slow interceptor to intercept a fast threat. This allows, for example, a Patriot interceptor to intercept a hypersonic missile. Social implications This helps defence of assets against short-range missiles. Originality/value The derivation of the intercept time formula of an RV by a ballistic interceptor is not simple due to the non-linearity of the gravitational force. Adding to the complexity is the non-zero detonation range: an interceptor explodes when it reaches a certain non-zero range to the RV. The anomaly of the intercept time is unexpected, as the usual wisdom may be to launch the interceptor as soon as a threat is detected. This would be true if trajectories were straight lines (in the absence of gravity).