Extension and Enhancement of the Allen–Eggers Analytical Ballistic Entry Trajectory Solution

The closed-form analytical solution to the equations of motion for ballistic entry developed by Allen and Eggers is presented using modern nomenclature and is extended and enhanced through several new developments. A method of choosing an appropriate constant flight-path angle is identified. Analytical limits are proposed that bound the domain of applicability of the approximation. Closed-form expressions for range and time are developed that are consistent with the assumptions in the Allen–Eggers approximation. Collectively, the improvements address key weaknesses in the original approximate solution. Assessment of the accuracy of the approximate solution relative to the planar equations of motion shows that the extended and enhanced Allen–Eggers solution provides good accuracy across a wide range of ballistic coefficients at Earth with initial flight-path angles steeper than about . In some instances, the expression developed for range-to-go may be accurate enough for use in onboard real-time guidance and targeting systems.