Universal Gravity Turn Trajectories

Abstract

One of the simplest trajectory programs for the powered flight of a missile through the atmosphere is the ``gravity turn,'' which results from simply keeping the propulsive thrust always parallel to the vector velocity. However, even for a ``point mass'' missile, in a uniform gravitational field with constant thrust and no aerodynamic forces, the differential equations for the motion are nonlinear and require numerical integration. To avoid the necessity of doing this computation anew for each missile preliminary design, a method has been found for integrating the equations for the singular case of zero initial velocity. When expressed in terms of appropriate dimensionless variables, the resulting solutions are ``universal'' in the sense that they constitute a good approximation to any gravity turn with a small, nearly vertical, initial velocity. The solutions depend upon two parameters, the initial thrust to weight ratio η and a parameter k which corresponds to the initial ``kick angle'' of nonsingular gravity turns.