Improved matched asymptotic solutions for three-dimensional atmospheric skip trajectories

An improved technique for matching the asymptotic solutions of non-linear differential equations is presented and successfully applied to the three-dimensional atmospheric skip trajectories. The classical method of matched asymptotic expansions (MAE) is generally applied to two-point-boundary value problems. When we apply the MAE method to initial value problems, due to error propagation, the resulting accuracy usually depends on the physical problems. In the proposed technique, the second-order solutions are obtained by first generating a set of equations for the small perturbations which are the discrepancies between the uniformly valid first-order solutions and the exact solutions. Then, the equations of the small perturbations are integrated separately near the outer and inner boundaries to obtain the perturbed outer and inner expansion solutions, respectively, for a secondorder matching. In addition, in this improved technique the end-point boundaries are artificially extended or constructed to strengthen the physical assumptions on the outer and inner expansions for the matching while in the evaluation of the constants of integration in the uniformly valid first-order solutions, the prescribed end-points are effectively enforced. In this paper, to show the applicability of the improved technique, we first apply it to the rectilinear restricted three-body problem. We men consider the threedimensional skip trajectory. Compared to the solutions obtained by numerical integration over a wide range of entry conditions, the second-order solutions obtained by this improved technique are very accurate. The trajectory elements at the lowest altitude and at exit as well as their accuracy are evaluated. Introduction With the advent of manned space explorations and the establishment of permanent space stations, the safe * Professor, Department of Aerospace Engineering. Member AIAA. t Associate Professor, Dept. of Mechanical Engineering. Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. recovery of an orbiting aerospace vehicle, or its orbital maneuver with minimum fuel consumption has been one of the most challenging technologies in space flight dynamics. During the atmospheric passage, there is a tremendous change in speed, kinetic energy, dynamic pressure and heating rate. It is then of interest to have explicit analytical solutions for the variations of the elements of the three-dimensional entry trajectory, since the heading change due to three-dimensional motion has promising applications in aeroassisted orbital transfer. A powerful method for analyzing dynamic systems governed by equations with the dominant forces varying widely between the two end-points is the method of matched asymptotic expansions (MAE). This technique, initiated by aerodynamicists', has been successfully applied to problems in astrodynamics-. By using this method, some analytical solutions for atmospheric re-entry problems have been obtained, but they are restricted to the first-order solutions''. In this paper, we propose an improved technique to go beyond the first-order solutions reported previously. In this improved technique, the perturbation equations are generated by considering the small discrepancies between the exact solutions and the uniformly valid first-order solutions. Then, the equations for the small perturbations arc integrated separately near the outer and inner boundaries to obtain the perturbed outer and inner expansion solutions, respectively, for a second-order matching. To illustrate the applicability of the improved method of matched asymptotic expansions (iMAE), the proposed technique was first applied to the rectilinear restricted three-body problem. Then, by using it to analyze the three-dimensional atmospheric re-entry problem, we obtain, besides the usual solutions for the altitude, speed and flight path angle variables, the secondorder solutions for the heading, latitude and longitude in explicit form with excellent accuracy. The explicit second-order solutions are compared with the pure numerical solutions and the errors incurred are assessed to show the region of validity for the application of the technique.