Improved matched asymptotic solutions for three-dimensional atmospheric skip trajectories.

Abstract

An improved technique for matching the asymptotic solutions of non-linear differential equations is presented and successfully applied to the atmospheric re-entry problems including ballistic entry, and three-dimensional skip trajectories. The classical method of matched asymptotic expansions is generally applied to two-point-boundary value problems. When it is applied to initial value problems, 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 small discrepancies between the uniformly valid first-order solutions and the exact solutions. Then, based on the first-order expansion solutions in conjunction with the matched asymptotic expansion technique, the perturbation equations of the small discrepancies are integrated separately near the outer and inner boundaries to obtain the perturbed outer and inner expansion solutions, respectively, for a second-order matching. Furthermore, 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. Compared with 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 explicit solutions for the planar skip re-entry, ballistic entry and three-dimensional skip trajectories are then used to analyze the effects of the initial speed and the initial entry angle at the peak deceleration and maximum heating rate during atmospheric entry. The critical altitudes, speeds and flight path angles where these maxima occur are accurately predicted by the second-order solutions. For skip trajectories the estimate exit elements and their accuracy are evaluated. The technique presented and its solutions for re-entry problems are undoubtedly useful in the planning of orbital operations involving aeroassist technology.