Higher-order analytic solutions for critical cases of ballistic entry

A new set of higher-order analytic solutions is introduced for the critical cases of: (a) ballistic skip trajectories with initial speed greater than the circular orbital speed (supercircular case); (b) shallow ballistic entry with entry speed less than the circular orbital speed (subcircular case); and (c) shallow ballistic entry from low circular orbits (circular case). Both non-circular cases admit the same set of analytic solutions, except that the supercircular case involves the regular error function of real argument, whereas in the subcircular case the solution depends on error functions of imaginary argument. Unlike previously obtained solutions, neither case now exhibits singularities in the flight path angle. For the circular case, a totally independent set of parametric expressions is obtained. In comparison with the numerical integration of the equations of motion, the analytic solutions exhibit a high degree of accuracy, representing a clear improvement over equivalent solutions currently available in the literature. Nomenclature B = small parameter specifying entry altitude and physical characteristics of vehicle B = B/E* CD.CL = coefficients of drag and lift, respectively CDo = zero-lift drag coefficient c = dimensionless flight path angle at entry D =drag E" = maximum lift-to-drag ratio g = magnitude of the acceleration due to gravity h = dimensionless altitude from the reference level K = induced drag factor k = auxiliary dimensionless parameter in the analysis of ballistic entry L =lift m = vehicle's mass r = radial distance from planet's center S = vehicle's characteristic area Professor of Aerospace Engineering. Member, AIAA. ' Research fellow. Member, AIAA. § Research fellow. * Professor Emeritus of Aerospace Engineering. Associate Fellow, AIAA. Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. t = time u = dimensionless speed in terms of the kinetic energy V = speed along the trajectory x = initial dimensionless flight path angle (noncircular cases) y = dimensionless altitude (density) variable z = auxiliary dimensionless variable in the computation of the altitude Greek letters: a = parameter defining trajectory type ft = inverse scale height 7 = flight path angle 5 =2(1a) 6 = range angle 'k = normalized hit coefficient v = dimensionless speed p = atmospheric density T = transformed range angle 0 = dimensionless flight path angle X = transformed longitude Subscripts: e : condition at entry 0 : reference trajectory Introduction After Chapman's classic first-order analytic solution, Loh, Yaroshevskii, and Longuski and Vinh, working independently, proposed different sets of second-order solutions for planetary entry trajectories, seeking greater accuracy for guidance purposes. But all those solutions left something to be desired. Loh's integration of the equations of motion was heuristic, introducing a step which finds no mathematical justification. On the other hand, inspired by Chapman's ideas, Yaroshevskii' s formulation is plagued by a strong singularity in the flight path angle, which is dealt with in a somewhat artificial way. Following a similar path, Longuski and Vinh integrated a system of simplified equations of motion, both numerically and analytically. Their work yields good results to within the validity of the simplifications made, except in the critical case of shallow entry. In physical applications, ballistic missiles typically reenter the dense atmosphere from subcircular orbital