Improved Perturbative Solution of Yaroshevskii's Planetary Entry Equation

Abstract

An improved approximate analytical solution is developed for Yaroshevskii's classical planetary entry equation for the ballistic entry of a spacecraft into planetary atmospheres at circular speed. Poincaré's method of small parameters is used to solve for the altitude and flight path angle as a function of the spacecraft's speed. From this solution, other important expressions are developed including deceleration, stagnation-point heat rate, and stagnation-point integrated heat load. The accuracy of the solution is assessed via numerical integration of the exact equations of motion. The solution is also compared to the classical solutions of Yaroshevskii and Allen and Eggers. The new second-order analytical solution is more accurate than Yaroshevskii's fifth-order solution for a range of shallow (−3 deg) to steep (up to −90 deg) entry flight path angles, thereby extending the range of applicability of the solution as compared to the classical Yaroshevskii solution, which is restricted to an entry flight path of approximately −40 deg.