New Look to the Constant Radial Acceleration Problem

Abstract

is revisited. Some closed-form analytical solutions for this problem exist: in particular, those regarding the acceleration level required for obtaining an escape condition and the value of the maximum amplitude of the radial oscillation when an escape does not occur. Other results are known to exist in terms of elliptic integrals; however, their practical use for mission design is difficult due to the lack of physical insight connected to these mathematical functions. This paper provides new theoretical results for the motion description and new accurate approximations forboththespacecrafttrajectoryandthetimeof flightusingsimpleandclosed-formrelationships.Thisispossibleby introducingasuitable regularizationofthemotion,forwhichthesolution isperiodicwithrespecttotheindependent angular variable, and therefore can be exactly expressed in terms of a Fourier series. The new results are conveniently applied to a spacecraft repositioning problem and to a sample-return mission. Nomenclature a = osculating orbit’s semimajor axis, distance unit b = dimensionless fitting coefficient c = dimensionless Fourier coefficient e = osculating orbit eccentricity h = osculating orbit’s angular momentum, � distance unit� 2 =time unit k = modulus of � N = number of complete radial oscillations n = parameter of � r = radial distance, distance unit