Analysis of approximate solutions with Edelbaum’s approximation for orbital maneuvers at periapsis/apoapsis

This paper proposes a series of estimated solutions with Electric Propulsion, based on Edelbaum’s approximation, for the preliminary calculation of the propellant consumption and the required orbital changes in periapsis and apoapsis maneuvers during an interplanetary mission to Near-Earth Asteroids. A set of 75 fictitious asteroids is considered to study the influence of the orbital parameters on the response of the method, with a revolution period close to one year and small eccentricity and inclination, and the assumed time of flight for the mission is three years, so three burning arcs are performed. The estimation does not need numerical integration, but only the resolution of two three-variable algebraic systems is necessary to define the whole transfer; thus, the computational time and cost are significantly reduced. Various modifications have been made to the orbital elements of the arrival orbit to observe the trends of the parameters useful for the calculation of the algebraic systems: the inclination of the orbit is initially considered linearly proportional to the eccentricity, while in the following cases it is fixed at a certain number of values; the right ascension of ascending node in the first two cases is null, with the periapsis and the ascending node coinciding, then is assumed equal to π⁄4 and π⁄2, monitoring the changes due to the distancing of the line of nodes and the line of apsides. The results generally follow the theoretical expectations, especially in particular conditions. Both in periapsis and apoapsis maneuvers, the estimation is accurate for the furthest asteroids from Earth and quite inaccurate as regards the closest asteroids to Earth, hence a subgroup of asteroids whose distance does not allow to perform long burning arcs. When the inclination is set at slightly higher but still small values, together with an obvious increase in consumption, there is a very accurate response on almost the entire group of asteroids for maneuvers at both apsides, except for the very first asteroids which always have some oddities. Furthermore, also the RAAN variations, even with a small increase of fuel consumed, evidently contribute to raise the overall quality of the estimation; this can be seen especially in the results of the cases at low inclination, in which the precise solution tends to include a larger number of asteroids if the periapsis and the ascending node are not coincident, but in quadrature. The analysis of the method in these different case studies has allowed to ascertain the goodness and relevance to theory of the results obtained, particularly in cases with a relatively high inclination (but always small to adhere to Edelbaum’s approximation) and in cases with low to null inclination, but RAAN different from 0 (even if the latter increase more propellant consumption). New studies and analysis may be focused on improving the accuracy of this estimation method in the planar problem, i.e., with null inclination, in which the periapsis and the ascending node are coincident, that is, Ω equal to 0.