Approximate solution of zero-effort-miss under gravitational acceleration inversely proportional to the cubic distance

In this paper, an approximate solution of zero-effort-miss distance in spherical earth model is obtained in Earth-centered inertial (ECI) coordinates. In this approach, the gravitational acceleration is assumed to be inversely proportional to the cubic distance of space vehicle from the earth center. The present solution gives an explicit formula in terms of initial conditions and flight time. Moreover, the piecewise solution, i.e., the connection of solutions, is utilized in order to increase the accuracy of the algorithm. For this propose, the total flight time is divided into several interval using middle points. In each interval, the gravitational acceleration is taken to be inversely proportional to the cubic distance, but the correction constant is updated for each interval. The accuracy of the proposed method and its computational burden are calculated for various numbers of time intervals. For comparison purpose, the piecewise solution is also applied to the linear gravity approximation. The simulation results and calculation of computational burden show that the presented method has better accuracy than linear solution in ECI reference and linearized solution for relative motion with the same computational burden.