A general method for N-order integral-form Gauss's variational equations under impulsive control

In this paper, an innovative derivatives-based method is proposed to derive any higher-order integral-form Gauss's Variational Equations (GVEs) of the orbital elements under impulsive control. In this new method, the Taylor series expansion is applied to express the variation of each orbital element as a sum of terms that are calculated from the values of the derivatives of orbital elements. The required N-order derivatives of orbital elements can be successively computed from the (N-1)-order differential-form GVEs by using the chain rule of calculus. Using this method, the final expressions of the second-order and third-order integral-form GVEs are provided in detail. To testify the accuracy of these new results, the Monte Carlo simulations are performed randomly and the histogram data are analyzed. It is demonstrated that the proposed higher-order GVEs are more accurate than the traditional lower-order ones. The method and the resultant higher-order GVEs can be potentially used on spacecraft impulsive orbital maneuver or formation flying.