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.

在本文中，提出了一种基于导数的创新方法，可以在脉冲控制下导出轨道元素的任何高阶积分形式的高斯变分方程（GVE）。在这种新方法中，采用泰勒级数展开式将每个轨道元素的变化表示为从轨道元素导数的值计算出的项之和。所需的轨道元素的N阶导数可从（N-1）利用微积分的链规则求微分形式GVE。使用此方法，详细提供了二阶和三阶整数形式的GVE的最终表达式。为了证明这些新结果的准确性，随机进行了蒙特卡洛模拟，并对直方图数据进行了分析。结果表明，提出的高阶GVE比传统的低阶GVE更准确。该方法和所得的高阶GVE可以潜在地用于航天器的脉冲轨道机动或编队飞行。