We present a new method for computing orbits in the perturbed two-body problem: the position and velocity vectors of the propagated object in Cartesian coordinates are replaced by eight orbital elements, i.e. constants of the unperturbed motion. The proposed elements are uniformly valid for any value of the total energy. Their definition stems from the idea of applying Sundman’s time transformation in the framework of the projective decomposition of motion, which is the starting point of the Burdet–Ferrándiz linearisation, combined with Stumpff’s functions. In analogy with Deprit’s ideal elements, the formulation relies on a special reference frame that evolves slowly under the action of external perturbations. We call it the intermediate frame, hence the name of the elements. Two of them are related to the radial motion, and the next four, given by Euler parameters, fix the orientation of the intermediate frame. The total energy and a time element complete the state vector. All the necessary formulae for extending the method to orbit determination and uncertainty propagation are provided. For example, the partial derivatives of the position and velocity with respect to the intermediate elements are obtained explicitly together with the inverse partial derivatives. Numerical tests are included to assess the performance of the proposed special perturbation method when propagating the orbit of comets C/2003 T4 (LINEAR) and C/1985 K1 (Machholz).

中文翻译：

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轨道计算的统一公式：中间元素

我们提出了一种计算扰动二体问题中轨道的新方法：笛卡尔坐标系中传播物体的位置和速度矢量被八个轨道元素代替，即未扰动运动的常数。建议的元素对于总能量的任何值都是一致有效的。他们的定义源于在运动的射影分解框架中应用 Sundman 的时间变换的想法，这是 Burdet-Ferrándiz 线性化的起点，结合 Stumpff 函数。与 Deprit 的理想元素类似，该公式依赖于在外部扰动作用下缓慢演化的特殊参考系。我们称它为中间框架，因此得名元素。其中两个与径向运动有关，接下来的四个，由欧拉参数给出，固定中间框架的方向。总能量和时间元素完成状态向量。提供了将方法扩展到轨道确定和不确定性传播的所有必要公式。例如，位置和速度相对于中间元素的偏导数与逆偏导数一起显式地获得。包括数值测试以评估在传播彗星 C/2003 T4 (LINEAR) 和 C/1985 K1 (Machholz) 的轨道时提出的特殊扰动方法的性能。例如，位置和速度相对于中间元素的偏导数与逆偏导数一起显式地获得。包括数值测试以评估在传播彗星 C/2003 T4 (LINEAR) 和 C/1985 K1 (Machholz) 的轨道时提出的特殊扰动方法的性能。例如，位置和速度相对于中间元素的偏导数与逆偏导数一起显式地获得。包括数值测试以评估在传播彗星 C/2003 T4 (LINEAR) 和 C/1985 K1 (Machholz) 的轨道时提出的特殊扰动方法的性能。