四艘航天器的描述，沿椭圆开普勒轨道运动
Description of 4 Spacecraft, Moving on Elliptic Kepler Orbits
AI总结：
本文提出了一种用于描述四航天器编队的新方法，通过解析解分析其在椭圆开普勒轨道上的运动，以提高测量太阳系重力场梯度的精度。
摘要：
四航天器编队对于测量各种物理场至关重要。将这种编队用于显著椭圆的日心开普勒轨道，可以测量太阳系中的重力场梯度。测量的准确性足以确认或反驳修改的引力理论。本文提出了一种新的描述这种编队的方法。通过线性化运动方程的解析解得到。这些解的特色是使用其中一艘航天器的笛卡尔坐标，称为主航天器。这些解具有明确的物理意义。证明了由航天器形成的四面体体积是主航天器笛卡尔坐标的三次多项式。多项式的系数是初始航天器坐标和速度的函数，并且线性依赖于时间。如果所有航天器绕太阳旋转的周期相同，体积是主航天器坐标的二次多项式，系数与时间无关。在这种情况下，体积可以在每个周期内从零到四次被抵消。所提方法可以显著简化测量各种星际场任务的规划。
英文摘要：
The four-spacecraft formation is essential for measurements of various physical fields. The use of this formation on substantially elliptical heliocentric Kepler orbits allows measuring gradients of gravitation field in Solar system. The accuracy of the measurements will be sufficient to confirm or to refute modified theories of gravity. In this paper a new approach for the description of this formation is presented. The analytical solutions of the linearized motion equations are obtained. The distinctive feature of the solutions is that they use Cartesian coordinates of one of the spacecraft, termed the chief. These solutions have a clear physical meaning. It is shown, that the volume of a tetrahedron formed by spacecraft is a polynomial of 3-rd degree of Cartesian coordinates of the chief. The polynomial's coefficients are functions of initial spacecraft coordinates and velocities and linearly depend on time. If all spacecraft have the same periods of rotation around the Sun, the volume is a polynomial of 2-nd degree of the chief coordinates with time-independent coefficients. In this case the volume can be zeroed from 0 to 4 times per the period. Suggested approach can significantly simplify planning missions for measurements of various interplanetary fields.
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