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Quaternion Equations of Disturbed Motion of an Artificial Earth Satellite

机译：人造地球卫星扰动运动的四元数方程

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摘要

The quaternion equations of disturbed motion of an artificial Earth satellite (AES) in Earth's gravitational field (with regard to its zonal, tesseral, and sectorial harmonics) are obtained in four-dimensional Kustaanheimo-Stiefel variables and in modified four-dimensional variables. In the latter variables the equations of satellite motion have a simpler and more symmetric structure, as compared to the equations of motion in the Kustaanheimo-Stiefel variables. The obtained equations are linear for undisturbed Keplerian motion and, in general, have the form of equations of motion of a disturbed four-dimensional oscillator. These equations, in contrast to classical equations, are regular (they do not contain specific points of singularity) for satellite motion in the central gravitational field of Earth under an effect of disturbing forces, the description of which does not contain negative, higher-than-first powers of the satellite distance to Earth's center. In these equations, the basic variables are Kustaanheimo-Stiefel variables, or modified four-dimensional variables proposed by the author, as well as satellite motion energy and time. The new independent variable is associated with time by the differential relation containing the satellite distance to Earth's center of mass. The equations are convenient for applying nonlinear mechanics methods and for high-precision numerical calculations. In the case of artificial satellite motion in Earth's gravitational field, the description of which does not take into account the tesseral and sectorial harmonics, but takes into account its zonal harmonics, the first integrals of the equations of satellite motion are found, and the replacements of variables and transformations of these equations are proposed, which allowed one to get, for studying satellite motion, closed systems of differential equations of lower dimension, in particular, the system of equations of fourth and third orders.
机译：在四维Kustaan​​heimo-Stiefel变量和经修改的四维变量中，获得了地球重力场中的人造地球卫星（AES）的扰动运动的四元数方程（关于其纬向，阶跃和扇形谐波）。在后一种变量中，与Kustaan​​heimo-Stiefel变量中的运动方程相比，卫星运动方程具有更简单，更对称的结构。对于不受干扰的开普勒运动，所获得的方程是线性的，并且通常具有扰动的四维振荡器的运动方程的形式。与经典方程式相反，这些方程式是规则的（它们不包含奇异点），用于在干扰力作用下在地球中心重力场中的卫星运动，其描述不包含负的，大于-卫星到地球中心的距离的第一功率。在这些方程式中，基本变量是Kustaan​​heimo-Stiefel变量或作者提出的修改后的三维变量，以及卫星运动的能量和时间。新的自变量通过包含卫星到地球质心的距离的微分关系与时间相关联。该方程式便于应用非线性力学方法和进行高精度数值计算。对于地球重力场中的人造卫星运动，其描述不考虑地谐和扇形谐波，而是考虑其纬向谐波，因此找到了卫星运动方程的第一个积分，并将其替换提出了这些方程的变量和变换的方法，这些变量和变换可以使人们获得用于研究卫星运动的低维微分方程的封闭系统，特别是四阶和三阶方程组。

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  • 来源
    《Cosmic research》 |2019年第2期|101-114|共14页
  • 作者

    Chelnokov Yu. N.;

  • 作者单位

    Russian Acad Sci, Inst Problems Precise Mech & Control, Saratov 410026, Russia;

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