Abstract

The motion of a point with zero mass under the action of attraction to the central body \(\mathcal{S}\) and perturbing acceleration \({\mathbf{P}}{\kern 1pt} ' = {\mathbf{P}}{\text{/}}{{r}^{2}}\), inversely proportional to the square of the distance to \(\mathcal{S}\), is considered. It is assumed that \({\mathbf{P}}{\kern 1pt} '\) is small in absolute value compared to the main acceleration, caused by the attraction of the central body. Further, the vector \({\mathbf{P}}\) components are constant in the reference frame used in astronomy, with the origin in the central body and the axes directed along the radius vector, the transversal (perpendicular to the radius vector in the osculating plane in the direction of motion), and the binormal (directed along the area vector). Earlier, we performed an averaging transformation of Euler-type equations of motion in osculating elements and obtained mean element evolutionary differential equations of motion in the first approximation in a small parameter. This article is devoted to solving the averaged equations, which are integrated completely. Moreover, the quadratures were expressed via elementary functions. The solution found has singularities at zero eccentricity and in the absence of the transverse acceleration. These and some other special cases are considered separately. There are at least two applications of the problem considered which are: an asteroid’s motion with allowance for the Yarkovsky–Radzievsky effect and a spacecraft’s motion with a solar sail. In both cases, the perturbation is inversely proportional to the squared distance from the Sun.

中文翻译：

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在参考半径相关的参考系中，根据反平方定律，具有扰动加速度的中心场运动

摘要

质量为零的点在吸引中心体\（\ mathcal {S} \）和扰动加速度\（{\ mathbf {P}} {\ kern 1pt}'的作用下的运动'= {\ mathbf {P }} {\ text {/}} {{r} ^ {2}} \），与到\（\ mathcal {S} \）的距离的平方成反比。假定\（{\ mathbf {P}} {\ kern 1pt}'\）的绝对值与主加速度相比较小，这是由中央物体的吸引力引起的。此外，向量\（{\ mathbf {P}} \）分量在用于天文学的参考系中是恒定的，其中心点的原点和轴线沿半径矢量，横向（垂直于在运动方向上的紧密平面中的半径矢量）和双法线（沿面积矢量定向）。早先，我们对振动单元中的Euler型运动方程进行了平均变换，并在较小的参数中以一阶近似值获得了运动的均值元素演化微分方程。本文致力于求解完全积分的平均方程。此外，通过基本函数表示正交。所找到的解在零偏心率和没有横向加速度的情况下具有奇异性。这些情况和其他一些特殊情况将分别考虑。考虑的问题至少有两个应用：小行星的运动允许Yarkovsky–Radzievsky效应和航天器的运动具有太阳帆。在这两种情况下，摄动与距太阳的平方距离成反比。