Displacement Norm in the Presence of an Inverse-Square Perturbing Acceleration in the Reference Frame Associated with the Radius Vector

IF 1.1 4区 物理与天体物理 Q3 ASTRONOMY & ASTROPHYSICS
T. N. Sannikova
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Abstract

The problem of motion of a zero-mass point under the influence of attraction to the central body and a small perturbing acceleration \({\mathbf{P}}{\kern 1pt} ' = {\mathbf{P}}{\text{/}}{{r}^{2}}\) is considered, where \(r\) is the distance to the attracting center, and components of vector \({\mathbf{P}}\) are assumed to be constant in a reference system with axes directed along the radius vector, the transversal, and the angular momentum vector. Previously, for this problem, we found equations of motion in the mean elements and formulas for the transition from the osculating elements to the mean elements in the first order of smallness; second-order quantities were neglected. In this study, the Euclidean (root-mean-square over the mean anomaly) displacement norm \({{\left\| {d{\mathbf{r}}} \right\|}^{2}}\) is obtained, where \(d{\mathbf{r}}\) represents the difference between the position vectors on the osculating and mean orbit. It turned out that \({{\left\| {d{\mathbf{r}}} \right\|}^{2}}\) depends only on the components of vector \({\mathbf{P}}\) (positive definite quadratic form), the semimajor axis (proportional to the second power), and the eccentricity of the osculating ellipse. The norm \({{\left\| {d{\mathbf{r}}} \right\|}^{2}}\) is obtained in the form of series in powers of \(\beta = e{\text{/}}(1 + \sqrt {1 - {{e}^{2}}} )\) and in powers of the eccentricity \(e\). The results are applied to the problem of the motion of asteroids under the influence of a perturbing acceleration inversely proportional to the square of the heliocentric distance, in particular, under the influence of the Yarkovsky effect.

Abstract Image

Abstract Image

半径矢量相关参照系中存在反平方扰动加速度时的位移规范
摘要 本文考虑了零质量点在中心体吸引力和小扰动加速度 \({\mathbf{P}}{\kern 1pt} ' = {\mathbf{P}}{text\{/}}{{r}^{2}}\) 作用下的运动问题、其中 \(r\)是到吸引中心的距离,矢量 \({\mathbf{P}}\)的分量假定在参考系中是恒定的,参考系的轴分别指向半径矢量、横向矢量和角动量矢量。在此之前,对于这个问题,我们找到了均值元素的运动方程,以及从一阶小循环元素过渡到均值元素的公式;二阶量被忽略了。在这项研究中,得到了欧几里得(均值异常的均方根)位移规范\({{\left\| {d\mathbf{r}}} \right\|}^{2}}\) ,其中\(d\{mathbf{r}}\)代表了在循环轨道和均值轨道上的位置向量之差。结果发现,\({{\left\| {d{mathbf{r}}} \right\|}^{2}}\) 只取决于矢量\({\mathbf{P}}\) 的分量(正定二次型)、半长轴(与二次幂成正比)和摆动椭圆的偏心率。规范 \({{\left\| {d{mathbf{r}}} \right\|}^{2}}\) 以 \(\beta = e{\text{/}}(1 + \sqrt {1 - {{e}^{2}}} )\) 的幂级数和偏心率 \(e\) 的幂级数的形式得到。这些结果被应用于小行星在与日心距离平方成反比的扰动加速度影响下的运动问题,特别是在雅尔科夫斯基效应影响下的运动问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Astronomy Reports
Astronomy Reports 地学天文-天文与天体物理
CiteScore
1.40
自引率
20.00%
发文量
57
审稿时长
6-12 weeks
期刊介绍: Astronomy Reports is an international peer reviewed journal that publishes original papers on astronomical topics, including theoretical and observational astrophysics, physics of the Sun, planetary astrophysics, radio astronomy, stellar astronomy, celestial mechanics, and astronomy methods and instrumentation.
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