Abstract—

The averaging method is widely used in celestial mechanics, in which a mean orbit is introduced and slightly deviates from an osculating one, as long as disturbing forces are small. The difference \(\delta {\mathbf{r}}\) in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm \(\left\| {\delta {\mathbf{r}}} \right\|\) for deviation is interesting to note. Earlier, the exact expression of the mean-square norm for one problem of celestial mechanics was obtained: a zero-mass point moves under the gravitation of a central body and a small perturbing acceleration \({\mathbf{F}}\). The vector \({\mathbf{F}}\) is taken to be constant in a co-moving coordinate system with axes directed along the radius vector, the transversal, and the angular momentum vector. Here, we solved a similar problem, assuming the vector \({\mathbf{F}}\) to be constant in the reference frame with axes directed along the tangent, the principal normal, and the angular momentum vector. It turned out that \({{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}\) is proportional to \({{a}^{6}}\), where \(a\) is the semi-major axis. The value \({{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}{{a}^{{ - 6}}}\) is the weighted sum of the component squares of \({\mathbf{F}}\). The quadratic form coefficients depend only on the eccentricity and are represented by the Maclaurin series in even powers of \(e\) that converge, at least for \(e < 1\). The series coefficients are calculated up to \({{e}^{4}}\) inclusive, so that the correction terms are of order \({{e}^{6}}\).

中文翻译：

---

动态天文学问题中天体位置偏移的范数

摘要—

该平均方法被广泛用于天体力学中，只要干扰力较小，就可以引入平均轨道，并且该平均轨道略微偏离振荡轨道。在平均轨道和振荡轨道中天体位置的差异\（\ delta {\ mathbf {r}} \）是时间的准周期函数。估计范数\（\ left \ | {\ delta {\ mathbf {r}}} \ right \ | \）的偏差很有趣。早先，获得了一个天体力学问题的均方范数的精确表达式：零质量点在中心物体的引力作用下移动，并且扰动加速度很小（\（{\ mathbf {F}} \）。向量\（{\ mathbf {F}} \）在共同移动的坐标系中，α是恒定的，其坐标轴沿半径向量，横向和角动量向量指向。在这里，我们解决了一个类似的问题，假设向量\（{\ mathbf {F}} \）在参考系中是恒定的，并且轴沿切线，主法线和角动量向量指向。原来\（{{\ left \ | {\ delta {\ mathbf {r}}} \ right \ |} ^ {2}} \）与\（{{a} ^ {6}} \ ），其中\（a \）是半长轴。值\（{{\ left \ | {\ delta {\ mathbf {r}}} \ right \ |} ^ {2}} {{a} ^ {{-6}}} \）是以下项的加权和\（{\ mathbf {F}} \）的分量平方。二次形式系数仅取决于偏心率，并且由Maclaurin级数表示，至少在\（e <1 \）时，以\（e \）的偶次幂收敛。序列系数的计算最高为\（{{e} ^ {4}} \），因此校正项的阶次为\（{{e} ^ {6}} \）。