Norm of the Position Shift of a Celestial Body in a Dynamical Astronomy Problem

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 $${{e}^{6}}$$.