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

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.