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

The problem of motion of a zero-mass-point under the action 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 the vector $${\mathbf{P}}$$ are assumed to be constant in a reference system with the axes directed along the velocity vector, the main normal, and the angular momentum vector. Previously, for this problem, equations of motion in the mean elements and formulas for the transition from osculating elements to the mean ones in the first order of smallness were found; second-order quantities are neglected. If the perturbing forces are small, then the osculating orbit slightly deviates from the mean one. The difference $$d{\mathbf{r}}$$ between the position vectors on the osculating and mean orbit is a quasi-periodic function of time. In this work, the Euclidean (root-mean-square over the mean anomaly) norm $${{\left\| {d{\mathbf{r}}} \right\|}^{2}}$$ of the displacement of the osculating orbit relative to the mean one is obtained. It turned out that $${{\left\| {d{\mathbf{r}}} \right\|}^{2}}$$ depends only on the components of the vector $${\mathbf{P}}$$ (positive definite quadratic form), the semi-major 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 a series in powers of eccentricity $$e$$ . The resulting expression holds up to $${{e}_{0}} \approx 0.995862$$ ; for $$e > {{e}_{0}}$$ , $$\varrho = \sqrt {{{{\left\| {d{\mathbf{r}}} \right\|}}^{2}}} $$ can take complex values. The results are applied to the problem of the motion of model bodies under the action of perturbing acceleration caused by the Yarkovsky effect. A comparison of the results with similar ones for the norm $${{\left\| {d{\mathbf{r}}} \right\|}^{2}}$$ in the reference system associated with the radius vector was also carried out.