Estimate of short-periodic perturbations in a problem of celestial mechanics

Abstract

Let us consider the following problem of celestial mechanics. A zero-mass point moves under a gravitational acceleration $\mathbf P_0$ to a central body of finite mass, and a disturbing acceleration $\mathbf P$. The last vector is constant in an accompanying reference frame with axes directed along the radius, the transversal, and the angular momentum vector. Earlier this problem has been transformed using averaging method. In more exact terms a change of variables excluding short-periodic harmonics has been found (in the first approximation with respect to the ratio $|\mathbf P|/|\mathbf P_0|$). So the differences between osculating and mean elements were obtained explicitly, as well as the equations of motion in mean elements. A problem of evaluating the magnitude of short-periodic harmonics arises. It is not difficult to evaluate them for each element. But we need to do it in the coordinate space, not in the space of elements. Meanwhile the standard estimate of a coordinates increment via an elements increment is drastically rough. In the present paper we succeed to obtain an exact estimate using Euclidean (mean-squared) norm of a variance. For this a relatively simple expression for the squared variance of the radius-vector via variances of elements was firstly derived. It was applied to estimate the norm $\|mathrm{d}\mathbf{r}\|$ (difference of position vectors on the osculating and mean orbit) in the above problem. It turns out that $\|mathrm{d}\mathbf r\|^2$ is a weighted sum of squared components of $\mathbf P$, and the corresponding coefficients depend on semi-major axis and eccentricity of the mean orbit only. The results are applied to the two real problems on the motion of sputniks, and of asteroids.