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 r in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm parallel to delta r parallel to 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 F. The vector 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 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 parallel to delta r parallel to(2) is proportional to a(6), where a is the semi-major axis. The value parallel to delta r parallel to(2) a(-6) is the weighted sum of the component squares of F. The quadratic form coefficients depend only on the eccentricity and are represented by the Maclaurin series in even powers of 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).