TY - JOUR

T1 - Norm of the Position Shift of a Celestial Body in a Dynamical Astronomy Problem

AU - Kholshevnikov, K. V.

AU - Batmunkh, N.

AU - Oskina, K. I.

AU - Titov, V. B.

N1 - Funding Information: This work was supported by the Russian Scientific Foundation (project 18-12-00050). Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - Abstract—: 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 δr in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm ||δr|| 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 ||δr|2 is proportional to a6, where a is the semi-major axis. The value ||δr|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 e that converge, at least for e < 1. The series coefficients are calculated up to e4 inclusive, so that the correction terms are of order e6.

AB - Abstract—: 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 δr in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm ||δr|| 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 ||δr|2 is proportional to a6, where a is the semi-major axis. The value ||δr|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 e that converge, at least for e < 1. The series coefficients are calculated up to e4 inclusive, so that the correction terms are of order e6.

UR - http://www.scopus.com/inward/record.url?scp=85085119205&partnerID=8YFLogxK

U2 - 10.1134/S1063772920040034

DO - 10.1134/S1063772920040034

M3 - Article

AN - SCOPUS:85085119205

VL - 64

SP - 369

EP - 373

JO - Astronomy Reports

JF - Astronomy Reports

SN - 1063-7729

IS - 4

ER -