TY - JOUR

T1 - The Laplace series of ellipsoidal figures of revolution

AU - Kholshevnikov, K. V.

AU - Milanov, D. V.

AU - Shaidulin, V. Sh

PY - 2017/10/1

Y1 - 2017/10/1

N2 - The theory of figures of equilibrium was extensively studied in the nineteenth century, when the reasons for which observed massive celestial bodies (such as the Sun, planets, and satellites) are almost ellipsoidal were discovered. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be represented by a Laplace series whose coefficients (the Stokes constants In) are determined by a certain integral operator. In the case of an ellipsoid of revolution with homothetic equidensites (surfaces of constant density), the general term of this series was found, and for some of the other mass distributions, the first few terms of the series were determined. In this paper, the general term of the series is found in the case where the equidensites are ellipsoids of revolution with oblateness increasing from the center to the surface. Simple estimates and asymptotics of the coefficients In are also found. It turns out that the asymptotics depends only on the mean density, the density on the surface of the outer ellipsoid, and the oblateness of the outer ellipsoid.

AB - The theory of figures of equilibrium was extensively studied in the nineteenth century, when the reasons for which observed massive celestial bodies (such as the Sun, planets, and satellites) are almost ellipsoidal were discovered. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be represented by a Laplace series whose coefficients (the Stokes constants In) are determined by a certain integral operator. In the case of an ellipsoid of revolution with homothetic equidensites (surfaces of constant density), the general term of this series was found, and for some of the other mass distributions, the first few terms of the series were determined. In this paper, the general term of the series is found in the case where the equidensites are ellipsoids of revolution with oblateness increasing from the center to the surface. Simple estimates and asymptotics of the coefficients In are also found. It turns out that the asymptotics depends only on the mean density, the density on the surface of the outer ellipsoid, and the oblateness of the outer ellipsoid.

UR - http://www.scopus.com/inward/record.url?scp=85038114622&partnerID=8YFLogxK

U2 - 10.3103/S1063454117040112

DO - 10.3103/S1063454117040112

M3 - Article

AN - SCOPUS:85038114622

VL - 50

SP - 406

EP - 413

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -