Laplace series for the level ellipsoid of revolution

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1 Citation (Scopus)

Abstract

The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity ε of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities ε and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients In of Laplace series representing the outer potential of T are uniquely determined by ε and q. In this paper, we have found explicit expressions for Stokes coefficients via ε and q, as well as their asymptotics when n→ ∞. If T does not coincide with a Maclaurin ellipsoid, then | In| ∼ Bεn/ n with a certain constant B. Let us compare this asymptotics with one of In for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: | In| ∼ B¯ εn/ (n2). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.

Original languageEnglish
Article number64
JournalCelestial Mechanics and Dynamical Astronomy
Volume130
Issue number10
DOIs
Publication statusPublished - 1 Oct 2018

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Ellipsoid of revolution
Angular velocity
Ellipsoid
ellipsoids
Laplace
eccentricity
Series
Stokes
Directly proportional
Eccentricity
Coefficient
Exception
parameter
coefficients
angular velocity
gravitational fields

Scopus subject areas

  • Modelling and Simulation
  • Mathematical Physics
  • Astronomy and Astrophysics
  • Space and Planetary Science
  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "Laplace series for the level ellipsoid of revolution",
abstract = "The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity ε of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities ε and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients In of Laplace series representing the outer potential of T are uniquely determined by ε and q. In this paper, we have found explicit expressions for Stokes coefficients via ε and q, as well as their asymptotics when n→ ∞. If T does not coincide with a Maclaurin ellipsoid, then | In| ∼ Bεn/ n with a certain constant B. Let us compare this asymptotics with one of In for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: | In| ∼ B¯ εn/ (n2). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.",
keywords = "Asymptotics, Gravitational potential, Laplace series, Level ellipsoid, Stokes coefficients",
author = "Kholshevnikov, {K. V.} and Milanov, {D. V.} and Shaidulin, {V. Sh}",
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journal = "Celestial Mechanics and Dynamical Astronomy",
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AU - Kholshevnikov, K. V.

AU - Milanov, D. V.

AU - Shaidulin, V. Sh

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N2 - The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity ε of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities ε and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients In of Laplace series representing the outer potential of T are uniquely determined by ε and q. In this paper, we have found explicit expressions for Stokes coefficients via ε and q, as well as their asymptotics when n→ ∞. If T does not coincide with a Maclaurin ellipsoid, then | In| ∼ Bεn/ n with a certain constant B. Let us compare this asymptotics with one of In for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: | In| ∼ B¯ εn/ (n2). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.

AB - The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity ε of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities ε and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients In of Laplace series representing the outer potential of T are uniquely determined by ε and q. In this paper, we have found explicit expressions for Stokes coefficients via ε and q, as well as their asymptotics when n→ ∞. If T does not coincide with a Maclaurin ellipsoid, then | In| ∼ Bεn/ n with a certain constant B. Let us compare this asymptotics with one of In for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: | In| ∼ B¯ εn/ (n2). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.

KW - Asymptotics

KW - Gravitational potential

KW - Laplace series

KW - Level ellipsoid

KW - Stokes coefficients

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