### 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 I_{n} 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 | I_{n}| ∼ Bε^{n}/ n with a certain constant B. Let us compare this asymptotics with one of I_{n} for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: | I_{n}| ∼ B¯ ε^{n}/ (n^{2}). 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 language | English |
---|---|

Article number | 64 |

Journal | Celestial Mechanics and Dynamical Astronomy |

Volume | 130 |

Issue number | 10 |

DOIs | |

Publication status | Published - 1 Oct 2018 |

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### 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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**Laplace series for the level ellipsoid of revolution.** / Kholshevnikov, K. V.; Milanov, D. V.; Shaidulin, V. Sh.

Research output

TY - JOUR

T1 - Laplace series for the level ellipsoid of revolution

AU - Kholshevnikov, K. V.

AU - Milanov, D. V.

AU - Shaidulin, V. Sh

PY - 2018/10/1

Y1 - 2018/10/1

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

UR - http://www.scopus.com/inward/record.url?scp=85053927878&partnerID=8YFLogxK

U2 - 10.1007/s10569-018-9851-7

DO - 10.1007/s10569-018-9851-7

M3 - Article

AN - SCOPUS:85053927878

VL - 130

JO - Celestial Mechanics and Dynamical Astronomy

JF - Celestial Mechanics and Dynamical Astronomy

SN - 0923-2958

IS - 10

M1 - 64

ER -