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Laplace series for the level ellipsoid of revolution. / Kholshevnikov, K. V.; Milanov, D. V.; Shaidulin, V. Sh.
In: Celestial Mechanics and Dynamical Astronomy, Vol. 130, No. 10, 64, 01.10.2018.Research output: Contribution to journal › Article › peer-review
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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
KW - laplace series
KW - level ellipsoid
KW - Gravitational potential,Level ellipsoid,Laplace se
KW - gravitational potential
KW - stokes coefficients
KW - LIAPUNOV SERIES
KW - CONVERGENCE
KW - FIGURES
UR - http://www.scopus.com/inward/record.url?scp=85053927878&partnerID=8YFLogxK
UR - https://doi.org/10.1007/s10569-018-9851-7
UR - http://www.mendeley.com/research/laplace-series-level-ellipsoid-revolution
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 -
ID: 35246004