Shapes of two‐layer models of rotating planets

Kong, Dali; Zhang, K.; Schubert, G.

doi:10.1029/2010je003720

Cited by 31 publications

(43 citation statements)

References 30 publications

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“…The rotation parameter β provides a measure of the centrifugal force. It is mathematically convenient (Zhang, Liao & Earnshaw 2004;Kong, Zhang & Schubert 2010) to adopt oblate spheroidal coordinates, (ξ , η, φ), defined by the coordinate transformation with Cartesian coordinates…”

Section: O D E L a N D G Ov E R N I N G E Q Uat I O N Smentioning

confidence: 99%

An exact solution for arbitrarily rotating gaseous polytropes with index unity

Kong

Zhang

Schubert

2015

Monthly Notices of the Royal Astronomical Society

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Many gaseous planets and stars are rapidly rotating and can be approximately described by a polytropic equation of state with index unity. We present the first exact analytic solution, under the assumption of the oblate spheroidal shape, for an arbitrarily rotating gaseous polytrope with index unity in hydrostatic equilibrium, giving rise to its internal structure and gravitational field. The new exact solution is derived by constructing the non-spherical Green's function in terms of the oblate spheroidal wavefunction. We then apply the exact solution to a generic object whose parameter values are guided by the observations of the rapidly rotating star α Eridani with its eccentricity E α = 0.7454, the most oblate star known. The internal structure and gravitational field of the object are computed from its assumed rotation rate and size. We also compare the exact solution to the three-dimensional numerical solution based on a finite-element method taking full account of rotation-induced shape change and find excellent agreement between the exact solution and the finite-element solution with about 0.001 per cent discrepancy.Key words: planets and satellites: gaseous planets -planets and satellites: interiors -stars: interiors. I N T RO D U C T I O NA fully compressible polytropic gas with index unity obeying the polytropic equation of state (EOS)where p * is the pressure, K is a constant and ρ * is the density, has been widely employed to study the physical properties of gaseous planets, exoplanets and stars (see for example, Chandrasekhar 1933; Roberts 1962;Hubbard 1973;Stevenson 1982;Dintrans & Ouyed 2001;Horedt 2004;Kong et al. 2014). In this paper, the superscript * is adopted to represent a dimensional variable and its corresponding dimensionless variable is denoted without the superscript. Many astrophysical gaseous bodies are rapidly rotating, causing significant departure from sphericity: the eccentricity at the one-bar surface is E S = 0.4316 for Saturn (Seidelmann et al. 2007) while the star α Eridani is marked by a much larger departure from sphericity with the approximate eccentricity E α = 0.7454 (Carciofi et al. 2008). The rotational effect on the shape and physical structure of a slowly rotating polytrope was first studied by Chandrasekhar (1933) using a perturbation analysis. For an isolated, non-rotating and selfgravitating body, the density distribution ρ * within the interior of E-mail: kzhang@ex.ac.uk the polytrope is spherically symmetric and described by the LaneEmden equation, a second-order ordinary differential equation that can be readily solved to determine the one-dimensional density distribution. For a polytropic body that is slowly rotating with small angular velocity such that its departure from sphericity is slight, Chandrasekhar (1933) introduced a small parameter ∼ 2 and was able to solve for the density distribution ρ * of the slowly rotating polytrope via a perturbation method in terms of the small expansion parameter . Without developing a small parameter expansion, R...

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Section: O D E L a N D G Ov E R N I N G E Q Uat I O N Smentioning

confidence: 99%

An exact solution for arbitrarily rotating gaseous polytropes with index unity

Kong

Zhang

Schubert

2015

Monthly Notices of the Royal Astronomical Society

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“…All three parameters are believed to be small for a typical planet: typical values of the Ekman number E for many planets and satellites are extremely small, with E O(10 −14 ); the Poincaré number Po is typically O(10 −4 ) (see e.g. Noir et al 2009, table 2); and the size of the eccentricity E is moderately small, typically with E = O(10 −1 ) (Kong, Zhang & Schubert 2010). In other words, many planets and satellites are typically marked by E 1 and Po 1 but with Po/E 1/2 1.…”

Section: Introduction and Formulationmentioning

confidence: 99%

Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

2012

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We consider a homogeneous fluid of viscosity ν confined within an oblate spheroidal cavity, x 2 /a 2 + y 2 /a 2 + z 2 /(a 2 (1 − E 2 )) = 1, with eccentricity 0 < E < 1. The spheroidal container rotates rapidly with an angular velocity Ω 0 , which is fixed in an inertial frame and defines a small Ekman number E = ν/(a 2 |Ω 0 |), and undergoes weak latitudinal libration with frequencyω|Ω 0 | and amplitude Po|Ω 0 |, where Po is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When |ω − 2/(2 − E 2 )| O(E 1/2 ), an asymptotic solution for E 1 and Po 1 in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial-temporal structure of the librating flow. In this case, the librationally driven flow is nonaxisymmetric with amplitude O(Po), and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When |ω − 2/(2 − E 2 )| O(E 1/2 ), the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes O(Po/E 1/2 ) at E 1 and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for E 1 describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at E 1.

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“…Kong et al (2010) discussed the particular case of a body formed by two homogeneous layers with same angular velocity. Hubbard (2013), with a recursive numerical form of the potential of a N-layers rotating planet, in hydrostatic equilibrium, showed a solution for the spheroidal shapes of the interfaces of the layers.…”

Section: Introductionmentioning

confidence: 99%

The flattenings of the layers of rotating planets and satellites deformed by a tidal potential

Folonier

Ferraz-Mello

Холшевников

2015

Celest Mech Dyn Astr

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We consider the Clairaut theory of the equilibrium ellipsoidal figures for differentiated nonhomogeneous bodies in non-synchronous rotation (Tisserand, Mécanique Céleste, t.II, Chap. 13 and 14) adding to it a tidal deformation due to the presence of an external gravitational force. We assume that the body is a fluid formed by n homogeneous layers of ellipsoidal shape and we calculate the external polar flattenings ǫ k , µ k and the mean radius R k of each layer, or, equivalently, their semiaxes a k , b k and c k . To first order in the flattenings, the general solution can be written as ǫ k = H k ǫ h and µ k = H k µ h , where H k is a characteristic coefficient for each layer which only depends on the internal structure of the body and ǫ h , µ h are the flattenings of the equivalent homogeneous problem. For the continuous case, we study the Clairaut differential equation for the flattening profile, using the Radau transformation to find the boundary conditions when the tidal potential is added. Finally, the theory is applied to several examples: i) a body composed of two homogeneous layers; ii) bodies with simple polynomial density distribution laws and iii) bodies following a polytropic pressure-density law.

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Shapes of two‐layer models of rotating planets

Cited by 31 publications

References 30 publications

An exact solution for arbitrarily rotating gaseous polytropes with index unity

An exact solution for arbitrarily rotating gaseous polytropes with index unity

Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

The flattenings of the layers of rotating planets and satellites deformed by a tidal potential

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