New exact solutions of magnetic ellipsoidal figures of equilibrium

Kawamura, Takumu; Taniguchi, Keisuke; Yoshida, Shin’ichirou; Eriguchi, Yoshiharu

doi:10.1111/j.1745-3933.2011.01101.x

Cited by 4 publications

(1 citation statement)

References 19 publications

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“…This may be interesting in the context of exoplanets and ocean planets. It could also be very interesting to consider the existence of nested figures involving prolate spheroidal surface, which may occur, for instance, in the presence of circulation of magnetic field (Kawamura et al 2011;Fujisawa & Eriguchi 2014). As shown here, rapidly rotating layers tend to produce quasi-spherical cores (see, for instance, configuration E).…”

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Nested spheroidal figures of equilibrium II. Generalization to L layers

Huré

2022

Preprint

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We present a vectorial formalism to determine the approximate solutions to the problem of a composite body made of L homogeneous, rigidly rotating layers bounded by spheroidal surfaces. The method is based on the 1st-order expansion of the gravitational potential over confocal parameters, thereby generalizing the method described in Paper I for L = 2. For a given relative geometry of the ellipses and a given set of mass-density jumps at the interfaces, the sequence of rotation rates and interface pressures is obtained analytically by recursion. A wide range of equilibria result when layers rotate in an asynchronous manner, although configurations with a negative oblateness gradient are more favorable. In contrast, states of global rotation (all layers move at the same rate), found by solving a linear system of L − 1 equations, are much more constrained. In this case, we mathematically demonstrate that confocal and coelliptical configurations are not permitted. Approximate formula for small ellipticities are derived. These results reinforce and prolongate known results and classical theorems restricted to small elliptiticities. Comparisons with the numerical solutions computed from the Self-Consistent-Field method are successful.

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confidence: 83%

Nested spheroidal figures of equilibrium II. Generalization to L layers

Huré

2022

Preprint

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Existence of slowly rotating bipolytropes with prolate cores

Staelen,

Huré,

Meunier

et al. 2024

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We report the existence of hydrostatic equilibrium states for a composite body made of two rigidly rotating, homogeneous layers bounded by spheroidal surfaces, where the core has a prolate shape. These new configurations require an oblate envelope that spins faster than the core. No solution with a prolate envelope is found. For some parameters, the prolate core can even be at rest. Numerical experiments based on the self-consistent field method support this result in the case of heterogeneous layers with polytropic equations of state. The possible cancellation of the first gravitational moment, $J_2$, is discussed.

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Nested spheroidal figures of equilibrium – II. Generalization to layers

Huré¹

2022

Monthly Notices of the Royal Astronomical Society

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We present a vectorial formalism to determine the approximate solutions to the problem of a composite body made of ${\cal L}$ homogeneous, rigidly rotating layers bounded by spheroidal surfaces. The method is based on the 1st-order expansion of the gravitational potential over confocal parameters, thereby generalizing the method described in Paper I for ${\cal L}=2$. For a given relative geometry of the ellipses and a given set of mass-density jumps at the interfaces, the sequence of rotation rates and interface pressures is obtained analytically by recursion. A wide range of equilibria result when layers rotate in an asynchronous manner, although configurations with a negative oblateness gradient are more favorable. In contrast, states of global rotation (all layers move at the same rate), found by solving a linear system of ${\cal L}-1$ equations, are much more constrained. In this case, we mathematically demonstrate that confocal and coelliptical configurations are not permitted. Approximate formula for small ellipticities are derived. These results reinforce and prolongate known results and classical theorems restricted to small elliptiticities. Comparisons with the numerical solutions computed from the Self-Consistent-Field method are successful.

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New exact solutions of magnetic ellipsoidal figures of equilibrium

Cited by 4 publications

References 19 publications

Nested spheroidal figures of equilibrium II. Generalization to L layers

Nested spheroidal figures of equilibrium II. Generalization to L layers

Existence of slowly rotating bipolytropes with prolate cores

Nested spheroidal figures of equilibrium – II. Generalization to layers

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