TY - JOUR

T1 - Shapes and gravitational fields of rotating two-layer Maclaurin ellipsoids

T2 - Application to planets and satellites

AU - Schubert, Gerald

AU - Anderson, John

AU - Zhang, Keke

AU - Kong, D.

AU - Helled, Ravit

N1 - Funding Information:
GS and RH acknowledge support from NSF 0909206. KZ is supported by UK NERC and Leverhulme Trust grants.

PY - 2011/8

Y1 - 2011/8

N2 - The exact solution for the shape and gravitational field of a rotating two-layer Maclaurin ellipsoid of revolution is compared with predictions of the theory of figures up to third order in the small rotational parameter of the theory of figures. An explicit formula is derived for the external gravitational coefficient J2 of the exact solution. A new approach to the evaluation of the theory of figures based on numerical integration of ordinary differential equations is presented. The classical Radau-Darwin formula is found not to be valid for the rotational parameter ε{lunate}2=Ω2/(2πGρ2)≥0.17 since the formula then predicts a surface eccentricity that is smaller than the eccentricity of the core-envelope boundary. Interface eccentricity must be smaller than surface eccentricity. In the formula for ε{lunate}2, Ω is the angular velocity of the two-layer body, ρ2 is the density of the outer layer, and G is the gravitational constant. For an envelope density of 3000kgm-3 the failure of the Radau-Darwin formula corresponds to a rotation period of about 3h. Application of the exact solution and the theory of figures is made to models of Earth, Mars, Uranus, and Neptune. The two-layer model with constant densities in the layers can provide realistic approximations to terrestrial planets and icy outer planet satellites. The two-layer model needs to be generalized to allow for a continuous envelope (outer layer) radial density profile in order to realistically model a gas or ice giant planet.

AB - The exact solution for the shape and gravitational field of a rotating two-layer Maclaurin ellipsoid of revolution is compared with predictions of the theory of figures up to third order in the small rotational parameter of the theory of figures. An explicit formula is derived for the external gravitational coefficient J2 of the exact solution. A new approach to the evaluation of the theory of figures based on numerical integration of ordinary differential equations is presented. The classical Radau-Darwin formula is found not to be valid for the rotational parameter ε{lunate}2=Ω2/(2πGρ2)≥0.17 since the formula then predicts a surface eccentricity that is smaller than the eccentricity of the core-envelope boundary. Interface eccentricity must be smaller than surface eccentricity. In the formula for ε{lunate}2, Ω is the angular velocity of the two-layer body, ρ2 is the density of the outer layer, and G is the gravitational constant. For an envelope density of 3000kgm-3 the failure of the Radau-Darwin formula corresponds to a rotation period of about 3h. Application of the exact solution and the theory of figures is made to models of Earth, Mars, Uranus, and Neptune. The two-layer model with constant densities in the layers can provide realistic approximations to terrestrial planets and icy outer planet satellites. The two-layer model needs to be generalized to allow for a continuous envelope (outer layer) radial density profile in order to realistically model a gas or ice giant planet.

KW - Maclaurin spheroid

KW - Planetary interiors

KW - Planetary shape

KW - Rotational flattening of planets

KW - Theory of figures

UR - http://www.scopus.com/inward/record.url?scp=80053197302&partnerID=8YFLogxK

U2 - 10.1016/j.pepi.2011.05.014

DO - 10.1016/j.pepi.2011.05.014

M3 - מאמר

AN - SCOPUS:80053197302

VL - 187

SP - 364

EP - 379

JO - Physics of the Earth and Planetary Interiors

JF - Physics of the Earth and Planetary Interiors

SN - 0031-9201

IS - 3-4

ER -