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Jacobi Ellipsoid
ellipsoid A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating, fluid body of uniform density rotates with a constant angular velocity.
The equatorial ( a, b) and polar ( c) semi-principal axes of a Jacobi ellipsoid and Maclaurin spheroid, as a function of normalized angular momentum, subject to abc= 1 (i.e. for constant volume of 4π/3).The broken lines are for the Maclaurin spheroid in the range where it has dynamic but not secular stability – it will relax into the Jacobi ellipsoid provided it can dissipate energy by virtue of a viscous constituent fluid.For an ellipsoid with equatorial semi-principal axes a,b{\displaystyle a,\ b} and polar semi-principal axis c{\displaystyle c}, the angular velocity Ω{\displaystyle \Omega } about c{\displaystyle c} is given by That is, each particle of the fluid of the Dedekind ellipsoid describes a similar elliptical circuit in the same period in which the Jacobi spheroid performs one rotation. In the special case of a=b{\displaystyle a=b}, the Jacobi and Dedekind ellipsoids (and the Maclaurin spheroid) become one and the same; bodily rotation and circular flow amount to the same thing.
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