Shape of a spinning body of self-gravitating fluid
A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for the shape of Earth in 1742. In fact the figure of the Earth is far less oblate than Maclaurin's formula suggests, since the Earth is not homogeneous, but has a dense iron core. The Maclaurin spheroid is considered to be the simplest model of rotating ellipsoidal figures in hydrostatic equilibrium since it assumes uniform density.
Maclaurin formula.
For a spheroid with equatorial semi-major axis formula_1 and polar semi-minor axis formula_2, the angular velocity formula_3 about formula_2 is given by Maclaurin's formula
formula_5
where formula_6 is the eccentricity of meridional cross-sections of the spheroid, formula_7 is the density and formula_8 is the gravitational constant. The formula predicts two possible equilibrium figures, one which approaches a sphere (formula_9) when formula_10 and the other which approaches a very flattened spheroid (formula_11) when formula_10. The maximum angular velocity occurs at eccentricity formula_13 and its value is formula_14, so that above this speed, no equilibrium figures exist. The angular momentum formula_15 is
formula_16
where formula_17 is the mass of the spheroid and formula_18 is the "mean radius", the radius of a sphere of the same volume as the spheroid.
Stability.
For a Maclaurin spheroid of eccentricity greater than 0.812670, a Jacobi ellipsoid of the same angular momentum has lower total energy. If such a spheroid is composed of a viscous fluid (or in the presence of gravitational radiation reaction), and if it suffers a perturbation which breaks its rotational symmetry, then it will gradually elongate into the Jacobi ellipsoidal form, while dissipating its excess energy as heat (or gravitational waves). This is termed "secular instability"; see Roberts–Stewartson instability and Chandrasekhar–Friedman–Schutz instability. However, for a similar spheroid composed of an inviscid fluid (or in the absence of radiation reaction), the perturbation will merely result in an undamped oscillation. This is described as "dynamic" (or "ordinary") "stability".
A Maclaurin spheroid of eccentricity greater than 0.952887 is dynamically unstable. Even if it is composed of an inviscid fluid and has no means of losing energy, a suitable perturbation will grow (at least initially) exponentially. Dynamic instability implies secular instability (and secular stability implies dynamic stability).
References.
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