AMaclaurin spheroid is an oblatespheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after theScottishmathematicianColin Maclaurin, who formulated it for the shape ofEarth in 1742.^[1] 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 inhydrostatic equilibrium since it assumes uniform density.

For aspheroid with equatorial semi-major axis${\displaystyle a}$ and polar semi-minor axis${\displaystyle c}$, the angular velocity${\displaystyle \mathrm{\Omega }}$ about${\displaystyle c}$ is given by Maclaurin's formula^[2]

${\displaystyle \frac{{\mathrm{\Omega }}^2}{\pi G\rho }=\frac{2\sqrt{1-e^2}}{e^3}(3-2e^2){\sin }^{-1}e-\frac{6}{e^2}(1-e^2),e^2=1-\frac{c^2}{a^2},}$

where${\displaystyle e}$ is theeccentricity of meridional cross-sections of the spheroid,${\displaystyle \rho }$ is the density and${\displaystyle G}$ is thegravitational constant. The formula predicts two possible equilibrium figures, one which approaches a sphere (${\displaystyle e\to 0}$) when${\displaystyle \mathrm{\Omega }\to 0}$ and the other which approaches a very flattened spheroid (${\displaystyle e\to 1}$) when${\displaystyle \mathrm{\Omega }\to 0}$. The maximum angular velocity occurs at eccentricity${\displaystyle e=0.92996}$ and its value is${\displaystyle {\mathrm{\Omega }}^2/(\pi G\rho )=0.449331}$, so that above this speed, no equilibrium figures exist. The angular momentum${\displaystyle L}$ is

${\displaystyle \frac{L}{\sqrt{G{M}^3\overline{a}}}=\frac{\sqrt{3}}{5}{\left(\frac{a}{\overline{a}}\right)}^2\sqrt{\frac{{\mathrm{\Omega }}^2}{\pi G\rho }},\overline{a}=(a^{2}c{)}^{1/3}}$

where${\displaystyle M}$ is the mass of the spheroid and${\displaystyle \overline{a}}$ is themean radius, the radius of a sphere of the same volume as the spheroid.

For a Maclaurin spheroid of eccentricity greater than 0.812670,^[3] aJacobi 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 (orgravitational waves). This is termedsecular instability; seeRoberts–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 asdynamic (orordinary)stability.

A Maclaurin spheroid of eccentricity greater than 0.952887^[3] 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).^[4]

• Jacobi ellipsoid
• Spheroid
• Ellipsoid

• Ellipsoids
• Astrophysics
• Fluid dynamics
• Effects of gravity

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