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Maclaurin spheroid

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

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

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

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

where ${\displaystyle M}$ is the mass of the spheroid and ${\displaystyle {\bar {a}}}$ is the mean 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 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 (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