# Maclaurin spheroid

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.[1] In fact the figure of the Earth is far less oblate than this, 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 equilibrium since it assumes uniform density.

## Maclaurin formula

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 when ${\displaystyle \Omega \rightarrow 0}$, one is a sphere (${\displaystyle e\rightarrow 0}$) and the other is a very flattened spheroid (${\displaystyle e\rightarrow 1}$). 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.

## Stability

For a Maclaurin spheroid of eccentricity greater than 0.812670,[3] a Jacobi ellipsoid of the same angular momentum has lower total energy. If such a spheroid is composed of a viscous fluid, 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. This is termed secular instability. However, for a similar spheroid composed of an inviscid fluid, 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[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]