“Maclaurin spheroid”的意思、由来-开放百科全书

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 spheroid with semi-major axis

and semi-minor axis

, the angular velocity

is given by the Maclaurin's formula^[2]

where

is the eccentricity of meridional cross-sections of the spheriod,

is the density and

is the Gravitational constant. The formula predicts two possible types of equilibrium figures when

, one is a sphere (

) and the other is a flattened spheroid (

). The maximum angular velocity occurs at eccentricity

and its value is

, so that above this speed, no equilibrium figures exist. The angular momentum

where

is the mass of the spheroid and

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, 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 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).^[5]

Maclaurin formula

Stability

See also

References