A Maclaurin spheroid is an oblate
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the
Scottish
Scottish usually refers to something of, from, or related to Scotland, including:
*Scottish Gaelic, a Celtic Goidelic language of the Indo-European language family native to Scotland
*Scottish English
*Scottish national identity, the Scottish ide ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
, who formulated it for the shape of
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
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
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
since it assumes uniform density.
Maclaurin formula
![Maclaurin spheroid](https://upload.wikimedia.org/wikipedia/commons/8/87/Maclaurin_spheroid.svg)
For a
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
with equatorial semi-major axis
and polar semi-minor axis
, the angular velocity
about
is given by Maclaurin's formula
[Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 10. New Haven: Yale University Press, 1969.]
:
where
is the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
of meridional cross-sections of the spheroid,
is the density and
is the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
. The formula predicts two possible equilibrium figures when
, one is a sphere (
) and the other is a very 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
is
:
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
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gu ...
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).
See also
*
Jacobi ellipsoid
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gu ...
*
Spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
*
Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
References
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, title = Gravity: Newtonian, Post-Newtonian, Relativistic
, year = 2014
, publisher = ]Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, isbn = 978-1107032866
, pages = 102–104
, url = https://books.google.com/books?id=lWBzAwAAQBAJ&pg=PA103
[{{cite book
, last1 = Lyttleton
, first1 = Raymond Arthur
, author-link1 = Raymond Lyttleton
, title = The Stability Of Rotating Liquid Masses
, year = 1953
, publisher = Cambridge University Press
, isbn = 9781316529911
, url = https://archive.org/details/stabilityofrotat032172mbp
]
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