View source for Astronomy:Maclaurin spheroid

{{short description| Shape of a spinning body of self-gravitating fluid}}
A '''Maclaurin spheroid''' is an oblate [[Spheroid|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 [[Biography:Mathematician|mathematician]] Colin Maclaurin, who formulated it for the shape of [[Astronomy:Earth|Earth]] in 1742.<ref>Maclaurin, Colin. A Treatise of Fluxions: In Two Books. 1. Vol. 1. Ruddimans, 1742.</ref>  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 [[Astronomy:Hydrostatic equilibrium|hydrostatic equilibrium]] since it assumes uniform density.

==Maclaurin formula==
[[File:Maclaurin spheroid.svg|thumb|Angular velocity for Maclaurin spheroid as a function of eccentricity]]
For a spheroid with equatorial semi-major axis <math>a</math> and polar semi-minor axis <math>c</math>, the angular velocity <math>\Omega</math> about <math>c</math> is given by Maclaurin's formula<ref>Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 10. New Haven: Yale University Press, 1969.</ref>

:<math>\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}, </math>

where <math>e</math> is the [[Eccentricity (mathematics)|eccentricity]] of meridional cross-sections of the spheroid, <math>\rho</math> is the density and <math>G</math> is the [[Physics:Gravitational constant|gravitational constant]]. The formula predicts two possible equilibrium figures when <math>\Omega\rightarrow 0</math>, one is a sphere (<math>e\rightarrow 0</math>) and the other is a very flattened spheroid (<math>e\rightarrow 1</math>). The maximum angular velocity occurs at eccentricity <math>e=0.92996</math> and its value is <math>\Omega^2/(\pi G\rho)=0.449331</math>, so that above this speed, no equilibrium figures exist. The angular momentum <math>L</math> is

:<math>\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^2c)^{1/3}</math>

where <math>M</math> is the mass of the spheroid and <math>\bar{a}</math> 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,<ref name="PoissonAndWill"/> a [[Astronomy:Jacobi ellipsoid|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<ref name="PoissonAndWill"/> 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).<ref name="Lyttleton1953"/>

==See also==
* [[Astronomy:Jacobi ellipsoid|Jacobi ellipsoid]]
* [[Spheroid]]
* [[Ellipsoid]]

==References==
{{Reflist|refs=
  <ref name="PoissonAndWill">{{cite book
    |last1     = Poisson
    |first1    = Eric
    |last2     = Will
    |first2    = Clifford
    |title     = Gravity: Newtonian, Post-Newtonian, Relativistic
    |year      = 2014
    |publisher = {{wipe|Cambridge University Press}}
    |isbn      = 978-1107032866
    |pages     = 102–104
    |url       = https://books.google.com/books?id=lWBzAwAAQBAJ&pg=PA103
  }}</ref>
  <ref name="Lyttleton1953">{{cite book
    |last1        = Lyttleton
    |first1       = Raymond Arthur
    |title        = The Stability Of Rotating Liquid Masses
    |year         = 1953
    |publisher    = Cambridge University Press
    |isbn         = 9781316529911
    |url          = https://archive.org/details/stabilityofrotat032172mbp
  }}</ref>
}}

[[Category:Quadrics]]
[[Category:Astrophysics]]
[[Category:Fluid dynamics]]
[[Category:Effects of gravitation]]

{{Sourceattribution|Maclaurin spheroid}}