User:Tohline/Apps/MaclaurinSpheroidSequence

Maclaurin Spheroid Sequence

Detailed Force Balance Conditions

The essential structural elements of each Maclaurin spheroid model are uniquely determined once we specify the system's axis ratio, <math>~c/a</math>, or by the system's meridional-plane eccentricity, <math>~e</math>, where

<math>~e</math>

<math>~\equiv</math>

<math>~\biggl[1 - \biggl(\frac{c}{a}\biggr)^2\biggr]^{1 / 2} \, ,</math>

which varies from e = 0 (spherical structure) to e = 1 (infinitesimally thin disk). According to our accompanying derivation, for a given choice of <math>~e</math>, the square of the system's equilibrium angular velocity is,

<math> ~ \omega_0^2 </math>

<math> ~= </math>

<math> 2\pi G \rho \biggl[ A_1 - A_3 (1-e^2) \biggr] \, , </math>

where,

<math> ~A_1 </math>	<math> ~= </math>	<math> \frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr](1-e^2)^{1/2} \, , </math>
<math> ~A_3 </math>	<math> ~= </math>	<math> \frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} \, . </math>

See Also

• Properties of Maclaurin Spheroids
• Excerpts from Maclaurin's (1742) A Treatise of Fluxions
• Properties of Homogeneous Ellipsoids

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation