User:Tohline/Apps/MaclaurinSpheroidSequence

Maclaurin Spheroid Sequence

Detailed Force Balance Conditions

Figure 1

Solid black curve also may be found as: Fig. 5 (p. 79) of [EFE]; Fig. 7.2 (p. 173) of [ST83]

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>

Figure 1 shows how the square of the angular velocity varies with eccentricity along the Maclaurin spheroid sequence; given the chosen normalization unit, <math>~\pi G\rho</math>, it is understood that the density of the configuration is held fixed as the eccentricity is varied. The small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:

<math>~\biggl[ e, \frac{\omega_0^2}{\pi G \rho} \biggr]</math>	<math>~\equiv</math>	<math>~\biggl[ 0.92995, 0.449331 \biggr] \, .</math>
[EFE], §32, p. 80, Eqs. (9) & (10)

Alternate Sequence Diagrams

Figure 2

Solid black curve also may be found as: Fig. 6 (p. 79) of [EFE]; Fig. 7.3 (p. 174) of [ST83]

The total angular momentum of each uniformly rotating Maclaurin spheroid is given by the expression,

<math>~L</math>

<math>~=</math>

<math>~I \omega_0 \, ,</math>

where, the moment of inertia <math>~(I)</math> and the total mass <math>~(M)</math> of a uniform-density spheroid are, respectively,

<math>~I</math>

<math>~=</math>

<math>~\biggl(\frac{2}{5}\biggr) M a^2 \, ,</math>

and,

<math>~M</math>

<math>~=</math>

<math>~\biggl( \frac{4\pi}{3} \biggr) \rho a^2c \, .</math>

Hence, we have,

<math>~L^2</math>	<math>~=</math>	<math>~ \frac{2^2 M^2 a^4}{5^2} \biggl[ A_1 - A_3 (1-e^2) \biggr] 2\pi G \biggl[ \frac{3}{2^2\pi} \cdot \frac{M}{a^2c} \biggr]</math>
	<math>~=</math>	<math>~ \frac{6GM^3 {\bar{a}}}{5^2} \biggl[ A_1 - A_3 (1-e^2) \biggr]\biggl(\frac{a}{c}\biggr)^{4/3} </math>
<math>~\Rightarrow ~~~ \frac{L}{(GM^3\bar{a})^{1 / 2}}</math>	<math>~=</math>	<math>~ \frac{6^{1 / 2}}{5} \biggl[ A_1 - A_3 (1-e^2) \biggr]^{1 / 2}(1 - e^2)^{-1 / 3} \, ,</math>
[EFE], §32, p. 78, Eq. (7) [T78], §4.5, p. 86, Eq. (54)
where,         <math>~\bar{a} \equiv (a^2 c)^{1 / 3} \, .</math>

Figure 2 shows how the system's angular momentum varies with eccentricity along the Maclaurin spheroid sequence; given the chosen normalization unit, <math>~(GM^3\bar{a})^{1 / 2}</math>, it is understood that the mass and the volume — hence, also the density — of the configuration are held fixed as the eccentricity is varied. Strictly speaking, along this sequence the angular momentum asymptotically approaches infinity as <math>~e \rightarrow 1</math>; by limiting the ordinate to a maximum value of 1.2, the plot masks this asymptotic behavior. The small solid-green square marker identifies the location along this sequence where the system with the maximum angular velocity resides (see Figure 1); this system is not associated with a turning point along this angular-momentum versus eccentricity sequence.

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