Editing Apps/MaclaurinToroid (section)

=Overview=


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<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, &hellip;  Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/|W| = 0.4574</math> &hellip; this is synonymous with saying that Maclaurin spheroids that have <math>T/|W| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations.  This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}&hellip; <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the  <math>P_4(\eta)</math> bifurcation point."</font>

<font color="darkgreen">"{{ Bardeen71 }}, however, did not examine the behavior of axisymmetric modes higher than <math>P_4.</math> {{ EH85 }} have tabulated bifurcation points along the Maclaurin spheroidal sequence for a number of axisymmetric modes higher than <math>P_4</math> and have found the point that occurs at the lowest value of <math>T/|W|</math> on the sequence to be the <math>P_6(\eta)</math> and not at the <math>P_4(\eta)</math> bifurcation point.  We suspect, therefore that when a general linear perturbation analysis is performed, the first dynamical axisymmetric (ring) mode instability will be found to set in at <math>T/|W| = 0.4512</math> and that the unstable mode will be of a <math>P_6(\eta)</math> geometric form."</font>
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&#8212; Drawn from (p. 598 of) {{ HTE87full }}
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Figure 10 extracted from p. 602 of<br />{{ HTE87figure }}
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[[File:HTE78Fig10.png|700px|HTE78Fig10]]
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NOTE:
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  <li>In constructing a variety of different ''incompressible'' equilibrium model sequences, {{ HTE87 }} adopted a specific angular momentum distribution given by the expression,
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<math>\dot\varphi\varpi^2</math>
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<math>=</math>
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<math>
(1 + q)\biggl(\frac{L}{M} \biggr)\biggl\{ 1 - [1 - m(\varpi) ]^{1 / q} \biggr\} \, ,
</math>
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{{ HTE87 }}, Eq. (8)
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When <math>q = 1.5</math>, this matches the distribution found in Maclaurin spheroids and in models along the so-called Maclaurin Toroid sequence &#8212; the curve labeled "1.5" in Fig. 10 of {{ HTE87hereafter }}.
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By combining Eqs. (4) and (7) in {{ HTE87 }}, we see that their Fig. 10 ordinate parameter, <math>F</math>, is related to the parameter, <math>L_*</math>, adopted a decade earlier by {{ MPT77 }} &#8212; and heavily used below &#8212; via the expression, 
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<math>F</math>
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<math>=</math>
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<math>
\pi \biggl(\frac{5^3}{6^2}\biggr)^2 \biggl[\biggl(\frac{5}{2}\biggr)^2 \biggl(\frac{L_*^2}{3}\biggr) \biggr]^{-3}
=
\biggl(\frac{4\pi}{3}\biggr)L_*^{-6} \, .
</math>
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