Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

===Roche Ellipsoids===
====Jeans (1919)====
From &sect; 51 (p. 47) of  [http://adsabs.harvard.edu/abs/1919pcsd.book.....J J. H. Jeans (1919)] &#8212; ''verbatum'' text in green:  <font color="green">
The simplest problem occurs when the secondary may be treated as a rigid sphere;  this is the special problem dealt with by Roche.  As in &sect; 47 the tide-generating potential acting on the primary may be supposed to be
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<math>~\frac{M^'}{R} + \frac{M^'}{R^2} x + \frac{M^'}{R^3}(x^2 - \tfrac{1}{2}y^2 - \tfrac{1}{2}z^2) + \cdots </math>
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<font color="green">We shall for the present be content to omit all terms beyond those written down.  The correction required by the neglect of these terms will be discussed later, and will be found to be so small that the results now to be obtained are hardly affected.</font>

<font color="green">On omitting these terms, and combining the two potentials &hellip; it appears that the primary may be supposed influenced by a statical field of potential
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<math>~\frac{M^'}{R} x\biggr(1 - \frac{\omega^2 R^3}{M + M^'}\biggr) + \frac{M^'}{R^3}(x^2 - \tfrac{1}{2}y^2 - \tfrac{1}{2}z^2) 
+ \tfrac{1}{2}\omega^2(x^2 + y^2)       \, .</math>
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<font color="green">The terms in <math>~x</math> may immediately be removed by supposing <math>~\omega</math> to have the appropriate value given by
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<math>~\omega^2</math>
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<math>~=</math>
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<math>~\frac{M+M^'}{R^3}</math>
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<font color="green">and the condition for equilibrium is now seen to be that we shall have, at every point of the surface,
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<math>~V_b + \mu (x^2 - \tfrac{1}{2}y^2 - \tfrac{1}{2}z^2) + \tfrac{1}{2}\omega^2(x^2 + y^2)     </math>
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<math>~=</math>
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constant
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<font color="green">where <math>~\mu</math> … stands for <math>~M^'/R^3</math> .
</font>

====Chandrasekhar (1969)====
From p. 190 of [<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; ''verbatum'' text in green:  <font color="green">
In Roche's particular problem, the secondary is treated as a rigid sphere.  Then, over the primary, the tide-generating potential, <math>~\mathfrak{B}^'</math> can be expanded in the form
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<math>~\mathfrak{B}^'</math>
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<math>~=</math>
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<math>~ \frac{GM^'}{R} \biggl( 1 + \frac{x_1}{R} + \frac{x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2}{R^2} + \cdots \biggr) \, ;</math>
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<font color="green">and the approximation which underlies this theory is to retain, in this expansion for <math>~\mathfrak{B}^'</math>, only the terms which have been explicitly written down and ignore all the terms which are of higher order.  On this assumption, the equation of motion becomes
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<math>~\frac{du_i}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial x_i}</math>
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<math>~=</math>
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<math>~
\frac{\partial}{\partial x_i} \biggl[ \mathfrak{B} + \tfrac{1}{2}\Omega^2(x_1^2 + x_2^2) + \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2)
+ \biggl( \frac{GM^'}{R^2} - \frac{M^' R}{M+M^'} ~\Omega^2 \biggr)x_1
\biggr] + 2\Omega \epsilon_{i\ell 3} u_\ell
</math>
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<font color="green">where we have introduced the abbreviation
</font>
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<math>~\mu = \frac{GM^'}{R^3} \, .</math>
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<font color="green">So far, we have left <math>~\Omega^2</math> unspecified.  If we now let <math>~\Omega^2</math> have the "Keplerian value"
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<math>~\Omega^2 = \frac{G(M+ M^')}{R^3} = \mu \biggl(1 + \frac{M}{M^'} \biggr) \, ,</math>
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<font color="green">the "unwanted" term in <math>~x_1</math>, on the right-hand side of</font> [this equation,] <font color="green">vanishes and we are left with
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<math>~\frac{du_i}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial x_i}</math>
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<math>~=</math>
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<math>~
\frac{\partial}{\partial x_i} \biggl[ \mathfrak{B} + \tfrac{1}{2}\Omega^2(x_1^2 + x_2^2) + \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2)
\biggr] + 2\Omega \epsilon_{i\ell 3} u_\ell \, .
</math>
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<font color="green">This is the basic equation of this theory;  and Roche's problem is concerned with the equilibrium and the stability of homogeneous masses governed by</font> [this relation].


====Tassoul (1978)====
From pp. 449-450 of [<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>] &#8212; ''verbatum'' text in green:  <font color="green">
In Roche's particular problem, the secondary is treated as a rigid sphere; hence, over the primary, the tide-generating potential can be expanded in the form
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<math>~ -\frac{GM^'}{d} \biggl( 1 + \frac{x_1}{d} + \frac{x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2}{d^2} + \cdots \biggr) \, .</math>
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<font color="green">The approximation that underlies the theory is to omit all terms beyond those written down.  On this assumption, we find that, apart from its own gravitation, the primary may be supposed to be acted upon by a total field of force derived from the potential
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<math>~-\tfrac{1}{2}\Omega^2(x_1^2 + x_x^2) - \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2) - \biggl(\mu - \frac{M^'}{M+M^'} \Omega^2\biggr) dx_1 \, ,</math>
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<font color="green">where</font>
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<math>~\mu</math>
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<math>~=</math>
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<math>~\frac{GM^'}{d^3} \, .</math>
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<font color="green">Further letting <math>~\Omega^2</math> have its "Keplerian value"</font>
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<math>~\Omega^2</math>
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<math>~=</math>
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<math>~\frac{G(M+M^')}{d^3} \, ,</math>
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<font color="green">we can thus write the conditions of relative equilibrium for the primary in the form</font>
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<math>~\frac{1}{\rho} \nabla p</math>
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<math>~=</math>
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<math>~
-\nabla [ V
-\tfrac{1}{2}\Omega^2(x_1^2 + x_x^2) - \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2) ] \, ,
</math>
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<font color="green">where <math>~V</math> is the self-gravitating potential of the primary.</font>