Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

====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
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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].