Editing SSC/Stability/BiPolytropes (section)

==Planned Approach(es)==

In an effort to answer the '''<font color="red">principal question</font>''' posed above, we have pursued each stability-analysis approach described in the introductory section of {{ B-KB74full }}.

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<font color="darkgreen"><span id="BKB74pt1">"Three different approaches are used in the study of the hydrodynamical stability of stars</span> and other gravitating objects &hellip;" &nbsp; 
<ul><li>"The first approach is based on the use of the equations of small oscillations.  In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations.  The solutions consist of a set of eigenfrequencies and eigenfunctions."</font> </li>

<li>Second, one can derive <font color="darkgreen">"a variational principle from the equations of small oscillations &hellip;</font>  <!-- This principle replaces the straightforward solution of these equations:</font>  In the context of rotating Newtonian systems, see, for example, [http://adsabs.harvard.edu/abs/1964ApJ...140.1045C Clement (1964)], [http://adsabs.harvard.edu/abs/1968ApJ...152..267C Chandrasekhar &amp; Lebovitz (1968)], [http://adsabs.harvard.edu/abs/1967MNRAS.136..293L Lynden-Bell and Ostriker (1967)], or [http://adsabs.harvard.edu/abs/1972ApJS...24..319S Schutz (1972)]. -->  <font color="darkgreen">With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations.  In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated &hellip;"</font></li>

<li>The third approach is what we usually refer to as a free-energy &#8212; and associated virial theorem &#8212; analysis.  <font color="darkgreen">"When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient.  The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability."</font> </li>
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<span id="BKB74pt2"><font color="darkgreen">"If one wants</span> to know from a stability analysis the answer to only one question &#8212; whether the model is stable or not &#8212; then the most straightforward procedure is to use the third, static method &hellip;  For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation."</font>

<font color="darkgreen">"Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point."</font>
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— Drawn from pp. 391 - 392 of {{ B-KB74 }} 
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