Editing SSC/Stability/BiPolytropes (section)

=In Search of Marginally Unstable (n<sub>c</sub>,n<sub>e</sub>) = (5,1) Bipolytropes=

<!-- Our aim is to determine whether or not there is a relationship between (1) equilibrium models at turning points along bipolytrope sequences and (2) bipolytropic models that are marginally (dynamically) unstable toward collapse (or dynamical expansion).-->


==Overview==
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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined]]Figure 1: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
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[[File:MassVsRadiusCombined02.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
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[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
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We expect the  content of this chapter &#8212; which examines the relative stability of bipolytropes &#8212; to parallel in many ways the content of an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|accompanying chapter in which we have successfully analyzed the relative stability of pressure-truncated polytopes]].  Figure 1, shown here on the right, has been copied from [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|a closely related discussion]].  The curves show the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>.  ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence.)  On each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass.  We have shown ''analytically'' that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero<sup>&dagger;</sup> for each one of these maximum-mass models.  As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.

<sup>&dagger;</sup>In each case, the fundamental-mode oscillation frequency is precisely zero if, and only if, the adiabatic index governing expansions/contractions is related to the underlying ''structural'' polytropic index via the relation, <math>\gamma_g = (n + 1)/n</math>, and if a constant surface-pressure boundary condition is imposed.

<font color="red">'''Key Realization:'''</font>  ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----


In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|accompanying chapter]], we have used purely analytic techniques to construct equilibrium sequences of spherically symmetric bipolytropes that have, <math>(n_c,n_e) = (5,1)</math>.   For a given choice of <math>~\mu_e/\mu_c</math> &#8212; the ratio of the mean-molecular weight of envelope material to the mean-molecular weight of material in the core &#8212; a physically relevant sequence of models can be constructed by steadily increasing the value of the dimensionless radius at the core/envelope interface, <math>\xi_i</math>, from zero to infinity.  Figure 2, whose content is essentially the same as [[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences|Figure 1 of this separate chapter]], shows how the fractional core mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, varies with the fractional core radius, <math>q \equiv r_\mathrm{core}/R</math>, along sequences having seven different values of <math>\mu_e/\mu_c</math>, as labeled: 1 (black), &frac12; (dark blue), 0.345 (brown), &#8531; (dark green), 0.316943 (purple), 0.309 (orange),  and &frac14; (light blue).  

When modeling bipolytropes, the default expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size &#8212; both the radius and the mass &#8212; of the core. This expectation is realized along the Figure 2 sequences that have the largest mean-molecular weight ratios:  <math>\mu_e/\mu_c</math> = 1 and &frac12;.  But the behavior is different along the other five illustrated sequences.  For sufficiently large <math>\xi_i</math>, the relative radius of the core begins to decrease; along each sequence, a solid purple circular marker identifies the location of this ''turning point'' in radius.  Furthermore, along sequences for which <math>\mu_e/\mu_c < \tfrac{1}{3}</math>, eventually the fractional mass of the core reaches a maximum and, thereafter, decreases even as the value of <math>\xi_i</math> continues to increase.  In Figure 2, a solid green circular marker identifies the location of this ''maximum mass turning point'' along each of these sequences; the analytically determined values of <math>\xi_i, q </math> and <math>\nu</math> that are associated with each of these ''turning points'' are provided in the table adjacent to Figure 2.  (Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence|yet another accompanying chapter]].)  

<span id="PrincipalQ">
<font color="red">'''The principal question is:'''</font>  ''Along bipolytropic sequences, are maximum-mass models associated with the onset of dynamical instabilities?''</span>


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'''Figure 2:  Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
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<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
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[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
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<math>\frac{\mu_e}{\mu_c}</math>
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<math>\xi_i</math>
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<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
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<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
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<math>\frac{1}{3}</math>
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<math>\infty</math> </td>
  <td align="center">0.0 </td>
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<math>\frac{2}{\pi}</math> </td>
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0.33
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24.00496  </td>
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0.038378833  </td>
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0.52024552  </td>
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0.316943
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10.744571  </td>
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0.068652714  </td>
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0.382383875  </td>
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0.31
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9.014959766  </td>
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0.0755022550  </td>
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0.3372170064  </td>
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0.3090
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8.8301772  </td>
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0.076265588  </td>
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0.331475715  </td>
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<math>\frac{1}{4}</math>
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4.9379256  </td>
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0.084824137  </td>
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0.139370157  </td>
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<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
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==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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==Supplemental Chapters==
<ol>
  <li>Contains [[SSC/Stability/InstabilityOnsetOverview#Displacement_Functions_Summary|Displacement Functions Summary]]</li>
  <li>[[SSC/Stability/BiPolytropes/PlannedApproach|Earlier Planned Approach]]</li>
  <li>[[SSC/Stability/BiPolytropes/HeadScratching|Headscratching]]</li>
  <li>[[SSC/Stability/BiPolytropes/SuccinctDiscussion|Succinct Discussions]]</li>
  <li>[[SSC/Stability/BiPolytropes/51Models|51Models]]</li>
</ol>