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(Begin new chapter that compares stability analyses of zero-zero bipolytropes)
 
(→‎Comparing Stability Analyses of Zero-Zero Bipolytropes: Provide links to various chapters that feed into this overview discussion)
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=Comparing Stability Analyses of Zero-Zero Bipolytropes=
=Comparing Stability Analyses of Zero-Zero Bipolytropes=
This chapter is an extension of two accompanying discussions:  [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|The original ''discovery'' and detailed derivation]]; and the [[User:Tohline/SSC/Stability/BiPolytrope0_0#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|more readable, summary]].
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In our [[User:Tohline/SSC/Stability/BiPolytrope0_0#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|accompanying summary]], we have demonstrated how analytically specified eigenvectors can be constructed for the mode labeled, <math>~(\ell, j) = (2,1)</math>.  This was done by specifying <math>~\gamma_e</math>, then solving a quartic equation for <math>~q</math>.  Shortly after completing this summary chapter, we noticed that an alternate approach may be to specify <math>~q</math>, then solve for <math>~\gamma_e</math>; and this path may be simpler because it may only involve solution of a quadratic equation. (Actually, we later have realized that the relevant equation is cubic, rather than quadratic.  This is nevertheless simpler than the quartic equation.)  If this proves to be the case, then it may also be possible to analytically construct eigenvectors of additional modes.  Let's see.


In separate chapters we have discussed the following interrelated aspects of Bipolytropes that have <math>~(n_c,n_e) = (0,0)</math>:
<ul>
  <li>Using a [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|detailed force-balance analysis to develop an analytic description of their equilibrium structure]]</li>
  <li>Using a [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Free_Energy|free-energy analysis]] to analytically identify the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Equilibrium_Condition|properties of equilibrium structures]]; see also, an explicit, analytic evaluation of the statement of ''[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Virial_Equilibrium|Virial Equilibrium]]''</li>
  <li>Developing the [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|Linear Adiabatic Wave Equation]] (LAWE) as it applies separately to the [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Core|core]] and to the [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Envelope|envelope]] of zero-zero bipolytropic configurations</li>
  <li>Identifying a [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Eureka_Regarding_Prasad.27s_1948_Paper|method to ''analytically'' solve the matching LAWEs]] for a certain subset of configurations</li>
    <ul>
    <li>A [[User:Tohline/SSC/Stability/BiPolytrope0_0#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|summary of this solution technique]], along with the [[User:Tohline/SSC/Stability/BiPolytrope0_0#Eigenvector|first illustrative analytic specification of an eigenvector]]</li>
    <li>The derivation of  [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes|analytically specifiable eigenvectors having a variety of mode quantum numbers]]</li>
    </ul>
  <li>A [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_Condition|free-energy analysis of the global stability]] of zero-zero bipolytropes</li>
</ul>


=Related Discussions=
=Related Discussions=

Revision as of 19:56, 6 January 2017

Comparing Stability Analyses of Zero-Zero Bipolytropes

In separate chapters we have discussed the following interrelated aspects of Bipolytropes that have <math>~(n_c,n_e) = (0,0)</math>:

Related Discussions

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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