User:Tohline/SSC/Stability/BiPolytrope0 0CompareApproaches

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>:

• Using a detailed force-balance analysis to develop an analytic description of their equilibrium structure
• Using a free-energy analysis to analytically identify the properties of equilibrium structures; see also, an explicit, analytic evaluation of the statement of Virial Equilibrium
• Developing the Linear Adiabatic Wave Equation (LAWE) as it applies separately to the core and to the envelope of zero-zero bipolytropic configurations
• Identifying a method to analytically solve the matching LAWEs for a certain subset of configurations
• A summary of this solution technique, along with the first illustrative analytic specification of an eigenvector
• The derivation of analytically specifiable eigenvectors having a variety of mode quantum numbers
• A free-energy analysis of the global stability of zero-zero bipolytropes

Building on these separate discussions, here we examine what might be learned from a comparison of the two traditional approaches to stability analysis, namely:  (1) solutions of the LAWE, and (2) a free-energy analysis.

Key Attributes of Equilibrium Configurations

Aside from specifying its radius, <math>~R</math>, and total mass, <math>~M_\mathrm{tot}</math>, there are three particularly interesting dimensionless parameters that characterize the internal structure of a bipolytrope having <math>~(n_c,n_e) = (0,0)</math>. They are, the radial location of the core/envelope interface,

<math>~q \equiv \frac{r_i}{R} \, ;</math>

the ratio of the density of the envelope material to the density of the core, <math>~\rho_e/\rho_c</math>; and the fraction of the total mass that is contained in the core,

<math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} \, .</math>

Identifying a unique bipolytropic configuration requires the specification of two of these three dimensionless parameters; the third parameter is, then, necessarily determined via the algebraic relation,

<math>~\frac{\rho_e}{\rho_c} </math>

<math>=</math>

<math>~\frac{q^3(1-\nu)}{\nu(1-q^3)} \, .</math>

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© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation