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  <td align="center"><b>Figure 1:</b><br />Equilibrium Sequences of Constant <math>~\rho_e/\rho_c</math></td>
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[[File:ConstDensitySequences.png|300px|Constant Density Sequences]]
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Across the two-dimensional, <math>~(q,\nu)</math> parameter space that is defined by the full range of physically viable values of <math>~q</math> and <math>~\nu</math>, namely,
Across the two-dimensional, <math>~(q,\nu)</math> parameter space that is defined by the full range of physically viable values of <math>~q</math> and <math>~\nu</math>, namely,
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<math>~\biggl[\frac{(1-q^3)}{q^3} \biggl( \frac{\rho_e}{\rho_c} \biggr) + 1\biggr]^{-1} \, .</math>
<math>~\biggl[\frac{(1-q^3)}{q^3} \biggl( \frac{\rho_e}{\rho_c} \biggr) + 1\biggr]^{-1} </math>
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&nbsp;
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<math>=</math>
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<math>~\frac{q^3}{q^3 + (1-q^3)(\rho_e/\rho_c)} </math>
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Revision as of 22:01, 6 January 2017

Comparing Stability Analyses of Zero-Zero Bipolytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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In separate chapters we have discussed the following interrelated aspects of Bipolytropes that have <math>~(n_c,n_e) = (0,0)</math>:

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>~0 \le \rho_e/\rho_c \le 1</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>

Figure 1:
Equilibrium Sequences of Constant <math>~\rho_e/\rho_c</math>

Constant Density Sequences

Across the two-dimensional, <math>~(q,\nu)</math> parameter space that is defined by the full range of physically viable values of <math>~q</math> and <math>~\nu</math>, namely,

<math>~0 \le q \le 1 \, ,</math>      and       <math>~0 \le \nu \le 1 \, ,</math>

an equilibrium model sequence can be defined by, for example, specifying that all models along the sequence have the same density jump at the interface. Drawing on the above constraint relation, each choice of <math>~\rho_e/\rho_c</math> will generate a sequence governed by the function,

<math>~\nu</math>

<math>=</math>

<math>~\biggl[\frac{(1-q^3)}{q^3} \biggl( \frac{\rho_e}{\rho_c} \biggr) + 1\biggr]^{-1} </math>

 

<math>=</math>

<math>~\frac{q^3}{q^3 + (1-q^3)(\rho_e/\rho_c)} </math>

Figure 1 displays several such equilibrium sequences across the <math>~(q,\nu)</math> plane — see also a related figure associated with our free-energy determination of stability.

Related Discussions

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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