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==Key Attributes of Equilibrium Configurations==
==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,
<div align="center">
<math>~q \equiv \frac{r_i}{R} \, ;</math>
</div>
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,
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<math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} \, .</math>
</div>
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,
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<table border="0" cellpadding="5" align="center">


<tr>
  <td align="right">
<math>~\frac{\rho_e}{\rho_c} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~\frac{q^3(1-\nu)}{\nu(1-q^3)} \, .</math>
  </td>
</tr>
</table>
</div>


=Related Discussions=
=Related Discussions=

Revision as of 20:26, 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>~\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>

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