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(→Structure:: Begin adding summary text for spherical "structure") 
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===Structure:===  ===Structure:===  
  Here we show how the set of principal governing equations (PGEs) may be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars  +  Here we show how the set of principal governing equations (PGEs) may be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars. After supplementing the PGEs by specifying an equation of state of the fluid, the system of equations is usually solved by employing one of three techniques to obtain a "detailed forcebalanced" model that provides the radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration — given its mass, <math>~M</math>, and central pressure, <math>~P_c</math>, for example — as well as details regarding the internal radial profiles of the massdensity and fluid pressure. As our various discussions illustrate, simply varying the powerlaw index, <math>~n</math>, in a ''polytropic'' equation of state, 
<div align="center">  <div align="center">  
<table border="0" cellpadding="5" align="center">  <table border="0" cellpadding="5" align="center">  
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gives rise to equilibrium configurations that have a wide variety of internal structural profiles.  gives rise to equilibrium configurations that have a wide variety of internal structural profiles.  
  If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a  +  If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniformdensity structure then identifying local extrema in the system's global free energy. In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such freeenergy extrema is often referred to as the ''scalar virial theorem''. Specifically, for ''isolated'' systems in virial equilibrium, the following relation between configuration parameters holds: 
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</table>  </table>  
</div>  </div>  
  where  +  where all three of the dimensionless ''structural form factors'', <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math>, are generically ''of order unity'' for detailed forcebalanced models having a wide range of internal structures and are all three exactly equal to unity under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout. Alternatively, if the polytropic constant, <math>~K</math> (which specifies the specific entropy of fluid elements throughout the system), instead of the central pressure, is held fixed while searching for extrema in the freeenergy, the virial equilibrium relation is, 
+  <div align="center">  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\biggl( \frac{G}{K} \biggr)^n M^{n1} R_\mathrm{eq}^{3n}</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~\frac{3}{4\pi} \biggl(\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr) \frac{1}{\mathfrak{f}_M} \, .  
+  \, ,</math>  
+  </td>  
+  </tr>  
+  </table>  
+  </div>  
+  
+  
+  If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, embedded in a hot, tenuous medium that exerts on the system a confining external pressure, <math>~P_e</math>, the configuration's equilibrium parameters will be related via the expression,  
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<td align="left">  <td align="left">  
<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A  \frac{P_e}{P_c} \biggr]  <math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A  \frac{P_e}{P_c} \biggr]  
  \, .</math>  +  \, ;</math> 
+  </td>  
+  </tr>  
+  </table>  
+  </div>  
+  or, fixing <math>~K</math> instead of <math>~P_c</math>,  
+  <div align="center">  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~P_e </math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}  
+   \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>  
</td>  </td>  
</tr>  </tr> 
Revision as of 15:22, 27 February 2015
Preface from the original version of this HyperText Book (H_Book):
November 18, 1994
Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, shortperiod binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations — most of which also are heavily utilized in studies of continuum flows in terrestrial environments — are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy. Although relatively simple in form, they prove to be very rich in nature... <more>
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Pictorial Table of Contents
Context
 Principal Governing Equations
 Continuity Equation
 Euler Equation
 1^{st} Law of Thermodynamics
 Poisson Equation
Applications
Spherically Symmetric Configurations
Introduction (Alternate Introduction)
Structure:
Here we show how the set of principal governing equations (PGEs) may be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars. After supplementing the PGEs by specifying an equation of state of the fluid, the system of equations is usually solved by employing one of three techniques to obtain a "detailed forcebalanced" model that provides the radius, , of the equilibrium configuration — given its mass, , and central pressure, , for example — as well as details regarding the internal radial profiles of the massdensity and fluid pressure. As our various discussions illustrate, simply varying the powerlaw index, , in a polytropic equation of state,



gives rise to equilibrium configurations that have a wide variety of internal structural profiles.
If one is not particularly concerned about details regarding the distribution of matter within the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniformdensity structure then identifying local extrema in the system's global free energy. In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such freeenergy extrema is often referred to as the scalar virial theorem. Specifically, for isolated systems in virial equilibrium, the following relation between configuration parameters holds:



where all three of the dimensionless structural form factors, , , and , are generically of order unity for detailed forcebalanced models having a wide range of internal structures and are all three exactly equal to unity under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout. Alternatively, if the polytropic constant, (which specifies the specific entropy of fluid elements throughout the system), instead of the central pressure, is held fixed while searching for extrema in the freeenergy, the virial equilibrium relation is,



If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, embedded in a hot, tenuous medium that exerts on the system a confining external pressure, , the configuration's equilibrium parameters will be related via the expression,



or, fixing instead of ,



We have introduced these structural form factors into our discussion of the free energy of spherically symmetric, selfgravitating configurations because
Solution Strategies: 
Detailed ForceBalance 
Virial Equilibrium 



Stability:
Solution Strategy Assuming Spherical Symmetry: 

Example Solutions: 
Dynamics:

TwoDimensional Configurations
 Introduction
Structure:
Solution Strategies 

Example Solutions:

Stability:
Dynamics:
ThreeDimensional Configurations
Structure:
Solution Strategies 
Example Solutions:

Stability:
 Lou & Bai (2011, MNRAS, 415, 925) — 3D perturbations in an isothermal selfsimilar flow
Dynamics:
Related Projects Underway
Appendices
See Also
 NIST Digital Library of Mathematical Functions; see also the related CUP Publication
© 2014  2020 by Joel E. Tohline 