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It is a virial expression specifically of this form <math>~(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbowcolored freeenergy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, displayed above. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>~P_e</math> is greater than some limiting value, <math>~P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>~P_e = P_\mathrm{crit}</math>, is marked by a red dot on the displayed freeenergy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[User:Tohline/SSC/Structure/BonnorEbert#PressureBounded_Isothermal_SphereBonnorEbert spheres]].  It is a virial expression specifically of this form <math>~(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbowcolored freeenergy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, displayed above. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>~P_e</math> is greater than some limiting value, <math>~P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>~P_e = P_\mathrm{crit}</math>, is marked by a red dot on the displayed freeenergy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[User:Tohline/SSC/Structure/BonnorEbert#PressureBounded_Isothermal_SphereBonnorEbert spheres]].  
+  
+  After rearranging terms, for any specified values of the parameters <math>~P_e</math> and <math>~K</math>, this virial equilibrium expression can also be viewed as a massradius relation of the form,  
+  <table border="0" align="right" cellpadding="5">  
+  <tr>  
+  <td align="center" bgcolor="white">  
+  [[File:MassRadiusVirialLabeled.png200pxrightborderVirial MassRadius Relation]]  
+  </td>  
+  </tr>  
+  </table>  
+  <div align="center">  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~a R_\mathrm{eq}^4  b M^{(n+1)/n} R_\mathrm{eq}^{(n3)/n} + c M^2</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~ 0\, ,</math>  
+  </td>  
+  </tr>  
+  </table>  
+  </div>  
+  where,  
+  <div align="center">  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~a</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>~ \frac{4\pi}{3} \cdot P_e \, ,</math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~b</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>~ \biggl( \frac{3}{4\pi} \biggr)^{1/n} \cdot K \mathfrak{f}_A \mathfrak{f}_M^{(n+1)/n} \, ,</math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~c</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>~ \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2} \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
+  </div>  
+  Using this virial equilibrium relation (and assuming a = b = c = 1), the curves drawn in the figure that is displayed here, on the right, show how the equilibrium radius of an embedded, pressuretruncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with BonnorEbert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>~M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>~n > 3</math>, two equilibrium configurations having different radii can be constructed for any system having a mass, <math>~M < M_\mathrm{max}</math>.  
Revision as of 19:53, 28 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) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds. 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 fluid density 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. An illustrative, undulating freeenergy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the freeenergy surface while white dots identify equilibria that lie along a "ridge" in the freeenergy surface.
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 unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically of order unity for detailed forcebalanced models having a wide range of internal structures. Alternatively, if the specific entropy of fluid elements (set by the value of ) throughout the system, rather than the central pressure, is held fixed while searching for extrema in the freeenergy, the virial equilibrium relation for isolated polytropes 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 , the relevant virial equilibrium expression is,



It is a virial expression specifically of this form with and that identifies extrema (e.g., valleys or ridges) in the rainbowcolored freeenergy surface, , displayed above. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if is greater than some limiting value, ; the equilibrium configuration associated with the limiting condition, , is marked by a red dot on the displayed freeenergy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, BonnorEbert spheres.
After rearranging terms, for any specified values of the parameters and , this virial equilibrium expression can also be viewed as a massradius relation of the form,



where,









Using this virial equilibrium relation (and assuming a = b = c = 1), the curves drawn in the figure that is displayed here, on the right, show how the equilibrium radius of an embedded, pressuretruncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with BonnorEbert spheres, for systems having , there is a mass, , above which equilibrium configurations do not exist; and, when , two equilibrium configurations having different radii can be constructed for any system having a mass, .
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 