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===Structure:===
===Structure:===
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Here we show how the set of principal governing equations (PGEs) may be solved to determine the equilibrium structure of spherically symmetric fluid configurations &#8212; such as individual, nonrotating stars or interstellar gas clouds having a rather idealized geometry.  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 force-balanced" model that provides the radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration &#8212; given its mass, <math>~M</math>, and central pressure, <math>~P_c</math>, for example &#8212; as well as details regarding the internal radial profiles of the mass-density and fluid pressure.  As our various discussions illustrate, even a simple ''polytropic'' equation of state, that is,
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<math>~P</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~K \rho^{(n+1)/n} \, ,</math>
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  </td>
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</tr>
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</table>
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gives rise to equilibrium configurations that have a wide variety of internal structural profiles.
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If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a reasonably good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying extrema in the system's global free energy, that is, by identifying properties that satisfy the ''scalar virial theorem''.  Specifically, for ''isolated'' systems in virial equilibrium, the following relation between configuration parameters holds:
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W} 
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\, ,</math>
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  </td>
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</table>
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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>, equal unity under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, while all three are ''of order'' unity otherwise.  If the configuration &#8212; such as an interstellar gas cloud &#8212; is not isolated but is, instead, embedded in a hot, tenuous external medium that exerts an external pressure, <math>~P_e</math>, the configuration's equilibrium parameters will be related via the expression,
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<div align="center">
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<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]
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\, .</math>
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  </td>
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</tr>
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</table>
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We have introduced these structural form factors into our discussion of the free energy of spherically symmetric, self-gravitating configurations because
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<table border="1" width="100%" cellspacing="2" cellpadding="8">
<table border="1" width="100%" cellspacing="2" cellpadding="8">

Revision as of 21:11, 26 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, short-period 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>

Whitworth's (1981) Isothermal Free-Energy Surface
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Pictorial Table of Contents

Context

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 or interstellar gas clouds having a rather idealized geometry. 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 force-balanced" model that provides the radius, ~R_\mathrm{eq}, of the equilibrium configuration — given its mass, ~M, and central pressure, ~P_c, for example — as well as details regarding the internal radial profiles of the mass-density and fluid pressure. As our various discussions illustrate, even a simple polytropic equation of state, that is,

~P

~=

~K \rho^{(n+1)/n} \, ,

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 reasonably good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying extrema in the system's global free energy, that is, by identifying properties that satisfy the scalar virial theorem. Specifically, for isolated systems in virial equilibrium, the following relation between configuration parameters holds:

~\frac{GM^2}{P_c R_\mathrm{eq}^4}

~=

~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}  
 \, ,

where, all three of the dimensionless structural form factors, ~\mathfrak{f}_M, ~\mathfrak{f}_W, and ~\mathfrak{f}_A, equal unity under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, while all three are of order unity otherwise. If the configuration — such as an interstellar gas cloud — is not isolated but is, instead, embedded in a hot, tenuous external medium that exerts an external pressure, ~P_e, the configuration's equilibrium parameters will be related via the expression,

~\frac{GM^2}{P_c R_\mathrm{eq}^4}

~=

~\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]
 \, .

We have introduced these structural form factors into our discussion of the free energy of spherically symmetric, self-gravitating configurations because


Solution Strategies:    

Detailed Force-Balance
(Introduction)

Virial Equilibrium
(Introduction)

Example Solutions:
  • Uniform-density sphere
        Isolated …
    Embedded in an External Medium …
  • Polytropes
    Isolated …
    Embedded in an External Medium …
  • Isothermal sphere
    Isolated …
    Embedded in an External Medium (Bonnor-Ebert Sphere) …
  • Zero-temperature White Dwarf — Overview
  • Power-law density distribution — Overview
  • BiPolytropes (also referred to as Composite Polytropes)
Overview Overview
        Core-Envelope Structure with ~(n_c,n_e) = (0,0)
        Core-Envelope Structure with ~(n_c,n_e) = (5,1)

Stability:

Background: Sound Waves

Solution Strategy Assuming Spherical Symmetry:

Virial Stability

Bipolytrope Generalization

Example Solutions:

Dynamics:

Spherical Collapse:


 
 
 

Two-Dimensional Configurations

Structure:

Solution Strategies

Virial Equilibrium

Example Solutions:

Stability:

Dynamics:

 
 
 

Three-Dimensional Configurations

Structure:

Solution Strategies

Example Solutions:

Stability:

Dynamics:

 
 
 

Related Projects Underway


Appendices


See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
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