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===Structure:===
===Structure:===
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Here we show how the set of principal governing equations (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations &#8212; 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 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 fluid density and fluid pressure.  As our various discussions illustrate, simply varying the power-law index in a ''polytropic'' equation of state gives rise to equilibrium configurations that have a wide variety of internal structural profiles.
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Here we show how the set of principal governing equations (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations &#8212; 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 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 fluid density and fluid pressure.  As our various discussions illustrate (see the table of contents, below), simply varying the power-law 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 uniform-density 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 free-energy extrema is often referred to as the ''scalar virial theorem'' &hellip; [[User:Tohline/SphericallySymmetricConfigurations/IntroductorySummary#Applications|<more>]]
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 uniform-density 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 free-energy extrema is often referred to as the ''scalar virial theorem'' &hellip; [[User:Tohline/SphericallySymmetricConfigurations/IntroductorySummary#Applications|<more>]]

Revision as of 16:11, 3 March 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) 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 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 fluid density and fluid pressure. As our various discussions illustrate (see the table of contents, below), simply varying the power-law 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 uniform-density 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 free-energy extrema is often referred to as the scalar virial theorem<more>

In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left — for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium." Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed Force-Balance." Insight into the properties of equilibrium systems that is revealed via an analysis of a system's free-energy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium."


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
  • Other Analytically Definable Models
  …  
  • 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)
        Core-Envelope Structure with ~(n_c,n_e) = (1,5)   …  
        Core-Envelope Structure with ~(n_c,n_e) = (\tfrac{3}{2},3)   …  

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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