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Spherically Symmetric Configurations Synopsis (Using Style Sheet)
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Structure
Tabular Overview
 

Equilibrium Structure  
① Detailed Force Balance  ③ FreeEnergy Identification of Equilibria  
Given a barotropic equation of state, , solve the equation of
for the radial density distribution, .  The FreeEnergy is,
Therefore, also,
Equilibrium configurations exist at extrema of the freeenergy function, that is, they are identified by setting . Hence, equilibria are defined by the condition,
 
② Virial Equilibrium  
Multiply the hydrostaticbalance equation through by and integrate over the volume:

Pointers to Relevant Chapters
⓪ Background Material:
·  Principal Governing Equations (PGEs) in most general form being considered throughout this H_Book 

·  PGEs in a form that is relevant to a study of the Structure, Stability, & Dynamics of spherically symmetric systems 
·  Supplemental relations — see, especially, barotropic equations of state 
① Detailed Force Balance:
·  Derivation of the equation of Hydrostatic Balance, and a description of several standard strategies that are used to determine its solution — see, especially, what we refer to as Technique 1 

② Virial Equilibrium:
·  Formal derivation of the multidimensional, 2^{nd}order tensor virial equations 

·  Scalar Virial Theorem, as appropriate for spherically symmetric configurations 
·  Generalization of scalar virial theorem to include the bounding effects of a hot, tenuous external medium 
Stability
Isolated & PressureTruncated Configurations
Stability Analysis: Applicable to Isolated & PressureTruncated Configurations  

④ Perturbation Theory  ⑦ FreeEnergy Analysis of Stability  
Given the radial profile of the density and pressure in the equilibrium configuration, solve the eigenvalue problem defined by the, LAWE: Linear Adiabatic Wave (or Radial Pulsation) Equation
to find one or more radially dependent, radialdisplacement eigenvectors, , along with (the square of) the corresponding oscillation eigenfrequency, . 
The second derivative of the freeenergy function is,
Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,
Note the similarity with ⑥.
the conditions for virial equilibrium and stability, may be written respectively as,
 
⑤ Variational Principle  
Multiply the LAWE through by , and integrate over the volume of the configuration gives the, Governing Variational Relation
Now, by setting , we can ensure that the pressure fluctuation is zero and, hence, at the surface, in which case this relation becomes,
 
⑥ Approximation: Homologous Expansion/Contraction  
If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radialdisplacement eigenfunction is, = constant, and the governing variational relation gives,

Bipolytropes
Stability Analysis: Applicable to Bipolytropic Configurations  

⑧ Variational Principle  ⑩ FreeEnergy Analysis of Stability  
Governing Variational Relation

As we have detailed in an accompanying discussion, the first derivative of the relevant freeenergy expression is,
where,
and the second derivative of that freeenergy function is,
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of freeenergy set to zero) will be identified by the model that exhibits the ratio,
See the accompanying discussion. If — based for example on ⑦ — we make the reasonable assumption that, in equilibrium, the statements,
hold separately, then we satisfy the virial equilibrium condition, namely,
and the second derivative of the relevant freeenergy function can be rewritten as,
Note the similarity with ⑨ — temporarily, see this discussion.  
⑨ Approximation: Homologous Expansion/Contraction  
If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radialdisplacement eigenfunction is, = constant, and the governing variational relation gives,

See Also
© 2014  2021 by Joel E. Tohline 