User:Tohline/SSC/Synopsis
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Spherically Symmetric Configurations Synopsis
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Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, 



Equilibrium Structure 

Detailed Force Balance 
FreeEnergy Analysis 



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,


Virial Equilibrium  


Stability Analysis 

Perturbation Theory 
FreeEnergy Analysis 

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,


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,


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  2020 by Joel E. Tohline 