User:Tohline/SSC/Structure/Polytropes/VirialSummary
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Virial Equilibrium of PressureTruncated Polytropes
Here we will draw heavily from …
 An accompanying Free Energy Synopsis;
 A detailed analysis of the Virial Equilibrium of Adiabatic Spheres.
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Groundwork
Basic Relation
In the context of spherically symmetric, pressuretruncated polytropic configurations, the relevant freeenergy expression is,






where, when written in terms of the trio of structural form factors, and the pair of constants,

and 

OftenReferenced Dimensionless Expressions
When rewritten in a suitably dimensionless form — see two useful alternatives, below — this expression becomes,



where is the configuration's dimensionless radius and , , and are constants. We therefore have,



and,



Virial equilibrium is obtained when , that is, when



And along an equilibrium sequence, the specific equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations — henceforth labeled as having the critical radius, — is identified by setting , that is, it is the configuration for which,






Inserting the adiabatic exponent in place of the polytropic index via the relation, , we have equivalently,



Useful Recognition
By comparing various terms in the first two algebraic Setup expressions, above, It is clear that,

and, 

Notice, then, that in every equilibrium configuration, we should find,






And, specifically in the critical configuration we should find that,






The equivalent of this last expression also appears at the end of subsection ⑦ of an accompanying Tabular Overview.
Equilibrium Sequences
In all of the polytropic configurations being considered here, is a constant — that is, the specific entropy of all fluid elements is assumed to be the same, both spatially and temporally.
Fix Mass While Varying External Pressure
In this case, we want to examine undulations of a twodimensional freeenergy surface that results from allowing and to vary while holding fixed. In our accompanying, more detailed discussion, this is referred to as Case M. Adopting the normalizations,






and, 



— which, as is detailed in an accompanying discussion, are similar but not identical to the normalizations adopted by Horedt (1970) and by Whitworth (1981) — the coefficients in the abovepresented Basic Relations become,






and, 



The relevant dimensionless freeenergy surface is, then, given by the expression,
Case M FreeEnergy Surface  


Fix External Pressure While Varying Mass
In this case, we want to examine undulations of a twodimensional freeenergy surface that results from allowing and to vary while holding fixed. In our accompanying, more detailed discussion, this is referred to as Case P. Motivated by Stahler's (1983) work, here we adopt the normalizations,






and,  



the coefficients in the abovepresented Basic Relations become,






and, 



where the pair of constants, and , have the same definitions in terms of the structural form factors as provided above. The relevant dimensionless freeenergy surface is, then, given by the expression,
Case P: FreeEnergy Surfaces  


Across this Case P freeenergy surface, extrema — and, hence, equilibrium configurations — will arise wherever the virialequilibrium condition is met, that is, wherever
Case P: Virial Equilibrium Sequences  


An equilibrium model sequence is thereby defined for each chosen polytropic index in the range, .
Along each equilibrium sequence for which , there is one specific equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations. For a given mass, the radius of this critical configuration is identified by the relation,



Below, we will examine the behavior of individual virialequilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, and . Here, in an effort to provide a broad overview, we set,
and
and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. Specifically, each curve in the lefthand panel of Figure 1 results from the massradius expression,



(As has been demonstrated in an accompanying discussion, we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass and radiusnormalizations — namely, and — instead of choosing these specific values of and .)
Figure 1: VirialAnalysisBased Equilibrium Sequences  

For comparison, the schematic diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of Stahler (1983). It seems that our derived, analytically prescribable, massradius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the massradius relationship for pressuretruncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.
Below, we will examine the behavior of individual virialequilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, and . Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,
and ,
and the resulting massradius relation of equilibrium sequences is governed by the algebraic massradius relation,



The lefthand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled. As has been demonstrated in an accompanying discussion, this governing massradius relation can be analytically manipulated into a form that provides either an explicit or relation for the cases labeled, and ; a generic rootfinding technique has been used to generate points along each of the other depicted equilibrium sequences.
Figure 1: VirialAnalysisBased Equilibrium Sequences  

For comparison, the schematic diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of Stahler (1983). It seems that our derived, analytically prescribable, massradius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the massradius relationship for pressuretruncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.
See Also
© 2014  2021 by Joel E. Tohline 