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Supporting Derivations for FreeEnergy PowerPoint Presentation
The derivations presented here are an extension of our accompanying freeenergy synopsis. These additional details proved to be helpful while developing an overarching PowerPoint presentation.
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General FreeEnergy Expression
We're considering a freeenergy function of the following form:



where,
As we have shown, setting,



generates a mathematical statement of virial equilibrium, namely,



And equilibrium configurations for which the second (as well as first) derivative of the free energy is zero are found at "critical" radii given by the expression,



PressureTruncated Polytropes
For pressuretruncated polytropes, set and let be the chosen polytropic index. In this case, the statement of virial equilibrium is,



and the critical equilibrium configuration has,



Case M
Set and constant and examine how the freeenergy behaves as a function of the coordinates, . In this case (see, for example, here),









where the structural form factors for pressuretruncated polytropes are precisely defined here. And (see, for example, here),






If we set all three structural formfactors to unity, we have,






Virial Equilibrium
Figure 1 

So the statement of virial equilibrium becomes,









The lightblue dots in Figure 1 trace the equilibrium sequence that is defined by this virial equilibrium function in the case of .
Dynamical Instability
Along the "Case M" equilibrium sequence, the transition from stable to unstable configurations occurs at,






which, in combination with the virial equilibrium condition gives,






The location of this critical configuration along the equilibrium sequence is marked by the red circular dot in Figure 1.
Turning Point
Let's examine the curvature of the equilibrium sequence.









Setting this derivative to zero let's us identify the location of the turning point that identifies






And, returning to the virial equilibrium expression, we find that, associated with this equilibrium radius,












Notice that, under the assumption that all three structural fillingfactors are unity, , that is, the location of the turning point coincides precisely with the point along the equilibrium sequence where the transition from stable to unstable equilibrium configurations occurs (marked by the red circular dot in Figure 1).
Case M Summary
Case M  

OrderofMagnitude Analysis: Assume  
Virial Equilibrium: 


Dynamical Instability:



Turning Point :


Case P
Set and constant and examine how the freeenergy behaves as a function of the coordinates, . In this case (see, for example, here),















where the structural form factors for pressuretruncated polytropes are precisely defined here. If we set all three structural formfactors to unity, we have,






Virial Equilibrium
So, the statement of virial equilibrium becomes,



Known Analytic LaneEmden Functions 

















Dynamical Instability
Along the "Case P" equilibrium sequence, the transition from stable to unstable configurations occurs at,






which, in combination with the "Case P" virial equilibrium expression gives,












Turning Points
Let's simplify the notation, defining,
The statement of virial equilibrium becomes,



where,
Differentiating gives,









One turning point occurs where the numerator is zero, that is,









Plugging this into the virial equilibrium expression gives,















The associated mass is,






Notice that, for ,



Another turning point occurs where the denominator is zero, that is,















Plugging this into the virial equilibrium expression gives,


















And the associated mass is,






Case P Summary
Case P  

OrderofMagnitude Analysis: Assume  
Virial Equilibrium: 


Dynamical Instability:



Turning Point :



Turning Point :


The righthand panel of Figure 2 presents substantial segments of Case P virial equilibrium sequences for a range of polytropic indexes (n = 1, 2, 2.8, 3, 3.5, 4, 5). For each sequence, the location of the and turning points — if they exist — are denoted by a yellow or red circular dot, respectively. The point along each sequence at which the transition from dynamically stable to dynamically unstable structures occurs coincides with the location of (i.e., with the red circular dot).
For display purposes, all normalized masses have been further normalized to the maximum mass on the n = 3 sequence. </div>
Detailed ForceBalance Models
Structural Form Factors
The following table of structural form factors has been drawn from here,
Structural Form Factors for Isolated Polytropes 
Structural Form Factors for PressureTruncated Polytropes 




and here,
Structural Form Factors for PressureTruncated n = 5 Polytropes  


Case M Equilibrium Conditions
Employing the renormalization factors,






we find from detailed forcebalance analyses that the equilibrium radius and corresponding external pressure for "Case M" configurations are,



which matches the expression derived in an ASIDE box found with our introduction of the LaneEmden equation, and



There are two turning points: One associated with a maximum in and one associated with a maximum in . According to Kimura's discussion, the first of these occurs in the configuration for which,



… 
For n = 5, this occurs when 
This point along the equilibrium sequence is identified by the dark green circular dot in Figure 3, below. The second occurs in the configuration for which,



… 
For n = 5, this occurs when 
This point along the equilibrium sequence is identified by the yellow circular dot in Figure 3, below. In addition, we have identified the point of dynamical instability.









But the equilibrium condition for n = 5 configurations is,















Putting the two expressions together gives,









This agrees with the expression derived in a separate ASIDE; as was pointed out in that context, the root of this equation is: , that is, . This point along the equilibrium sequence is identified by the red circular dot in Figure 3, below. It is almost — but definitely not — coincident with the configuration along the sequence (marked by the yellow circular dot) that is associated with the minimumradius turning point.
Now for a movie!
Case P Equilibrium Conditions
The equilibrium radius and corresponding configuration mass from a "Case P" analysis are,






According to our review of, especially, Kimura's work, the turning point associated with occurs where,



… 
For n = 5, this occurs when 
And a turning point associated with occurs where,



For configurations, this means,















FiveOne Bipolytropes
Basic Properties
For bipolytropes, in general, let and . The statement of virial equilibrium is, then,



And the critical equilibrium configuration has,



Here we choose to set and . Hence, these two conditions become, respectively,



and,



Comparing the general freeenergy expression at the beginning of this chapter with the freeenergy expression provided via our accompanying summary discussion of fiveone bipolytropes, we find that,









where,












and,






In rewriting this last expression, we have made use of the two relations,






Consistent with our generic discussion of the stability of bipolytropes and the specific discussion of the stability of bipolytropes having , it can straightforwardly be shown that is satisfied by setting ; that is, the equilibrium condition is,
Furthermore, the equilibrium configuration is unstable whenever , that is, the transition from stable to unstable configurations whenever,



Table 1 of an accompanying chapter — and the reddashed curve in the figure adjacent to that table — identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the meanmolecular weight ratio, .
Coincidence Between Points of Secular and Dynamical Instability
From the accompanying graphical display of equilibrium sequences, it seems that a turning point will only exist in fiveone bipolytropes for less than some value — call it, — which is less than but approximately equal to . As we move along any sequence for which , in the direction of increasing , it is fair to ask whether the system becomes dynamically unstable (at ) before or after it encounters the point of secular instability marked by .
© 2014  2021 by Joel E. Tohline 