User:Tohline/SSC/Stability BoundedCompositePolytropes
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Instabilities in Bounded and Composite Polytropes
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Unbounded, Complete Polytropes
FreeEnergy Function and Its Derivatives
The freeenergy function that is relevant to a discussion of the structure and stability of unbounded configurations having polytropic index, , has the form,



where identifies the radius of the configuration and is an arbitrary constant. If the coefficients, , and , are held constant while varying the configuration's size, we see that,



and,



In terms of the system's mass and its structural form factors, , , and , the two relevant coefficients are,






Equilibrium Radius
A configuration's equilibrium radius, , can be determined in one of two ways:
Extrema in the Free Energy
Equilibria are identified by extrema in the freeenergy function. Setting , we find,









If one assumes that the equilibrium system has no internal structure — that is, that the interior density and temperature are uniform throughout — then and this derived expression gives a good estimate of the equilibrium radius, given any choice of the pair of parameters, and .
Detailed Force Balance
Alternatively, a solution of the,
gives information regarding the detailed interior structure of the equilibrium system via knowledge of the properties of the LaneEmden function, , as well as an exact expression for the equilibrium radius, namely,
where,
Together
Now, once the LaneEmden function, , is known from a detailed forcebalance model, the other two structural form factors also can be straightforwardly determined. For unbounded, complete polytropes, the relevant expressions are,






Plugging the appropriate ratio of these two functions, namely,



into the expression for the equilibrium radius obtained from the freeenergy analysis gives precisely the same answer as was obtained from the detailed forcebalance analysis. Using either method of determination we conclude, therefore, that,
Stability
The procedure that has been used to obtain a detailed forcebalanced model of unbounded polytropes cannot readily be extended to provide a stability analysis of such systems. However, the freeenergy analysis can be readily extended. If the first derivative of the freeenergy function is zero — that is, if you have identified an equilibrium configuration — and the second derivative of the freeenergy function for that configuration is positive, then the equilibrium system is dynamically stable. If, however, the second derivative is negative, then the equilibrium system is dynamically unstable.
Using the expression for the second derivative of the freeenergy function derived above, we deduce that equilibrium configurations are dynamically unstable when,












Bounded (PressureTruncated) Polytropes
Throughout this section we will rely on the following definitions of two normalization constants:
FreeEnergy Function and Its Derivatives
The freeenergy function that is relevant to a discussion of the structure and stability of a pressuretruncated configuration having polytropic index, , has the form,



where identifies the radius of the configuration and is an arbitrary constant. If the coefficients, , and , are held constant while varying the configuration's size, we see that,






and,



In terms of the system's mass and its structural form factors, , , and , the three relevant coefficients are,









Equilibrium Radius
A configuration's equilibrium radius, , can be determined in one of two ways:
Extrema in the Free Energy
Equilibria are identified by extrema in the freeenergy function, that is, by setting . Hence, is given by the root(s) of the polynomial expression that is often referred to as the,
Scalar Virial Theorem



Plugging the definitions of the three freeenergy coefficients into this virial expression gives the massradius relationship for pressuretruncated, polytropic equilibrium configurations, namely,



If one assumes that the equilibrium system has no internal structure — that is, that the interior density and temperature are uniform throughout — then and this derived expression provides a good, approximate massradius relationship for configurations having a given that are truncated by a surrounding, massless medium exerting a pressure, , on the system.
Detailed Force Balance
From the detailed forcebalance analysis presented in Appendix B of Steven W. Stahler (1983), we see that the mass, , associated with the equilibrium radius, , of bounded (pressuretruncated) polytropic spheres is given through the following pair of parametric relations:






Together
As was realized in the case of unbounded polytropes, once the LaneEmden function, , and its radial derivative, , are known from a detailed forcebalance analysis, all three structural form factors can be straightforwardly determined. For bounded (pressuretruncated) polytropes, the relevant expressions are,









We have noticed, as well, that the relationship between and is relatively simple, namely,



As it turns out, it is sufficient to use this relationship — rather than the separate, explicit definitions of and — in order to obtain identical expressions for the equilibrium radius from both the detailed forcebalance and freeenergy analyses. By way of demonstration, let's plug this simpler expression for into the virial equilibrium relation.









where, we have temporarily adopted the shorthand notation,



Regrouping terms from this last virial expression, we also can write,



Now, this relation will be satisfied without having to plug in the fully explicit definition of if we demand that the expressions inside of the curly braces on both sides of the equation are zero, independently. This demand implies that (from the LHS),



and, independently (from the RHS),



This pair of conditions, in turn, imply that,






and,






which match exactly the pair of parametric equations relating to that Stahler derived via his detailed forcebalance analysis. We note as well that, after eliminating the parameter, — which appears in the definition of both normalizations, and — using either method of determination we have,
Stability
As before, if the first derivative of the freeenergy function is zero — that is, if you have identified an equilibrium configuration — and the second derivative of the freeenergy function for that configuration is positive, then the equilibrium system is dynamically stable. If, however, the second derivative is negative, then the equilibrium system is dynamically unstable.
Using the expression for the second derivative of the freeenergy function derived above, we deduce that equilibrium configurations are dynamically unstable when,


















Plugging in the expressions for the freeenergy coefficients, and , this in turn implies that an equilibrium system will be dynamically unstable if,









Composite Polytropes (Bipolytropes)
Throughout this section we will rely on the following definitions of three normalization constants:









FreeEnergy Function and Its Derivatives
The freeenergy function that is relevant to a discussion of the structure and stability of a composite polytropic configurations of index, , for the core and, , for the envelope, has the form,



where identifies the radius of the configuration and is an arbitrary constant. If the coefficients, , and , are held constant while varying the configuration's size, we see that,






and,



The four relevant coefficients are,












Specifically for analytically definable composite polytropes having ,















where,










… 








Let's try putting these terms all together in the context of the virial theorem.





















where the last step has been made by appreciating that,









and we have adopted the shorthand notation,






Adding these two expressions together and setting the result equal to zero (as prescribed by the virial theorem) gives,









Related Discussions
 Constructing BiPolytropes
 Analytic description of BiPolytrope with (n_{c},n_{e}) = (5,1)
 BonnorEbert spheres
 SchönbergChandrasekhar limiting mass
 Relationship between BonnorEbert and SchönbergChandrasekhar limiting masses
 Wikipedia introduction to the LaneEmden equation
 Wikipedia introduction to Polytropes
© 2014  2019 by Joel E. Tohline 