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Virial Equilibrium of Adiabatic Spheres
Highlights of the rather detailed discussion presented below have been summarized in an accompanying chapter of this H_Book.
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Review
Adopted Normalizations
In an introductory discussion of the virial equilibrium structure of spherically symmetric configurations, we adopted the following physical parameter normalizations for adiabatic systems.
Adopted Normalizations for Adiabatic Systems  


Virial Equilibrium
Also in our introductory discussion — see especially the section titled, Energy Extrema — we deduced that an adiabatic system's dimensionless equilibrium radius,
is given by the root(s) of the following equation:
where the definitions of the various coefficients are,















(The dimensionless structural form factors, that appear in these expressions are defined for isolated polytropes in our accompanying introductory discussion and are discussed further, below.) Once the pressure exerted by the external medium (), and the configuration's mass (), angular momentum (), and specific entropy (via ) have been specified, the values of all of the coefficients are known and can be determined.
Isolated Nonrotating Adiabatic Configuration
For a nonrotating configuration that is not influenced by the effects of a bounding external medium , the statement of virial equilibrium is,
Hence, one equilibrium state exists for each value of and it occurs where,



Two Points of View 


In terms of and 
In terms of and 



According to the solution shown in the lefthand column, for fluid with a given specify entropy content, the equilibrium massradius relationship for adiabatic configurations is,
We see that, for , the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, for , the mass of the configuration is independent of the radius. For or , configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for in the range, , configurations with larger mass have smaller equilibrium radii. (Note that the related result for isothermal configurations can be obtained by setting in this adiabatic solution, because when .)
Role of Structural Form Factors
When employing a virial analysis to determine the radius of an equilibrium configuration, it is customary to set the structural form factors, , and , to unity and accept that the expression derived for is an estimate of the configuration's radius that is good to within a factor of order unity. As has been demonstrated in our related discussion of the equilibrium of uniformdensity spheres, these form factors can be evaluated if/when the internal structural profile of an equilibrium configuration is known from a complementary detailed forcebalance analysis. In the case being discussed here of isolated, spherical polytropes, solutions to the,
can provide the desired internal structural information. Here we draw on Chandrasekhar's [C67] discussion of the structure of spherical polytropes to show precisely how our structural form factors can be expressed in terms of the LaneEmden function, , dimensionless radial coordinate, , and the function derivative, .
Mass
We note, first, that Chandrasekhar [C67] — see his Equation (78) on p. 99 — presents the following expression for the meantocentral density ratio:



where the notation at the bottom of the closing square bracket means that everything inside the square brackets should be, "evaluated at the surface of the configuration," that is, at the radial location, , where the LaneEmden function, , first goes to zero. But, as we pointed out when defining the structural form factors, the form factor associated with the configuration mass, , is equivalent to the meantocentral density ratio. We conclude, therefore, that,



Gravitational Potential Energy
Second, we note that Chandrasekhar's [C67] expression for the gravitational potential energy — see his Equation (90), p. 101 — is,



whereas our analogous expression is,



We conclude, therefore, that,






MassRadius Relationship
Third, Chandrasekhar [C67] shows — see his Equation (72), p. 98 — that the general massradius relationship for isolated spherical polytropes is,



which we choose to rewrite as,






By comparison, the expression for the equilibrium radius that has been derived, above, from an analysis of extrema in the free energy function — specifically, see the last expression in the lefthand column of the table titled "Two Points of View" — we obtain,



Hence, it appears as though, quite generally,



Or, taking into account the expressions for and that have just been uncovered, we conclude that,












Central Pressure
It is also worth pointing out that Chandrasekhar [C67] — see his Equations (80) & (81), p. 99 — introduces a dimensionless structural form factor, , for the central pressure via the expression,



and demonstrates that,



It is therefore clear that a spherical polytrope's central pressure is expressible in terms of our structural form factor, , as,



Alternate Derivation of Gravitational Potential Energy
As has been discussed elsewhere, we have learned from Chandrasekhar's discussion of polytropic spheres [C67] — see his Equation (16), p. 64 — that if a spherically symmetric system is in hydrostatic balance, the total gravitational potential energy can be obtained from the following integral:



Using "technique #3" to solve the differential equation that governs the statement of hydrostatic balance, we know that in any polytropic sphere, is related to the configuration's radial enthalpy profile, , via the algebraic expression,



where, , is an integration constant. At the surface of the equilibrium configuration, and , so the integration constant is,



which implies,



Now, from our general discussion of barotropic relations, we can write,



Hence,



and,


















We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of and , namely,



and,



Plugging these into our newly derived expression for the gravitational potential energy gives,












As it should, this agrees with the expression for the ratio, , that was derived in our above discussion of the gravitational potential energy.
Summary
In summary, expressions for the three structural form factors associated with isolated, spherically symmetric polytropes are as follows:
Structural Form Factors for Isolated Polytropes  


Nonrotating Adiabatic Configuration Embedded in an External Medium
For a nonrotating configuration that is embedded in, and is influenced by the pressure of, an external medium, the statement of virial equilibrium is,
Solution Expressed in Terms of K and M (Whitworth's 1981 Relation)
This is precisely the same condition that derives from setting equation (3) to zero in Whitworth's (1981, MNRAS, 195, 967) discussion of the Global Gravitational Stability for Onedimensional Polytropes. The overlap with Whitworth's narative is clearer after introducing the algebraic expressions for the coefficients , , and , to obtain,



dividing the equation through by ,






and inserting expressions for the parameter normalizations as defined in our accompanying introductory discussion to obtain,






If the structural form factors are set equal to unity, this exactly matches equation (5) of Whitworth, which reads:
Equation and accompanying sentence extracted without modification from p. 970 of Whitworth (1981)
"Global Gravitational Stability for Onedimensional Polytropes"
Monthly Notices of the Royal Astronomical Society, vol. 195, pp. 967977 © Royal Astronomical Society 
Notice that, when , this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of and (see the lefthand column of our table titled "Two Points of View").
Solution Expressed in Terms of M and Central Pressure
Beginning again with the relevant statement of virial equilibrium, namely,
but adopting the alternate expression for the coefficient, , given above, that is,
we can write,












Again notice that, when , this expression reduces to the solution we obtained for an isolated polytrope, but this time expressed in terms of and (see the righthand column of our table titled "Two Points of View").
Contrast with Detailed ForceBalanced Solution
As has just been demonstrated, the virial theorem provides a mathematical expression that allows us to relate the equilibrium radius of a configuration to the applied external pressure, once the configuration's mass and either its specific entropy or central pressure are specified. In contrast to this, as has been discussed in detail in another chapter, Horedt (1970), Whitworth (1981) and Stahler (1983) have each derived separate analytic expressions for and — given in terms of the LaneEmden function, , and its radial derivative — without demonstrating how the equilibrium radius and external pressure directly relate to one another. That is to say, solution of the detailed forcebalanced equations provides a pair of equilibrium expressions that are parametrically related to one another through the LaneEmden function. For example — see our related discussion for more details — Horedt derives the following set of parametric equations relating the configuration's dimensionless radius, , to a specified dimensionless bounding pressure, :






where,






It is important to appreciate that, in the expressions for and , the tilde indicates that the LaneEmden function and its derivative are to be evaluated, not at the radial coordinate, , that is traditionally associated with the "first zero" of the LaneEmden function and therefore with the surface of the isolated polytrope, but at the radial coordinate, , where the internal pressure of the isolated polytrope equals and at which the embedded polytrope is to be truncated. The coordinate, , therefore identifies the surface of the embedded — or, pressuretruncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to in both defining relations because it is clear that Hoerdt intended for the normalization mass to be the mass of the pressuretruncated object, not the mass of the associated isolated (and untruncated) polytrope. In anticipation of further derivations, below, we note here the ratio of Hoerdt's normalization parameters to ours, assuming :












Next, we demonstrate that this pair of parametric relations satisfies the virial theorem and, in so doing, demonstrate how and may be directly related to each other. Given that Hoerdt's chosen normalization radius and normalization pressure are defined in terms of and , we begin with the virial theorem derived above in terms of and , setting .



After setting , a bit of algebraic manipulation shows that the first term on the righthand side of the virial equilibrium expression becomes,



while the second term on the righthand side becomes,



But, using Horedt's expression for , the lefthand side of the virial equilibrium equation becomes,



Hence, the statement of virial equilibrium is,









or, multiplying through by and rearranging terms,

Now, Hoerdt has given analytic expressions for and in terms of the LaneEmden function and its first derivative. The question is, what should the expressions for our structural form factors be in order for this virial expression to hold true for all pressuretruncated polytropic structures? As has been summarized above, in the case of an isolated polytrope, whose surface is located at and whose global properties are defined by evaluation of the LaneEmden function at , we know that (see the above summary),
Structural Form Factors for Isolated Polytropes  


These same expressions may or may not work for pressuretruncated polytropes, even if the evaluation radius is shifted from to . Let's see …
January 13, 2015: As is noted in our accompanying outline of work, I no longer believe that and have the same expressions as in isolated polytropes. Hence, all of the material that follows is suspect and needs to be reworked.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
 Go Home 
Inserting Hoerdt's expressions for and into the viral equilibrium expression, we have,












If we assume that both of the structural form factors, and , have the same functional expressions as in the case of isolated polytropes (but evaluated at instead of at ), the virial relation further reduces to the form,









This all seems to make a great deal of sense. Only the structural parameter that is derived from an integral over the pressure distribution, , gets modified when the polytropic configuration is truncated. Notice, as well, that the term that has been added to the definition of naturally goes to zero in the limit of , that is, for an isolated polytrope. We should definitely go back to the original definitions of all three structural parameters and prove that this is the case. But, in the meantime, here is the summary:
WRONG!! For the correct formfactor expressions, go here.  

Structural Form Factors for PressureTruncated Polytropes  

WRONG!! For the correct formfactor expressions, go here.
Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three formfactors, .
Example
Outline
Let's identify an equilibrium configuration numerically, using the freeenergy expression. From our introductory discussion, the relevant expression is,
where,












For later use, note that,
and,

Now, we could just blindly start setting values of the three leading coefficients, , , and , then plot to look for extrema. But let's accept a little guidance from this chapter's virial analysis before choosing the coefficient values. For embedded polytropes, we know that the structural form factors are,









Hence, the coefficient expressions become,









Strategy
Generic setup:
 Choose the polytropic index, , which also sets the value of the adiabatic index, .
 Fix and , so that the radial and pressure normalizations are fixed; specifically,

and 

 Fix , and let it be the normalization mass; that is, set .
 As a result of the above choices, the value of is set, and fixed; specifically,
Case I:
 Fix , which fixes the external pressure; specifically,
 Choose a variety of values of the remaining coefficient, ; then, for each value, plot and locate one or more extrema along with the value of that is associated with each free energy extremum. This identifies the equilibrium value of the mean pressure inside the pressuretruncated polytrope via the expression,
 In order to check whether we've identified the correct value of , we have to relate it to the radial coordinate, , used in the analytic solution of the LaneEmden equation. As has been explained in our discussion of detailed forcebalanced models of polytropes, generically,


where, 















Hence,









But, from above, we also know that,



where,



Hence we can write,



or,



This last expression may be useful because the numerical value of the righthandside will be known once an extremum of a freeenergy plot has been identified, while the function on the lefthand side can be evaluated separately, from knowledge of the internal structure of detailed forcebalanced, isolated polytropes.
Strategy2
 Pick the desired polytropic index, , and a radial coordinate within the isolated polytropic model, , that will serve as the truncated edge of the embedded polytrope.
 Knowledge of the isolated polytrope's internal structure will give the value of the LaneEmden function, , and its radial derivative, , at this truncated edge of the structure.
 According to Horedt (1970) — see our accompanying discussion of detailed forcebalanced models — the physical radius and external pressure that corresponds to this choice of the truncated edge is given by the expressions,






 Using the chosen value of and its associated function values, and , determine the values of the three relevant structural form factors via the following analytic relations:









 Using these values of the structural form factors, determine the values of the three freeenergy coefficients (set for the time being) via the expressions:












 Plot the following freeenergy function and see if the value of associated with the extremum is equal to the dimensionless equilibrium radius, , as predicted by Hoerdt's expression, above:






 Virial equilbrium — that is, an extremum in the free energy function — occurs when , that is, where,



Note that if the coefficient, , is written in terms of the normalized central pressure, the statement of virial equilibrium becomes,









But, in this situation, , so virial equilibrium implies,












Compare With Detailed Force Balanced Solution
In a separate discussion, we presented the detailed forcebalanced model of an polytrope that is embedded in an external medium. We showed that, for an applied external pressure given by,
the associated equilibrium radius of the pressureconfined configuration is,
Flipping this around, after we use a plot of the freeenergy expression to identify the equilibrium radius, , the corresponding dimensionless radius as used in the LaneEmden equation should be,






Keep in mind that, for an isolated polytrope, the (zero pressure) surface is identified by . Hence we should expect freeenergy extrema to occur at values of .
Renormalization
Grunt Work
Returning to the dimensionless form of the virial expression and multiplying through by , we obtain,
or, after plugging in definitions of the coefficients, , , and , and rewriting explicitly as ,



This relation can be written in a more physically concise form, as follows. First, normalize to a new pressure scale — call it — and multiply through by in order to normalizing to a new length scale,:






or,



where,






By demanding that the leading coefficients of both terms on the righthandside of the expression are simultaneously unity — that is, by demanding that,



and,



we obtain the expressions for and as shown in the following table.
Renormalization for Adiabatic (ad) Systems  


Using these new normalizations, we arrive at the desired, concise virial equilibrium relation, namely,



or,



For the sake of completeness, we should develop expressions for both and that are entirely in terms of the LaneEmden function, , its derivative, , and the associated dimensionless radial coordinate, , at which the function and its derivative are to be evaluated. (Adopting a unified notation, we will set .)









Inserting the functional expressions for Hoerdt's and our structural form factors, , gives,












And,









Inserting the functional expressions for Hoerdt's and our structural form factors, , gives,












(Not a particularly simple or transparent expression!)
Summary
Defining,


and, 



the expressions for the dimensionless equilibrium radius and the dimensionless external pressure may be written as, respectively,


and, 



Using these expressions, it is easy to demonstrate that the virial equilibrium relation is satisfied, namely,



Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
 Go Home 
PV Diagram for Unity Form Factors
Writing the coefficient, B, in terms of the average sound speed and setting the radial scale factor equal to the equilibrium radius of an isolated adiabatic sphere, that is, setting,
the equation governing the radii of adiabatic equilibrium states becomes,
where,
As in the isothermal case, for a given choice of configuration mass and sound speed, this parameter, Π_{a}, can be viewed as a dimensionless external pressure. Alternatively, for a given choice of P_{e} and , can represent a dimensionless mass; or, for a given choice of M and P_{e}, can represent a dimensionless sound speed. Here we will view it as a dimensionless external pressure.
Unlike the isothermal case, for an arbitrary value of the adiabatic exponent, γ_{g}, it isn't possible to invert this equation to obtain an analytic expression for χ as a function of Π_{a}. But we can straightforwardly solve for Π_{a} as a function of χ. The solution is,
For physically relevant solutions, both χ and Π_{a} must be nonnegative. Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,
Figure 4: Equilibrium Adiabatic PV Diagram 

The curves trace out the function,
for six different values of γ_{g} (, as labeled) and show the dimensionless external pressure, Π_{a}, that is required to construct a nonrotating, selfgravitating, uniform density, adiabatic sphere with an equilibrium radius χ. The mathematical solution becomes unphysical wherever the pressure becomes negative. The solid red curve, drawn for the case γ_{g} = 1, is identical to the solid black (isothermal) curve displayed above in Figure 1. 
Each of the Π_{a}(χ) curves drawn in Figure 4 exhibits an extremum. In each case this extremum occurs at a configuration radius, χ_{extreme}, given by,
that is, where,
For each value of γ_{g}, the corresponding dimensionless pressure is,
Note, first, that for γ_{g} > 4 / 3, an equilibrium configuration with a positive radius can be constructed for all physically realistic — that is, for all positive — values of Π_{a}. Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant — i.e., positive — pressures only for values of γ_{g} < 4 / 3; and in each case it represents a maximum limiting pressure.
Maximum Mass
n = 5 Polytropic
When γ_{a} = 6 / 5 — which corresponds to an n = 5 polytropic configuration — we obtain,
which corresponds to a maximum mass for pressurebounded n = 5 polytropic configurations of,
This result can be compared to other determinations of the BonnorEbert mass limit.
More Precise Form Factors
Here we attempt to determine proper expressions for several form factors such that the equilibrium configurations determined from virial analysis will precisely match the pressuretruncated polytropic configurations that have been determined from detailed forcebalanced models that have been derived and published separately by Horedt (1970), by Whitworth (1981) and by Stahler (1983). It seems simplest to begin with the freeenergy expressions that we have already generalized in the context of bipolytropic configurations, properly modified to embed the "core" in an external medium of pressure, , rather then inside an envelope that has a different polytropic index. Specifically,






where,









Virial equilibrium occurs where , that is, when,



n = 5 Polytropic
From our analysis of the free energy of bipolytropes, we deduce that the coefficient, , that quantifies the gravitational potential energy of a pressuretruncated polytrope is,



the coefficient, , that quantifies the thermal energy content of a pressuretruncated polytrope is,



and the coefficient, — which can be obtained by setting in expressions drawn from out analysis of the thermal energy content of an bipolytrope — is,



Hence, virial equilibrium occurs when,



that is, when,


















So this will always be true!
The detailed forcebalanced analysis of polytropes shows that the pair of equations defining the equilibrium (truncated) radius for a specified external pressure are,
.
Applying our chosen normalizations, this pair of defining equations becomes,






Relationship to Detailed Force Balance Solution
Let's plug these formfactors into our expressions for the dimensionless equilibrium radius, , and dimensionless surface pressure, , that have been derived from the identification of extrema in the freeenergy function and see how they compare to the dimensionless radius, , and dimensionless pressure, , given by Horedt's (1970) detailed forcebalance models. Note that expressions for and are given in our accompanying discussion of embedded polytropic spheres and that the conversion from Hoerdt's scaling parameters to our normalization parameters, and , can be found in our introductory discussion of the virial equilibrium of spherical configurations.
























Now we insert the formfactor expressions from above, to obtain,
























Notice that the bracketed term that is to be evaluated at is identical in both expressions. After renormalization, as drived above, the statement of virial equilibrium for embedded polytropes is,



or, setting ,



This relation can now be written in terms of Horedt's (1970) dimensionless radius and pressure.






where, from our separate summary of Hoerdt's presentation,






Hence, the virial relation becomes,









This needs to be checked, perhaps with specific applications to the cases and .
See Also
© 2014  2021 by Joel E. Tohline 