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Virial Equilibrium of Adiabatic Spheres (Summary)
The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressuretruncated polytropes.
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Detailed ForceBalanced Solution
As has been discussed in detail in another chapter, Horedt (1970), Whitworth (1981) and Stahler (1983) have separately derived what the equilibrium radius, , is of a polytropic sphere that is embedded in an external medium of pressure, . Their solution of the detailed forcebalanced equations provides a pair of analytic expressions for and that are parametrically related to one another through the LaneEmden function, , and its radial derivative. For example — see our related discussion for more details — from Horedt's work we obtain the following pair of equations:






where we have introduced the normalizations,






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 Horedt intended for the normalization mass to be the mass of the pressuretruncated object, not the mass of the associated isolated (and untruncated) polytrope.
From these previously published works, it is not obvious how — or even whether — this pair of parametric equations can be combined to directly show how the equilibrium radius depends on the value of the external pressure. Our examination of the freeenergy of these configurations and, especially, an application of the viral theorem shows this direct relationship. Foreshadowing these results, we note that,



or, given that , this can be rewritten as,



Free Energy Function and Virial Theorem
The variation with size of the normalized free energy, , of pressuretruncated adiabatic spheres is described by the following,
Algebraic FreeEnergy Function
In this expression, the size of the configuration is set by the value of the dimensionless radius, ; as is clarified, below, the values of the coefficients, and , characterize the relative importance, respectively, of the gravitational potential energy and the internal thermal energy of the configuration; is the exponent (from the adopted equation of state) that identifies the adiabat along which the configuration heats or cools upon expansion or contraction; and the relative importance of the imposed external pressure is expressed through the freeenergy expression's third constant coefficient, specifically,
When examining a range of physically reasonable configuration sizes for a given choice of the constants , a plot of versus will often reveal one or two extrema. Each extremum is associated with an equilibrium radius, .
Equilibrium radii may also be identified through an algebraic relation that originates from the scalar virial theorem — a theorem that, itself, is derivable from the freeenergy expression by setting . In our accompanying detailed analysis of the structure of pressuretruncated polytropes, we use the virial theorem to show that the equilibrium radii that are identified by extrema in the freeenergy function always satisfy the following,
Algebraic Expression of the Virial Theorem
where, after setting ,


and, 



The curves shown in the accompanying "pressureradius" diagram trace out this derived virialtheorem function for six different values of the adiabatic exponent, , as labeled. They show the dimensionless external pressure, , that is required to construct a nonrotating, selfgravitating, adiabatic sphere with a dimensionless equilibrium radius . The mathematical solution becomes unphysical wherever the pressure becomes negative.
If we multiply the above free=energy function through by an appropriate combination of the coefficients, and , and make the substitution, , it also takes on a particularly simple form featuring the newly defined dimensionless external pressure, , and the newly identified dimensionless radius, . Specifically, we obtain the,
Renormalized FreeEnergy Function
Relationship to Detailed ForceBalanced Models
Structural Form Factors
In our accompanying detailed analysis, we demonstrate that the expressions given above for the freeenergy function and the virial theorem are correct in sufficiently strict detail that they can be used to precisely match — and assist in understanding — the equilibrium of embedded polytropes whose structures have been determined from the set of detailed forcebalance equations. In order to draw this association, it is only necessary to realize that, very broadly, the constant coefficients, and , in the above algebraic freeenergy expression are expressible in terms of three structural form factors, , , and , as follows:






and that, specifically in the context of spherically symmetric, pressuretruncated polytropes, we can write …









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 
After plugging these nontrivial expressions for and into the righthand sides of the above equations for and and, simultaneously, using Horedt's detailed forcebalanced expressions for and to specify, respectively, and in these same equations, we find that,






where the newly identified, key physical parameter,



It is straightforward to show that this more compact pair of expressions for and satisfy the virial theorem presented above.
Physical Meaning of Parameter
As defined in our above discussion, is the ratio of the two terms that are summed together in the definition of the structural form factor, . It is worth pointing out what physical quantities are associated with these two terms.
At any radial location within a polytropic configuration, the LaneEmden function, , is defined in terms of a ratio of the local density to the configuration's central density, specifically,
Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,
we see that,
Hence, the quantity, , which appears as the second term in our definition of , is the ratio, , evaluated at the surface of the truncated polytropic sphere. But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
In our accompanying detailed analysis, we have employed the virial theorem expression to demonstrate that the first term in our definition of provides a measure the configuration's normalized central pressure. Specifically, we show that,



We conclude, therefore, that quite generally,






and that,



Desired PressureRadius Relation
It is now clear from our review, above, of Horedt's detailed forcebalanced solution, that



Hence, the pair of parametric equations obtained via a solution of the detailed forcebalanced equations satisfy our, slightly rearranged,
Algebraic Expression of the Virial Theorem
More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by Horedt, , and the dimensionless applied external pressure as defined by Horedt, , that was not apparent from the original pair of parametric relations. Horedt's parameters, and , can be directly associated to our parameters, and , via two new normalizations, and , defined through the relations,

and 

Specifically in terms of the coefficients in the freeenergy expression,



and,



while, in terms of the structural form factors,



and,



Discussion
Model Sequences
After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), , the functional form of the virial theorem expression, , is known and, hence, the equilibrium model sequence can be plotted. Halfadozen such model sequences are shown in the figure near the beginning of this discussion. Each curve can be viewed as mapping out a singleparameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, , over the physically relevant range, . To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.









When is zero
For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, , can be zero for one of two reasons: Either ; or . In the latter case, all curves converge on the same point, that is, . This corresponds to the case of no external medium and, hence, the associated equilibrium configuration is the familiar isolated polytropic sphere. As can be deduced from our above discussion of the algebraic expression of the virial theorem, because , the equilibrium radius of such a configuration is,



As is demonstrated in an accompanying discussion and also mentioned above, after inserting the relevant expressions for the freeenergy coefficients, and , this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,



As we have reviewed elsewhere — see also our detailed discussion of isolated polytropes — this is a familiar relationship, appearing prominently in Chapter IV (p. 99, equations 80 and 81) of Chandrasekhar [C67] in association with his discussion of the dimensionless coefficient, , and the central pressure of polytropes.
In the former case — that is, in the case where because the chosen polytropic index is, — it must be the case that along the entire sequence (see the green curve labeled in the accompanying figure). This means that the expression for the central pressure,



does not explicitly depend on the size of the applied external pressure. But the central pressure does depend on the radial location at which the configuration is truncated, via the parameter , which is evaluated at , rather than at .
Stability
Analysis of the freeenergy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressuretruncated polytropic configurations, but also the relative stability of these configurations. We begin by repeating the,
Renormalized FreeEnergy Function
The first and second derivatives of , with respect to the dimensionless radius, , are, respectively,






As alluded to, above, equilibrium radii are identified by values of that satisfy the equation, . Specifically, marking equilibrium radii with the subscript "ad", they will satisfy the
Algebraic Expression of the Virial Theorem
Dynamical stability then depends on the sign of the second derivative of , evaluated at the equilibrium radius; specifically, configurations will be stable if,


(stable) 
and they will be unstable if, upon evaluation at the equilibrium radius, the sign of the second derivative is less than zero. Hence, isolated polytropes as well as pressuretruncated polytropic configurations will be stable if,











(stable) 
Reference to this stability condition proves to be simpler if we define the limiting configuration size as,
and write the stability condition as,
(stable)
When examining the equilibrium sequences found in the upperrighthand quadrant of the figure at the top of this page — each corresponding to a different value of the polytropic index, or — we find that corresponds to the location along each sequence where the dimensionless external pressure, , reaches a maximum. (Keeping in mind that the virial theorem defines each of these sequences, this statement of fact can be checked by identifying where the condition, , occurs according to the algebraic expression of the virial theorem.) Hence, we conclude that, along each sequence, no equilibrium configurations exist for values of the dimensionless external pressure that are greater than,












In the context of a general examination of the freeenergy of pressuretruncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, , that appears in the free energy function. Specifically, we will not expect to find any extrema in the free energy if,



Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, always corresponds to,
Summary  


Try Polytropic Index of 4
Groundwork
In an effort to more fully understand what can be learned from an examination of the freeenergy, let's play with polytropic models. First, let's plot using a specific, trial value of the coefficient, , keeping in mind that,









At the top of the table, shown below, we display a plot of the,
Renormalized FreeEnergy Function
where we have set , and . Reading quantities off of the plot, the left and right extrema identify equilibria having the following approximate dimensionless radii: and . Upon closer examination (plots not shown), we have determined that, and . In accordance with our stability analysis, these values of fall on either side of the demarcation value, , with the one on the left being a local maximum in the free energy — indicating an unstable equilibrium — while the one on the right is a local minimum — indicating a stable equilibrium. Next, let's check to see if both extrema satisfy the,
Algebraic Expression of the Virial Theorem
For the unstable equilibrium configuration, we calculate,
;
while, for the stable equilibrium we calculate,
.
Because we inserted a value of into the freeenergy expression, we conclude that, as desired, both identified extrema satisfy the virial relation to the measured accuracy. These parameter values, and the corresponding values of many other related physical parameters are summarized in the following table, along with the algebraic relations that were used to calculate them.
First Table
Determined from Plot of Renormalized FreeEnergy with 


Maximum  Minimum  
Immediate Implications from Virial Theorem 

Associated Detailed ForceBalanced Model Parameters obtained via interpolation of tabulated numbers on p. 399 of Horedt (1986, ApJS, vol. 126) 

(approx.)  
(approx.)  
(approx.)  
(check)  
and, hence, Implied Structural Form Factors & Coefficients & 

Given , , and , we obtain 

Compare with Horedt's Equilibrium Parameters obtained from DFB Models 

Now, we are convinced that both extrema identify perfectly valid equilibrium configurations. However, in the context of astrophysics, the two identified equilibria are not connected to one another in any meaningful way. In particular, two of the freeenergy coefficients, and , have different values in the two cases; and, by inference, the normalized external pressure, , is different in the two cases. So the plotted freeenergy curve does not represent a "constant pressure" evolutionary trajectory. How do we identify two equilibria that are associated with the same normalized external pressure? And how do we identify the freeenergy "evolutionary trajectory" that connects the two states?
Second Table
Here, we have decided to look for a stable equilibrium state that is bounded by the same external pressure as the unstable state that has been identified in the above figure and table. Rather than going straight to the freeenergy expression in search of the desired stable configuration, we cheated a bit. Using the properties of an polytrope, as tabulated on p. 399 of Horedt (1986, ApJS, vol. 126), in conjunction with the algebraic expression found in the nexttolast row of the above table, namely,
we examined how varies with . We found that at , which is almost identical to the value of the normalized external pressure that we determined was associated with the unstable equilibrium state (at ) above. As is illustrated by the figure and table that follows, we determined that the stable equilibrium state associated with this normalized external pressure is the minimum that occurs on the free energy curve having parameters, .
Determined from Plot of Renormalized FreeEnergy with 


Maximum  Minimum  
Immediate Implications from Virial Theorem 

Associated Detailed ForceBalanced Model Parameters obtained via interpolation of tabulated numbers on p. 399 of Horedt (1986, ApJS, vol. 126) 

(approx.)    
(approx.)    
(approx.)    
(check)    
and, hence, Implied Structural Form Factors & Coefficients & 

  
  
  
  
Given , , and , we obtain 

  
  
Compare with Horedt's Equilibrium Parameters obtained from DFB Models 

  
 
Summary
The algebraic freeenergy function associated with pressuretruncated polytropes is,
and the corresponding renormalized freeenergy function is,
As has been demonstrated, above, the two equilibrium states that are supported by the same external pressure of, , are associated with extrema found in the following freeenergy curves: The unstable equilibrium appears as a relative maximum in the freeenergy curves having the coefficient values,
or
The stable equilibrium appears as a relative minimum in the freeenergy curves having the coefficient values,
or
Configurations Sharing the Same External Pressure 


ASIDE: In retrospect, it is obvious that pairs of truncated equilibrium configurations of a given polytropic index that are bounded by the same external pressure — and, hence, that may share a physical evolutionary connection — will share the same value of Horedt's dimensionless pressure,

The implication is that a single freeenergy curve with constant coefficients cannot connect the two equilibrium states. There are certainly two separate equilibrium states that can be supported by the specified external pressure, but these two states exhibit somewhat different values of the structural form factors, which leads to different values of the coefficient, . The righthand plot in the following figure shows how varies with the applied external pressure in polytropes.
Variation of Various Physical Parameters along the Sequence of PressureTruncated Polytropes [Structural data obtained from the table provided on p. 399 of Horedt (1986, ApJS, vol. 126)] 

Left: This loglog plot displays the variation with applied external pressure, (increasing to the right along the horizontal axis), of the renormalized pressure, (light blue diamonds), the renormalized equilibrium radius, (light green triangles), and the key physical parameter, (maroon circles). As the diagram illustrates, each parameter is doublevalued, demonstrating that, for any choice of the dimensionless external pressure (as long as the pressure is less than a welldefined limiting value), there are two available equilibrium states. Along all three curves, parameter values associated with the stable equilibrium are traced by the upper portion of the curve. The red vertical line has been drawn at the value of , corresponding to the external pressure examined in the above two tables. This red line intersects the curve at (unstable state examined above) and at (stable state examined above). Right: This plot (linear scale on both axes) shows how (curve outlined by light blue diamonds) varies with the applied external pressure, , in polytropes. The curve bends back on itself, showing that at any value of , below some limiting value, two equilibrium configurations exist and they have different values of . The vertical red line identifies the value of the external pressure that has been used as an example in the above two tables to illustrate how a pair of physically associated equilibrium states can be identified. This red line intersects the displayed curve at (unstable state examined above) and at (stable state examined above). 
Curiosity
The figure displayed here, on the right, is a magnification of a segment of the curve (light blue diamonds) shown in the lefthand panel of the preceding figure, although here we have used a linear, rather than a log, scale on both axes. The quantity being plotted along both axes is the external pressure, but normalized in different ways. The quantity, (horizontal axis), provides a direct measure of the physical external (hence, also, surface) pressure, while the quantity, (vertical axis), is the external pressure renormalized by a specific combination of the freeenergy coefficients. Our stability analysis has been conducted assuming that the freeenergy coefficients — which are expressible in terms of structural form factors — are constants, that is, they do not vary with the size of the configuration. Hence, it is the limiting value of , specifically,



that identifies the demarcation between stable and unstable states. This limiting value is identified by the horizontal reddashed line in the figure; and the relevant demarcation point appears where this tangent line touches the curve. According to our stability analysis, equilibrium configurations to the left of this demarcation point are stable while configurations to the right are unstable.
In the context of our discussion of the lefthand diagram in the preceding figure — see especially the relevant figure caption — we claimed that, for each physically allowed value of the external pressure, , the parameter, , was doublevalued and that configurations along the upper segment of its curve were stable. After studying a magnification of this parameter curve near its turning point, a bit of clarification is required. It appears as though equilibrium models lying along the short upper segment of the curve that falls between the demarcation/tangent point at and the maximum value of are unstable. This means that, even though two equilibrium configurations can be constructed at each value of in this region near and including the turning point, both configurations are dynamically unstable. We conclude, therefore, that stable configurations only exist for values of that are less than the value associated with .
MassRadius Relation
Up to this point in our discussion, we have focused on an analysis of the pressureradius relationship that defines the equilibrium configurations of pressuretruncated polytropes. In effect, we have viewed the problem through the same lens as did Horedt (1970) and, separately, Whitworth (1981), defining variable normalizations in terms of the polytropic constant, , and the configuration mass, , which were both assumed to be held fixed throughout the analysis. Here we switch to the approach championed by Stahler (1983), defining variable normalizations in terms of and , and examining the massradius relationship of pressuretruncated polytropes.
Detailed ForceBalanced Solution
As has been summarized in our accompanying review of detailed forcebalanced models of pressuretruncated polytropes, Stahler (1983) found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:






where,
Mapping from Above Discussion
Looking back on the definitions of and that we introduced in connection with our initial concise algebraic expression of the virial theorem, we can write,












The first of these two expressions can be flipped around to give an expression for in terms of and, then, as normalized to . Specifically,









This means, as well, that we can rewrite the equilibrium radius as,





















Flipping both of these expressions around, we see that,



and,









Hence, our earlier derived compact expression for the virial theorem becomes,









Or, rearranged,

After adopting modified length and massnormalizations, and , such that,






we obtain the
Virial Theorem in terms of Mass and Radius
This analytic function is plotted for seven different values of the polytropic index, , as indicated, in the lefthand diagram of the following table.
Figure 17 extracted^{†}from p. 184 of S. W. Stahler (1983)
"The Equilibria of Rotating Isothermal Clouds. II. Structure and Dynamical Stability"
ApJ, vol. 268, pp. 165184 © American Astronomical Society 

^{†}Figure displayed here, as a digital image, has been modified from the original publication only via the addition of the word "SCHEMATIC". 
Now that we have this very general, yet concise, algebraic expression for the massradius relationship of all pressuretruncated polytropes, let's replace the new "mod" normalizations with Stahler's original normalizations, and , to understand more completely how this general expression should be viewed in relation to the parametric relations provided by solutions of the detailed forcebalanced models. We will henceforth use the notation,






In the virial theorem expression we will make the following replacements:






Next, we recognize that, in order to graphically display the massradius relation derived from the virial theorem in the plane, we must write out the expressions for the freeenergy coefficients. After setting in the above summary expressions, we obtain,






In an effort not to be caught dividing by zero while investigating the specific case of polytropes, we will adopt as shorthand notation,
Hence, the replacements become,












and, in particular, the cross term in the virial theorem expression becomes,



Via these replacements the concise and general Virial Theorem expression derived above morphs into the,
Virial Theorem written in terms of , , and
where the leading coefficient is,



In carrying out this last derivation we could be accused of reinventing the wheel, as the expression inside the curly braces is simply times the virial expression presented inside an outlined box, above, just before we introduced the modified normalization parameters, and .
Relating and Reconciling Two MassRadius Relationships for n = 5 Polytropes
Now, let's examine the case of pressuretruncated, polytropes. As we have discussed in the context of detailed forcebalanced models, Stahler (1983) has deduced that all equilibrium configurations obey the massradius relationship,



where, as reviewed above, the mass and radius normalizations, and , may be treated as constants once the parameters and are specified. In contrast to this, the massradius relationship that we have just derived from the virial theorem for pressuretruncated, polytropes is,
where the mass and radius normalizations,






depend, not only on and via the definitions of and , but also on the structural form factors via the freeenergy coefficients, and . While these two separate massradius relationships are similar, they are not identical. In particular, the middle term involving the crossproduct of the mass and radius contains different exponents in the two expressions. It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.
This apparent discrepancy is reconciled as follows: The structural form factors — and, hence, the freeenergy coefficients — vary from equilibrium configuration to equilibrium configuration. So it does not make sense to discuss evolution along the sequence that is defined by the second of the two polynomial expressions. If you want to know how a given system's equilibrium radius will change as its mass changes, the first of the two polynomials will do the trick. However, the equilibrium radius of a given system can be found by looking for extrema in the freeenergy function while holding the freeenergy coefficients, and , constant; more importantly, the relative stability of a given equilibrium system can be determined by analyzing the behavior of the system's free energy while holding the freeenergy coefficients constant. Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression. It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other.
Plotting Stahler's Relation
Switching, again, to the shorthand notation,






the equilibrium massradius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways. One way is to recognize that the polynomial is a quadratic equation whose solution is,



In the figure shown here on the right — see also the bottom panel of Figure 2 in our accompanying discussion of detailed forcebalance models — Stahler's massradius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the positive root while the segment derived from the negative root is shown in orange. The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at
We note that, when , . Along the entire sequence, the maximum value of occurs at the location where along the segment of the curve corresponding to the positive root. This occurs along the upper segment of the curve where , at the location,
The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations. Drawing partly from our above discussion and partly from a separate discussion where we provide a tabular summary of the properties of pressuretruncated polytropes, these are,






The entire sequence will be traversed by varying the LaneEmden parameter, , from zero to infinity. Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, , is associated with an embedded polytrope whose truncated, dimensionless LaneEmden radius is,
Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, , is associated with an embedded polytrope whose truncated, dimensionless LaneEmden radius is, precisely,
Plotting the Virial Theorem Relation
The relevant relation is obtained by plugging into the general massradius relation derived above, repeated here for clarity:
where,

We will begin by plugging into these expressions everywhere except for the coefficient , which we will leave unresolved, for the time being, in order to better appreciate the interplay of various terms. We obtain,



where, if they are to be assigned values that are actually associated with a particular detailed forcebalance model having truncation radius, ,






Now, if we plug into the remaining unresolved coefficient, the righthand side goes to zero and the massradius relationship provided by the virial theorem becomes,









Hence, for a given value of the structural form factor(s) — which implies a specific value of the constant coefficient, — the scalar virial theorem defines a relationship where the normalized mass varies as the square root of the normalized radius . On the other hand, if we demand that the expression inside the square brackets on the righthand side of the virial theorem relation go to zero on its own — without relying on the leading coefficient to knock it zero — the massradius relationship provided by the virial theorem becomes,









Relating and Reconciling Two MassRadius Relationships for n = 4 Polytropes
For pressuretruncated polytropes, Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed forcebalance models (summarized above), we appreciate that the governing pair of parametric relations is,






On the other hand, the polynomial that results from plugging into the general massradius relation that is obtained via the virial theorem is,
where,
[For the record we note that, throughout the structure of an polytrope, is a number of order unity. Its value is never less than , which pertains to the center of the configuration; its maximum value of occurs at ; and at its (zero pressure) surface, . A plot showing the variation with of the closely allied parameter, is presented in the righthand panel of the above parameter summary figure.]
In both panels of the following figure, the blue curve displays the massradius relation for pressuretruncated polytropes, , that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of Horedt (1986, ApJS, vol. 126)] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangleshaped points traces the massradius relation, , that is obtained via the virial theorem, assuming that the coefficient, , is constant along the sequence. The "green" sequence in the lefthand panel results from setting , which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at ; the "orange" sequence in the righthand panel results from setting , which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at .
Comparing Two Separate MassRadius Relations for PressureTruncated n = 4 Polytropes 

According to Horedt's (1986)] tabulated data, the surface of an isolated , spherically symmetric, polytrope occurs at the dimensionless (LaneEmden) radius, . In both panels of the above figure, this isolated configuration is identified by the discrete (blue diamond) point at the origin, that is, at . As we begin to examine pressuretruncated models and is steadily decreased from , the massradius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) massradius relation. A maximum mass of (corresponding to a radius of ) is reached from the left as drops to a value of approximately . As continues to decrease, the massradius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at — corresponding to — then decreasing in radius until, once again, the origin is reached, but this time because drops to zero.
If we set (corresponding to a choice of ), the virial theorem massradius relation maps onto the "Stahler" massradius coordinate plane as depicted by the set of green, triangleshaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue massradius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed forcebalanced model having . In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangleshaped points shows how the virial theorem massradius relation maps onto the "Stahler" massradius coordinate plane when we set (corresponding to a choice of ); it intersects the blue massradius relation precisely at the coordinate location, — again, emphasized by a black filled circle — that corresponds to a detailed forcebalanced model having . Hence, the two relations give the same massradius coordinates when the value of that is plugged into the virial theorem matches the value of that reflects the structural form factor that is properly associated with a detailed forcebalanced model.
When we mapped the virial theorem massradius relation onto Stahler's massradius coordinate plane using a value of (as traced by the orange triangleshaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where , for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, — also highlighted by a black filled circle — that corresponds to a detailed forcebalanced model having . This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (LaneEmden) radius, , that assures precise agreement between the two different massradius expressions.
As is detailed in our above discussion of the dynamical stability of pressuretruncated polytropes, an examination of freeenergy variations can not only assist us in identifying the properties of equilibrium configurations (via a freeenergy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our above groundwork derivations, for polytropic structures, the critical point is identified by the dimensionless parameters,
and 
In the context of the above figure, independent of the chosen value of , this critical point always corresponds to the maximum mass that occurs along the massradius relationship established via the virial theorem. In both panels of the figure, a horizontal reddotted line has been drawn tangent to this critical point and identifies the corresponding critical value of ; a vertical reddashed line drawn through this same point helps identify the corresponding critical value of . We have deduced (details of the derivation not shown) that, for pressuretruncated polytropes, the coordinates of this critical point in Stahler's plane depends on the choice of as follows:






In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, — it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two massradius relations. If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable. On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable. We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is stable; however, both configurations identified by filled black circles in the righthand panel are unstable.
It is significant that the critical point identified by our freeenergybased stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence. One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximummass point are stable. But the righthand panel of our above figure contradicts this expectation. While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies to the right of the maximummass point along the blue "Stahler" sequence. This finding is related to the curiosity raised earlier in our discussion of the structural properties of pressuretruncated, polytropes.
Relating and Reconciling Two MassRadius Relationships for n = 3 Polytropes
For pressuretruncated polytropes, Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed forcebalance models (summarized above), we appreciate that the governing pair of parametric relations is,






On the other hand, the polynomial that results from plugging into the general massradius relation that is obtained via the virial theorem is,
where,
Summary Comments
From our above, detailed analysis of the massradius relation for pressuretruncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximummass configuration. Instead, the onset of dynamical instability is associated with the critical point on the massradius relation that arises from the freeenergybased virial theorem. In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure constant, but also assuming that the configuration's structural form factors are invariable.
This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary dynamically when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different and desirable set of form factors. In support of this argument, we point to the paper by Goldreich & Weber (1980) which explicitly derives a selfsimilar solution for the homologous collapse of stellar cores that can be modeled as polytropes; an associated chapter of this H_Book details the Goldreich & Weber derivation. Goldreich & Weber use linear perturbation techniques to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressuretruncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of Schwarzschild (1941, ApJ, 94, 245), who has evaluated radial modes, and of Cowling (1941, MNRAS, 101, 367), who has obtained eigenvalues of some loworder nonradial modes.)
In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors. In order to test this underlying assumption, following Goldreich & Weber (1980), it would be desirable to carry out a fullblown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation. Ideally, we should be using the structural form factors associated with this pulsationmode eigenfunction in our freeenergy analysis of stability. Better yet, the sign of the eigenfrequency associated with the system's pulsationmode eigenvector should signal whether the system is dynamically stable or unstable.
Serious Concern
Statement of Concern
Throughout our discussion of embedded (pressuretruncated) polytropes — both on this "summary" page and in an accompanying chapter where critical background derivations are presented — we have used expressions for the structural form factors that include an overall leading factor of . For clarity, the form factors that we have used for isolated polytropes is reprinted on the lefthand side of the following table while the ones that we have used for pressuretruncated polytropes is reprinted on the righthand side of the table.
Structural Form Factors for Isolated Polytropes 
Structural Form Factors for PressureTruncated Polytropes 




This factor seemed destined to become a nuisance in the specific case of polytropic structures. But we did not let its appearance in these expressions deter us from using a freeenergy analysis to study the equilibrium and stability of spherical polytropes because, after all, the factor of appears in Chandrasekhar's [C67] expression for the gravitational potential energy of isolated polytropes — see his Equation (90), p. 101. In retrospect, its appearance in the structural form factors for isolated polytropes did not prove to be a problem because, via a freeenergy and virial theorem analysis, the derived expression for the configuration's equilibrium radius depends on the ratio of to , so the awkward factor of cancels out.
However, in our discussion of pressuretruncated polytropic structures, the factor of did not conveniently cancel out at the appropriate time and we were forced to carry out some logical contortions as we tried to compare the massradius relation obtained from the virial theorem to Stahler's massradius relation, which was derived from detailed forcebalance arguments. This leads us to seriously question whether our, rather casually derived, expressions for the structural form factors in pressuretruncated polytropes are correct.
Further Evaluation of n = 5 Polytropic Structures
Throughout most of this subsection, we will adopt the shorthand notation,

This will not only simplify the appearance of some expressions, it will facilitate direct comparison with an expression for the freeenergy coefficient, , that has been derived in a companion chapter following a different train of logic and with an expression for the normalized gravitational potential energy that has been derived via a bruteforce integration in association with our discussion of bipolytropic configurations where the variable, , has the same definition as .
FreeEnergy Expression
From our general review of the topic, to within an additive constant, the freeenergy of a nonrotating, pressuretruncated polytrope comes from the sum of three principal energy terms, namely,
Furthermore, as has been shown in our extended introductory discussion of free energy, the corresponding normalized free energy (applied to or configurations) is,
where,












Also note that the relevant normalizations are,
Virial Theorem
The traditional expression for the virial theorem in this context is,
From our introductory discussion of the thermodynamic energy reservoir, we know that, for configurations,
So, making this substitution and recognizing that , the (normalized) virial theorem expression becomes,
Furthermore, by comparing terms in the first freeenergy expression, above, with the second (normalized) freeenergy expression, we see that,
Hence, the normalized virial theorem may be written as,
For sake of consistency, let's check this by holding the coefficients , , and constant and setting equal to zero:






This, in turn, implies that the expression inside the square brackets sums to zero, which identically matches the (normalized) traditional virial theorem expression. Excellent!
Borrowing from Bipolytrope Discussion
In an accompanying chapter that presents the detailed forcebalanced models of bipolytropes we explicitly show that, for configurations with the correct equilibrium radius, the virial theorem is satisfied. In the case of bipolytropes, which are not embedded in an external medium, the relevant normalized virial theorem states that,
In the bipolytrope, the (truncated) core is confined by an envelope; in addition to demanding that the relevant virial theorem be satisfied, there is also a constraint that the pressure at the inner edge of the envelope be equal to the pressure at the (truncated) outer edge of the core. As we have just discussed, for a (truncated) polytrope that is confined by a hot, tenuous external medium instead of by an enveloping envelope, the relevant normalized virial theorem is,









where, as discussed/defined in an accompanying chapter of this H_Book, we have adopted the normalization radius, , first introduced by Steven W. Stahler (1983). For configurations, its definition is,
As has also been discussed in the accompanying chapter, we can deduce from Stahler's detailed forcebalanced models that the equilibrium radius of embedded, polytropes is given in terms of the dimensionless, truncated LaneEmden radius, — and our corresponding variable, — by the expression,






Hence, upon careful evaluation of the thermal energy and gravitational potential energy of truncated polytropes, we should find that,






Well, it just so happens that, in our accompanying chapter that presents the detailed forcebalanced models of bipolytropes, we explicitly carried out the volume integrals defining these two key components of the free energy expression with the results being,



Realizing that the variable, , in that context is the same as , in the present context, we see that the two separately derived results are identical to one another.
Determining Expressions for FreeEnergy Coefficients
We should be able to convert the separately derived expression for into an expression for the freeenergy coefficient, , in equilibrium configurations. As noted above, for a fixed value of ,
Therefore, in an equilibrium configuration, we can write,






Now, from immediately above, we know that,



and, from our accompanying discussion of the freeenergy of bipolytropic configurations, we know that,



So, again realizing that and are interchangeable, we have,






Finally, we need to determine an expression for the ratio, . Drawing the definition of from our introductory chapter on the virial equilibrium, we have,



From our tabular summary of Stahler's derived mass & radius relationships for truncated, polytropes we have,



In addition, from our review of Stahler's defined normalizations, we see that,



and, 


which, when combined to cancel gives,






Hence, we can write,






In combination with the expression for , then, we have,



which means that, for truncated polytropes, the expression for the freeenergy coefficient is,






Finally, drawing from our accompanying derivation of expressions for the structural form factors in this case, we know that,



which gives,



This exactly matches the expression for the freeenergy coefficient, , that we derived separately in conjunction with our derivation of expressions for the structural form factors.
Take Care Comparing Gravitational Potential Energies
Does this derived relation for the coefficient, , make sense? Well, we've derived the relation by comparing two separate expressions for the gravitational potential energy that were normalized in slightly different ways, so the leading numerical coefficient may not be correct. We need to repeat the derivation, checking the relative normalizations carefully. But before doing this, let's determine what we expected the relation to be, based on the expressions for the structural form factors that we have been using.
From the leadin paragraphs of this subsection, we have previously assumed that,



According to the line of reasoning presented above, the coefficient, , is related to the gravitational potential energy via the expression,









From our introductory layout of the freeenergy function for polytropes — see, also, p. 64, Equation (12) of Chandrasekhar [C67] — the gravitational potential energy is,



where,



Now, independent of the chosen normalization, if we use to represent the total mass of an isolated polytrope, then from an earlier review, we have,



and we can write, in terms of the LaneEmden dimensionless radius, ,
Virial Chapter
Now, in our discussion of the virial equilibrium of embedded polytropes, we used the normalizations specified above and wrote,






We can replace with by recognizing that,





















Hence,



Also,















Hence,






So the energy integral becomes,






This needs to be compared with the integral that we previously have handled in the chapter discussing bipolytrope models.
© 2014  2021 by Joel E. Tohline 