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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 pressure-truncated polytropes.

Whitworth's (1981) Isothermal Free-Energy Surface
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Detailed Force-Balanced Solution

As has been discussed in detail in another chapter, Horedt (1970), Whitworth (1981) and Stahler (1983) have separately derived what the equilibrium radius, ~R_\mathrm{eq}, is of a polytropic sphere that is embedded in an external medium of pressure, ~P_e. Their solution of the detailed force-balanced equations provides a pair of analytic expressions for ~R_\mathrm{eq} and ~P_e that are parametrically related to one another through the Lane-Emden function, ~\theta, and its radial derivative. For example — see our related discussion for more details — from Horedt's work we obtain the following pair of equations:


~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = r_a \cdot \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)

~=~


\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)} \, ,


~\frac{P_\mathrm{e}}{P_\mathrm{norm}} = p_a \cdot \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)

~=~


\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} 
\biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)}  \, ,

where we have introduced the normalizations,

~R_\mathrm{norm}

~\equiv

~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,

~P_\mathrm{norm}

~\equiv

~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, .

In the expressions for ~r_a and ~p_a, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, ~\xi_1, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the isolated polytrope, but at the radial coordinate, ~\tilde\xi, where the internal pressure of the isolated polytrope equals ~P_e and at which the embedded polytrope is to be truncated. The coordinate, ~\tilde\xi, therefore identifies the surface of the embedded — or, pressure-truncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to ~M in both defining relations because it is clear that Horedt intended for the normalization mass to be the mass of the pressure-truncated 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 free-energy of these configurations and, especially, an application of the viral theorem shows this direct relationship. Foreshadowing these results, we note that,

~\biggl[ \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr) \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4\biggr]_\mathrm{Horedt}

~=


\biggl[ \frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}} \biggr] 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, ;

or, given that ~P_\mathrm{norm}R_\mathrm{norm}^4 = GM_\mathrm{tot}^2, this can be rewritten as,

~\biggl[ \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt}

~=


\frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}}   \, .


Free Energy Function and Virial Theorem

The variation with size of the normalized free energy, ~\mathfrak{G}^*, of pressure-truncated adiabatic spheres is described by the following,

Algebraic Free-Energy Function


\mathfrak{G}^* = 
-3\mathcal{A} \chi^{-1} +~ \frac{1}{(\gamma - 1)} \mathcal{B} \chi^{3-3\gamma} +~ \mathcal{D}\chi^3 \, .

In this expression, the size of the configuration is set by the value of the dimensionless radius, ~\chi \equiv R/R_\mathrm{norm}; as is clarified, below, the values of the coefficients, ~\mathcal{A} and ~\mathcal{B}, characterize the relative importance, respectively, of the gravitational potential energy and the internal thermal energy of the configuration; ~\gamma 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 free-energy expression's third constant coefficient, specifically,

~\mathcal{D} \equiv \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .

When examining a range of physically reasonable configuration sizes for a given choice of the constants ~(\gamma, \mathcal{A}, \mathcal{B}, \mathcal{D}), a plot of ~\mathfrak{G}^* versus ~\chi will often reveal one or two extrema. Each extremum is associated with an equilibrium radius, ~\chi_\mathrm{eq} \equiv R_\mathrm{eq}/R_\mathrm{norm}.

Equilibrium Adiabatic Pressure-Radius Diagram

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 free-energy expression by setting ~\partial\mathfrak{G}^*/\partial\chi = 0. In our accompanying detailed analysis of the structure of pressure-truncated polytropes, we use the virial theorem to show that the equilibrium radii that are identified by extrema in the free-energy function always satisfy the following,

Algebraic Expression of the Virial Theorem


\Pi_\mathrm{ad} = \frac{(\Chi_\mathrm{ad}^{4-3\gamma} - 1)}{\Chi_\mathrm{ad}^4} \, ,

where, after setting ~\gamma = (n+1)/n,

~\Pi_\mathrm{ad}

~=


~\mathcal{D} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)}
\, ,
        and,

~\Chi_\mathrm{ad}

~=


~\chi_\mathrm{eq} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .

The curves shown in the accompanying "pressure-radius" diagram trace out this derived virial-theorem function for six different values of the adiabatic exponent, ~\gamma, as labeled. They show the dimensionless external pressure, ~\Pi_\mathrm{ad}, that is required to construct a nonrotating, self-gravitating, adiabatic sphere with a dimensionless equilibrium radius ~\Chi_\mathrm{ad}. 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, ~\mathcal{A} and ~\mathcal{B}, and make the substitution, ~\gamma \rightarrow (n+1)/n, it also takes on a particularly simple form featuring the newly defined dimensionless external pressure, ~\Pi_\mathrm{ad}, and the newly identified dimensionless radius, ~\Chi \equiv \chi(\mathcal{B}/\mathcal{A})^{n/(n-3)}. Specifically, we obtain the,

Renormalized Free-Energy Function


\mathfrak{G}^{**} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} = 
-3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, .


Relationship to Detailed Force-Balanced Models

Structural Form Factors

In our accompanying detailed analysis, we demonstrate that the expressions given above for the free-energy 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 force-balance equations. In order to draw this association, it is only necessary to realize that, very broadly, the constant coefficients, ~\mathcal{A} and ~\mathcal{B}, in the above algebraic free-energy expression are expressible in terms of three structural form factors, ~\tilde\mathfrak{f}_M, ~\tilde\mathfrak{f}_W, and ~\tilde\mathfrak{f}_A, as follows:

~\mathcal{A}

~\equiv

\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W \, ,

~\mathcal{B}

~\equiv


\frac{4\pi}{3} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  
\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A 
= \frac{4\pi}{3} 
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3(n+1)/n} \biggr]_\mathrm{eq} 
\cdot \tilde\mathfrak{f}_A \, ;

and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write …

~\tilde\mathfrak{f}_M

~=

~ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \, ,

\tilde\mathfrak{f}_W

~=

~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \, ,

\tilde\mathfrak{f}_A

~=


\frac{3(n+1) }{(5-n)} ~\biggl[ \tilde\theta^' \biggr]^2  + \tilde\theta^{n+1} \, .


January 13, 2015: As is noted in our accompanying outline of work, I no longer believe that ~\mathfrak{f}_W and ~\mathfrak{f}_A 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 sign is under development and therefore may contain incorrect mathematical equations and/or physical misinterpretations.    |   Go Home   |


After plugging these nontrivial expressions for ~\mathcal{A} and ~\mathcal{B} into the righthand sides of the above equations for ~\Pi_\mathrm{ad} and ~\Chi_\mathrm{ad} and, simultaneously, using Horedt's detailed force-balanced expressions for ~r_a and ~p_a to specify, respectively, ~\chi_\mathrm{eq} and ~P_e/P_\mathrm{norm} in these same equations, we find that,

~\Pi_\mathrm{ad}

~=

~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,

~\Chi_\mathrm{ad}

~=

~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,

where the newly identified, key physical parameter,

~\eta_\mathrm{ad}

~\equiv

~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} \, .

It is straightforward to show that this more compact pair of expressions for ~\Pi_\mathrm{ad} and ~\Chi_\mathrm{ad} satisfy the virial theorem presented above.

Physical Meaning of Parameter ~\eta_\mathrm{ad}

As defined in our above discussion, ~\eta_\mathrm{ad} is the ratio of the two terms that are summed together in the definition of the structural form factor, ~\tilde\mathfrak{f}_A. It is worth pointing out what physical quantities are associated with these two terms.

At any radial location within a polytropic configuration, the Lane-Emden function, ~\theta, is defined in terms of a ratio of the local density to the configuration's central density, specifically,

\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .

Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,

P = K\rho^{(n+1)/n} \, ,

we see that,

\frac{P}{P_c} = \theta^{n+1} \, .

Hence, the quantity, ~\tilde\theta^{n+1}, which appears as the second term in our definition of ~\tilde\mathfrak{f}_A, is the ratio, ~(P/P_c)_{\tilde\xi}, 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,

\tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .

In our accompanying detailed analysis, we have employed the virial theorem expression to demonstrate that the first term in our definition of ~\tilde\mathfrak{f}_A provides a measure the configuration's normalized central pressure. Specifically, we show that,

~\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}

~=

~[3 (n+1) (\tilde\theta^')^2]^{-1} \, .

We conclude, therefore, that quite generally,

(5-n) \tilde\mathfrak{f}_A

~=


\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4}  + (5-n) \frac{P_e}{P_c}

 

~=


\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4} \biggl[1  + \eta_\mathrm{ad} \biggr] \, ,

and that,

~\eta_\mathrm{ad}

~=

\biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \, .

Desired Pressure-Radius Relation

It is now clear from our review, above, of Horedt's detailed force-balanced solution, that

\frac{4\pi (5-n)}{3}\biggl[\frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt}

~=

~\eta_\mathrm{ad} \, .

Hence, the pair of parametric equations obtained via a solution of the detailed force-balanced equations satisfy our, slightly rearranged,

Algebraic Expression of the Virial Theorem


\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, .

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, ~r_a, and the dimensionless applied external pressure as defined by Horedt, ~p_a, that was not apparent from the original pair of parametric relations. Horedt's parameters, ~r_a and ~p_a, can be directly associated to our parameters, ~\Chi_\mathrm{ad} and ~\Pi_\mathrm{ad}, via two new normalizations, ~r_n and ~p_n, defined through the relations,

~\Chi_\mathrm{ad} = \frac{r_a}{r_n}

     and     

~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .

Specifically in terms of the coefficients in the free-energy expression,

~r_n^{n-3}

~\equiv

~
\frac{(n+1)^n}{4\pi} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^n
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{-n} \, ,

and,

~p_n^{n-3}

~\equiv

~
\frac{3^{n-3}}{(4\pi)^4 (n+1)^{3(n+1)}} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^{-3(n+1)}
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{4n} \, ;

while, in terms of the structural form factors,

~r_n^{n-3}

~\equiv

~
\frac{1}{3} \biggl[ \frac{(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n \mathfrak{f}_M^{1-n}
\, ,

and,

~p_n^{n-3}

~\equiv

~
\frac{1}{(4\pi)^8} \biggl[ \frac{3\cdot 5^3}{(n+1)^3} \cdot \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W^3} \biggr]^{n+1} \mathfrak{f}_A^{4n}
\, .


Discussion

Model Sequences

Equilibrium Adiabatic Pressure-Radius Diagram

After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), ~\gamma = (n+1)/n, the functional form of the virial theorem expression, ~\Pi_\mathrm{ad}(\chi_\mathrm{ad}), is known and, hence, the equilibrium model sequence can be plotted. Half-a-dozen such model sequences are shown in the figure near the beginning of this discussion. Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, ~\eta_\mathrm{ad}, over the physically relevant range, 0 \le \eta_\mathrm{ad} < \infty. To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.

~\eta_\mathrm{ad}

~\equiv

~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} = \biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}\, ,

~\Pi_\mathrm{ad}

~=

~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,

~\Chi_\mathrm{ad}

~=

~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,

When ~\eta_\mathrm{ad} is zero

For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, ~\eta_\mathrm{ad}, can be zero for one of two reasons: Either ~n=5; or ~\tilde\theta \rightarrow \theta_{\xi_1} = 0. In the latter case, all curves converge on the same point, that is, ~(\Chi_\mathrm{ad}, \Pi_\mathrm{ad}) = (1, 0). This corresponds to the case of no external medium ~(P_e = 0) 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 ~\Chi_\mathrm{ad} = 1, the equilibrium radius of such a configuration is,

~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}

~=

~\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .

As is demonstrated in an accompanying discussion and also mentioned above, after inserting the relevant expressions for the free-energy coefficients, ~\mathcal{A} and ~\mathcal{B}, this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,

~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}

~=

~\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .

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, ~W_n, and the central pressure of polytropes.

In the former case — that is, in the case where ~\eta_\mathrm{ad} \rightarrow 0 because the chosen polytropic index is, ~n=5 — it must be the case that ~\Chi_\mathrm{ad} = 1 along the entire sequence (see the green curve labeled ~\gamma = (n+1)/n = 6/5 in the accompanying figure). This means that the expression for the central pressure,

~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}

~=

~\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,

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 ~\tilde\theta^', which is evaluated at ~\tilde\xi, rather than at ~\xi_1.

Stability

Analysis of the free-energy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressure-truncated polytropic configurations, but also the relative stability of these configurations. We begin by repeating the,

Renormalized Free-Energy Function


\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, .

The first and second derivatives of ~\mathfrak{G}^{**}, with respect to the dimensionless radius, ~\Chi, are, respectively,

~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}

~=

~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,

~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}

~=

~-6 \Chi^{-3} + \frac{3(n+3)}{n} \Chi^{-(2n+3)/n} + 6\Pi_\mathrm{ad} \Chi \, .

As alluded to, above, equilibrium radii are identified by values of ~\Chi that satisfy the equation, \partial\mathfrak{G}^{**}/\partial\Chi = 0. Specifically, marking equilibrium radii with the subscript "ad", they will satisfy the

Algebraic Expression of the Virial Theorem


\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .

Dynamical stability then depends on the sign of the second derivative of ~\mathfrak{G}^{**}, evaluated at the equilibrium radius; specifically, configurations will be stable if,

~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}

~>

~0 \, ,      (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 pressure-truncated polytropic configurations will be stable if,

~0

~<

~3 \Chi_\mathrm{ad}^{-3} \biggl[ - 2 + \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 \biggr]

 

~<

~3 \Chi_\mathrm{ad}^{-3} \biggl\{ \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2[\Chi_\mathrm{ad}^{(n-3)/n} -1] - 2\biggr\}

 

~<

~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n}  - 4\biggr]

\Rightarrow~~~~\Chi_\mathrm{ad}

~>

~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .      (stable)

Reference to this stability condition proves to be simpler if we define the limiting configuration size as,

~\Chi_\mathrm{min} \equiv \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, ,

and write the stability condition as,

~\Chi_\mathrm{ad} > \Chi_\mathrm{min} \, .      (stable)

When examining the equilibrium sequences found in the upper-righthand quadrant of the figure at the top of this page — each corresponding to a different value of the polytropic index, ~n > 3 or ~n < 0 — we find that ~\Chi_\mathrm{min} corresponds to the location along each sequence where the dimensionless external pressure, ~\Pi_\mathrm{ad}, 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, ~\partial\Pi_\mathrm{ad}/\partial\Chi_\mathrm{ad} = 0, 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,

~\Pi_\mathrm{max}

~\equiv

~\Chi_\mathrm{min}^{-4}  \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr]

 

~=

~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)}  - 1 \biggr]

 

~=

~\biggl\{ \biggl[ \frac{3(n+1)}{4n} \biggr]^{4n} \biggl[\frac{n-3}{3(n+1)} \biggr]^{n-3} \biggr\}^{1/(n-3)}

~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}

~=

~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .

In the context of a general examination of the free-energy of pressure-truncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, ~\mathcal{D}, that appears in the free energy function. Specifically, we will not expect to find any extrema in the free energy if,

~\mathcal{D} > \mathcal{D}_\mathrm{max}

~\equiv

~(n-3) \biggl\{ \biggl[ \frac{\mathcal{B}}{4n} \biggr]^{4n}~\biggl[ \frac{3(n+1)}{\mathcal{A}} \biggr]^{3(n+1)} ~\biggr\}^{1/(n-3)} \, .

Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, ~(\Chi_\mathrm{min}, \Pi_\mathrm{max}) always corresponds to,

~\eta_\mathrm{ad} = \eta_\mathrm{crit} \equiv \frac{n-3}{3(n+1)} \, .

Summary

~\eta_\mathrm{crit}

~\equiv

~ \frac{n-3}{3(n+1)}

~\Pi_\mathrm{max}

~\equiv

~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)}

~\Chi_\mathrm{min}

~\equiv


\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}

Try Polytropic Index of 4

Groundwork

In an effort to more fully understand what can be learned from an examination of the free-energy, let's play with ~n=4 polytropic models. First, let's plot ~\mathfrak{G}^{**}(\Chi) using a specific, trial value of the coefficient, ~\Pi_\mathrm{ad}, keeping in mind that,

~\eta_\mathrm{crit}\biggr|_{n=4}

~=

~\frac{1}{15} = 0.066667 \, ;

~\Pi_\mathrm{max}\biggr|_{n=4}

~=

~\frac{15^{15}}{16^{16}} = 0.02373828 \, ;

~\Chi_\mathrm{min}\biggr|_{n=4}

~=

~\biggl( \frac{16}{15} \biggr)^4 = 1.294538 \, .

At the top of the table, shown below, we display a plot of the,

Renormalized Free-Energy Function


\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, ,

where we have set ~n = 4, and ~\Pi_\mathrm{ad} = 0.01. Reading quantities off of the plot, the left and right extrema identify equilibria having the following approximate dimensionless radii: ~\Chi_\mathrm{left} \approx 1.03 and ~\Chi_\mathrm{right} \approx 2.13. Upon closer examination (plots not shown), we have determined that, ~\Chi_\mathrm{left} \approx 1.0494 and ~\Chi_\mathrm{right} \approx 2.13905. In accordance with our stability analysis, these values of ~\Chi_\mathrm{ad} fall on either side of the demarcation value, ~\Chi_\mathrm{min} = (16/15)^4, 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


\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .

For the unstable equilibrium configuration, we calculate,

\Pi_\mathrm{ad} \approx [(1.0494)^{1/4} - 1]/(1.0494)^4 = 1.000024 \times 10^{-2};

while, for the stable equilibrium we calculate,

\Pi_\mathrm{ad} \approx [(2.13905)^{1/4} - 1]/(2.13905)^4 = 1.000018 \times 10^{-2}.

Because we inserted a value of ~\Pi_\mathrm{ad} = 0.01 into the free-energy 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

Dimensionless Free-Energy Curve

Determined from Plot of Renormalized Free-Energy

with ~(n, \Pi_\mathrm{ad}) = (4, 0.01)

    Maximum Minimum
  ~\Chi ~1.0494 ~2.13905

Immediate Implications from Virial Theorem

~\Chi^{1/4} - 1 ~\eta_\mathrm{ad} ~0.012128 ~0.20936
~(\Chi^{1/4} - 1)\cdot \Chi^{-4} ~\Pi_\mathrm{ad} ~1.000024 \times 10^{-2} ~1.000018 \times 10^{-2}

Associated Detailed Force-Balanced Model Parameters

obtained via interpolation of tabulated numbers on p. 399 of Horedt (1986, ApJS, vol. 126)

  ~\tilde\xi (approx.) ~4.81 ~1.624
  ~\tilde\theta (approx.) ~0.251 ~0.709
  ~- \tilde\theta^' (approx.) ~0.0727 ~0.239
~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2} ~\eta (check) ~0.0126 ~0.2091

and, hence, Implied Structural Form Factors & Coefficients ~\mathcal{B} & ~\mathcal{A}

~3(-\tilde\theta^')/\tilde\xi ~\mathfrak{f}_M ~0.0453 ~0.4415
~5[3(-\tilde\theta^')/\tilde\xi]^2 ~\mathfrak{f}_W ~0.01028 ~0.975
~15(-\tilde\theta^')^2 + \tilde\theta^5 ~\mathfrak{f}_A ~0.08028 ~1.036
~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A ~\mathcal{B} ~2.682 ~2.0122
~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} ~\mathcal{A} ~1 ~1

Given ~\Pi_\mathrm{ad}, ~\Chi, and ~\mathcal{B}, we obtain

~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} ~\frac{P_e}{P_\mathrm{norm}} ~1.71 \times 10^4 ~1.72 \times 10^2
~\Chi \mathcal{B}^{-4} ~\chi_\mathrm{eq} ~0.0203 ~0.1305

Compare with Horedt's Equilibrium Parameters obtained from DFB Models

\biggl[  \biggl( \frac{5^3}{4\pi} \biggr) 
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} 
~\frac{P_e}{P_\mathrm{norm}} ~1.76 \times 10^4 ~1.73 \times 10^2

\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}  
~\chi_\mathrm{eq} ~0.0203 ~0.130

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 free-energy coefficients, ~\mathcal{B} and ~\mathcal{D}, have different values in the two cases; and, by inference, the normalized external pressure, ~P_e/P_\mathrm{norm}, is different in the two cases. So the plotted free-energy 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 free-energy "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 free-energy expression in search of the desired stable configuration, we cheated a bit. Using the properties of an ~n=4 polytrope, as tabulated on p. 399 of Horedt (1986, ApJS, vol. 126), in conjunction with the algebraic expression found in the next-to-last row of the above table, namely,


\frac{P_e}{P_\mathrm{norm}} = \biggl[  \biggl( \frac{5^3}{4\pi} \biggr) \tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} \, ,

we examined how ~P_e varies with ~\tilde\xi. We found that ~P_e/P_\mathrm{norm} = 1.71\times 10^4 at ~\tilde\xi = 2.6, which is almost identical to the value of the normalized external pressure that we determined was associated with the unstable equilibrium state (at ~\tilde\xi = 4.81) 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, ~(n, \Pi_\mathrm{ad}) = (4, 0.02369).


Dimensionless Free-Energy Curve

Determined from Plot of Renormalized Free-Energy

with ~(n, \Pi_\mathrm{ad}) = (4, 0.02369)

    Maximum Minimum
  ~\Chi ~1.274 ~1.317

Immediate Implications from Virial Theorem

~\Chi^{1/4} - 1 ~\eta_\mathrm{ad} ~0.0624 ~0.0713
~(\Chi^{1/4} - 1)\cdot \Chi^{-4} ~\Pi_\mathrm{ad} ~0.02369 ~0.02369

Associated Detailed Force-Balanced Model Parameters

obtained via interpolation of tabulated numbers on p. 399 of Horedt (1986, ApJS, vol. 126)

  ~\tilde\xi (approx.) ---- ~2.6
  ~\tilde\theta (approx.) ---- ~0.5048
  ~- \tilde\theta^' (approx.) ---- ~0.175
~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2} ~\eta (check) ---- ~0.0714

and, hence, Implied Structural Form Factors & Coefficients ~\mathcal{B} & ~\mathcal{A}

~3(-\tilde\theta^')/\tilde\xi ~\mathfrak{f}_M ---- ~0.2019
~5[3(-\tilde\theta^')/\tilde\xi]^2 ~\mathfrak{f}_W ---- ~0.2039
~15(-\tilde\theta^')^2 + \tilde\theta^5 ~\mathfrak{f}_A ---- ~0.4922
~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A ~\mathcal{B} ---- ~2.542
~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} ~\mathcal{A} ~1 ~1

Given ~\Pi_\mathrm{ad}, ~\Chi, and ~\mathcal{B}, we obtain

~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} ~\frac{P_e}{P_\mathrm{norm}} ---- ~1.71 \times 10^4
~\Chi \mathcal{B}^{-4} ~\chi_\mathrm{eq} ---- ~0.0316

Compare with Horedt's Equilibrium Parameters obtained from DFB Models

\biggl[  \biggl( \frac{5^3}{4\pi} \biggr) 
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} 
~\frac{P_e}{P_\mathrm{norm}} ---- ~1.71 \times 10^4

\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}  
~\chi_\mathrm{eq} ---- ~0.0316

Summary

The algebraic free-energy function associated with pressure-truncated ~n=4 polytropes is,


\mathfrak{G}^*\biggr|_{n=4} = 
-3\mathcal{A} \chi^{-1} +~ 4\mathcal{B} \chi^{-3/4} +~ \mathcal{D}\chi^3 \, ,

and the corresponding renormalized free-energy function is,


\mathfrak{G}^{**}\biggr|_{n=4} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} = 
-3 \Chi^{-1} +~ 4\Chi^{-3/4} +~ \Pi_\mathrm{ad}\Chi^3 \, .

As has been demonstrated, above, the two equilibrium states that are supported by the same external pressure of, ~P_e/P_\mathrm{norm} = 1.71 \times 10^4, are associated with extrema found in the following free-energy curves: The unstable equilibrium appears as a relative maximum in the free-energy curves having the coefficient values,

~\Pi_\mathrm{ad} = 0.01      or      (\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.682, 7.16\times 10^4) \, .

The stable equilibrium appears as a relative minimum in the free-energy curves having the coefficient values,

~\Pi_\mathrm{ad} = 0.02369      or      (\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.542, 7.16\times 10^4) \, .

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,


~p_a

~\equiv


\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}  \, .


The implication is that a single free-energy 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, ~\mathcal{B}. The righthand plot in the following figure shows how ~\mathcal{B} varies with the applied external pressure in ~n=4 polytropes.


Variation of Various Physical Parameters along the Sequence of Pressure-Truncated ~n=4 Polytropes

[Structural data obtained from the table provided on p. 399 of Horedt (1986, ApJS, vol. 126)]

Parameters for n = 4 Embedded Polytropes

Left: This log-log plot displays the variation with applied external pressure, ~p_a (increasing to the right along the horizontal axis), of the renormalized pressure, ~\Pi_\mathrm{ad} (light blue diamonds), the renormalized equilibrium radius, ~\Chi_\mathrm{ad} (light green triangles), and the key physical parameter, ~\eta_\mathrm{ad} (maroon circles). As the diagram illustrates, each parameter is double-valued, demonstrating that, for any choice of the dimensionless external pressure (as long as the pressure is less than a well-defined 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 ~{p_a} = 0.176, corresponding to the external pressure ~(P_e/P_\mathrm{norm} = 1.71\times 10^4) examined in the above two tables. This red line intersects the ~\Pi(p_a) curve at ~\Pi = 0.01 (unstable state examined above) and at ~\Pi = 0.02369 (stable state examined above).

Right: This plot (linear scale on both axes) shows how ~\mathcal{B} (curve outlined by light blue diamonds) varies with the applied external pressure, ~P_e/P_\mathrm{norm}, in ~n=4 polytropes. The curve bends back on itself, showing that at any value of ~P_e, below some limiting value, two equilibrium configurations exist and they have different values of ~\mathcal{B}. The vertical red line identifies the value of the external pressure ~(P_e/P_\mathrm{norm} = 1.71\times 10^4) 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 ~\mathcal{B} = 2.682 (unstable state examined above) and at ~\mathcal{B} = 2.542 (stable state examined above).


Curiosity

Pressure vs. pressure plot

The figure displayed here, on the right, is a magnification of a segment of the ~\Pi(p_a) 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, ~p_a (horizontal axis), provides a direct measure of the physical external (hence, also, surface) pressure, while the quantity, ~\Pi (vertical axis), is the external pressure renormalized by a specific combination of the free-energy coefficients. Our stability analysis has been conducted assuming that the free-energy 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 ~\Pi_\mathrm{ad}, specifically,

~\Pi_\mathrm{max}\biggr|_{n=4}

~=

~\frac{15^{15}}{16^{16}} = 0.02373828 \, ,

that identifies the demarcation between stable and unstable states. This limiting value is identified by the horizontal red-dashed 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, ~p_a, the parameter, ~\Pi, was double-valued 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 ~\Pi_\mathrm{max} and the maximum value of ~p_a are unstable. This means that, even though two equilibrium configurations can be constructed at each value of ~p_a 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 ~p_a that are less than the value associated with ~\Pi_\mathrm{max}.

Mass-Radius Relation

Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated 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, ~K, and the configuration mass, ~M_\mathrm{tot}, 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 ~K and ~P_e, and examining the mass-radius relationship of pressure-truncated polytropes.

Detailed Force-Balanced Solution

As has been summarized in our accompanying review of detailed force-balanced models of pressure-truncated polytropes, Stahler (1983) found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:


~\frac{M_\mathrm{tot}}{M_\mathrm{SWS} }

~=~


\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, ,


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }

~=~


\biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{(n-1)/2} \, ,

where,

M_\mathrm{SWS} \equiv 
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,


R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .

Mapping from Above Discussion

Looking back on the definitions of ~\Pi_\mathrm{ad} and ~\Chi_\mathrm{ad} that we introduced in connection with our initial concise algebraic expression of the virial theorem, we can write,

~P_e

~=


~P_\mathrm{norm} \biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}

 

~=


~\biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, ,

~R_\mathrm{eq}

~=


~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}

 

~=


~\Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} 
\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, .

The first of these two expressions can be flipped around to give an expression for ~M_\mathrm{tot} in terms of ~P_e and, then, as normalized to ~M_\mathrm{SWS}. Specifically,

~ M_\mathrm{tot}^{2(n+1)}

~=


~\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]

 

~=


~M_\mathrm{SWS}^{2(n+1)} \biggl( \frac{n}{n+1} \biggr)^{3(n+1)} 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]

~ \Rightarrow~~~ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}

~=


~\biggl( \frac{n}{n+1} \biggr)^{3/2} 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)/[2(n+1)]}  \biggl[ \frac{\mathcal{B}^{2n/(n+1)}}{\mathcal{A}^{3/2}} \biggr] \, .

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

~R_\mathrm{eq}^{n-3}

~=


~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} 
\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1}

 

~=


~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} 
\biggl( \frac{G}{K} \biggr)^n \biggl\{   
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] 
\biggr\}^{(n-1)/[2(n+1)]}

 

~=


~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} 
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{(n-1)/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl( \frac{G}{K} \biggr)^n 
\biggl\{  
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] 
\biggr\}^{(n-1)/[2(n+1)]}

 

~=


~\Chi_\mathrm{ad}^{n-3} \biggl\{ \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{2n(n+1)} 
\biggl[ \frac{\mathcal{B}^{4n(n-1)}}{\mathcal{A}^{3(n+1)(n-1)}} \biggr]\biggr\}^{1/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} 
\biggl\{ \biggl( \frac{G}{K} \biggr)^{2n(n+1)} 
\biggl[ \frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}P_e^{(n-3)(n-1)} } \biggr] 
\biggr\}^{1/[2(n+1)]}

 

~=


~\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} 
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
\biggl[ G^{(3-n)(n+1)} K^{2n(n-3)} P_e^{(n-3)(1-n)} \biggr]^{1/[2(n+1)]}

 

~=


~R_\mathrm{SWS}^{n-3} \biggl( \frac{n}{n+1} \biggr)^{(n-3)/2} 
\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} 
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}

~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }

~=


~\biggl( \frac{n}{n+1} \biggr)^{1/2} 
\Chi_\mathrm{ad} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-1)/[2(n+1)]} 
\biggl[ \frac{\mathcal{B}^{n/(n+1)}}{\mathcal{A}^{1/2}} \biggr] \, .

Flipping both of these expressions around, we see that,

~\Pi_\mathrm{ad}

~=


~\frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}   \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} 
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \, ,

and,

~\Chi_\mathrm{ad}

~=


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2} 
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr] 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(1-n)/[2(n+1)]}

 

~=


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2} 
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr] 
\biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}   \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} 
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{(1-n)/[2(n+1)(n-3)]}

 

~=


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} 
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .

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

~1

~=


\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} 
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^{(n-3)/n}

 

 


-~ \frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}   \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} 
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} 
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} 
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^4

 

~=


\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/n}  \biggl( \frac{n}{n+1} \biggr)
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
-~ \frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4 \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{-2} \biggl( \frac{n}{n+1} \biggr)
\frac{1}{\mathcal{A}} \, .

Or, rearranged,

\frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4  - 
\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2} = 0 \, .

After adopting modified length- and mass-normalizations, ~R_\mathrm{mod} and ~M_\mathrm{mod}, such that,

~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}

~\equiv

~\biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, ,

~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}

~\equiv

~\biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,

we obtain the

Virial Theorem in terms of Mass and Radius


\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n} 
+ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2 = 0 
\, .

This analytic function is plotted for seven different values of the polytropic index, ~n, as indicated, in the lefthand diagram of the following table.


Virial Theorem Mass-Radius Relation

Figure 17 extractedfrom p. 184 of S. W. Stahler (1983)

"The Equilibria of Rotating Isothermal Clouds.     II. Structure and Dynamical Stability"

ApJ, vol. 268, pp. 165-184 © American Astronomical Society

Stahler (1983) Figure 17 (edited)
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 mass-radius relationship of all pressure-truncated polytropes, let's replace the new "mod" normalizations with Stahler's original normalizations, ~R_\mathrm{SWS} and ~M_\mathrm{SWS}, to understand more completely how this general expression should be viewed in relation to the parametric relations provided by solutions of the detailed force-balanced models. We will henceforth use the notation,

~\mathcal{X}

~\equiv

~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,

~\mathcal{Y}

~\equiv

~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, .

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

~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}

~=

~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,

~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}

~=

~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, .

Next, we recognize that, in order to graphically display the mass-radius relation derived from the virial theorem in the ~\mathcal{X}-\mathcal{Y} plane, we must write out the expressions for the free-energy coefficients. After setting ~M_\mathrm{limit}/M_\mathrm{tot} = 1 in the above summary expressions, we obtain,

~\mathcal{A}

~=

\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  = \frac{1}{5-n} \, ,

~\mathcal{B}

~\equiv


\biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} =
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .

In an effort not to be caught dividing by zero while investigating the specific case of ~n=5 polytropes, we will adopt as shorthand notation,

\mathfrak{b}_n \equiv \biggl[ (4\pi)^{1/n} (5-n)\mathcal{B}\biggr] 
= \biggl[ (n+1) (\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .

Hence, the replacements become,

~\frac{R_\mathrm{eq}}{R_\mathrm{mod}}

~~~ \rightarrow ~~~

~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} 
(5-n)^{(n-1)/[2(n+1)]} (4\pi)^{1/(n+1)} \mathfrak{b}_n^{-n/(n+1)}

 

~=

~\mathcal{X} (5-n)^{(n-1)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{1/2} (4\pi)^{1/2} \cdot 3^{(1-n)/[2(n+1)]} \cdot
\mathfrak{b}_n^{-n/(n+1)} \, ,

~\frac{M_\mathrm{tot}}{M_\mathrm{mod}}

~~~ \rightarrow ~~~

~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} 
(5-n)^{(n-3)/[2(n+1)]} (4\pi)^{2/(n+1)} \mathfrak{b}_n^{-2n/(n+1)}

 

~=

~\mathcal{Y} (5-n)^{(n-3)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{3/2} (4\pi)^{1/2} \cdot 3^{(3-n)/[2(n+1)]} \cdot \mathfrak{b}_n^{-2n/(n+1)}  \, ,

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

~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}}  \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}

~=

~\mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl( \frac{n+1}{n} \biggr)^{2} (4\pi)^{(n-1)/n} 
\cdot \mathfrak{b}_n^{(1-3n)/(n+1)}  \, .

Via these replacements the concise and general Virial Theorem expression derived above morphs into the,

Virial Theorem written in terms of ~\mathcal{X}, ~\mathcal{Y}, and ~\mathfrak{b}_n


k_\xi \biggl\{
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}  (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr)
\biggr\}  = 0 \, ,

where the leading coefficient is,

~k_\xi

~\equiv

~ 4\pi
\biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl(\frac{n+1}{n} \biggr)^2 \mathfrak{b}_n^{-4n/(n+1)}  \, .

In carrying out this last derivation we could be accused of reinventing the wheel, as the expression inside the curly braces is simply ~(5-n) times the virial expression presented inside an outlined box, above, just before we introduced the modified normalization parameters, ~R_\mathrm{mod} and ~M_\mathrm{mod}.

Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes

Now, let's examine the case of pressure-truncated, ~n=5 polytropes. As we have discussed in the context of detailed force-balanced models, Stahler (1983) has deduced that all ~n=5 equilibrium configurations obey the mass-radius relationship,

~\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4

~=

~0 \, ,

where, as reviewed above, the mass and radius normalizations, ~M_\mathrm{SWS} and ~R_\mathrm{SWS}, may be treated as constants once the parameters ~K and ~P_e are specified. In contrast to this, the mass-radius relationship that we have just derived from the virial theorem for pressure-truncated, ~n=5 polytropes is,


\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{2/5} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{6/5} 
+ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 = 0 \, ,

where the mass and radius normalizations,

~M_\mathrm{mod}\biggr|_{n=5}

~=

~M_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi} \biggr)^{5/3} \biggl[ \frac{2\cdot 5\pi}{3^2 \mathcal{A}} \biggr]^{3/2} 
\, ,

~R_\mathrm{mod}\biggr|_{n=5}

~=

~R_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi}\biggr)^{5/6} \biggl[ \frac{2\cdot 5\pi}{3^2\mathcal{A}} \biggr]^{1/2} \, ,

depend, not only on ~K and ~P_e via the definitions of ~M_\mathrm{SWS} and ~R_\mathrm{SWS}, but also on the structural form factors via the free-energy coefficients, ~\mathcal{A} and ~\mathcal{B}. While these two separate mass-radius relationships are similar, they are not identical. In particular, the middle term involving the cross-product 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 free-energy 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 free-energy function while holding the free-energy coefficients, ~\mathcal{A} and ~\mathcal{B}, 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 free-energy 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

Pressure vs. pressure plot
Switching, again, to the shorthand notation,

~\mathcal{X}

~\equiv

~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,

~\mathcal{Y}

~\equiv

~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, ,

the equilibrium mass-radius 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,

~\mathcal{Y}

~=

~\frac{5}{2} \mathcal{X} \biggl\{ 1 \pm \biggl[ 1 - \biggl( \frac{2^4\cdot \pi}{3\cdot 5} \biggr) \mathcal{X}^2 \biggr]^{1/2} \biggr\} \, .

In the figure shown here on the right — see also the bottom panel of Figure 2 in our accompanying discussion of detailed force-balance models — Stahler's mass-radius 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

\mathcal{X}_\mathrm{max} \equiv \biggl[ \frac{3\cdot 5}{2^4 \pi} \biggr]^{1/2} \approx 0.54627 \, .

We note that, when ~\mathcal{X} = \mathcal{X}_\mathrm{max}, ~\mathcal{Y} = (5\mathcal{X}_\mathrm{max}/2) \approx 1.36569. Along the entire sequence, the maximum value of ~\mathcal{Y} occurs at the location where ~d\mathcal{Y}/d\mathcal{X} = 0 along the segment of the curve corresponding to the positive root. This occurs along the upper segment of the curve where ~\mathcal{X}/\mathcal{X}_\mathrm{max} = \sqrt{3}/2, at the location,

\mathcal{Y}_\mathrm{max} \equiv \biggl[ \frac{3^3 \cdot 5^2}{2^6 } \biggr]^{1/2} \mathcal{X}_\mathrm{max} 
= \biggl[ \frac{3^4 \cdot 5^3}{2^{10} \pi } \biggr]^{1/2}  \approx 1.77408 \, .


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 pressure-truncated ~n=5 polytropes, these are,


~\mathcal{X}\biggr|_{n=5}

~=~


\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} =
\biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, ,


~\mathcal{Y}\biggr|_{n=5}

~=~


\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') =
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, .

The entire sequence will be traversed by varying the Lane-Emden parameter, ~\tilde\xi, 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, ~\mathcal{X}_\mathrm{max}, is associated with an embedded ~n=5 polytrope whose truncated, dimensionless Lane-Emden radius is,


~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = \frac{1}{5^{1/2}} \biggl[ 2^5\pi - 15 + 2^3\pi^{1/2}(2^4\pi-15)^{1/2} \biggr]^{1/2} 
\approx 5.8264 \, .

Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, ~\mathcal{Y}_\mathrm{max}, is associated with an embedded ~n=5 polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,


~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, .

Plotting the Virial Theorem Relation

The relevant relation is obtained by plugging ~n = 5 into the general mass-radius relation derived above, repeated here for clarity:


\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}  (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr) 
= 0

where,

\mathfrak{b}_n = \biggl[ (n+1) (-\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n}

We will begin by plugging ~n = 5 into these expressions everywhere except for the coefficient ~(5-n), which we will leave unresolved, for the time being, in order to better appreciate the interplay of various terms. We obtain,

~
\frac{6}{5} \mathcal{Y}^2 - \biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}

~=

~
(n-5)\biggl[ \biggl(\frac{4\pi}{3} \biggr) \mathcal{X}^4 -  \biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} \biggr] \, ,

where, if they are to be assigned values that are actually associated with a particular detailed force-balance model having truncation radius, ~\tilde\xi,

~
\mathfrak{b}_I^5

~\equiv

~
\biggl[ (n+1) (-\tilde\theta^')^2 \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5} 
= 2^5\cdot 3^5 \biggl[ (-\tilde\theta^')^4 {\tilde\xi}^6 \biggr]  
= 2^5 \cdot 3 \cdot \tilde\xi^{10} (1+\tilde\xi^2/3)^{-6}\, ,

~
\mathfrak{b}_{II}^5

~\equiv

~
\biggl[ \frac{1}{3} ~ \tilde\theta^{n+1}  \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5}
= \frac{1}{3^5} ~ \tilde\theta^{30}  \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6} 
= 3(1+\tilde\xi^2/3)^{-6} \, .

Now, if we plug ~n=5 into the remaining unresolved ~(n-5) coefficient, the righthand side goes to zero and the mass-radius relationship provided by the virial theorem becomes,

~\frac{6}{5}  \mathcal{Y}^2

~=

\biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}

~\Rightarrow ~~~~ \mathcal{Y}^4

~=

\mathcal{X}^{2} \biggl[ \biggl( \frac{5^5}{2^5\cdot 3^5}\biggr) \frac{\mathfrak{b}^5_{I}}{4\pi} \biggr]

~\Rightarrow ~~~~ \mathcal{Y}

~=

\mathcal{X}^{1/2} \biggl( \frac{5^5 \mathfrak{b}^5_{I}}{2^7\cdot 3^5 \pi}\biggr)^{1/4} \, .

Hence, for a given value of the structural form factor(s) — which implies a specific value of the constant coefficient, ~\mathfrak{b}_I — the scalar virial theorem defines a relationship where the normalized mass ~(\mathcal{Y}) varies as the square root of the normalized radius ~(\mathcal{X}). 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 mass-radius relationship provided by the virial theorem becomes,

~\frac{4\pi}{3} ~ \mathcal{X}^4

~=

\biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}

~\Rightarrow ~~~~ \mathcal{Y}^6

~=

\mathcal{X}^{18} \biggl[ \biggl( \frac{4\pi}{3}\biggr)^5 \frac{4\pi}{\mathfrak{b}^5_{II}} \biggr]

~\Rightarrow ~~~~ \mathcal{Y}

~=

4\pi \mathcal{X}^{3} ( 3^5\mathfrak{b}^5_{II} )^{-1/6} \, .

Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes

For pressure-truncated ~n=4 polytropes, Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models (summarized above), we appreciate that the governing pair of parametric relations is,


~\mathcal{X}

~=~


\biggl( \frac{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, ,


~\mathcal{Y}

~=~


\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .

On the other hand, the polynomial that results from plugging ~n=4 into the general mass-radius relation that is obtained via the virial theorem is,


\frac{4\pi}{3} \mathcal{X}^4   - \biggl[ \frac{\mathcal{X} \mathcal{Y}^{5}}{4\pi}\biggr]^{1/4} \mathfrak{b}_{n=4} + \frac{5}{4} \mathcal{Y}^2 
= 0 \, ,

where,

\mathfrak{b}_{n=4} = \biggl[ 5 (-\tilde\theta^')^2 + \frac{1}{3} \tilde\theta^{5} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{5/4} \, .

[For the record we note that, throughout the structure of an ~n=4 polytrope, ~\mathfrak{b}_{n=4} is a number of order unity. Its value is never less than ~3^{1/4}, which pertains to the center of the configuration; its maximum value of \approx 5.098 occurs at ~\tilde\xi \approx 4.0; and ~\mathfrak{b}_{n=4} \approx 3.946 at its (zero pressure) surface, ~\tilde\xi = \xi_1 \approx 14.97. A plot showing the variation with ~P_e of the closely allied parameter, ~\mathcal{B}|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4} is presented in the righthand panel of the above parameter summary figure.]

In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated ~n=4 polytropes, ~\mathcal{Y}(\mathcal{X}), 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, triangle-shaped points traces the mass-radius relation, ~\mathcal{Y}(\mathcal{X}), that is obtained via the virial theorem, assuming that the coefficient, ~\mathfrak{b}_{n=4}, is constant along the sequence. The "green" sequence in the lefthand panel results from setting ~\mathfrak{b}_{n=4} = 3.4205, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at ~\tilde\xi = 1.4; the "orange" sequence in the righthand panel results from setting ~\mathfrak{b}_{n=4} = 4.8926, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at ~\tilde\xi = 2.8.

Comparing Two Separate Mass-Radius Relations for Pressure-Truncated n = 4 Polytropes

Comparison of Two Mass-Radius Relations

According to Horedt's (1986)] tabulated data, the surface of an isolated ~(P_e = 0), spherically symmetric, ~n=4 polytrope occurs at the dimensionless (Lane-Emden) radius, ~\xi_1 = 14.9715463. In both panels of the above figure, this isolated configuration is identified by the discrete (blue diamond) point at the origin, that is, at ~(\mathcal{X}, \mathcal{Y}) = (0, 0). As we begin to examine pressure-truncated models and ~\tilde\xi is steadily decreased from ~\xi_1, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation. A maximum mass of ~\mathcal{Y} \approx 2.042 (corresponding to a radius of ~\mathcal{X} \approx 0.4585) is reached from the left as ~\tilde\xi drops to a value of approximately ~3.4. As ~\tilde\xi continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at ~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554) — corresponding to ~\tilde\xi \approx 2.0 — then decreasing in radius until, once again, the origin is reached, but this time because ~\tilde\xi drops to zero.


If we set ~\mathfrak{b}_{n=4} = 3.4205 (corresponding to a choice of ~\tilde\xi = 1.4), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius 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 force-balanced model having ~\tilde\xi = 1.4. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set ~\mathfrak{b}_{n=4} = 4.8926 (corresponding to a choice of ~\tilde\xi = 2.8); it intersects the blue mass-radius relation precisely at the coordinate location, ~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965) — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having ~\tilde\xi = 2.8. Hence, the two relations give the same mass-radius coordinates when the value of ~\mathfrak{b}_{n=4} that is plugged into the virial theorem matches the value of ~\mathfrak{b}_{n=4} that reflects the structural form factor that is properly associated with a detailed force-balanced model.


When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of ~\mathfrak{b}_{n=4} = 4.8926 (as traced by the orange triangle-shaped 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 ~\tilde\xi = 2.8, for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, ~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67) — also highlighted by a black filled circle — that corresponds to a detailed force-balanced model having ~\tilde\xi \approx 6.0. This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, ~\tilde\xi, that assures precise agreement between the two different mass-radius expressions.


As is detailed in our above discussion of the dynamical stability of pressure-truncated polytropes, an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy 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 ~n = 4 polytropic structures, the critical point is identified by the dimensionless parameters,

\eta_\mathrm{crit}\biggr|_{n=4}~=~\frac{1}{15} \, ;          \Pi_\mathrm{max}\biggr|_{n=4}~=~\frac{15^{15}}{16^{16}} \, ;      and      \Chi_\mathrm{min}\biggr|_{n=4}~=~\biggl( \frac{16}{15} \biggr)^4 \, .

In the context of the above figure, independent of the chosen value of ~\mathfrak{b}_{n=4}, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem. In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of ~\mathcal{Y}; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of ~\mathcal{X}. We have deduced (details of the derivation not shown) that, for pressure-truncated ~n=4 polytropes, the coordinates of this critical point in Stahler's ~\mathcal{X}-\mathcal{Y} plane depends on the choice of ~\mathfrak{b}_{n=4} as follows:

~\mathcal{X}_\mathrm{crit}

~=

~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,

~\mathcal{Y}_\mathrm{crit}

~=

~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .

In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, \mathfrak{b}_{n=4} — 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 mass-radius 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 free-energy-based 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 maximum-mass 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 maximum-mass point along the blue "Stahler" sequence. This finding is related to the curiosity raised earlier in our discussion of the structural properties of pressure-truncated, ~n=4 polytropes.

Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes

For pressure-truncated ~n=3 polytropes, Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models (summarized above), we appreciate that the governing pair of parametric relations is,


~\mathcal{X}

~=~


\biggl( \frac{3}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta \, ,


~\mathcal{Y}

~=~


\biggl( \frac{3^3}{4\pi} \biggr)^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .

On the other hand, the polynomial that results from plugging ~n=3 into the general mass-radius relation that is obtained via the virial theorem is,


\frac{2^3 \pi}{3} \mathcal{X}^4   - \biggl[ \frac{\mathcal{Y}^{4}}{4\pi}\biggr]^{1/3} \mathfrak{b}_{n=3} + \frac{4}{3} \mathcal{Y}^2 
= 0 \, ,

where,

\mathfrak{b}_{n=3} = \biggl[ 4 (-\tilde\theta^')^2 + \frac{2}{3} \tilde\theta^{4} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{4/3} \, .

Summary Comments

From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration. Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based 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 ~(P_e) 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 self-similar solution for the homologous collapse of stellar cores that can be modeled as ~n=3 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 pressure-truncated 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 low-order 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 full-blown 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 pulsation-mode eigenfunction in our free-energy analysis of stability. Better yet, the sign of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.


Serious Concern

Statement of Concern

Throughout our discussion of embedded (pressure-truncated) 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 ~(5-n)^{-1}. 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 pressure-truncated polytropes is reprinted on the righthand side of the table.

Structural Form Factors for Isolated Polytropes

Structural Form Factors for Pressure-Truncated Polytropes

~\mathfrak{f}_M

~=

~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1}

\mathfrak{f}_W

~=

~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1}

\mathfrak{f}_A

~=


\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}

~\tilde\mathfrak{f}_M

~=

~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi}

\tilde\mathfrak{f}_W

~=

~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi}

\tilde\mathfrak{f}_A

~=


\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}  + \tilde\Theta^{n+1}

This factor seemed destined to become a nuisance in the specific case of ~n=5 polytropic structures. But we did not let its appearance in these expressions deter us from using a free-energy analysis to study the equilibrium and stability of spherical polytropes because, after all, the factor of ~(5-n)^{-1} 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 free-energy and virial theorem analysis, the derived expression for the configuration's equilibrium radius depends on the ratio of ~f_W to ~f_A, so the awkward factor of ~(5-n)^{-1} cancels out.

However, in our discussion of pressure-truncated ~n=5 polytropic structures, the factor of ~(5-n)^{-1} 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 mass-radius relation obtained from the virial theorem to Stahler's mass-radius relation, which was derived from detailed force-balance arguments. This leads us to seriously question whether our, rather casually derived, expressions for the structural form factors in pressure-truncated polytropes are correct.

Further Evaluation of n = 5 Polytropic Structures

Throughout most of this subsection, we will adopt the shorthand notation,

~\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~~~\Rightarrow ~~~~~ \ell^2 = \frac{\tilde\xi^2}{3} \, .

This will not only simplify the appearance of some expressions, it will facilitate direct comparison with an expression for the free-energy coefficient, ~\mathcal{A}, 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 brute-force integration in association with our discussion of bipolytropic configurations where the variable, ~x_i, has the same definition as ~\ell.

Free-Energy Expression

From our general review of the topic, to within an additive constant, the free-energy of a nonrotating, pressure-truncated polytrope comes from the sum of three principal energy terms, namely,


\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_e V \, .

Furthermore, as has been shown in our extended introductory discussion of free energy, the corresponding normalized free energy (applied to ~n=5 or ~\gamma_g = 6/5 configurations) is,


\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}} = 
-3A\chi^{-1} + 5B \chi^{-3/5} +~ D\chi^3 \, ,

where,

~\chi

~\equiv

\frac{R}{R_\mathrm{norm}}  \, ,

~A

~\equiv

\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,

~B

~\equiv


\biggl(\frac{3}{4\pi} \biggr)^{1/5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{6/5}  \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{6/5}} \, ,

~D

~\equiv


\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .

Also note that the relevant normalizations are,

~E_\mathrm{norm} \equiv \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \, ;        ~R_\mathrm{norm} \equiv \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \, ;        ~P_\mathrm{norm} \equiv \frac{K^{10}}{G^9 M_\mathrm{tot}^{6}}  \, .       

Virial Theorem

The traditional expression for the virial theorem in this context is,


~2S_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{E_\mathrm{norm}} = 0 \, .

From our introductory discussion of the thermodynamic energy reservoir, we know that, for ~\gamma_g=6/5 configurations,


~S_\mathrm{therm} = \frac{3}{2}(\gamma_g-1) \mathfrak{S}_\mathrm{therm} = \frac{3}{10} \mathfrak{S}_\mathrm{therm}\, .

So, making this substitution and recognizing that ~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3, the (normalized) virial theorem expression becomes,


~\frac{3}{5}\mathfrak{S}_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{P_\mathrm{norm}R_\mathrm{norm}^3} = 0 \, .

Furthermore, by comparing terms in the first free-energy expression, above, with the second (normalized) free-energy expression, we see that,

~W_\mathrm{grav}^* \equiv \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \rightarrow -3A\chi^{-1} \, ;        ~\mathfrak{S}_\mathrm{therm}^* \equiv \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}} \rightarrow 5B\chi^{-3/5} \, ;        ~\frac{P_e V}{E_\mathrm{norm}} \rightarrow D\chi^{3}  \, .       

Hence, the normalized virial theorem may be written as,


3B \chi^{-3/5} -3A\chi^{-1}  -~ 3D\chi^3 =0\, .

For sake of consistency, let's check this by holding the coefficients ~A, ~B, and ~D constant and setting ~d\mathfrak{G}^*/d\chi equal to zero:

~\frac{d\mathfrak{G}^*}{d\chi} = - 3B \chi^{-8/5} + 3A\chi^{-2}  +~ 3D\chi^2

~=

~0

~\Rightarrow ~~~ - \chi^{-1} \biggl[ 3B \chi^{-3/5} - 3A\chi^{-1} -~ 3D\chi^3 \biggr]

~=

~0  \, .

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 force-balanced models of ~(n_c, n_e) = (5, 1) 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,


~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} + (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{env} = 0 \, .

In the bipolytrope, the (truncated) ~n=5 core is confined by an ~n=1 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) ~n=5 polytrope that is confined by a hot, tenuous external medium instead of by an enveloping envelope, the relevant normalized virial theorem is,

~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} - \frac{3P_e V_\mathrm{eq}}{E_\mathrm{norm}}

~=

~0

~\Rightarrow ~~~ (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core}

~=

~\frac{3P_e }{(K^5/G^3)^{1/2}} \biggl( \frac{4\pi}{3} R_\mathrm{eq}^3 \biggr)

 

~=

~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2}  \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^3 \, ,

where, as discussed/defined in an accompanying chapter of this H_Book, we have adopted the normalization radius, ~R_\mathrm{SWS}, first introduced by Steven W. Stahler (1983). For ~n=5 configurations, its definition is,


R_\mathrm{SWS}\biggr|_{n=5} = \biggl( \frac{2\cdot 3}{5} \biggr)^{1/2} \biggl[ \frac{(K^{5}/G^3)^{1/2}}{P_\mathrm{e}}\biggr]^{1/3} \, .

As has also been discussed in the accompanying chapter, we can deduce from Stahler's detailed force-balanced models that the equilibrium radius of embedded, ~n=5 polytropes is given in terms of the dimensionless, truncated Lane-Emden radius, ~\tilde\xi — and our corresponding variable, \ell — by the expression,


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}

~=~


\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^2 
= \biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-1}

 

~=~


\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1} \, .

Hence, upon careful evaluation of the thermal energy and gravitational potential energy of truncated ~n=5 polytropes, we should find that,

~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core}

~=

~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2}  \biggl[ \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggr]^3

 

~=

~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \ell^3 (1 + \ell^2)^{-3} \, .

Well, it just so happens that, in our accompanying chapter that presents the detailed force-balanced models of ~(n_c, n_e) = (5, 1) bipolytropes, we explicitly carried out the volume integrals defining these two key components of the free energy expression with the results being,

~(2S^* + W^*)_\mathrm{core}

~=~

~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \biggl[ x_i ^3 (1 + x_i^2)^{-3} \biggr] \, .

Realizing that the variable, ~x_i, in that context is the same as ~\ell, in the present context, we see that the two separately derived results are identical to one another.

Determining Expressions for Free-Energy Coefficients

We should be able to convert the separately derived expression for ~W_\mathrm{grav}^* into an expression for the free-energy coefficient, ~A, in equilibrium configurations. As noted above, for a fixed value of ~A,

~W_\mathrm{grav}^* ~~\rightarrow ~~ -3A\chi^{-1} \, .

Therefore, in an equilibrium configuration, we can write,


~W_\mathrm{grav}^*

~=~


-3A\chi_\mathrm{eq}^{-1} = -3A \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)
= -3A \biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{eq}}\biggr)\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)


~\Rightarrow ~~~ A

~=~


-\frac{1}{3} W_\mathrm{grav}^* \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .

Now, from immediately above, we know that,


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}

~=~


\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}  \, ;

and, from our accompanying discussion of the free-energy of bipolytropic configurations, we know that,

~W^*_\mathrm{core}

~=~

~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} 
\biggl[ x_i\biggl(x_i^4 - \frac{8}{3}x_i^2 - 1\biggr) (1 + x_i^2)^{-3}  + \tan^{-1}(x_i) \biggr] \, .

So, again realizing that ~x_i and ~\ell are interchangeable, we have,

~A

~=~

~ \frac{1}{3}
\biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)

 

~=~

~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .

Finally, we need to determine an expression for the ratio, ~R_\mathrm{SWS}/R_\mathrm{norm}. Drawing the definition of ~R_\mathrm{norm} from our introductory chapter on the virial equilibrium, we have,

~R_\mathrm{norm}\biggr|_{\gamma = 6/5}

~=

~\biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{2-6/5} \biggr]^{1/(4-18/5)}
= \biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{4/5} \biggr]^{5/2}
= \biggl( \frac{G}{K}\biggr)^{5/2} M_\mathrm{tot}^{2} \, .

From our tabular summary of Stahler's derived mass & radius relationships for truncated, ~n=5 polytropes we have,

~M_\mathrm{limit}^2

~=

~
M_\mathrm{SWS}^2 \biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] \, .

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

~M_\mathrm{SWS}^2

~=

~
\biggl( \frac{2^3 \cdot 3^3}{5^3} \biggr) G^{-3} K^{10/3} P_e^{-1/3} \, ,

and,    ~R_\mathrm{SWS}

~=

~
\biggl( \frac{2 \cdot 3}{5} \biggr)^{1/2} G^{-1/2} K^{5/6} P_e^{-1/3} \, ,

which, when combined to cancel ~P_e gives,

~M_\mathrm{SWS}^2

~=

~
\biggl( \frac{2^6 \cdot 3^6}{5^6} \biggr)^{1/2} G^{-3} K^{10/3} 
\biggl( \frac{5}{2 \cdot 3} \biggr)^{1/2} G^{1/2} K^{-5/6} R_\mathrm{SWS}

 

~=

~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} R_\mathrm{SWS} \, .

Hence, we can write,

~M_\mathrm{limit}^2

~=

~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] R_\mathrm{SWS}

~\Rightarrow ~~~~ \frac{1}{R_\mathrm{SWS}}

~=

~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \frac{1}{M_\mathrm{limit}^2} \, .

In combination with the expression for ~R_\mathrm{norm}, then, we have,

~\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}

~=

~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2} 
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \biggl(\frac{M_\mathrm{tot}}{M_\mathrm{limit}}\biggr)^{2} \, ,

which means that, for truncated ~n=5 polytropes, the expression for the free-energy coefficient is,

~A

~=~

~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \biggl( \frac{\pi^2}{2\cdot 3^7 \cdot 5} \biggr)^{1/2}
\ell (1 + \ell^2)^{-1} \cdot \ell^{-6} (1+\ell^2)^{4}  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2}

 

~=~

~ 
2^{-4} \ell^{-5} (1+\ell^2)^{3}  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, .

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

~\frac{M_\mathrm{limit}}{M_\mathrm{tot} }

~=


\ell^3 (1+\ell^2)^{-3/2} \, ,

which gives,

~A

~=~

~ 
\frac{\ell}{2^{4} }  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] \, .

This exactly matches the expression for the free-energy coefficient, ~\mathcal{A}, 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, ~A, 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 lead-in paragraphs of this subsection, we have previously assumed that,

~\mathcal{A}

~\equiv

\frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} 
= \frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \biggr\}
\biggl\{ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \biggr\}^{-2} = \frac{1}{(5-n)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \, .

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

~W_\mathrm{grav}

~=

~-3A\chi^{-1} E_\mathrm{norm}

 

~=

~-3A \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \frac{1}{R}

 

~=

~-3A \biggl(\frac{GM_\mathrm{tot}^2}{R} \biggr) \, .

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

~W_\mathrm{grav}

~=

~ - \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr \, ,

where,

~M_r

~\equiv

~\int_0^r 4\pi r^2 \rho dr \, .

Now, independent of the chosen normalization, if we use ~M_\mathrm{tot} to represent the total mass of an isolated ~n=5 polytrope, then from an earlier review, we have,

~M_\mathrm{tot}

~=

~\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  \rho_c^{-1/5} \, ,

and we can write, in terms of the Lane-Emden dimensionless radius, ~\xi,


\frac{M_r}{M_\mathrm{tot}} = \xi^3 (3 + \xi^2)^{-3/2}  \, .

Virial Chapter

Now, in our discussion of the virial equilibrium of embedded polytropes, we used the normalizations specified above and wrote,

~W_\mathrm{grav}

~=


- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} \biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^*

 

~=


- E_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} 3\biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .

We can replace ~r^* \equiv r/R_\mathrm{norm} with ~\xi \equiv r/a_5 by recognizing that,

~M_\mathrm{tot}

~=

~
\biggl(  \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}

~\Rightarrow ~~~ \rho_c^{-2/5}

~=

~
\biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)  M_\mathrm{tot}^2 \, ;

~a_5

~=

~
\biggl(  \frac{3K}{2\pi G} \biggr)^{1/2} \rho_c^{-2/5}

 

~=

~
\biggl(  \frac{3K}{2\pi G} \biggr)^{1/2} \biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)  M_\mathrm{tot}^2

 

~=

~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2}  \biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^2 \, ;

~R_\mathrm{norm}

~=

~
\biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2}

~\Rightarrow~~~ \frac{a_5}{R_\mathrm{norm}}

~=

~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2}  \, .

Hence,

~r^*

~=

~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .

Also,

~\rho_\mathrm{norm}

~=

~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} M_\mathrm{tot}^{-5}

 

~=

~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl[ \biggl(  \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}\biggr]^{-5}

 

~=

~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)^{5/2} \rho_c

 

~=

~
\biggl(\frac{3^2}{2^4\pi^2} \biggr)^{1/2} \biggl(  \frac{\pi^5 }{2^5 \cdot 3^{20}} \biggr)^{1/2} \rho_c

 

~=

~
\biggl(  \frac{\pi^3 }{2^9 \cdot 3^{18}} \biggr)^{1/2} \rho_c \, .

Hence,

~\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}}

~=

~
\biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \frac{\rho}{\rho_c}

 

~=

~
\biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} \, .

So the energy integral becomes,

~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }

~=


- 3 \biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr) \biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} 
\int_0^{\tilde\xi} \biggl[\xi^3 (3 + \xi^2)^{-3/2}  \biggr]  \xi \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} d\xi

 

~=


- \biggl(  \frac{2^3 \cdot 3^{6}}{\pi } \biggr)^{1/2} 
\int_0^{\tilde\xi} \xi^4 \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-4} d\xi \, .

This needs to be compared with the ~W_\mathrm{grav}^* integral that we previously have handled in the chapter discussing bipolytrope models.

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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