Difference between revisions of "User:Tohline/SSC/Virial/PolytropesEmbeddedOutline"
(→ASIDE: Isothermal Configurations: Finished caption) 
(→ASIDE: Isothermal Configurations: Changed sign on \mathcal{G}_0 normalization + a few other wording changes) 

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====ASIDE: Isothermal Configurations====  ====ASIDE: Isothermal Configurations====  
While our focus in this chapter is on polytropic systems, it is advantageous to review [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] discussion of pressuretruncated isothermal configurations because that discussion includes presentation of a freeenergy surface — see, specifically, Whitworth's Figure 2, which is reproduced in the lowerright quadrant of our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurfacecomposite figure, below]]. Setting <math>~\delta_{1\eta} = 1</math> in Whitworth's freeenergy expression (his equation 10  While our focus in this chapter is on polytropic systems, it is advantageous to review [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] discussion of pressuretruncated isothermal configurations because that discussion includes presentation of a freeenergy surface — see, specifically, Whitworth's Figure 2, which is reproduced in the lowerright quadrant of our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurfacecomposite figure, below]]. Setting <math>~\delta_{1\eta} = 1</math> in Whitworth's freeenergy expression (his equation 10, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Coefficient_Definitionsreprinted above]]) gives,  
<div align="center" id="WhitworthFreeEnergyExpression">  <div align="center" id="WhitworthFreeEnergyExpression">  
<math>  <math>  
\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}} = \frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{1}  \frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}} = \frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{1}  
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3  +~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3  
2\ln\biggl( \frac{R}{R_\mathrm{rf}}\biggr)  2\ln\biggl( \frac{R}{R_\mathrm{rf}}\biggr)  \mathcal{G}_0 \, ,  
</math>  </math>  
</div>  </div>  
where, as earlier, we have  where, as earlier, we have inserted the additional constant, <math>\mathcal{G}_0</math>, to accommodate normalization. A segment of the freeenergy surface defined by this function is displayed in the righthand panel of the figure shown at the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spherestop of this page]]. In constructing this figure, <math>~\mathcal{G}_0</math> has been set to a value that ensures that <math>~\mathcal{U}</math> is everywhere positive over the displayed domain: <math>~0.1 \le P_e/P_\mathrm{rf} \le 1.1</math> and <math>~0.3 \le R/R_\mathrm{rf} \le 3.0</math>.  
<span id="ScalarVirialTheorem">  <span id="ScalarVirialTheorem">  
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</div>  </div>  
and <math>~[d^2\mathcal{U}/dR^2]_\mathrm{eq} > 0</math>  and equilibria are stable — that is, <math>~[d^2\mathcal{U}/dR^2]_\mathrm{eq} > 0</math> — if <math>~R_\mathrm{eq}/R_\mathrm{rf} > 1</math>. For physically realistic systems, of course, <math>~P_e</math> must be positive — which means that all equilibria have <math>~R_\mathrm{eq}/R_\mathrm{rf} > 3/4</math>. But, from this algebraic virial theorem expression, it is also clear that physically realistic equilibrium configurations only exist when <math>~P_e/P_\mathrm{rf} \le 1</math>. The sequence of small, colored spherical dots in the righthand panel of the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#IntroFiguresabove figure]] identify parametervalue pairs, <math>~(R_\mathrm{eq}, P_e)</math>, associated with fourteen different equilibrium configurations: Blue dots — tracing out the valley of the freeenergy surface — identify stable configurations; white dots — balanced along the crest of the surface ridge — identify dynamically unstable configurations; and the lone red dot identifies the critical neutral equilibrium state, which is also associated with the maximum allowable value of <math>~P_e</math> along the equilibrium model sequence.  
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<td align="left" colspan="5">  <td align="left" colspan="5">  
Graphical depictions of the freeenergy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressuretruncated, uniformdensity isothermal configurations — see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpressionabove]].  Graphical depictions of the freeenergy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressuretruncated, uniformdensity isothermal configurations — see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpressionabove]].  
*''Lowerright quadrant'': A reproduction of the black & white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(  *''Lowerright quadrant'': A reproduction of the black & white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>.  
*''Upperright quadrant'': The undulating freeenergy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red), External Pressure (green), and Free Energy (blue). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(  *''Upperright quadrant'': The undulating freeenergy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red), External Pressure (green), and Free Energy (blue). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>.  
*''Table of numbers'': Coordinates <math>(X, Y, Z) = (R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathcal{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the freeenergy surface.  *''Table of numbers'': Coordinates <math>(X, Y, Z) = (R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathcal{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the freeenergy surface. [Actually, two of the three attending color plots display only fifteen dots because the position of the (unstable) equilibrium configuration of highest energy falls outside the boundaries of the plot.] Seven of the nine equilibrium configurations that are identified in Whitworth's Figure 2 are among the sixteen equilibria that are identified here, including the critical neutral equilibrium state, which is highlighted in red. For completeness, the value of the corresponding normalization energy, <math>~\mathcal{G}_0(P_e)</math>, is also tabulated.  
*''Upperleft quadrant'': The twodimensional image that results from viewing the freeenergy surface "from above," along a  *''Upperleft quadrant'': The twodimensional image that results from viewing the freeenergy surface "from above," along a line of sight that is parallel to the freeenergy <math>~(Z)</math> axis and looking directly down onto the radiuspressure <math>~(XY)</math> plane. From this vantage point, the sixteen small colored dots cleanly trace out the <math>~P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#ScalarVirialTheoremalgebraic expression of the scalar virial theorem]].  
*''Lowerleft quadrant'': The twodimensional image that results from viewing the freeenergy surface "from underneath," along a light of sight that is parallel to the externalpressure <math>~(Y)</math> axis and looking directly up at the radiusfreeenergy <math>~(XZ)</math> plane. This image can be directly compared with Whitworth's Figure 2. Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image. For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is peaking from behind the "Radius" axis — that is, the unstable equilibrium configuration that has a freeenergy value of zero — corresponds to the open circle in Whitworth's plot that is labeled as <math>~P_e = 0.632</math> (see also the coordinate values given in our accompanying table).  *''Lowerleft quadrant'': The twodimensional image that results from viewing the freeenergy surface "from underneath," along a light of sight that is parallel to the externalpressure <math>~(Y)</math> axis and looking directly up at the radiusfreeenergy <math>~(XZ)</math> plane. This image can be directly compared with Whitworth's Figure 2. Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image. For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is peaking from behind the "Radius" axis — that is, the unstable equilibrium configuration that has a freeenergy value of zero — corresponds to the open circle in Whitworth's plot that is labeled as <math>~P_e = 0.632</math> (see also the coordinate values given in our accompanying table).  
</td>  </td> 
Revision as of 20:24, 13 February 2015
Virial Equilibrium of Embedded Polytropic Spheres
FreeEnergy Surfaces for PressureTruncated Structures 


<math>\mathcal{G}(M,R)</math> surface for <math>n = 5</math> polytropic configurations 
<math>\mathcal{G}(P_e,R)</math> surface for isothermal configurations (see further elaboration below) 
Overview
The freeenergy function that is relevant to a discussion of the structure and stability of a pressuretruncated configuration having polytropic index, <math>~n</math>, has the form,
<math>~\mathcal{G}(x)</math> 
<math>~=</math> 
<math> ax^{1} +b x^{3/n} + c x^3 + \mathcal{G}_0 \, , </math> 
where <math>~x</math> identifies the size of the configuration and <math>\mathcal{G}_0</math> is an arbitrary constant. (As is explained more fully, below, the lefthand panel of the figure at the top of this page displays a freeenergy surface of this form for the case, <math>~n=5</math>.) If the coefficients, <math>~a, b</math>, and <math>~c</math>, are held constant while varying the configuration's size, we see that,
<math>~\frac{d\mathcal{G}}{dx}</math> 
<math>~=</math> 
<math> ax^{2}  \frac{3b}{n}\cdot x^{(3+n)/n} + 3c x^2 </math> 

<math>~=</math> 
<math> x^{2} \biggl[ a  \frac{3b}{n}\cdot x^{(n3)/n} + 3c x^4 \biggr] \, , </math> 
and,
<math>~\frac{d^2\mathcal{G}}{dx^2}</math> 
<math>~=</math> 
<math> x^{3} \biggl[ 2a + \frac{3(3+n)b}{n^2}\cdot x^{(n3)/n} + 6c x^4 \biggr] \, . </math> 
Equilibrium Configurations
The size, <math>~x_\mathrm{eq}</math>, of each equilibrium configuration is determined by setting, <math>d\mathcal{G}/dx = 0</math>. Hence, <math>~x_\mathrm{eq}</math> is given by the root(s) of the polynomial expression that is often referred to as the,
Scalar Virial Theorem
<math>~x^{(n3)/n}_\mathrm{eq} </math> 
<math>~=</math> 
<math>\frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] \, . </math> 
(The equilibrium radii of <math>~n = 5</math> polytropic configurations having a variety of different masses are identified by the sequence of a dozen, small colored spherical dots in the above figure.)
Stability
The relative stability of each equilibrium configuration is determined by the sign of the second derivative of the freeenergy function, evaluated at the specified equilibrium radius. Specifically, the systems being considered here are stable if the second derivative is positive, but they are unstable if the second derivative is negative. Evaluating the second derivative in this manner gives,
<math>~\biggl[ x^{3} \cdot \frac{d^2\mathcal{G}}{dx^2}\biggr]_\mathrm{eq}</math> 
<math>~=</math> 
<math> 2a + \frac{3(3+n)}{n} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] + 6c x^4_\mathrm{eq} </math> 

<math>~=</math> 
<math> 2a + \frac{(3+n)a}{n} + \frac{3(3+n)c}{n} \cdot x^4_\mathrm{eq} + 6c x^4_\mathrm{eq} </math> 

<math>~=</math> 
<math>\frac{9(n+1)c}{n}\cdot x^4_\mathrm{eq}  \frac{a(n3)}{n} \, . </math> 
Defining <math>~x_\mathrm{crit}</math> as the equilibrium radius at which this function goes to zero gives,
<math>~x_\mathrm{crit} </math> 
<math>~\equiv</math> 
<math> \biggl[ \frac{a(n3)}{3^2c(n+1)} \biggr]^{1/4} \, . </math> 
(The small red spherical dot in the above figure identifies the equilibrium configuration at <math>~x_\mathrm{crit} </math>.) We conclude, therefore, that pressuretruncated, equilibrium polytropic configurations having <math>~n > 3</math> are stable if,
<math>~x_\mathrm{eq}</math> 
<math>~></math> 
<math> ~x_\mathrm{crit} \, , </math> 
while they are unstable if,
<math>~x_\mathrm{eq}</math> 
<math>~<</math> 
<math> ~x_\mathrm{crit} \, . </math> 
The Physics
The above mathematical statements, ostensibly defining the freeenergy function, the scalar virial theorem, and stability, cannot be interpreted in physical terms until the definitions of the various coefficients have been provided. In the discussion that follows, we will focus on sequences of equilibrium configurations that have a polytropic index <math>~n > 3</math> because, as has been foreshadowed in the above overview, such sequences include both stable and unstable equilbria and are therefore of considerable interest in an astrophysical context. Isothermal sequences — corresponding to <math>~n = \infty</math> — are of particular astrophysical interest; however, we will devote a great deal of attention to <math>~n=5</math> configurations because their structures can be defined entirely in terms of analytic expressions.
In order to determine the equilibrium radius, <math>~R_\mathrm{eq}</math>, of any pressuretruncated polytropic configuration, we must specify the configuration's mass, <math>~M_\mathrm{limit}</math>, its polytropic constant, <math>~K</math>, and the pressure, <math>~P_e</math>, of the external medium in which the configuration is embedded, and locate extrema in the resulting <math>~\mathcal{G}(R)</math> function. Then, a sequence of equilibria can be identified if, for example:
 Case P: <math>~P_e</math> is varied while holding <math>~K</math> and <math>~M_\mathrm{limit}</math> fixed; or
 Case M: <math>~M_\mathrm{limit}</math> is varied while holding <math>~K</math> and <math>~P_e</math> fixed.
In the first case, the analysis reveals how <math>~R_\mathrm{eq}</math> varies with the applied external pressure and usually is displayed as a <math>~P_e(R_\mathrm{eq})</math> function. The second case identifies a massradius relationship for the polytropic sequence under consideration and is usually displayed as a <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> function. (In the figure at the top of this page, a "Case M" massradius relation for pressuretruncated, <math>~n = 5</math> polytropic configurations is traced by the sequence of a dozen, small colored spherical dots that each reside at an extremum in the displayed freeenergy function.)
Whitworth's (1981) Case P Analysis of UniformDensity Configurations
Coefficient Definitions
Whitworth's (1981) Case P analysis of pressuretruncated polytropic spheres produces the following governing freeenergy function — referred to by Whitworth as the "global potential function":
After setting <math>~\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic index, Whitworth's freeenergy expression becomes,
<math> \frac{2\mathcal{U}}{3M_0 K_1} = \frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{1} +~ \frac{2n}{3}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{3/n} +~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 \, . </math>
This expression is identical to the freeenergy function given above if the following coefficient and variable substitutions are made:
Whitworth's (1981) Case P Analysis  

where (see an accompanying ASIDE),
<math>~R_\mathrm{rf}</math> 
<math>~\equiv</math> 
<math> \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{n} G^{n} M_\mathrm{limit}^{n1} \biggr]^{1/(n3)} \, , </math> 
<math>~P_\mathrm{rf}</math> 
<math>~\equiv</math> 
<math>~ \biggl[ 2^{2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{3(n+1)} M_\mathrm{limit}^{2(n+1)}\biggr]^{1/(n3)} \, , </math> 
<math>~\mathcal{U}_\mathrm{rf} \equiv (M_0K_1)_\mathrm{Whitworth}</math> 
<math>~=</math> 
<math>~\biggl[ \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr) K^n G^{3} M_\mathrm{limit}^{n5}\biggr]^{1/(n3)} \, .</math> 
Virial Equilibrium
Plugging these coefficient assignments into the above mathematical prescription of the virial theorem gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressuretruncated polytropic configurations,
<math>~\frac{2}{3}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n3)/n} </math> 
<math>~=</math> 
<math>\frac{1}{2} + \frac{1}{6} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 \, . </math> 
A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
<math>\biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr) </math> 
<math>~=</math> 
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n3)/n} 3\biggr] \, ,</math> 

<math>~=</math> 
<math>~4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{3(n+1)/n} 3\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{4} \, ,</math> 
or,
<math>~P_e </math> 
<math>~=</math> 
<math>~ 4P_\mathrm{rf} R_\mathrm{rf}^{3(n+1)/n} R_\mathrm{eq}^{3(n+1)/n} 3P_\mathrm{rf} R_\mathrm{rf}^4 R_\mathrm{eq}^{4} </math> 

<math>~=</math> 
<math>~ 4R_\mathrm{eq}^{3(n+1)/n}\biggl[ 2^{2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{3(n+1)} M_\mathrm{limit}^{2(n+1)}\biggr]^{1/(n3)} \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{n} G^{n} M_\mathrm{limit}^{n1} \biggr]^{3(n+1)/[n(n3)]} </math> 


<math>~ 3R_\mathrm{eq}^{4}\biggl[ 2^{2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{3(n+1)} M_\mathrm{limit}^{2(n+1)}\biggr]^{1/(n3)} \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{n} G^{n} M_\mathrm{limit}^{n1} \biggr]^{4/(n3)} </math> 

<math>~=</math> 
<math>~ 2^2 R_\mathrm{eq}^{3(n+1)/n}\biggl[ 2^{2n(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n(n+1)} K^{4n^2} G^{3n(n+1)} M_\mathrm{limit}^{2n(n+1)}\cdot \frac{\pi^{3(n+1)}}{5^{3n(n+1)}} \biggl( \frac{2^2}{3}\biggr)^{3(n+1)(n+1)} K^{3n(n+1)} G^{3n(n+1)} M_\mathrm{limit}^{3(n1)(n+1)} \biggr]^{1/[n(n3)]} </math> 


<math>~ 3R_\mathrm{eq}^{4}\biggl[ 2^{2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{3(n+1)} M_\mathrm{limit}^{2(n+1)}\cdot \frac{\pi^4}{5^{4n}} \biggl( \frac{2^2}{3}\biggr)^{4(n+1)} K^{4n} G^{4n} M_\mathrm{limit}^{4(n1)} \biggr]^{1/(n3)} </math> 

<math>~=</math> 
<math>~ R_\mathrm{eq}^{3(n+1)/n} \biggl( \frac{3}{2^2\pi}\biggr)^{(n+1)/n} K M_\mathrm{limit}^{(n+1)/n}  R_\mathrm{eq}^{4} \biggl(\frac{3}{2^2\cdot 5\pi}\biggr) G M_\mathrm{limit}^2 </math> 

<math>~=</math> 
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n}  \frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \, .</math> 
Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of Whitworth, which reads:
Stability
Similarly, according to the abovederived stability criterion, pressuretruncated polytropic configurations will only be stable if,
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 </math> 
<math>~></math> 
<math>\frac{(n3)}{(n+1)} \biggl( \frac{P_e}{P_\mathrm{rf}} \biggr)^{1} \, . </math> 
Or, given that <math>~P_\mathrm{rf}R_\mathrm{rf}^4 = (G M_\mathrm{limit}^2)/(20\pi)</math>, the criterion for stability may be written as,
<math>~P_e </math> 
<math>~></math> 
<math>~ \frac{(n3)}{20\pi(n+1)} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr) \, . </math> 
ASIDE: Isothermal Configurations
While our focus in this chapter is on polytropic systems, it is advantageous to review Whitworth's (1981) discussion of pressuretruncated isothermal configurations because that discussion includes presentation of a freeenergy surface — see, specifically, Whitworth's Figure 2, which is reproduced in the lowerright quadrant of our composite figure, below. Setting <math>~\delta_{1\eta} = 1</math> in Whitworth's freeenergy expression (his equation 10, reprinted above) gives,
<math> \frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}} = \frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{1} +~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 2\ln\biggl( \frac{R}{R_\mathrm{rf}}\biggr)  \mathcal{G}_0 \, , </math>
where, as earlier, we have inserted the additional constant, <math>\mathcal{G}_0</math>, to accommodate normalization. A segment of the freeenergy surface defined by this function is displayed in the righthand panel of the figure shown at the top of this page. In constructing this figure, <math>~\mathcal{G}_0</math> has been set to a value that ensures that <math>~\mathcal{U}</math> is everywhere positive over the displayed domain: <math>~0.1 \le P_e/P_\mathrm{rf} \le 1.1</math> and <math>~0.3 \le R/R_\mathrm{rf} \le 3.0</math>.
For a given choice of <math>~P_e</math>, equilibrium radii are identified by setting <math>~d\mathcal{U}/dR = 0</math>, that is, they are defined by the (scalar virial theorem) relation,
<math>~\frac{P_e}{P_\mathrm{rf}} </math> 
<math>~=</math> 
<math>~ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr)^{4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}}\biggr)  3 \biggr] \, ; </math> 
and equilibria are stable — that is, <math>~[d^2\mathcal{U}/dR^2]_\mathrm{eq} > 0</math> — if <math>~R_\mathrm{eq}/R_\mathrm{rf} > 1</math>. For physically realistic systems, of course, <math>~P_e</math> must be positive — which means that all equilibria have <math>~R_\mathrm{eq}/R_\mathrm{rf} > 3/4</math>. But, from this algebraic virial theorem expression, it is also clear that physically realistic equilibrium configurations only exist when <math>~P_e/P_\mathrm{rf} \le 1</math>. The sequence of small, colored spherical dots in the righthand panel of the above figure identify parametervalue pairs, <math>~(R_\mathrm{eq}, P_e)</math>, associated with fourteen different equilibrium configurations: Blue dots — tracing out the valley of the freeenergy surface — identify stable configurations; white dots — balanced along the crest of the surface ridge — identify dynamically unstable configurations; and the lone red dot identifies the critical neutral equilibrium state, which is also associated with the maximum allowable value of <math>~P_e</math> along the equilibrium model sequence.
Virial PressureRadius Relation for UniformDensity, PressureTruncated Isothermal Configurations 
Equilibria 


<math>~\frac{P_e}{P_\mathrm{rf}}</math> 
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{rf}}</math> 
<math>~\frac{2\mathcal{U}}{3\mathcal{U}_\mathrm{rf}}</math> 
<math>~\mathcal{G}_0</math> 

0.200 
2.395 
0.03205 
1.8831 

0.30946 
2.008 
0.04507 
1.6786 

0.400 
1.800 
0.05548 
1.5646 

0.50364 
1.622 
0.06730 
1.4666 

0.632 
1.452 
0.08213 
1.3743 

0.72543 
1.347 
0.09327 
1.3205 

0.800 
1.270 
0.10252 
1.2835 

0.93516 
1.127 
0.12070 
1.2263 

0.97564 
1.071 
0.12681 
1.2111 

1.000 
1.000 
0.13086 
1.2025 

0.9897 
0.9612 
0.12880 
1.2061 

0.92636 
0.9061 
0.11370 
1.2297 

0.800 
0.859 
0.07423 
1.2835 

0.632 
0.822 
0.000 
1.3743 

0.400 
0.789 
+0.17021 
1.5646 

0.200 
0.767 
+0.47300 
1.8831 

Graphical depictions of the freeenergy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressuretruncated, uniformdensity isothermal configurations — see equation (10) of Whitworth (1981) or our restatement of this equation, above.

Figure 2 from Whitworth (1981) — reproduced in the lowerright quadrant of the composite image shown immediately above — does an excellent job of conveying many of the essential elements of this isothermal freeenergy surface within the constraints imposed by a twodimensional black & white line plot. In order to construct this compact plot, Whitworth employed a different freeenergy normalization parameter for each selected value of the external pressure. Specifically, he used,
<math>~\mathcal{G}_0(P_e) = \frac{2}{3}\biggl[ 1 + \ln\biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)\biggr]  \frac{3}{2} \biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)^{1/3} \, .</math>
In an effort to quantitatively compare (and check for accuracy) our results with Whitworth's, we have adopted the same <math>~\mathcal{G}_0(P_e)</math> normalization function when generating the multicolored, threedimensional freeenergy surface that is displayed (with three different projections) in the above composite image, along with the reproduction of Whitworth's Figure 2. Aside from this variable normalization parameter, the surface drawn for comparison with Whitworth's Figure 2 is identical to the one displayed in the righthand panel of the figure at the top of this page.
Our Case P Analysis Allowing for Nonuniform Density Structures
Coefficient Definitions
As has been both summarized and detailed in an accompanying discussion, our Case P analysis has demonstrated that the following,
Algebraic FreeEnergy Function
<math> \mathfrak{G}^* = 3\mathcal{A} \chi^{1} +~ n\mathcal{B} \chi^{3/n} +~ \mathcal{D}\chi^3 \, , </math>
properly governs the equilibrium structure and stability of pressuretruncated polytropic configurations. This algebraic expression is identical to the freeenergy function given above if the following coefficient and variable substitutions are made:
Our Case P Analysis  

where — see, for example, our accompanying review,
<math>~R_\mathrm{norm}</math> 
<math>~=</math> 
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n1} \biggr]^{1/(n3)} \, ,</math> 
<math>~P_\mathrm{norm}</math> 
<math>~=</math> 
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n3)} \, , </math> 
<math>~E_\mathrm{norm}</math> 
<math>~=</math> 
<math>~ \biggl[ K^n G^{3}M_\mathrm{tot}^{n5} \biggr]^{1/(n3)} \, ,</math> 
and, in terms of the structural form factors, <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_A</math>, and <math>~\tilde\mathfrak{f}_W</math>,
<math>~\mathcal{A}</math> 
<math>~=</math> 
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, , </math> 
<math>~\mathcal{B}</math> 
<math>~=</math> 
<math> \biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n+1)/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, . </math> 
Virial Equilibrium
Plugging these coefficient assignments into the above mathematical prescription of the virial theorem gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressuretruncated polytropic configurations,
<math>~\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n3)/n} </math> 
<math>~=</math> 
<math>\mathcal{A} + \frac{4\pi}{3} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{norm}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \, , </math> 
which matches the statement of virial equilibrium presented in our accompanying, more detailed analysis. A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
<math>~ \frac{P_\mathrm{e}}{P_\mathrm{norm}} </math> 
<math>~=</math> 
<math>~\frac{3}{4\pi} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4} \biggl[ \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n3)/n} \mathcal{A} \biggr] \, , </math> 
or (see the accompanying derivation for details),
<math>~P_e </math> 
<math>~=</math> 
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}  \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math> 
Notice that this equilibrium relation exactly matches the one derived by Whitworth — and rederived above — when all three structural form factors are set to unity. This is as it should be because all of Whitworth's results were derived assuming uniformdensity configurations. Also notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the lefthand column of our table titled "Two Points of View").
Stability
Similarly, according to the abovederived stability criterion, pressuretruncated polytropic configurations will only be stable if,
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math> 
<math>~></math> 
<math>\frac{(n3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{1} \, . </math> 
Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,
<math>~P_e </math> 
<math>~></math> 
<math>~ \frac{(n3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) =\frac{(n3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr) \, . </math> 
Our Case M Analysis
Coefficient Definitions
As has been detailed in an accompanying discussion, our Case M analysis has demonstrated that the freeenergy expression governing the equilibrium structure and stability of pressuretruncated polytropic configurations can also be written as,
<math>~\mathfrak{G}^*_\mathrm{SWS}</math> 
<math>~=</math> 
<math>~ 3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{1} +~ n\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3/n} +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \, . </math> 
Written in this form, the expression highlights the functional dependence on the configuration's mass while assuming that the external pressure is held fixed. Still, the expression is identical to the freeenergy function given above, but viewed in this manner the appropriate coefficient and variable substitutions are:
Our Case M Analysis  

where, drawing from Steven W. Stahler's (1983) work — see also our accompanying discussion,
<math>~M_\mathrm{SWS}</math> 
<math>~\equiv</math> 
<math>~\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3n)/[2(n+1)]} \, ,</math> 
<math>~R_\mathrm{SWS}</math> 
<math>~\equiv</math> 
<math>~ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1n)/[2(n+1)]} \, ,</math> 
<math>~E_\mathrm{SWS} \equiv \biggl( \frac{n}{n+1}\biggr)\frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}} </math> 
<math>~\equiv</math> 
<math>~ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5n)/[2(n+1)]} \, ,</math> 
and, in terms of the structural form factors, <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_A</math>, and <math>~\tilde\mathfrak{f}_W</math>,
<math>~\mathcal{A}_{M_\ell}</math> 
<math>~\equiv</math> 
<math>~ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} = \frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,</math> 
<math>~\mathcal{B}_{M_\ell}</math> 
<math>~\equiv</math> 
<math>~\mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n+1)/n} = \biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, . </math> 
Virial Equilibrium
Plugging these coefficient assignments into the above mathematical prescription of the virial theorem gives the massradius relationship for pressuretruncated, polytropic equilibrium configurations, namely,
<math>~\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n3)/n} </math> 
<math>~=</math> 
<math>\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} + \frac{4\pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 \, , </math> 
which matches the statement of virial equilibrium presented in our accompanying, more detailed analysis.
Stability
Similarly, according to the abovederived stability criterion, pressuretruncated polytropic configurations will only be stable if,
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} </math> 
<math>~></math> 
<math>\biggl[ \frac{\mathcal{A}_{M_\ell}(n3)}{4\pi n}\biggr]^{1/4} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{1/2} \, . </math> 
Or, flipped around, the criterion for stability may be written as,
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} </math> 
<math>~<</math> 
<math>~ \biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n3)}\biggr]^{1/2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^2 \, . </math> 
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
 Go Home 
Outline of Detailed Investigations Leading to Above Summary
First Effort
My first attempt to analytically define the free energy, and then the virial equilibrium, of pressuretruncated (embedded) polytropic spheres was developed as a direct extension of my description of the virial equilibrium of isolated polytropes. An important outcome of this "first effort" was the unveiling of analytic expressions for the key structural form factors, both for isolated polytropes and, separately, for pressuretruncated polytropic structures.
I am very confident that the formfactor expressions presented for isolated polytropes are all correct because they have been crosschecked with expressions for closely related "integral" parameters discussed by Chandrasekhar [C67]. Although the formfactor expressions derived for pressuretruncated polytropes make some sense — they look very similar to the ones presented for isolated polytropes and seem to behave properly for models which, based on detailed forcebalanced analysis, are known to be in equilibrium — I have much less confidence that they are correct. A couple of strategies were developed in an effort to demonstrate the validity and utility of these more general formfactor expressions, resulting in the derivation of a concise virial equilibrium relation,
<math>\Pi_\mathrm{ad} = \chi_\mathrm{ad}^{3\gamma}  \chi_\mathrm{ad}^{4} \, ,</math>
that incorporates the newly defined normalization parameters, <math>~R_\mathrm{ad}</math> and <math>~P_\mathrm{ad}</math>. But subsequent derivations aimed at more conclusively demonstrating the correctness of the more general formfactor expressions were messy and got bogged down.
Second Effort
My second attempt to analytically define the free energy, and then the virial equilibrium, of pressuretruncated (embedded) polytropic spheres built upon my first effort and, for a couple of different polytropic indexes, focused on comparing the massradius relationship embodied in detailed forcebalanced models against the massradius relationship implied by the virial theorem. A lot of reasonable results seem to have arisen from a discussion of models (done numerically using Excel) with <math>~n=4</math> polytropic index. And there are some nice aspects of models with an <math>~n=5</math> index, but these models raise some serious concerns related to the fact that two of our "derived" formfactor expressions involve division by the factor, <math>~(5n)</math>, that is, division by zero.
Third Effort
In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically, specifically, for indexes <math>~n=5</math> and <math>~n=1</math>. The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> do not match the general formfactor expressions that were rather cavalierly "derived" during our first effort. Oddly enough, as we discovered while fiddling around with the new results, the ratio of these form factors appears to be the same as before, namely,
<math>~\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_A  \tilde\theta^{n+1}}</math> 
<math>~=</math> 
<math>~ \biggl[ \frac{3\cdot 5}{(n+1) \tilde\xi^2 } \biggr] \, . </math> 
It is worth noting that, as a result of this more thorough "third effort" examination, we have confirmed that the third key form factor,
<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[ \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>
which is the same as before and the same as for isolated polytropes. We also have determined that,
<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} = \biggl(\frac{\tilde\xi^2 \tilde\theta^'}{\xi_1 \theta^'_1} \biggr)\biggl[ \frac{\tilde\xi}{3\tilde\theta^'}\biggr] =  \frac{\tilde\xi^3 }{3\xi_1 \theta^'_1} \, , </math>
except in the case of <math>~n=5</math> structures, for which we have determined,
<math>~\biggl[\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_{n=5} = \ell^3 = \biggl( \frac{\tilde\xi^2}{3} \biggr)^{3/2} \, . </math>
First Effort, Second Time Around
In an accompanying chapter, we reproduce the discussion associated with our "First Effort", as referenced above, but correct expressions for <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, as identified in our "Third Effort" and, accordingly, rederive various affected expressions that follow.
Second Effort, Second Time Around
In an accompanying chapter, we reproduce the discussion associated with our "Second Effort", as referenced above, but revise key sections to incorporate corrected expressions for the structural form factors.
© 2014  2021 by Joel E. Tohline 