Editing SSC/Virial/Polytropes/Pt2 (section)

====Solution Expressed in Terms of K and M (Whitworth's 1981 Relation)====
This is precisely the same condition that derives from setting equation (3) to zero in the {{ Whitworth81full }} discussion of the ''Global Gravitational Stability for One-dimensional Polytropes''.  The overlap with Whitworth's narative is clearer after introducing the algebraic expressions for the coefficients <math>~A</math>, <math>~B</math>, and <math>~D</math>, to obtain,
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<math>~4\pi \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi_\mathrm{eq}^3 </math>
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<math>~=</math>
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<math>~3 \biggl( \frac{4\pi}{3} \biggr)^{1-\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot \chi_\mathrm{eq}^{3 -3\gamma_g} 
~-~\frac{3}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \cdot \chi_\mathrm{eq}^{-1} \, ;</math>
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dividing the equation through by <math>~(4\pi \chi_\mathrm{eq}^3/P_\mathrm{norm})</math>, 
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<math>~P_e </math>
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<math>~=</math>
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<math>~P_\mathrm{norm} \biggl[ \biggl( \frac{3}{4\pi} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} 
\cdot \chi_\mathrm{eq}^{-3\gamma_g} ~- \biggl(\frac{3}{20\pi} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \cdot \chi_\mathrm{eq}^{-4} \biggr] </math>
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&nbsp;
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<math>~=</math>
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<math>~P_\mathrm{norm} R_\mathrm{norm}^4\biggl[ \biggl( \frac{3}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot R_\mathrm{norm}^{3\gamma_g-4} 
- \biggl(\frac{3}{20\pi R_\mathrm{eq}^4} \biggr)\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \biggr] \, ;</math>
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and inserting expressions for the [[SSCpt1/Virial#Normalizations|parameter normalizations]] as defined in our accompanying [[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|introductory discussion]] to obtain,
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<math>~P_e </math>
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<math>~=</math>
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<math>~GM_\mathrm{tot}^2\biggl[ \biggl( \frac{3}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \cdot \frac{K M_\mathrm{tot}^{\gamma_g-2}}{G} 
- \biggl(\frac{3}{20\pi R_\mathrm{eq}^4} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \biggr] </math>
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&nbsp;
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<math>~=</math>
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<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} 
- \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>
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If the structural form factors are set equal to unity, this exactly matches equation (5) of {{ Whitworth81 }}, which reads:

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Equation and accompanying sentence drawn directly from p. 970 of<br />
{{ Whitworth81figure }}<br />
&copy; Royal Astronomical Society
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<!--
[[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
[[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]]
-->
The general equilibrium condition, <math>(d\mathcal{U}/dR)_{R_0} = 0</math>, reduces to
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  <td align="right" width="25%"><math>R_0 \rightarrow R_\mathrm{eq}\, ,</math></td>
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<math>P_\mathrm{ex} = K(3M_0/4\pi R^3_\mathrm{eq})^\eta - 3GM_0^2/20\pi R^4_\mathrm{eq}</math>
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  <td align="right" width="8%">(5)</td>
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(subscript 'eq' for equilibrium).
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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 left-hand column of our [[SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]).