Editing SSC/Virial/Isothermal (section)

====Quartic Solution====
In the above <math>~P-V</math> diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give <math>~\Pi</math> for any chosen value of <math>~\chi</math>.  Alternatively, the four roots of the quartic equation &#8212; <math>~\chi_1</math>, <math>~\chi_2</math>, <math>~\chi_3</math> and <math>~\chi_4</math> in the presentation that follows &#8212; will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, <math>~\Pi</math>, assuming the roots are real.  
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Roots of the quartic equation:  <math>\chi^4 - \chi \Pi^{-1}+ \Pi^{-1} = 0 </math>
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<math>~\chi_1</math>
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<math>~=</math>
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<math>~
+\frac{1}{2} y_r^{1/2} + \frac{1}{2} D_q \, ;
</math>
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<math>~\chi_2</math>
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<math>~=</math>
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<math>~
+\frac{1}{2} y_r^{1/2} - \frac{1}{2} D_q \, ;
</math>
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<math>~\chi_3</math>
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<math>~=</math>
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<math>~
-\frac{1}{2} y_r^{1/2} + \frac{1}{2} E_q \, ;
</math>
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<math>~\chi_4</math>
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<math>~=</math>
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<math>~
-\frac{1}{2} y_r^{1/2} - \frac{1}{2} E_q \, ,
</math>
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where,
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<math>~D_q</math>
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<math>~\equiv</math>
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<math>~
y_r^{1/2} \biggl[ \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2}  \, ,
</math>
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<math>~E_q</math>
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<math>~\equiv</math>
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<math>~
y_r^{1/2} \biggl[ - \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2}  \, ,
</math>
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and,
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<math>~
y_r \equiv \biggl( \frac{1}{2\Pi^2} \biggr)^{1/3} \biggl\{ \biggl[ 1 + \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3} + \biggl[ 1 - \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3}  \biggr\} \, ,
</math>
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is the real root of the cubic equation,
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<math>~
y^3 - \frac{4y}{\Pi} - \frac{1}{\Pi^{2}} = 0 \, .
</math>
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Because <math>~\Pi</math> must be positive in physically realistic solutions, we conclude that the two roots involving <math>~E_q</math> &#8212; that is, <math>~\chi_3</math> and <math>~\chi_4</math> &#8212; are imaginary and, hence, unphysical.  The other two roots  &#8212; <math>~\chi_1</math> and <math>~\chi_2</math> &#8212; will be real only if the arguments inside the radicals in the expression for <math>~y_r</math> are positive.  That is, <math>~\chi_1</math> and <math>~\chi_2</math> will be real only for values of the dimensionless external pressure,
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<math>~\Pi \leq \Pi_\mathrm{max} \equiv \frac{3^3}{2^8} \, .</math>
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This is the same upper limit on the external pressure that was derived above, via a different approach, and translates into a maximum mass for a pressure-bounded isothermal configuration of,
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<math>~M_\mathrm{max} = \Pi_\mathrm{max}^{1/2}  \biggl(\frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2}
= \biggl(\frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, .</math> 
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When combined, a plot of <math>~\chi_1</math> versus <math>~\Pi</math> and <math>~\chi_2</math> versus <math>~\Pi</math> will reproduce the solid black curve shown in Figure 2, but with the axes flipped. The top-right quadrant of Figure 3 presents such a plot, but in logarithmic units along both axes; also <math>~\Pi</math> is normalized to <math>~\Pi_\mathrm{max}</math> and <math>~\chi</math> is normalized to the equilibrium radius <math>~(4/3)</math> at that pressure.  This is the manner in which [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) chose to present this result for uniform-density, spherical isothermal <math>~(\gamma_\mathrm{g}=1)</math> configurations.  Our solid and dashed curve segments &#8212; identifying, respectively, the <math>~\chi_1(\Pi)</math> and <math>~\chi_2(\Pi)</math> solutions to the above quadratic equation &#8212; precisely match the solid and dashed curve segments labeled "1" in Whitworth's Figure 1a (replicated here in the bottom-right quadrant of Figure 3). 

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'''Figure 3:''' <font color="darkblue">Equilibrium R-P Diagram </font> 
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''Top:'' The solid curve traces the function <math>~\chi_1(\Pi)</math> and the dashed curve traces the function <math>~\chi_2(\Pi)</math>, where <math>~\chi_1</math> and <math>~\chi_2</math> are the two real roots of the quartic equation,
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<math>~
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, .
</math>
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Logarithmic units are used along both axes; <math>~\Pi</math> is normalized to <math>~\Pi_\mathrm{max}</math>; and <math>~\chi</math> is normalized to the equilibrium radius <math>~(4/3)</math> at <math>~\Pi_\mathrm{max}</math>. 

''Bottom:'' A reproduction of Figure 1a from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967).  The solid and dashed segments of the curve labeled "1" identify the equilibrium radii, <math>~R_\mathrm{eq}</math>, that result from embedding a uniform-density, isothermal <math>~(\gamma_\mathrm{g} = 1)</math> gas cloud in an external medium of pressure <math>~P_\mathrm{ex}</math>.  

''Comparison:'' The curve shown above that traces out <math>~\chi_1(\Pi)</math> and <math>~\chi_2(\Pi)</math> should be identical to the "Whitworth" curve labeled "1".
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[[File:WhitworthLogFig1a_norm.jpg|450px|center|To be compared with Whitworth (1981)]]
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[[File:WhitworthFig1aCopy.jpg|450px|center|Whitworth (1981) Figure 1a]]
<!-- [[Image:AAAwaiting01.png|450px|center|Whitworth (1981) Figure 1a]] -->
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