Editing SSCpt1/Virial (section)

=====Summary of Normalized Expressions=====
Hence, our normalized expressions become,
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Normalized Expressions
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<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
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<math>=</math>
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<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, ,</math>
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<math>\frac{P_e V}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)  \chi^3 \, ,</math>
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<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx 
</math>
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&nbsp;
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<math>=</math>
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<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx\biggr\}  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,
</math>
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<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1}  3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\} \, ,</math>
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<math>\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>\int_0^{1} 
\biggl\{ \ln \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] -3\ln \biggl[ \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr]  \biggr\} 
3 x^2 \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] dx </math>
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&nbsp;
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<math>=</math>
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<math>-3 \ln \chi  + \mathrm{constant} \, ,
</math>
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<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \biggl( \frac{\rho_c}{\bar\rho}  \biggr)_\mathrm{eq} 
\int_0^{1} \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{edge}} \biggr]^2 w^3  dw
\int_{-\sqrt{1 - w^2}}^{\sqrt{1 - w^2}}  \biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr]   d\zeta \, .
</math>
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<font color="red">NOTE to self (21 September 2014)<b></b></font>:  The expressions for <math>\mathfrak{S}_I</math> and <math>T_\mathrm{rot}</math> may not properly account for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>.
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It should be emphasized that the coefficient involving the density ratio, <math>(\rho_c/\bar\rho)</math>, that lies outside of the integral in most of these expressions depends only on the internal structure, and not the overall size, of the configuration.  It can therefore be evaluated at any time.  We usually will choose to evaluate this coefficient in an equilibrium state, that is, when <math>R_\mathrm{limit} \rightarrow R_\mathrm{eq}</math>.  Accordingly, the subscript "eq" has been attached to this coefficient.  The inverse of this density ratio can be obtained from the integral expression for <math>M_r</math> by recognizing that <math>M_r \rightarrow M_\mathrm{limit}</math> when the upper limit on the integral <math>x \rightarrow 1</math>.  Hence,
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<math>\biggl(\frac{\rho_c}{\bar\rho} \biggr)^{-1}_\mathrm{eq}  </math>
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<math>=</math>
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<math>\int_0^{1}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]_\mathrm{eq}  dx \, .</math>
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This coefficient also may be rewritten in terms of the central pressure in the equilibrium state; specifically, using a sequence of steps similar to the ones that were used, above, in rewriting <math>P^*</math>, we can write,
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<math>
\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
</math>
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<math>=</math>
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<math>\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .</math>
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