Editing VE (section)

===Scalar Virial Theorem===
====Standard Presentation [the Virial of Clausius (1870)]====
The trace of the tensor virial equation (TVE), which is obtained by identifying the trace of each term in the TVE, produces the scalar virial equation, which is widely referenced and used by the astrophysics community.  More specifically, setting, 
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Description
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&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]
Reference
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<math>I = \sum\limits_{i=1,3} I_{ii}</math>
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<math>=</math>
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<math>\int\limits_V \rho (\vec{x}) |\vec{x}|^2 d^3x  </math>
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=
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scalar moment of inertia
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eqs. (3) &amp; (5), p. 16]
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<math>T_\mathrm{kin} = \sum\limits_{i=1,3} \mathfrak{T}_{ii}</math>
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<math>=</math>
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<math>\frac{1}{2} \int\limits_V \rho |\vec{v}|^2  d^3x </math>
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=
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total (ordered) kinetic energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eq. (8), p. 16]
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<math>W_\mathrm{grav} = \sum\limits_{i=1,3} \mathfrak{W}_{ii}</math>
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<math>\equiv</math>
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<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_i} d^3x </math>
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=
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gravitational potential energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eq. (18), p. 18]
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<math>S_\mathrm{therm} = \frac{1}{2} \sum\limits_{i=1,3} \delta_{ii}\Pi</math>
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<math>=</math>
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<math>\frac{3}{2} \int\limits_V P d^3x </math>
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=
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total thermal (random kinetic) energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eq. (7), p. 16]
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the scalar virial equation is,
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<math>\frac{1}{2} \frac{d^2 I}{dt^2}</math>
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<math>=</math>
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<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} \, ;</math>
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and, for a stationary state, we have the equilibrium condition that is broadly referred to as the,
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<span id="TVE"><font color="#770000">'''Scalar Virial Theorm'''</font></span><br />
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<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} </math>
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<math>=</math>
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<math>0 \, .</math>
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-79)
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(In a footnote to their Equation 4-79, [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] point out that the ''scalar virial theorem'' was first proved by R. Clausius in 1870; see various links to this work under our [[VE#Related_Discussions|"Related Discussions" subsection, below]].)

====Generalization====
Chapter 24 in Volume II (''Gas Dynamics'') of [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] presents a generalization of the scalar virial theorem that includes the effects of (a) a magnetic field that threads through a self-gravitating fluid system, and (b) an imposed surface pressure, <math>P_e</math>, when the configuration is embedded in a hot, tenuous external medium.  Text that appears in an orange font in the following paragraph has been drawn ''verbatim'' from this reference.

[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] begins by adding a term to the Euler equation that accounts for <font color="orange">the Maxwell stress tensor, <math>T_{ik}</math>, associated with the ambient magnetic field</font>, <math>~\vec{B}</math>, where,
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<math>
T_{ik} = \frac{B_i B_k}{4\pi} - \frac{|\vec{B}|^2}{8\pi} \delta_{ik} \, .
</math>
<br />&nbsp;<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 329, Eq. (24.3)
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Drawing from Equation (24.1), the associated modified Euler equation is,
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<math>\rho \frac{dv_i}{dt} </math>
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<math>=</math>
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<math>- \frac{\partial P}{\partial x_i} - \rho \frac{\partial \Phi}{\partial x_i} + \frac{\partial T_{ik}}{\partial x_k}  \, .</math>
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[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>, Vol. II, pp. 329-330] <font color="orange">If we were to multiply [this modified Euler equation] by <math>~x_m</math> and integrate over volume <math>V</math>, we would get the [appropriately modified] ''tensor virial theorem'', the off-diagonal elements of which carry information concerning angular-momentum conservation (see </font> [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] <font color="orange">for an exposition).  [Here] we shall be more interested in the trace of the tensor equation, which we may derive by simply multiplying [the modified Euler equation] by <math>~x_i</math> (with an implicit summation over repeated indices) and integrating over <math>V</math></font>.  The resulting relation governing the equilibrium of stationary states (see [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] for derivation details), as we shall reference it, is the
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<span id="GenTVE"><font color="#770000">'''Generalized Scalar Virial Theorem'''</font></span><br />
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<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} + \mathcal{M}</math>
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<math>=</math>
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<math> P_e \oint \vec{x}\cdot \hat{n} dA - \oint \vec{x}\cdot \overrightarrow{T}\hat{n} dA   \, ,</math>
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[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 331, Eq. (24.12)
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<font color="orange">where <math>\mathcal{M}</math> equals the magnetic energy contained in volume <math>V</math></font>,
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<math>
\mathcal{M} \equiv \int\limits_V \frac{|\vec{B}|^2}{8\pi} d^3x \, .
</math>
<br />&nbsp;<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], p. 165, Eq. (7.1.18)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 330, Eq. (24.9)
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[It should be noted that {{ CF53full }} and {{ MS56full }} provide early discussions of virial equilibrium conditions that take into account the energy associated with a magnetic field.]