Editing VE (section)

==Virial Equations (Inertial Frame)==
Most of the material presented here has been drawn from Chandrasekhar's ''Ellipsoidal Figures of Equilibrium'' &#8212; hereafter [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; first published in 1969.  Relying heavily on [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] in-depth treatment of the topic, our aim is to highlight key aspects of the tensor-virial equations and to present them in a form that serves as a foundation for our separate discussions of the equilibrium and stability of self-gravitating fluid systems.  Strong parallels are drawn between the [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] presentation and our own so that it will be relatively straightforward for the reader to consult the [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] publication to obtain details of the various derivations. Text that appears in a green font has been drawn ''verbatim'' from this reference. 

===Setting the Stage===

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;8, p. 15] <font color="#007700">A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations.  The ''virial method'' &hellip; is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account.  The ''virial equations'' of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations.</font>  In this context, Chandrasekhar's focus is on two of the four [[PGE#Principal_Governing_Equations|principal governing equations]] that serve as the foundation of our entire H_Book, namely, the

<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
('''Momentum Conservation''')

{{ Template:Math/EQ_Euler01 }}
</div>
and the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Template:Math/EQ_Poisson01}}
[[File:OriginButton.jpg|125px|link=PGE/PoissonOrigin]]
</div>

In [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the Euler equation first appears in &sect;11 (p. 20) as equation (38) and is written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho \frac{du_i}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i} \, ,</math>
  </td>
</tr>
</table>
</div>
and the Poisson equation appears in &sect;10 (p. 20) &#8212; specifically, the left-most component of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>'s] equation (37) &#8212; as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\nabla^2 \mathfrak{B}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 4\pi G \rho \, .</math>
  </td>
</tr>
</table>
</div>
It is clear, therefore, that Chandrasekhar uses the variable <math>\vec{u}</math> instead of <math>\vec{v}</math> to represent the inertial velocity field.  More importantly, he adopts a different variable name ''and a different sign convention'' to represent the gravitational potential, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>- \Phi = \mathfrak{B} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
</div>
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] into this H_Book.

===First-Order Virial Equations===
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;11(a), p. 21] <font color="#007700">The [virial] equations of the first order are obtained by simply integrating [the Euler equation] over the instantaneous volume, <math>V</math>, occupied by the fluid</font>.  Specifically, using our H_Book variable notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\int\limits_V \rho \frac{dv_i}{dt} d^3x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int\limits_V \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
  </td>
</tr>
</table>
</div>
leads to (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d^2 I_i}{dt^2} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
where the <span id="MomentOfInertia">moments of inertia</span> about the three separate principal axes <math>(i = 1,2,3)</math> are defined by the expressions,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>I_i</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho x_i d^3x \, .</math>
  </td>
</tr>
</table>
</div>
Thus, the first-order virial equation(s) <font color="#007700">expresses the uniform motion of the center of mass of the system</font>.

===Second-Order Tensor Virial Equations===
In discussing the origin of the second-order (tensor) virial equation, [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] will continue to serve as our primary reference.  However, in &sect;4.3 of their widely referenced textbook titled, "Galactic Dyamics," Binney &amp; Tremaine (1987) &#8212; hereafter [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] &#8212; also present a detailed derivation of the second-order virial equation, which they refer to as the ''tensor virial theorem.''  Because their presentation is set in the context of discussions of the structure of ''stellar dynamic'' systems, the [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] derivation fundamentally originates from the collisionless Boltzmann equation. In what follows we will identify where various key equations appear in [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], as well as in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], because it can sometimes be useful to compare derivations made from the stellar-dynamic versus the fluid-dynamic perspective.

====Derivation====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;11(b), p. 22] The second-order (tensor) virial equations <font color="#007700">are obtained by multiplying [the Euler equation] by <math>x_j</math> and integrating over the volume, <math>V</math></font>.  Specifically, again using our H_Book variable notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\int\limits_V \rho \frac{dv_i}{dt} x_j d^3x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int\limits_V x_j \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho x_j \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 211, Eq. (4-72)
  </td>
</tr>
</table>
</div>
or, separating the term on the left-hand side into two physically distinguishable components &#8212; see equation 44 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; this can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d}{dt} \int\limits_V \rho v_i x_j d^3x - 2 \mathfrak{T}_{ij}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\delta_{ij}\Pi + \mathfrak{W}_{ij} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 22, Eq. (47)
  </td>
</tr>
</table>
</div>
where, by definition,
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
  <td colspan="6">
&nbsp;
  </td>
  <th align="center">
References
  </th>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{T}_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho v_i v_j  d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is the (ordered) kinetic energy tensor
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (9)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
  </td>
</tr>

<tr>
  <td align="right">
<math>\Pi</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V P d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is &#x2154; of the total thermal (''i.e.,'' random kinetic) energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (7)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{W}_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho \Phi_{ij} d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (15)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 68, Eq. (2-126)
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_j} d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is the gravitational potential energy tensor
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 18, Eq. (18)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 67, Eq. (2-123)
  </td>
</tr>
</table>
</div>
Note that, in the definition of the gravitational potential energy tensor, Chandrasekhar has introduced a tensor generalization of the gravitational potential [see his Eq. (14), p. 17], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>- \Phi_{ij} = \mathfrak{B}_{ij}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>G\int\limits_V \rho(\vec{x}^') \frac{ (x_i - x_i^')(x_j - x_j^') }{|\vec{x} - \vec{x}^{~'}|^3} d^3x^' \, ;</math>
  </td>
</tr>
</table>
</div>
this same potential energy tensor appears explicitly as part of the expression for <math>\mathfrak{W}_{ij}</math> that is presented as Equation (2-126), on p. 67 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].


The antisymmetric part of this tensor expression gives (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d}{dt} \int\limits_V \rho (v_ix_j - v_j x_i) d^3x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>.  The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details) the,
<div align="center">
<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)
  </td>
</tr>
</table>
</div>
<span id="MOItensor">where,</span>
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
  <td colspan="6">
&nbsp;
  </td>
  <th align="center">
References
  </th>
</tr>

<tr>
  <td align="right">
<math>I_{ij}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho x_i  x_j d^3x </math>
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="left">
is the moment of inertia tensor
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (4)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-76)
  </td>
</tr>
</table>
</div>

====Steady State (Virial Equilibrium)====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \delta_{ij}\Pi \, .</math>
  </td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:#9390DB;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Scalar<br />Virial<br />Theorem</b>]]</font>
|}
===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, 
<div align="center">
<table border="0" cellpadding="2" align="center">
<tr>
  <td colspan="4">
&nbsp;
  </td>
  <th align="center">
Description
  </th>
  <td colspan="1">
&nbsp;
  </td>
  <th align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]
Reference
  </th>
</tr>

<tr>
  <td align="right">
<math>I = \sum\limits_{i=1,3} I_{ii}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int\limits_V \rho (\vec{x}) |\vec{x}|^2 d^3x  </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
scalar moment of inertia
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eqs. (3) &amp; (5), p. 16]
  </td>
</tr>

<tr>
  <td align="right">
<math>T_\mathrm{kin} = \sum\limits_{i=1,3} \mathfrak{T}_{ii}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho |\vec{v}|^2  d^3x </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
total (ordered) kinetic energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eq. (8), p. 16]
  </td>
</tr>

<tr>
  <td align="right">
<math>W_\mathrm{grav} = \sum\limits_{i=1,3} \mathfrak{W}_{ii}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_i} d^3x </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
gravitational potential energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eq. (18), p. 18]
  </td>
</tr>

<tr>
  <td align="right">
<math>S_\mathrm{therm} = \frac{1}{2} \sum\limits_{i=1,3} \delta_{ii}\Pi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{2} \int\limits_V P d^3x </math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
total thermal (random kinetic) energy
  </td>
  <td align="center">
&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
  </td>
  <td align="center">
[Eq. (7), p. 16]
  </td>
</tr>
</table>
</div>
the scalar virial equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{1}{2} \frac{d^2 I}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} \, ;</math>
  </td>
</tr>
</table>
</div>
and, for a stationary state, we have the equilibrium condition that is broadly referred to as the,
<div align="center">
<span id="TVE"><font color="#770000">'''Scalar Virial Theorm'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-79)
  </td>
</tr>
</table>
</div>
(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,
<div align="center">
<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)
</div>
Drawing from Equation (24.1), the associated modified Euler equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\rho \frac{dv_i}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial P}{\partial x_i} - \rho \frac{\partial \Phi}{\partial x_i} + \frac{\partial T_{ik}}{\partial x_k}  \, .</math>
  </td>
</tr>
</table>
</div>
[<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
<div align="center">
<span id="GenTVE"><font color="#770000">'''Generalized Scalar Virial Theorem'''</font></span><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} + \mathcal{M}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> P_e \oint \vec{x}\cdot \hat{n} dA - \oint \vec{x}\cdot \overrightarrow{T}\hat{n} dA   \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. II, p. 331, Eq. (24.12)
  </td>
</tr>
</table>
</div>
<font color="orange">where <math>\mathcal{M}</math> equals the magnetic energy contained in volume <math>V</math></font>,
<div align="center">
<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)
</div>

[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.]