Editing VE (section)

===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">
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<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>
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  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 211, Eq. (4-72)
  </td>
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</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,
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<table border="0" cellpadding="5" align="center">
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  <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>
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</table>
</div>
where, by definition,
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&nbsp;
  </td>
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References
  </th>
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  <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>
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  <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>
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  <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>
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  <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>
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</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,
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<math>- \Phi_{ij} = \mathfrak{B}_{ij}</math>
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  <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>
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</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">
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<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,
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<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
<table border="0" cellpadding="5" align="center">
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<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>
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</table>
</div>
<span id="MOItensor">where,</span>
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  <td colspan="6">
&nbsp;
  </td>
  <th align="center">
References
  </th>
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<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>
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</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>.

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