Editing SSCpt1/Virial (section)

====Volume Integrals====
We note, first, that the mass enclosed within each interior radius, <math>r</math>, is
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>M_r(r) = \int\limits_V dm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^r  4\pi r^2 \rho dr  \, .</math>
  </td>
</tr>
</table>
</div>
Hence, if the volume of the configuration extends out to a radius denoted by <math>R_\mathrm{limit}</math>, the configuration mass is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>M_\mathrm{limit}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{R_\mathrm{limit}}  4\pi r^2 \rho dr  \, .</math>
  </td>
</tr>
</table>
</div>

<table align="center" border="1" width="80%" cellpadding="8">
<tr><td align="left">
NOTE:  The following considerations have led us to formally draw a distinction between <math>M_\mathrm{limit}</math> and the "total" mass, <math>M_\mathrm{tot}</math>, that we use (see below) for normalization.  

<font color="maroon"><b>Isolated Polytropes</b></font>:  For [[SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|isolated polytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will be the natural edge of the configuration, where the pressure and mass-density drop to zero.  In this case, <math>M_\mathrm{limit}</math> quite naturally corresponds to the total mass of the configuration.  

<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>:  But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|configuration embedded in an external medium]] of pressure, <math>P_e</math>, will have a (pressure-truncated) surface whose radius, <math>R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>P_e</math>.  In this case as well, one might choose to refer to <math>M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>M_\mathrm{limit}</math> from <math>M_\mathrm{tot}</math>, continuing to rely on <math>M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.  

<font color="maroon"><b>BiPolytropes</b></font>:  When discussing [[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization|bipolytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope.  In this case, <math>M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
</td></tr>
</table>

<font color="red">Confinement by External Pressure:</font>  For spherically symmetric configurations, the energy term due to confinement by an external pressure can be expressed, simply, in terms of the configuration's radius, <math>R_\mathrm{limit}</math>, as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_e V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>P_e \int_0^{R_\mathrm{limit}}  4\pi r^2 dr  = \frac{4\pi}{3} P_e R_\mathrm{limit}^3 \, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Gravitational Potential Energy:</font>  From our discussion of the [[VE#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; see, specifically, the reference to Equation (18), on p. 18 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the gravitational potential energy is given by the expression,
<div align="center">
<math>
W_\mathrm{grav} = - \int\limits_V \rho x_i \frac{\partial\Phi}{\partial x_i} d^3 x
= - \int\limits_V \vec{r} \cdot \nabla\Phi dm = - \int_0^{R_\mathrm{limit}} \biggl( r \frac{d\Phi}{dr} \biggr) dm \, .
</math>
</div>
For spherically symmetric systems, the

<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{ Template:Math/EQ_Poisson01 }}
</div>
becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{1}{r^2} \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi G \rho(r) \, , </math>
  </td>
</tr>
</table>
</div>
which implies,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>r^2 \frac{d\Phi}{dr} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^r 4\pi G \rho(r) r^2 dr = GM_r(r) \, .</math>
  </td>
</tr>
</table>
</div>
<span id="Wgrav">Hence</span> &#8212; see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; the desired expression for the gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) dm = - \int_0^{R_\mathrm{limit}} \frac{G}{r}\biggl[\int_0^r 4\pi r^2 \rho dr \biggr] 4\pi r^2 \rho dr \, .</math>
  </td>
</tr>
</table>
</div>


<div id="AlternateGravPotEnergy">
<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
Also, as pointed out by [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; see p. 64, Equation (16) &#8212; it may sometimes prove advantageous to recognize that, if a spherically symmetric system is in hydrostatic balance, an alternate expression for the total gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>+ \frac{1}{2} \int_0^{R_\mathrm{limit}} \Phi(r) dm \, .</math>
  </td>
</tr>
</table>
</div>
</td></tr>
</table>
</div>

<div>
<font color="red">Rotational Kinetic Energy:</font>  We will also consider a system that is rotating with a specified [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple'' angular velocity profile]], <math>\dot\varphi(\varpi)</math>, in which case, from our discussion of the [[VE#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; see, specifically, the reference to Equation (8), on p. 16 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the (ordered) kinetic energy,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>T_\mathrm{kin}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho |\vec{v} |^2 d^3x = \frac{1}{2} \int\limits_V |\vec{v} |^2 dm  \, ,</math>
  </td>
</tr>
</table>
</div>
is entirely rotational kinetic energy, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>T_\mathrm{kin} = T_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\int\int \dot\varphi^2 \varpi^2 dm
= \frac{1}{2} \int_0^{R_\mathrm{limit}} \dot\varphi^2 \varpi^2   \int_{-\sqrt{{R_\mathrm{limit}}^2 - \varpi^2}}^{\sqrt{{R_\mathrm{limit}}^2 - \varpi^2}}  \rho(r(\varpi,z)) 2\pi \varpi d\varpi dz\, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Reservoir of Thermodynamic Energy:</font>  As has been explained in [[VE#Reservoir_of_Thermodynamic_Energy|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally.  For [[VE#Isothermal_Systems|isothermal systems]],
<div align="center" id="Reservoir">
<math>
\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~\mathfrak{S}_I 
= + \int\limits_V c_s^2  \ln \biggl(\frac{\rho}{\rho_\mathrm{norm}}\biggr) dm
= c_s^2  \int_0^{R_\mathrm{limit}} \ln \biggl(\frac{\rho}{\rho_\mathrm{norm}}\biggr) 4\pi r^2 \rho dr \, ,
</math>
</div>
where, <math>c_s</math> is the isothermal sound speed and <math>\rho_\mathrm{norm}</math> is a (as yet unspecified) reference mass density; while, for [[VE#Adiabatic_Systems|adiabatic systems]],
<div align="center">
<math>
\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~ \mathfrak{S}_A 
= + \int\limits_V  \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) dm
= \frac{1}{({\gamma_g}-1)}  \int_0^{R_\mathrm{limit}}  4\pi r^2 P dr 
 \, ,</math>
</div>
where, <math>P(r)</math> is the system's pressure distribution and <math>\gamma_g</math> is the specified adiabatic index.