Editing SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt1 (section)

===Alternate Derivation of Gravitational Potential Energy===

<!-- BEGINNING OF DELETION:  The following subsubsection is redundant, now that the "Two Points of View" table has been included.

====Central Pressure====
As is pointed out, above, in our [[SSC/Virial/Polytropes#Mass-Radius_Relationship|discussion of the mass-radius relationship]], the expression for the equilibrium radius that has been derived from our analysis of extrema in the free energy function can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{3}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \, .
</math>
  </td>
</tr>
</table>
</div>
Via the polytropic equation of state, we can relate <math>~K</math> to the central pressure as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~K \rho_c^{1+1/n} = K \bar\rho^{1+1/n} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{1+1/n}
= K \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr]^{(n+1)/n} \biggl( \frac{1}{\mathfrak{f}_M} \biggr)^{(n+1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ K^{n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
P_c^{n} \biggl[ \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr]^{(n+1)} \mathfrak{f}_M^{(n+1)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, in the mass-radius relationship we can replace <math>~K</math> with this expression to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{P_c}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)} 
\biggl[ \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr]^{(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~3 \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M^2}{\mathfrak{f}_W} \biggr)^n 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl( \frac{G M_\mathrm{tot}^2}{P_c R_\mathrm{eq}^4}\biggr)^n 
\biggl( \frac{3}{4\pi } \biggr)^{n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M^2}{\mathfrak{f}_W} \biggr)^n 
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
<math>
\Rightarrow ~~~~ P_c = \frac{3 }{20\pi }\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2}  \, .
</math>
  </td>
</tr>
</table>
</div>
Now, from our above [[SSC/Virial/Polytropes#Gravitational_Potential_Energy|examination of the expression for the gravitational potential energy]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{5-n} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, our expression for the central pressure becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="3">
<math>
~P_c = \frac{3 }{4\pi (5-n)}\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
\cdot \frac{1}{\mathfrak{f}_A }  \, .
</math>
  </td>
</tr>
</table>
</div>
As it should, this precisely matches [[SSC/Virial/Polytropes#Central_Pressure|the expression that we derived, above]], starting from [[Appendix/References|Chandrasekhar's [C67]]] presentation.
====Gravitational Potential Energy====


END OF DELETION  -->

As has been [[SSCpt1/Virial#AlternateGravPotEnergy|discussed elsewhere]], we have learned from [[Appendix/References|Chandrasekhar's discussion of polytropic spheres [C67]]] &#8212; see his Equation (16), p. 64 &#8212; that if a spherically symmetric system is in hydrostatic balance, the total gravitational potential energy can be obtained from the following integral:
<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 \Phi(r) dm \, .</math>
  </td>
</tr>
</table>
</div>
Using "[[SSCpt2/SolutionStrategies#Technique_3|technique #3]]" to solve the differential equation that governs the statement of hydrostatic balance, we know that in any polytropic sphere, <math>~\Phi(r)</math> is related to the configuration's radial enthalpy profile, <math>~H(r)</math>, via the algebraic expression,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi(r) + H(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_B \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~C_B</math>, is an integration constant.  At the surface of the equilibrium configuration, <math>~H = 0</math> and <math>~\Phi = - GM_\mathrm{tot}/R_\mathrm{eq}</math>, so the integration constant is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~C_B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, ,</math>
  </td>
</tr>
</table>
</div>
which implies,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi(r) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - H(r) - \frac{GM_\mathrm{tot}}{R_\mathrm{eq}}  \, .</math>
  </td>
</tr>
</table>
</div>
Now, from [[SR#Barotropic_Structure|our general discussion of barotropic relations]], we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~H(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n+1) \frac{P(r)}{\rho(r)} \, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~-\Phi(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n+1) \frac{P(r)}{\rho(r)} + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<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 \biggl[ (n+1) \frac{P(r)}{\rho(r)} + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \biggr] 4\pi \rho(r) r^2 dr </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\pi \biggl\{ 
(n+1) \int_0^R P(r) r^2 dr 
+ \frac{GM_\mathrm{tot}}{R_\mathrm{eq}}  \int_0^R  \rho(r) r^2 dr 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\pi \biggl\{ 
\frac{1}{3} (n+1)P_c R_\mathrm{eq}^3 \int_0^1 3\biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx 
+ \frac{GM_\mathrm{tot}}{3R_\mathrm{eq}}  \biggl( \rho_c R_\mathrm{eq}^3  \biggr) \int_0^1  3 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] x^2 dx 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\pi \biggl\{ 
\frac{1}{3} (n+1)P_c R_\mathrm{eq}^3 \mathfrak{f}_A 
+ \frac{GM_\mathrm{tot}}{3R_\mathrm{eq}}  \biggl( \rho_c R_\mathrm{eq}^3  \biggr) \mathfrak{f}_M 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggl\{ 
\frac{2\pi}{3} (n+1) \biggl[ \frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} \biggr] \mathfrak{f}_A 
+  \frac{1}{2} \biggl[ \frac{4\pi \rho_c R_\mathrm{eq}^3}{3M_\mathrm{tot}}  \biggr] \mathfrak{f}_M 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{1}{2} \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggl\{ 
\frac{4\pi}{3} (n+1) \biggl[ \frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} \biggr] \mathfrak{f}_A 
+  1 
\biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of <math>~W</math> and <math>~P_c</math>, namely,
<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>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~P_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3 }{20\pi }\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \, .</math>
  </td>
</tr>
</table>
</div>
Plugging these into our newly derived expression for the gravitational potential energy gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~- \frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{1}{2} \biggl\{ 
\frac{4\pi}{3} (n+1) \biggl[ \frac{3 }{20\pi }\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \biggr] \mathfrak{f}_A 
+  1 \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ (2\cdot 3) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
(n+1) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} +  5 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ (5 - n) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  5 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \frac{5}{5-n} \, . </math>
  </td>
</tr>
</table>
</div>
As it should, this agrees with the expression for the ratio, <math>\mathfrak{f}_W/\mathfrak{f}_M^2</math>, that was [[SSC/Virial/Polytropes#Gravitational_Potential_Energy|derived in our above discussion of the gravitational potential energy]].