Editing SSC/Structure/OtherAnalyticRamblings (section)

===Examine Structural Pressure-Density Relation===

====Derivation====

One striking property exhibited by the [[SSC/Structure/OtherAnalyticModels#Examples|example configurations tabulated above]] is the ''structural'' relationship between the chosen function, <math>~\rho_0(\chi_0)</math>, and the corresponding radial pressure distribution, <math>~P_0(\chi_0)</math>, that is  [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|dictated by]], 

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} = - \frac{GM_r}{r^2} </math> ,
</div>

As has been detailed in the last column of [[SSC/Structure/OtherAnalyticModels#Table1b|Table 1b]], in all four cases, the ratio, <math>~(P_0/P_c)(\rho_0/\rho_c)^{-2}</math>, is an analytically prescribed polynomial expression.  That is, the pressure is "evenly divisible" by the square of the density.  Let's examine how broadly reliable this behavior is.  [Note that, for simplicity in typing, hereafter throughout this subsection we will drop the subscript zero and, rather than <math>~\chi_0</math>, we will use <math>~z \equiv r/R</math> to denote the dimensionless radial coordinate.] 


Assume a mass-distribution given by the general quadratic function,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - a z - b z^2 \, ,</math>
  </td>
</tr>
</table>
</div>
where both coefficients, <math>~a</math> and <math>~b</math>, are positive.


<div align="center" id="Surface">
<table border="1" width="75%" align="center" cellpadding="5">
<tr>
<td align="center">ASIDE:  Surface Location</td>
</tr>
<tr><td align="left">
The surface of the configuration will be defined by the radial location, <math>~z_s</math>, at which the density first goes to zero.  If <math>~b = 0</math>, then the surface will be located at <math>~z_s = a^{-1}</math>; and if <math>~a = 0</math>, it will be located at <math>~z_s = b^{-1/2}</math>.  More generally, however, the roots of the quadratic equation that results from setting <math>~\rho/\rho_c</math> to zero are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{a}{2b}\biggl[1 \mp \biggl(1+\frac{4b}{a^2}\biggr)^{1/2}  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Because only the <math>~z_+</math> solution provides positive roots, we conclude that, when both <math>~a</math> and <math>~b</math> are nonzero, the radial coordinate of the surface is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_s = z_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{2b}\biggl[\biggl(1+\frac{4b}{a^2}\biggr)^{1/2}  - 1\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

We acknowledge, as well, that the density profile can now be written in terms of these roots; specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~b(z_+ - z)(z - z_-) \, .</math>
  </td>
</tr>
</table>
</div>

In the discussion, below, it may be advantageous to adopt the following notation:
<div align="center">
<math>~\ell^2 \equiv \frac{4b}{a^2} ~~~~~\Rightarrow ~~~~~ \ell = \frac{2b^{1/2}}{a} \, ,</math>
</div>
in which case,
<div align="center">
<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr]</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~b^{1/2}z_s = \frac{1}{\ell}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr] \, .</math>
</div>

</td></tr>
</table>
</div>


This specified density profile implies a mass distribution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R^3 \rho_c \int_0^z (1 - a z - b z^2)z^2 dz</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>

The hydrostatic balance condition therefore implies,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{R\rho_c} \biggl( \frac{\rho}{\rho_c} \biggr)^{-1} \frac{dP}{dz}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{G}{R^2 z^2} \biggl[ 4\pi R^3 \rho_c \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\biggl[ \frac{1}{4\pi G R^2 \rho_c^2 } \biggr] \frac{dP}{dz}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr)\biggl( \frac{\rho}{\rho_c} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
- \biggl( \frac{z}{3} - \frac{a z^2}{4} - \frac{b z^3}{5} \biggr) 
+ a \biggl( \frac{z^2}{3} - \frac{a z^3}{4} - \frac{b z^4}{5} \biggr) 
+ b \biggl( \frac{z^3}{3} - \frac{a z^4}{4} - \frac{b z^5}{5} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{a}{4} + \frac{a}{3}  \biggr) + z^3 \biggl( \frac{b}{5} - \frac{a^2}{4} + \frac{b}{3}  \biggr) 
+ z^4 \biggl( - \frac{ab}{5} - \frac{ab}{4}  \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{7a}{12} \biggr) + z^3 \biggl( \frac{8b}{15} - \frac{a^2}{4}\biggr) 
- z^4 \biggl( \frac{9ab}{20} \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{P}{P_n} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int \biggl[ -\biggl(\frac{1}{3}\biggr)z + z^2 \biggl( \frac{7a}{12} \biggr) + z^3 \biggl( \frac{8b}{15} - \frac{a^2}{4}\biggr) 
- z^4 \biggl( \frac{9ab}{20} \biggr) - z^5 \biggl(\frac{b^2}{5}   \biggr) \biggr] dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -\biggl(\frac{1}{6}\biggr)z^2 + z^3 \biggl( \frac{7a}{36} \biggr) + z^4 \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) 
- z^5 \biggl( \frac{9ab}{100} \biggr) - z^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] + C \, .
</math>
  </td>
</tr>
</table>
</div>

where, <math>~C</math>, is an integration constant, and,
<div align="center">
<math>~P_n \equiv 4\pi G \rho_c^2 R^2 \, .</math>
</div>
The integration constant &#8212; which also proves to be the normalized central pressure &#8212; is determined by ensuring that the pressure goes to zero at [[SSC/Structure/OtherAnalyticModels#Surface|the surface of the configuration]], <math>~z_s</math>.  That is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C = \frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -
\biggl[ -\biggl(\frac{1}{6}\biggr)z_s^2 + z_s^3 \biggl( \frac{7a}{36} \biggr) + z_s^4 \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) 
- z_s^5 \biggl( \frac{9ab}{100} \biggr) - z_s^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the pressure profile is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) - \biggl( \frac{2b}{15} - \frac{a^2}{16}\biggr) (z_s^4 - z^4)
+ \biggl( \frac{9ab}{100} \biggr)(z_s^5 - z^5) + \biggl(\frac{b^2}{30}   \biggr)(z_s^6 - z^6) \, .
</math>
  </td>
</tr>
</table>
</div>

Let's check this general expression against the specific cases described above.

====Example1====

First, let's set <math>~b=0</math>, but leave <math>~a</math> general.  As described in the [[SSC/Structure/OtherAnalyticModels#Surface|above ASIDE]], this means that <math>~z_s=a^{-1}</math>.  So the pressure distribution is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{7a}{36} \biggr)(z_s^3 - z^3) + \biggl( \frac{a^2}{16}\biggr) (z_s^4 - z^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(a^{-2} - z^2) - \biggl( \frac{7a}{36} \biggr)(a^{-3} - z^3) + \biggl( \frac{a^2}{16}\biggr) (a^{-4} - z^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^{-2} \biggl\{
\frac{1}{6}[1 - (az)^2] - \frac{7}{36} [1 - (az)^3] + \frac{1}{16} [1 - (az)^4]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And from the expression for the integration constant, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
\biggl(\frac{1}{6}\biggr)a^{-2}  - a^{-3} \biggl( \frac{7a}{36} \biggr) + a^{-4} \biggl( \frac{a^2}{16}\biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^{-2} \biggl[\frac{1}{6} - \frac{7}{36}  + \frac{1}{16} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{2^4\cdot 3^2 a^2} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, dividing one expression by the other, we obtain,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}\biggl\{ 24[1 - (az)^2] - 28 [1 - (az)^3] + 9 [1 - (az)^4] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Let's check to see if this "general linear" pressure distribution is evenly divisible by the square of the density distribution which, in this case, is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - a z  \, .</math>
  </td>
</tr>
</table>
</div>
Strategically rewriting the expression for the pressure distribution gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}\biggl\{ 24[1 - (az)][1 + (az)] - 28 [1 - (az)][1 + (az) + (az)^2] + 9 [1 - (az)][1 + (az)][1 + (az)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{ 24[1 + (az)] - 28 [1 + (az) + (az)^2] + 9 [1 + (az)][1 + (az)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 24(az) - 28 [(az) + (az)^2] + 9 [(az) + (az)^2 + (az)^3] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl\{5+ 5(az) - 19 (az)^2 + 9 (az)^3 \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And, as luck would have it, the expression inside the curly braces can be "divided evenly" by the quantity, <math>~[1-(az)]</math>, one more time.  Specifically, the expression becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [5+ 10(az) - 9 (az)^2 ] \, .
</math>
  </td>
</tr>
</table>
</div>


====Example2====

First, let's set <math>~a=0</math>, but leave <math>~b</math> general.  As described in the [[SSC/Structure/OtherAnalyticModels#Surface|above ASIDE]], this means that <math>~z_s=b^{-1/2}</math>.  So the pressure distribution is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(z_s^2 - z^2) - \biggl( \frac{2b}{15} \biggr) (z_s^4 - z^4) + \biggl(\frac{b^2}{30}   \biggr)(z_s^6 - z^6) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{6}\biggr)(b^{-1} - z^2) - \biggl( \frac{2b}{15} \biggr) (b^{-2} - z^4) + \biggl(\frac{b^2}{30}   \biggr)(b^{-3} - z^6) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{b}\biggl\{
\biggl(\frac{1}{6}\biggr)[1 - bz^2] - \biggl( \frac{2}{15} \biggr) [1 - (bz^2)^2] + \biggl(\frac{1}{30}   \biggr)[1 - (bz^2)^3] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

And from the expression for the integration constant, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \biggl(\frac{1}{6}\biggr)z_s^2 - z_s^4 \biggl( \frac{2b}{15}\biggr) + z_s^6 \biggl(\frac{b^2}{30}   \biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{b}
\biggl[ \frac{1}{6} -  \frac{2}{15} + \frac{1}{30} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{3 \cdot 5b} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, dividing one expression by the other, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl\{
5 [1 - bz^2] - 4  [1 - (bz^2)^2] +  [1 - (bz^2)^3] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Let's check to see if this "general parabolic" pressure distribution is evenly divisible by the square of the density distribution which, in this case, is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - b z^2  \, .</math>
  </td>
</tr>
</table>
</div>
Strategically rewriting the expression for the pressure distribution gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl\{ 5[1 - (bz^2)] - 4[1 - (bz^2)][1 + (bz^2)] + [1 - (bz^2)][1 + (bz^2) + (bz^2)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl\{ 5 - 4 [1 + (bz^2)] + [1 + (bz^2) + (bz^2)^2] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl[ 2 - 3 (bz^2) + (bz^2)^2 \biggr] 
</math>
  </td>
</tr>
</table>
</div>

Again, as luck would have it, the expression inside the square brackets can be "divided evenly" by the quantity, <math>~[1-(bz^2)]</math>, one more time.  Specifically, the expression becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl(\frac{\rho}{\rho_c}\biggr)^2 [2-(bz^2) ] \, .
</math>
  </td>
</tr>
</table>
</div>

====Example3====
In the most general quadratic case, we should rewrite the pressure distribution as,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P}{P_n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2^4\cdot 3^2\cdot 5^2}\biggl\{
2^3\cdot 3\cdot 5^2 (z_s^2 - z^2) - 2^2\cdot 5^2 \cdot 7 a(z_s^3 - z^3) - [2^5\cdot 3\cdot 5 b - 3^2\cdot 5^2 a^2] (z_s^4 - z^4)
+ 2^2\cdot 3^4 ab (z_s^5 - z^5) + 2^3\cdot 3\cdot 5b^2 (z_s^6 - z^6) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{z_s^2}{2^4\cdot 3^2\cdot 5^2}\biggl\{
600 (1 - \zeta^2) - 700 (a z_s) (1 - \zeta^3) - [480 (b z_s^2) - 225 (a z_s)^2] (1 - \zeta^4)
+ 324 (az_s)(b z_s^2) (1 - \zeta^5) + 120(b z_s^2)^2 (1 - \zeta^6) 
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>

where, <math>~\zeta \equiv z/z_s</math>.  Similarly, let's rewrite the integration constant as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c}{P_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{z_s^2}{2^4\cdot 3^2\cdot 5^2}\biggl\{
600 - 700 (a z_s)  - [480 (b z_s^2) - 225 (a z_s)^2] 
+ 324 (az_s)(b z_s^2)  + 120(b z_s^2)^2  
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

So the pressure can be written as,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ \frac{2^4\cdot 3^2\cdot 5^2}{z_s^2} \biggr] \biggl[ \frac{P}{P_n} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~ \biggl[ \frac{2^4\cdot 3^2\cdot 5^2}{z_s^2} \biggr] \biggl[ \frac{P_c}{P_n}\biggr]
- 600 \zeta^2 + 700 (a z_s) \zeta^3 + [480 (b z_s^2) - 225 (a z_s)^2] \zeta^4
- 324 (az_s)(b z_s^2) \zeta^5 - 120(b z_s^2)^2 \zeta^6  \, .
</math>
  </td>
</tr>
</table>
</div>

The question that remains to be answered is:  Is this expression for the pressure distribution "evenly divisible" by the square (or even the ''first'' power) of the normalized density distribution which, [[SSC/Structure/OtherAnalyticModels#Derivation|as defined above]] for the general quadratic case, is,
<div align="center">
<math>\frac{\rho}{\rho_c} = 1 - az - bz^2 = 1 - (az_s)\zeta - (bz_s^2)\zeta^2 \, .</math>
</div>

In attempting to answer this question, it may prove advantageous to refer back to the [[SSC/Structure/OtherAnalyticModels#Surface|above ASIDE discussion]] of the roots of this quadratic function and, in particular, that,
<div align="center">
<math>~az_s = \frac{2}{\ell^2}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr]</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~b^{1/2}z_s = \frac{1}{\ell}\biggl[\biggl(1+\ell^2\biggr)^{1/2}  - 1\biggr] \, ,</math>
</div>
where, <math>~\ell^2 \equiv 4b/a^2</math>.


<!--  HIDE TABLE 1C  
<table border="1" cellpadding="5" align="center" width="90%">
<tr>
  <th align="center" colspan="5"><font size="+1">Table 1c:  Properties of Analytically Defined Equilibrium Structures</font></th>
</tr>
<tr>
  <td align="center" colspan="5">
<math>~\frac{\rho}{\rho_c} = 1 - a\chi_0 - b\chi_0^2</math>
  </td>
</tr>
<tr>
  <td align="center" width="10%">Model</td>
  <td align="center"><math>~a</math></td>
  <td align="center"><math>~b</math></td>
  <td align="center"><math>~C = \frac{P_c}{P_n}</math></td>
  <td align="center"><math>~\frac{P_0}{P_n}</math></td>
</tr>
<tr>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{1}{6}</math>
  </td>
  <td align="center">
<math>C - \frac{1}{6}\chi_0^2 </math> 
  </td>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{1}{6}\biggl[1 - \biggl( \frac{7}{6} \biggr) + \biggl( \frac{15}{40} \biggr) \biggr] = \frac{5}{2^4\cdot 3^2}</math></td>
  <td align="center">
<math>~
C + \biggl[ -\biggl(\frac{1}{6}\biggr)\chi_0^2 + \chi_0^3 \biggl( \frac{7}{36} \biggr) - \chi_0^4 \biggl( \frac{1}{16}\biggr) \biggr]
= C + \frac{1}{2^4\cdot 3^2}\biggl[ -24\chi_0^2 +  28\chi_0^3  - 9 \chi_0^4\biggr]
</math> 
  </td>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~\frac{1}{6}\biggl[1 - \frac{32}{40}  + \frac{1}{5}  \biggr] = \frac{1}{15}</math></td>
  <td align="center">
<math>~
C + \biggl[ -\biggl(\frac{1}{6}\biggr)\chi_0^2 + \chi_0^4 \biggl( \frac{8}{60} \biggr) - \chi_0^6 \biggl(\frac{1}{30}   \biggr) \biggr]
= C + \frac{1}{30}\biggl[ -5\chi_0^2 + 4 \chi_0^4  - \chi_0^6 \biggr]
</math> 
  </td>
</tr>
<tr>
  <td align="center">General Linear</td>
  <td align="center"><math>~a</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~\frac{1}{2^4\cdot 3^2}\biggl[
24 - 28a  + 9a^2  \biggr]
</math>
  </td>
  <td align="center">
<math>~
C + \frac{1}{2^4\cdot 3^2}\biggl[ -24 \chi_0^2 + 28a \chi_0^3 -9a^2 \chi_0^4  \biggr] 
</math> 
  </td>
</tr>
</table>
END HIDING of TABLE 1C -->