Editing ParabolicDensity/Spheres/Structure (section)

==Spherically Symmetric Equilibrium Structure==
In an article titled, "Radial Oscillations of a Stellar Model," {{ Prasad49full }} investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,
<div align="center">
<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>
</div>
where, <math>\rho_c</math> is the central density and <math>R</math> is the radius of the star.  

===Radial Profiles===
In a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|related discussion]] we have derived the following expressions that describe analytically various structural properties of this equilibrium configuration.

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

<tr>
  <td align="right">
<math>M_r(r)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi\rho_c r^3}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2 \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~g_0(r) \equiv \frac{G M_r(r) }{r^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi G \rho_c r}{3}  
\biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2\biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Phi_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{G M_\mathrm{tot}}{8R} \biggl\{- 15 + 10 \biggl(\frac{r}{R}\biggr)^2- 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P(r)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi G\rho_c^2 R^2}{15} 
\biggl[1-\biggl(\frac{r}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr] 
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>H(r)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{GM_\mathrm{tot}}{8 R} \biggl[7 - 10\biggl(\frac{r}{R}\biggr)^2 + 3\biggl(\frac{r}{R}\biggr)^4\biggr] 
\, .
</math>
  </td>
</tr>
</table>

Note that the total mass is obtained by setting <math>r = R</math> in the expression for <math>M_r(r )</math>, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi\rho_c R^3}{3} \biggl[\frac{2}{5}\biggr] 
=
\frac{8\pi\rho_c R^3}{15} 
</math>
&nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; 
<math>
2\pi \rho_c = \frac{15 M_\mathrm{tot}}{4R^3}
\, .
</math>
  </td>
</tr>
</table>

===Effective Barotropic Relations===

By replacing <math>r/R</math> with <math>\rho/\rho_c</math>, we obtain analytic expression for, respectively, the pressure-density and enthalpy-density (effective [[SR#Barotropic_Structure|barotropic]]) relations that are relevant in this ''parabolic'' configuration.  Specifically,

<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>\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl(\frac{r}{R}\biggr)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ 1 - \biggl(\frac{\rho}{\rho_c} \biggr) \biggr]</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>P_c</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{4\pi G\rho_c^2 R^2}{15} \, .</math>
  </td>
</tr>
</table>
And,

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

<tr>
  <td align="right"><math>\frac{H(\rho)}{H_\mathrm{norm}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
7 - 10\biggl[ 1 - \biggl(\frac{\rho}{\rho_c} \biggr) \biggr] + 3\biggl[ 1 - \biggl(\frac{\rho}{\rho_c} \biggr) \biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
4\biggl(\frac{\rho}{\rho_c} \biggr) + 3\biggl(\frac{\rho}{\rho_c} \biggr)^2
\, , 
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>H_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{GM_\mathrm{tot}}{8 R}  \, .</math>
  </td>
</tr>
</table>