Editing ParabolicDensity/Axisymmetric/Structure (section)

====Isobaric Surfaces====

By design, the mass within our oblate-spheroidal configuration is distributed in such a way that iso-density surfaces are concentric spheroids.  As stated earlier, the relevant mathematically prescribed density distribution is,

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

<tr>
  <td align="right">
<math>\frac{\rho(\chi, \zeta)}{\rho_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] 
\, .</math>
  </td>
</tr>
</table>

In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids.  (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure|barotropic]] &#8212; but not necessarily a polytropic &#8212; equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)  In an effort to make this determination for our <math>e = 0.6</math> spheroid, we first examine the iso-density surface for which <math>\rho/\rho_c = 0.3</math>.  Via the expression,

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

<tr>
  <td align="right">
<math>\zeta^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1-e^2)\biggl[1 - \chi^2 - \frac{\rho}{\rho_c} \biggr] 
=
0.64 \biggl[1 - \chi^2 - 0.3 \biggr]
\, ,</math>
  </td>
</tr>
</table>

we can immediately determine that our three chosen radial cuts <math>(\chi = 0.0, 0.6, 0.75)</math> intersect this iso-density surface at the vertical locations, respectively, <math>\zeta = 0.66933, 0.46648, 0.29665</math>; these numerical values have been recorded in the following table.  The table also contains coordinates for the points where our three cuts intersect the <math>(e = 0.6)</math> iso-density surface for which <math>\rho/\rho_c = 0.6</math>.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" rowspan="2">diamond<br />marker<br />color</td>
  <td align="center" rowspan="2">chosen<br /><math>\rho/\rho_c</math></td>
  <td align="center" rowspan="2">chosen<br /><math>\chi</math></td>
  <td align="center" colspan="2">resulting &hellip;</td>
</tr>
<tr>
  <td align="center">&nbsp; &nbsp; <math>\zeta</math> &nbsp; &nbsp;</td>
  <td align="center">normalized<br />pressure</td>
</tr>
<tr>
  <td align="center" rowspan="3"><font color="darkgreen">green</font></td>
  <td align="center" rowspan="3"><math>0.3</math></td>
  <td align="center" rowspan="1"><math>0.00</math></td>
  <td align="center" rowspan="1"><math>0.66933</math></td>
  <td align="center" rowspan="1"><math>0.060466</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.60</math></td>
  <td align="center" rowspan="1"><math>0.46648</math></td>
  <td align="center" rowspan="1"><math>0.057433</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.75</math></td>
  <td align="center" rowspan="1"><math>0.29665</math></td>
  <td align="center" rowspan="1"><math>0.055727</math></td>
</tr>
<tr>
  <td align="center" rowspan="3"><font color="purple">purple</font></td>
  <td align="center" rowspan="3"><math>0.6</math></td>
  <td align="center" rowspan="1"><math>0.00</math></td>
  <td align="center" rowspan="1"><math>0.50596</math></td>
  <td align="center" rowspan="1"><math>0.292493</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.60</math></td>
  <td align="center" rowspan="1"><math>0.16000</math></td>
  <td align="center" rowspan="1"><math>0.280361</math></td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>0.75</math></td>
  <td align="center" rowspan="1">n/a</td>
  <td align="center" rowspan="1">n/a</td>
</tr>
</table>
For each of these five <math>(\chi,\zeta)</math> coordinate pairs, we have used our above derived expression for <math>P^*_\mathrm{deduced}/P^*_c</math> to calculate the "normalized pressure" at the relevant point inside the configuration.  These results appear in the last column of the table; they also have been marked in the accompanying figure: dark green diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.3</math> and purple diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.6</math>. Notice that the normalized density is everywhere lower than <math>0.6</math> along the <math>\chi = 0.75</math> cut, so the final row in the table has been marked "n/a" (not applicable).

The dark green diamond-shaped markers in the figure  &#8212; along with the associated tabular data &#8212; show that at three separate points along the <math>\rho/\rho_c = 0.3</math> iso-density surface, the normalized pressure is ''nearly'' &#8212; but not exactly &#8212; the same; its value is approximately <math>0.057</math>.  Similarly, the purple diamond-shaped markers show that at two separate points along the <math>\rho/\rho_c = 0.6</math> iso-density surface, the normalized pressure is nearly the same; in this case its value is approximately <math>0.28</math>.  This seems to indicate that, throughout our configuration, the isobaric surfaces are almost &#8212; but not exactly &#8212; aligned with iso-density surfaces.