Editing SSC/Structure/BiPolytropes/AnalyzeStepFunction (section)

===Establish Fidelity of Finite-Difference Model===
We know the analytic structure of the equilibrium configuration.  Let's choose a Lagrangian grid that is labeled by <math>(r_0)_j</math> and the corresponding enclosed mass, <math>m_j(r_0)</math>, where the center of the spherical bipolytrope is labeled by <math>j=0</math> while each subsequent grid "line" is labeled <math>j</math>.  We will identify the mean density of each mass shell by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>{\bar\rho}_{j-1/2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{\Delta m}{\mathrm{Volume}}\biggr]_{j-1/2}
=
\biggl[m_j - m_{j-1}\biggr]\biggl[\frac{4\pi}{3} (r_j^3 - r_{j-1}^3)  \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
The pressure can be determined from knowledge of the density via knowledge of the (fixed) specific entropy, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P_{j-1/2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
({\bar\rho}_{j-1/2})^{\gamma} \cdot \exp\biggl[ \frac{\mu (\gamma-1)s}{\mathfrak{R}} \biggr]
\, .
</math>
  </td>
</tr>
</table>
These two expressions, effectively, originate from the continuity equation and the adiabatic form of the first law of thermodynamics, respectively.  They are relations that allow the determination of the mass density and the pressure, given fixed mass shells but varying mass-shell radial locations.

<font color="red"><b>STEP 1:</b></font> &nbsp; From the analytically known equilibrium structure of the <math>(n_c, n_e) = (5, 1)</math> bipolytrope, create a table that documents how the radial location of each mass shell, <math>r_j</math>, varies with the enclosed mass, <math>m_j</math>.

<font color="red"><b>STEP 2:</b></font> &nbsp; Determine how <math>{\bar\rho}_{j-1/2}</math> varies with radial shell location, using the above continuity-equation relation.  (Plot <math>{\bar\rho}</math> versus <math>m</math> obtained in this discrete manner &#8212; see the red-dotted curve in the following figure &#8212; then also plot how <math>\rho</math> varies with <math>m</math> according to the analytic equilibrium structure &#8212; see the dark-blue-dotted curve; the plotted curves should be nearly, but not exactly, the same.) 

<font color="red"><b>STEP 3A:</b></font> &nbsp; Given <math>{\bar\rho}_{j-1/2}</math>, determine how <math>{P}_{j-1/2}</math> varies with radial shell location, using the above 1<sup>st</sup> Law relation.  (Plot the pressure, as determined in this discrete manner, versus <math>m</math> &#8212; see the light-green-dotted curve in the following figure &#8212; then also plot how <math>P</math> varies with <math>m</math> according to the analytic equilibrium structure &#8212; see the purple-dotted curve; the plotted curves should be nearly, but not exactly, the same.)

<font color="red"><b>STEP 3B:</b></font> &nbsp; Determine (and plot) how <math>(dP/dm)_{j}</math> varies with <math>m</math>.  

Now, what can we learn from the "Euler + Poisson Equation"?  Well, for the equilibrium state, we should find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP}{dm}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{Gm}{4\pi r^4}
\, .
</math>
  </td>
</tr>
</table>

<font color="red"><b>STEP 4:</b></font> &nbsp; Show how <math>dP/dm</math> varies with <math>m</math>, according to this analytic prescription, and compare it against the pressure gradient behavior obtained in STEP 3.  Do they match?

<table border="1" width="80%" align="center" cellpadding="5">
<tr>
  <td align="center">[[File:FiniteDifferenceLAWE01.png|600px|Finite-Difference Representation of n = 5 Core]]</td>
</tr>
<tr>
  <td align="left">
On a semi-log plot &hellip;
* As prescribed in <font color="red">STEP 2</font>: The red-dotted curve shows how the ''discrete'' density, <math>{\bar\rho}_{j-1/2}</math>, varies with enclosed mass, and the dark-blue-dotted curve shows how the analytically determined density varies with enclosed mass. 
* As prescribed in <font color="red">STEP 3A</font>: The light-green-dotted curve shows how the ''discrete'' pressure, <math>{P}_{j-1/2}</math>, varies with enclosed mass, and the purple-dotted curve shows how the analytically determined pressure varies with enclosed mass. 
* As prescribed in <font color="red">STEP 3B</font>: The light-blue-dotted curve shows how the ''discrete'' pressure gradient, <math>(dP/dm)_j</math>, varies with enclosed mass, and the orange-dotted curve shows how the analytically determined quantity, <math>-Gm/(4\pi r^4)</math>, varies with enclosed mass. Note that, in each case, we have added "1" to the logarithm of the specified quantity in order to shift both curves up in the plot and thereby unclutter the diagram.
  </td>
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</table>