Editing ThreeDimensionalConfigurations/Challenges (section)

====Strategy====

<font color="red">'''STEP 0:'''</font>&nbsp; Choose the pair of model-sequence parameters, <math>~(C_0, C_1)</math>, that are associated with the function, <math>~F_B(\Psi)</math>.  Hold these fixed during iterations.

<font color="red">'''STEP 1:'''</font>&nbsp; Guess a density distribution, <math>~\rho(x,y)</math>.  For example, pick the equatorial-plane (uniform) density distribution of a Riemann S-type ellipsoid with an equatorial-plane axis-ratio, <math>~b/a</math> and meridional-plane axis-ratio, <math>~c/a</math>; use the same <math>~b/a</math> ratio to define two points on the configuration's surface throughout the iteration cycle.

<font color="red">'''STEP 2:'''</font>&nbsp; Given <math>~\rho(x,y)</math>, solve the Poisson equation to obtain, <math>~\Phi_\mathrm{grav}(x,y)</math>.  In the first iteration, this should exactly match the <math>~A_1, A_2, A_3</math> values associated with the chosen Riemann S-type ellipsoid.

<font color="red">'''STEP 3:'''</font>&nbsp; Guess a value of <math>~\Omega_f</math> &#8212; perhaps the spin-frequency associated with your "initial guess" Riemann ellipsoid &#8212; then solve the following two-dimensional, elliptic PDE to obtain <math>~\Psi(x,y)</math> &hellip;
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  <td align="right">
<math>~ 
2\Omega_f - C_0 \rho
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~~C_1 \rho \Psi + \frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x}  \biggr] + \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y}  \biggr] \, .</math>
  </td>
</tr>
</table>
<table border="1" cellpadding="10" width="70%" align="center"><tr><td align="left">
<div align="center">'''Boundary Condition'''</div>

Moving along various rays from the center of the configuration, outward, the surface is determined by the location along each ray where <math>~H(x,y)</math> goes to zero for the first time.  We set <math>~\Psi = 0</math> at these various surface locations.  At each of these locations, the velocity vector must be tangent to the surface.  This requirement also, then, sets the value of <math>~\partial \Psi/\partial y</math> and <math>~\partial \Psi/\partial x</math> at each location.
</td></tr></table>

<font color="red">'''STEP 4:'''</font>&nbsp; Determine (rotating-frame) velocity from knowledge of <math>~\Psi(x,y)</math> and <math>~\rho(x,y)</math>.
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  <td align="right">
<math>~\bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho}\biggl\{ \boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[  \frac{\partial \Psi}{\partial x}\biggr] \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ u^2 = \bold{u} \cdot \bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho^2}\biggl\{ \biggl[ \frac{\partial \Psi}{\partial y} \biggr]^2
+ \biggl[  \frac{\partial \Psi}{\partial x}\biggr]^2 \biggr\} \, .</math>
  </td>
</tr>
</table>

<font color="red">'''STEP 5:'''</font>&nbsp; Using the "scalar Euler equation,"

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<tr>
  <td align="right">
<math>~
H + \Phi_\mathrm{grav} + C_0 \Psi + \frac{1}{2} C_1 \Psi^2 + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2)  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>C_B \, , </math>
  </td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], &sect;4.3, p. 85, Eq. (4.23)</td></tr>
</table>
<ul>
<li>Set <math>~H = 0</math> at two different points on the surface of the configuration &#8212; usually at <math>~(x,y) = (a,0)</math> and <math>~(x,y) = (0,b)</math> &#8212; to determine values of the two constants, <math>~\Omega_f^2</math> and <math>~C_B</math>.</li>
<li>At all points inside the configuration, determine <math>~H(x,y)</math>.</li>
</ul>
<font color="red">'''STEP 6:'''</font>&nbsp; Use the barotropic equation of state to determine the "new" mass-density distribution from the knowledge of the enthalpy, <math>~H(x,y)</math>.