Editing Apps/OstrikerBodenheimerLyndenBell66 (section)

===Further Implementation by OB68===

If we define an effective potential,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{eff}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\Phi + \Psi \, ,</math>
  </td>
</tr>
</table>
and recall that, for a [[SR#Barotropic_Structure|barotropic equation of state]], we can make the substitution, <math>~\nabla P \rightarrow \rho\nabla H</math>, where <math>~H</math> is the fluid enthalpy, then OBL66's hydrostatic balance equation &#8212; their equation (3) &#8212; can be rewritten as,
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<tr>
  <td align="right">
<math>~\nabla [H + \Phi_\mathrm{eff} ]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
<td align="center">&nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp;</td>
  <td align="right">
<math>~H + \Phi_\mathrm{eff}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_B \, , </math>
  </td>
</tr>
</table>
where, <math>~C_B</math> is a constant (i.e., independent of position). Rotationally flattened, steady-state (equilibrium) configurations can be constructed by finding spatial density distributions that simultaneously satisfy the Poisson equation and this deceptively simple algebraic relation.  [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract Ostriker &amp; Bodenheimer (1968; hereinafter, OB68)] used this "self-consistent field" technique to obtain models of rotationally flattened white dwarfs; it is a technique of choice that we [[AxisymmetricConfigurations/SolutionStrategies#Technique|broadly promote]] as well.


In the specific case of a zero-temperature Fermi (degenerate electron) gas &#8212; see our [[SR#Barotropic_Structure|related discussion of barotropic structures]] &#8212; to within an additive constant, the enthalpy associated with <math>~P_\mathrm{deg}</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (1 + \chi^2 )^{1 / 2} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl\{ \biggl[1 + \biggl(\frac{\rho}{B_\mathrm{F}}\biggr)^{2/3} \biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="6">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (4)</td>
</tr>
</table>
Note that, using this expression, the enthalpy at the surface <math>~(\rho = 0)</math> is <math>~H_s = 8A_\mathrm{F}/B_\mathrm{F}</math>.   ([[SR#Barotropic_Structure|Our tabulated expression for the enthalpy]] has been shifted by this constant value so that the enthalpy naturally goes to zero at the surface.)  If we use <math>~\Phi_\mathrm{eff,s}</math> to denote the surface value of the effective potential, the constant in the algebraic hydrostatic-balance expression must be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~H_s + \Phi_\mathrm{eff,s}</math>
  </td>
</tr>
</table>
Then, at every other spatial location, <math>~\vec{x}</math>, we must have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H(\vec{x}) - H_s + \Phi_\mathrm{eff}(\vec{x}) - \Phi_\mathrm{eff,s}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{8A_\mathrm{F}}{B_\mathrm{F}} \Biggl[ \biggl\{ \biggl[1 + \biggl(\frac{\rho(\vec{x})}{B_\mathrm{F}}\biggr)^{2/3} \biggr]^{1 / 2} \biggr\}  - 1 \Biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}</math>
  </td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\rho(\vec{x})}{B_\mathrm{F}}    </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Biggl[ \biggl\{ \frac{B_\mathrm{F}}{8A_\mathrm{F}} \biggl[- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}\biggr] + 1 \biggr\}^2 - 1 \Biggr]^{3/2} \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (5)</td>
</tr>
</table>
(Note that, as with OBL66, a different sign convention was adopted by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68] for the effective potential than we have used; that is, <math>~\mathfrak{B} = - \Phi_\mathrm{eff}</math>.)