Editing Apps/OstrikerBodenheimerLyndenBell66 (section)

===Approach Outlined by OBL66===

* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer &amp; D. Lynden-Bell (1966; hereinafter OBL66)], Phys. Rev. Letters, 17, 816:  ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">&hellip; work by Roxburgh (1965, Z. Astrophys., 62, 134), Anand (1965, Proc. Natl. Acad. Sci. U.S., 54, 23), and James (1964, ApJ, 140, 552) shows that the</font> [Chandrasekhar (1931, ApJ, 74, 81)] <font color="green">mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, &hellip;</font>

<font color="green">In this Letter we demonstrate that white-dwarf models with masses considerably greater than  <math>~M_3</math> are possible if differential rotation is allowed &hellip; models are based on the physical assumption of an axially symmetric, completely degenerate, self-gravitating fluid, in which the effects of viscosity, magnetic fields, meridional circulation, and relativistic terms in the hydrodynamical equations have been neglected.</font>
</td></tr></table>

====Their Equation (4)====

One can immediately appreciate that, independent of the chosen coordinate base, the first expression listed among our trio of governing PDEs derives from the ''differential representation'' of the Poisson equation as [[AxisymmetricConfigurations/PoissonEq#Overview|discussed elsewhere]] and as has been reprinted here as Table 2.

<div align="center">
<table border="1" cellpadding="8" align="center" width="70%">
<tr><th align="center" colspan="2"><font size="+0">Table 2: &nbsp;Poisson Equation</font></th></tr>
<tr>
  <th align="center">Integral Representation</th>
  <th align="center">Differential Representation </th>
</tr>
<tr>
  <td align="center">
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">
{{Math/EQ_Poisson01}}
  </td>
</tr>
</table>
</div>


[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] chose, instead, to use the ''integral representation'' of the Poisson equation to evaluate the gravitational potential; specifically, they write,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_g(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (4)
  </td>
</tr>
</table>
(Note that, in defining <math>~\Phi_g</math>, [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] have adopted a sign convention for the gravitational potential that is the opposite of ours; that is, <math>~\Phi_g = - \Phi</math>.)  

====Their Equations (3) &amp; (5)====

The two relevant components of the Euler equation that are identified, above, result from imposing a ''steady-state'' condition on the,

<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\cancel{\frac{\partial \vec{v}}{\partial t} } + (\vec{v} \cdot \nabla)\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho} \nabla P - \nabla \Phi \, ,
</math>
  </td>
</tr>
</table>
</div>
<!-- {{Math/EQ_Euler02}} -->
and adopting a steady-state rotational velocity field in which the angular velocity is either constant or is only a function of the cylindrical-coordinate radius, <math>~\varpi</math>; that is,

<div align="center">
<math>~\vec{v} = \hat{e}_\varphi [v_\varphi]  = \hat{e}_\varphi [\varpi \dot\varphi (\varpi)] \, .</math>
</div>


As we have demonstrated in [[AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Solution_Strategies.29|an accompanying discussion]], for any of a number of astrophysically relevant [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple rotation profiles'']] of this form, the [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|convective operator]] on the left-hand side of this steady-state Euler equation gives (most conveniently written here in a cylindrical-coordinate base),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla)\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\hat{e}_\varpi \biggl[\frac{v_\varphi^2}{\varpi} \biggr] = -~\hat{e}_\varpi \biggl[ \varpi {\dot\varphi}^2(\varpi) \biggr] = -~\hat{e}_\varpi \biggl[\frac{j^2(\varpi)}{\varpi^3} \biggr] \, ,</math>
  </td>
</tr>
</table>
where, <math>~j \equiv \varpi^2 \dot\varphi</math> is the (radially dependent) specific angular momentum measured relative to the symmetry (rotation) axis.  As we have pointed out in an [[AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Solution_Strategies.29|accompanying discussion]], this last expression can be rewritten in terms of the gradient of a scalar (centrifugal) potential; specifically, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\nabla \Psi \, ,</math>
  </td>
</tr>
</table>
if the centrifugal potential is defined such that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \int_0^\varpi \frac{j^2(\varpi^')}{(\varpi^')^3} d\varpi^' \, .</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (5)
  </td>
</tr>
</table>
(Note that [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] adopted a sign convention for the centrifugal potential that is the opposite of ours; that is, <math>~\Phi_c = - \Psi</math>.)  Hence, assuming that our intent is to construct a rotationally flattened equilibrium configuration whose rotation profile is of the form, <math>~\vec{v} = \hat{e}_\varphi [\varpi \dot\varphi (\varpi)] </math>, the ''steady-state'' Euler equation can be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla \Phi - \nabla \Psi \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (3)
  </td>
</tr>
</table>

====Their Adopted Angular-Momentum Distribution====

In what follows, text that has been extracted directly from p. 817 of [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] is presented using a dark green font.  

"<font color="darkgreen">The angular-velocity distribution in the model is determined through the specification of a distribution of angular momentum per unit mass <math>~j(m)</math>, where <math>~m</math> is a Lagrangian coordinate equal to the fraction of the total mass interior to a cylindrical surface around the axis of rotation.  The specification of <math>~j(m)</math> rather than <math>~\dot\varphi(\varpi)</math> permits the construction of equilibrium models for a given choice of [total] angular momentum <math>~J</math>.  The angular-momentum distribution chosen for the computed models is that of a uniformly rotating polytrope of index <math>~\tfrac{3}{2}</math>.</font>"  

Later papers refer to models with OBL66's specified angular momentum profile as belonging to an <math>~n^' = \tfrac{3}{2}</math> sequence.  It cannot be described by a closed-form analytic expression.  But, as a point of reference and drawing from [http://adsabs.harvard.edu/abs/1965ApJ...142..208S Stoeckly's (1965)] work, in an [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_.28n.27_.3D_0.29|accompanying discussion]] we derive the analytic expression for the angular momentum distribution of models that lie along a so-called  <math>~n^' = 0</math> sequence.

====Their Adopted Barotropic Equation of State====

Because they were interested in constructing equilibrium models of rotationally flattened white dwarfs, [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] chose a barotropic equation of state that describes a ''zero-temperature Fermi (degenerate electron) gas''.  As has been documented in our [[SR#Barotropic_Structure|accompanying discussion of barotropic equations of state]], the set of key relations that define this equation of state is,

<div align="center">
{{Math/EQ_ZTFG01}}
Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C S. Chandraskehar (1935)]<p></p>
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (2)
</div>


As we also have [[SSC/Structure/WhiteDwarfs#ChandrasekharMass|reviewed elsewhere]], the ''Chandrasekhar limiting mass'' that is associated with this equation of state in ''nonrotating'' stars is given by the expression,
<div align="center">
<math>
\frac{M_\mathrm{Ch}}{M_\odot} = \frac{5.742}{\mu_e^2} .
</math><p></p>
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 816, Eq. (1)
</div>