Editing Apps/ReviewStahler83 (section)

==Governing Equations==

[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] states that the <font color="darkgreen">equilibrium configuration is found by solving the equation for momentum balance together with Poisson's equation for the gravitational potential</font>, <math>~\Phi_g</math>.  Stahler  chooses to use the ''integral'' form of Poisson's equation to define the gravitational potential, namely (see his equation 10, but note the sign change and "pink comment" shown here on the right),
<div align="center" id="GravitationalPotential">
[[File:CommentButton02.png|right|100px|Note from J. E. Tohline:  As is written here and in eq. (A1) of Stahler's (1983a) ''Appendix A'', a negative sign explicitly appears on the right-hand-side of the integral form of Poisson's equation.  In contrast to this, the right-hand side of Stahler's eq. (10) is explicitly positive.  We suspect that the absence of negative sign in Stahler's eq. (10) is a typographical error.]]
<table border="0" cellpadding="5" 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>
</table>
</div>
As is clear from our [[PGE/PoissonOrigin#Step_1|separate discussion of the origin of Poisson's equation]], this matches the expression for the scalar gravitational potential that is widely used in astrophysics.  

Working in cylindrical coordinates <math>~(\varpi, z)</math> &#8212; as we have [[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|explained elsewhere]], the assumption of axisymmetry eliminates the azimuthal angle &#8212; Stahler states that <font color="darkgreen">the momentum equation is</font> (see his equation 2):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\nabla P}{\rho} + \nabla\Phi_g + \nabla\Phi_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~\nabla \equiv (\partial/\partial\varpi, \partial/\partial z)</math>, and the centrifugal potential is given by (see Stahler's equation 3, but note the sign change and "pink comment" shown here on the right):
<div align="center">
[[File:CommentButton02.png|right|100px|Note from J. E. Tohline:  As is written here and in the definition of &Psi; that accompanies our separate discussion of ''simple rotation profiles'', a negative sign explicitly appears in the integral expression for the centrifugal potential.  In contrast to this, the right-hand side of Stahler's eq. (3) is explicitly positive.  We suspect that the absence of a negative sign in Stahler's eq. (3) is a typographical error.]]
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_c(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-
\int_0^\varpi \frac{j^2(\varpi^') d\varpi^'}{(\varpi^')^3} \, ,
</math>
  </td>
</tr>
</table>
</div>
where <math>j</math> is the z-component of the angular momentum per unit mass.  This last expression is precisely the same expression for the centrifugal potential that we have defined in the context of our discussion of  [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple rotation profiles'']].  As Stahler stresses, by adopting a centrifugal potential of this form, he is implicitly assuming <font color="darkgreen">that <math>j</math> is not a function of <math>z</math></font>; this builds in the physical constraint enunciated by the [[2DStructure/AxisymmetricInstabilities#Poincar.C3.A9-Wavre_Theorem|Poincar&eacute;-Wavre theorem]], <font color="darkgreen">which guarantees that rotational velocity is constant on cylinders for the equilibrium of any barotropic fluid</font>.

As we have demonstrated in our [[AxisymmetricConfigurations/SolutionStrategies#2DgoverningEquations|overview discussion of axisymmetric configurations]], the equations that govern the equilibrium properties of axisymmetric structures are,

<div align="center">
<table border="1" cellpadding="8" align="center"><tr><td align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho \, .</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>

Let's compare this set of governing equations with the ones used by [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)].