View source for User:Tohline/AxisymmetricConfigurations/SolutionStrategies

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=Axisymmetric Configurations (Structure &#8212; Part II)=
[[Image:LSU_Structure_still.gif|74px|left]]
Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE|identified set of simplified governing equations]].  We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:



<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span>

<math>\cancel{\frac{\partial\rho}{\partial t}} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] 
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br />


<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span>
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\cancel{\frac{\partial \dot\varpi}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr]  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}  
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\cancel{\frac{\partial \dot{z}}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr]
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>
</table>

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
\biggl\{\cancel{\frac{\partial \epsilon}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} +
P \biggl\{\cancel{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr)} + 
\biggl[ \dot\varpi \frac{\partial }{\partial\varpi}\biggl(\frac{1}{\rho}\biggr) \biggr] + 
\biggl[ \dot{z} \frac{\partial }{\partial z}\biggl(\frac{1}{\rho}\biggr) \biggr] \biggr\} = 0
</math>


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho . 
</math><br />
</div>


The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_\varphi (\varpi \dot\varphi)</math>.  That is, <math>\dot\varpi = \dot{z} = 0</math> but, in general, <math>\dot\varphi</math> is not zero and can be an arbitrary function of <math>\varpi</math> and <math>z</math>, that is, <math>\dot\varphi = \dot\varphi(\varpi,z)</math>.  We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity <math>\dot\varphi(\varpi,z)</math>, or of the specific angular momentum, <math>j(\varpi,z) = \varpi^2 \dot\varphi(\varpi,z)</math>.


After setting the radial and vertical velocities to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>4^\mathrm{th}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> &amp; <math>3^\mathrm{rd}</math> (Euler) and <math>5^\mathrm{th}</math> (Poisson) give, respectively,

<div 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">
=
  </td>
  <td align="left">
0
  </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">
=
  </td>
  <td align="left">
0
  </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">
=
  </td>
  <td align="left">
<math>4\pi G \rho</math>
  </td>
</tr>
</table>

</div>

As has been outlined in our discussion of [[User:Tohline/SR#Time-Independent_Problems|supplemental relations for time-independent problems]], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}. 


==Solution Strategies==
Equilibrium axisymmetric structures &#8212; that is, solutions to the above set of simplified governing equations &#8212; can be found for specified angular momentum distributions that display a wide range of variations across both of the spatial coordinates, <math>\varpi</math> and <math>z</math>.  Experience has shown, however, that the derived structures tend to be dynamically unstable unless the angular velocity is uniform on cylinders, that is, unless the angular velocity is independent of <math>z</math>.  With this in mind, we will focus here on a solution strategy that is designed to construct structures with a

<div align="center">
<span id="SimpleRotation"><font color="#770000">'''Simple Rotation Profile'''</font></span>

<math>\dot\varphi(\varpi,z) = \dot\varphi(\varpi) ,</math>
</div>

which of course means that we will only be examining axisymmetric structures with specific angular momentum distributions of the form <math>j(\varpi,z) = j(\varpi)</math>.  (We will find that even this ''simple rotation'' profile does not guarantee dynamical stability; for example, unstable structures will arise if <math>j</math> is a decreasing function of the radial coordinate, <math>\varpi</math>.)  

===Simple Rotation and Barotropic EOS===
Adopting a simple rotation profile along with a barotropic equation of state &#8212; in which case <math>dP/\rho</math> can be replaced by <math>dH</math> &#8212; we can combine the two components of the Euler equation shown above back into a single vector equation of the form,

<div align="center">
<math>
\nabla \biggl[ H + \Phi_\mathrm{eff} \biggr] = 0 ,
</math> 
</div>
where it is understood that, [[User:Tohline/AxisymmetricConfigurations/PGE|as displayed earlier]], here the gradient represents a two-dimensional operator appropriate for axisymmetric configurations, namely,
<div align="center">
<math>
\nabla f = {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] +  {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] ,
</math>
</div>
and the effective potential,
<div align="center">
<math>
\Phi_\mathrm{eff} \equiv \Phi + \Psi ,
</math>
</div>
has been written in terms of a centrifugal potential,
<div align="center">
<math>
\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi ~.
</math>
</div>


<table align="center" border="1" cellpadding="5">
<tr>
  <th align="center" colspan="7">
<font color="maroon">
''Simple Rotation Profiles'' <br />Found in the Published Literature
</font>
  </th>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <th align="center">
<b><math>\dot\varphi(\varpi)</math></b>
  </th>
  <th align="center">
<b><math>v_\varphi(\varpi)</math></b>
  </th>
  <th align="center">
<b><math>j(\varpi)</math></b>
  </th>
  <th align="center">
<b><math>\frac{j^2}{\varpi^3}</math></b>
  </th>
  <th align="center">
<b><math>\Psi(\varpi)</math></b>
  </th>
  <th align="center">
Refs.
  </th>
</tr>

<tr>
  <td align="center">
<font color="maroon"><b>Power-law</b></font><br />(any <math>q \neq 1</math>)
  </td>
  <td align="center">
<math>\frac{j_0}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(q-2)}</math>
  </td>
  <td align="center">
<math>\frac{j_0}{\varpi_0} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(q-1)}</math>
  </td>
  <td align="center">
<math>j_0\biggl( \frac{\varpi}{\varpi_0} \biggr)^{q}</math>
  </td>
  <td align="center">
<math>\frac{j_0^2}{\varpi_0^3} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(2q-3)}</math>
  </td>
  <td align="center">
<math>- \frac{1}{2(q-1)} \biggl[ \frac{j_0^2}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{2(q-1)} \biggr]</math>
  </td>
  <td align="center">
d
  </td>
</tr>

<tr>
  <td align="center">
<font color="maroon"><b>Uniform rotation</b></font><br /><math>(q = 2)</math>
  </td>
  <td align="center">
<math>\omega_0</math>
  </td>
  <td align="center">
<math>\varpi \omega_0</math>
  </td>
  <td align="center">
<math>\varpi^2 \omega_0</math>
  </td>
  <td align="center">
<math>\varpi \omega_0^2</math>
  </td>
  <td align="center">
<math>- \frac{1}{2} \varpi^2 \omega_0^2</math>
  </td>
  <td align="center">
a, f
  </td>
</tr>

<tr>
  <td align="center">
<font color="maroon"><b>Uniform</b></font> <math>v_\varphi</math><br /><math>(q = 1)</math>
  </td>
  <td align="center">
<math>\frac{v_0}{\varpi}</math>
  </td>
  <td align="center">
<math>v_0</math>
  </td>
  <td align="center">
<math>\varpi v_0</math>
  </td>
  <td align="center">
<math>\frac{v_0^2}{\varpi}</math>
  </td>
  <td align="center">
<math> - v_0^2 \ln\biggl( \frac{\varpi}{\varpi_0} \biggr)</math>
  </td>
  <td align="center">
e
  </td>
</tr>


<tr>
  <td align="center">
<font color="maroon"><b>Keplerian</b></font><br /><math>(q = 1/2)</math>
  </td>
  <td align="center">
<math>\omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-3/2}</math>
  </td>
  <td align="center">
<math>\varpi_0 \omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-1/2}</math>
  </td>
  <td align="center">
<math>\varpi_0^2 \omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{1/2}</math>
  </td>
  <td align="center">
<math>\varpi_0 \omega_K^2 \biggl( \frac{\varpi}{\varpi_0} \biggr)^{-2}</math>
  </td>
  <td align="center">
<math>+ \frac{\varpi_0^3 \omega_K^2}{\varpi}  </math>
  </td>
  <td align="center">
d
  </td>
</tr>


<tr>
  <td align="center">
<font color="maroon"><b>Uniform specific <br />angular momentum</b></font><br /><math>(q = 0)</math>
  </td>
  <td align="center">
<math>\frac{j_0}{\varpi^2}</math>
  </td>
  <td align="center">
<math>\frac{j_0}{\varpi}</math>
  </td>
  <td align="center">
<math>j_0</math>
  </td>
  <td align="center">
<math>\frac{j_0^2}{\varpi^3}</math>
  </td>
  <td align="center">
<math>+ \frac{1}{2} \biggl[ \frac{j_0^2}{\varpi^2} \biggr]</math>
  </td>
  <td align="center">
c
  </td>
</tr>


<tr>
  <td align="center">
<font color="maroon"><b>j-constant <br />rotation</b></font>
  </td>
  <td align="center">
<math>\omega_c \biggl[ \frac{A^2}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
<math>\omega_c \biggl[ \frac{A^2 \varpi}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
<math>\omega_c \biggl[ \frac{A^2 \varpi^2}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
<math>\omega_c^2 \biggl[ \frac{A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr]</math>
  </td>
  <td align="center">
<math>+ \frac{1}{2} \biggl[ \frac{\omega_c^2 A^4}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
a,b
  </td>
</tr>

<tr>
  <td align="left" colspan="7">
<sup>a</sup>Hachisu, I. [http://adsabs.harvard.edu/abs/1986ApJS...61..479H 1986, ApJS, 61, 479-507] 
(especially &sect;II.c)<br />
<sup>b</sup>Ou, S. &amp; Tohline, J.E. [http://iopscience.iop.org/0004-637X/651/2/1068/pdf/0004-637X_651_2_1068.pdf 2006, ApJ, 651, 1068-1078] 
(especially &sect;2.1)<br />
<sup>c</sup>Woodward, J.W., Tohline, J.E. &amp; Hachisu, I. [http://adsabs.harvard.edu/abs/1994ApJ...420..247W 1994, ApJ, 420, 247-267]<br />
<sup>d</sup>Tohline, J.E. &amp; Hachisu, I. [http://adsabs.harvard.edu/abs/1990ApJ...361..394T 1990, ApJ, 361, 394-407]<br />
<sup>e</sup>Hayashi, C., Narita, S. &amp; Miyama, S.M. [http://adsabs.harvard.edu/abs/1982PThPh..68.1949H 1982, ''Progress of Theoretical Physics'', 68, 1949-1966]<br />
<sup>f</sup>Maclaurin, C. 1742, ''A Treatise of Fluxions''
  </td>
</tr>
</table>


===Structures===

When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where {{User:Tohline/Math/VAR_Density01}} is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,
<div align="center">
<math>dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,
</div>
where we have pulled from the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates] to define the spherical surface element <math>dS</math>.  Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,
<div align="center">
<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
</div>
We can also state that,
<div align="center">
{{User:Tohline/Math/EQ_SSmassConservation01}}
</div>
This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.  

===Technique 3===
As in Technique #2, we replace <math>dP/\rho</math> by d{{User:Tohline/Math/VAR_Enthalpy01}} in the hydrostatic balance relation, but this time we realize that the resulting expression can be written in the form,
<div align="center">
<math>\frac{d}{dr}(H+\Phi) = 0</math> .
</div>
This means that, throughout our configuration, the functions {{User:Tohline/Math/VAR_Enthalpy01}}({{User:Tohline/Math/VAR_Density01}}) and {{User:Tohline/Math/VAR_NewtonianPotential01}}({{User:Tohline/Math/VAR_Density01}}) must sum to a constant value, call it <math>C_\mathrm{B}</math>.  That is to say, the statement of hydrostatic balance reduces to the ''algebraic'' expression,
<div align="center">
<math>H + \Phi = C_\mathrm{B}</math> .
</div>
This relation must be solved in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr)  = 4\pi G \rho </math> ,
</div>
giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}}

=See Also=
* Part I of ''Axisymmetric Configurations'': [[User:Tohline/AxisymmetricConfigurations/PGE|Simplified Governing Equations]]



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