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=Massless Polytropic Tori=
(''aka'' Papaloizou-Pringle Tori)


[[Image:LSU_Structure_still.gif|74px|left]]
In a seminal paper that focused on an analysis of nonaxisymmetric instabilities in accretion disks, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84) began by constructing equilibrium structures of axisymmetric, polytropic tori that reside in (orbit about) a Newtonian point-mass potential.  The derived structures have analytic prescriptions.  Although the tori constructed by PP84 were not self-gravitating &#8212; ''i.e.'', the tori were ''massless'' &#8212; it is nevertheless instructive for us to examine how these equilibrium structures were derived.

It is not widely appreciated that the equilibrium structure of Papaloizou-Pringle tori was first derived by [http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone &amp; Moncrief (1976), ApJ, 207, 962] in a paper whose primary aim was to analyze the properties of thick disks orbiting around Kerr black holes.  In &sect;IV of their paper, Fishbone &amp; Moncrief show what thick-disk structures result ''in the Newtonian limit'' for homentropic systems with uniform specific angular momentum. These toroidal structures are identical to the ones whose stability was analyzed in PP84.  After reviewing the PP84 derivation, [[#Fishbone_.26_Moncrief_.281976.29|we present the Fishbone &amp; Moncrief (1976) description]] of these equilibrium systems.

==Governing Relations==

As has been derived [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies|elsewhere]], for axisymmetric configurations that obey a barotropic equation of state, hydrostatic balance is governed by the following ''algebraic'' expression:
<div align="center">
<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B} ,</math>
</div>
where <math>~C_\mathrm{B}</math> is the Bernoulli constant,
<div align="center">
<math>\Phi_\mathrm{eff} \equiv \Phi + \Psi ,</math>
</div>
and <math>~\Psi</math> is the relevant centrifugal potential.  For self-gravitating configurations, this algebraic expression must be satisfied in concert with a self-consistent solution of the Poisson equation, but for the ''massless'' PP84 toroidal structures, {{User:Tohline/Math/VAR_NewtonianPotential01}} is just the Newtonian potential presented by a point-like object of mass <math>~M_\mathrm{pt}</math>, namely, 
<div align="center">
<math>\Phi = - \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} .</math>
</div>
Hence, the enthalpy distribution <math>H(\varpi,z)</math> throughout the torus can be described analytically for any one of a number of different [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#SRPtable|''simple rotation profiles'']] by plugging the appropriate expression for the associated centrifugal potential, <math>\Psi(\varpi)</math>, into the governing relation,
<div align="center">
<math>
H = -\Phi_\mathrm{eff} + C_\mathrm{B} = \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} - \Psi + C_\mathrm{B} .
</math>
</div>

==Supplemental Relations==
Following PP84, we supplement the above-specified set of governing equations with a polytropic equation of state, as defined in our overview of [[User:Tohline/SR#Time-Independent_Problems|supplemental relations for time-independent problems]].  Specifically, we will assume that {{User:Tohline/Math/VAR_Density01}} is related to {{User:Tohline/Math/VAR_Enthalpy01}} through the relation, 

<div align="center">
{{User:Tohline/Math/EQ_Polytrope02}} ,
</div>
or we can set,
<div align="center"><font size="+1">
<math> 
H = (n+1) \frac{P}{\rho} .
</math> 
</font></div>
Inverting the first of these two expressions allows us to immediately derive the density distribution <math>\rho(\varpi,z)</math> throughout the torus from the derived algebraic expression for <math>H(\varpi,z)</math>.  Different density distributions will be obtained for different selected polytropic indexes and/or for different selected functional forms of the centrifugal potential. 

Here, also following PP84, we impose a steady-state velocity flow-field that is described by a fluid with uniform specific angular momentum, <math>~j_0</math>.  Drawing from our table of example [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#SRPtable|''simple rotation profiles'']], the centrifugal potential that describes this chosen flow-field is given by the expression,
<div align="center"><font size="+1">
<math> 
\Psi(\varpi) = \frac{j_0^2}{2\varpi^2} .
</math> 
</font></div>

==Summary==
Inserting the two supplemental relations into the governing algebraic expression for <math>H(\varpi,z)</math>, we conclude that the polytropic tori studied by PP84 must have structures that satisfy the equation,
<div align="center">
<math>
(n+1)\frac{P}{\rho} = \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} - \frac{j_0^2}{2\varpi^2}  + C_\mathrm{B} .
</math>
</div> 
This is identical to Eq. (2.8) of PP84.


==Solution==

As PP84 point out, for these toroidal structures it is convenient to normalize all lengths to the position in the equatorial plane, <math>~\varpi_0</math>, at which a fluid particle with specific angular momentum <math>~j_0</math> has an angular orbital frequency, <math>~\dot\varphi = j_0/\varpi^2</math>, that matches the Keplerian orbital frequency associated with the point-mass, <math>~M_\mathrm{pt}</math>, 
<div align="center">
<math>
\omega_K \equiv \biggl[ \frac{GM_\mathrm{pt}}{\varpi^3} \biggr]^{1/2} .
</math>
</div>
That is, it is convenient to define the dimensionless coordinates,
<div align="center">
<math>
\chi \equiv \frac{\varpi}{\varpi_0} ~~~~~\mathrm{and}~~~~~ \zeta \equiv \frac{z}{\varpi_0} ,
</math>
</div>
where,
<div align="center">
<math>
\varpi_0 \equiv \frac{j_0^2}{GM_\mathrm{pt}} .
</math>
</div>

The algebraic relation defining the equilibrium structure then becomes,
<div align="center">
<math>
H = (n+1)\frac{P}{\rho} = \frac{GM_\mathrm{pt}}{\varpi_0} \biggl[ (\chi^2 + \zeta^2)^{-1/2} - \frac{1}{2}\chi^{-2}  - C_\mathrm{B}^' \biggr] ,
</math>
</div> 
where the normalized Bernoulli constant,
<div align="center">
<math>
C_\mathrm{B}^' \equiv - \frac{C_\mathrm{B} \varpi_0}{GM_\mathrm{pt}} .
</math>
</div>

It is worth noting that, in terms of the dimensionless radial coordinate <math>~\chi</math>, the angular velocity and azimuthal velocity at any point inside the torus are, respectively,
<div align="center">
<math>
\dot\varphi = \chi^{-2} \omega_K |_0 \, ,
</math>
</div>
and,
<div align="center">
<math>
v_\varphi \equiv \varpi\dot\varphi = \chi^{-1} v_\varphi |_0 \, ,
</math>
</div>
where <math>\omega_K|_0 = (GM_\mathrm{pt} \varpi_0^{-3})^{1/2}</math> is the Keplerian frequency at <math>\varpi_0</math> and <math>v_\varphi|_0 = (GM_\mathrm{pt} \varpi_0^{-1})^{1/2}</math> is the azimuthal velocity at <math>\varpi_0</math>.

===Pressure Maximum===
It is instructive to examine at what location inside the torus the enthalpy (and density and gas pressure) assumes its maximum value.  At any radial location, <math>~\chi_i</math>, <math>~H</math> has a vertical maximum where  <math>~\partial H/\partial\zeta = 0.</math>  That is, the maximum occurs where,
<div align="center">
<math>
\frac{\partial}{\partial\zeta} \biggl[ (\chi_i^2 + \zeta^2)^{-1/2} - \frac{1}{2}\chi_i^{-2}  - C_\mathrm{B}^' \biggr] 
= -\zeta (\chi_i^2 + \zeta^2)^{-3/2} = 0 .
</math>
</div> 
For all values of <math>~\chi_i</math>, this relation will be satisfied only at <math>~\zeta = 0</math>.  Hence, vertically, the enthalpy maximum will be located in the equatorial plane of the torus.  Examining the radial enthalpy profile in the equatorial plane, we realize that the enthalpy maximum will occur where,
<div align="center">
<math>
\frac{\partial}{\partial\chi} \biggl[ \chi^{-1} - \frac{1}{2}\chi^{-2}  - C_\mathrm{B}^' \biggr] 
= -\chi^{-2} + \chi^{-3} = 0 .
</math>
</div> 
This relation will be satisfied only at <math>~\chi = \varpi/\varpi_0 = 1</math>.  Hence, <math>~\varpi_0</math> is not only the radial location (in the equatorial plane) where the angular frequency of the torus equals the Keplerian frequency, it is also the radial location of the pressure maximum.  

Note that the enthalpy at the pressure maximum &#8212; that is, at the location <math>~(\chi,\zeta) = (1,0)</math> &#8212; is,
<div align="center">
<math>
H_0 = (n+1)\biggl[ \frac{P}{\rho} \biggr]_0 = \frac{GM_\mathrm{pt}}{\varpi_0} \biggl[ \frac{1}{2}  - C_\mathrm{B}^' \biggr]  = \frac{(v_\varphi |_0)^2}{2} [1 - 2C_\mathrm{B}^' ] \, .
</math>
</div> 

===Boundary Conditions===

In the equatorial plane, the locations of the inner and outer edges of the torus are obtained by setting both <math>~H</math> and <math>~\zeta</math> to zero, that is, they are obtained from the roots of the equation,
<div align="center">
<math>
~\chi^{-2} - 2\chi^{-1} + 2C_\mathrm{B}^' = 0  ,
</math>
</div>
that is, the inner <math>~(\chi_-)</math> and outer <math>~(\chi_+)</math> edges are located at,
<div align="center">
<math>
~\chi_\pm  =  \frac{1}{1 \mp \sqrt{1-2C_\mathrm{B}^'}} .
</math>
</div>
Clearly, the relative cross-sectional size &#8212; or, as defined in PP84, the dimensionless distortion parameter <math>~\delta \equiv (\chi_+ + \chi_-)/2 = (2C_\mathrm{B}^')^{-1}</math> &#8212; is determined by the choice of the dimensionless Bernoulli constant, which must lie in the range, <math>~\tfrac{1}{2} \geq C_\mathrm{B}^' \geq 0</math>.  For later reference, we will also note that the ratio of the inner edge to the outer edge is,
<div align="center">
<math>
~\frac{\chi_{-}}{\chi_{+}}  =  \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} } .
</math>
</div>

Indeed, more generally, at all vertical locations above and below the equatorial plane, the surface of the torus occurs where the enthalpy goes to zero.  The meridional <math>~(\chi,\zeta)</math> cross-section through three such surfaces are displayed in the left panel of Figure 1.  The displayed surfaces correspond to <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (green curve).  Carrying forward the above discussion, we note that in all three cases the pressure maximum is located at <math>(\chi,~\zeta) = (1,~0)</math>.  Properties of these tori are described in more detail in an accompanying discussion entitled, [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Papaloizou-Pringle_Tori|''Toroidal Configurations and Related Coordinate Systems'']].  

We should perhaps emphasize that the shape of the surface of each torus, as shown in the left-hand panel of Figure 1, is independent of the chosen polytropic index.


<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<tr><th align="center" colspan="2">Figure 1:  Papaloizou-Pringle Torus</th></tr>
<tr><td align="center">
[[File:LSU_PP84Tori.jpg|350px|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]  
</td>
<td align="center" bgcolor="#D0FFFF">
[[File:TorusMovie1.gif|350px|Pivoting PP Torus]]
</td></tr>
<tr>
  <td align="left">''Left Panel'' &#8212; A meridional cross-section outlining the surface of three separate PP tori:  <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (green curve).</td>
  <td align="left">''Right Panel'' &#8212; A three-dimensional, cutaway rendering illustrating the geometric structure of one PP torus having <math>~C_\mathrm{B}^' = 0.3239,~</math><math>~x_\mathrm{in} = \chi_{-}/\chi_{+} = 0.2551, ~</math>&nbsp;<math>~x_0 = 0.4065.</math> The separately colored contours illustrate four surfaces of constant enthalpy; the red contour identifies the toroidal surface.</td>
</tr>
</table>
</div>

===Model as Described by Kojima===
From the above expression for <math>~\chi_\pm</math>, we see that,
<div align="center">
<math>
\sqrt{1-2C_\mathrm{B}^'} = \frac{1}{\chi_-} - 1 = 1 - \frac{1}{\chi_+} \, .
</math>
</div>
Hence,
<div align="center">
<math>
\chi_+ = \biggl[ 2 - \frac{1}{\chi_-}\biggr]^{-1} = \frac{\chi_-}{2\chi_- - 1} \, .
</math>
</div>
This agrees with the relation between <math>~\chi_+</math> (written as <math>~r_+</math>) and <math>~\chi_-</math> (written as <math>~r_-</math>) presented as equation (7) by [http://http://ptp.ipap.jp/link?PTP/75/251/ Kojima] (1986, Progress of Theoretical Physics, 75, 251-261), who also has examined the properties of Papaloizou-Pringle tori.  In his equation (8), Kojima points out that the inner edge of the torus lies within the range, <math>~\tfrac{1}{2} < \chi_- < 1</math>, which is consistent with stating that the value of PB84's Bernoulli constant falls within the range, <math> ~\tfrac{1}{2} \geq C_\mathrm{B}^' \geq 0</math>.  

Using the just-derived relationship between <math>~C_\mathrm{B}^'</math> and <math>~\chi_-</math>, we deduce that the enthalpy at the pressure maximum depends only on the choice of <math>~\chi_-</math> as follows:
<div align="center">
<math>
H_0 = (n+1)\biggl[ \frac{P}{\rho} \biggr]_0 = \frac{(v_\varphi |_0)^2}{2} \biggl[ \frac{1}{\chi_-} - 1 \biggr]^2 \, .
</math>
</div> 
Under adiabatic conditions, the sound speed of a gas is given by the algebraic relation,
<div align="center">
<math>
c_s^2 = \gamma \biggl[\frac{P}{\rho}\biggr] \, ,
</math>
</div>
where <math>~\gamma</math> is the specified adiabatic exponent.  Hence, the expression just derived for <math>~H_0</math> also gives us the sound speed at the pressure maximum of the torus, namely,
<div align="center">
<math>
(c_s |_0)^2 = \frac{\gamma}{(n + 1)}  \frac{(v_\varphi |_0)^2}{2} \biggl[ \frac{1}{\chi_-} - 1 \biggr]^2 \, ,
</math>
</div> 
that is,
<div align="center">
<math>
\frac{c_s|_0}{v_\varphi |_0} = \biggl[ \frac{\gamma}{2(n + 1)} \biggr]^{1/2} \biggl[ \frac{1}{\chi_-} - 1 \biggr] \, .
</math>
</div> 
This matches Kojima's equation (11) with <math>~\chi_-</math> written in place of <math>~r_-</math>, namely, 
<div align="center">
<math>
\biggl( \frac{c_s}{v_\varphi} \biggr)_0 = \frac{1}{\sqrt{6}} \biggl(\frac{1}{\chi_-} - 1 \biggr) \, ,
</math>
</div>
when one adopts Kojima's choice of <math>~\gamma = 4/3</math> and <math>~n = (\gamma - 1)^{-1} = 3</math>.

===Hachisu Self-Consistent Field Technique===
For the sake of contextual continuity, it is useful to relate the above result to the Hachisu self-consistent field (HSCF) technique, which we rely upon heavily to derive the equilibrium structure of self-gravitating systems.  The HSCF technique begins by identifying the relevant algebraic expression for the enthalpy as derived above, namely,
<div align="center">
<math>
H = \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} - \frac{j_0^2}{2\varpi^2}  + C_\mathrm{B} .
</math>
</div> 
In the HSCF technique, it is customary to normalize all lengths to the distance <math>\varpi_{+}</math> from the origin to the point where the outer edge of the torus touches the equatorial plane.  From this perspective, then, the relevant dimensionless coordinates are,
<div align="center">
<math>
x_\mathrm{HSCF} \equiv \frac{\varpi}{\varpi_{+}} ~~~~~\mathrm{and}~~~~~ z_\mathrm{HSCF} \equiv \frac{z}{\varpi_{+}} ,
</math>
</div>
and the equation for the dimensionless enthalpy becomes,
<div align="center">
<math>
H_\mathrm{HSCF} \equiv \frac{H}{(GM_\mathrm{pt}/\varpi_{+})} = (x_\mathrm{HSCF}^2 + z_\mathrm{HSCF}^2)^{-1/2} - \frac{1}{2} (j_\mathrm{HSCF}^2) x_\mathrm{HSCF}^{-2}  - C_\mathrm{HSCF} ,
</math>
</div> 
where the normalized Bernoulli constant,
<div align="center">
<math>
C_\mathrm{HSCF} \equiv - \frac{C_\mathrm{B} \varpi_{+}}{GM_\mathrm{pt}} ,
</math>
</div>
and the normalized specific angular momentum is,
<div align="center">
<math>
j_\mathrm{HSCF} \equiv \frac{j_0}{(GM_\mathrm{pt} \varpi_{+})^{1/2}} .
</math>
</div>
This expression for the dimensionless enthalpy has two unknowns:  <math>~j_\mathrm{HSCF}</math> and <math>~C_\mathrm{HSCF}</math>.  Hence, two boundary points must be specified before a solution for <math>~H_\mathrm{HSCF}(\varpi,z)</math> can be obtained.  In the HSCF technique, it is customary to specify the location of two points on the surface of the configuration, where by design <math>~H=0</math>.  For toroidal configurations, it is furthermore customary for that specification to be where the inner and outer edges of the torus touch the equatorial plane.  By design, the outer edge is at <math>~(x_\mathrm{HSCF}, ~z_\mathrm{HSCF}) = (1, ~0)</math>; the inner edge will be at <math>(x_\mathrm{HSCF}, ~z_\mathrm{HSCF}) = (x_{in}, ~0)</math>.  Setting <math>~H_\mathrm{HSCF}=0</math> at both of these locations gives the following two algebraic relations,
<div align="center">
<math>
C_\mathrm{HSCF} = 1 - \frac{1}{2} (j_\mathrm{HSCF}^2) ,
</math>
</div> 
and
<div align="center">
<math>
C_\mathrm{HSCF} = x_{in}^{-1} - \frac{1}{2} (j_\mathrm{HSCF}^2) x_{in}^{-2}  ,
</math>
</div> 
which can be used in combination to solve for the two unknowns. The result is,
<div align="center">
<math>
C_\mathrm{HSCF} = \frac{1}{1 + x_{in}} ,
</math>
</div> 
and,
<div align="center">
<math>
j_\mathrm{HSCF} = \biggl[ \frac{2x_{in}}{1 + x_{in}} \biggr]^{1/2} .
</math>
</div>

At what radial position in the equatorial plane,  <math>~x_0</math>, is the pressure maximum located?  The pressure maximum is also the enthalpy maximum, so the answer is given by looking in the equatorial plane <math>~(z_\mathrm{HSCF} = 0)</math> for the location where <math>~dH_\mathrm{HSCF}/dx_\mathrm{HSCF} = 0</math>.  Since,
<div align="center">
<math>
\frac{dH_\mathrm{HSCF}}{dx_\mathrm{HSCF}} = - x_\mathrm{HSCF}^{-2} + (j_\mathrm{HSCF}^2) x_\mathrm{HSCF}^{-3} \, ,
</math>
</div>
the pressure maximum must be located at,
<div align="center">
<math>
x_\mathrm{HSCF} = x_0 = j_\mathrm{HSCF}^2 \, .
</math>
</div>
Hence, in terms of <math>~x_{in}</math>,
<div align="center">
<math>
x_0 =  \frac{2x_{in}}{1 + x_{in}} .
</math>
</div>

===Comparison===
The solution derived by PP84 (presented above) is normalized such that the pressure maximum is always in the equatorial plane at radial coordinate <math>~\chi = 1</math>, and tori of different thicknesses are constructed by specifying different values of the dimensionless Bernoulli constant, <math>~C_\mathrm{B}^'</math>.  Using the HSCF approach, the position of the outer edge of the torus is fixed and tori of different thicknesses are selected by moving the inner edge, <math>~x_{in}</math>, around.  In order to examine whether or not the two solutions are, in fact, the same, we will re-write the expression given above for <math>~\chi_{-}/\chi_{+}</math> in terms of <math>~x_{in}</math>.  In order to do this, we need to convert <math>~C_\mathrm{HSCF}</math> to <math>~C_\mathrm{B}^'</math>.  Based on their definitions, the ratio
<div align="center">
<math>
\frac{C_\mathrm{B}^'}{C_\mathrm{HSCF} } = \frac{\varpi_0}{\varpi_{+}} = x_0 .
</math>
</div>
Hence,
<div align="center">
<math>
C_\mathrm{B}^' = x_0~C_\mathrm{HSCF} =  \biggl( \frac{2x_{in}}{1+x_{in}} \biggr) \biggl( \frac{1}{1+x_{in}} \biggr) = \frac{2x_{in}}{(1+x_{in})^2}.
</math>
</div>
Therefore,
<div align="center">
<math>
1 - 2C_\mathrm{B}^' = 1 - \frac{4x_{in}}{(1+x_{in})^2} = \frac{ (1+x_{in})^2 - 4x_{in} }{(1+x_{in})^2} = \frac{ (1-x_{in})^2 }{(1+x_{in})^2} \, ,
</math>
</div>
and,
<div align="center">
<math>
\frac{\chi_{-}}{\chi_{+}} =  \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} }  =  \frac{ 1 - \frac{1-x_{in}}{1+x_{in}} }{1 + \frac{1-x_{in}}{1+x_{in}}}  =  \frac{  (1+x_{in}) - (1-x_{in})  }{  (1+x_{in}) + (1-x_{in}) } = x_{in}.
</math>
</div>
Hence, both the PP84 derivation and the HSCF derivation produce the same result.

==Blaes (1985)==

In a [[User:Tohline/Appendix/Ramblings/PPTori#Blaes85|separate chapter]], we review the analytic analysis of unstable nonaxisymmetric eigenmodes in geometrically thin PP tori that has been presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985), MNRAS, 216, 553].  Our review includes an explanation of how the [[#Solution|equations derived here]] to describe the equilibrium structure of PP Tori are straightforwardly transformed into the equations and notation adopted by Blaes.



{{LSU_WorkInProgress}}
==Fishbone &amp; Moncrief (1976)==
[http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone &amp; Moncrief (1976), ApJ, 207, 962] derived the identical Papaloizou-Pringle torus structure using a spherical rather than cylindrical coordinate system.

===Derived Equilibrium Structure ===
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c^2 \frac{\partial w}{\partial r} \equiv \frac{1}{(nm)} \frac{\partial P}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial r}\biggl(\frac{GM}{r}\biggr) + \omega^2  r^2 \sin^2\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c^2 \frac{\partial w}{\partial \theta} \equiv \frac{1}{(nm)} \frac{\partial P}{\partial \theta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial \theta}\biggl(\frac{GM}{r}\biggr) + \omega^2  r^2 \sin\theta \cos\theta \, .</math>
  </td>
</tr>
</table>
</div>

As presented in their equation (4.4), imposing the condition of uniform specific angular momentum leads to an angular velocity profile of the form,
<div align="center">
<math>~\omega = \frac{\ell}{r^2 \sin^2\theta} \, .</math>
</div>


===Comparison===

=Key References=

* [http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone &amp; Moncrief (1976), ApJ, 207, 962] &#8212; ''Relativistic fluid disks in orbit around Kerr black holes''
* [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle (1984), MNRAS, 208, 721] &#8212; ''The dynamical stability of differentially rotating discs with constant specific angular momentum''
* [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985), MNRAS, 213, 799] &#8212; ''The dynamical stability of differentially rotating discs. &nbsp; II.''
* [http://adsabs.harvard.edu/abs/1987MNRAS.225..267P Papaloizou &amp; Pringle (1987), MNRAS, 225, 267] &#8212; ''The dynamical stability of differentially rotating discs. &nbsp; III.''
* [http://adsabs.harvard.edu/abs/1985MNRAS.213P...7G Goldreich &amp; Narayan (1985), MNRAS, 213, 7] &#8212; ''Non-axisymmetric instability in thin discs''
* [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldrich, Goodman &amp; Narayan (1986), MNRAS, 221, 339] &#8212; ''The stability of accretion tori. &nbsp; I. -- Long-wavelength modes of slender tori''
* [http://adsabs.harvard.edu/abs/1987MNRAS.225..695G Goodman, Narayan, &amp; Goldreich (1987), MNRAS, 225, 695] &#8212; ''The stability of accretion tori. &nbsp; II -- Non-linear evolution to discrete planets''
* [http://adsabs.harvard.edu/abs/1988MNRAS.231...97G Goodman &amp; Narayan (1988), MNRAS, 231, 97] &#8212; ''The stability of accretion tori. &nbsp; III -- The effect of self-gravity''
* [http://adsabs.harvard.edu/abs/1992ApJ...388..451C Christodoulou &amp; Narayan (1992), ApJ, 388, 451] &#8212; ''The stability of accretion tori. &nbsp; IV. Fission and fragmentation of slender, self-gravitating annuli''
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=Related Wikipedia Discussions=
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