Editing Apps/HayashiNaritaMiyama82 (section)

==Structural Properties==

By analogy with the solution that was derived for a [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|spherically symmetric isothermal structure with a power-law density distribution]], we can associate the scale length <math>~\varpi_0</math> with the characteristic density <math>~\rho_0</math> at that location through the relation,
<div align="center">
<math>
~\rho_0 = \frac{c_s^2}{2\pi G \varpi_0^2 } ,
</math>
</div>
in which case we can write,
<div align="center">
<math>
~g(\zeta) = \frac{\gamma^2}{\cosh^2(\gamma\zeta)} .
</math>
</div>
With this definition in hand, the equilibrium models discovered by HNM82 exhibit the following properties.

* <span id="Density"><font color="red">Density Distribution</font></span>: 
:Confirming the expression presented as Eq. (3.1) in HNM82, the 2D density distribution for models with different values of the dimensionless parameter <math>~\gamma</math> is, 
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\rho(\varpi,z) 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\rho_0 g(\varpi,z) \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{c_s^2}{2\pi G \varpi_0^2} \biggr]\biggl(\frac{\varpi_0}{\varpi}\biggr)^{2} \frac{\gamma^2}{\cosh^2(\gamma\zeta)}  .
</math>
  </td>
</tr>
</table>

* <span id="Potential"><font color="red">Gravitational Potential</font></span>: 
:As presented in Eq. (3.4) of HNM82, to within an additive constant the 2D potential distribution for models with different values of the dimensionless parameter <math>~\gamma</math> is, 
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\frac{\Phi(\varpi,z)}{2c_s^2} 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{C_\mathrm{B}}{2c_s^2} - \frac{1}{2} \ln g(\varpi,z) + \biggl(1 + \frac{v_\varphi^2}{2c_s^2} \biggr) \ln\biggl(\frac{\varpi}{\varpi_0}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{C_\mathrm{B}}{2c_s^2} - \frac{1}{2} \ln \biggl[ \frac{\gamma^2}{\cosh^2(\gamma\zeta)} \biggr] + \gamma \ln\biggl(\frac{\varpi}{\varpi_0}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{C_\mathrm{B}}{2c_s^2} + \ln \biggl[ \frac{1}{\gamma} \biggr] + \ln \biggl[ \cosh (\gamma\zeta) \biggr] + \ln\biggl(\frac{\varpi}{\varpi_0}\biggr)^\gamma 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~C' + \ln \biggl[ \varpi^\gamma \cosh (\gamma\zeta) \biggr] .
</math>
  </td>
</tr>
</table>

:HNM82 point out that this last expression also can be written in terms of <math>~r</math> and <math>~z</math> as follows:

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\frac{\Phi(\varpi,z)}{2c_s^2} 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~C' + \ln \biggl[ \frac{1}{2}(r+z)^\gamma + \frac{1}{2}(r-z)^\gamma \biggr] .
</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="5" width="420px">
<tr>
  <th align="center">3D Renderings of Isothermal Disks from [https://ui.adsabs.harvard.edu/abs/1982PThPh..68.1949H/abstract Hayashi, Narita &amp; Miyama (1982)]</th>
</tr>
<tr><td align="center">
[[File:HNM82ThreeDisksAB.png|400px|Three HNM82 disks]]
</td></tr>
<tr>
  <td align="left">(Top) Edge-on view of three isothermal disks; (Bottom) Cut-away view of highest-density region of the same three disks.  Values of flattening parameter, <math>~\gamma</math>, shown in bottom-right corner of each frame.  
</table>