Editing Apps/HayashiNaritaMiyama82 (section)

===HNM82 Derivation===

HNM82 ''guess'' a density distribution of the form (see their Eq.2.3),
<div align="center">
<math>
~\frac{\rho(\varpi,z)}{\rho_0} = g(\varpi,z) \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-2} .
</math>
</div>
The algebraic expression defining hydrostatic balance then becomes,
<div align="center">
<math>
~\Phi(\varpi,z) = C_\mathrm{B} - c_s^2 \ln g(\varpi,z) + (2c_s^2 + v_\varphi^2) \ln\biggl(\frac{\varpi}{\varpi_0}\biggr) ;
</math>
</div>
and, after multiplying both sides by <math>~\varpi^2</math>, the Poisson equation becomes,
<div align="center">
<math>
~\varpi \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi(\varpi,z)}{\partial\varpi} \biggr] + \varpi \frac{\partial}{\partial z} \biggl[ \varpi \frac{\partial \Phi(\varpi,z)}{\partial z} \biggr] = (4\pi G\rho_0 \varpi_0^2) g(\varpi,z) . 
</math><br />
</div>
Plugging the expression for <math>~\Phi</math> into the Poisson equation gives (see Eq. 2.4 of HNM82),
<div align="center">
<math>
~\varpi \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \ln g(\varpi,z)}{\partial\varpi} \biggr] + \varpi \frac{\partial}{\partial z} \biggl[ \varpi \frac{\partial \ln g(\varpi,z)}{\partial z} \biggr] = - \biggl[ \frac{4\pi G\rho_0 \varpi_0^2}{c_s^2} \biggr] g(\varpi,z) . 
</math><br />
</div>
HNM82 realized that since this equation "is invariant to the scale changes of both <math>~\varpi</math> and <math>~z</math>, it has a conformal solution such that <math>~g</math> is a function of <math>~z/\varpi</math> alone."  In particular, as HNM82 pointed out, by making the substitution,
<div align="center">
<math>
~\zeta \equiv \sinh^{-1}\biggl(\frac{z}{\varpi}\biggr)= \ln \biggl[ \frac{r+z}{\varpi} \biggr] ,
</math>
</div>
the above, 2D elliptic PDE (Poisson equation) can be written as the following, 1D second-order ODE:
 <div align="center">
<math>
~\frac{d^2\ln g(\zeta)}{d\zeta^2} = - \biggl[ \frac{4\pi G\rho_0 \varpi_0^2}{c_s^2} \biggr] g(\zeta) .
</math>
</div>
As presented by HNM82, the solution to this Poisson equation that meets the most physically reasonable boundary conditions at <math>~\zeta = 0</math> (''i.e.,'' <math>~g</math> is finite and <math>~dg/d\zeta = 0</math>) is,
<div align="center">
<math>
~g(\zeta) = \biggl[ \frac{c_s^2}{2\pi G\rho_0 \varpi_0^2} \biggr] \frac{\gamma^2}{\cosh^2(\gamma\zeta)} ,
</math>
</div>
where <math>~\gamma = [1 + v_\varphi^2/(2c_s^2)]</math> is a parameter that identifies an individual equilibrium structure from the ''family'' of allowed solutions.