Editing Apps/HayashiNaritaMiyama82 (section)

==Solution==
===Rationalizing the Approach===
HNM82 provide a very clear and detailed description of the approach that they took to solving the above-identified set of simplified governing relations.  (Note that Hayashi is credited with deriving the analytic solution.)  In very general terms, one can understand the thought process that must have been going on in Hayashi's mind:
# For an isothermal gas cloud, the enthalpy is necessarily a logarithmic function of the density.
# For spherically symmetric, isothermal configurations, a solution to the governing relations exists in which {{ Math/VAR_Density01 }} can be expressed as a [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|power-law function]] of the radius, specifically, <math>~\rho \propto r^{-2}</math>; hence, the enthalpy displays a logarithmic dependence on the distance.
# Although one could attempt to derive equilibrium structures having a wide range of [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|''Simple rotation profiles'']], it would seem wisest to select a centrifugal potential function that at least has the same ''form'' as the enthalpy; hence, the choice was made to impose <math>~v_\varphi = \mathrm{constant}</math> so that, like the enthalpy, the centrifugal potential would exhibit a logarithmic dependence on the distance.
# Three birds, so to speak, can be killed with one stone by ''guessing'' a 2D equilibrium density profile of the form <math>~\rho(\varpi,z) = g(\varpi,z)/\varpi^{2}</math>:  The <math>~\varpi^{-2}</math> dependence can be combined strategically with <math>~\varpi^{-2}</math> dependence of the centrifugal potential; although it depends on <math>\varpi</math> instead of <math>r</math>, the density profile will at least ''resemble'' the spherical solution; and &#8212; certainly the most critical realization &#8212; the Poisson equation, which for this problem is a 2D elliptic PDE, can be rewritten as a 1D ODE and solved analytically!

===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.