Editing SSC/Structure/BonnorEbert (section)

==Governing Relation==
The equilibrium structure of an ''isolated'' isothermal sphere, as derived by [http://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden] (1907), has been [[SSC/Structure/IsothermalSphere#Isothermal_Sphere_(structure)|discussed elsewhere]].  From this separate discussion we appreciate that the governing ODE is,
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<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =-  \frac{4\pi G}{c_s^2} \rho \, ,</math>
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where,
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<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>
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is the square of the isothermal sound speed.  In their studies of ''pressure-bounded'' isothermal spheres, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955, ZA, 37, 217) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) both started with this governing ODE, but developed its solution in different ways.  Here we present both developments while highlighting transformations between the two.

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  <td align="center" width="40%">Derivation by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] (edited)</td>
  <td align="center" width="20%">''translation''</td>
  <td align="center">Derivation by [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert (1955)] (edited)</td>
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<!-- [[File:BonnorDerivation01.jpg|300px|center|Bonnor (1956, MNRAS, 116, 351)]] -->
<!-- [[Image:AAAwaiting01.png|300px|center|Bonnor (1956, MNRAS, 116, 351)]] -->
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<math>~\frac{1}{r^2} \frac{d}{dr}\biggl(\frac{r^2}{\rho} \frac{d\rho}{dr}\biggr) = - \frac{4\pi Gm\rho}{kT} \, .</math>
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  <td align="right">(2.3)</td>
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Let us now transform (2.3) by making the following substitutions:

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<math>~\rho = \lambda e^{-\psi} \, ,</math> &nbsp; <math>r = \beta^{1 / 2} \lambda^{-1 / 2} \xi \, ,</math>
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  <td align="right">(2.6)</td>
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where <math>~\lambda</math> is an arbitrary constant [we choose <math>~\lambda = \rho_c</math>], and 

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<math>~\beta = \frac{kT}{4\pi Gm} \, .</math>
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  <td align="right">(2.7)</td>
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Then (2.3) becomes

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<math>~\xi^{-2} \frac{d}{d\xi}\biggl(\xi^2 \frac{d\psi}{d\xi} \biggr) = e^{-\psi} \, .</math>
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  <td align="right">(2.8)</td>
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  </td>
  <td align="center"><math>G \Leftrightarrow \gamma</math></td>
  <td align="left" rowspan="5">
<!-- [[File:EbertDerivation01.jpg|300px|center|Ebert (1955, ZA, 37, 217)]] -->
<!-- [[Image:AAAwaiting01.png|300px|center|Ebert (1955, ZA, 37, 217)]] -->

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<math>~\ell_0 = \biggl(\frac{\Re T_0}{4\pi \mu \gamma \rho_0} \biggr)^{1 / 2} \, .</math>
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  <td align="right">(4)</td>
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Wir setzen unter Verwendung von (4) <math>~r = \ell_0\xi</math> und <math>~\rho = \rho_0\eta</math> mit <math>~\rho_0 = \rho(0)</math>.  F&uuml;r <math>~\eta</math> ergibt sich die Differential-gleichung der isothermen Gaskugel:

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<math>~\eta^{''} - \frac{(\eta^')^2}{\eta} + \frac{2\eta^'}{\xi} + \eta^2 = 0 \, .</math>
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  <td align="right">(17)</td>
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Der Strich bezeichnet die Differentiation nach <math>~\xi</math>.
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  <td align="center"><math>\rho_c \Leftrightarrow \rho_0</math></td>
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  <td align="center"><math>\frac{kT}{m} \Leftarrow c_s^2 \Rightarrow \frac{\Re T_0}{\mu}</math></td>
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  <td align="center"><math>\beta^{1/2}\lambda^{-1/2} \Leftrightarrow l_0</math></td>
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  <td align="center"><math>e^{-\psi} \Leftrightarrow \eta</math></td>
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Both of these dimensionless governing ODEs &#8212; Bonnor's Eq. (2.8) and Ebert's Eq. (17) &#8212; are identical to the dimensionless expression derived by Emden (see the [[SSC/Structure/IsothermalSphere#Governing_Relations|presentation elsewhere]]), namely,
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<math>
\frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{d\mathfrak{r}_1} + e^{v_1} = 0 \, .
</math> 
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The translation from Emden-to-Bonnor-to-Ebert is straightforward:
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<math>
\mathfrak{r}_1 = \xi|_\mathrm{Bonner} = \xi|_\mathrm{Ebert}~~~~\mathrm{and}~~~~e^{v_1} = e^{-\psi} = \eta \, .
</math>
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In much of what follows, we will use Bonnor's <math>~(\xi, \psi)</math> notation rather than Emden's <math>~(\mathfrak{r}_1, v_1)</math> notation.  This means that we will be referring to the ''isothermal Lane-Emden function'', <math>~\psi(\xi)</math>, which provides a solution to the governing,
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<font color="maroon">'''Isothermal Lane-Emden Equation'''</font> <p></p>
{{ Math/EQ_SSLaneEmden02 }}
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