Editing SSC/Structure/IsothermalSphere (section)

==Governing Relations==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{Math/VAR_Density01}} and {{Math/VAR_Enthalpy01}}:

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<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =-  4\pi G \rho</math> .
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Using the {{Math/VAR_Enthalpy01}}-{{Math/VAR_Density01}} relationship for an isothermal gas presented above, this can be rewritten entirely in terms of the density as,

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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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<span id="keyExpression">or, equivalently,</span>
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<math>
\frac{d^2\ln\rho}{dr^2} +\frac{2}{r} \frac{d\ln\rho}{dr} + \beta^2 \rho = 0 \, ,
</math> 
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where,

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<math>
\beta^2 \equiv \frac{4\pi G}{c_s^2} \, .
</math>
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This matches the governing ODE whose derivation was published on p. 131 of the book by {{ Emden07full }}. 

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Derivation Appearing on p. 131 of {{ Emden07 }} (edited)
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[[File:EmdenBookCover1907.png|240px|center|Emden (1907)]]
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<!-- [[File:EmdenIsothermalDerivation.jpg|500px|center|Emden (1907)]] -->
&sect;2.  Wir gehen wieder aus von der Gleichung (59)
<div align="center"><math>\frac{d}{dr}\biggl(\frac{r^2}{\rho} \frac{dp}{dr}\biggr) = -4\pi G\rho r^2 \, .</math></div>
Da wir haben <math>p = \rho H T, T = ~\mathrm{konst.}</math>, so ergibt sich
<div align="center"><math>\frac{dp}{\rho} = HT \frac{d\rho}{\rho} = HT d\log\rho \, ,</math></div>
und setzen wir
<div align="center"><math>\beta^2 = \frac{4\pi G}{HT}</math> (gramm<sup>-1</sup> cent)</div>
und f&uuml;hren die Differentiation aus, so ergibt sich die
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<font color="maroon"><b>''Differentialgleichung der isothermal Gaskugel''</b></font><br />
<math>\frac{d^2 ~\mathrm{lg}\rho}{dr^2} + \frac{2}{r} \frac{d~\mathrm{lg}\rho}{r} + \beta^2 \rho = 0 \, .</math>
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Note that, in Emden's derivation, <math>H</math> is not enthalpy but, rather, the effective gas constant, <math>H = c_s^2/T</math>.
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By adopting the following dimensionless variables,

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<math>
\mathfrak{r}_1 \equiv \rho_c^{1/2} \beta r \, , ~~~~\mathrm{and}~~~~v_1 \equiv \ln(\rho/\rho_c) \, ,
</math>
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where <math>~\rho_c</math> is the configuration's central density, the governing ODE can be rewritten in dimensionless form as,

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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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which is exactly the equation numbered (II"a) that can be found on p. 133 of {{ Emden07 }}.
Emden numerically determined the behavior of the function <math>~v_1(\mathfrak{r}_1)</math>, its first derivative with respect to <math>~\mathfrak{r}_1</math>, <math>~v_1'</math>, along with <math>~e^{v_1}</math> and several other useful products, and published his results as Table 14, on p. 135 of his book.  This table has been reproduced [[#Emden.27s_Numerical_Solution|immediately below]], primarily for historical purposes.

Note that a somewhat more extensive tabulation of the structural properties of isothermal spheres is provided by {{ CW49full }}.  In this published work as well as in &sect;22 of Chapter IV in [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chandrasekhar has written the governing ODE in a form that we will refer to as the,
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<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />
{{ Math/EQ_SSLaneEmden02 }}
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It is straightforward to show that this is identical to Emden's governing expression after making the variable substitutions:
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<math>~\mathfrak{r}_1 \rightarrow \xi</math>&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~v_1 \rightarrow -\psi </math>.
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Across the astrophysics community, Chadrasekhar's notation has been widely &#8212; although not universally &#8212; adopted as the standard.