Editing SSC/Structure/IsothermalSphere (section)

==Emden's Numerical Solution==
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[[File:EmdenTable14Corrected.jpg|600px|center|Emden's (1907) Table 14]]
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Note:  The entry highlighted in blue in the <math>3^\mathrm{rd}</math> column must be a typesetting error.
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A more recent and more extensive tabulation of the structural properties of isothermal spheres is provided by: 
* {{ CW49full }}: &nbsp; ''The Isothermal Function''
* {{ Horedt86full }}:&nbsp; ''Seven-Digit Tables of Lane-Emden Functions'' &#8212; See, in particular, pp. 405-406 (Sphere of polytropic index <math>~n = \pm \infty</math>).

An analytic &#8212; but ''approximate'' &#8212; solution to the isothermal Lane-Emden equation can be found:
* [http://adsabs.harvard.edu/abs/1996MNRAS.281.1197L F. K. Liu (1996, MNRAS, 281, 1197-1205)]: &nbsp; ''Polytropic Gas Spheres:  An Approximate Analytic Solution of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1997MNRAS.286..268N Priyamvada Natarajan &amp; Donald Lynden-Bell (1997, MNRAS, 286, 268-270)]: &nbsp; '' An Analytic Approximation to the Isothermal Sphere''
* [http://adsabs.harvard.edu/abs/2013RMxAA..49...63R A. C. Raga, J. C. Rodr&iacute;guez-Ram&iacute;rez, M. Villasante, A. Rodr&iacute;guez-Gonz&aacute;lez, &amp; V. Lora (2013, Revista Mexicana de Astronom&iacute;a y Astrof&iacute;sica, 49, 63-69)]: &nbsp; ''A New Analytic Approximation to the Isothermal, Self-Gravitating Sphere''

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A plot of <math>v_1</math> versus <math>\ln\mathfrak{r}_1</math>, as shown below in Figure 1a, translates into a log-log plot of the equilibrium configuration's <math>~\rho(r)</math> density profile.  Notice that this isolated isothermal configuration extends to infinity and that, at large radii, the density profile displays a simple power-law behavior &#8212; specifically, <math>~ \rho \propto r^{-2}</math>.  This is consistent with our general discussion, [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|presented elsewhere]], of power-law density distributions as solutions of the Lane-Emden equation.  


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'''Figure 1:  Emden's Numerical Solution'''
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[[File:IsothermalDensityPlot.png|350px|center|Plotted from Emden's (1907) tabulated data]]
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[[File:EmdenMassProfile.png|350px|center|Plotted from Emden's (1907) tabulated data]]
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(a) The <math>~(x,y)</math> locations of the data points plotted in blue are drawn directly from column 1 and column 3 of Emden's Table 14 &#8212; specifically, <math>~x = \ln(\mathfrak{r}_1)</math> and <math>~y = v_1</math>.  The dashed red line has a slope of <math>~-2</math> and serves to illustrate that, at large radii, the [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State|isothermal density profile tends toward a <math>~\rho \propto r^{-2}</math> distribution]].
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(b) The <math>~(x,y)</math> locations of the data points plotted in purple are drawn directly from column 1 and column 7 of Emden's Table 14 &#8212; specifically, <math>~x = \ln(\mathfrak{r}_1)</math> and <math>~y = \mathfrak{r}_1^2 v_1'</math>.  The dashed green line has a slope of <math>~+1</math> and serves to illustrate that, at large radii, the isothermal <math>~M(r)</math> distribution tends toward a <math>~M_r \propto r</math> distribution.
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