Editing SSC/Structure/IsothermalSphere (section)

==Summary==
</font>

Based on the above derivations, the internal structural properties of an equilibrium isothermal sphere can be described in terms of the tabulated quantities provided in Emden's Table 14 as follows:
* <font color="red">Radial Coordinate Position</font>: 
: Given the isothermal sound speed, <math>c_s</math>, and the central density, <math>\rho_c</math>, the radial coordinate is,
<div align="center">
<math>r = ( \rho_c \beta^2 )^{-1/2} \mathfrak{r}_1 = \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \mathfrak{r}_1 </math> .
</div>

* <font color="red">Density &amp; Pressure</font>: 
: As a function of the radial coordinate, <math>r(\mathfrak{r}_1)</math>, the density profile is,
<div align="center">
<math>\rho(r(\mathfrak{r}_1))= \rho_c e^{v_1(\mathfrak{r}_1)}</math>;
</div>
: and the pressure profile is,
<div align="center">
<math>P(r(\mathfrak{r}_1))= (c_s^2 \rho_c) e^{v_1(\mathfrak{r}_1)}</math>.
</div>
: As has been explicitly pointed out in the above discussion associated with Figure 1a, the density profile &#8212; and, hence, also the pressure profile &#8212; extends to infinity and, at large radii, behaves as a power law; specifically, <math>\rho \propto r^{-2}</math>. 

* <font color="red">Mass</font>:  
: Given <math>c_s</math> and <math>\rho_c</math>, the natural mass scale is,
<div align="center">
<math>M_0 \equiv \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2}  </math> ;
</div>
: and, expressed in terms of <math>M_0</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>
M_r = M_0 [ - \mathfrak{r}_1^2 v_1' ] \, .
</math>
</div>
: As discussed above in the context of Figure 1b, at large radii, the mass increases linearly with <math>r</math>.  Because the density and pressure profiles extend to infinity, this means that the mass of an isolated isothermal sphere is infinite.

* <font color="red">Enthalpy &amp; Gravitational Potential</font>: 
: To within an additive constant, the enthalpy distribution is,
<div align="center">
<math>H(r(\mathfrak{r}_1))= c_s^2  [- v_1(\mathfrak{r}_1)]</math>;
</div>
: and the gravitational potential is,
<div align="center">
<math>\Phi(r(\mathfrak{r}_1)) = - H(r(\mathfrak{r}_1))= c_s^2  v_1(\mathfrak{r}_1)</math>.
</div>

* <font color="red">Mean-to-Local Density Ratio</font>:
: The ratio of the configuration's mean density, inside a given radius, to its local density at that radius is,
<div align="center">
<math>\frac{\bar{\rho}}{\rho} = \frac{3M_r}{4\pi r^3 \rho} = 3\biggl[- \frac{v_1'}{\mathfrak{r}_1 e^{v_1}} \biggr] </math> .
</div>
: As Figure 2 shows, at large <math>r</math> this density ratio goes to the value of 3, which means that the term inside the square brackets goes to unity at large <math>r</math>.  This behavior is consistent with the limiting power-law behavior of both <math>M_r</math> and <math>\rho</math>, discussed above.

<div align="center">
<table border="0" cellpadding="5" width="360">
<tr>
  <td align="center">
'''Figure 2:  From Emden's Tabulated Data'''
  </td>
</tr>
<tr>
  <td align="center">
[[File:PlotMeanToLocalDensity.png|350px|center|Plot based on data from Emden's (1907) Table 14]]
  </td>
</tr>
<tr>
  <td align="left">
The blue curve displays an evaluation of the density ratio, <math>[- 3v_1'/(\mathfrak{r}_1 e^{v_1}) ]</math>, as a function of <math>\ln (\mathfrak{r}_1)</math>, as determined from the data presented in Emden's Table 14, shown above.
  </td>
</tr>
</table>
</div>