Editing SSC/Structure/IsothermalSphere (section)

==Mass Profile==
The mass enclosed within a given radius, <math>~M_r</math>, can be determined by performing an appropriate volume-weighted integral over the density distribution.  Specifically, based on the key expression for,

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<span id="HydrostaticBalance"><font color="#770000">'''Mass Conservation'''</font></span><br />

{{Math/EQ_SSmassConservation01}}
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in spherically symmetric configurations, the relevant integral is,
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<math>
~M_r = \int_0^r 4\pi r^2 \rho(r) dr \, .
</math>
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But <math>~M_r</math> also can be determined from the information provided in column 7 of Emden's Table 14 &#8212; that is, from knowledge of the first derivative of <math>~v_1</math>.  The appropriate expression can be obtained from the mathematical prescription for

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<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

{{Math/EQ_SShydrostaticBalance01}}
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in a spherically symmetric configuration.  Since, for an isothermal equation of state (see above),

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<math>
~\frac{dP}{\rho} = c_s^2 {d\ln\rho} \, ,
</math>
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the statement of hydrostatic balance can be rewritten as,

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<math>
~M_r = \frac{c_s^2}{G} \biggl[ - r^2 \frac{d\ln\rho}{dr} \biggr] = \frac{c_s^2}{G \rho_c^{1/2} \beta} \biggl[ - \mathfrak{r}_1^2 \frac{dv_1}{d\mathfrak{r}_1} \biggr]
= \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2}  \biggl[ - \mathfrak{r}_1^2 v_1' \biggr] \, .
</math>
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The quantity tabulated in column 7 of Emden's Table 14 is precisely the dimensionless term inside the square brackets of this last expression; having units of mass, the coefficient out front sets the mass scale of the equilibrium configuration and depends only on the choice of central density and isothermal sound speed.  Hence, a plot of <math>~\ln(\mathfrak{r}_1^2 v_1')</math> versus <math>~\ln\mathfrak{r}_1</math>, as shown above in Figure 1b, translates into a log-log plot of the equilibrium configuration's <math>~M_r</math> mass profile.  Notice that, along with the radius, the mass of this isolated isothermal configuration extends to infinity and that, at large radii, the mass profile displays a simple power-law behavior &#8212; specifically, <math> ~M_r \propto r^{+1}</math>. 

As was realized independently by {{ Ebert55full }} and {{ Bonnor56full }}, a spherically symmetric isothermal equilibrium configuration of finite radius and finite mass can be constructed if the system is embedded in a pressure-confining external medium.  We discuss their findings [[SSC/Structure/BonnorEbert|elsewhere]].

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