Editing SSC/Structure/BonnorEbert (section)

==Limiting Pressure and Maximum Mass==
No matter how you look at the bounded isothermal sphere sequence &#8212; whether plotted as a curve in Bonnor's P-V diagram or as a curve in Whitworth's R-P diagram &#8212; it is clear that, for a given choice of the sound speed and the mass, there is a value of the pressure above which no equilibrium configuration exists.  (Alternatively, for a given sound speed and external pressure, there is a limiting mass above which no equilibrium configuration exists; see below.)  The configuration identifying this limiting pressure resides at the position along Bonnor's P-V diagram sequence where <math>~dP_e/dV</math> &#8212; or, in Whitworth's discussion, <math>~dP_e/dR</math> &#8212; first goes to zero.

===Following Bonnor's Presentation===
As is shown in his equation (3.1), [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] determined that, at a fixed mass and sound speed, <math>~dP/dV</math> goes to zero when,
<div align="center">
<math>~1-\frac{1}{2} e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2 = 0 \, .</math>
</div>
In the following figure, the function defined by the left-hand-side of this expression is plotted versus <math>~\ln\xi</math> using [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) tabulated data]].  As Bonnor noted, the function first crosses zero when <math>~\xi \approx 6.5</math> <math>~(\ln\xi = 1.87)</math>.

<div align="center">
<table border="1">
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  <td align="center">
[[File:Bonnor1956DPDVplot.jpg|400px|center|Bonnor (1956, MNRAS, 116, 351)]]
  </td>
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</table>
</div>

An isothermal sphere that is truncated at this location will have a radius,
<div align="center">
<math>
R = \biggl(  \frac{c_s^2}{G\rho_c} \biggr)^{1/2} \frac{6.5}{\sqrt{4\pi}} = 1.83 \biggl(  \frac{c_s^2}{G\rho_c} \biggr)^{1/2} \, ,
</math>
</div>
which matches equation (3.3) of [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)]; and, drawing on function values provided in [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) Table 14]], it will have a total mass,  
<div align="center">
<math>~M_R = \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[\xi^2 \frac{d\psi}{d\xi} \biggr]_e \approx 
\biggl( \frac{c_s^6}{G^3 \rho_c} \biggr)^{1/2} \biggl[ \frac{(14.353+17.214)/2}{\sqrt{4\pi}} \biggr] = 
4.45 \biggl( \frac{c_s^6}{G^3 \rho_c} \biggr)^{1/2}\, .</math>
</div>
Dividing this expression for <math>~M_R</math> by the expression for <math>~R</math> gives,
<div align="center">
<math>
\frac{M_R}{R} = 2.43 \biggl(  \frac{c_s^2}{G} \biggr)  \, ,
</math>
</div>
which matches equation (3.6) of [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)].  In order to maintain an equilibrium structure while truncating the isothermal model at this radius requires applying an external pressure of the following magnitude:
<div align="center">
<math>P_e = c_s^2 \rho_c e^{-\psi} \approx c_s^2 \rho_c  (0.08493 + 0.05833)/2 = 0.0716~ c_s^2 \rho_c \, ,</math>
</div>
where, again, numerical values have been drawn from [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) Table 14]].

<span id="cychenMaryland">
Finally, using this relation to eliminate <math>~\rho_c</math> from the expression for <math>~M_R</math> gives,</span>
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<table border="0" cellpadding="5">
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<math>~M_R</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>\biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2}\biggl[ \frac{1}{4\pi}\biggr]^{1/2} \biggl[\xi^2  e^{-\psi/2} \biggl(\frac{d\psi}{d\xi} \biggr) \biggr]_e</math>
  </td>
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<tr>
  <td align="center" colspan="3"><font color="red">On 3/2/2026, we changed the exponential from (incorrect) <math>e^{+\psi/2}</math> to (correct) <math>e^{-\psi/2}</math>.</font></td>
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&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
\biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \frac{1}{2\pi^{1/2}} \biggl\{ \frac{1}{2} \biggl[ 14.353 (0.08493)^{1/2}  + 
17.214 (0.05833)^{1/2}  \biggr] \biggr\}
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center"><math>\approx</math></td>
  <td align="left">
<math>
1.18 \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>

This is the mass upper limit for a stable, pressure-bounded isothermal sphere &#8212; the so-called Bonnor-Ebert mass; see, for example, equation (1) of [http://adsabs.harvard.edu/abs/1977ApJ...214..488S Shu (1977)].  
[[SSC/Structure/LimitingMasses#Bounded_Isothermal_Sphere_.26_Bonnor-Ebert_Mass|In a separate discussion]], we compare this result to the determinations of other related mass upper-limits.

===Ebert's Corresponding Presentation===
The expression derived by Bonnor for the ratio <math>~M_R/R</math> in the limiting configuration can be inverted to give,
<div align="center">
<math>
~R \approx 0.41 \biggl(  \frac{G M_R}{c_s^2} \biggr)  \, .
</math>
</div>
This matches the expression for the critical radius, <math>~R_\mathrm{kr}</math>, that appears as equation (23) in [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert's (1955) published derivation].  Similarly, the above relation that expresses the Bonnor-Ebert limiting mass in terms of <math>~c_s</math>, <math>~G</math>, and <math>~P_e</math> can be inverted to give,
<div align="center">
<math>
~P_e \approx 4\pi (1.18)^2 \biggl( \frac{c_s^8}{4\pi G^3 M_R^2} \biggr) = 17.5 \biggl( \frac{c_s^8}{4\pi G^3 M_R^2} \biggr) \, ,
</math>
</div>
which agrees with the limiting pressure that was derived by [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert (1955)] and that is also presented in his equation (23).