Editing SSC/Structure/LimitingMasses (section)

==Relationship Between the Bonnor-Ebert and Sch&ouml;nberg-Chandrasekhar Critical Masses==
[[SSC/Structure/BiPolytropes/Analytic51#Limit_when_m3_=_0|As we have shown elsewhere]], in the limit <math>m_3 \rightarrow 0</math>, the physically relevant root of the above analytic relation is <math>\xi_i = 3</math> and the mass contained in the core is, 
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<math>
M_\mathrm{core} = \biggl( \frac{1}{2} \biggr)^{3/10}\biggl( \frac{3^{7}}{2^{8}\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, .</math>
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While the pressure at the base of the envelope in this model is satisfactorily confining the core to a finite radius, the condition <math>m_3 \rightarrow 0</math> implies that the mass-density at the base of &#8212; indeed, throughout &#8212; the envelope is zero.  [[SSC/Structure/BiPolytropes#The_Tohline_Generalization|As we have pointed out elsewhere]], this can be achieved by introducing a temperature discontinuity rather than a molecular weight discontinuity across the interface.  Hence, there is no mass in the envelope and the resulting model is, effectively, a truncated <math>~n=5</math> polytrope embedded in a hot, tenuous external medium.  It therefore should not come as a surprise that this critical Sch&ouml;nberg-Chandrasekhar-type configuration has a mass that is precisely equal to the Bonnor-Ebert limiting mass, as defined by a "bounded <math>~n=5</math> polytrope."  (See the second row from the bottom in the above table.)

It should be clear from this analysis that the Bonnor-Ebert critical mass is not distinct from the Sch&ouml;nberg-Chandrasekhar critical mass.  It can be derived from the Sch&ouml;nberg-Chandrasekhar mass in the limit when <math>\mu_e/\mu_c \rightarrow 0</math>.  Had {{ SC42 }} examined their model in this limit, they would have "discovered" the Bonnor-Ebert mass a decade prior to both Bonnor's and Ebert's published works.

{{ SGFworkInProgress }}

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<font color="darkblue">Equilibrium Mass-Radius Diagrams </font> 
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<!-- [[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]] -->
<!-- [[Image:AAAwaiting01.png|300px|center|Stahler (1983) Title Page]] -->
Figure 17 from <br />{{ Stahler83figure }}
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Virial-Determined Mass-Radius Sequences for Embedded Polytropes, as detailed in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation|accompanying discussion]]
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<div align="center">Detailed Force-Balanced Determination of Mass-Radius Sequences for Embedded Polytropes</div>
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''Top:''&nbsp; Sequences with n = 1, 2.5, 3, 3.05, 3.5, 5, 6. <br />
''Bottom:'' &nbsp; Same as ''top'' without n = 1, 2.5 while adding isothermal (n = &infin;).
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Equilibrium Sequences for <br /><math>(n_c, n_e) = (5, 1)</math> Bipolytropes<br />having various values of <math>\mu_e/\mu_c</math>
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[[File:Stahler_MRdiagram1.png|250px|center|Stahler (1983) Figure 17]]
<!-- [[Image:AAAwaiting01.png|300px|center|Stahler (1983) Figure 17 (edited)]] -->
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[[File:VirialDeterminedMRsequencesLabeled.png|300px|center]]
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<!-- [[File:Stahler1983Comparison.png|250px|center|To be compared with Stahler (1983)]] -->
[[File:DFBsequenceB.png|250px|center]]
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<!-- [[Image:PlotSequencesBest02.png|250px|center]] -->
[[File:TurningPoints51Bipolytropes.png|250px|center]]
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&nbsp;
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&nbsp;
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<!-- [[Image:MassRadiusVirialLabeled.png|250px|center]] -->
[[Image:MassVsRadiusCombined02.png|250px|center]]
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&nbsp;
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