Editing SSC/Structure/LimitingMasses (section)

==Bounded Isothermal Sphere & Bonnor-Ebert Mass==
In the mid-1950s, {{ Ebert55 }} and {{ Bonnor56 }} independently realized that an isothermal gas cloud can be stabilized by embedding it in a hot, tenuous external medium.  The relevant mathematical model is constructed by chopping off the isothermal sphere at some finite radius &#8212; call it, <math>\xi_e</math> &#8212; and imposing an externally applied pressure, <math>~P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere.  But for a given mass and temperature, there is a value of <math>P_e</math> below which the truncated isothermal sphere is dynamically unstable, like its isolated and untruncated counterpart.  Viewed another way, given the value of <math>P_e</math> and the isothermal sound speed, <math>c_s</math>, a bounded isothermal sphere will be dynamically stable only if its mass is below a critical value, 
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'''<font color="red">Bonnor-Ebert Mass</font>'''
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<math>M_\mathrm{max} = \alpha \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2}</math>
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Coefficient <math>\alpha</math> for Pressure-Bounded Configurations
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<math>\alpha</math>
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Context
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Source
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<math>1.18</math>
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Bounded Isothermal Sphere
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(numerically derived)
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<font size="-2">Discovery Paper</font>
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[http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956)
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<font size="-2">(see also [[SSC/Structure/BonnorEbert#Maximum_Mass|here]])</font>
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<math>\biggl( \frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2}</math>
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Isothermal Virial Analysis
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(exact)
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[[SSCpt1/Virial#Bounded_Isothermal|Here]]
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<math>\biggl( \frac{1}{2} \biggr)^{3/10}\biggl( \frac{3^{7}}{2^{8}\pi} \biggr)^{1/2}</math>
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Bounded <math>~n=5</math> Polytrope
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(exact)
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[[SSC/Structure/PolytropesEmbedded#n_.3D_5_Polytrope|Here]]
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<math>\biggl( \frac{3^{19}}{2^{12}\cdot 5^7\pi} \biggr)^{1/2}</math>
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<math>~n=5</math> Virial Analysis
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(exact)
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[[SSCpt1/Virial#Bounded_Adiabatic|Here]]
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where <math>~\alpha</math> is a dimensionless coefficient of order unity.  This limiting mass is often referred to as the Bonnor-Ebert mass.  It appears most frequently in the astrophysics literature in discussions of star formation because that is the arena in which both Bonnor and Ebert were conducting research when they made their discoveries.  

As is [[SSC/Structure/BonnorEbert#Maximum_Mass|reviewed in a related discussion]] and as is documented in the table accompanying the expression for <math>~M_\mathrm{max}</math>, above, {{ Bonnor56 }} used the [[SSC/Structure/IsothermalSphere#Emden's_Numerical_Solution|tabulated properties of an isothermal sphere]] that were published by {{ Emden07full }}  to determine that the dimensionless radius of this limiting configuration is <math>\xi_e \approx 6.5</math> and that the leading coefficient, <math>\alpha \approx 1.18</math>.  It is worth noting that a [[SSCpt1/Virial#BonnorEbertMass|global virial analysis of the stability of bounded isothermal spheres]] produces the same expression for <math>M_\mathrm{max}</math> with a leading coefficient that has an exact, analytic prescription, namely, <math>\alpha = (3^4 \cdot 5^3/2^{10}\pi)^{1/2} \approx 1.77408</math>.  While it can be advantageous to reference this analytic prescription of <math>\alpha</math>, the virial analysis must be considered more approximate than Bonnor's analysis because it does not require the construction of models that are in detailed force balance.  

Our [[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|detailed force-balance analysis of truncated and pressure-bounded, <math>n=5</math> polytropes]] identifies a physically analogous limiting mass.  If the average isothermal sound speed, <math>~\bar{c_s}</math>, [[SSCpt1/Virial#average|as defined elsewhere]], is used in place of <math>c_s</math>, the mathematical expression for <math>~M_\mathrm{max}</math> has exactly the same form as in the isothermal case.  But for the <math>~n=5</math> polytrope we know that the limiting configuration has a dimensionless radius given precisely by <math>\xi_e = 3</math>; and, as a result, the leading coefficient in the definition of <math>M_\mathrm{max}</math> is prescribable analytically, namely, <math>\alpha = 2^{-3/10} \cdot (3^7/2^8 \pi)^{1/2} \approx 1.33943</math>.  As is documented in the table accompanying the relation for <math>~M_\mathrm{max}</math>, above, in the case of a truncated <math>~n=5</math> polytrope, [[SSCpt1/Virial#Maximum_Mass|the simpler and more approximate virial analysis gives]], <math>\alpha = (3^{19}/2^{12}\cdot 5^7\pi)^{1/2} \approx 1.07523</math>.