Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

==Sch&ouml;nberg-Chandrasekhar Mass==

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Figures from (left) [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich &amp; Chandraskhar (1941)]
and (center) [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)]
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Bipolytropes with <math>~(n_c,n_e) = (5,1)</math>
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[[Image:HenrichChandra41b.jpg|100px|center]]
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[[Image:SC42_Fig1.jpg|250px|center]]
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[[File:SC_42EvolutionArrows.png|300px|center]]
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Figures from (left) [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich &amp; Chandraskhar (1941)]
and (center) [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)]
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Bipolytropes with <math>~(n_c,n_e) = (5,1)</math>
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[[Image:HenrichChandra41b.jpg|100px|center]]
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[[Image:SC42_Fig1.jpg|200px|center]]
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[[File:Qvsnu51b2.png|300px|center]]
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In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich &amp; Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value.  They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index &#8212; say, <math>~n_c</math> &#8212; and the star's envelope is described by a polytropic equation of state of a different index &#8212; say, <math>~n_e</math>.  In [[SSC/Structure/BiPolytropes#BiPolytropes|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs.  As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well. 

[http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich &amp; Chandraskhar (1941)] built structures of uniform composition having an isothermal core (<math>~n_c = \infty</math>) and an <math>~n_e = 3/2</math> polytropic envelope and found that equilibrium models exist only for values of <math>~\nu \le \nu_\mathrm{max} \approx 0.35</math>.  [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] extended this analysis to include structures in which the mean molecular weight of the gas changes discontinuously across the interface.  Specifically, they used the same values of <math>~n_c</math> and <math>~n_e</math> as Henrich &amp; Chandrasekhar, but they constructed models in which the ratio of the molecular weight in the core to the molecular weight in the envelope is <math>~\mu_c/\mu_e = 2</math>. This was done to more realistically represent stars as they evolve off the main sequence; they have inert, isothermal helium cores and envelopes that are rich in hydrogen.  Note that introducing a discontinuous drop in the mean molecular weight at the core-envelope interface also introduces a discontinuous drop in the gas density across the interface.  As the following excerpt from p. 168 of their article summarizes, in these models, [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] found that <math>~\nu_\mathrm{max} \approx 0.101</math>.  This is commonly referred to as the Sch&ouml;nberg-Chandrasekhar mass limit, although it was Henrich &amp; Chandrasekhar who were the first to identify the instability.
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Text excerpt from [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)]
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In an effort to develop a more complete appreciation of the onset of the instability associated with the Sch&ouml;nberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Sch&ouml;nberg &amp; Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface.  [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, &amp; &#x017B;ytkow] (2012, MNRAS, 421, 2713)].  Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions. 

[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core.  For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:

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\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2] -
m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3] = 0 \, ,
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where,
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\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
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m_3 \equiv 3 \biggl( \frac{\mu_c}{\mu_e} \biggr)^{-1} \, ;
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\Lambda_i \equiv \frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2]  \, .
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