Editing SSC/Structure/BiPolytropes/Analytic51/Pt3 (section)

==Derivation by Eggleton, Faulkner, and Cannon (1998)==

The analytically prescribable sequence of bipolytropic models having <math>(n_c, n_e) = (5, 1)</math> displays an interesting behavior that extends beyond identification of a Sch&ouml;nberg-Chandrasekhar-like mass limit. After reaching a maximum value of <math>q</math> but before reaching the maximum value of <math>\nu</math>, the sequence bends back on itself.  This means that, even though the fraction of mass enclosed in the core is steadily increasing, the total radius of the configuration is increasing faster than the radius of the core.  Qualitatively, at least, this mimics the behavior exhibited by normal stars as they evolve off the main sequence and up the red giant branch.

As I pondered whether or not to probe this analogy in more depth, I recalled &#8212; even dating back to my years as a graduate student at Lick Observatory &#8212; hearing John Faulkner profess that he finally understood why stars become red giants.  I also recalled the following passage from  [<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>]'s textbook on ''Stellar Interiors'':
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Excerpt from &sect;2.3, p. 55 of [<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>]
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<font color="darkgreen">"The enormous increase in radius that accompanies hydrogen exhaustion moves the star into the red giant region of the H-R diagram.  While the transition to red giant dimensions is a fundamental result of all evolutionary calculations, a convincing yet intuitively satisfactory explanation of this dramatic transformation has not been formulated.  Our discussion of this phenomenon follows that of [http://adsabs.harvard.edu/abs/1984PhR...105..329I Iben and Renzini (1984)] although we must state that it is not the whole story."</font><sup>&dagger;</sup>
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<sup>&dagger;</sup><font color="darkgreen">"Other attempts include:  {{ EF81 }}; [http://adsabs.harvard.edu/abs/1983A%26A...127..411W Weiss (1983)]; [http://adsabs.harvard.edu/abs/1985ApJ...296..554Y Yahil &amp; Van den Horn (1985)]; [http://adsabs.harvard.edu/abs/1988ApJ...329..803A Applegate (1988)]; [http://adsabs.harvard.edu/abs/1989MNRAS.236..505W Whitworth (1989)]; [http://adsabs.harvard.edu/abs/1992ApJ...400..280R Renzini et al. (1992)]. [http://adsabs.harvard.edu/abs/1991ApJ...372..592B Bhaskar &amp; Nigam (1991)] use an interesting set of dimensional arguments plus notions from polytrope theory.  We suspect the answers may lie in their paper but someone has yet to come along and translate the mathematics into an easily comprehensible physical picture."</font>
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While examining the set of authors who more recently have cited the work by {{ EF81 }}, I discovered a paper by {{ EFC98full }} with the following abstract:

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[[Image:EagletonFaulknerCannon98.jpg|600px|center|Eggleton, Faulkner, &amp; Cannon (1998, MNRAS, 298, 831)]]
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<div align="center">{{ EFC98figure }}</div>
Abstract: &nbsp;<font color="darkgreen">"We present a simple analytic model of a composite polytropic star, which exhibits a limiting Sch&ouml;nberg-Chandrasekhar core mass fraction strongly analogous to the classic numerical result for an isothermal core, a radiative envelope and a &mu;-jump (i.e. a molecular weight jump) at the interface.  Our model consists of an n<sub>c</sub> = 5 core, an n<sub>e</sub> = 1 envelope and a &mu;-jump by a factor &ge; 3; the core mass fraction cannot exceed 2/&pi;.  We use the classic ''U, V'' plane to show that composite models will exhibit a Sch&ouml;nberg-Chandrasekhar limit only if the core is 'soft', i.e. has n<sub>c</sub> &ge; 5, and the envelope is 'hard', i.e. has n<sub>c</sub> < 5; in the critical case (n<sub>c</sub> = 5), the limit only exists if the &mu;-jump is sufficiently large, &ge; 6/(n<sub>e</sub> + 1)."</font>
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This paper uses analytic techniques to derive precisely the same sequence of <math>(n_c, n_e) = (5, 1)</math> bipolytropic models that we have presented above.