Editing SSC/Structure/BiPolytropes (section)

==Background==
While [[SSC/Structure/Polytropes#Polytropic_Spheres|polytropic spheres]] can give us useful ''general'' insight into the internal structure of stars, they do not faithfully represent the detailed structure of most stars, in part, because the equation of state that is most relevant to the densest regions of a star usually is different from the equation of state that is relevant to the star's envelope.  As is highlighted in the following excerpt from [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C Cowling (1931)], {{ Milne30full }} was among the first to suggest that more realism could be achieved by constructing what is now commonly referred to as a bipolytrope: a composite model in which the envelope, obeying one equation of state &#8212; described by polytropic index <math>n_e</math> &#8212; is fitted to a core obeying a different equation of state &#8212; described by polytropic index <math>n_c</math>. 

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Text extracted<sup>&dagger;</sup> from p. 472 of [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C T. G. Cowling (1931)]<p></p>
"''Note on the Fitting of Polytropic Models in the Theory of Stellar Structure''"<p></p>
MNRAS, 91, 472-478 &copy; Royal Astronomical Society
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[[Image:CowlingRE_Milne1931.jpg|500px|center|Cowling (1931)]]
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<tr><td align="left"><sup>&dagger;</sup>Text displayed here, as a single digital image, with presentation order &amp; layout modified from the original publication.</td></tr>
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{{ Milne30 }} envisioned that the temperature, {{Math/VAR_Temperature01}}, and pressure, {{Math/VAR_Pressure01}}, should be "continuous across the surface of fit."  Given that {{Math/VAR_Temperature01}} and {{Math/VAR_Pressure01}} are continuous, he envisioned that the gas density, {{Math/VAR_Density01}}, should be continuous across the surface of fit as well.  {{ SC42full }} later pointed out &#8212; see the following journal article excerpt &#8212; that, if the core and envelope have different molecular weights, it is the ratio {{Math/VAR_Density01}}/{{Math/MP_MeanMolecularWeight}} that should be continuous across the interface.  A discontinuous jump in {{Math/MP_MeanMolecularWeight}} at the interface &#8212; for example, switching from a pure helium core to an envelope whose dominant element is hydrogen &#8212; will therefore lead to a discontinuous jump in {{Math/VAR_Density01}} at the interface. 

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Text excerpts<sup>&dagger;</sup> from &sect;2 (pp. 161-162) of <br />{{ SC42figure }}<br />&copy; American Astronomical Society
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[[Image:SchonbergChandra1942.jpg|600px|center|Sch&ouml;nberg &amp; Chandrasekhar (1942)]]
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{{ SC42 }} furthermore pointed out &#8212; again, see the relevant article excerpt &#8212; that, when constructing a bipolytropic model, the mathematical function  that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''core'', <math>M(r)|_\mathrm{c}</math>, must give the same value as the mathematical function  that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''envelope'', <math>M(r)|_\mathrm{e}</math> ''at the radial location of the interface'', <math>r_i</math>.