Editing SSC/Structure/BiPolytropes/Analytic51/Pt2 (section)

===Model Sequences===

For a given choice of <math>\mu_e/\mu_c</math> a physically relevant sequence of models can be constructed by steadily increasing the value of <math>\xi_i</math> from zero to infinity &#8212; or at least to some value, <math>\xi_i \gg 1</math>.  Figure 1 shows how the fractional core mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, varies with the fractional core radius, <math>q \equiv r_\mathrm{core}/R</math>, along sequences having six different values of <math>\mu_e/\mu_c</math>, as detailed in the figure caption.  The natural expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size &#8212; both the radius and the mass &#8212; of the core. This expectation is realized along the sequences marked by blue diamonds (<math>\mu_e/\mu_c = 1</math>) and by red squares (<math>\mu_e/\mu_c = </math>&frac12;).  But the behavior is different along the other four illustrated sequences.  For sufficiently large <math>\xi_i</math>, the relative radius of the core begins to decrease; then, as <math>\xi_i</math> is pushed to even larger values, eventually the relative core mass begins to decrease.  Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence|an accompanying chapter]].

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[[Image:PlotSequencesBest02.png|500px|center]]
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'''Figure 1:'''  Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), &frac12; (red squares), 0.345 (dark purple crosses), &#8531; (pink triangles), 0.309 (light green dashes),  and &frac14; (purple asterisks).  Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values:  0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
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The variation of <math>\nu</math> with <math>q</math> for a seventh analytically determined model sequence &#8212; one for which <math>\mu_e/\mu_c = 1/5</math> &#8212; is mapped out by a string of blue diamond symbols in the left-hand side of Figure 2. It behaves in an analogous fashion to the <math>\mu_e/\mu_c = </math>&frac14; (purple asterisks) sequence displayed in Figure 1.  It also quantitatively, as well as qualitatively, resembles the sequence that was numerically constructed by {{ SC42full }} for models with an isothermal core (<math>n_c = \infty</math>) and an <math>n_e=3/2</math> envelope; Fig. 1 from their paper has been reproduced here on the right-hand side of Figure 2.

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Figure 2: Relationship to Sch&ouml;nberg-Chandrasekhar Mass Limit
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Analytic BiPolytrope with <math>n_c=5</math>, <math>n_e = 1</math>, and <math>\mu_e/\mu_c = 1/5</math>
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Edited excerpt from [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)]
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Figure from [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich &amp; Chandraskhar (1941)]
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[[Image:SC_42Comparison.jpg|400px|center]]
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[[Image:SC42_Fig1.jpg|400px|center]]
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[[Image:HenrichChandra41b.jpg|200px|center]]
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(''Above'') Plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for the analytic bipolytrope having <math>\mu_e/\mu_c = 1/5</math>.  The behavior of this analytically defined model sequence resembles the behavior of the numerically constructed isothermal core models presented by (''center'') {{ SC42 }} and by (''far right'') {{ HC41full }}.
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