Editing SSC/Structure/BiPolytropes/Analytic15/Pt2 (section)

===Interface Mapping===

As has already been stated in our [[SSC/Structure/BiPolytropes/Analytic15#Steps_2_.26_3|above description of the <math>~n_c = 1</math> core]] of these bipolytropic configurations, the structure of the core will be physically viable as long as the radial location of the interface, <math>~\xi_i</math>, between the core and the envelope is positioned somewhere within the range,
<div align="center">
<math>~0 \le \xi_i \le \pi \, .</math>
</div>

Similarly, our [[SSC/Structure/Polytropes#Srivastava.27s_F-Type_Solution|discussion of Srivastava's Lane-Emden function]], which is being used to define the envelope of these bipolytropic configurations, makes it clear that the envelope will have a physically viable structure as long as the parameter, <math>~\Delta_i</math>, associated with the radial location of the interface is positioned somewhere within the range,
<div align="center">
<math>~\eta_\mathrm{crit}  < e^{2\Delta_i} < e^{2\pi} \, ,</math>
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where,
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<math>\eta_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} = 10.05836783\, ,</math>
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and, in keeping with the definition provided above,
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<math>\Delta_i = \tan^{-1}(y_\mathrm{root}) + m\pi \, .</math>
</div>
This coordinate range for the physically viable envelope can be rewritten as,
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<math>~\tan^{-1}(1+2^{1/3})  \le \Delta_i \le \pi \, .</math>
</div>

As is illustrated in the following figure, titled "Interface Mapping," our analytic solution defines a smooth, one-to-one mapping of the radial coordinate that defines the outer edge of the <math>~n_c=1</math> core, <math>~\xi_i</math>, to the parameter, <math>~\Delta_i</math>,  that defines the inner edge of the <math>~n_e = 5</math> envelope.  


<div align="center">
<table border="1" cellpadding="0" align="center">
<tr>
  <td align="center">
[[File:InterfaceMap02.png|center|500px|Illustration of Interface Mapping]]
  </td>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center"><math>~\mathrm{Model}</math></td>
<td align="center"><math>~\xi_i</math></td>
<td align="center"><math>~m</math></td>
<td align="center"><math>~\Delta_i</math></td>
</tr>
<tr>
<td align="center">
----</td>
<td align="center">
----</td>
<td align="center">
----</td>
<td align="center">
----</td>
</tr>
<tr>
<td align="center">&hellip;</td>
<td align="center"><math>~0</math></td>
<td align="center"><math>~0</math></td>
<td align="center"><math>~\tan^{-1}(1+2^{1/3})</math></td>
</tr>
<tr>
<td align="center"><math>~(2)</math></td>
<td align="center"><math>~0.8154</math></td>
<td align="center"><math>~0</math></td>
<td align="center"><math>~1.24287</math></td>
</tr>
<tr>
<td align="center"><math>~(3)</math></td>
<td align="center"><math>~1.6598</math></td>
<td align="center"><math>~0</math></td>
<td align="center"><math>~1.49179</math></td>
</tr>
<tr>
<td align="center"><math>~(4)</math></td>
<td align="center"><math>~2.0914</math></td>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1.73281</math></td>
</tr>
<tr>
<td align="center"><math>~(5)</math></td>
<td align="center"><math>~2.7302</math></td>
<td align="center"><math>~1</math></td>
<td align="center"><math>~2.87493</math></td>
</tr>
<tr>
<td align="center">&hellip;</td>
<td align="center"><math>~\pi</math></td>
<td align="center"><math>~1</math></td>
<td align="center"><math>~\pi</math></td>
</tr>
</table>
  </td>
</tr>
</table>
</div>



In the figure, the green curve traces the segment of the E-Type Lane-Emden function for the <math>~n_c = 1</math> polytropic core, <math>~\theta_{1E}</math> &#8212; that is, the segment of the "sinc" function &#8212; that covers the range <math>~[0,\pi]</math> for the interface parameter, <math>~\xi_i</math>;  the blue curve traces the segment of the F-Type Lane-Emden function for the <math>~n_e = 5</math> polytropic envelope, <math>~\phi_{5F}</math>, that covers this same range <math>~[0,\pi]</math>, but for the interface parameter, <math>~\Delta_i</math>.  (This blue curve is also displayed, and its properties described in more depth, in a [[SSC/Structure/Polytropes#Example_Interval|separate discussion devoted to Srivastava's Lane-Emden function]] for <math>~n=5</math> polytropes.)  This entire displayed parameter range, <math>~[0,\pi]</math>, is associated with physically viable interface values for the core (green curve), but for the envelope (blue curve), only a subset of this range can be associated with physically viable interface values &#8212; namely, the range <math>~[\tan^{-1}(1+2^{1/3}),\pi]</math> over which <math>~\phi_{5F}</math> is positive but its slope is either negative or zero.


The purple dashed line segment labeled "(2)" in the above "Interface Mapping" figure intersects the green curve at the value, <math>~\xi_i = 0.8154</math>, and it intersects the blue curve at the interface value, <math>~\Delta_i = 1.24287</math>.  [These two numerical values are listed  in the row labeled "Model (2)" of the table that has been placed immediately to the right of the figure.]   This is intended to illustrate that an equilibrium bipolytropic configuration with <math>~(n_c, n_e) = (1, 5)</math> can be constructed by truncating the <math>~n_c = 1</math> core at a radius, <math>~\xi_i = 0.8154</math>, and matching it to an <math>~n_e = 5</math> envelope whose innermost radius is associated with the parameter, <math>~\Delta_i = 1.24287</math>.  (The corresponding, analytically determined values of the coefficients <math>~A_0</math> and <math>~B_0</math> dictate the manner in which the blue curve should be stretched both horizontally and vertically to complete a smooth attachment of the envelope to the core.)  In an analogous fashion, the dashed line segments labeled "(3)", "(4)", and "(5)" &#8212; and their corresponding coordinate values as listed in the accompanying table &#8212; illustrate how three additional equilibrium bipolytropic configurations with progressively larger cores can be constructed with the proper interface mapping.  We have specifically chosen to illustrate the interface-mapping of these four bipolytropic models because their cores are truncated at the same values of <math>~\xi_i</math> that [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] used for the models numbered "(2)", "(3)", "(4)", and "(5)" in his Table 3.  (See the [[SSC/Structure/BiPolytropes/Analytic15#Murphy.27s_Example_Model_Characteristics|much more in-depth discussion, below]]; and note that the edge of the core is labeled by the parameter, <math>~\zeta_J</math>, rather than by <math>~\xi_i</math>, in Murphy's Table 3.)


For clarity we note that, the analytically determined values of <math>~\Delta_i</math> that are listed in the table that sits to the right of the above "Interface Mapping" figure have been shifted in phase by <math>~m\pi</math>, where the relevant value of the integer, <math>~m</math>, is also listed in the table.  In the figure, the two ''purple'' dashed-line segments are associated with models for which <math>~m=0</math>, while the two ''orange'' dashed-line segments are associated with models for which <math>~m=1</math>.


In his Table 3, [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] also tabulates the characteristics of two additional models:  The core of his "model 1" is truncated at a radius much less than <math>~\pi</math>, namely, <math>~\xi_i = 0.032678</math>; as a result, the bipolytropic configuration has a very small core and its structure is almost entirely that of an <math>~n = 5</math> polytrope.  At the other extreme, the core of his "model 6" is truncated at a radius that is almost, but not quite, equal to <math>~\pi</math>, namely, <math>~\xi_i = 3.1415</math>; the resulting bipolytropic configuration has a very tiny envelope and its structure is almost entirely an <math>~n = 1</math> polytrope.  Our analytic solution permits us to set <math>~\xi_i</math> to either of the two limiting values, <math>~0</math> or <math>~\pi</math>, and to show that, in these limits, <math>~\Delta_i</math> exactly equals, respectively, <math>~\tan^{-1}(1 + 2^{1/3})</math> and <math>~\pi</math>.  The mapping of the first of these two limits is illustrated by the red dashed line segment in the above figure; the second limit is illustrated simply by the intersection of the two curves at the coordinate location, <math>~\pi</math>.  It appears, therefore, that in these two limits Murphy's bipolytrope can be used, respectively, to define the structure of an isolated <math>~n=5</math> polytrope or an isolated <math>~n=1</math> polytrope.


'''<font color="red">ASIDE:</font>'''  It has previously been thought that Srivastava's Lane-Emden function, <math>~\phi_{5F}</math>, cannot be used on its own to define the structure of an ''isolated'' <math>~n=5</math> polytrope because the function's amplitude grows without bound and oscillates more and more rapidly between positive and negative values as the governing radial coordinate gets smaller and smaller.  From the results presented here, it now appears as though an isolated <math>~n=5</math> polytrope of this type ''can'' be constructed by letting the interface parameter <math>~\xi_i \rightarrow 0</math> &#8212; and, hence the alternative parameter <math>~\Delta_i \rightarrow \tan^{-1}(1+2^{1/3})</math> &#8212; in Murphy's bipolytrope.  It should be interesting to determine the values of the coefficients, <math>~A_0</math> and <math>~B_0</math>, that arise in this limit, and to examine in detail the structure of the complete <math>~n=5</math> model that results.  Does its radial density profile resemble &#8212; or, perhaps, exactly match &#8212; the radial density profile of the well-known ''isolated'' <math>~n=5</math> ploytrope?  Does the new structure have a finite radius as well as a finite central density?  In this context it is worth noting that, in two separate papers &#8212;  [http://adsabs.harvard.edu/abs/1980PASAu...4...37M Murphy (1980a)] and [http://adsabs.harvard.edu/abs/1981PASAu...4..205M Murphy (1981)] &#8212; Murphy has constructed and discussed the physical characteristics of equilibrium models that obey the <math>~n=5</math> polytropic equation of state all the way from the center to a surface which is of finite radius.  He accomplishes this by piecing together a core that is defined by the familiar, analytically specified, <math>~\phi_{5E}</math> Lane-Emden function and an envelope that is defined by Srivastava's <math>~\phi_{5F}</math> Lane-Emden function.