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

==Analytic Solution of Key Interface Relation==

Returning to our [[SSC/Structure/BiPolytropes/Analytic15#KeyInterfaceRelation|previously derived]],
<div align="center" id="KeyInterfaceRelation">

<font color="#770000">'''Key Nonlinear Interface Relation'''</font>
<br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, ,</math>
  </td>
</tr>
</table>

</div>

and, as in our [[SSC/Structure/Polytropes#Example_Interval|separate discussion of the properties of Srivastava's function]], adopting the shorthand notation,
<div align="center">
<math>~y_i \equiv \tan\Delta_i \, ,</math>
</div>

this key interface condition becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\kappa_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3 - 2y_i^2(1+y_i^2)^{-1}- 3y_i^{-1}  }{3-2y_i^2(1+y_i^2)^{-1}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3y_i(1+y_i^2)- 2y_i^3 -3(1+y_i^2)}{3y_i(1+y_i^2)-2y_i^3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{y_i^3 -3y_i^2 + 3y_i -3  }{3y_i+y_i^3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ \kappa_i(3y_i+y_i^3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_i^3 -3y_i^2 + 3y_i -3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~  y_i^3(1-\kappa_i) -3 y_i^2 + 3(1-\kappa_i)y_i -3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>
</div>

This is a cubic equation whose solution, <math>~y_\mathrm{root}(\kappa_i)</math>, will also immediately supply the desired interface angle, <math>~\Delta_i</math>, and an interface coordinate root, <math>~(A_0\eta)_\mathrm{root}</math>.  

<div align="center">
<table border="1" cellpadding="8" width="80%">
<tr>
  <td align="center">
<font color="red">'''ASIDE:'''</font>  Analytic Solution of Cubic Equation
  </td>
</tr>
<tr>
  <td align="left">
As is well known and documented &#8212; see, for example [http://mathworld.wolfram.com/CubicFormula.html Wolfram MathWorld] or [http://en.wikipedia.org/wiki/Cubic_function Wikipedia's discussion] of the topic &#8212; the roots of any cubic equation can be determined analytically.  In order to evaluate the root(s) of our particular cubic equation, we have drawn from the utilitarian [http://www.math.vanderbilt.edu/~schectex/courses/cubic/ online summary provided by Eric Schechter at Vanderbilt University].  For a cubic equation of the general form,
<div align="center">
<math>~ay^3 + by^2 + cy + d = 0 \, ,</math>
</div> 
a real root is given by the expression,
<div align="center">
<math>~
y = p + \{q + [q^2 + (r-p^2)^3]^{1/2}\}^{1/3} + \{q - [q^2 + (r-p^2)^3]^{1/2}\}^{1/3}
\, ,</math>
</div> 
where,
<div align="center">
<math>~p \equiv -\frac{b}{3a} \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~q \equiv \biggl[p^3 + \frac{bc-3ad}{6a^2} \biggr] \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~r=\frac{c}{3a} \, .</math>
</div>
In our particular case,
<div align="center">
<math>~a =(1-\kappa_i)\, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~b =-3\, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~c = 3(1-\kappa_i) \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~d = - 3 \, .</math>
</div>
Hence,
<div align="center">
<math>~p = \frac{1}{(1-\kappa_i)} \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~r=+1 \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~q = p^3 = \frac{1}{(1-\kappa_i)^3} \, ,</math>
</div>
which implies that the real root, <math>~y_\mathrm{root}</math>, is given by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_\mathrm{root}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p + \{p^3 + [p^6 + (1-p^2)^3]^{1/2}\}^{1/3} + \{p^3 - [p^6 + (1-p^2)^3]^{1/2}\}^{1/3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p + \{p^3 + p^3[1 + (p^{-2}-1)^3]^{1/2}\}^{1/3} + \{p^3 - p^3[1 + (p^{-2}-1)^3]^{1/2}\}^{1/3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ p^{-1} y_\mathrm{root}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \{1 + [1 + (p^{-2}-1)^3]^{1/2}\}^{1/3} + \{1 - [1 + (p^{-2}-1)^3]^{1/2}\}^{1/3}
\, .
</math>
  </td>
</tr>
</table>
</div>

(There is also a pair of imaginary roots, but they are irrelevant in the context of our overarching astrophysical discussion.)
  </td>
</tr>
</table>
</div>

In summary, then, 
* Once the location, <math>~\xi_i</math>, of the outer edge of the core has been specified, which determines <math>~\theta_i</math> and <math>~\theta^'_i</math> as well, the value of the parameters, <math>~\kappa_i</math> and <math>~p</math>, are known via the expressions,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa_i \equiv - \frac{2\theta_i^' \xi_i}{3\theta_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{3} \biggl(1 - \xi_i \cot\xi_i  \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p \equiv (1-\kappa_i)^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{3-2(\mu_e/\mu_c)(1-\xi_i \cot\xi_i)}  \, .</math>
  </td>
</tr>
</table>
</div>

* The value of <math>~y_\mathrm{root}</math> is determined from the just-derived solution to the governing cubic equation, which then gives the interface value of the envelope parameter (adjusted by a <math>~\pi m</math> phase angle, where <math>~m</math> is an, as yet unspecified, integer),
<div align="center">
<math>~\Delta_i = \tan^{-1}(y_\mathrm{root}) +\pi m \, .</math>
</div>
* This, in turn, can be interpreted as a coordinate root &#8212; which we shall refer to as <math>~\eta_\mathrm{root}</math> &#8212; via the expression,
<div align="center">
<math>~(A_0\eta)_\mathrm{root} = e^{2\Delta_i} = e^{2\pi m} \cdot e^{2\tan^{-1}(y_\mathrm{root})}  \, .</math>
</div>


<div align="center" id="Caution">
<table border="1" cellpadding="8" width="80%">
<tr>
  <td align="center">
<font color="red">'''CAUTION:'''</font>  Solution Behavior When <math>~\kappa_i = 1</math> and <math>~\Delta_i = \pi/2 \, .</math>
  </td>
</tr>
<tr>
  <td align="left">
In building a ''sequence'' of bipolytropic configurations having <math>~(n_c, n_e) = (1, 5)</math>, it will make sense to steadily increase the value of the parameter that marks the edge of the core, <math>~\xi_i</math>, from zero &#8212; meaning no mass in the core &#8212; to its maximum allowed value, <math>~\pi</math> &#8212; meaning no mass in the envelope.  (A more complete discussion of physically viable parameter values is [[SSC/Structure/BiPolytropes/Analytic15#Physically_Viable_Parameter_Values|presented below]].)  As <math>~\xi_i</math> steadily increases from zero, for a while the parameter, <math>~\kappa_i</math>, will steadily increase from zero as well. As the value of <math>~\kappa_i</math> crosses through the value of "one," the associated parameter, <math>~p = (1-\kappa_i)^{-1}</math>, as well as the root of the cubic equation, <math>~y_\mathrm{root}</math>, will flip from positive infinity to negative infinity.  From the standpoint of the construction of physically realistic models, this does not pose a problem.  It simply reflects the fact that, when <math>~\kappa_i = 1</math>, the governing interface angle, <math>~\Delta_i = \pi/2</math> &#8212; or, allowing for the aforementioned phase shift, <math>~\Delta_i = \pi/2 + m\pi</math>.  


From a practical standpoint, however, it is useful to keep in mind that the root of the governing cubic equation will change abruptly from <math>~+ \infty</math> to <math>~- \infty</math> and its value will be quite sensitive to the choice of <math>~\xi_i</math> in the vicinity of <math>~\kappa_i = 1</math>.  Given the definition of the function, <math>~\kappa_i(\xi_i)</math>, this means that the abrupt transition will occur at an interface location, <math>~\xi_\mathrm{trans}</math>, whose value satisfies the condition,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\xi_\mathrm{trans}}{\tan(\xi_\mathrm{trans})}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1-\frac{3}{2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \, .</math>
  </td>
</tr>

</table>
</div>

For the case, <math>~\mu_e/\mu_c = 1 \, ,</math> we have determined empirically (to a dozen significant digits), that,
<div align="center">
<math>~\xi_\mathrm{trans} = 1.836597203152 \, .</math>
</div>

Just for fun &#8212; and, again, for the case, <math>~\mu_e/\mu_c = 1 </math> &#8212;  we also have determined that <math>~\xi_\mathrm{trans}</math> satisfies the series,
<div align="center">
<math>~
0 = \sum_{n=0}^{\infty} (-1)^{n+1} \biggl(\frac{4n+3}{2n+1} \biggr) \frac{\xi_\mathrm{trans}^{2n}}{(2n)!} 
\, .</math>
</div>

The black, vertical, long-dashed line in the following figure identifies where <math>~\xi_i = \xi_\mathrm{trans}</math>.  


Because the function <math>~\tan(\xi_\mathrm{trans})</math> is periodic, other values of <math>~\xi_\mathrm{trans}</math> will also satisfy the condition,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\xi_\mathrm{trans}}{\tan(\xi_\mathrm{trans})}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1-\frac{3}{2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \, .</math>
  </td>
</tr>
</table>
</div>
But the one whose value we have determined here is the only one that falls within the physically viable range of interface location values, <math>~0 \le \xi_i \le \pi </math>.

  </td>
</tr>
<tr>
  <td align="left">
<font color="red">NOTE as well:</font> &nbsp; A very similar expression arises in the [[Appendix/Ramblings/51BiPolytropeStability/RethinkEnvelope#Setup|accompanying discussion where we ''Rethink Handling of n = 1 Envelope'']].  Specifically,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\tan\Delta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Delta + B \, .</math>
  </td>
</tr>
</table>
I'm not sure whether this is relevant information or not!
  </td>
</tr>

</table>
</div>


<span id="MurphyF2Function">The following log-log plot shows what value(s) of <math>~(A_0\eta)_\mathrm{root} </math> (vertical axis) result from a wide range of interface coordinate specifications, <math>~\xi_i</math> (horizontal axis), under the simplifying assumption that <math>~(\mu_e/\mu_c) = 1</math>:  Solid blue diamonds identify roots resulting from setting <math>~m=0</math> in the phase-shift specification, solid green triangles identify roots for which <math>~m=-1</math>, and solid purple squares identify roots for which <math>~m=-2</math>.  </span>  As has just been discussed in the context of a cautionary note, as <math>~\xi_i</math> is increased from zero for any given value of <math>~m</math> &#8212; that is, as one traverses the plot, moving from the lefthand edge toward the right along a curve of ''constant color'' &#8212; the solution, <math>~(A_0\eta)_\mathrm{root} </math>, makes an abrupt jump when <math>~\xi_i</math> crosses the value, <math>~\xi_\mathrm{trans}</math>.   This identifies the root of the cubic equation for which <math>~\Delta_i = \pi/2 + m\pi</math>.  One can continue to move in a smooth, continuous fashion along a single &#8212; but multi-colored &#8212; solution curve by letting <math>~m \rightarrow (m+1)</math> as <math>~\xi_i</math> crosses <math>~\xi_\mathrm{trans}</math>.


<div align="center">
<table border="1" cellpadding="5">
<tr>
  <td align="center" colspan="2">
Examination of F2 Function Discussed by  
[http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)]
  </td>
</tr>
<tr>
  <td align="center">
[[File:MurphyF2roots05.png|center|500px|F2 Roots by Murphy (1983)]]
  </td>
  <td align="center">
<table border="0" cellpadding="0">
<tr>
  <td align="center">
Murphy's<p></p>
Roots of<p></p>
F2 = 0<p></p>
Function<p></p>
----
  </td>
</tr>
<tr>
  <td align="left">
<font face="Courier">3.5076E-05</font>
  </td>
</tr>
<tr>
  <td align="left">
<font face="Courier">1.8785E-02</font>
  </td>
</tr>
<tr>
  <td align="left">
<font face="Courier">4.2993E+00</font>
  </td>
</tr>
<tr>
  <td align="left">
<font face="Courier">7.6882E+00</font>
  </td>
</tr>
<tr>
  <td align="left">
<font face="Courier">1.0913E+01</font>
  </td>
</tr>
<tr>
  <td align="left">
<font face="Courier">1.4101E+01</font>
  </td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

The red line in the figure shows where the coordinate value along the vertical axis equals the coordinate value along the horizontal axis.  Hence the points where this red line intersects the other curves in the figure can be interpreted as identifying solutions to the cubic equation for which,
<div align="center">
<math>~(A_0\eta)_\mathrm{root} = \xi_i \, .</math>
</div>

The points of intersection that are highlighted in this figure are also what [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] refers to as roots of his <math>~F2 = 0</math> function.  The values of ten successive coordinate roots of this function are listed in the right-hand column of Table 1 in [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)].  We have re-listed the values of six of these roots in a column immediately to the right of the above figure and have marked with orange circles the locations of these six points in the figure.  As they should, Murphy's identified roots lie precisely at the intersection of the red line with the other curves.