Editing SSC/Stability/BiPolytrope00Details (section)

===Solve Cubic Equation===

Here we draw from a [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution|separate discussion of solutions to a cubic equation]].

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Using <math>~y</math> in place of <math>~c_0</math>, this "derivative matching" relation can be written in the form of a standard cubic equation.  Specifically,
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<math>~a y^3 + b y^2 + c y + d</math>
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<math>~=</math>
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<math>~
0 \, ,
</math>
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where,
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<math>~a</math>
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<math>~\equiv</math>
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<math>~
4 + 4\Chi(\Chi - 2)\, ,
</math>
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<math>~b</math>
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<math>~\equiv</math>
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<math>~
( 26 - 4Q_{21}) + 4\Chi[ 13\Chi  -14  - Q_{21}  (\Chi-2) ] +  22\Chi (\Chi  - 2)\, ,
</math>
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<math>~c</math>
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<math>~\equiv</math>
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<math>~
 [40  - 26 Q_{21}]+ 4\Chi[42\Chi  - Q_{21} (7\Chi-8) -24]  + 22\Chi[13\Chi  -14  - Q_{21}  (\Chi-2) ] \, ,
</math>
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<math>~d</math>
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<math>~\equiv</math>
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<math>~
- 40 Q_{21}+ 22\Chi[42\Chi  -Q_{21} (7\Chi-8) -24] \, .
</math>
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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,
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<math>~ay^3 + by^2 + cy + d = 0 \, ,</math>
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a real root is given by the expression,
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<math>~
y = p + \{z + [z^2 + (r-p^2)^3]^{1/2}\}^{1/3} + \{z - [z^2 + (r-p^2)^3]^{1/2}\}^{1/3}
\, ,</math>
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where,
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<math>~p \equiv -\frac{b}{3a} \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~z \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>
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(There is also a pair of imaginary roots, but they are irrelevant in the context of our overarching astrophysical discussion.)

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Upon evaluation, we have found that the expression inside of the square root is negative over the region of parameter space that is of most physical interest.  Hence, we need to call upon [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution|a separate discussion]] in which the cube root of complex numbers was discussed.


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We'll shift to Wolfram's notation; specifically,
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<math>~a_0 </math>
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<math>~=</math>
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<math>~\frac{d}{a}\, ,</math>
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<math>~a_1 </math>
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<math>~=</math>
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<math>~\frac{c}{a}\, ,</math>
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<math>~a_2 </math>
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<math>~=</math>
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<math>~\frac{b}{a}\, ,</math>
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<math>~R </math>
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<math>~\equiv</math>
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<math>~\frac{3^2 a_2 a_1 - 3^3a_0 - 2a_2^3}{2\cdot 3^3}</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{bc - 3ad }{6 a^2} - \biggl( \frac{ b}{3 a} \biggr)^3 = z \, ,</math>
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<math>~Q </math>
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<math>~\equiv</math>
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<math>~\frac{3 a_1 - a_2^2}{3^2}</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{ c }{3a} - \biggl(\frac{b}{3a}\biggr)^2 = r-p^2 \, .</math>
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Then, after defining,
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<math>~D \equiv Q^3 + R^2 = z^2 + (r-p^2)^3 \, ;</math>
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~S^3 \equiv R+ \sqrt{D} \, ;</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~T^3 \equiv R- \sqrt{D} \, ;</math>
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Wolfram states that the three roots of the cubic equation are (the first one being identical to the "real" root identified above),
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<math>~y_1</math>
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<math>~=</math>
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<math>~p + (S + T) \, ,</math>
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<math>~y_2</math>
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<math>~=</math>
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<math>~p - \frac{1}{2}(S + T) + \frac{1}{2} i \sqrt{3}(S-T) \, ,</math>
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<math>~y_2</math>
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<math>~=</math>
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<math>~p - \frac{1}{2}(S + T) - \frac{1}{2} i \sqrt{3}(S-T) \, .</math>
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Now, whenever <math>~D</math> is intrinsically negative, we need to treat both <math>~S^3</math> and <math>~T^3</math> as complex numbers.  If we define,
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<math>~\mathfrak{r} \equiv (R^2 + |D|)^{1 / 2} \, ,</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<math>~\theta \equiv \tan^{-1}\biggl[ \frac{\sqrt{|D|}}{R} \biggr] \, ,</math>
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then we can write,
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<math>~S^3 = \mathfrak{r} e^{+i\theta} \, ,</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<math>~T^3 = \mathfrak{r} e^{-i\theta} \, .</math>
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As is explained in [http://math.stackexchange.com/questions/8760/what-are-the-three-cube-roots-of-1 this online resource], both <math>~S</math> and <math>~T</math> must formally submit to three separate roots tagged by the integer index, <math>~(j=0,1,2)</math>.  Working only with the <math>~j=0</math> root for both, we find that the above expressions for the three roots of our cubic equation become,
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<math>~y_1</math>
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<math>~=</math>
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<math>~p + 2 \mathfrak{r}^{1 / 3} \cos\biggl(\frac{\theta}{3}\biggr) \, ,</math>
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<math>~y_2</math>
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<math>~=</math>
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<math>~p - \mathfrak{r}^{1 / 3} \biggl[ \cos\biggl(\frac{\theta}{3}\biggr) + \sqrt{3} \sin\biggl(\frac{\theta}{3}\biggr) \biggr] \, ,</math>
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<math>~y_3</math>
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<math>~=</math>
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<math>~p - \mathfrak{r}^{1 / 3} \biggl[ \cos\biggl(\frac{\theta}{3}\biggr) - \sqrt{3} \sin\biggl(\frac{\theta}{3}\biggr) \biggr] \, .</math>
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We have deduced empirically that <math>~y_3</math> is the root that is physically relevant in our case.  That is to say, for a given <math>~0 < q < 1</math>,
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<math>~c_0 = p - \mathfrak{r}^{1 / 3} \biggl[ \cos\biggl(\frac{\theta}{3}\biggr) - \sqrt{3} \sin\biggl(\frac{\theta}{3}\biggr) \biggr] \, .</math>
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In turn, for a given value of <math>~q</math>, the corresponding value of <math>~\alpha_e</math> is obtained via the relation, <math>~\alpha_e = c_0(c_0 + 2)</math>.

The right-hand panel of Figure 1 presents a plot of this <math>~c_0(q)</math> function; actually what has been plotted is the inverted relation, <math>~q(\alpha_e)</math>.  The open, red circular markers trace the portion of the function that provides physically viable solutions, in the sense that the corresponding value of <math>~\alpha_c</math> lies between the values, negative one and three; the filled, light=blue circular markers identify roots of the cubic equation that are not physically viable.

In the left-hand panel of Figure 1, we re-display a plot that has been discussed in an [[SSC/Stability/BiPolytrope00#Figure1|accompanying chapter]].  It contains a plot (blue markers) of the same same <math>~q(\alpha_e)</math> function but, this time, as determined from the root of a quartic equation.  In order to illustrate more clearly that the two curves are the same, we have plotted the quartic solution (small, purple circular markers) ''on top of'' the cubic solution in the right-hand panel.  

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Figure 1:&nbsp; Comparing Roots to Quartic and Cubic Equations
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[[File:Quartic21Solution02Corrected.png|300px|quartic solution]]
[[File:Cubic21Solution01Corrected.png|300px|cubic solution]]
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