User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors

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Numerically Determined Eigenvectors of a Zero-Zero Bipolytrope

Here we build on the analytic foundation summarized in an accompanying chapter and attempt to numerically construct a variety of eigenvectors that describe radial oscillations of bipolytropes for which, <math>~(n_c, n_e) = (0,0)</math>.

Whitworth's (1981) Isothermal Free-Energy Surface
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Setup

We'll begin with the linear-adiabatic wave equations that describe oscillations of the core and envelope, separately. We also will immediately restrict our investigation to configurations for which,

<math>~g^2 = \mathcal{B} </math>         <math>~\Rightarrow</math>         <math>~g^2 = \frac{1+8q^3}{ (1+2q^3)^2 } \, ,</math>         and,         <math>~q^3 = \mathcal{D} = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr] </math>         <math>~\Rightarrow</math>         <math>~\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3} \, .</math>

For the core we have,

<math>~0</math>

<math>~=</math>

<math>~ (1 - \eta^2)\frac{d^2x}{d\eta^2} + ( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx}{d\eta} + \mathfrak{F}_\mathrm{core} x \, , </math>

where,

<math>~\eta \equiv \frac{\xi}{g} \, ,</math>         and         <math>~\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c\, .</math>

And, for the envelope we have,

<math>~0</math>

<math>~=</math>

<math>~ ( 1 - q^3 \xi^3 ) \frac{d^2x}{d\xi^2} + ( 3 - 6q^3 \xi^3 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ q^3 \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e \biggr]\frac{x}{\xi^2} \, , </math>

where,

<math>~\mathfrak{F}_\mathrm{env}</math>

<math>~\equiv</math>

<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} - 2\alpha_e \, . </math>


Initial Focus

Properties of 21Analytic Solution

Evidently, one analytic solution with quantum numbers, <math>~(\ell,j) = (2,1)</math>, is available for a zero-zero bipolytrope that has the following properties:

<math>~q</math>

<math>~\approx</math>

<math>~0.6840119</math>

<math>~\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3}</math>

<math>~\approx</math>

<math>~0.3902664</math>

<math>~\gamma_e = \frac{4}{3+0.35}</math>

<math>~\approx</math>

<math>~1.1940299</math>

<math>~\gamma_c </math>

<math>~\approx</math>

<math>~1.845579</math>

<math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G \rho_c} = 20\gamma_c - 8 </math>

<math>~\approx</math>

<math>~28.91158 \, .</math>

This means, as well, that,

<math>~c_0 \equiv \sqrt{1+\alpha_e} - 1</math>

<math>~=</math>

<math>~\sqrt{0.65}-1 \approx - 0.1937742</math>

<math>~g^2 \equiv \frac{1+8q^3}{(1+2q^3)^2}</math>

<math>~\approx</math>

<math>~1.3236092</math>

<math>~\mathfrak{F}_\mathrm{core} \equiv \frac{\sigma_c^2 + 8}{\gamma_c} - 6</math>

<math>~=</math>

<math>~14</math>

<math>~\mathfrak{F}_\mathrm{env} \equiv \frac{1}{\gamma_e} \biggl[ \sigma_c^2 \biggl(\frac{\rho_c}{\rho_e} \biggr) + 8\biggr]- 6</math>

<math>~=</math>

<math>~(c_0^2 + 17c_0 +66) = 62.743385</math>


In the envelope, the analytically defined eigenfunction is given by the expression,

<math>~x_{\ell=2} |_\mathrm{env}</math>

<math>~=</math>

<math>~ \xi^{c_0}\biggl[ \frac{ 1 + q^3 A_{21} \xi^{3} + q^6 A_{21}B_{21}\xi^{6} }{ 1 + q^3 A_{21} + q^6 A_{21}B_{21}}\biggr] \, , </math>

where,

<math>~A_{21}</math>

<math>~\equiv</math>

<math>~-\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr) \approx -4.6016533 \, ,</math>

<math>~B_{21}</math>

<math>~\equiv</math>

<math>~-\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr) \approx -0.8940912 \, ; </math>

and in the core, it is,

<math>~x_{j=1} |_\mathrm{core}</math>

<math>~=</math>

<math>~ \frac{5(1+8q^3) - 7 (1+2q^3)^2 \xi^2}{5(1+8q^3)-7(1+2q^3)^2} \, .</math>

More succinctly we have,

<math>~x_\mathrm{core}</math>

<math>~=</math>

<math>~-17.326820 + 18.326820~\xi^2 \, ;</math>

and,

<math>~a \cdot x_\mathrm{env}</math>

<math>~=</math>

<math>~ \xi^{- 0.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] \, , </math>

where,

<math>~a \equiv -[ 1 + q^3 A_{21} + q^6 A_{21}B_{21}] \approx - 0.05128445 \, .</math>

Demonstrate Core Solution

This means that,

<math>~\frac{dx_\mathrm{core}}{d\xi}</math>

<math>~=</math>

<math>~36.65364~\xi \, ,</math>

and,

<math>~\frac{d^2x_\mathrm{core}}{d\xi^2}</math>

<math>~=</math>

<math>~36.65364 \, .</math>

Therefore, the LAWE for the core becomes,

<math>~[\mathrm{LAWE}]_\mathrm{core}</math>

<math>~=</math>

<math>~ (1 - \eta^2)\frac{d^2x_\mathrm{core}}{d\eta^2} + ( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx_\mathrm{core}}{d\eta} + \mathfrak{F}_\mathrm{core} x_\mathrm{core} </math>

 

<math>~=</math>

<math>~ (g^2 - \xi^2)\frac{d^2x_\mathrm{core}}{d\xi^2} + ( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx_\mathrm{core}}{d\xi} + \mathfrak{F}_\mathrm{core} x_\mathrm{core} </math>

 

<math>~=</math>

<math>~ 36.65364(1.3236092 - \xi^2) + 36.65364( 5.2944368 - 6\xi^2 ) + 14( -17.326820 + 18.326820~\xi^2) </math>

 

<math>~=</math>

<math>~ 36.65364(1.3236092 ) + 36.65364( 5.2944368 ) + 14( -17.326820 ) + [36.65364(-1) + 36.65364( - 6 ) + 14( 18.326820)]\xi^2 </math>

 

<math>~=</math>

<math>~ 36.65364(6.618046 ) - 14( 17.326820 ) + 36.65364 [-1 - 6 + 7]\xi^2 </math>

 

<math>~=</math>

<math>~ 0 \, . </math>

Q.E.D.


Demonstrate Envelope Solution

Given that,

<math>~a\cdot x_\mathrm{env}</math>

<math>~=</math>

<math>~ \xi^{- 0.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] \, , </math>

we deduce that,

<math>~a\cdot \frac{dx_\mathrm{env}}{d\xi}</math>

<math>~=</math>

<math>~ -0.1937742~ \xi^{- 1.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] + \xi^{- 0.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] \, , </math>

and,

<math>~a \cdot \frac{d^2x_\mathrm{env}}{d\xi^2}</math>

<math>~=</math>

<math>~ 0.2313226~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] - 2 \times 0.1937742~ \xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] </math>

 

 

<math>~ + \xi^{- 0.1937742}\biggl[ -8.8360086~ \xi + 12.641508~\xi^{4} \biggr] \, . </math>

Therefore, the LAWE for the envelope becomes,

<math>~a\cdot [\mathrm{LAWE}]_\mathrm{env}</math>

<math>~=</math>

<math>~ a( 1 - q^3 \xi^3 ) \frac{d^2x}{d\xi^2} + a( 3 - 6q^3 \xi^3 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + a\biggl[ q^3 \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e \biggr]\frac{x}{\xi^2} \, , </math>

 

<math>~=</math>

<math>~ a\biggl\{ \frac{d^2x}{d\xi^2} + \frac{3}{\xi} \cdot \frac{dx}{d\xi} -\alpha_e \biggl(\frac{x}{\xi^2} \biggr) \biggr\} - a q^3 \xi^3 \biggl\{ \frac{d^2x}{d\xi^2} + \frac{6}{\xi} \cdot \frac{dx}{d\xi} - \mathfrak{F}_\mathrm{env} \biggl(\frac{x}{\xi^2} \biggr)\biggr\} \, . </math>

Now, the first of these sub-expressions gives,

<math>~ a\biggl\{ \frac{d^2x}{d\xi^2} + \frac{3}{\xi} \cdot \frac{dx}{d\xi} -\alpha_e \biggl(\frac{x}{\xi^2} \biggr) \biggr\} </math>

<math>~=</math>

<math>~ 0.2313226~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] - 2 \times 0.1937742~ \xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] </math>

 

 

<math>~ + \xi^{- 0.1937742}\biggl[ -8.8360086~ \xi + 12.641508~\xi^{4} \biggr] </math>

 

 

<math>~ -0.5813226~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] + 3\xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] </math>

 

 

<math>~+0.35 \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] </math>

 

<math>~=</math>

<math>~ (0.2313226 -0.5813226 + 0.35) ~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] </math>

 

 

<math>~ + (3 - 2 \times 0.1937742)~ \xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] + \xi^{- 0.1937742}\biggl[ -8.8360086~ \xi + 12.641508~\xi^{4} \biggr] </math>

 

<math>~=</math>

<math>~ (3 - 2 \times 0.1937742)~ \xi^{- 2.1937742}\biggl[ - 4.4180043~ \xi^{3} + 2.5283016~\xi^{6} \biggr] + \xi^{- 2.1937742}\biggl[ -8.8360086~ \xi^3 + 12.641508~\xi^{6} \biggr] </math>

 

<math>~=</math>

<math>~ 

\xi^{- 2.1937742}\biggl[  -20.37783~ \xi^3 + 19.24657~\xi^{6} \biggr]

</math>

And the sub-expression inside the second set of curly braces gives,

<math>~ a\biggl\{ \frac{d^2x}{d\xi^2} + \frac{6}{\xi} \cdot \frac{dx}{d\xi} -\mathfrak{F}_\mathrm{env} \biggl(\frac{x}{\xi^2} \biggr) \biggr\} </math>

<math>~=</math>

<math>~ 0.2313226~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] - 2 \times 0.1937742~ \xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] </math>

 

 

<math>~ + \xi^{- 0.1937742}\biggl[ -8.8360086~ \xi + 12.641508~\xi^{4} \biggr] </math>

 

 

<math>~ -1.1626452~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] + 6\xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] </math>

 

 

<math>~- 62.74339 \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] </math>

 

<math>~=</math>

<math>~ (0.2313226 -1.1626452 - 62.74339)~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] </math>

 

 

<math>~ + (6- 2 \times 0.1937742)~ \xi^{- 1.1937742}\biggl[ - 4.4180043~ \xi^{2} + 2.5283016~\xi^{5} \biggr] + \xi^{- 0.1937742}\biggl[ -8.8360086~ \xi + 12.641508~\xi^{4} \biggr] </math>

 

<math>~=</math>

<math>~ -63.67471~ \xi^{- 2.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] </math>

 

 

<math>~ + 5.612452~ \xi^{- 2.1937742}\biggl[ - 4.4180043~ \xi^{3} + 2.5283016~\xi^{6} \biggr] + \xi^{- 2.1937742}\biggl[ -8.8360086~ \xi^3 + 12.641508~\xi^{6} \biggr] </math>

 

<math>~=</math>

<math>~ \xi^{- 2.1937742}\biggl[ -63.67471 + 93.77171~ \xi^{3} -26.83148~\xi^{6} \biggr] </math>

 

 

<math>~ + ~ \xi^{- 2.1937742}\biggl[ - 24.79584~ \xi^{3} + 14.18997~\xi^{6} \biggr] + \xi^{- 2.1937742}\biggl[ -8.8360086~ \xi^3 + 12.641508~\xi^{6} \biggr] </math>

 

<math>~=</math>

<math>~ \xi^{- 2.1937742}\biggl[ -63.67471 + (93.77171-24.79584- 8.8360086) ~ \xi^{3} + (-26.83148+14.18997 + 12.641508)~\xi^{6} \biggr] </math>

 

<math>~=</math>

<math>~ \xi^{- 2.1937742}\biggl[ -63.67471 + 60.13986 ~ \xi^{3} \biggr] </math>

<math>~\Rightarrow~~~ a(q^3\xi^3) \biggl\{ \frac{d^2x}{d\xi^2} + \frac{6}{\xi} \cdot \frac{dx}{d\xi} -\mathfrak{F}_\mathrm{env} \biggl(\frac{x}{\xi^2} \biggr) \biggr\} </math>

<math>~=</math>

<math>~ \xi^{- 2.1937742}\biggl[ -20.37783 +19.24657 ~ \xi^{6} \biggr] \, . </math>

But these two sub-expressions cancel precisely, which means that our eigenfunction satisfies the LAWE! Q.E.D.

Boundary Conditions

Notice that for this particular eigenfunction solution, the value and first radial derivative at the center <math>~(\xi=0)</math> of the configuration is,

<math>~x_\mathrm{core}</math>

<math>~=</math>

<math>~-17.326820 + 18.326820~\cancelto{0}{\xi^2} = -17.326820 \, ;</math>

and,

<math>~\frac{dx_\mathrm{core}}{d\xi}</math>

<math>~=</math>

<math>~36.65364~\cancelto{0}{\xi} = 0 \, .</math>

And, at the surface <math>~(\xi = q^{-1}) </math> the value and first radial derivative are,

<math>~a \cdot x_\mathrm{env}</math>

<math>~=</math>

<math>~ \biggl\{\xi^{- 0.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] \biggr\}_{\xi=1/q} </math>

 

<math>~\approx</math>

<math>~ 0.47627246\, , </math>

where,

<math>~a \approx - 0.05128445 \, ;</math>

and,

<math>~\frac{d\ln x_\mathrm{env}}{d\ln \xi} </math>

<math>~=</math>

<math>~ \frac{\xi}{a\cdot x_\mathrm{env}} \biggl[ a\cdot \frac{dx_\mathrm{env}}{d\xi} \biggr]</math>

 

<math>~=</math>

<math>~ \frac{ -0.1937742~\xi^{- 0.1937742}[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} ] + \xi^{- 0.1937742}[ - 4.4180043~ \xi^{3} + 2.5283016~\xi^{6} ] }{\xi^{- 0.1937742} [ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} ] } </math>

 

<math>~=</math>

<math>~ -0.1937742 + \frac{~ [ - 4.4180043~ \xi^{3} + 2.5283016~\xi^{6} ] }{ [ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} ] } </math>

<math>~\Rightarrow ~~~ \biggl\{ \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr\}_{\xi=1/q} </math>

<math>~=</math>

<math>~ -0.1937742 + \biggl\{ \frac{[ - 4.4180043~ \xi^{3} + 2.5283016~\xi^{6} ] }{ [ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} ] } \biggr\}_{\xi=1/q} </math>

 

<math>~=</math>

<math>~ -0.1937742 + 21.22492 = 21.03115 \, . </math>


Whitworth's (1981) Isothermal Free-Energy Surface

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