User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors

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>.

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

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>~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>

That is,

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

<math>~=</math>

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

and,

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

<math>~=</math>

<math>~- 19.499089~ \xi^{- 0.1937742}\biggl[ 1 - 1.4726681~ \xi^{3} + 0.4213836~\xi^{6} \biggr] \, . </math>

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation