Appendix/Ramblings/51BiPolytropeStability/BetterInterface

Better Interface for 51BiPolytrope Stability Study

Content Pointing to Previous Work

Tilded Menu Pointers

1. Murphy & Fiedler (1985b): SSC/Stability/MurphyFiedler85
   1. Interface Conditions as promoted by Ledoux & Walraven (1958)
   2. Numerical Integration
      1. General Approach
      2. Special Handling at the Center
      3. Special Handling at the Interface
   3. Reconcile Approaches
2. Excellent Foundation (no pointer from Tiled Menu): SSC/Stability/Biipolytropes
3. Our Broader Analysis: SSC/Stability/BiPolytropes/HeadScratching
4. Succinct Discussion: SSC/Stability/BiPolytropes/SuccinctDiscussion

Ramblings: Analyzing Five-One Bipolytropes

1. Assessing the Stability of Spherical, BiPolytropic Configurations
2. Searching for Analytic EigenVector for (5,1) Bipolytropes
3. See (below) Discussing Patrick Motl's 2019 BiPolytrope Simulations
4. Continue Search
5. Renormalize Structure
6. Renormalize Structure (Part 2)
7. More Carefully Exam Step Function Behavior
8. More Focused Search for Analytic EigenVector if (5,1) Bipolytropes
9. Do Not Confine Search to Analytic Eigenvector
10. Clean, Methodical Examination
11. Rethink Handling of n = 1 Envelope
12. Improved Treatment of Core-Envelope Interface

Solid Foundation

Here we pull primarily from the chapters labeled II and III, above.

Entire Configuration

Beginning with the familiar,

Adiabatic Wave (or Radial Pulsation) Equation

$\frac{d^{2}x}{d{r}_0^2}+[\frac{4}{r_0}-(\frac{g_0{\rho }_0}{P_0}\left)\right]\frac{dx}{d{r}_0}+\left(\frac{{\rho }_0}{{\gamma }_{\mathrm{g}}{P}_0}\right)[{\omega }^2+(4-3{\gamma }_{\mathrm{g}})\frac{g_0}{r_0}]x=0$

where,

$g_0$

$=$

$\frac{G{M}_r^{*}}{(r^{*}{)}^2}\left[{\rho }_c^{3/5}\right(\frac{K_c}{G}{)}^{1/2}],$

if we adopt the variable normalizations,

${\rho }^{*}$	$\equiv$	$\frac{{\rho }_0}{{\rho }_c}$	;	$r^{*}$	$\equiv$	$\frac{r_0}{[K_c^{1/2}/(G^{1/2}{\rho }_c^{2/5})]}$
$P^{*}$	$\equiv$	$\frac{P_0}{K_c{\rho }_c^{6/5}}$	;	$M_r^{*}$	$\equiv$	$\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_c^{1/5})]}$

the LAWE takes the form,

$0$

$=$

$\frac{d^{2}x}{dr{*}^2}+\frac{{\mathscr{H}}}{r^{*}}\frac{dx}{dr*}+\left[\right(\frac{{\sigma }_c^2}{{\gamma }_g}){{𝒦}}_1-{\alpha }_g{{𝒦}}_2]x,$

where,

${\mathscr{H}}$	$\equiv$	$\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}$	,	${{𝒦}}_1$	$\equiv$	$\frac{2\pi }{3}\left(\frac{{\rho }^{*}}{P^{*}}\right)$	and	${{𝒦}}_2$	$\equiv$	$\left(\frac{{\rho }^{*}}{P^{*}}\right)\frac{M_r^{*}}{(r^{*}{)}^3},$
${\sigma }_c^2$	$\equiv$	$\frac{3{\omega }^2}{2\pi G{\rho }_c}$	,	${\alpha }_g$	$\equiv$	$(3-\frac{4}{{\gamma }_g}).$

Core

Given that, in the core, ${\gamma }_g=6/5$ and,

$r^{*}$

$=$

$(\frac{3}{2\pi }{)}^{1/2}\xi ,$

we can rewrite the LAWE to read,

$0$

$=$

$\frac{d^{2}x}{d{\xi }^2}+\frac{{\mathscr{H}}}{\xi }\frac{dx}{d\xi }+\left(\frac{1}{4\pi }\right)[5{\sigma }_c^2{{𝒦}}_1+2{{𝒦}}_2]x,$

where,

${{𝒦}}_1$	$=$	$\frac{2\pi }{3}(1+\frac{1}{3}{\xi }^2{)}^{1/2},$
${\mathscr{H}}$	$=$	$4-2{\xi }^2(1+\frac{1}{3}{\xi }^2{)}^{-1},$
${{𝒦}}_2$	$=$	$\left(\frac{4\pi }{3}\right)(1+\frac{1}{3}{\xi }^2{)}^{-1}.$

Structure at the Interface

Once ${\mu }_e/{\mu }_c$ and ${\xi }_i$ have been specified, other parameter values at the interface are:

${\theta }_i$	$=$	$(1+\frac{1}{3}{\xi }_i^2{)}^{-1/2},$
${\eta }_i$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)\sqrt{3}{\theta }_i^2{\xi }_i,$
${\mathrm{\Lambda }}_i$	$=$	$\frac{1}{{\eta }_i}-\frac{{\xi }_i}{\sqrt{3}},$
$A$	$=$	${\eta }_i(1+{\mathrm{\Lambda }}_i^2{)}^{1/2},$
$B$	$=$	${\eta }_i-\frac{\pi }{2}+{\tan }^{-1}({\mathrm{\Lambda }}_i),$
${\eta }_s$	$=$	$B+\pi .$

Linearized Perturbation at the Interface

The three spatially dependent quantities — $p,d,$ and $x$ — are related to one another via the set of linearized governing relations, namely,

Linearized Equation of Continuity $r_0\frac{dx}{d{r}_0}=-3x-d,$ Linearized Euler + Poisson Equations $\frac{P_0}{{\rho }_0}\frac{dp}{d{r}_0}=(4x+p)g_0+{\omega }^2{r}_{0}x,$ Linearized Adiabatic Form of the First Law of Thermodynamics $p={\gamma }_{\mathrm{g}}d.$

Envelope

Given that, throughout the envelope ${\gamma }_g=2$ and,

$r^{*}$

$=$

$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta ,$

we can rewrite the LAWE to read,

$0$

$=$

$\frac{d^{2}x}{d{\eta }^2}+\frac{{\mathscr{H}}}{\eta }\frac{dx}{d\eta }+\frac{1}{2\pi {\theta }_i^4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right[\left(\frac{{\sigma }_c^2}{2}\right){{𝒦}}_1-{{𝒦}}_2]x,$

where,

${{𝒦}}_1$	$=$	$\frac{2\pi }{3}\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{-1}\left[\frac{\eta }{A\sin (\eta -B)}\right],$
${\mathscr{H}}$	$=$	$2[1+\frac{\eta }{\tan (\eta -B)}],$
${{𝒦}}_2$	$=$	$4\pi \left(\frac{{\mu }_e}{{\mu }_c}{)}^2{\theta }_i^4\right[1-\frac{\eta }{\tan (\eta -B)}]\frac{1}{{\eta }^2}.$

See Also