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

At all radial locations throughout the equilibrium configuration, the three spatially dependent quantities — $p\equiv \delta p/P^{*},d\equiv \delta \rho /{\rho }^{*},$ and $x\equiv \delta r/r^{*}$ — 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 $p={\gamma }_{\mathrm{g}}d.$

Combining the 2^nd and 3^rd equations, we find,

$(4x+{\gamma }_{g}d)g_0+{\omega }^2{r}_{0}x$

$=$

$\frac{P_0}{{\rho }_0}\frac{d({\gamma }_{g}d)}{d{r}_0}$

---

At the interface, presumably the dimensional structural variables, $P^{*}$ and $r^{*}$ have the same values, whether viewed from the perspective of the core or from the perspective of the envelope. But ${\rho }^{*}$ has a different value, depending on the point of view. Specifically,

${\rho }^{*}{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}$

$=$

$\left(\frac{{\mu }_e}{{\mu }_c}\right){\rho }^{*}{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}.$

Hence, from the perspective of the core, the linearized equation of continuity may be written as,

$[r_0\cdot \frac{dx}{d{r}_0}+3x{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$-\frac{\delta \rho }{[{\rho }^{*}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}};$

while, from the perspective of the envelope, the linearized equation of continuity may be written as,

$[r_0\cdot \frac{dx}{d{r}_0}+3x{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$

$=$

$-\frac{\delta \rho }{[{\rho }^{*}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}$

---

Try again

From here, we know …

$\frac{P}{{\rho }^{{\gamma }_g}}$

$=$

$\exp \left[\frac{s({\gamma }_g-1)}{\mathfrak{\Re }/\overline{\mu }}\right].$

And, from my discussions with Patrick Motl, we find …

CORE:   Throughout the core,

$P^{*}$

$=$

$(1+\frac{{\xi }^2}{3}{)}^{-3}$

and

${\rho }^{*}$

$=$

$(1+\frac{{\xi }^2}{3}{)}^{-5/2}$

and

${\gamma }_g$

$=$

$\frac{6}{5}.$

Hence, independent of the radial location, $\xi$, throughout the core,

$\frac{s}{\mathrm{\Re }/\overline{\mu }}{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$5\ln (5).$

ENVELOPE:   Throughout the envelope,

$P^{*}$

$=$

${\theta }_i^6[\varphi (\eta ){]}^2$

and

${\rho }^{*}$

$=$

$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5[\varphi (\eta )]$

and

${\gamma }_g$

$=$

$2.$

Hence, independent of the radial location, $\eta$, throughout the envelope,

$\frac{s}{\mathrm{\Re }/\overline{\mu }}{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}$

$=$

$\ln \left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}]$

$=$

$\ln \left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}(1+\frac{{\xi }_i^2}{3}{)}^2].$

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,

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

Entropy as a Step Function

Useful Chapters:

• Appendix/Mathematics/Stepfunction
• Step-Function Behavior of Specific Entropy

Review

The unit — or, Heaviside — step function, $H(x)$, is defined such that,

$H(x)=\{\begin{array}{ll}0;& x<0\\ 1;& x>0\end{array}$ [MF53], Part I, §2.1 (p. 123), Eq. (2.1.6)

In evaluating this function at $x=0$, we will adopt the half-maximum convention and set $H(0)=\frac{1}{2}$. As has been pointed out in, for example, a relevant Wikipedia discussion, the derivative of the unit step function is,

$\frac{dH(x)}{dx}$

$=$

$\delta (x),$

where, $\delta (x)$ is the Dirac Delta function.

Perturbed Density

Let,

$\zeta$	$\equiv$	$\frac{r}{r_i}-1$
$\Rightarrow [{\rho }^{*}{]}_{\mathrm{e}\mathrm{q}}$	$\equiv$	$H(\zeta )\cdot {\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}+H(-\zeta )\cdot {\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*};$

and, more generally after a perturbation, $\delta \rho (\zeta )$,

${\rho }^{*}=[{\rho }^{*}{]}_{\mathrm{e}\mathrm{q}}+\delta \rho$

$=$

$[{\rho }^{*}{]}_{\mathrm{e}\mathrm{q}}\{1+\frac{\delta \rho }{[{\rho }^{*}{]}_{\mathrm{e}\mathrm{q}}}\}=[{\rho }^{*}{]}_{\mathrm{e}\mathrm{q}}\{1+d{e}^{i\omega t}\}.$

Hence, in the linearized version of the continuity equation, we recognize that,

$d{e}^{i\omega t}$

$=$

$\frac{\delta \rho }{[{\rho }^{*}{]}_{\mathrm{e}\mathrm{q}}}.$

CORE:

${\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$

$=$

$(1+\frac{{\xi }^2}{3}{)}^{-5/2}$

and

${\gamma }_g$

$=$

$\frac{6}{5}$

and

${{𝒮}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}\equiv \frac{s({\gamma }_g-1)}{\mathrm{\Re }/\overline{\mu }}{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$\ln (5).$

ENVELOPE:

${\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}$

$=$

$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5[\varphi (\eta )]$

and

${\gamma }_g$

$=$

$2$

and

${{𝒮}}_{\mathrm{e}\mathrm{n}\mathrm{v}}\equiv \frac{s({\gamma }_g-1)}{\mathrm{\Re }/\overline{\mu }}{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}$

$=$

$\ln \left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}(1+\frac{{\xi }_i^2}{3}{)}^2].$

Perturbed Pressure

$P^{*}$

$=$

$({\rho }^{*}{)}^{{\gamma }_g}\cdot \exp [\frac{s({\gamma }_g-1)}{\mathrm{\Re }/\overline{\mu }}]$

$[P^{*}{]}_{\mathrm{e}\mathrm{q}}$

$\equiv$

$H(\zeta )\cdot P_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}+H(-\zeta )\cdot P_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*};$

and, more generally after a perturbation, $\delta P(\zeta )$,

$P^{*}=[P^{*}{]}_{\mathrm{e}\mathrm{q}}+\delta P$

$=$

$[P^{*}{]}_{\mathrm{e}\mathrm{q}}\{1+\frac{\delta P}{[P^{*}{]}_{\mathrm{e}\mathrm{q}}}\}=[P^{*}{]}_{\mathrm{e}\mathrm{q}}\{1+p{e}^{i\omega t}\}.$

Hence, in the linearized version of the first law of thermodynamics, we recognize that,

$p{e}^{i\omega t}$

$=$

$\frac{\delta P}{[P^{*}{]}_{\mathrm{e}\mathrm{q}}}.$

Obtaining Perturbed Density from Perturbed Pressure

Given that, quite generally,

$\frac{P}{{\rho }^{{\gamma }_g}}$

$=$

$\exp \left[\frac{s({\gamma }_g-1)}{\mathfrak{\Re }/\overline{\mu }}\right],$

let's define the density-like quantity,

${{𝒬}}^{*}$

$\equiv$

$e^{-{𝒮}/{{\gamma }}_{{ℊ}}}[P^{*}{]}^{1/{\gamma }_g},$

in which case,

${\rho }_{\mathrm{e}\mathrm{q}}^{*}$

$\equiv$

$H(\zeta )\cdot [{{𝒬}}_{\mathrm{e}\mathrm{q}}^{*}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}+H(-\zeta )\cdot [{{𝒬}}_{\mathrm{e}\mathrm{q}}^{*}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}.$

What happens if we perturb the pressure? In either region (core or envelope),

${{𝒬}}^{*}$

$=$

$e^{-{𝒮}/{{\gamma }}_{{ℊ}}}[P_{\mathrm{e}\mathrm{q}}^{*}{]}^{1/{\gamma }_g}[1+\frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}{]}^{1/{\gamma }_g}\approx e^{-{𝒮}/{{\gamma }}_{{ℊ}}}[P_{\mathrm{e}\mathrm{q}}^{*}{]}^{1/{\gamma }_g}[1+\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]=[Q_{\mathrm{e}\mathrm{q}}^{*}][1+\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}].$

As a result,

${\rho }^{*}={\rho }_{\mathrm{e}\mathrm{q}}^{*}+\delta \rho$	$\equiv$	$H(\zeta )\cdot [{{𝒬}}^{*}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}+H(-\zeta )\cdot [{{𝒬}}^{*}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
	$=$	$H(\zeta )\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][1+\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}+H(-\zeta )\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][1+\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
	$=$	$H(\zeta )\cdot [Q_{\mathrm{e}\mathrm{q}}^{*}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}+H(-\zeta )\cdot [Q_{\mathrm{e}\mathrm{q}}^{*}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
		$+H(\zeta )\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}+H(-\zeta )\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
	$=$	${\rho }_{\mathrm{e}\mathrm{q}}^{*}+H(\zeta )\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}+H(-\zeta )\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
$\Rightarrow \frac{\delta \rho }{{\rho }_{\mathrm{e}\mathrm{q}}^{*}}$	$=$	$\frac{H(\zeta )}{{\rho }_{\mathrm{e}\mathrm{q}}^{*}}\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}+\frac{H(-\zeta )}{{\rho }_{\mathrm{e}\mathrm{q}}^{*}}\cdot \{[Q_{\mathrm{e}\mathrm{q}}^{*}][\frac{1}{{\gamma }_g}\cdot \frac{\delta P}{P_{\mathrm{e}\mathrm{q}}^{*}}]{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

See Also