Appendix/Ramblings/51BiPolytropeStability/NoAnalytic

Analog of Bonnor-Ebert Limiting Pressure

As has been demonstrated in an accompanying discussion, the mass of a pressure-truncated, n = 5 polytrope is,

$\frac{M}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$

$=$

$\left[\right(\frac{3\cdot 5^3}{2^2\pi })\frac{({\xi }_e^2/3{)}^3}{(1+{\xi }_e^2/3{)}^4}{]}^{1/2}$

where,

$M_{\mathrm{S}\mathrm{W}\mathrm{S}}$	$\equiv$	$\left[\right(\frac{n+1}{nG}{)}^{3/2}{K}_n^{2n/(n+1)}{P}_{\mathrm{e}}^{(3-n)/[2(n+1)]}{]}_{n=5}$
	$\equiv$	$(\frac{2\cdot 3}{5G}{)}^{3/2}{K}^{5/3}{P}_{\mathrm{e}}^{-1/6}$

Now, as we have recorded in an accompanying summary table, the maximum (critical) mass arises precisely at ${\xi }_e=3$. That is,

$M_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^6$	$=$	$M_{\mathrm{S}\mathrm{W}\mathrm{S}}^6\left[\right(\frac{3\cdot 5^3}{2^2\pi })\frac{({\xi }_e^2/3{)}^3}{(1+{\xi }_e^2/3{)}^4}{]}_{{\xi }_e=3}^3$
	$=$	$\left[\right(\frac{2\cdot 3}{5G}{)}^{3/2}{K}^{5/3}{P}_{\mathrm{e}}^{-1/6}{]}^6\left[\right(\frac{3\cdot 5^3}{2^2\pi })\frac{3^3}{2^8}{]}^3$
	$=$	$\left(\frac{2\cdot 3}{5}{)}^9\frac{K^{10}}{G^9{P}_{\mathrm{e}}}\right(\frac{3^4\cdot 5^3}{2^{10}\pi }{)}^3$
	$=$	$\left[\right(\frac{3}{2}{)}^7\frac{1}{\pi }{]}^3\frac{K^{10}}{G^9{P}_{\mathrm{e}}}.$

According to our accompanying renormalization of the equilibrium configuration (also see below),

$P$	$=$	$\left(\frac{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^6{K}_c^{10}{G}^{-9}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}(1+\frac{1}{3}{\xi }^2{)}^{-3};$
$M_r$	$=$	$\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}}\right)\left(\frac{{\mu }_e}{{\mu }_c}{)}^2\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}].$

That is,

$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^6\cdot P_i$	$=$	$\left(\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}}{)}^6\right(\frac{{\mu }_e}{{\mu }_c}{)}^{12}\left(\frac{2\cdot 3}{\pi }{)}^3\right[{\xi }_i^3(1+\frac{1}{3}{\xi }_i^2{)}^{-3/2}{]}^6\cdot (\frac{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{)}^6{K}_c^{10}{G}^{-9}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}\right(1+\frac{1}{3}{\xi }_i^2{)}^{-3}$
	$=$	$\left(\frac{2\cdot 3}{\pi }{)}^3\right[{\xi }_i^3(1+\frac{1}{3}{\xi }_i^2{)}^{-2}{]}^6\frac{K_c^{10}}{G^9}.$

Setting these expressions equal to one another means,

$\frac{3^2}{2^4}$

$=$

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

What do I make of this?

Do Not Confine Search to Analytic Eigenvector

Maximum Mass Fraction (ν)

Maximum Mass Fraction $(\nu )$
$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	${\mathrm{\Lambda }}_i$	$\nu$	$q$
$\frac{1}{3}$	$\mathrm{\infty }$	---	$\frac{2}{\pi }$	0.0
0.3300	24.00496	0.2128753	0.52024552	0.038378833
0.316943	10.744571	0.4903393	0.382383875	0.068652715
0.3100†	9.0149598	0.5983505	0.213039696	0.153835
0.3090	8.83017723	0.6130669	0.331475715	0.076265588
$\frac{1}{4}$	4.9379256	1.4179907	0.139370157	0.084824137
†This model also used in a related discussion where we investigate the relevance of the 📚 Bisnovatyi-Kogan & Blinnikov (1974) conjecture.

Overview

STEP01:
Develop an algorithm (for Excel) that numerically integrates the LAWEs from the center to the surface of a $(n_c,n_e)=(5,1)$ bipolytrope, for an arbitrary specification of the three parameters:   ${\mu }_e/{\mu }_c,{\xi }_i$, and ${\sigma }_c^2$.

• Enforce the proper interface matching condition(s) at the interface location, ${\xi }_i$.
• Note that in general, for an arbitrarily chosen set of the three parameter values, the resulting surface displacement function will not match the desired boundary condition.

STEP02:
Fix your chosen value of the parameter pair, $({\mu }_e/{\mu }_c,{\xi }_i)$, and vary ${\sigma }_c^2$ until the proper surface boundary condition is realized.

• In an accompanying discussion, we claim to have identified at what point along various ${\mu }_e/{\mu }_c$ sequences the fundamental mode of radial oscillation becomes unstable — that is, when ${\sigma }_c^2=0$. For a given choice of ${\mu }_e/{\mu }_c$, it would be wise to begin our eigenvector search at a value of ${\xi }_i<[{\xi }_i{]}_{\mathrm{F}\mathrm{M}}$, as specified in the following table:

  Marginally Unstable Fundamental Modes
  $\frac{{\mu }_e}{{\mu }_c}$	$[{\xi }_i{]}_{\mathrm{F}\mathrm{M}}$
  1	1.6686460157
  $\frac{1}{2}$	2.27925811317
  0.345	2.560146865247
  $\frac{1}{3}$	2.582007485476
  0.309	2.6274239687695
  $\frac{1}{4}$	2.7357711469398
  See orange-colored triangular markers in the associated Figure 4

• Keep steadily raising the value of the interface location until you find the 1^st overtone mode; a related discussion (with animation) shows the results of this type of search in the context of isolated n = 1 polytropes. Our expectation is that, if this mode is unstable, the model will coincide with the turning point along the equilibrium sequence and its eigenvector will essentially overlap with the eigenvector found using the B-KB74 conjecture. At the same time, the square-of-the-eigenfrequency for the fundamental mode will be very negative.

STEP03:
Regarding analytically specified eigenvectors that satisfy the governing LAWES …

• If we force ${\sigma }_c^2=0$ in the core, we have shown that a parabolic-shaped eigenfunction satisfies the LAWE of the core. We expect this eigenfunction to precisely overlay the numerically determined, marginally unstable displacement function in both the case of the unstable fundamental mode and the case of the unstable 1^st overtone.
• If we force ${\sigma }_c^2=0$ in the envelope, we have derived a different — $\cos (\eta -B)$ dependent — eigenfunction that satisfies the LAWE of the envelope. However, this proves to be irrelevant in the context of our bipolytrope because the derived eigenfunction does not match the physically relevant surface boundary condition.

Renormalized LAWE

As presented, for example, in a parallel discussion, in terms of our original ( * ) parameter normalizations, the polytropic LAWE takes the form,

$0$	$=$	$\frac{d^{2}x}{dr{*}^2}+\{\frac{4}{r^{*}}-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*}{)}^2}\right\}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)\{\frac{{\omega }^2}{{\gamma }_{\mathrm{g}}G{\rho }_c}+(\frac{4}{{\gamma }_{\mathrm{g}}}-3\left)\frac{M_r^{*}}{(r^{*}{)}^3}\right\}x$
	$=$	$\frac{d^{2}x}{dr{*}^2}+\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}\frac{1}{r^{*}}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}\}x,$

where, ${\alpha }_g\equiv (3-4/{\gamma }_g)$, and

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

New Normalization $\tilde{\rho }$ $\equiv$ $\rho \left[\right(\frac{K_c}{G}{)}^{3/2}\frac{1}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{]}^{-5};$ $\tilde{P}$ $\equiv$ $P\left[K_c^{-10}{G}^9{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^6\right];$ $\tilde{r}$ $\equiv$ $r\left[\right(\frac{K_c}{G}{)}^{5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-2}],$ ${\tilde{M}}_r$ $\equiv$ $\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}};$ $\tilde{H}$ $\equiv$ $H\left[K_c^{-5/2}{G}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right].$

Switching to the new normalization, where it is understood that,

$(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}$

$=$

$(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}},$

where,

${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$

$\equiv$

$\left(\frac{2}{\pi }{)}^{1/2}{\theta }_i^{-1}\right(-{\eta }^2\frac{d\varphi }{d\eta }{)}_s=(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i},$

we find the following relevant relations:

$(r^{*}{)}^2$	$=$	$r^2\left[\frac{G{\rho }_0^{4/5}}{K_c}\right]={\tilde{r}}^2\left[\right(\frac{K_c}{G}{)}^{5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-2}{]}^{-2}\left[\frac{G{\rho }_0^{4/5}}{K_c}\right]$
	$=$	${\tilde{r}}^2\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^4;$
$\frac{{\rho }^{*}}{P^{*}}$	$=$	$\left(\frac{\rho }{{\rho }_0}\right)\left[\frac{K_c{\rho }_0^{6/5}}{P}\right]=\left(\frac{\rho }{P}\right)\left[K_c{\rho }_0^{1/5}\right]$
	$=$	$\frac{\tilde{\rho }}{\tilde{P}}\left[\right(\frac{K_c}{G}{)}^{3/2}\frac{1}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{]}^5\left[K_c^{-10}{G}^9{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^6\right]\left[K_c{\rho }_0^{1/5}\right]$
	$=$	$\frac{\tilde{\rho }}{\tilde{P}}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}];$
$\frac{M_r^{*}}{r^{*}}$	$=$	$\frac{M_r}{r}\left\{\frac{[K_c^{1/2}/(G^{1/2}{\rho }_0^{2/5})]}{[K_c^{3/2}/(G^{3/2}{\rho }_0^{1/5})]}\right\}=\frac{M_r}{r}\left[\right(\frac{G}{K_c}\left){\rho }_0^{-1/5}\right]$
	$=$	$\frac{{\tilde{M}}_r}{\tilde{r}}\left[\right(\frac{K_c}{G}{)}^{5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-2}\left]\right[\left(\frac{G}{K_c}\right){\rho }_0^{-1/5}]M_{\mathrm{t}\mathrm{o}\mathrm{t}}$
	$=$	$\frac{{\tilde{M}}_r}{\tilde{r}}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-1}.$

Therefore, in terms of the renormalized variables the LAWE becomes,

$0$	$=$	$\frac{d^{2}x}{dr{*}^2}+\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}\frac{1}{r^{*}}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}\}x$
	$=$	$\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-4}\frac{d^{2}x}{d{\tilde{r}}^2}+\{4-(\frac{\tilde{\rho }}{\tilde{P}}\left)\frac{{\tilde{M}}_r}{\tilde{r}}\right\}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-4}\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}+\frac{\tilde{\rho }}{\tilde{P}}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}\left]\right\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{\tilde{M}}_r}{(\tilde{r}{)}^3}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-5}\}x.$

---

Now let's change how "time" — and, hence, how frequency — is normalized. Specifically, we employ the mapping,

$\left(\frac{2\pi }{3}\right)\frac{{\sigma }_c^2}{{\gamma }_g}$

$\to$

$\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-5}\left(\frac{2\pi }{3}\right)\frac{{\tilde{\sigma }}^2}{{\gamma }_g},$

where,

${\tilde{\sigma }}^2$

$\equiv$

$\frac{3{\omega }^2}{2\pi }\left[K_c^{-15/2}{G}^{+13/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^5\right].$

That is, the renormalized LAWE becomes,

$0$

$=$

$\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-4}\frac{d^{2}x}{d{\tilde{r}}^2}+\{4-(\frac{\tilde{\rho }}{\tilde{P}}\left)\frac{{\tilde{M}}_r}{\tilde{r}}\right\}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-4}\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}+\frac{\tilde{\rho }}{\tilde{P}}\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^{-4}\{\frac{2\pi {\tilde{\sigma }}^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{\tilde{M}}_r}{(\tilde{r}{)}^3}\}x.$

---

After multiplying through by

$\left[\right(\frac{G}{K_c}{)}^{3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\rho }_0^{1/5}{]}^4$

$=$

$\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{]}^4$

we have,

$0$

$=$

$\frac{d^{2}x}{d{\tilde{r}}^2}+\stackrel{\tilde{{\mathscr{H}}}}{\overbrace{\{4-(\frac{\tilde{\rho }}{\tilde{P}}\left)\frac{{\tilde{M}}_r}{\tilde{r}}\right\}}}\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}}+\underset{\widetilde{{𝒦}}}{\underbrace{\frac{\tilde{\rho }}{\tilde{P}}\{\frac{2\pi {\tilde{\sigma }}^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{\tilde{M}}_r}{(\tilde{r}{)}^3}\}}}x.$

From above, note that, $\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{]}^5$ $=$ $\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-10}\right(\frac{2}{\pi }{)}^{5/2}[\frac{A{\eta }_s}{{\theta }_i}{]}^5;$ and, according to the parameter values associated with the bipolytropic equilibrium configuration, $\left(\frac{{\mu }_e}{{\mu }_c}\right)\frac{{\rho }_c}{\overline{\rho }}$ $=$ $\frac{{\eta }_s^2}{3A{\theta }_i^5}.$

---

CAUTION:

Numerical Integration

Here, the finite-difference representation of the LAWE parallel the approach used in a closely related discussion.

General Approach

The 2^nd-order ODE that must be integrated to obtain the desired eigenvectors has the generic form,

$0$

$=$

$x^{″}+\frac{\tilde{{\mathscr{H}}}}{\tilde{r}}{x}^{\prime }+\widetilde{{𝒦}}x,$

where,

$x^{\prime }$

$=$

$\frac{dx}{d\tilde{r}}$

and

$x^{″}$

$=$

$\frac{d^{2}x}{d{\tilde{r}}^2}.$

Adopting the same approach as before when we integrated the LAWE for pressure-truncated polytropes, we will enlist the finite-difference approximations,

$x^{\prime }$

$\approx$

$\frac{x_{+}-x_{-}}{2\delta \tilde{r}}$

and

$x^{″}$

$\approx$

$\frac{x_{+}-2x_j+x_{-}}{(\delta \tilde{r}{)}^2}.$

The finite-difference representation of the LAWE is, therefore,

$\frac{x_{+}-2x_j+x_{-}}{(\delta \tilde{r}{)}^2}$	$=$	$-\frac{\tilde{{\mathscr{H}}}}{\tilde{r}}\left[\frac{x_{+}-x_{-}}{2\delta \tilde{r}}\right]-\widetilde{{𝒦}}{x}_j$
$\Rightarrow x_{+}-2x_j+x_{-}$	$=$	$-\frac{\delta \tilde{r}}{2\tilde{r}}[x_{+}-x_{-}]\tilde{{\mathscr{H}}}-(\delta \tilde{r}{)}^2\widetilde{{𝒦}}{x}_j$
$\Rightarrow x_{j+1}[1+(\frac{\delta \tilde{r}}{2\tilde{r}}\left)\tilde{{\mathscr{H}}}\right]$	$=$	$[2-(\delta \tilde{r}{)}^2\widetilde{{𝒦}}]x_j-[1-\left(\frac{\delta \tilde{r}}{2\tilde{r}}\right)\tilde{{\mathscr{H}}}]x_{j-1}.$

In what follows we will also find it useful to rewrite ${𝒦}$ in the form,

$\widetilde{{𝒦}}\to \left(\frac{{\tilde{\sigma }}^2}{{\gamma }_{\mathrm{g}}}\right){\widetilde{{𝒦}}}_1-{\alpha }_{\mathrm{g}}{\widetilde{{𝒦}}}_2.$

The relevant coefficient expressions for all regions of the configuration are,

$\tilde{{\mathscr{H}}}$

$\equiv$

$\{4-(\frac{\tilde{\rho }}{\tilde{P}}\left)\frac{{\tilde{M}}_r}{\tilde{r}}\right\}$

,

${\widetilde{{𝒦}}}_1$

$\equiv$

$\frac{2\pi }{3}\left(\frac{\tilde{\rho }}{\tilde{P}}\right)$

and

${\widetilde{{𝒦}}}_2$

$\equiv$

$\left(\frac{\tilde{\rho }}{\tilde{P}}\right)\frac{{\tilde{M}}_r}{{\tilde{r}}^3}.$

See Also