SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2

BiPolytrope with (n_c, n_e) = (5, 1) Renormalized

Part I:   (5, 1) Analytic Renormalize

Part II:  Envelope

III:  Interface Pressure Gradient

Numerical Integration Through Envelope

In an effort to numerically determine the eigenfunction of the envelope, we will follow the procedure described in an accompanying stability analysis of pressure-truncated polytropes to integrate the $n=1$ envelope from the core/envelope interface to the surface. In a closely related chapter titled, Radial Oscillations of n = 1 Polytropic Spheres, we have tried to find analytic expressions for the eigenvector of marginally unstable configurations.

Setup

Continuous Form of LAWE

We begin by writing our generic version of the polytropic LAWE,

	$0=\frac{d^{2}x}{d{\xi }^2}+[4-(n+1)Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+(n+1)\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_g})\frac{{\xi }^2}{\theta }-\alpha Q]\frac{x}{{\xi }^2}$
where:    $Q(\xi )\equiv -\frac{d\ln \theta }{d\ln \xi },$    ${\sigma }_c^2\equiv \frac{3{\omega }^2}{2\pi G{\rho }_c},$     and,     $\alpha \equiv (3-\frac{4}{{\gamma }_{\mathrm{g}}})$

then focus on the $n=1$ case — setting ${\gamma }_g=1+1/n=2$ and $\alpha =+1$ — the relevant LAWE becomes,

$0$

$=$

$\frac{d^{2}x}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{{\eta }^2}{\varphi }-Q\}\frac{x}{{\eta }^2},$

📚 Chatterji (1951) — STEP 1

If we focus on the $n=1$ case but leave $\gamma$ (and, hence, $\alpha$) unspecified, the relevant LAWE becomes, $0$ $=$ $\frac{d^{2}x}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+2\left\{\right(\frac{{\sigma }_c^2}{6\gamma })\frac{{\eta }^2}{\varphi }-\alpha Q\}\frac{x}{{\eta }^2}.$ If, in addition, we make the notation substitutions, $Q$ $\to$ $\frac{\mu }{n+1}=\frac{\mu }{2}$       and,       $\frac{{\sigma }_c^2}{3\gamma }$ $\to$ ${\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2,$ the relevant LAWE becomes, $0$ $=$ $\frac{d^{2}x}{d{\eta }^2}+\{4-\mu \}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\left\{\right(\frac{{\sigma }_c^2}{3\gamma })\frac{{\eta }^2}{\varphi }-\alpha \mu \}\frac{x}{{\eta }^2}$   $=$ $\frac{d^{2}x}{d{\eta }^2}+\left[\frac{4-\mu }{\eta }\right]\frac{dx}{d\eta }+[\frac{{\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2}{\varphi }-\frac{\alpha \mu }{{\eta }^2}]x,$ which, apart from notation, is identical to equation (1) of 📚 Chatterji (1951).

Now, in the broadest context (see our related discussion),

$\varphi (\eta )$	$=$	$A\left[\frac{\sin (\eta -B)}{\eta }\right].$
Beech88, §3, p. 221, Eq. (6)

Therefore, also, we have,

$-\frac{d\varphi }{d\eta }$	$=$	$\frac{A}{{\eta }^2}[\sin (\eta -B)-\eta \cos (\eta -B)],$
$\Rightarrow Q(\eta )\equiv -\frac{d\ln \varphi }{d\ln \eta }$	$=$	$\frac{A}{{\eta }^2}[\sin (\eta -B)-\eta \cos (\eta -B)]\cdot \frac{{\eta }^2}{A\sin (\eta -B)}$
	$=$	$[1-\eta \cot (\eta -B)].$

📚 Chatterji (1951) — STEP 2

In the context of an isolated, $n=1$ polytrope, the appropriate parameter values are, $A=1$ and $B=0$, in which case, $\varphi (\eta )$ $=$ $\frac{\sin \eta }{\eta },$       and,       $\mu =2Q$ $=$ $2(1-\eta \cot \eta ),$ and the relevant LAWE becomes, $0$ $=$ $\frac{d^{2}x}{d{\eta }^2}+\left[\frac{4+2(\eta \cot \eta -1)}{\eta }\right]\frac{dx}{d\eta }+[\frac{\eta {\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2}{\sin \eta }+\frac{2\alpha (\eta \cot \eta -1)}{{\eta }^2}]x$   $=$ $\frac{d^{2}x}{d{\eta }^2}+\left[\frac{2(1+\eta \cot \eta )}{\eta }\right]\frac{dx}{d\eta }+[\frac{\eta {\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2}{\sin \eta }+\frac{2\alpha (\eta \cot \eta -1)}{{\eta }^2}]x,$ which, again apart from notation, is identical to equation (2) of 📚 Chatterji (1951); see also the (unnumbered) equation in the middle of the left-hand-column of p. 223 in 📚 Murphy & Fiedler (1985b).

If we set $\gamma =(n+1)/n=2$ (and correspondingly set $\alpha =[3-4/\gamma ]=+1)$, the $n=1$ LAWE we becomes,

$0$	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-2[1-\eta \cot (\eta -B)\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{{\eta }^3}{A\sin (\eta -B)}-[1-\eta \cot (\eta -B)\left]\right\}\frac{x}{{\eta }^2}$
	$=$	$\frac{d^{2}x}{d{\eta }^2}+2\{1+\eta \cot (\eta -B)\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{{\eta }^3}{A\sin (\eta -B)}-1+\eta \cot (\eta -B)\}\frac{x}{{\eta }^2}.$

Multiplying through by $\varphi$, we can write,

$\left[\frac{A\sin (\eta -B)}{\eta }\right]\frac{d^{2}x}{d{\eta }^2}$

$=$

$-\frac{2A}{\eta }\{\sin (\eta -B)+\eta \cos (\eta -B)\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\frac{2A}{\eta }\{\sin (\eta -B)-\eta \cos (\eta -B)\}\frac{x}{{\eta }^2}-2\left(\frac{{\sigma }_c^2}{12}\right)x.$

Discrete Form of LAWE

In order to integrate this 2^nd-order ODE numerically, we will build from the more general expression for polytropes used in our separate development of a finite-difference scheme, namely,

${\theta }_i{x_i}^{″}$

$=$

$-[4{\theta }_i-(n+1){\xi }_i(-{\theta }^{\text{'}}{)}_i]\frac{{x_i}^{\prime }}{{\xi }_i}-(n+1)[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{{\xi }_i}(-{\theta }^{\text{'}}{)}_i]x_i.$

Making the notation substitutions, $(\xi ,\theta )\to (\eta ,\varphi )$, we have instead,

${\varphi }_i{x_i}^{″}$

$=$

$-[4{\varphi }_i-(n+1){\eta }_i(-{\varphi }^{\text{'}}{)}_i]\frac{{x_i}^{\prime }}{{\eta }_i}-(n+1)[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{{\eta }_i}(-{\varphi }^{\text{'}}{)}_i]x_i.$

Now, adopting the finite-difference expressions,

${x_i}^{\prime }$	$\approx$	$\frac{x_{+}-x_{-}}{2{\mathrm{\Delta }}_{\eta }},$    and,
${x_i}^{″}$	$\approx$	$\frac{x_{+}-2x_i+x_{-}}{{\mathrm{\Delta }}_{\eta }^2},$

the discrete form of the LAWE becomes,

$x_{+}\stackrel{\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{1}}{\overbrace{[2{\varphi }_i+\frac{4{\mathrm{\Delta }}_{\eta }{\varphi }_i}{{\eta }_i}-{\mathrm{\Delta }}_{\eta }(n+1)(-{\varphi }^{\text{'}}{)}_i]}}$

$=$

$x_{-}\stackrel{\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{2}}{\overbrace{[\frac{4{\mathrm{\Delta }}_{\eta }{\varphi }_i}{{\eta }_i}-{\mathrm{\Delta }}_{\eta }(n+1)(-{\varphi }^{\text{'}}{)}_i-2{\varphi }_i]}}+x_i\stackrel{\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{3}}{\overbrace{\{4{\varphi }_i-\frac{{\mathrm{\Delta }}_{\eta }^2(n+1)}{3}[\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha (-\frac{3{\varphi }^{\text{'}}}{\eta }{)}_i]\}}}.$

When applied specifically to an $n=1$, polytropic configuration, we should insert the following specific expressions:

${\gamma }_g$	$=$	$1+\frac{1}{n}=2,$
$\alpha$	$=$	$3-\frac{4}{{\gamma }_g}=+1,$
${\varphi }_i$	$=$	$A\left[\frac{\sin ({\eta }_i-B)}{{\eta }_i}\right],$
$(-{\varphi }^{\prime }{)}_i$	$=$	$\frac{A}{{\eta }_i^2}[\sin ({\eta }_i-B)-{\eta }_i\cos ({\eta }_i-B)].$

Pressure-Truncated n = 1 Polytrope

In the case of an isolated $n=1$ polytrope, we must set $B=0$; in addition, it is customary to set $A=1$. The relevant LAWE is, then,

$0$

$=$

$\frac{d^{2}x}{d{\eta }^2}+2\{1+\eta \cot (\eta )\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{{\eta }^3}{\sin (\eta )}-1+\eta \cot (\eta )\}\frac{x}{{\eta }^2}.$

Review of Trial Analytic Eigenfunction

This is the same 2^nd-order ODE that we derived in a separate discussion; there it was accompanied by the surface boundary condition,

$-\frac{d\ln x}{d\ln \xi }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	=	$\left(\frac{3-n}{n+1}\right)+\frac{n{\sigma }_c^2}{6(n+1)}[\frac{\xi }{{\theta }^{\prime }}{]}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$
$\Rightarrow -\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	=	$1+\frac{{\sigma }_c^2}{12}[\frac{{\eta }^3}{(\eta \cos \eta -\sin \eta )}{]}_{\eta =\pi }=1-\frac{{\pi }^2{\sigma }_c^2}{12}.$

From, for example, a separate succinct demonstration, we appreciate that if the displacement function is assumed to be,

$x_P$

$=$

$\frac{3}{{\eta }^2}[1-\eta \cot \eta ]$

… that is, $x_P$ $=$ $\frac{3}{{\eta }^2}-\frac{3\cos \eta }{\eta \sin \eta },$ in which case, $\frac{d{x}_P}{d\eta }$ $=$ $-\frac{6}{{\eta }^3}+3[\frac{\cos \eta }{{\eta }^2\sin \eta }+\frac{1}{\eta }+\frac{{\cos }^2\eta }{\eta {\sin }^2\eta }],$ and, $\frac{d^2{x}_P}{d{\eta }^2}$ $=$ $+\frac{18}{{\eta }^4}+3\frac{d}{d\eta }\left[\frac{\cos \eta }{{\eta }^2\sin \eta }\right]+3\frac{d}{d\eta }\left[\frac{1}{\eta }\right]+3\frac{d}{d\eta }\left[\frac{{\cos }^2\eta }{\eta {\sin }^2\eta }\right]$   $=$ $+\frac{18}{{\eta }^4}-3[\frac{2\cos \eta }{{\eta }^3\sin \eta }+\frac{{\cos }^2\eta }{{\eta }^2{\sin }^2\eta }+\frac{\sin \eta }{{\eta }^2\sin \eta }]-\left[\frac{3}{{\eta }^2}\right]-3[\frac{{\cos }^2\eta }{{\eta }^2{\sin }^2\eta }+\frac{2{\cos }^3\eta }{\eta {\sin }^3\eta }+\frac{2\cos \eta }{\eta \sin \eta }].$ Hence, LAWE $=$ $\{\frac{2}{\eta }+\frac{2\cos \eta }{\sin \eta }\}\{-\frac{6}{{\eta }^3}+[\frac{3\cos \eta }{{\eta }^2\sin \eta }+\frac{3}{\eta }+\frac{3{\cos }^2\eta }{\eta {\sin }^2\eta }\left]\right\}+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{{\eta }^3}{\sin (\eta )}-1+\frac{\eta \cos \eta }{\sin \eta }\}[\frac{3}{{\eta }^4}-\frac{3\cos \eta }{{\eta }^3\sin \eta }]$     $+\frac{18}{{\eta }^4}-3[\frac{2\cos \eta }{{\eta }^3\sin \eta }+\frac{{\cos }^2\eta }{{\eta }^2{\sin }^2\eta }+\frac{\sin \eta }{{\eta }^2\sin \eta }]-\left[\frac{3}{{\eta }^2}\right]-3[\frac{{\cos }^2\eta }{{\eta }^2{\sin }^2\eta }+\frac{2{\cos }^3\eta }{\eta {\sin }^3\eta }+\frac{2\cos \eta }{\eta \sin \eta }]$   $=$ $[\frac{2}{\eta }+2\cot \eta ][-\frac{6}{{\eta }^3}+\frac{3\cot \eta }{{\eta }^2}+\frac{3}{\eta }+\frac{3{\cot }^2\eta }{\eta }]+\left\{\right(\frac{{\sigma }_c^2}{6}\left)\frac{{\eta }^3}{\sin \eta }\right\}[\frac{3}{{\eta }^4}-\frac{3\cot \eta }{{\eta }^3}]+[-2+2\eta \cot \eta ][\frac{3}{{\eta }^4}-\frac{3\cot \eta }{{\eta }^3}]$     $+\frac{18}{{\eta }^4}-[\frac{6\cot \eta }{{\eta }^3}+\frac{3{\cot }^2\eta }{{\eta }^2}+\frac{3}{{\eta }^2}]-\left[\frac{3}{{\eta }^2}\right]-[\frac{3{\cot }^2\eta }{{\eta }^2}+\frac{6{\cot }^3\eta }{\eta }+\frac{6\cot \eta }{\eta }]$   $=$ $\left\{\right(\frac{{\sigma }_c^2}{2}\left)\frac{1}{\eta \sin \eta }\right\}[1-\eta \cot \eta ]+[-\frac{12}{{\eta }^4}+\frac{6}{{\eta }^2}]+[2\cot \eta ][\frac{3}{{\eta }^3}+\frac{3\cot \eta }{{\eta }^2}]+[2\cot \eta ][-\frac{6}{{\eta }^3}+\frac{3\cot \eta }{{\eta }^2}+\frac{3}{\eta }+\frac{3{\cot }^2\eta }{\eta }]$     $+[-\frac{6}{{\eta }^4}+\frac{6\cot \eta }{{\eta }^3}]+[2\cot \eta ][\frac{3}{{\eta }^3}-\frac{3\cot \eta }{{\eta }^2}]+\frac{18}{{\eta }^4}-\frac{6}{{\eta }^2}+[2\cot \eta ][-\frac{3{\cot }^2\eta }{\eta }-\frac{3\cot \eta }{{\eta }^2}-\frac{3}{\eta }-\frac{3}{{\eta }^3}]$   $=$ $\left\{\right(\frac{{\sigma }_c^2}{2}\left)\frac{1}{\eta \sin \eta }\right\}[1-\eta \cot \eta ]+[2\cot \eta ][-\frac{3}{{\eta }^3}+\frac{6\cot \eta }{{\eta }^2}+\frac{3}{\eta }+\frac{3{\cot }^2\eta }{\eta }]$     $+[2\cot \eta ][\frac{3}{{\eta }^3}-\frac{3{\cot }^2\eta }{\eta }-\frac{6\cot \eta }{{\eta }^2}-\frac{3}{\eta }]$   $=$ $\left\{\right(\frac{{\sigma }_c^2}{2}\left)\frac{1}{\eta \sin \eta }\right\}[1-\eta \cot \eta ].$

the $n=1$ LAWE reduces to …

LAWE

$=$

$\left\{\right(\frac{{\sigma }_c^2}{2}\left)\frac{1}{\eta \sin \eta }\right\}[1-\eta \cot \eta ].$

ASSESSMENT: 

1. If we set ${\sigma }_c^2=0$, the right-hand-side of this expression goes to zero — and, hence, the $n=1$ LAWE is satisfied — for any chosen truncation radius in the range, $0<{\eta }_i<\pi$. (We have not included the isolated $n=1$ polytrope because $x_P$ blows up at its surface, ${\eta }_i=\pi$.)
2. At the surface, ${\eta }_i$, the slope of this trial eigenfunction is,

   $\frac{1}{3}\cdot \frac{d{x}_P}{d\eta }{|}_i$

   $=$

   $[\frac{\cot {\eta }_i}{{\eta }_i^2}+\frac{1}{{\eta }_i}+\frac{{\cot }^2{\eta }_i}{{\eta }_i}]-\frac{2}{{\eta }_i^3}.$

   By contrast, as stated above, the eigenvalue problem will be properly solved only if the surface slope is,

   $-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	=	$1+\frac{{\xcancel{{\sigma }_c^2}}^0}{12}[\frac{{\eta }^3}{(\eta \cos \eta -\sin \eta )}{]}_{\eta =\pi }=1$
   $\Rightarrow \frac{1}{3}\cdot \frac{d{x}_P}{d\eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	=	$-\frac{x_P}{3\eta }{|}_i$
   	=	$-\frac{1}{{\eta }_i^3}[1-{\eta }_i\cot {\eta }_i]=\frac{\cot {\eta }_i}{{\eta }_i^2}-\frac{1}{{\eta }_i^3}.$

   These two slopes do not appear to be the same, for any allowed choice of ${\eta }_i$. We conclude, therefore, that no model along the sequence of pressure-truncated $n=1$ polytropes is marginally unstable.

Determining Discrete Representation of Eigenfunction

Let's numerically integrate the discrete form of the $n=1$ LAWE over the radial coordinate range, $0\le {\eta }_i\le {\eta }_s$. Following our discussion of the more general polytropic case, we will kickstart integration from the center, outward, via the expression,

$x_2$

$=$

$x_1[1-\frac{(n+1)\mathfrak{𝔉}{\mathrm{\Delta }}_{\eta }^2}{60}],$

where,

$\mathfrak{𝔉}$

$\equiv$

$[\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha ].$

Here, we will restrict our investigation to the case where ${\gamma }_g=(n+1)/n=2$, in which case, $\alpha =(3-4/{\gamma }_g)=+1$, $\mathfrak{𝔉}=({\sigma }_c^2-4)/2$, and

$x_2$

$=$

$x_1[1-\frac{({\sigma }_c^2-4){\mathrm{\Delta }}_{\eta }^2}{60}].$

EXAMPLE:   $A=1$, $B=0$, ${\mathrm{\Delta }}_{\eta }=\pi /99=0.031733259$; evaluated over range, $0\le {\eta }_i\le \pi$. ${\sigma }_c^2$ $\mathfrak{𝔉}$ $x_2$ ${\varphi }_2$ $(-{\varphi }^{\prime }{)}_2$ TERM1 TERM2 TERM3 $x_3$ 5 +0.5 0.999983217 0.999832175 0.010576686 5.998321784 1.998993085 3.998992898 0.999932829

Isolated n = 1 Polytrope

If we integrate all the way out to the natural, zero-pressure surface of our $n=1$ polytrope, then ${\eta }_s=\pi$ and — as derived in our discussion of the equilibrium structure of n = 1 polytropes — $({\rho }_c/\overline{\rho })={\pi }^2/3$. In line with our discussion of Schwarzschild's model of oscillations in $n=3$ polytropes, we therefore expect the boundary condition at the surface of our $n=1$ configurations to be given by the expression,

$-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	$=$	$-\frac{1}{2}\left\{\right[\mathfrak{𝔉}+2\alpha \left]\right(\frac{{\rho }_c}{\overline{\rho }})-2\alpha \}$
	$=$	$1-\left(\frac{{\sigma }_c^2{\pi }^2}{12}\right),$

as reviewed immediately above.

📚 Chatterji (1951) — STEP 3

Here we examine what the boundary condition should be at the surface of an isolated $n=1$ polytrope. Given that, quite generally in the context of isolated polytropes, $\alpha$ $=$ $3-\frac{4}{\gamma },$       and,       $\mathfrak{𝔉}$ $=$ $\frac{{\sigma }_c^2}{\gamma }-2\alpha ,$ the surface boundary condition is, $-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$ $=$ $-\frac{1}{2}\left\{\right[\mathfrak{𝔉}+2\alpha \left]\right(\frac{{\rho }_c}{\overline{\rho }})-2\alpha \}$   $=$ $-\frac{{\sigma }_c^2}{2\gamma }\left(\frac{{\rho }_c}{\overline{\rho }}\right)+3-\frac{4}{\gamma }$   $=$ $3-\frac{1}{\gamma }[4+\frac{{\sigma }_c^2}{2}(\frac{{\rho }_c}{\overline{\rho }}\left)\right].$ Considerations: For an isolated $n=1$ polytrope, the central-to-mean density is, ${\rho }_c/\overline{\rho }={\pi }^2/3$. Hence, $\{-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{\}}_{n=1}$ $=$ $3-\frac{1}{\gamma }[4+\frac{{\pi }^2{\sigma }_c^2}{6}].$ If, in addition, we set $\gamma =(n+1)/n=2\Rightarrow \alpha =+1$, then, $\{-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{\}}_{n=1,\alpha =+1}$ $=$ $3-\frac{1}{2}[4+\frac{{\pi }^2{\sigma }_c^2}{6}]=1-\frac{{\pi }^2{\sigma }_c^2}{12}.$ If we set $\gamma =[4/(3-\alpha )]$ for all other values of $\alpha$, we can write, $\{-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{\}}_{n=1,\alpha }$ $=$ $3-(3-\alpha )[1+\frac{{\pi }^2{\sigma }_c^2}{24}]=\alpha -(3-\alpha )\left[\frac{{\pi }^2{\sigma }_c^2}{24}\right].$ At the bottom of p. 469 of his article, 📚 Chatterji (1951) states that, "… the condition for the Node to fall at the surface of the star is, $3f+z\frac{df}{dz}$ $=$ $0,$ which we interpret to mean, $\{-\frac{d\ln f}{d\ln z}{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{\}}_{n=1,\alpha }$ $=$ $3.$ He goes on to say, "As the adiabatic approximation breaks down near the boundary we have not strictly followed this condition."

This should be compared with the finite-difference representation of the logarithmic derivative, namely,

$+\frac{\mathrm{\Delta }\ln x}{\mathrm{\Delta }\ln \xi }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$

$\approx$

$\frac{{\xi }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{x_N}\left[\frac{x_{N+1}-x_{N-1}}{2{\mathrm{\Delta }}_{\xi }}\right].$

CAUTION!     Because, for each guess of ${\sigma }_c^2$, the eigenfunction climbs (or plummets) rapidly as we approach the surface, in practice we evaluated the finite-difference representation of the logarithmic derivative at a zone location that is a bit inside of the actual surface; for example, when we divided the equilibrium configuration into $N=100$ grid zones, we evaluated the "surface" derivative at zone number 97.

Here we have adopted an analysis that closely resembles our discussion of the analysis of $n=3$ polytropes that was published by 📚 Schwarzschild (1941). Here we have divided our model into $N=100$ radial zones and, using this algorithm, integrated the LAWE from the center of the configuration to the surface, for $\alpha =+1$, and approximately 40 different chosen values of the frequency parameter across the range, $-2\le \mathfrak{𝔉}\le +18$. The radial displacement functions resulting from these integrations are presented in the following figure as an animation sequence. The specified value of $\mathfrak{𝔉}$ is displayed at the top of each animation frame, and the resulting displacement function, $x(\eta )$, is traced by the small, red circular markers in each frame.

file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = n1Oscillations Four Modes of Oscillation of an Isolated, $n=1$ Polytrope
Mode	${\sigma }_c^2$	Neg. Slope $1-({\sigma }_c^2{\pi }^2/12)$	$\mathfrak{𝔉}=\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha$
Fundamental	2.2405295	3.1287618	-0.879735
1st Overtone	6.340767	-32.06757	1.1703835
2nd Overtone	13.694927	-153.2545	4.8474635
3rd Overtone	28.462829	-665.3074	12.231415
file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n1Eigenvectors/MovieFrames/KeyAnimation/n1osc06.gif

L. D. Chatterji (1951) Radial Oscillations of a Gaseous Star of Polytropic Index I Publication of the Indian National Science Academy, Vol. 17, No. 6, pp. 467 - 470
Amplitudes and frequencies of the displacement functions for three modes, assuming $\alpha =0.6$ and $\gamma =5/3$
Amplitudes extracted from Chatterji's Table II (p. 468)	Plot for comparison with Chatterji's Fig. 1 (p. 469)	Eigenfrequencies extracted from Chatterji's Table I (p. 468)
1st 2nd Abscissa Fundamental Overtone Overtone 0.0 1.000000 1.000000 1.000000 0.1 1.000169 0.998882 0.996820 0.2 1.000677 0.995518 0.987281 0.3 1.001525 0.989874 0.971386 0.4 1.002716 0.981890 0.949136 0.5 1.004251 0.971495 0.920555 0.6 1.006143 0.958596 0.885671 0.7 1.008389 0.943052 0.844515 0.8 1.011009 0.924713 0.797168 0.9 1.014006 0.903389 0.743728 1.0 1.017390 0.878859 0.684350 1.1 1.021175 0.850865 0.619257 1.2 1.025374 0.819105 0.548757 1.3 1.030004 0.783221 0.473277 1.4 1.035102 0.742868 0.393396 1.5 1.040853 0.697654 0.309941 1.6 1.047084 0.646920 0.223844 1.7 1.053810 0.590042 0.136419 1.8 1.061066 0.526318 0.049379 1.9 1.068886 0.454939 -0.035065 2.0 1.077309 0.374974 -0.114079 2.1 1.086374 0.285357 -0.184050 2.2 1.096138 0.184876 -0.240444 2.3 1.106766 0.072271 -0.277903 2.4 1.118204 -0.054152 -0.288854 2.5 1.130496 -0.196154 -0.264401 2.6 1.143708 -0.355598 -0.193931 2.7 1.157893 -0.534282 -0.065351 2.8 1.173099 -0.733397 0.133422 2.9 1.189288 -0.951614 0.408698 3.0 1.206063 -1.175893 0.734701 3.1 1.218532 -1.251806 0.780444		Mode ${\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$ ${\sigma }_c^2$ Fundamental 0.231 1.155 1st Overtone 1.517 7.585 2nd Overtone 3.580 17.900
The solid circular markers in the plot (center panel) show how the amplitude of the displacement function varies with radius $0\le \eta <\pi$ for three separate radial modes, according to the data provided in Table II of 📚 Chatterji (1951), which has been reproduced here in the scrollable left-hand panel. In the plot, blue is the fundamental mode, red is the 1st overtone, green is the 2nd overtone. The (square of the) eigenfrequency corresponding to each mode, according to Table I of 📚 Chatterji (1951), is provided in the column of the right-hand panel that is (highlighted in pink and) labeled ${\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$; also listed are the corresponding values of ${\sigma }_c^2=3\gamma {\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$.   The smooth, solid curves in the middle-panel plot are not fits to Chatterji's data. Rather, they result from our own, independent numerical integration of the relevant LAWE, assuming that Chatterji's published values of the (square of the) eigenfrequency are correct for all three modes. In all three cases for the specified eigenfrequency, there is excellent agreement between our determination of the radial eigenfunction and the determination obtained by 📚 Chatterji (1951).

We are exceptionally pleased to find that, for each of the three modes of oscillation, the displacement function obtained via our integration of the LAWE (solid curves in the figure) runs through the discrete points recorded by 📚 Chatterji (1951) (solid circular markers in the figure). But in doing so, we find from our higher resolution model that there is an inflection point just inside the surface of the model; this is not the smooth behavior that is expected as the surface is approached. In an effort to correct this behavior, we have changed the constraint that is applied while integrating the LAWE from the center, outward: Instead of forcing ${\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$ to match the value published by 📚 Chatterji (1951), we have let the value of this oscillation frequency vary while enforcing the surface boundary condition describe above as CONSIDERATION "C". The resulting "improved" solution is shown in the figure that follows.

Our Imposed Surface Boundary Condition: $\{-\frac{d\ln x}{d\ln \eta }{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{\}}_{n=1,\alpha }$ $=$ $\alpha -(3-\alpha )\left[\frac{{\pi }^2{\sigma }_c^2}{24}\right].$
Amplitudes and frequencies of the displacement functions for three modes, assuming $\alpha =0.6$ and $\gamma =5/3$
Amplitudes determined from our numerical integration of the LAWE	Our version of Chatterji's Fig. 1	Eigenfrequencies determined from our integration
Abscissa Fundamental 1st Overtone 2nd Overtone 0.000000000 1.000000000 1.000000000 1.000000000 0.031733259 1.000047844 0.999943498 0.999786141 0.063466518 1.000089016 0.999601959 0.998867632 0.095199777 1.000174709 0.999061021 0.997382877 0.126933037 1.000294726 0.998302963 0.995304066 0.158666296 1.000449107 0.997327028 0.992631032 0.190399555 1.000637906 0.996132244 0.989363571 0.222132814 1.000861191 0.994717421 0.985501447 0.253866073 1.001119039 0.993081149 0.981044389 0.285599332 1.001411544 0.991221792 0.975992105 0.317332591 1.001738809 0.989137492 0.970344285 0.34906585 1.002100951 0.986826157 0.964100608 0.38079911 1.002498098 0.984285466 0.957260757 0.412532369 1.002930394 0.981512860 0.949824425 0.444265628 1.003397993 0.978505538 0.94179133 0.475998887 1.003901064 0.975260454 0.933161228 0.507732146 1.004439786 0.971774312 0.923933924 0.539465405 1.005014356 0.968043557 0.914109296 0.571198664 1.005624981 0.964064374 0.903687306 0.602931923 1.006271884 0.959832677 0.892668024 0.634665183 1.006955299 0.955344103 0.881051648 0.666398442 1.007675478 0.950594005 0.868838527 0.698131701 1.008432684 0.945577446 0.856029188 0.72986496 1.009227198 0.940289184 0.842624363 0.761598219 1.010059312 0.934723671 0.828625017 0.793331478 1.010929336 0.928875035 0.814032388 0.825064737 1.011837596 0.922737076 0.798848013 0.856797996 1.01278443 0.916303251 0.783073774 0.888531256 1.013770197 0.909566661 0.766711937 0.920264515 1.01479527 0.902520043 0.749765195 0.951997774 1.015860038 0.895155752 0.732236722 0.983731033 1.016964908 0.887465751 0.714130222 1.015464292 1.018110307 0.879441592 0.695449985 1.047197551 1.019296676 0.8710744 0.676200956 1.07893081 1.020524478 0.862354861 0.656388792 1.110664069 1.021794193 0.853273198 0.636019943 1.142397329 1.023106322 0.843819155 0.615101726 1.174130588 1.024461383 0.833981977 0.593642407 1.205863847 1.025859919 0.82375039 0.5716513 1.237597106 1.027302491 0.813112573 0.549138858 1.269330365 1.028789682 0.802056142 0.526116785 1.301063624 1.030322099 0.790568118 0.502598146 1.332796883 1.03190037 0.778634906 0.478597499 1.364530142 1.033525148 0.766242261 0.454131023 1.396263402 1.035197111 0.753375263 0.429216666 1.427996661 1.03691696 0.740018284 0.403874305 1.45972992 1.038685425 0.726154954 0.378125913 1.491463179 1.04050326 0.711768126 0.351995745 1.523196438 1.042371249 0.696839839 0.325510537 1.554929697 1.044290203 0.681351278 0.298699722 1.586662956 1.046260964 0.665282732 0.271595661 1.618396215 1.048284403 0.64861355 0.244233898 1.650129475 1.050361424 0.631322093 0.216653434 1.681862734 1.052492964 0.613385687 0.188897019 1.713595993 1.054679993 0.594780567 0.161011472 1.745329252 1.056923518 0.575481822 0.133048033 1.777062511 1.059224581 0.555463336 0.105062731 1.80879577 1.061584262 0.534697725 0.077116794 1.840529029 1.064003681 0.513156272 0.049277088 1.872262289 1.066484 0.490808851 0.021616595 1.903995548 1.069026422 0.467623859 -0.005785076 1.935728807 1.071632194 0.443568129 -0.032841123 1.967462066 1.074302612 0.418606853 -0.059456951 1.999195325 1.077039017 0.392703483 -0.085529513 2.030928584 1.079842802 0.365819646 -0.110946599 2.062661843 1.082715411 0.337915031 -0.135586067 2.094395102 1.085658345 0.308947292 -0.159315006 2.126128362 1.08867316 0.278871926 -0.181988834 2.157861621 1.091761474 0.247642155 -0.203450318 2.18959488 1.094924965 0.215208794 -0.223528513 2.221328139 1.098165381 0.181520118 -0.242037616 2.253061398 1.101484536 0.146521711 -0.258775724 2.284794657 1.104884319 0.110156314 -0.273523491 2.316527916 1.108366697 0.072363659 -0.286042681 2.348261175 1.111933717 0.033080298 -0.296074601 2.379994435 1.115587515 -0.007760587 -0.303338425 2.411727694 1.11933032 -0.050229374 -0.307529379 2.443460953 1.12316446 -0.094400211 -0.308316806 2.475194212 1.127092372 -0.140351229 -0.305342096 2.506927471 1.131116609 -0.188164774 -0.298216493 2.53866073 1.135239852 -0.237927637 -0.286518776 2.570393989 1.139464924 -0.289731302 -0.269792846 2.602127248 1.143794808 -0.343672193 -0.247545241 2.633860508 1.148232666 -0.39985192 -0.219242641 2.665593767 1.152781872 -0.458377516 -0.184309461 2.697327026 1.157446048 -0.519361653 -0.142125667 2.729060285 1.162229122 -0.582922804 -0.092025093 2.760793544 1.167135405 -0.649185328 -0.033294642 2.792526803 1.172169712 -0.718279394 0.034824895 2.824260062 1.17733755 -0.790340637 0.113135186 2.855993321 1.18264542 -0.865509304 0.202471616 2.887726581 1.188101346 -0.943928474 0.303685139 2.91945984 1.193715839 -1.025740391 0.417605887 2.951193099 1.199503802 -1.111078809 0.54496891 2.982926358 1.205488688 -1.200051936 0.686253524 3.014659617 1.2117128 -1.292700149 0.841298052 3.046392876 1.21826791 -1.388871861 1.008209416 3.078126135 1.225416475 -1.487741753 1.179292479 3.109859394 1.234427405 -1.584566325 1.314705175		Mode ${\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$ ${\sigma }_c^2$ Fundamental 0.2298579 1.1492896 1st Overtone 1.4733124 7.366562 2nd Overtone 3.3484654 16.742327
The solid circular markers in the plot (center panel) show how the amplitude of the displacement function varies with radius $0\le \eta <\pi$ for three separate radial modes, according to the data provided in Table II of 📚 Chatterji (1951), which has been reproduced here in the scrollable left-hand panel. In the plot, blue is the fundamental mode, red is the 1st overtone, green is the 2nd overtone. The (square of the) eigenfrequency corresponding to each mode, according to Table I of 📚 Chatterji (1951), is provided in the column of the right-hand panel that is (highlighted in pink and) labeled ${\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$; also listed are the corresponding values of ${\sigma }_c^2=3\gamma {\omega }_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{i}}^2$.   The smooth, solid curves in the middle-panel plot are not fits to Chatterji's data. Rather, they result from our own, independent numerical integration of the relevant LAWE, assuming that Chatterji's published values of the (square of the) eigenfrequency are correct for all three modes. In all three cases for the specified eigenfrequency, there is excellent agreement between our determination of the radial eigenfunction and the determination obtained by 📚 Chatterji (1951).

Pressure-Truncated n = 1 Polytrope

Drawing from an accompanying discussion, if the polytropic configuration is truncated by the pressure, $P_e$, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,

$-\frac{d\ln x}{d\ln \eta }$

$=$

$3$     at     $\eta =\tilde{\eta }$.

Bipolytropic Envelope (Trial Simplification)

For the $n=1$ envelope of a $(n_c,n_e)=(5,1)$ bipolytrope, the relevant LAWE is,

LAWE	$=$	$\frac{d^{2}x}{d{\eta }^2}$
		$+2\{1+\frac{\eta \cos (\eta -B)}{\sin (\eta -B)}\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }$
		$+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{{\eta }^3}{A\sin (\eta -B)}-1+\frac{\eta \cos (\eta -B)}{\sin (\eta -B)}\}\frac{x}{{\eta }^2}.$

Three terms in this expression blow up at the surface, where $(\eta -B)\to \pi$ and, hence, $\sin (\eta -B)\to 0$. We can improve the behavior of this LAWE expression by assuming that the eigenfunction is of the form,

$x$

$=$

$f(\eta )[\sin (\eta -B){]}^m,$

in which case,

$\frac{dx}{d\eta }$	$=$	$[\sin (\eta -B){]}^m\frac{df}{d\eta }+mf(\eta )[\sin (\eta -B){]}^{m-1}\cos (\eta -B);$     and,
$\frac{d^{2}x}{d{\eta }^2}$	$=$	$[\sin (\eta -B){]}^m\frac{d^{2}f}{d{\eta }^2}+2m[\sin (\eta -B){]}^{m-1}\cos (\eta -B)\cdot \frac{df}{d\eta }+m(m-1)f(\eta )[\sin (\eta -B){]}^{m-2}{\cos }^{}(\eta -B)-mf(\eta )[\sin (\eta -B){]}^{m-1}\sin (\eta -B).$

This gives,

LAWE	$=$	$[\sin (\eta -B){]}^m\frac{d^{2}f}{d{\eta }^2}+2m[\sin (\eta -B){]}^{m-1}\cos (\eta -B)\cdot \frac{df}{d\eta }+m(m-1)f(\eta )[\sin (\eta -B){]}^{m-2}{\cos }^{}(\eta -B)-mf(\eta )[\sin (\eta -B){]}^{m-1}\sin (\eta -B)$
		$+2\{1+\frac{\eta \cos (\eta -B)}{\sin (\eta -B)}\}\frac{1}{\eta }\cdot \left\{\right[\sin (\eta -B){]}^m\frac{df}{d\eta }+mf(\eta )[\sin (\eta -B){]}^{m-1}\cos (\eta -B)\}$
		$+2\left\{\right(\frac{{\sigma }_c^2}{12})\frac{\eta }{A\sin (\eta -B)}-\frac{1}{{\eta }^2}+\frac{\cos (\eta -B)}{\eta \sin (\eta -B)}\}f(\eta )[\sin (\eta -B){]}^m$
	$=$	$[\sin (\eta -B){]}^m\frac{d^{2}f}{d{\eta }^2}+2\{\frac{1}{\eta }+\frac{\cos (\eta -B)}{\sin (\eta -B)}\left\}\right[\sin (\eta -B){]}^m\frac{df}{d\eta }+2m[\sin (\eta -B){]}^{m-1}\cos (\eta -B)\cdot \frac{df}{d\eta }$
		$+m(m-1)[\sin (\eta -B){]}^{m-2}{\cos }^{}(\eta -B)\cdot f(\eta )-m[\sin (\eta -B){]}^{m-1}\sin (\eta -B)\cdot f(\eta )$
		$+2m\{\frac{1}{\eta }+\frac{\cos (\eta -B)}{\sin (\eta -B)}\}[\sin (\eta -B){]}^{m-1}\cos (\eta -B)\cdot f(\eta )+2\{\left(\frac{{\sigma }_c^2}{12}\right)\frac{\eta }{A\sin (\eta -B)}-\frac{1}{{\eta }^2}+\frac{\cos (\eta -B)}{\eta \sin (\eta -B)}\left\}\right[\sin (\eta -B){]}^{m}f(\eta )$
	$=$	$[\sin (\eta -B){]}^m\frac{d^{2}f}{d{\eta }^2}+2\{\frac{1}{\eta }[\sin (\eta -B)]+(m+1)\cos (\eta -B)\left\}\right[\sin (\eta -B){]}^{m-1}\frac{df}{d\eta }$
		$+\{m(m-1){\cos }^2(\eta -B)-m[\sin (\eta -B){]}^2\left\}\right[\sin (\eta -B){]}^{m-2}\cdot f(\eta )$
		$+\left\{\frac{2m}{\eta }\right[\sin (\eta -B)]\cos (\eta -B)+2m{\cos }^{}(\eta -B)+[\left(\frac{{\sigma }_c^2}{6}\right)\frac{\eta }{A}\left]\right[\sin (\eta -B)]-\frac{2}{{\eta }^2}[\sin (\eta -B){]}^2+\frac{2\cos (\eta -B)}{\eta }[\sin (\eta -B)]\left\}\right[\sin (\eta -B){]}^{m-2}\cdot f(\eta ).$

Try m = 1 and m = 2

If we set $m=1$, there are still terms in the LAWE expression that blow up at the surface, where $(\eta -B)\to \pi$ and, hence, $\sin (\eta -B)\to 0$. Instead, let's try $m=2$:

$\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}{|}_{m=2}$	$=$	$[\sin (\eta -B){]}^2\frac{d^{2}f}{d{\eta }^2}+2\{\frac{1}{\eta }[\sin (\eta -B)]+3\cos (\eta -B)\left\}\right[\sin (\eta -B)]\frac{df}{d\eta }$
		$+\{2{\cos }^2(\eta -B)-2[\sin (\eta -B){]}^2\}\cdot f(\eta )$
		$+\left\{\frac{4}{\eta }\right[\sin (\eta -B)]\cos (\eta -B)+4{\cos }^{}(\eta -B)+[\left(\frac{{\sigma }_c^2}{6}\right)\frac{\eta }{A}\left]\right[\sin (\eta -B)]-\frac{2}{{\eta }^2}[\sin (\eta -B){]}^2+\frac{2\cos (\eta -B)}{\eta }[\sin (\eta -B)]\}\cdot f(\eta ),$

which, at the surface $\eta \to {\eta }_s=(\pi +B)$, reduces to …

$\{\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}{|}_{m=2}{\}}_{{\eta }_s}$

$=$

$6f({\eta }_s).$

Hence, this LAWE will be satisfied for any function, $f(\eta )$, that goes to zero at the surface.

Try m = 3

Setting $m=3$, we obtain,

$\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}{|}_{m=3}$	$=$	$[\sin (\eta -B){]}^3\frac{d^{2}f}{d{\eta }^2}+2\{\frac{1}{\eta }[\sin (\eta -B)]+4\cos (\eta -B)\left\}\right[\sin (\eta -B){]}^2\frac{df}{d\eta }$
		$+\{6{\cos }^2(\eta -B)-3[\sin (\eta -B){]}^2\left\}\right[\sin (\eta -B)]\cdot f(\eta )$
		$+\left\{\frac{6}{\eta }\right[\sin (\eta -B)]\cos (\eta -B)+6{\cos }^{}(\eta -B)+[\left(\frac{{\sigma }_c^2}{6}\right)\frac{\eta }{A}\left]\right[\sin (\eta -B)]-\frac{2}{{\eta }^2}[\sin (\eta -B){]}^2+\frac{2\cos (\eta -B)}{\eta }[\sin (\eta -B)]\left\}\right[\sin (\eta -B)]\cdot f(\eta ).$

which trivially reduces to zero at the surface because, $\eta \to {\eta }_s=(\pi +B)\Rightarrow \sin (\eta -B)\to 0$. For all other relevant radial positions in the envelope, ${\eta }_i\le \eta <{\eta }_s$, we can divide through by ${\sin }^3(\eta -B)$ to obtain,

$[\sin (\eta -B){]}^{-3}\times \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}{|}_{m=3}$

$=$

$\frac{d^{2}f}{d{\eta }^2}+2\{\frac{1}{\eta }+4\cot (\eta -B)\}\cdot \frac{df}{d\eta }+\{12{\cot }^{}(\eta -B)-3+\frac{8}{\eta }[\cot (\eta -B)]+[\left(\frac{{\sigma }_c^2}{6}\right)\frac{\eta }{A\sin (\eta -B)}]-\frac{2}{{\eta }^2}\}\cdot f(\eta ),$

Boundary Condition

In addition, there is a (boundary condition) constraint on the slope of the eigenfunction at the surface. So, let's examine …

$\frac{d\ln x}{d\ln \eta }{|}_{m=2}\equiv [\frac{\eta }{x}\cdot \frac{dx}{d\eta }{]}_{m=2}$	$=$	$\frac{\eta }{f(\eta ){\sin }^2(\eta -B)}\{{\sin }^2(\eta -B)\frac{df}{d\eta }+2f(\eta )\sin (\eta -B)\cos (\eta -B)\}$
	$=$	$\frac{d\ln f}{d\ln \eta }+2\eta \cot (\eta -B)$

Now, from above, we appreciate that when $\varphi =A\sin (\eta -B)/\eta$,

$\eta \cot (\eta -B)$	$=$	$1+\frac{d\ln \varphi }{d\ln \xi }$
$\Rightarrow \frac{d\ln x}{d\ln \eta }{|}_{m=2}$	$=$	$\frac{d\ln f}{d\ln \eta }+2[1+\frac{d\ln \varphi }{d\ln \xi }].$

It therefore appears as though we should adopt the function relation,

$\frac{d\ln f}{d\ln \eta }$

$=$

$-2\frac{d\ln \varphi }{d\ln \eta }$

$x$

$=$

$f(\eta )[\sin (\eta -B){]}^m,$

---

Let's now examine "model A" from above, for which, $A=0.200812422$ and $B=-0.859270052$. If we set ${\sigma }_c^2=0$, this LAWE becomes,

$0$

$=$

$\frac{d^{2}x}{d{\eta }^2}+[4-2Q]\frac{1}{\eta }\cdot \frac{dx}{d\eta }-2\left[\alpha Q\right]\frac{x}{{\eta }^2}.$

Discrete Determination of Bipolytropic Envelope

Here we focus on the specific $(n_c,n_e)=(5,1)$ equilibrium model sequence that has $({\mu }_e/{\mu }_c)=0.31$; and along this sequence, we attempt to analyze the dynamical stability of "model A" from above, which sits along the sequence at the maximum-core-mass turning point for which …

$(n_c,n_e)=(5,1)$ and $({\mu }_e/{\mu }_c)=0.31$
Model	$A$	$B$	${\xi }_i$	${\theta }_i=(1+{\xi }_i^2/3{)}^{-1/2}$	${\eta }_i=3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\xi }_i{\theta }_i^2$	${\eta }_s=B+\pi$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}$
A	0.200812422	- 0.859270052	9.0149598	0.188679805	0.17232050	2.28232260	1.9381270

Key Parameter-Parameter Ratios
$\frac{\xi }{\tilde{r}}=\left(\frac{2\pi }{3}{)}^{1/2}{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^2\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-4}$	$\frac{\eta }{\tilde{r}}={{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^2(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\theta }_i^2(2\pi {)}^{1/2}$	$\frac{\eta }{\xi }$
588.6362811	11.25175286	0.019114950

As presented above, when ${\sigma }_c^2=0$, the eigenfunction for the core that we have deduced via the B-KB74 conjecture appears to be well represented by the expressions,

$x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$C_0[1-\frac{{\xi }^2}{15}],$       and,       $\frac{dx}{d\xi }{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=-\frac{2C_0\xi }{15},$       with,       $C_0=-0.0011,$

over the radial-parameter range, $0\le \xi \le {\xi }_i.$

At the Core/Envelope Interface (as viewed from the core)
$x_i$	$\frac{dx}{d\xi }{|}_i$	${\tilde{r}}_i$	$\frac{dx}{d\tilde{r}}{|}_i$	$[\frac{d\ln x}{d\ln \xi }{]}_i=[\frac{d\ln x}{d\ln \tilde{r}}{]}_i$
+ 0.004859763	+ 0.001322194	0.015314992	+ 0.778291359	+ 2.4526969

Copying from our earlier discussion of the envelope for "model A", the range of the radial parameter is,

$({\eta }_i=0.1723205)\le \eta \le ({\eta }_s=2.282322601).$

SLOPE:  As we have detailed elsewhere, we expect that the slope of the function, $x_{\mathrm{e}\mathrm{n}\mathrm{v}}(\tilde{r})$, is related to the slope of $x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}(\tilde{r})$ at the interface via the expression, $\{\frac{d\ln x}{d\ln r}{|}_i{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}=\{\frac{d\ln x}{d\ln \eta }{|}_i{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$ $=$ $3(\frac{{\gamma }_c}{{\gamma }_e}-1)+\frac{{\gamma }_c}{{\gamma }_e}\{\frac{d\ln x}{d\ln \xi }{|}_i{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}.$ In our case, ${\gamma }_c=6/5$ and ${\gamma }_e=2\Rightarrow {\gamma }_c/{\gamma }_e=3/5$. Hence, from the point of view of the envelope displacement function, at the interface, $[\frac{\tilde{r}}{x_{\mathrm{e}\mathrm{n}\mathrm{v}}}\cdot \frac{d{x}_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\tilde{r}}{]}_i$ $=$ $\frac{3}{5}\left\{\right[\frac{d\ln (x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}})}{d\ln \xi }{]}_i-2\}$   $=$ $\frac{3}{5}\left\{\right[2.452697-2\}=0.271618.$ Now, at the interface of any bipolytrope, the ratio $\tilde{r}/x$ should have the same numerical value whether it is viewed from the point of view of the core or the envelope. Given that, for our particular "model A", $[\frac{\tilde{r}}{x}{]}_i=\frac{0.015315}{0.00485976}=3.15139,$ we should expect the slope of the envelope's displacement function at the interface to be, $\frac{d{x}_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\tilde{r}}{|}_i$ $=$ $0.08619.$

As above, we will integrate the discrete LAWE outward using the finite-difference expression,

$x_{+}\stackrel{\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{1}}{\overbrace{[2{\varphi }_i+\frac{4{\mathrm{\Delta }}_{\eta }{\varphi }_i}{{\eta }_i}-{\mathrm{\Delta }}_{\eta }(n+1)(-{\varphi }^{\text{'}}{)}_i]}}$

$=$

$x_{-}\stackrel{\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{2}}{\overbrace{[\frac{4{\mathrm{\Delta }}_{\eta }{\varphi }_i}{{\eta }_i}-{\mathrm{\Delta }}_{\eta }(n+1)(-{\varphi }^{\text{'}}{)}_i-2{\varphi }_i]}}+x_i\stackrel{\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{3}}{\overbrace{\{4{\varphi }_i-\frac{{\mathrm{\Delta }}_{\eta }^2(n+1)}{3}[\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha (-\frac{3{\varphi }^{\text{'}}}{\eta }{)}_i]\}}}.$

When we started the integration at the center of the configuration, we kickstarted the process by, first, setting $x_1=1$; then, second, setting,

$x_2$

$=$

$x_1[1-\frac{(n+1)\mathfrak{𝔉}{\mathrm{\Delta }}_{\eta }^2}{60}],$

where,

$\mathfrak{𝔉}$

$\equiv$

$[\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha ].$

Having obtained $x_1\to x_{-}$ and $x_2\to x_i$, we then used the finite-difference expression to calculate $x_{+}\to x_3$, as well as all subsequent "$x_{+}$" values, all the way to the surface.

Here, instead, we want to start the envelope integration at the core/envelope interface as follows:

1. The displacement function for the core gives us the value of the displacement function, $x_i=+0.004859763$, at ${\xi }_i=9.0149598$, that is, at ${\eta }_i=0.17232050$; we recognize that this value of $x_i$ (at the interface) also furnishes the value of $x_i$ in the first integration step of the finite-difference expressions.
2. We will then "guess" the slope of the envelope's displacement function, $q\equiv [dx/d\eta {]}_i$, at the interface.
3. Our discrete representation of this first derivative permits us to write,

   $q$	$\approx$	$\frac{x_{+}-x_{-}}{2{\mathrm{\Delta }}_{\eta }}$
   $\Rightarrow x_{-}$	$\approx$	$x_{+}-2q{\mathrm{\Delta }}_{\eta }.$

   Inserting this expression into the finite-difference approximation to the LAWE gives for the first integration step only!

   $x_{+}\cdot \mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{1}$	$=$	$(x_{+}-2q{\mathrm{\Delta }}_{\eta })\cdot \mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{2}+x_i\cdot \mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{3}$
   $\Rightarrow x_{+}\cdot [\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{1}-\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{2}]$	$=$	$x_i\cdot \mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{3}-2q{\mathrm{\Delta }}_{\eta }\cdot \mathrm{T}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{2}.$

NOTE:   Judging by the behavior of the B-KB74 generated displacement function, at the interface we expect the slope, $[d\ln x/d\ln \tilde{r}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$, from the envelope's perspective to be shallower than the slope, $[d\ln x/d\ln \tilde{r}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, from the core's perspective. That is to say, we expect to "guess" values of $q$ such that at the interface,

$0$	$<$	$[\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-[\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$
$\Rightarrow [\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$<$	$[\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
$\Rightarrow [\frac{d\ln x}{d\ln \eta }{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$<$	$[\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
$\Rightarrow [\frac{\eta }{x}\cdot \frac{dx}{d\eta }{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$<$	$[\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
$\Rightarrow [\frac{dx}{d\eta }{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$<$	$\frac{x_i}{{\eta }_i}\cdot [\frac{d\ln x}{d\ln \tilde{r}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
$\Rightarrow q$	$<$	$\left[\frac{+0.004859763}{+0.1723205}\right]\cdot 2.4526969=+0.06917068.$

See Also

• Prasad, C. (1953), Proc. Natn. Inst. Sci. India, Vol. 19, 739, Radial Oscillations of a Composite Model.
• Singh, Manmohan, (1969), Proc. Natn. Inst. Sci., India, Part A, Vol. 35, pp. 586 - 589, Radial Oscillations of Composite Polytropes — Part I
• Singh, Manmohan, (1969), Proc. Nat. Inst. Sci., India, Part A, Vol. 35, pp. 703 - 708, Radial Oscillations of Composite Polytropes — Part II
• Kumar, S., Saini, S., Singh, K. K., Bhatt, V., & Vashishta, L. (2021), Astronomical & Astrophysical Transactions, Vol. 32, Issue 4, pp. 371-382, Radial Pulsations of distorted Polytropes of Non-Uniform Density.