SSC/Stability/n1PolytropeLAWE/Pt3

Radial Oscillations of n = 1 Polytropic Spheres (Pt 3)

Part I:   Search for Analytic Solutions

Part II:  New Ideas

Part III:  What About Bipolytropes?

Part IV:  Most General Structural Solution

What About Bipolytropes?

Here we will try to find an analytic expression for the radial displacement function, $x$, for a bipolytropic envelope whose polytropic index is, $n_e=1$. As in the above succinct derivation, the relevant LAWE is,

LAWE	$=$	$\frac{d^{2}x}{d{\xi }^2}+[4-(n+1)Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+(n+1)\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_g})\frac{{\xi }^2}{\theta }-\alpha Q]\frac{x}{{\xi }^2}$
	$=$	$\frac{d^{2}x}{d{\xi }^2}+[4-2Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+\left[\right(\frac{{\sigma }_c^2}{6})\frac{{\xi }^3}{\sin \xi }-2Q]\frac{x}{{\xi }^2}$
	$=$	$\frac{d^{2}x}{d{\xi }^2}+[2+\frac{2\xi \cos \xi }{\sin \xi }]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+[-2+\frac{2\xi \cos \xi }{\sin \xi }]\frac{x}{{\xi }^2}+\left[\right(\frac{{\sigma }_c^2}{6}\left)\frac{\xi }{\sin \xi }\right]x$

First Attempt

Let's try,

$x$

$=$

$A+\frac{B}{(\xi -F{)}^2}[1-(\xi -D)\cot (\xi -C)].$

First, note that,

$\frac{d}{d\xi }[\cot (\xi -C)]$	$=$	$\frac{d}{d\xi }\left[\frac{\cos (\xi -C)}{\sin (\xi -C)}\right]$
	$=$	$-[1+{\cot }^2(\xi -C)].$

Hence,

$\frac{dx}{d\xi }$	$=$	$-\frac{2B}{(\xi -F{)}^3}[1-(\xi -D)\cot (\xi -C)]-\frac{B}{(\xi -F{)}^2}\{\cot (\xi -C)-(\xi -D)[1+{\cot }^{}(\xi -C)]\}$
	$=$	$-\frac{B}{(\xi -F{)}^3}\left\{\right[2-2(\xi -D)\cot (\xi -C)]-(\xi -F)[\cot (\xi -C)-(\xi -D)[1+{\cot }^{}(\xi -C)]\left]\right\}$
	$=$	$-\frac{B}{(\xi -F{)}^3}\{2-\cot (\xi -C)[2(\xi -D)+(\xi -F)]+(\xi -F)(\xi -D)[1+{\cot }^{}(\xi -C)]\}$
	$=$	$-\frac{B}{(\xi -F{)}^3}\{2-[3\xi -(2D+F)]\cot (\xi -C)+[{\xi }^2-(D+F)\xi +FD]+[{\xi }^2-(D+F)\xi +FD]{\cot }^{}(\xi -C)\}$
	$=$	$-\frac{B}{(\xi -F{)}^3}\{[{\xi }^2-(D+F)\xi +FD+2]-[3\xi -(2D+F)]\cot (\xi -C)+[{\xi }^2-(D+F)\xi +FD]{\cot }^{}(\xi -C)\}.$

And,

$\frac{d^{2}x}{d{\xi }^2}$	$=$	$\frac{3B}{(\xi -F{)}^4}\{[{\xi }^2-(D+F)\xi +FD+2]-[3\xi -(2D+F)]\cot (\xi -C)+[{\xi }^2-(D+F)\xi +FD]{\cot }^{}(\xi -C)\}$
		$-\frac{B}{(\xi -F{)}^3}\{[2\xi -(D+F)]-3\cot (\xi -C)-[3\xi -(2D+F)]\frac{d\cot (\xi -C)}{d\xi }+[2\xi -(D+F)]{\cot }^{}(\xi -C)+[{\xi }^2-(D+F)\xi +FD\left]\frac{d{\cot }^2(\xi -C)}{d\xi }\right\}$
$\Rightarrow \left[\frac{(\xi -F{)}^4}{B}\right]\frac{d^{2}x}{d{\xi }^2}$	$=$	$3[{\xi }^2-(D+F)\xi +FD+2]-3[3\xi -(2D+F)]\cot (\xi -C)+3[{\xi }^2-(D+F)\xi +FD]{\cot }^{}(\xi -C)$
		$-(\xi -F)[2\xi -(D+F)]+3(\xi -F)\cot (\xi -C)-(\xi -F)[2\xi -(D+F)]{\cot }^{}(\xi -C)$
		$+(\xi -F)[3\xi -(2D+F)]\frac{d\cot (\xi -C)}{d\xi }-(\xi -F)[{\xi }^2-(D+F)\xi +FD]\frac{d{\cot }^2(\xi -C)}{d\xi }$
	$=$	$3[{\xi }^2-(D+F)\xi +FD+2]-(\xi -F)[2\xi -(D+F)]+\{3(\xi -F)-3[3\xi -(2D+F)\left]\right\}\cot (\xi -C)$
		$+\{3[{\xi }^2-(D+F)\xi +FD]-(\xi -F)[2\xi -(D+F)]\}{\cot }^2(\xi -C)$
		$-(\xi -F)[3\xi -(2D+F)][1+{\cot }^2(\xi -C)]+(\xi -F)[{\xi }^2-(D+F)\xi +FD]2\cot (\xi -C)[1+{\cot }^{}(\xi -C)]$
	$=$	$3[{\xi }^2-(D+F)\xi +FD+2]-(\xi -F)[2\xi -(D+F)]-(\xi -F)[3\xi -(2D+F)]+\{3(\xi -F)-3[3\xi -(2D+F)]+2(\xi -F)[{\xi }^2-(D+F)\xi +FD]\}\cot (\xi -C)$
		$+\{3[{\xi }^2-(D+F)\xi +FD]-(\xi -F)[2\xi -(D+F)]-(\xi -F)[3\xi -(2D+F)]\}{\cot }^2(\xi -C)+2(\xi -F)[{\xi }^2-(D+F)\xi +FD]{\cot }^3(\xi -C).$

Let's set $C=D=F$ and see if these expressions match the ones above.

$\frac{dx}{d\xi }{|}_{C=D=F}$

$=$

$-\frac{B}{{\xi }^3}\{2+{\xi }^2-3\xi \cot \xi +{\xi }^2{\cot }^{}\xi \}.$

$\frac{{\xi }^4}{B}\cdot \frac{d^{2}x}{d{\xi }^2}{|}_{C=D=F}$	$=$	$3[{\xi }^2+2]-(\xi )[2\xi ]-\xi [3\xi ]+\{3(\xi )-3[3\xi ]+2\xi [{\xi }^2]\}\cot (\xi )$
		$+\{3[{\xi }^2]-\xi [2\xi ]-\xi [3\xi ]\}{\cot }^2\xi +2\xi [{\xi }^2]{\cot }^3\xi$
	$=$	$3{\xi }^2+6-2{\xi }^2-3{\xi }^2+[3\xi -9\xi +2{\xi }^3]\cot (\xi )+[3{\xi }^2-2{\xi }^2-3{\xi }^2]{\cot }^{}\xi +2{\xi }^3{\cot }^{}\xi$
	$=$	$6-2{\xi }^2+[-6\xi +2{\xi }^3]\cot (\xi )-2{\xi }^2{\cot }^{}\xi +2{\xi }^3{\cot }^{}\xi$

Second Attempt

Up to this point we have been rather cavalier about the use of $\xi$ (and ${\xi }_i$) to represent the envelope's dimensionless radius (and interface location). Let's switch to $\eta$,

$r^{*}$

$=$

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

$0$

$=$

$\frac{d^{2}x}{d{\eta }^2}+\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\frac{1}{2\pi {\theta }_i^4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{{\rho }^{*}}{P^{*}}\left)\right\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}\}x.$

and, throughout the envelope we have,

$\frac{{\rho }^{*}}{P^{*}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{-1}\varphi (\eta {)}^{-1};$
$\frac{M_r^{*}}{r^{*}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta {]}^{-1}=2(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i\eta (-\frac{d\varphi }{d\eta }).$

Hence, the LAWE relevant to the envelope is,

$0$	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-[\frac{{\rho }^{*}}{P^{*}}\left]\right[\frac{M_r^{*}}{(r^{*})}\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\frac{1}{2\pi {\theta }_i^4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right[\frac{{\rho }^{*}}{P^{*}}\left]\right\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_e}{(r^{*}{)}^2}\left[\frac{M_r^{*}}{r^{*}}\right]\}x$
	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-[\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{-1}\varphi (\eta {)}^{-1}]\left[2\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i\eta (-\frac{d\varphi }{d\eta })\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\frac{1}{2\pi {\theta }_i^4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right[\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{-1}\varphi (\eta {)}^{-1}]\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-{\alpha }_e[(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta {]}^{-2}\left[2\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i\eta (-\frac{d\varphi }{d\eta })\left]\right\}x$
	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-[\frac{2\eta }{\varphi }(-\frac{d\varphi }{d\eta })\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\frac{1}{2\pi {\theta }_i^5\varphi }\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-{\alpha }_e\left[\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^5(4\pi ){\eta }^{-1}](-\frac{d\varphi }{d\eta })\}x$
	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-[\frac{2\eta }{\varphi }(-\frac{d\varphi }{d\eta })\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+\frac{1}{2\pi {\theta }_i^5\varphi }\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}\}x-{\alpha }_e[\frac{2\eta }{\varphi }(-\frac{d\varphi }{d\eta })]\frac{x}{{\eta }^2}.$

If we assume that, ${\alpha }_e=(3-4/2)=1$ and ${\sigma }_c^2=0$, then the relevant envelope LAWE is,

$0$

$=$

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

where,

$Q\equiv -\frac{d\ln \varphi }{d\ln \eta }.$

Now consider the,

Precise Solution to the Polytropic LAWE
$x_P$	$=$	$\frac{b(n-1)}{2n}[1+(\frac{n-3}{n-1}\left)\right(\frac{1}{\eta {\varphi }^n}\left)\frac{d\varphi }{d\eta }\right]$
	$=$	$-b\left[\right(\frac{1}{\eta \varphi }\left)\frac{d\varphi }{d\eta }\right]$
	$=$	$\frac{b}{{\eta }^2}[-\frac{d\ln \varphi }{d\ln \eta }]$
	$=$	$\frac{bQ}{{\eta }^2}.$

From our accompanying discussion, we recall that the most general solution to the $n=1$ Lane-Emden equation can be written in the form,

$\varphi =A\left[\frac{\sin (\eta -B)}{\eta }\right],$

where $A$ and $B$ are constants whose values can be obtained from our accompanying parameter table. The first derivative of this function is,

$\frac{d\varphi }{d\eta }=\frac{A}{{\eta }^2}[\eta \cos (\eta -B)-\sin (\eta -B)].$

Hence,

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

What is this in terms of the dimensionless radius, $r^{*}/R^{*}$? Well,

$\frac{r^{*}}{R^{*}}$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta \left[\frac{\sqrt{2\pi }{\theta }_i^2}{{\eta }_s}\right(\frac{{\mu }_e}{{\mu }_c}\left)\right]$
	$=$	$\frac{\eta }{{\eta }_s}=\frac{\eta }{(\pi +B)}$
$\Rightarrow \eta$	$=$	$\frac{r^{*}}{R^{*}}(\pi +B).$

Also,

$\eta -B$	$=$	$\frac{r^{*}}{R^{*}}(\pi +B)-B=\pi \left(\frac{r^{*}}{R^{*}}\right)-B[1-(\frac{r^{*}}{R^{*}}\left)\right]$
	$=$	$\pi +\pi \left[\right(\frac{r^{*}}{R^{*}})-1]-B[1-(\frac{r^{*}}{R^{*}}\left)\right]$
	$=$	$\pi -(\pi +B)[1-(\frac{r^{*}}{R^{*}}\left)\right].$

[12 January 2019]: Here's what appears to work pretty well, empirically:

$x_P$ $=$ $\frac{1}{{\eta }^2}\{1-\eta \cot [\eta -(\pi -0.8)]\}.$ $\eta$ $=$ $\frac{r^{*}}{R^{*}}(\pi -0.6\pi ).$

Let's work through the analytic derivatives again. Keeping in mind that,

$\frac{d}{d\eta }[\cot (\eta -B)]$

$=$

$-[1+{\cot }^2(\eta -B)],$

and starting with the guess,

$x_P$

$=$

$\frac{b}{{\eta }^2}[1-\eta \cot (\eta -B)],$

we have,

$\frac{d{x}_P}{d\eta }$	$=$	$-\frac{2b}{{\eta }^3}[1-\eta \cot (\eta -B)]-\frac{b}{{\eta }^2}\{\cot (\eta -B)-\eta [1+{\cot }^{}(\eta -B)\left]\right\}$
$\Rightarrow \left(\frac{{\eta }^3}{b}\right)\frac{d{x}_P}{d\eta }$	$=$	$-[2-2\eta \cot (\eta -B)]-\{\eta \cot (\eta -B)-{\eta }^2[1+{\cot }^{}(\eta -B)\left]\right\}$
	$=$	${\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B).$

The second derivative then gives,

$\frac{d^2{x}_P}{d{\eta }^2}$	$=$	$\frac{d}{d\eta }\left\{\frac{b}{{\eta }^3}\right[{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)\left]\right\}$
	$=$	$-\frac{3b}{{\eta }^4}[{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)]$
		$+\frac{b}{{\eta }^3}\{2\eta +\cot (\eta -B)+2\eta {\cot }^{}(\eta -B)+\eta \frac{d}{d\eta }[\cot (\eta -B)]+2{\eta }^2\cot (\eta -B)\frac{d}{d\eta }[\cot (\eta -B)\left]\right\}$
$\Rightarrow \frac{d^2{x}_P}{d{\eta }^2}$	$=$	$\frac{b}{{\eta }^4}[6-3{\eta }^2-3\eta \cot (\eta -B)-3{\eta }^2{\cot }^{}(\eta -B)]$
		$+\frac{b}{{\eta }^4}\{2{\eta }^2+\eta \cot (\eta -B)+2{\eta }^2{\cot }^{}(\eta -B)-{\eta }^2[1+{\cot }^{}(\eta -B)]-2{\eta }^3\cot (\eta -B)[1+{\cot }^{}(\eta -B)\left]\right\}$
$\Rightarrow \frac{{\eta }^4}{b}\cdot \frac{d^2{x}_P}{d{\eta }^2}$	$=$	$6-3{\eta }^2-3\eta \cot (\eta -B)-3{\eta }^2{\cot }^{}(\eta -B)$
		$+2{\eta }^2+\eta \cot (\eta -B)+2{\eta }^2{\cot }^{}(\eta -B)-{\eta }^2-{\eta }^2{\cot }^{}(\eta -B)-2{\eta }^3\cot (\eta -B)-2{\eta }^3{\cot }^{}(\eta -B)$
	$=$	$2[3-{\eta }^2-(\eta +{\eta }^3)\cot (\eta -B)-{\eta }^2{\cot }^{}(\eta -B)-{\eta }^3{\cot }^{}(\eta -B)].$

Recalling that,

$Q=[1-\eta \cot (\eta -B)],$

plugging these expressions into the relevant envelope LAWE gives,

LAWE	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }-2Q\cdot \frac{x}{{\eta }^2}$
	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-2[1-\eta \cot (\eta -B)\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }-[1-\eta \cot (\eta -B)]\frac{2x}{{\eta }^2}$
	$=$	$\frac{b}{{\eta }^4}\{\frac{{\eta }^4}{b}\cdot \frac{d^{2}x}{d{\eta }^2}+[1+\eta \cot (\eta -B)]\frac{2{\eta }^3}{b}\cdot \frac{dx}{d\eta }-[1-\eta \cot (\eta -B)\left]\frac{2{\eta }^{2}x}{b}\right\}$
	$=$	$\frac{2b}{{\eta }^4}\{3-{\eta }^2-(\eta +{\eta }^3)\cot (\eta -B)-{\eta }^2{\cot }^{}(\eta -B)-{\eta }^3{\cot }^{}(\eta -B)$
		$+[1+\eta \cot (\eta -B)][{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)]-[1-\eta \cot (\eta -B)][1-\eta \cot (\eta -B)]\}$
	$=$	$\frac{2b}{{\eta }^4}\{3-{\eta }^2-(\eta +{\eta }^3)\cot (\eta -B)-{\eta }^2{\cot }^{}(\eta -B)-{\eta }^3{\cot }^{}(\eta -B)+[{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)]-[1-\eta \cot (\eta -B)]$
		$+\eta \cot (\eta -B)[{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)]+\eta \cot (\eta -B)[1-\eta \cot (\eta -B)]\}$
	$=$	$\frac{2b}{{\eta }^4}\{-(\eta +{\eta }^3)\cot (\eta -B)-{\eta }^3{\cot }^{}(\eta -B)+2\eta \cot (\eta -B)$
		$+{\eta }^3\cot (\eta -B)-2\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)+{\eta }^3{\cot }^{}(\eta -B)+\eta \cot (\eta -B)-{\eta }^2{\cot }^{}(\eta -B)\}$
	$=$	$\frac{2b}{{\eta }^4}\{[-\eta \cot (\eta -B)-\eta \cot (\eta -B)+2\eta \cot (\eta -B)]+[{\eta }^3\cot (\eta -B)-{\eta }^3\cot (\eta -B)]$
		$+[{\eta }^2{\cot }^2(\eta -B)-{\eta }^2{\cot }^2(\eta -B)]+[{\eta }^3{\cot }^3(\eta -B)-{\eta }^3{\cot }^3(\eta -B)]\}$
	$=$	$0.$

Okay. Now let's determine at what value of $\eta$ the logarithmic derivative of $x_P$ goes to negative one.

$\frac{d\ln x_P}{d\ln \eta }=\frac{\eta }{x_P}\cdot \frac{d{x}_P}{d\eta }$	$=$	$\frac{{\eta }^3}{b}[1-\eta \cot (\eta -B){]}^{-1}\cdot \frac{d{x}_P}{d\eta }$
	$=$	$[1-\eta \cot (\eta -B){]}^{-1}[{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)].$

Setting this to negative one, we have,

$-[1-\eta \cot (\eta -B)]$	$=$	$[{\eta }^2-2+\eta \cot (\eta -B)+{\eta }^2{\cot }^{}(\eta -B)]$
$\Rightarrow 1$	$=$	${\eta }^2[1+{\cot }^2(\eta -B)]$
	$=$	${\eta }^2\left[\frac{1}{{\sin }^2(\eta -B)}\right]$
$\Rightarrow 1$	$=$	$\frac{{\eta }^2}{{\sin }^2(\eta -B)}.$

And this occurs when,

$(\frac{A}{\varphi }{)}^2=1.$

Third Attempt

Prior to the Brute-Force Trial Fit

Let's work through the analytic derivatives again. Keeping in mind that,

$\frac{d}{d\eta }[\cot (\eta -C)]$

$=$

$-[1+{\cot }^2(\eta -C)],$

and starting with the guess,

$x_P$

$=$

$\frac{b}{{\eta }^2}[1-\eta \cot (\eta -C)],$

we have,

$\left(\frac{{\eta }^3}{b}\right)\frac{d{x}_P}{d\eta }$

$=$

${\eta }^2-2+\eta \cot (\eta -C)+{\eta }^2{\cot }^{}(\eta -C),$

and,

$\frac{{\eta }^4}{b}\cdot \frac{d^2{x}_P}{d{\eta }^2}$

$=$

$2[3-{\eta }^2-(\eta +{\eta }^3)\cot (\eta -C)-{\eta }^2{\cot }^{}(\eta -C)-{\eta }^3{\cot }^{}(\eta -C)].$

Note that the relevant logarithmic derivative is, $\frac{d\ln x_P}{d\ln \eta }$ $=$ $\left(\frac{b}{{\eta }^2}\right)[{\eta }^2-2+\eta \cot (\eta -C)+{\eta }^2{\cot }^{}(\eta -C)]x_P^{-1}$   $=$ $[{\eta }^2-2+\eta \cot (\eta -C)+{\eta }^2{\cot }^{}(\eta -C)][1-\eta \cot (\eta -C){]}^{-1}$ If we know the logarithmic slope and the value of $\eta$ at the interface, then we can solve for $y_i\equiv {\eta }_i\cot ({\eta }_i-C),$ via the quadratic relation, $(1-y_i)[\frac{d\ln x_P}{d\ln \eta }{]}_i$ $=$ ${\eta }_i^2-2+y_i+y_i^2$ $\Rightarrow 0$ $=$ ${\eta }_i^2-2+y_i+y_i^2-(1-y_i)[\frac{d\ln x_P}{d\ln \eta }{]}_i$   $=$ $y_i^2+y_i\{1+[\frac{d\ln x_P}{d\ln \eta }{]}_i\}+\{{\eta }_i^2-2-\left[\frac{d\ln x_P}{d\ln \eta }{]}_i\right\}.$ (In practice it appears as though the "plus" solution to this quadratic equation is desired if the quantity inside the last set of curly braces is positive; and the "minus" solution is desired if this quantity is negative.) Once the value of $y_i$ is known, we can solve for the key coefficient, $C$, via the relation, $\tan ({\eta }_i-C)$ $=$ $\frac{{\eta }_i}{y_i}$ $\Rightarrow C$ $=$ ${\eta }_i-{\tan }^{-1}\left(\frac{{\eta }_i}{y_i}\right).$

Recalling that,

$Q=[1-\eta \cot (\eta -B)],$

plugging these expressions into the relevant envelope LAWE gives,

LAWE	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }-2Q\cdot \frac{x}{{\eta }^2}$
	$=$	$\frac{d^{2}x}{d{\eta }^2}+\{4-2[1-\eta \cot (\eta -B)\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }-[1-\eta \cot (\eta -B)]\frac{2x}{{\eta }^2}$
	$=$	$\frac{b}{{\eta }^4}\{\frac{{\eta }^4}{b}\cdot \frac{d^{2}x}{d{\eta }^2}+[1+\eta \cot (\eta -B)]\frac{2{\eta }^3}{b}\cdot \frac{dx}{d\eta }-[1-\eta \cot (\eta -B)\left]\frac{2{\eta }^{2}x}{b}\right\}$
	$=$	$\frac{2b}{{\eta }^4}\{3-{\eta }^2-(\eta +{\eta }^3)\cot (\eta -C)-{\eta }^2{\cot }^{}(\eta -C)-{\eta }^3{\cot }^{}(\eta -C)$
		$+[1+\eta \cot (\eta -B)][{\eta }^2-2+\eta \cot (\eta -C)+{\eta }^2{\cot }^{}(\eta -C)]-[1-\eta \cot (\eta -B)][1-\eta \cot (\eta -C)]\}$
	$=$	$\frac{2b}{{\eta }^4}\{3-{\eta }^2-(\eta +{\eta }^3)\cot (\eta -C)-{\eta }^2{\cot }^{}(\eta -C)-{\eta }^3{\cot }^{}(\eta -C)+[{\eta }^2-2+\eta \cot (\eta -C)+{\eta }^2{\cot }^{}(\eta -C)]-[1-\eta \cot (\eta -C)]$
		$+\eta \cot (\eta -B)[{\eta }^2-2+\eta \cot (\eta -C)+{\eta }^2{\cot }^{}(\eta -C)]+\eta \cot (\eta -B)[1-\eta \cot (\eta -C)]\}$
	$=$	$\frac{2b}{{\eta }^4}\{(\eta -{\eta }^3)\cot (\eta -C)-{\eta }^3{\cot }^{}(\eta -C)+\eta \cot (\eta -B)[{\eta }^2-1+{\eta }^2{\cot }^{}(\eta -C)\left]\right\}$
	$=$	$\frac{2b}{{\eta }^4}\{(\eta -{\eta }^3)[\cot (\eta -C)-\cot (\eta -B)]+{\eta }^3{\cot }^{}(\eta -C)[\cot (\eta -B)-\cot (\eta -C)]\}$
	$=$	$\frac{2b}{{\eta }^4}[\cot (\eta -C)-\cot (\eta -B)][\eta -{\eta }^3-{\eta }^3{\cot }^{}(\eta -C)]$
	$=$	$\frac{2b}{{\eta }^3}[\cot (\eta -C)-\cot (\eta -B)]\{1-{\eta }^2[1+{\cot }^{}(\eta -C)\left]\right\}.$

This will go to zero if $C=(B-2m\pi ),$ where $m$ is a positive integer. When $m=1$, for example,

$\cot (\eta -C)$

$=$

$\cot [\eta -(B-2\pi )]=\cot (\eta -B).$

Okay. Now let's determine at what value of $\eta$ the logarithmic derivative of $x_P$ goes to negative one.

Brute-Force Trial Fit

Using a couple of separate Excel spreadsheets — FaulknerBipolytrope2.xlsx/mu100Mode0 and AnalyticTrialBipolytropeA.xlsx/Sheet2, both stored in a DropBox account under the folder Wiki_edits/Bipolytrope/LinearPerturbation — we used an inelegant and inefficient trial & error technique in search of an eigenfunction that had the same analytic form as the one represented above for $x_P$, but that, when plotted, appeared to qualitatively match the numerically determined envelope eigenfunction. Then, on a whiteboard — see the photo, here on the right — we formulated a concise expression for a trial function that seemed to work pretty well. Our primary finding was that $\alpha$, appearing as the argument to the $\tan \alpha$ function, needed to be shifted by something like $-3\pi /4$.   THIS SPACE INTENTIONALLY LEFT BLANK

Following Up on the Brute-Force Trial Fit

In an accompanying discussion — see especially Attempt #2 — we have determined by visual inspection that a decent fit to the envelope's eigenfunction is given by the expression,

$x_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}$	$=$	$\frac{b_0}{{\mathrm{\Lambda }}^2}\{1-\mathrm{\Lambda }[\frac{\tan ({\eta }_i-\mathrm{\Lambda }-3\pi /4)+f_{\alpha }}{1-f_{\alpha }\cdot \tan ({\eta }_i-\mathrm{\Lambda }-3\pi /4)}\left]\right\}-a_0$
	$=$	$\frac{b_0}{{\mathrm{\Lambda }}^2}\{1-\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)\}-a_0,$

Limiting Parameter Values
	min	max	$\alpha ={\alpha }_s$
${\eta }_{\mathrm{F}}$	${\eta }_i$	${\eta }_s$	$\frac{8}{\pi }({\eta }_s-{\eta }_i{)}^2+2{\eta }_s-{\eta }_i$
$\alpha$	$-\frac{\pi }{2}$	$-\frac{5\pi }{8}$	${\eta }_i-{\eta }_s-\frac{3\pi }{4}$
$\mathrm{\Lambda }$	${\eta }_i-\frac{\pi }{4}$	${\eta }_i-\frac{\pi }{8}$	${\eta }_s$

where, over the range, ${\eta }_i\le \eta \le {\eta }_s,$

$E$	$\equiv$	${\eta }_i-\frac{5\pi }{4}+{\tan }^{-1}{f}_{\alpha },$
$\mathrm{\Lambda }(\eta )$	$\equiv$	${\eta }_i+g_{\mathrm{F}}[{\eta }_i-2{\eta }_s+\eta ]={\mathrm{\Lambda }}_0+g_{\mathrm{F}}\eta ,$
$\frac{1}{f_{\alpha }}=\tan ({\alpha }_s)$	$\equiv$	$\tan [-({\eta }_s-{\eta }_i+\frac{3\pi }{4})],$
$g_{\mathrm{F}}$	$\equiv$	$\frac{\pi }{8({\eta }_s-{\eta }_i)}.$

---

Here, we reference a separate discussion of the bipolytrope's underlying equilibrium structure

$B={\eta }_i-\frac{\pi }{2}+{\tan }^{-1}f$	$E={\eta }_i-\frac{5\pi }{4}+{\tan }^{-1}{f}_{\alpha }$
$\Rightarrow \cot ({\eta }_i-B)$ $=$ $\tan [\frac{\pi }{2}-({\eta }_i-B)]$   $=$ $\tan [\frac{\pi }{2}-(\frac{\pi }{2}-{\tan }^{}f)]$   $=$ $f$ $\Rightarrow f$ $=$ $\tan (B+\frac{\pi }{2}-{\eta }_i)$	$\Rightarrow \cot ({\eta }_i-E)$ $=$ $\tan [\frac{\pi }{2}-({\eta }_i-E)]$   $=$ $\tan [\frac{\pi }{2}-(\frac{5\pi }{4}-{\tan }^{}{f}_{\alpha })]$   $=$ $\tan ({\tan }^{}{f}_{\alpha }-\frac{3\pi }{4})$   $=$ $-\tan (\frac{3\pi }{4}-{\tan }^{}{f}_{\alpha })$   $=$ $-\cot ({\tan }^{}{f}_{\alpha })$   $=$ $-\frac{1}{f_{\alpha }}$
Hence …     $\cot (\eta -B)$ $=$ $\tan [\frac{\pi }{2}-(\eta -B)]$   $=$ $\tan [\frac{\pi }{2}-\eta +{\eta }_i-\frac{\pi }{2}+{\tan }^{}f]$   $=$ $\tan [{\eta }_i-\eta +{\tan }^{}f]$   $=$ $\frac{\tan ({\eta }_i-\eta )+f}{1-f\cdot \tan ({\eta }_i-\eta )}$	Hence …     $\cot (\mathrm{\Lambda }-E)$ $=$ $\tan [\frac{\pi }{2}-(\mathrm{\Lambda }-E)]$   $=$ $\tan [{\eta }_i-\mathrm{\Lambda }-\frac{3\pi }{4}+{\tan }^{}{f}_{\alpha }]$   $=$ $\frac{\tan ({\eta }_i-\mathrm{\Lambda }-\frac{3\pi }{4})+f_{\alpha }}{1-f_{\alpha }\cdot \tan ({\eta }_i-\mathrm{\Lambda }-\frac{3\pi }{4})}$
Also …    $B={\eta }_s-\pi$	$$
$\Rightarrow f=\cot ({\eta }_i-B)$ $=$ $\cot ({\eta }_i-{\eta }_s+\pi )$ $\Rightarrow \frac{1}{f}$ $=$ $\tan ({\eta }_i-{\eta }_s+\pi )$	$$

---

Let's examine the first and second derivatives of this trial eigenfunction, recognizing that,

$\frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\eta }=\frac{d\mathrm{\Lambda }}{d\eta }\cdot \frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\mathrm{\Lambda }}=g_{\mathrm{F}}\cdot \frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\mathrm{\Lambda }}$

and

$\frac{d^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d{\eta }^2}=\frac{d\mathrm{\Lambda }}{d\eta }\cdot \frac{d}{d\mathrm{\Lambda }}[g_{\mathrm{F}}\cdot \frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\mathrm{\Lambda }}]=g_{\mathrm{F}}^2\cdot \frac{d^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d{\mathrm{\Lambda }}^2}.$

and drawing from the derivative expressions already derived, above. For the first derivative, we have,

$\frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\eta }$

$=$

$g_{\mathrm{F}}\left(\frac{b_0}{{\mathrm{\Lambda }}^3}\right)[{\mathrm{\Lambda }}^2-2+\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)+{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)].$

And the second derivative gives,

$\frac{d^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d{\eta }^2}$

$=$

$g_{\mathrm{F}}^2\left(\frac{2b_0}{{\mathrm{\Lambda }}^4}\right)[3-{\mathrm{\Lambda }}^2-(\mathrm{\Lambda }+{\mathrm{\Lambda }}^3)\cot (\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^3{\cot }^{}(\mathrm{\Lambda }-E)].$

Hence,

LAWE	$=$	$\frac{d^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\eta }-2Q\cdot \frac{x_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{{\eta }^2}$
	$=$	$\frac{d^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d{\eta }^2}+\{4-2[1-\eta \cot (\eta -B)\left]\right\}\frac{1}{\eta }\cdot \frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\eta }-[1-\eta \cot (\eta -B)]\frac{2x_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{{\eta }^2}$
	$=$	$\frac{b_0}{{\eta }^4}\{\frac{{\eta }^4}{b_0}\cdot \frac{d^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d{\eta }^2}+[1+\eta \cot (\eta -B)]\frac{2{\eta }^3}{b_0}\cdot \frac{d{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{d\eta }-[1-\eta \cot (\eta -B)\left]\frac{2{\eta }^2{x}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}{b_0}\right\}$
	$=$	$\frac{b_0}{{\eta }^4}\{\frac{{\eta }^4}{b_0}\cdot g_{\mathrm{F}}^2(\frac{2b_0}{{\mathrm{\Lambda }}^4}\left)\right[3-{\mathrm{\Lambda }}^2-(\mathrm{\Lambda }+{\mathrm{\Lambda }}^3)\cot (\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^3{\cot }^{}(\mathrm{\Lambda }-E)]$
		$+[1+\eta \cot (\eta -B)]\frac{2{\eta }^3}{b_0}\cdot g_{\mathrm{F}}\left(\frac{b_0}{{\mathrm{\Lambda }}^3}\right)[{\mathrm{\Lambda }}^2-2+\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)+{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)]$
		$-[1-\eta \cot (\eta -B)]\frac{2{\eta }^2}{b_0}\cdot \left[\frac{b_0}{{\mathrm{\Lambda }}^2}\right\{1-\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)\}-a_0]\}$
	$=$	$\frac{b_0}{{\eta }^4}\left\{g_{\mathrm{F}}^2\right(\frac{2{\eta }^4}{{\mathrm{\Lambda }}^4}\left)\right[3-{\mathrm{\Lambda }}^2-(\mathrm{\Lambda }+{\mathrm{\Lambda }}^3)\cot (\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^3{\cot }^{}(\mathrm{\Lambda }-E)]$
		$+[1+\eta \cot (\eta -B)]\cdot g_{\mathrm{F}}\left(\frac{2{\eta }^3}{{\mathrm{\Lambda }}^3}\right)[{\mathrm{\Lambda }}^2-2+\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)+{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)]$
		$-[1-\eta \cot (\eta -B)][\frac{2{\eta }^2}{{\mathrm{\Lambda }}^2}[1-\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)]-\frac{2{\eta }^2{a}_0}{b_0}]\}$
	$=$	$\frac{2b_0}{{\mathrm{\Lambda }}^4{\eta }^2}\left\{g_{\mathrm{F}}^2{\eta }^2\right[3-{\mathrm{\Lambda }}^2-(\mathrm{\Lambda }+{\mathrm{\Lambda }}^3)\cot (\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^3{\cot }^{}(\mathrm{\Lambda }-E)]$
		$+[1+\eta \cot (\eta -B)]\cdot g_{\mathrm{F}}\mathrm{\Lambda }\eta [{\mathrm{\Lambda }}^2-2+\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)+{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)]$
		$-[1-\eta \cot (\eta -B)][{\mathrm{\Lambda }}^2[1-\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)]-\frac{a_0{\mathrm{\Lambda }}^4}{b_0}]\}$

$\Rightarrow \left(\frac{{\mathrm{\Lambda }}^4}{2b_0}\right)\cdot$  LAWE	$=$	$g_{\mathrm{F}}^2[3-{\mathrm{\Lambda }}^2-(\mathrm{\Lambda }+{\mathrm{\Lambda }}^3)\cot (\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)-{\mathrm{\Lambda }}^3{\cot }^{}(\mathrm{\Lambda }-E)]$
		$+\frac{g_{\mathrm{F}}\mathrm{\Lambda }}{\eta }[{\mathrm{\Lambda }}^2-2+\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)+{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)]-\left(\frac{\mathrm{\Lambda }}{\eta }{)}^2\right[1-\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)-\frac{a_0{\mathrm{\Lambda }}^2}{b_0}]$
		$+[\eta \cot (\eta -B)]\left\{\frac{g_{\mathrm{F}}\mathrm{\Lambda }}{\eta }\right[{\mathrm{\Lambda }}^2-2+\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)+{\mathrm{\Lambda }}^2{\cot }^{}(\mathrm{\Lambda }-E)]+(\frac{\mathrm{\Lambda }}{\eta }{)}^2[1-\mathrm{\Lambda }\cot (\mathrm{\Lambda }-E)-\frac{a_0{\mathrm{\Lambda }}^2}{b_0}]\}$

Fourth Attempt

XXXX

If we assume that, ${\alpha }_e=(3-4/2)=1$ and ${\sigma }_c^2=0$, then the relevant envelope LAWE is,

$0$

$=$

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

where,

$Q\equiv -\frac{d\ln \varphi }{d\ln \eta }$

$=$

$[1-\eta \cot (\eta -B_0)].$

Let's work through the analytic derivatives again. Keeping in mind that,

$\frac{d}{d\eta }[\cot (\eta -B)]$

$=$

$-[1+{\cot }^2(\eta -B)];$

and that the,

Precise Solution to the Polytropic LAWE
$x_P$	$=$	$\frac{b(n-1)}{2n}[1+(\frac{n-3}{n-1}\left)\right(\frac{1}{\eta {\varphi }^n}\left)\frac{d\varphi }{d\eta }\right]$
	$=$	$-b\left[\right(\frac{1}{\eta \varphi }\left)\frac{d\varphi }{d\eta }\right]$
	$=$	$\frac{b}{{\eta }^2}[-\frac{d\ln \varphi }{d\ln \eta }]$
	$=$	$\frac{bQ}{{\eta }^2}$
$\Rightarrow x_P$	$=$	$\frac{b}{{\eta }^2}[1-\eta \cot (\eta -B_0)].$

As we have already tried once, above, let's try a more general form of this expression, namely,

$x_Q$

$=$

$A+\frac{C}{(\eta -F{)}^2}[1-(\eta -D)\cot (\eta -B)].$

Hence,

$\frac{d{x}_Q}{d\eta }$	$=$	$[1-(\eta -D)\cot (\eta -B)]\frac{d}{d\eta }\left[\frac{C}{(\eta -F{)}^2}\right]-\frac{C}{(\eta -F{)}^2}\frac{d}{d\eta }[(\eta -D)\cot (\eta -B)]$
	$=$	$[1-(\eta -D)\cot (\eta -B)]\left[\frac{-2C}{(\eta -F{)}^3}\right]-\frac{C}{(\eta -F{)}^2}[\cot (\eta -B)]+\frac{C(\eta -D)}{(\eta -F{)}^2}[1+{\cot }^{}(\eta -B)]$
	$=$	$\frac{C}{(\eta -F{)}^3}\{-2+2(\eta -D)\cot (\eta -B)-(\eta -F)[\cot (\eta -B)]+(\eta -D)(\eta -F)[1+{\cot }^{}(\eta -B)\left]\right\}$
	$=$	$\frac{C}{(\eta -F{)}^3}\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}.$

And,

$\frac{d^2{x}_Q}{d{\eta }^2}$	$=$	$\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}\frac{d}{d\eta }\left[\frac{C}{(\eta -F{)}^3}\right]$
		$+\frac{C}{(\eta -F{)}^3}\cdot \frac{d}{d\eta }\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}$
	$=$	$\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}\left[\frac{-3C}{(\eta -F{)}^4}\right]$
		$+\frac{C}{(\eta -F{)}^3}\{[2\eta -(D+F)]+\cot (\eta -B)-(\eta -2D+F)[1+{\cot }^{}(\eta -B)]+[2\eta -(D+F)]{\cot }^{}(\eta -B)-2[{\eta }^2-\eta (D+F)+DF]\cot (\eta -B)[1+{\cot }^{}(\eta -B)\left]\right\}$
	$=$	$\frac{C}{(\eta -F{)}^4}\{-3[(\eta -D)(\eta -F)-2]-3(\eta -2D+F)\cot (\eta -B)-3(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}$
		$+\frac{C}{(\eta -F{)}^3}\{[2\eta -(D+F)]-(\eta -2D+F)+\cot (\eta -B)-(\eta -2D+F){\cot }^{}(\eta -B)+[2\eta -(D+F)]{\cot }^{}(\eta -B)-2[{\eta }^2-\eta (D+F)+DF]\cot (\eta -B)-2[{\eta }^2-\eta (D+F)+DF]{\cot }^{}(\eta -B)\}$

YYYY

And,

$\frac{d^2{x}_Q}{d{\eta }^2}$	$=$	$\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}\frac{d}{d\eta }\left[\frac{C}{(\eta -F{)}^3}\right]$
		$+\frac{C}{(\eta -F{)}^3}\cdot \frac{d}{d\eta }\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}$
	$=$	$\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}\left[\frac{-3C}{(\eta -F{)}^4}\right]$
		$+\frac{C}{(\eta -F{)}^3}\cdot \left\{\frac{d}{d\eta }\right[(\eta -2D+F)\cot (\eta -B)]+\frac{d}{d\eta }[(\eta -D)(\eta -F){\cot }^{}(\eta -B)\left]\right\}$
	$=$	$\frac{C}{(\eta -F{)}^4}\{-3[(\eta -D)(\eta -F)-2]-3(\eta -2D+F)\cot (\eta -B)-3(\eta -D)(\eta -F){\cot }^{}(\eta -B)$
		$+(\eta -F)\frac{d}{d\eta }[(\eta -2D+F)\cot (\eta -B)]+(\eta -F)\frac{d}{d\eta }[(\eta -D)(\eta -F){\cot }^{}(\eta -B)]\}$
	$=$	$\frac{C}{(\eta -F{)}^4}\{-3[(\eta -D)(\eta -F)-2]-3(\eta -2D+F)\cot (\eta -B)-3(\eta -D)(\eta -F){\cot }^{}(\eta -B)$
		$+(\eta -F)\cot (\eta -B)\frac{d}{d\eta }[(\eta -2D+F)]+(\eta -F)(\eta -2D+F)\frac{d}{d\eta }[\cot (\eta -B)]$
		$+(\eta -F){\cot }^2(\eta -B)\frac{d}{d\eta }[{\eta }^2-\eta (D+F)+DF]+(\eta -F)(\eta -D)(\eta -F)\frac{d}{d\eta }[{\cot }^2(\eta -B)]\}$
	$=$	$\frac{C}{(\eta -F{)}^4}\{-3[(\eta -D)(\eta -F)-2]-3(\eta -2D+F)\cot (\eta -B)-3(\eta -D)(\eta -F){\cot }^{}(\eta -B)$
		$+(\eta -F)\cot (\eta -B)-(\eta -F)(\eta -2D+F)[1+{\cot }^{}(\eta -B)]$
		$+(\eta -F){\cot }^2(\eta -B)[2\eta -(D+F)]-2(\eta -F)(\eta -D)(\eta -F)\cot (\eta -B)[1+{\cot }^{}(\eta -B)]\}$
	$=$	$\frac{C}{(\eta -F{)}^4}\{-3[(\eta -D)(\eta -F)-2]-(\eta -F)(\eta -2D+F)$
		$+[(\eta -F)-3(\eta -2D+F)-2(\eta -F)(\eta -D)(\eta -F)]\cot (\eta -B)$
		$+[(\eta -F)[2\eta -(D+F)]-3(\eta -D)(\eta -F)-2(\eta -F)(\eta -D)(\eta -F)\cot (\eta -B)-(\eta -F)(\eta -2D+F)]{\cot }^{}(\eta -B)\}$

So the envelope LAWE becomes,

$\frac{(\eta -F{)}^4}{C}\cdot \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}$	$=$	$\frac{(\eta -F{)}^4}{C}\cdot \frac{d^2{x}_Q}{d{\eta }^2}+\frac{(\eta -F{)}^4}{C}[1+\eta \cot (\eta -B_0)]\frac{2}{\eta }\cdot \frac{d{x}_Q}{d\eta }-\frac{(\eta -F{)}^4}{C}[1-\eta \cot (\eta -B_0)]\frac{2x_Q}{{\eta }^2}$
	$=$	$\{-3[(\eta -D)(\eta -F)-2]-(\eta -F)(\eta -2D+F)+[(\eta -F)-3(\eta -2D+F)-2(\eta -F)(\eta -D)(\eta -F)]\cot (\eta -B)$
		$+[(\eta -F)[2\eta -(D+F)]-3(\eta -D)(\eta -F)-2(\eta -F)(\eta -D)(\eta -F)\cot (\eta -B)-(\eta -F)(\eta -2D+F)]{\cot }^{}(\eta -B)\}$
		$+(\eta -F)[1+\eta \cot (\eta -B_0)]\frac{2}{\eta }\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)+(\eta -D)(\eta -F){\cot }^{}(\eta -B)\}$
		$-\frac{(\eta -F{)}^4}{C}[1-\eta \cot (\eta -B_0)]\frac{2}{{\eta }^2}\{A+\frac{C}{(\eta -F{)}^2}[1-(\eta -D)\cot (\eta -B)\left]\right\}$
	$=$	$-3[(\eta -D)(\eta -F)-2]-(\eta -F)(\eta -2D+F)+[(\eta -F)-3(\eta -2D+F)-2(\eta -F)(\eta -D)(\eta -F)]\cot (\eta -B)$
		$+(\eta -F)[1+\eta \cot (\eta -B_0)]\frac{2}{\eta }\{[(\eta -D)(\eta -F)-2]+(\eta -2D+F)\cot (\eta -B)\}$
		$+[(\eta -F)[2\eta -(D+F)]-3(\eta -D)(\eta -F)-2(\eta -F)(\eta -D)(\eta -F)\cot (\eta -B)-(\eta -F)(\eta -2D+F)]{\cot }^{}(\eta -B)$
		$+(\eta -F)[1+\eta \cot (\eta -B_0)]\frac{2}{\eta }[(\eta -D)(\eta -F){\cot }^{}(\eta -B)]-(\eta -F{)}^2[1-\eta \cot (\eta -B_0)\left]\frac{2}{{\eta }^2}\right[1-(\eta -D)\cot (\eta -B)]$
		$-\frac{(\eta -F{)}^4}{C}[1-\eta \cot (\eta -B_0)]\frac{2A}{{\eta }^2}.$

What does this reduce to if $A=D=F=0$.

$\frac{{\eta }^4}{C}\cdot \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}$	$=$	$-3[(\eta )(\eta )-2]-(\eta )(\eta )+[(\eta )-3(\eta )-2(\eta )(\eta )(\eta )]\cot (\eta -B)$
		$+(\eta )[1+\eta \cot (\eta -B_0)]\frac{2}{\eta }\{[(\eta )(\eta )-2]+(\eta )\cot (\eta -B)\}$
		$+[(\eta )[2\eta ]-3(\eta )(\eta )-2(\eta )(\eta )(\eta )\cot (\eta -B)-(\eta -)(\eta )]{\cot }^{}(\eta -B)$
		$+(\eta )[1+\eta \cot (\eta -B_0)]\frac{2}{\eta }[(\eta )(\eta ){\cot }^{}(\eta -B)]-(\eta {)}^2[1-\eta \cot (\eta -B_0)\left]\frac{2}{{\eta }^2}\right[1-(\eta )\cot (\eta -B)]$
		$-\frac{(\eta {)}^4}{C}[1-\eta \cot (\eta -B_0)]\frac{2A}{{\eta }^2}.$
	$=$	$6-4{\eta }^2-2(\eta +{\eta }^3)\cot (\eta -B)+2[1+\eta \cot (\eta -B_0)][{\eta }^2-2+\eta \cot (\eta -B)]-2[{\eta }^2+{\eta }^3\cot (\eta -B)]{\cot }^{}(\eta -B)$
		$+2[1+\eta \cot (\eta -B_0)][{\eta }^2{\cot }^{}(\eta -B)]-2[1-\eta \cot (\eta -B_0)][1-\eta \cot (\eta -B)]-\frac{2A{\eta }^2}{C}[1-\eta \cot (\eta -B_0)]$
	$=$	$6-4{\eta }^2-2(\eta +{\eta }^3)\cot (\eta -B)+2{\eta }^2-4+2\eta \cot (\eta -B)+2{\eta }^3\cot (\eta -B_0)-4\eta \cot (\eta -B_0)+2{\eta }^2\cot (\eta -B_0)\cot (\eta -B)-2{\eta }^2{\cot }^{}(\eta -B)-2{\eta }^3{\cot }^{}(\eta -B)$
		$+2{\eta }^2{\cot }^2(\eta -B)+2{\eta }^3\cot (\eta -B_0){\cot }^{}(\eta -B)-2+4\eta \cot (\eta -B_0)-2{\eta }^2{\cot }^{}(\eta -B_0)-\frac{2A{\eta }^2}{C}[1-\eta \cot (\eta -B_0)]$
	$=$	$-2{\eta }^2-2{\eta }^3[\cot (\eta -B)+{\cot }^{}(\eta -B)]+\cot (\eta -B_0)[2{\eta }^3+2{\eta }^2\cot (\eta -B)+2{\eta }^3{\cot }^{}(\eta -B)]-2{\eta }^2{\cot }^{}(\eta -B_0)-\frac{2A{\eta }^2}{C}[1-\eta \cot (\eta -B_0)]$
	$=$	$-2{\eta }^2+2{\eta }^3[\cot (\eta -B_0)-\cot (\eta -B)][1+{\cot }^{}(\eta -B)]+2{\eta }^2\cot (\eta -B_0)[\cot (\eta -B)-\cot (\eta -B_0)]-\frac{2A{\eta }^2}{C}[1-\eta \cot (\eta -B_0)].$

See Also

• Radial Oscillations of Uniform-density sphere
• Radial Oscillations of Isolated Polytropes
  • Setup
  • n = 1:  Attempt at Formulating an Analytic Solution
  • n = 3:  Numerical Solution to compare with 📚 M. Schwarzschild (1941, ApJ, Vol. 94, pp. 245 - 252)
  • n = 5:  Attempt at Formulating an Analytic Solution

• In an accompanying Chapter within our "Ramblings" Appendix, we have played with the adiabatic wave equation for polytropes, examining its form when the primary perturbation variable is an enthalpy-like quantity, rather than the radial displacement of a spherical mass shell. This was done in an effort to mimic the approach that has been taken in studies of the stability of Papaloizou-Pringle tori.

• $n=3$ …
  • 📚 A. S. Eddington (1918, MNRAS, Vol. 79, pp. 2 - 22), On the Pulsations of a Gaseous Star and the Problem of the Cepheid Variables.   Part I.
  • 📚 M. Schwarzschild (1941, ApJ, Vol. 94, pp. 245 - 252), Overtone Pulsations of the Standard Model: This work is referenced in §38.3 of [KW94]. It contains an analysis of the radial modes of oscillation of $n=3$ polytropes, assuming various values of the adiabatic exponent.

• $n=2$ …
  • 📚 J. C. P. Miller (1929, MNRAS, Vol. 90, pp. 59 - 64), The Effect of Distribution of Density on the Period of Pulsation of a Star
  • 📚 C. Prasad & H. S. Gurm (1961, MNRAS, Vol. 122, pp. 409 - 411), Radial Pulsations of the Polytrope, n = 2

• $n=\frac{3}{2}$ … D. Lucas (1953, Bul. Soc. Roy. Sci. Liege, 25, 585) … Citation obtained from the Prasad & Gurm (1961) article.

• $n=1$ … Citation also appears at the beginning of this chapter, and in the Prasad & Gurm (1961) article.
  • 📚 L. D. Chatterji (1951, PINSA, Vol. 17, No. 6, pp. 467 - 470), Radial Oscillations of a Gaseous Star of Polytropic Index I
  • 📚 L. D. Chatterji (1952, PINSA, Vol. 18, No. 3, pp. 187 - 191), Anharmonic Pulsations of a Polytropic Model of Index Unity

• Composite Polytropes … M. Singh (1968, MNRAS, 140, 235-240), Effect of Central Condensation on the Pulsation Characteristics

• Summary of Known Analytic Solutions … R. Stothers (1981, MNRAS, 197, 351-361), Analytic Solutions of the Radial Pulsation Equation for Rotating and Magnetic Star Models

• Interesting Composite! … C. Prasad (1948, MNRAS, 108, 414-416), Radial Oscillations of a Particular Stellar Model