SSC/Stability/BiPolytropes/RedGiantToPN/Pt2

Main Sequence to Red Giant to Planetary Nebula (Part 2)

Part I:  Background & Objective

Part II:

Part III:

Part IV:

Foundation

In an accompanying discussion, we derived the so-called,

Adiabatic Wave (or Radial Pulsation) Equation

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

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.

Introducing the dimensionless frequency-squared, ${\sigma }_c^2\equiv 3{\omega }^2/(2\pi G{\rho }_c)$, we can rewrite this LAWE as, $0$ $=$ $\frac{d^{2}x}{d{r}_0^2}+[4-(\frac{g_0}{r_0}\cdot \frac{{\rho }_0{r}_0^2}{P_0}\left)\right]\frac{1}{r_0}\cdot \frac{dx}{d{r}_0}+\left(\frac{{\rho }_0{r}_0^2}{P_0}\right)[\frac{2\pi G{\rho }_c{\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{g_0}{r_0}\right]\frac{x}{r_0^2},$ where, as a reminder, $g_0\equiv GM(r_0)/r_0^2$. Now, for our $(n_c,n_e)=(5,1)$ bipolytrope, we have found it useful to adopt the following four dimensionless variables: ${\rho }^{*}$ $\equiv$ $\frac{{\rho }_0}{{\rho }_c}$ ;     $r^{*}$ $\equiv$ $\frac{r_0}{[K_c^{1/2}/(G^{1/2}{\rho }_c^{2/5})]}$ $P^{*}$ $\equiv$ $\frac{P_0}{K_c{\rho }_c^{6/5}}$ ;     $M_r^{*}$ $\equiv$ $\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_c^{1/5})]}$ This means that, $\frac{g_0}{r_0}=\frac{GM(r_0)}{r_0^3}$ $=$ $G{M}_r^{*}\left[K_c^{3/2}{G}^{-3/2}{\rho }_c^{-1/5}\right]r_{*}^{-3}\left[K_c^{-3/2}{G}^{3/2}{\rho }_c^{6/5}\right]=\left[G{\rho }_c\right]M_r^{*}{r}_{*}^{-3};$ $\frac{{\rho }_0{r}_0^2}{P_0}$ $=$ ${\rho }^{*}{\rho }_c(r^{*}{)}^2[K_c{G}^{-1}{\rho }_c^{-4/5}](P^{*}{)}^{-1}\left[K_c^{-1}{\rho }_c^{-6/5}\right]=\left[G^{-1}{\rho }_c^{-1}\right]{\rho }^{*}(r^{*}{)}^2(P^{*}{)}^{-1};$ $\frac{g_0}{r_0}\cdot \frac{{\rho }_0{r}_0^2}{P_0}$ $=$ $\left[G{\rho }_c\right]M_r^{*}{r}_{*}^{-3}\cdot \left[G^{-1}{\rho }_c^{-1}\right]{\rho }^{*}(r^{*}{)}^2(P^{*}{)}^{-1}=\frac{M_r^{*}{\rho }^{*}}{P^{*}{r}^{*}}.$ Making these substitutions, the LAWE can be rewritten as, $0$ $=$ $\frac{d^{2}x}{d{r}_0^2}+[4-\frac{M_r^{*}{\rho }^{*}}{P^{*}{r}^{*}}]\frac{1}{r_0}\cdot \frac{dx}{d{r}_0}+\frac{1}{G{\rho }_c}\left[\frac{{\rho }^{*}(r^{*}{)}^2}{P^{*}}\right][\frac{2\pi G{\rho }_c{\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{G{\rho }_c{M}_r^{*}}{(r^{*}{)}^3}\right]\frac{x}{r_0^2};$ then, multiplying through by $[K{G}^{-1}{\rho }_c^{-4/5}]$ allows us to everywhere switch from $(r_0{)}^2$ to $(r^{*}{)}^2$, namely, $0$ $=$ $\frac{d^{2}x}{d(r^{*}{)}^2}+[4-\frac{M_r^{*}{\rho }^{*}}{P^{*}{r}^{*}}]\frac{1}{r^{*}}\cdot \frac{dx}{d(r^{*})}+\left[\frac{{\rho }^{*}(r^{*}{)}^2}{P^{*}}\right][\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{M_r^{*}}{(r^{*}{)}^3}\right]\frac{x}{(r^{*}{)}^2}.$

In shorthand, we can rewrite this equation in the form,

$0$

$=$

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

where,

$x^{\prime }$

$=$

$\frac{dx}{d{r}^{*}}$

and

$x^{″}$

$=$

$\frac{d^{2}x}{d(r^{*}{)}^2};$

and,

${𝒦}\equiv \left(\frac{{\rho }^{*}}{P^{*}}\right)\left[\right(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}})\frac{2\pi }{3}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{M_r^{*}}{(r^{*}{)}^3}\right];$

and,

${\mathscr{H}}$

$\equiv$

$\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}.$

Specific Case of (n_c, n_e) = (5,1)

Drawing from our "Table 2" profiles, let's evaluate ${\mathscr{H}}$ and ${𝒦}$ for the two separate regions of bipolytrope model.

The n_c = 5 Core

${\displaystyle }$

$r^{*}=(\frac{3}{2\pi }{)}^{1/2}\xi$

$\frac{{\rho }^{*}}{P^{*}}$	$=$	$(1+\frac{{\xi }^2}{3}{)}^{-5/2}(1+\frac{{\xi }^2}{3}{)}^{6/2}=(1+\frac{{\xi }^2}{3}{)}^{1/2}$
$\frac{M^{*}}{r^{*}}$	$=$	$\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-3/2}](\frac{2\pi }{3}{)}^{1/2}{\xi }^{-1}=2{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{-3/2}$
$\Rightarrow {\mathscr{H}}$	$=$	$4-2{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{-3/2}(1+\frac{{\xi }^2}{3}{)}^{1/2}=4-2{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{-1}.$

Also,

$\Rightarrow {𝒦}$

$=$

$(1+\frac{{\xi }^2}{3}{)}^{1/2}\{\left(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}}\right)\frac{2\pi }{3}-(3-\frac{4}{{\gamma }_{\mathrm{g}}})2{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{-3/2}(\frac{2\pi }{3}\left){\xi }^{-2}\right\}=\frac{2\pi }{3}(1+\frac{{\xi }^2}{3}{)}^{1/2}\{\left(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}}\right)-2(3-\frac{4}{{\gamma }_{\mathrm{g}}})(1+\frac{{\xi }^2}{3}{)}^{-3/2}\}$

Hence, the LAWE becomes,

$0$

$=$

$\left[\frac{2\pi }{3}\right]\frac{d^{2}x}{d{\xi }^2}+\left[{\mathscr{H}}\right]\frac{2\pi }{3}{\xi }^{-1}\frac{dx}{d\xi }+\frac{2\pi }{3}(1+\frac{{\xi }^2}{3}{)}^{1/2}\{\left(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}}\right)-2(3-\frac{4}{{\gamma }_{\mathrm{g}}})(1+\frac{{\xi }^2}{3}{)}^{-3/2}\}x.$

Multiplying through by $3{\xi }^2/(2\pi )$ gives,

$0$

$=$

${\xi }^2\frac{d^{2}x}{d{\xi }^2}+[4\xi -2{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-1}]\frac{dx}{d\xi }+{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{1/2}\left\{\right(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}})-2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right(1+\frac{{\xi }^2}{3}{)}^{-3/2}\}x.$

Let's compare this with the equivalent expression presented separately, namely, Polytropic LAWE (linear adiabatic wave equation) $0=\frac{d^{2}x}{d{\xi }^2}+[4-(n+1)Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+(n+1)\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_g})\frac{{\xi }^2}{\theta }-\alpha Q]\frac{x}{{\xi }^2}$ 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,     $\alpha \equiv (3-\frac{4}{{\gamma }_{\mathrm{g}}})$ The primary E-type solution for n = 5 polytropes states that, ${\theta }_{n=5}$ $=$ $[1+\frac{{\xi }^2}{3}{]}^{-1/2}.$ $\Rightarrow Q=-\frac{\xi }{\theta }\frac{d\theta }{d\xi }$ $=$ $-\xi [1+\frac{{\xi }^2}{3}{]}^{1/2}\frac{d}{d\xi }[1+\frac{{\xi }^2}{3}{]}^{-1/2}$   $=$ $+\xi [1+\frac{{\xi }^2}{3}{]}^{1/2}\{\frac{\xi }{3}[1+\frac{{\xi }^2}{3}{]}^{-3/2}\}$   $=$ $\frac{{\xi }^2}{3}[1+\frac{{\xi }^2}{3}{]}^{-1}.$ Hence, the LAWE may be written as, $0$ $=$ $\frac{d^{2}x}{d{\xi }^2}+[4-6Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+6\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_{\mathrm{g}}})\frac{{\xi }^2}{\theta }-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)Q\right]\frac{x}{{\xi }^2}$   $=$ $\frac{d^{2}x}{d{\xi }^2}+\{4-2{\xi }^2[1+\frac{{\xi }^2}{3}{]}^{-1}\}\frac{1}{\xi }\cdot \frac{dx}{d\xi }+\{\left(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}}\right){\xi }^2[1+\frac{{\xi }^2}{3}{]}^{1/2}-2{\xi }^2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right[1+\frac{{\xi }^2}{3}{]}^{-1}\}\frac{x}{{\xi }^2}$   $=$ $\frac{d^{2}x}{d{\xi }^2}+\{4-2{\xi }^2[1+\frac{{\xi }^2}{3}{]}^{-1}\}\frac{1}{\xi }\cdot \frac{dx}{d\xi }+\{\left(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}}\right)[1+\frac{{\xi }^2}{3}{]}^{1/2}-2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right[1+\frac{{\xi }^2}{3}{]}^{-1}\}x$ Versus above, $0$ $=$ ${\xi }^2\frac{d^{2}x}{d{\xi }^2}+[4\xi -2{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-1}]\frac{dx}{d\xi }+{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{1/2}\left\{\right(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}})-2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right(1+\frac{{\xi }^2}{3}{)}^{-3/2}\}x.$

If we set ${\gamma }_{\mathrm{g}}=6/5$ and we set ${\sigma }_c^2=0$, this becomes,

$0$

$=$

${\xi }^2\frac{d^{2}x}{d{\xi }^2}+[4\xi -2{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-1}]\frac{dx}{d\xi }+\frac{2}{3}{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{-1}x$

Next, try the solution, $x=(1-{\xi }^2/15)\Rightarrow dx/d\xi =-2\xi /15$ and $d^{2}x/d{x}^2=-2/15$:

LAWE	$=$	$-\frac{2}{15}{\xi }^2-[4\xi -2{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-1}]\frac{2\xi }{15}+\frac{2}{3}{\xi }^2(1+\frac{{\xi }^2}{3}{)}^{-1}[1-{\xi }^2/15]$
$\Rightarrow 15(1+\frac{{\xi }^2}{3})$  LAWE	$=$	$-2{\xi }^2(1+\frac{{\xi }^2}{3})-\left[60\xi \right(1+\frac{{\xi }^2}{3})-30{\xi }^3]\frac{2\xi }{15}+10{\xi }^2[1-{\xi }^2/15]$
	$=$	$-2{\xi }^2-\frac{2{\xi }^4}{3}-[60{\xi }^2-10{\xi }^4]\frac{2}{15}+\frac{10}{15}[15{\xi }^2-{\xi }^4]$
	$=$	$-2{\xi }^2-\frac{2{\xi }^4}{3}-[4{\xi }^2-\frac{2}{3}{\xi }^4]2+[10{\xi }^2-\frac{2}{3}{\xi }^4]$
	$=$	$-2{\xi }^2-\frac{2{\xi }^4}{3}-8{\xi }^2+\frac{4}{3}{\xi }^4+[10{\xi }^2-\frac{2}{3}{\xi }^4]$
	$=$	$0.$

The n_e = 1 Envelope

Throughout the envelope we have,

$r^{*}$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$
$\frac{{\rho }^{*}}{P^{*}}$	$=$	$\left[\right(\frac{{\mu }_e}{{\mu }_c}\left){\theta }_i^5\varphi \right]\left[{\theta }_i^{-6}{\varphi }^{-2}\right]=\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{-1}\varphi (\eta {)}^{-1};$
$\frac{M_r^{*}}{r^{*}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta {]}^{-1}=2(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\theta }_i}{\eta }(-{\eta }^2\frac{d\varphi }{d\eta }).$

Hence,

${\mathscr{H}}$

$\equiv$

$\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}=4-\left[\right(\frac{{\mu }_e}{{\mu }_c}\left){\theta }_i^{-1}{\varphi }^{-1}\right]\left[2\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\theta }_i}{\eta }(-{\eta }^2\frac{d\varphi }{d\eta })]=4-2(-\frac{d\ln \varphi }{d\ln \eta });$

and,

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

Let's compare this with the equivalent expression presented separately, namely, Polytropic LAWE (linear adiabatic wave equation) $0=\frac{d^{2}x}{d{\xi }^2}+[4-(n+1)Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+(n+1)\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_g})\frac{{\xi }^2}{\theta }-\alpha Q]\frac{x}{{\xi }^2}$ 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,     $\alpha \equiv (3-\frac{4}{{\gamma }_{\mathrm{g}}})$ The equilibrium, off-center equilibrium solution for n = 1 polytropes states that, ${\varphi }_{n=1}$ $=$ $-\frac{A}{\eta }\cdot \sin (B-\eta );$ $\frac{d\varphi }{d\eta }$ $=$ $-A\frac{d}{d\eta }\{{\eta }^{-1}\cdot \sin (B-\eta )\}$   $=$ $A\{{\eta }^{-2}\cdot \sin (B-\eta )+{\eta }^{-1}\cdot \cos (B-\eta )\};$ $\Rightarrow Q=-\frac{\eta }{\varphi }\frac{d\varphi }{d\eta }$ $=$ $-\eta [-\frac{A}{\eta }\cdot \sin (B-\eta ){]}^{-1}A\{{\eta }^{-2}\cdot \sin (B-\eta )+{\eta }^{-1}\cdot \cos (B-\eta )\}$   $=$ $\eta \left[\frac{\eta }{\sin (B-\eta )}\right]\{{\eta }^{-2}\cdot \sin (B-\eta )+{\eta }^{-1}\cdot \cos (B-\eta )\}$   $=$ $[1+\eta \cdot \cot (B-\eta )]$ Hence, the LAWE may be written as, $0$ $=$ $\frac{d^{2}x}{d{\eta }^2}+\{4-2[1+\eta \cdot \cot (B-\eta )\left]\right\}\frac{1}{\eta }\cdot \frac{dx}{d\eta }+2\left\{\right(\frac{{\sigma }_c^2}{6{\gamma }_{\mathrm{g}}})\frac{{\eta }^2}{\varphi }-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right[1+\eta \cdot \cot (B-\eta )\left]\right\}\frac{x}{{\eta }^2}$

Blind Alleys

Reminder

From a separate discussion, we have demonstrated that the LAWE relevant to the envelope is,

$0$

$=$

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

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,

$Q\equiv -\frac{d\ln \varphi }{d\ln \eta }=[1-\eta \cot (\eta -B)]=[1+\eta \cot (B-\eta )].$

Also separately, we have derived the following,

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

First Try

$x$

$=$

${\eta }^{-m}\Rightarrow \frac{dx}{d\eta }=-m{\eta }^{-m-1}$       and     $\frac{d^{2}x}{d{\eta }^2}=-m(-m-1){\eta }^{-m-2},$

in which case,

LAWE	$=$	$-m(-m-1){\eta }^{-m-2}+\{4-2[1+\eta \cdot \cot (B-\eta )\left]\right\}\frac{1}{\eta }\cdot [-m{\eta }^{-m-1}]+2\left\{\right(\frac{{\sigma }_c^2}{6{\gamma }_{\mathrm{g}}})\frac{{\eta }^2}{\varphi }-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right[1+\eta \cdot \cot (B-\eta )\left]\right\}{\eta }^{-m-2}$
$\Rightarrow {\eta }^{m+2}\times \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}$	$=$	$m(m+1)-m\{4-2[1+\eta \cdot \cot (B-\eta )\left]\right\}+2\left\{\right(\frac{{\sigma }_c^2}{6{\gamma }_{\mathrm{g}}})\frac{{\eta }^2}{\varphi }-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right[1+\eta \cdot \cot (B-\eta )\left]\right\}.$

Now set ${\sigma }_c^2=0$ and set ${\gamma }_{\mathrm{g}}=2$:

$\Rightarrow {\eta }^{m+2}\times \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}$	$=$	$m(m+1)-m\{4-2[1+\eta \cdot \cot (B-\eta )\left]\right\}-2\left\{\right[1+\eta \cdot \cot (B-\eta )\left]\right\}$
	$=$	$m(m+1)-4m+2m[1+\eta \cdot \cot (B-\eta )]-2[1+\eta \cdot \cot (B-\eta )]$

We see that the complexity of the LAWE reduces substantially if we set $m=+1$; specifically, this choice gives,

$[{\eta }^{m+2}\times \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}{]}_{m\to 1}$

$=$

$-2.$

Close, but no cigar!

Second Try

Next, let's set ${\sigma }_c^2=0$ but let's leave ${\gamma }_{\mathrm{g}}$ unspecified:

$\Rightarrow {\eta }^{m+2}\times \mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}$	$=$	$m(m+1)-m\{4-2[1+\eta \cdot \cot (B-\eta )\left]\right\}-2\left\{\right(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right[1+\eta \cdot \cot (B-\eta )\left]\right\}$
	$=$	$m(m+1)-4m+2m[1+\eta \cdot \cot (B-\eta )]-2(3-\frac{4}{{\gamma }_{\mathrm{g}}})[1+\eta \cdot \cot (B-\eta )]$
	$=$	$m(m-3)+\{2m-2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right\}[1+\eta \cdot \cot (B-\eta )].$

The first term goes to zero if we set $m=3$; then, in order for the second term to go to zero, we need …

$0$	$=$	$\{6-2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right\}$
$\Rightarrow {\gamma }_{\mathrm{g}}$	$=$	$\mathrm{\infty }.$

This means that the envelope is incompressible.

Third Try

Note that,

$Q\equiv -\frac{d\ln \varphi }{d\ln \eta }=[1-\eta \cot (\eta -B)]=[1+\eta \cot (B-\eta )],$

and that,

$\frac{d}{d\eta }[\cot (B-\eta )]$	$=$	$+[\sin (B-\eta ){]}^{-2}$
$\Rightarrow \frac{d^2}{d{\eta }^2}[\cot (B-\eta )]$	$=$	$+2[\cos (B-\eta ){]}^{-3}$

Let's try …

$x=x_1+x_2$

$=$

$\frac{b}{{\eta }^2}+\frac{c}{\eta }\cdot \cot (B-\eta ).$

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

${\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_1$	$=$	$\frac{d^2{x}_1}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{d{x}_1}{d\eta }-\left[2Q\right]\frac{x_1}{{\eta }^2};$
${\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_2$	$=$	$\frac{d^2{x}_2}{d{\eta }^2}+\{4-2Q\}\frac{1}{\eta }\cdot \frac{d{x}_2}{d\eta }-\left[2Q\right]\frac{x_2}{{\eta }^2}.$

One at a time:

---

$\frac{d{x}_1}{d\eta }$	$=$	$-\frac{2b}{{\eta }^3};$
$\frac{d^2{x}_1}{d{\eta }^2}$	$=$	$\frac{6b}{{\eta }^4}.$
$\Rightarrow {\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_1$	$=$	$\frac{6b}{{\eta }^4}+\{4-2Q\}[-\frac{2b}{{\eta }^4}]-\left[2Q\right]\frac{b}{{\eta }^4}$
	$=$	$\frac{1}{{\eta }^4}\{6b-2b[4-2Q]-[2bQ\left]\right\}$
	$=$	$\frac{2b}{{\eta }^4}[Q-1].$

---

$\frac{d{x}_2}{d\eta }=\frac{d}{d\eta }[\frac{c}{\eta }\cdot \cot (B-\eta )]$	$=$	$-\frac{c}{{\eta }^2}\cdot \cot (B-\eta )+\frac{c}{\eta }\cdot [\sin (B-\eta ){]}^{-2}$
	$=$	$\frac{c}{{\eta }^2}\{-\frac{\cos (B-\eta )}{\sin (B-\eta )}+\frac{\eta }{{\sin }^2(B-\eta )}\}$
	$=$	$\frac{c}{{\eta }^2}\{\eta -\sin (B-\eta )\cos (B-\eta )\}[\sin (B-\eta ){]}^{-2};$
$\frac{d^2{x}_2}{d{\eta }^2}$	$=$	$\frac{d}{d\eta }\{-\frac{c}{{\eta }^2}\cdot \cot (B-\eta )\}+\frac{d}{d\eta }\{\frac{c}{\eta }\cdot [\sin (B-\eta ){]}^{-2}\}$
	$=$	$\{\frac{2c}{{\eta }^3}\cdot \cot (B-\eta )-\frac{c}{{\eta }^2}\cdot [\sin (B-\eta ){]}^{-2}\}+\{-\frac{c}{{\eta }^2}\cdot [\sin (B-\eta ){]}^{-2}+\frac{2c}{\eta }\cdot [\sin (B-\eta ){]}^{-3}\cos (B-\eta )\}$
	$=$	$\frac{2c}{{\eta }^3}\{\cot (B-\eta )-\eta \cdot [\sin (B-\eta ){]}^{-2}+{\eta }^2\cdot [\sin (B-\eta ){]}^{-3}\cos (B-\eta )\}$
	$=$	$\frac{2c}{{\eta }^3}[\sin (B-\eta ){]}^{-3}\{{\sin }^{}(B-\eta )\cos (B-\eta )-\eta \cdot [\sin (B-\eta )]+{\eta }^2\cdot \cos (B-\eta )\}$
$\Rightarrow {\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_2$	$=$	$\frac{d^2{x}_2}{d{\eta }^2}$
		$+\{4-2Q\}\frac{1}{\eta }\cdot \frac{d{x}_2}{d\eta }-\left[2Q\right]\frac{x_2}{{\eta }^2}$
	$=$	$\frac{2c}{{\eta }^3}[\sin (B-\eta ){]}^{-3}\{{\sin }^{}(B-\eta )\cos (B-\eta )-\eta \cdot [\sin (B-\eta )]+{\eta }^2\cdot \cos (B-\eta )\}$
		$+\{4-2Q\}\cdot \frac{c}{{\eta }^3}\{\eta -\sin (B-\eta )\cos (B-\eta )\}[\sin (B-\eta ){]}^{-2}-[2Q]\frac{c}{{\eta }^3}\cdot \cot (B-\eta )$
	$=$	$\frac{2c}{{\eta }^3}[\sin (B-\eta ){]}^{-3}\{{\sin }^{}(B-\eta )\cos (B-\eta )-\eta \cdot [\sin (B-\eta )]+{\eta }^2\cdot \cos (B-\eta )\}$
		$+\frac{2c}{{\eta }^3}[\sin (B-\eta ){]}^{-3}\{(2-Q)\cdot [\eta -\sin (B-\eta )\cos (B-\eta )][\sin (B-\eta )]-Q[\sin (B-\eta ){]}^3\cot (B-\eta )\}$
	$=$	$\frac{2c}{{\eta }^3}[\sin (B-\eta ){]}^{-3}\{{\eta }^2\cdot \cos (B-\eta )-[{\sin }^{}(B-\eta )\cos (B-\eta )]+(1-Q)\cdot [\eta \cdot \sin (B-\eta )]\}$

---

Hence,

${\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_1+{\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_2$	$=$	$\frac{2b}{{\eta }^4}[Q-1]+\frac{2c}{{\eta }^3}[\sin (B-\eta ){]}^{-3}\{{\eta }^2\cdot \cos (B-\eta )-[{\sin }^{}(B-\eta )\cos (B-\eta )]+(1-Q)\cdot [\eta \cdot \sin (B-\eta )]\}$
	$=$	$\frac{2(b-c)}{{\eta }^3}[\cot (B-\eta )]$

Fourth Try

Try adding an additional term that was discussed above under "First Try", namely,

$x_3$

$=$

$\frac{d}{\eta },$

in which case,

${\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_3$

$=$

$-\frac{2d}{{\eta }^3},$

and,

${\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_1+{\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_2+{\mathrm{L}\mathrm{A}\mathrm{W}\mathrm{E}}_3$

$=$

$\frac{2}{{\eta }^3}[(b-c)\cot (B-\eta )-d].$

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