SSC/Stability/BiPolytropes/RedGiantToPN/Pt3

Main Sequence to Red Giant to Planetary Nebula (Part 3)

Part I:  Background & Objective

Part II:

Yabushita68-Motivated Analysis

Part IV:

Yabushita68-Motivated Analysis

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

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,     $\alpha \equiv (3-\frac{4}{{\gamma }_{\mathrm{g}}})$

Motivated by the derivation presented by 📚 S. Yabushita (1968, MNRAS, Vol. 140, pp. 109 - 120), let's now insert an integration constant, $C_0$, into this 2^nd-order ODE to obtain what we henceforth will refer to as the,

Yabushita68-Motivated Polytropic LAWE

$C_0=\frac{d^{2}x}{d{\eta }^2}+[4-(n+1)Q]\frac{1}{\eta }\cdot \frac{dx}{d\eta }+(n+1)\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_g})\frac{{\eta }^2}{\theta }-\alpha Q]\frac{x}{{\eta }^2}$

After setting ${\sigma }_c^2=0$, let's guess an eigenfunction of the form,

$x$

$=$

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

in which case we find that,

$C_0$	$=$	$-m(-m-1){\eta }^{-m-2}+[4-(n+1)Q]\frac{1}{\eta }[-m{\eta }^{-m-1}]+(n+1)[-\alpha Q]{\eta }^{-m-2}$
	$=$	$\{m(m+1)-[4-(n+1)Q\left]\right[m]+(n+1)[-\alpha Q\left]\right\}{\eta }^{-m-2}$
	$=$	$\{m(m+1)-4m+[m(n+1)]Q-[\alpha (n+1)\left]Q\right\}{\eta }^{-m-2}$
	$=$	$\{m^2-3m+(n+1)(m-\alpha )Q\}{\eta }^{-m-2}.$

The dependence on $Q$ can be eliminated if we set $m=\alpha$; in which case we obtain,

$C_0$

$=$

$\{{\alpha }^2-3\alpha \}{\eta }^{-m-2}.$

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