SSC/Stability/BiPolytropes/Pt4

Reconciliation

Part I:   The Search

Part II:  Review of MF85b

III:  (5,1) Radial Oscillations

IV:  Reconciliation

Figure 7: Conflicting Instability Regions

Figure 7, shown here on the right, is identical to the right-hand panel of Figure 4, as displayed above. From a standard global, free-energy analysis — such as the one summarized above — we have determined that the red-dashed curve shown in the right panel of Figure 6 divides the $q-\nu$ plane into dynamically stable (below and to the right) and unstable (above and to the left) regions.

Variational Principle

Setup

Let's follow the guidelines of the variational principle. Instead of starting with the form of the LAWE given above, namely,

Adiabatic Wave (or Radial Pulsation) Equation

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

we will start with a form that is more amenable to the variational principle, namely,

$0$

$=$

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

ASIDE: Let's show that these two expressions are equivalent. Remembering that, $\frac{d{P}_0}{d{r}_0}$ $=$ $-g_0{\rho }_0=-\left(\frac{G{M}_r}{r_0^2}\right){\rho }_0,$ the second expression becomes, $0$ $=$ $r_0^4\gamma P_0\frac{d^{2}x}{d{r}_0^2}+\gamma \frac{dx}{d{r}_0}[4r_0^3{P}_0-r_0^4{g}_0{\rho }_0]+[{\omega }^2{\rho }_0{r}_0^4-(3\gamma -4)r_0^3{g}_0{\rho }_0]x$   $=$ $r_0^4\gamma P_0\{\frac{d^{2}x}{d{r}_0^2}+[\frac{4}{r_0}-\frac{g_0{\rho }_0}{P_0}]\frac{dx}{d{r}_0}+\frac{{\rho }_0}{\gamma P_0}[{\omega }^2-(3\gamma -4)\frac{g_0}{r_0}\left]x\right\}.$ Hence, we must multiply the first expression through by $r_0^4\gamma P_0$ in order to obtain the second expression.

From above, we realize that multiplying the second expression through by $(K_c/G){\rho }_c^{-4/5}$ gives,

$0$	$=$	$r_0^4\gamma P_0\{\frac{d^{2}x}{dr{*}^2}+[4-\left(\frac{{\rho }^{*}}{P^{*}}\right)\frac{M_r^{*}}{(r^{*})}]\frac{1}{r^{*}}\frac{dx}{dr*}+(\frac{{\rho }^{*}}{P^{*}}\left)\right[\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}\left]x\right\}$
	$=$	$(r^{*}{)}^4[K_c^{1/2}/(G^{1/2}{\rho }_c^{2/5}){]}^4\gamma P^{*}[K_c{\rho }_c^{6/5}]\{\frac{d^{2}x}{dr{*}^2}+[4-\left(\frac{{\rho }^{*}}{P^{*}}\right)\frac{M_r^{*}}{(r^{*})}]\frac{1}{r^{*}}\frac{dx}{dr*}+(\frac{{\rho }^{*}}{P^{*}}\left)\right[\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}\left]x\right\}$
	$=$	$\left[\frac{K_c^3}{G^2{\rho }_c^{2/5}}\right](r^{*}{)}^4\gamma P^{*}\{\frac{d^{2}x}{dr{*}^2}+[4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right]\frac{1}{r^{*}}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)[\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}]x\}.$

That is, multiplying the second expression through by, $(K_c/G){\rho }_c^{-4/5}\cdot G^2{\rho }_c^{2/5}/K_c^3=G/(K_c^2{\rho }_c^{2/5})$ , should give a desirable, totally dimensionless version of the LAWE. Remembering that,

${\rho }^{*}$	$\equiv$	$\frac{{\rho }_0}{{\rho }_c}$	;	$r^{*}$	$\equiv$	$\frac{r_0}{[K_c^{1/2}/(G^{1/2}{\rho }_c^{2/5})]}$
$P^{*}$	$\equiv$	$\frac{P_0}{K_c{\rho }_c^{6/5}}$	;	$M_r^{*}$	$\equiv$	$\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_c^{1/5})]}$
$\frac{d{P}^{*}}{d{r}^{*}}$	$=$	$-\frac{M_r^{*}{\rho }^{*}}{(r^{*}{)}^2}$	;	$E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$	$=$	$[\frac{K_c^5}{G^3}{]}^{1/2}$

let's try it.

$0$	$=$	$G{K}_c^{-2}{\rho }_c^{-2/5}\left\{\frac{d}{d{r}_0}\right[r_0^4\gamma P_0\frac{dx}{d{r}_0}]+[{\omega }^2{\rho }_0{r}_0^4+(3\gamma -4)r_0^3\frac{d{P}_0}{d{r}_0}\left]x\right\}$
	$=$	$G{K}_c^{-2}{\rho }_c^{-2/5}\left\{\right(\frac{K_c^2{\rho }_c^{2/5}}{G}\left)\frac{d}{d{r}^{*}}\right[(r^{*}{)}^4\gamma P^{*}\frac{dx}{d{r}^{*}}]+\left[{\omega }^2\right(\frac{K_c^2}{G^2{\rho }_c^{8/5}}){\rho }_c{\rho }^{*}(r^{*}{)}^4+(3\gamma -4)\left(\frac{K_c}{G{\rho }_c^{4/5}}\right)K_c{\rho }_c^{6/5}(r^{*}{)}^3\frac{d{P}^{*}}{d{r}^{*}}]x\}$
	$=$	$\frac{d}{d{r}^{*}}[(r^{*}{)}^4\gamma P^{*}\frac{dx}{d{r}^{*}}]+[\left(\frac{{\omega }^2}{G{\rho }_c}\right){\rho }^{*}(r^{*}{)}^4+(3\gamma -4)(r^{*}{)}^3\frac{d{P}^{*}}{d{r}^{*}}]x.$

Now, guided by the accompanying summary, if we multiply through by $4\pi xd{r}^{*}$ and integrate over the entire volume, we obtain the governing variational relation, namely,

$0$	$=$	$\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}4\pi x\cdot d[(r^{*}{)}^4{\gamma }_c{P}^{*}\frac{dx}{d{r}^{*}}]+\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}[\left(\frac{{\omega }^2}{G{\rho }_c}\right){\rho }^{*}(r^{*}{)}^4+(3{\gamma }_c-4)(r^{*}{)}^3\frac{d{P}^{*}}{d{r}^{*}}]4\pi x^{2}d{r}^{*}$
		$+\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}4\pi x\cdot d[(r^{*}{)}^4{\gamma }_e{P}^{*}\frac{dx}{d{r}^{*}}]+\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}[\left(\frac{{\omega }^2}{G{\rho }_c}\right){\rho }^{*}(r^{*}{)}^4+(3{\gamma }_e-4)(r^{*}{)}^3\frac{d{P}^{*}}{d{r}^{*}}]4\pi x^{2}d{r}^{*}$
	$=$	$[4\pi x(r^{*}{)}^4{\gamma }_c{P}^{*}\frac{dx}{d{r}^{*}}{]}_0^{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}-\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}4\pi [(r^{*}{)}^4{\gamma }_c{P}^{*}\left(\frac{dx}{d{r}^{*}}{)}^2\right]d{r}^{*}-\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}[(3{\gamma }_c-4)M_r^{*}{\rho }^{*}]4\pi x^2{r}^{*}d{r}^{*}$
		$+[4\pi x(r^{*}{)}^4{\gamma }_e{P}^{*}\frac{dx}{d{r}^{*}}{]}_{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}^{R^{*}}-\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}4\pi [(r^{*}{)}^4{\gamma }_e{P}^{*}\left(\frac{dx}{d{r}^{*}}{)}^2\right]d{r}^{*}-\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}[(3{\gamma }_e-4)M_r^{*}{\rho }^{*}]4\pi x^2{r}^{*}d{r}^{*}+\underset{0}{\overset{R^{*}}{\int }}\left[\right(\frac{{\omega }^2}{G{\rho }_c}){\rho }^{*}(r^{*}{)}^4]4\pi x^{2}d{r}^{*}$
	$=$	$-\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^2{\gamma }_{c}4\pi (r^{*}{)}^2{P}^{*}d{r}^{*}+\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}(3{\gamma }_c-4)x^2(-\frac{M_r^{*}}{r^{*}})4\pi {\rho }^{*}(r^{*}{)}^{2}d{r}^{*}-[4\pi x^2(r^{*}{)}^3{\gamma }_c{P}^{*}(-\frac{d\ln x}{d\ln r^{*}}){]}_0^{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}$
		$-\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^2{\gamma }_{e}4\pi (r^{*}{)}^2{P}^{*}d{r}^{*}+\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}(3{\gamma }_e-4)x^2(-\frac{M_r^{*}}{r^{*}})4\pi {\rho }^{*}(r^{*}{)}^{2}d{r}^{*}-[4\pi x^2(r^{*}{)}^3{\gamma }_e{P}^{*}(-\frac{d\ln x}{d\ln r^{*}}){]}_{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}^{R^{*}}+\underset{0}{\overset{R^{*}}{\int }}4\pi (\frac{{\omega }^2}{G{\rho }_c}){\rho }^{*}(r^{*}{)}^4{x}^{2}d{r}^{*}.$

Energy Normalization:

$(\gamma -1)d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}$	$=$	$4\pi r^{2}Pdr$
	$=$	$4\pi [\frac{K_c^{1/2}}{G^{1/2}{\rho }_c^{2/5}}{]}^3{K}_c{\rho }_c^{6/5}(r^{*}{)}^2{P}^{*}d{r}^{*}$
	$=$	$4\pi [\frac{K_c^5}{G^3}{]}^{1/2}(r^{*}{)}^2{P}^{*}d{r}^{*}$
$\Rightarrow E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$	$\equiv$	$[\frac{K_c^5}{G^3}{]}^{1/2}$
$d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$	$=$	$-\left(\frac{G{M}_r}{r}\right)4\pi r^2\rho dr$
	$=$	$-\left(\frac{G{M}_r^{*}}{r^{*}}\right)4\pi (r^{*}{)}^2{\rho }^{*}d{r}^{*}[\frac{K_c}{G{\rho }_c^{4/5}}\left]{\rho }_c\right[\frac{K_c^{3/2}}{G^{3/2}{\rho }_c^{1/5}}]$
	$=$	$-\left(\frac{M_r^{*}}{r^{*}}\right)4\pi (r^{*}{)}^2{\rho }^{*}d{r}^{*}{E}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$

Hence, the dimensionless governing variational relation becomes,

$\underset{0}{\overset{R^{*}}{\int }}\left(\frac{2\pi }{3}\right){\sigma }_c^2(r^{*}{)}^2{x}^{2}d{M}_r^{*}$	$=$	$\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{\gamma }_c({\gamma }_c-1)x^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}-\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}(3{\gamma }_c-4)x^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}+[4\pi x^2(r^{*}{)}^3{\gamma }_c{P}^{*}(-\frac{d\ln x}{d\ln r^{*}}){]}_0^{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}$
		$+\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{\gamma }_e({\gamma }_e-1)x^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}-\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}(3{\gamma }_e-4)x^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}+[4\pi x^2(r^{*}{)}^3{\gamma }_e{P}^{*}(-\frac{d\ln x}{d\ln r^{*}}){]}_{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}^{R^{*}}$
	$=$	$\frac{2{\gamma }_c}{3}\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}-(3{\gamma }_c-4)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}+4\pi x_i^2(r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}{)}^3{\gamma }_c{P}_i^{*}\{-\frac{d\ln x}{d\ln r^{*}}{|}_i{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$
		$+\frac{2{\gamma }_e}{3}\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}-(3{\gamma }_e-4)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}-4\pi x_i^2(r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}{)}^3{\gamma }_e{P}_i^{*}\{-\frac{d\ln x}{d\ln r^{*}}{|}_i{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$
	$=$	$\frac{2{\gamma }_c}{3}\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}-(3{\gamma }_c-4)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$
		$+\frac{2{\gamma }_e}{3}\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}-(3{\gamma }_e-4)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}-4\pi x_i^2(r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}{)}^3{P}_i^{*}\left[{\gamma }_c\right\{\frac{d\ln x}{d\ln r^{*}}{|}_i{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-{\gamma }_e\left\{\frac{d\ln x}{d\ln r^{*}}{|}_i{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}\right]$
	$=$	$\frac{2{\gamma }_c}{3}\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}-(3{\gamma }_c-4)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$
		$+\frac{2{\gamma }_e}{3}\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}-(3{\gamma }_e-4)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}-4\pi x_i^2(r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}{)}^3{P}_i^{*}[3({\gamma }_e-{\gamma }_c)],$

where,

$d{M}_r^{*}$	$=$	$4\pi (r^{*}{)}^2{\rho }^{*}d{r}^{*},$
$d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}$	$=$	$\frac{1}{(\gamma -1)}[4\pi (r^{*}{)}^2{P}^{*}d{r}^{*}]=[\frac{2}{3(\gamma -1)}]d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*},$
$d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$	$=$	$-\left(\frac{M_r^{*}}{r^{*}}\right)4\pi (r^{*}{)}^2{\rho }^{*}d{r}^{*}.$

Or, for inclusion in our accompanying Tabular Overview,

$\left(\frac{2\pi }{3}\right){\sigma }_c^2\underset{0}{\overset{R^{*}}{\int }}(x{r}^{*}{)}^{2}d{M}_r^{*}$	$=$	${\gamma }_c({\gamma }_c-1)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}-(3{\gamma }_c-4)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$
		$+{\gamma }_e({\gamma }_e-1)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}-(3{\gamma }_e-4)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$
		$+3^2({\gamma }_c-{\gamma }_e)x_i^2{P}_i^{*}{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}.$

Implementation

Table 3
$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	Core						Envelope						Virial
		Integrals over $d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}$			Integrals over $d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$			Integrals over $d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}$			Integrals over $d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$			Numerical $[\frac{2{\mathfrak{𝔰}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|{\mathfrak{𝔴}}_{\mathrm{t}\mathrm{o}\mathrm{t}}|}-1]$
		Analytic ${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	Numerical ${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	TERM1	Analytic ${\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	Numerical ${\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	TERM2	Analytic ${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	Numerical ${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	TERM3	Analytic ${\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	Numerical ${\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	TERM4
1	1.6686460157	3.021916335	3.021921	0.116389175	-3.356583022	-3.35666	-2.649752079	1.47780476	1.47791	1.0720821	-5.642859167	-5.642820	-1.91142893	0.000020
$\frac{1}{2}$	2.27925811317	4.241287117	4.241410819	0.440878529	-6.074241035	-6.074317546	-4.150731169	4.284931508	4.28547195	1.44651932	-10.97819621	-10.97847622	-0.92598634	0.000057
0.345	2.560146865247	4.639705843	4.6399114	0.6794857	-7.125754184	-7.125854025	-4.5487829	11.72861751	11.730381	1.51410084	-25.61089252	-25.6115597	-0.4496513	0.000097
$\frac{1}{3}$	2.582007485476	4.667042505	4.667254935	0.700414598	-7.200966267	-7.201068684	-4.57274936	13.15887139	13.1608467	1.51408246	-28.45086152	-28.45153761	-0.4170461	0.000101
0.309	2.6274239687695	4.722277318	4.722504339	0.744964507	-7.354156963	-7.3542507	-4.61961058	17.1374434	17.1399773	1.51055838	-36.36528446	-36.36591543	-0.3524855	0.000110
$\frac{1}{4}$	2.7357711469398	4.84592201	4.846185027	0.857001395	-7.70305421	-7.703178009	-4.7163542	37.84289623	37.8479208	1.47966673	-77.67458196	-77.67408155	-0.2194152	0.000128
${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ $\equiv$ $\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}$           ;           TERM1 $\equiv$ $\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}$ ${\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ $\equiv$ $\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$           ;           TERM2 $\equiv$ $\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$ ${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$ $\equiv$ $\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}$           ;           TERM3 $\equiv$ $\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{S}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}^{*}$ ${\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$ $\equiv$ $\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$           ;           TERM4 $\equiv$ $\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$ NOTE: In all integrals, the fractional radial-displacement function, $x$, has been normalized to unity at the center of the spherical model.

$[{\sigma }_c^2{]}_{{\mu }_e/{\mu }_c=1}^{\mathrm{V}\mathrm{P}}$	$=$	$\frac{3}{2\pi \cdot \mathbf{𝐓}\mathbf{𝐄}\mathbf{𝐑}\mathbf{𝐌}\mathbf{𝟓}}\{\frac{2{\gamma }_c}{3}[\mathbf{𝐓}\mathbf{𝐄}\mathbf{𝐑}\mathbf{𝐌}\mathbf{𝟏}]-(3{\gamma }_c-4)[\mathbf{𝐓}\mathbf{𝐄}\mathbf{𝐑}\mathbf{𝐌}\mathbf{𝟐}]+\frac{2{\gamma }_e}{3}[\mathbf{𝐓}\mathbf{𝐄}\mathbf{𝐑}\mathbf{𝐌}\mathbf{𝟑}]-(3{\gamma }_e-4)[\mathbf{𝐓}\mathbf{𝐄}\mathbf{𝐑}\mathbf{𝐌}\mathbf{𝟒}]-4\pi x_i^2(r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}{)}^3{P}_i^{*}[3({\gamma }_e-{\gamma }_c)]\}$
	$=$	$0.036312577\times [0.093111340-1.059900832+1.4294428-(-3.82285786)-1.782200484\times (2.4)]=0.036312577\times [0.00819]$
	$=$	$0.0002973.$
$[{\sigma }_c^2{]}_{{\mu }_e/{\mu }_c=1/2}^{\mathrm{V}\mathrm{P}}$	$=$	$0.020704186\times [0.352702823-1.660292468+1.928692426-(-1.85197268)-1.029029184\times (2.4)]=0.020704186\times [0.003405]$
	$=$	$0.0000705.$
$[{\sigma }_c^2{]}_{{\mu }_e/{\mu }_c=0.345}^{\mathrm{V}\mathrm{P}}$	$=$	$0.019377703\times [0.54358856-1.81951316+2.01880112-(-0.8993026)-0.68747441\times (2.4)]=0.019377703\times [0.003585]$
	$=$	$0.0000695.$
$[{\sigma }_c^2{]}_{{\mu }_e/{\mu }_c=1/3}^{\mathrm{V}\mathrm{P}}$	$=$	$0.019467503\times [0.560331678-1.81099744+2.018776613-(-0.8340922)-0.658354814\times (2.4)]=0.019467503\times [1.602203052-1.580051555]$
	$=$	$0.00043123.$
$[{\sigma }_c^2{]}_{{\mu }_e/{\mu }_c=0.309}^{\mathrm{V}\mathrm{P}}$	$=$	$0.019781992\times [0.595971606-1.847844232+2.01407784-(-0.704971)-0.60905486\times (2.4)]=0.019781992\times [1.467176214-1.461731665]$
	$=$	$0.0001077.$
$[{\sigma }_c^2{]}_{{\mu }_e/{\mu }_c=1/4}^{\mathrm{V}\mathrm{P}}$	$=$	$0.021344909\times [0.685601116-1.88654168+1.972888973-(-0.4388304)-0.499262049\times (2.4)]=0.021344909\times [1.210778809-1.198228918]$
	$=$	$0.000267876.$

Table 4
$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	Analytic ${\mathfrak{𝔪}}_{\mathrm{t}\mathrm{o}\mathrm{t}}$	Numerical ${\mathfrak{𝔪}}_{\mathrm{t}\mathrm{o}\mathrm{t}}$	TERM5	$x_i$	$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$P_i^{*}$	$\{-\frac{d\ln x}{d\ln r^{*}}{|}_i{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$\{-\frac{d\ln x}{d\ln r^{*}}{|}_i{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$[{\sigma }_c^2{]}^{\mathrm{V}\mathrm{P}}$
1	1.6686460157	4.818155928	4.818145	13.14874521	0.814374698	1.153014872	0.139506172	+0.455871977	+1.47352	0.0002973
$\frac{1}{2}$	2.27925811317	9.020985415	9.021084268	23.0612702	0.653665497	1.574940686	0.049058481	+1.059668912	+1.835801347	0.0000705
0.345	2.560146865247	17.41399388	17.4141672	24.63990825	0.563043202	1.769031527	0.030957085	+1.552125296	+2.131275177	0.0000695
$\frac{1}{3}$	2.582007485476	18.8449906	18.8451614	24.52624951	0.555549156	1.78413696	0.029889634	+1.600041467	+2.16002488	0.0004312
0.309	2.6274239687695	22.61541791	22.6155686	24.13633675	0.539776219	1.815519219	0.027798189	+1.705239188	+2.223143513	0.0001077
$\frac{1}{4}$	2.7357711469398	39.12088278	39.1208075	22.36902623	0.501037082	1.890385851	0.023427588	+1.991720516	+2.39503231	0.0002679
${\mathfrak{𝔪}}_{\mathrm{t}\mathrm{o}\mathrm{t}}$ $\equiv$ $\underset{0}{\overset{R^{*}}{\int }}d{M}_r^{*}$           ;           $I_{\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}}^{*}$ $\equiv$ $\underset{0}{\overset{R^{*}}{\int }}(r^{*}{)}^{2}d{M}_r^{*}$           ;           TERM5 $\equiv$ $\underset{0}{\overset{R^{*}}{\int }}(r^{*}{)}^2{x}^{2}d{M}_r^{*}$ NOTE: In the TERM5 integral (as elsewhere), the fractional radial-displacement function, $x$, has been normalized to unity at the center of the spherical model.

Revised Free-Energy Analysis

If we set $x$ = constant in the variational principle relation, we have,

$\left(\frac{2\pi }{3}\right){\sigma }_c^2\underset{0}{\overset{R^{*}}{\int }}(r^{*}{)}^{2}d{M}_r^{*}$	$=$	$-(3{\gamma }_c-4)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}-(3{\gamma }_e-4)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}-4\pi (r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}{)}^3{P}_i^{*}[3({\gamma }_e-{\gamma }_c)]$
$\Rightarrow \left(\frac{2\pi }{3}\right){\sigma }_c^2{I}_{\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}}^{*}$	$=$	$(4-3{\gamma }_c){\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(4-3{\gamma }_e){\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}}+P_i^{*}{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}[3^2({\gamma }_c-{\gamma }_e)],$

Notice the similarity between this last expression and the pair of expressions — numbered ⑥ and ⑦ — that arise in the context of pressure-truncated polytropes.

See Also

• K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206), A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.