SSC/Stability/BiPolytropes/RedGiantToPN

Main Sequence to Red Giant to Planetary Nebula

Part I:  Background & Objective

Part II:

Part III:

Part IV:

Preface

Go here for Stahler schematic.

ApJ reference Stahler Schematic

file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2 Figure 1:  Equilibrium Sequences of Pressure-Truncated Polytropes

As has been detailed in an accompanying chapter, we have successfully analyzed the relative stability of pressure-truncated polytopes. The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range $1\le n\le 6$. (Another version of this figure includes the isothermal sequence, for which $n=\mathrm{\infty }$.)

 
Along each sequence for which $n\ge 3$, the green filled circle identifies the model with the largest mass. This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by 📚 Ebert (1955) and 📚 Bonnor (1956) in the context of studies of pressure-truncated isothermal equilibrium configurations.

1. The maximum-mass model's position along each sequence has been determined analytically by setting $\partial M_{\mathrm{t}\mathrm{o}\mathrm{t}}/\partial \xi {|}_{\tilde{\xi }}=0.$
2. By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
   1. Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;
   2. To our surprise, we also have been able to determine analytically an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence $(n\ge 3)$; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is precisely zero.
   As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.

Key Realization: Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.

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Figure 2: Equilibrium Sequences of Bipolytropes with $(n_c,n_e)=(5,1)$ and Various ${\mu }_e/{\mu }_c$

Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have $(n_c,n_e)=(5,1)$. Our analytically constructed equilibrium model sequences replicate the ones originally presented by 📚 Eggleton, Faulkner, & Cannon (1998) for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by 📚 Henrich & Chandrasekhar (1941) and 📚 Schönberg & Chandrasekhar (1942) for bipolytropic configurations that have $(n_c,n_e)=(\mathrm{\infty },\frac{3}{2})$.

Following HC41 and SC42, we have found it particularly useful to label each equilibrium model according to the key structural parameters, $q\equiv r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/R_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$ and $\nu \equiv M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$. The curves shown here on the right in Figure 2 graphically present the $q-\nu$ relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, $\frac{1}{4}\le {\mu }_e/{\mu }_c\le 1$, at the core-envelope interface, as labeled. Along each Fig. 2 sequence for which ${\mu }_e/{\mu }_c\le \frac{1}{3}$, the green filled circle identifies the model with the largest mass ratio, $\nu$. This maximum-mass model is a polytropic analogue of the Schönberg-Chandrasekhar mass limit, which was identified by HC41 and SC42 in the context of their studies of stars with isothermal cores.

1. The maximum-mass model's position along each sequence has been determined analytically by setting $\partial \nu /\partial q=0.$
2. By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.
   1. Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;
   2. To our surprise, we also have been able to determine analytically an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence $(n\ge 3)$; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is precisely zero.
   As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.

indexes, as labeled, over the range $1\le n\le 6$.

Figure 2: Equilibrium Sequences of Bipolytropes with $(n_c,n_e)=(5,1)$ and Various ${\mu }_e/{\mu }_c$	Analytically Determined Parameters† for Models that have the Maximum Fractional Core Mass (solid green circular markers) Along Various Equilibrium Sequences
	$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$
	$\frac{1}{3}$	$\mathrm{\infty }$	0.0	$\frac{2}{\pi }$
	0.33	24.00496	0.038378833	0.52024552
	0.316943	10.744571	0.068652714	0.382383875
	0.31	9.014959766	0.0755022550	0.3372170064
	0.3090	8.8301772	0.076265588	0.331475715
	$\frac{1}{4}$	4.9379256	0.084824137	0.139370157
	†Additional model parameters can be found here.

In terms of mass $(m)$, length $(\ell )$, and time $(t)$, the units of various physical constants and variables are: Mass-density $\sim$ $m{\ell }^{-3}$ Pressure (energy-density) $\sim$ $m{\ell }^{-1}{t}^{-2}$ Newtonian gravitational constant, $G$ $\sim$ $m^{-1}{\ell }^3{t}^{-2}$ The core's polytropic constant, $K_c$ $\sim$ $[m^{-1}{\ell }^{13}{t}^{-10}{]}^{1/5}$ The envelope's polytropic constant, $K_e$ $\sim$ $m^{-1}{\ell }^5{t}^{-2}$ As a result, for example (see details below), if we hold the central-density $({\rho }_0)$ — as well as $G$ and $K_c$ — constant along an equilibrium sequence, mass will scale as … Mass $\sim$ $\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]$   $\sim$ $\left\{\right[m^{-1}{\ell }^{13}{t}^{-10}{]}^{1/5}{\}}^{3/2}\left[m^{-1}{\ell }^3{t}^{-2}{]}^{-3/2}\right[m{\ell }^{-3}{]}^{-1/5}$   $\sim$ $\left[m^{-3/10+3/2-1/5}\right]\left[{\ell }^{39/10-9/2+3/5}\right]\left[t^{-3+3}\right]$   $\sim$ $\left[m^{+1}\right]\left[{\ell }^0\right]\left[t^0\right].$ If instead (see details below) we hold $K_e$ — as well as $G$ and $K_c$ — constant along an equilibrium sequence, mass will scale as … Mass $\sim$ $[K_c^5{K}_e{G}^{-6}{]}^{1/4}$   $\sim$ $\left\{\right[m^{-1}{\ell }^{13}{t}^{-10}\left]\right[m^{-1}{\ell }^5{t}^{-2}\left]\right[m^{-1}{\ell }^3{t}^{-2}{]}^{-6}{\}}^{1/4}$   $\sim$ $\left\{\right[m^{-1-1+6}\left]\right[{\ell }^{13+5-18}\left]\right[t^{-10-2+12}]{\}}^{1/4}$   $\sim$ $\left[m^{+1}\right]\left[{\ell }^0\right]\left[t^0\right].$

Original Model Construction

Fixed Central Density

From Examples, we find,

$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]$	$=$	$\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right](\frac{6}{\pi }{)}^{1/2}({\xi }_i{\theta }_i{)}^3;$
$M_{\mathrm{t}\mathrm{o}\mathrm{t}}=M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]$	$=$	$\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i};$
$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right]$	$=$	$\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right](\frac{3}{2\pi }{)}^{1/2}{\xi }_i;$
$R=r_s^{*}\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right]$	$=$	$\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right](\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\eta }_s}{\sqrt{2\pi }{\theta }_i^2};$

where, rewriting the relevant expressions in terms of the parameters,

${\ell }_i\equiv \frac{{\xi }_i}{\sqrt{3}};$       and       $m_3\equiv 3\left(\frac{{\mu }_e}{{\mu }_c}\right),$

we find,

${\theta }_i$	$=$	$(1+{\ell }_i^2{)}^{-1/2}$
${\eta }_i$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)\sqrt{3}{\theta }_i^2{\xi }_i=m_3\left(\frac{{\ell }_i}{1+{\ell }_i^2}\right)$
${\mathrm{\Lambda }}_i$	$=$	$\frac{1}{{\eta }_i}-\frac{{\xi }_i}{\sqrt{3}}=\left[\frac{1+{\ell }_i^2}{m_3{\ell }_i}\right]-{\ell }_i=\frac{1}{m_3{\ell }_i}[1+(1-m_3){\ell }_i^2]$
$A^2$	$=$	${\eta }_i^2(1+{\mathrm{\Lambda }}_i^2)$
	$=$	$m_3^2(\frac{{\ell }_i}{1+{\ell }_i^2}{)}^2\cdot \frac{(1+{\ell }_i^2)}{m_3^2{\ell }_i^2}[1+(1-m_3{)}^2{\ell }_i^2]$
	$=$	$\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]$
$\Rightarrow (1+{\mathrm{\Lambda }}_i^2)$	$=$	${\eta }_i^{-2}\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]=\frac{1}{m_3^2}\left[\frac{(1+{\ell }_i^2{)}^2}{{\ell }_i^2}\right]\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]=\frac{(1+{\ell }_i^2)}{m_3^2{\ell }_i^2}[1+(1-m_3{)}^2{\ell }_i^2]$
${\eta }_s$	$=$	$\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)$

Fixed Interface Pressure

Equilibrium Sequence Expressions

From the relevant interface conditions, we find,

$\left(\frac{K_e}{K_c}\right)$

$=$

${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}.$

Inverting this last expression gives,

${\rho }_0^{4/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}\right(\frac{K_e}{K_c}{)}^{-1}$
$\Rightarrow {\rho }_0^{1/5}$	$=$	$\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1/2}{\theta }_i^{-1}\left(\frac{K_e}{K_c}{)}^{-1/4}\right].$

Hence, keeping $K_c$ and $K_e$ constant, we have,

$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\left[K_c^{3/2}{G}^{-3/2}\right(\frac{K_e}{K_c}{)}^{1/4}\left]\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\right](\frac{6}{\pi }{)}^{1/2}({\xi }_i{\theta }_i{)}^3$
	$=$	$\left[K_c^5{K}_e{G}^{-6}{]}^{1/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}(\frac{6}{\pi }{)}^{1/2}{\xi }_i^3{\theta }_i^4;$
$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$\left[K_c^{3/2}{G}^{-3/2}\right(\frac{K_e}{K_c}{)}^{1/4}\left]\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}$
	$=$	$\left[K_c^5{K}_e{G}^{-6}{]}^{1/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}(\frac{2}{\pi }{)}^{1/2}A{\eta }_s;$
$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\left[K_e{G}^{-1}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}\left)\right(\frac{3}{2\pi }{)}^{1/2}{\xi }_i{\theta }_i^2;$
$R$	$=$	$[K_e{G}^{-1}{]}^{1/2}\frac{{\eta }_s}{\sqrt{2\pi }};$
${\rho }_0$	$=$	$\left[\frac{K_e}{K_c}{]}^{-5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5};$
$P_i$	$=$	$\left[K_c\right]\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\theta }_i^{-6}\left(\frac{K_e}{K_c}{)}^{-3/2}\right]{\theta }_i^6$
	$=$	$\left[K_c^5{K}_e^{-3}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}.$

This last expression shows that if $K_c$ and $K_e$ are both held fixed, then the interface pressure, $P_i$, will be constant along the sequence of equilibrium models.

Note also:

$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	$\left\{\right[K_c^5{K}_e{G}^{-6}{]}^{1/4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}\right(\frac{6}{\pi }{)}^{1/2}{\xi }_i^3{\theta }_i^4\left\}\right\{\left[K_c^5{K}_e{G}^{-6}{]}^{1/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}(\frac{2}{\pi }{)}^{1/2}A{\eta }_s{\}}^{-1}$
	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^2\frac{\sqrt{3}{\xi }_i^3{\theta }_i^4}{A{\eta }_s};$
$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$=$	$\left\{\right[K_e{G}^{-1}{]}^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right)\left(\frac{3}{2\pi }{)}^{1/2}{\xi }_i{\theta }_i^2\right\}\left\{\right[K_e{G}^{-1}{]}^{1/2}\frac{{\eta }_s}{\sqrt{2\pi }}{\}}^{-1}$
	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)\frac{\sqrt{3}{\xi }_i{\theta }_i^2}{{\eta }_s}.$

Fixed Interface Pressure Sequence Plots

A plot of $M_{\mathrm{t}\mathrm{o}\mathrm{t}}[K_c^5{K}_e{G}^{-6}{]}^{-1/4}$ versus $R[K_e{G}^{-1}{]}^{-1/2}$ at fixed interface pressure will be generated via the relations,

Ordinate: $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$		Abscissa: $R$
$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}A{\eta }_s$	vs	$\frac{{\eta }_s}{\sqrt{2\pi }}.$

Alternatively, a plot of $M_{\mathrm{t}\mathrm{o}\mathrm{t}}[K_c^5{K}_e{G}^{-6}{]}^{-1/4}$ versus ${\log }_{10}({\rho }_0)[\frac{K_e}{K_c}{]}^{5/4}$ at fixed interface pressure will be generated via the relations,

Ordinate: $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$		Abscissa: ${\log }_{10}({\rho }_0)$
$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}A{\eta }_s$	vs	${\log }_{10}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}]$

The expression for $d{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}/d{\ell }_i$ is … $[K_c^5{K}_e{G}^{-6}{]}^{-1/4}\frac{d{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{d{\ell }_i}$ $=$ $\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}\frac{d}{d{\ell }_i}\left[A{\eta }_s\right]$   $=$ $\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}\frac{d}{d{\ell }_i}\left[A{\eta }_s\right]$   $=$ $\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}\frac{d}{d{\ell }_i}\left\{\right[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\}$ $\Rightarrow \left(\frac{{\mu }_e}{{\mu }_c}{)}^{3/2}\right(\frac{2}{\pi }{)}^{-1/2}[K_c^5{K}_e{G}^{-6}{]}^{-1/4}\frac{d{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{d{\ell }_i}$ $=$ $\frac{1}{2}[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{-1/2}\frac{d}{d{\ell }_i}\right\{\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]\}$     $+\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\right\{\frac{d{\eta }_i}{d{\ell }_i}\}+[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\left\{\frac{d}{d{\ell }_i}\right[{\tan }^{-1}({\mathrm{\Lambda }}_i)\left]\right\}$   $=$ $\frac{1}{2}[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{-1/2}\frac{d}{d{\ell }_i}\right\{\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]\}$     $+\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\right\{\frac{d}{d{\ell }_i}\left[m_3\right(\frac{{\ell }_i}{1+{\ell }_i^2}\left)\right]\}+[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\left[\frac{1}{1+{\mathrm{\Lambda }}_i^2}\right]\cdot \frac{d{\mathrm{\Lambda }}_i}{d{\ell }_i}$ The extremum in $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ occurs when the LHS of this expression is zero, that is, when … ${\eta }_i$ $=$ $m_3{\ell }_i(1+{\ell }_i^2{)}^{-1}$ ${\mathrm{\Lambda }}_i$ $=$ $\frac{[1+(1-m_3){\ell }_i^2]}{m_3{\ell }_i}$ $(1+{\mathrm{\Lambda }}_i^2{)}^{-1}$ $=$ $\frac{m_3^2{\ell }_i^2}{(1+{\ell }_i^2)}[1+(1-m_3{)}^2{\ell }_i^2{]}^{-1}$ $2\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\right\{\frac{d}{d{\ell }_i}\left[m_3\right(\frac{{\ell }_i}{1+{\ell }_i^2}\left)\right]\}+2[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\left[\frac{1}{1+{\mathrm{\Lambda }}_i^2}\right]\cdot \frac{d{\mathrm{\Lambda }}_i}{d{\ell }_i}$ $=$ $-[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{-1/2}\frac{d}{d{\ell }_i}\right\{\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]\}$ $\Rightarrow 2m_3\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\frac{d}{d{\ell }_i}\right\{{\ell }_i(1+{\ell }_i^2{)}^{-1}\}+\frac{2}{m_3}\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{1/2}\right[\frac{1}{1+{\mathrm{\Lambda }}_i^2}]\cdot \frac{d}{d{\ell }_i}\{{\ell }_i^{-1}[1+(1-m_3){\ell }_i^2]\}$ $=$ $-[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}{]}^{-1/2}\frac{d}{d{\ell }_i}\right\{(1+{\ell }_i^2{)}^{-1}[1+(1-m_3{)}^2{\ell }_i^2]\}$ $\Rightarrow 2m_3\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]\{(1+{\ell }_i^2{)}^{-1}-2{\ell }_i^2(1+{\ell }_i^2{)}^{-2}\}+\frac{2}{m_3}\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]\left[\frac{1}{1+{\mathrm{\Lambda }}_i^2}\right]\{-{\ell }_i^{-2}[1+(1-m_3){\ell }_i^2]+[2(1-m_3)\left]\right\}$ $=$ $-[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\{-2{\ell }_i(1+{\ell }_i^2{)}^{-2}[1+(1-m_3{)}^2{\ell }_i^2]+(1+{\ell }_i^2{)}^{-1}[2{\ell }_i(1-m_3{)}^2\left]\right\}$ $\Rightarrow 2m_3\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{(1+{\ell }_i^2{)}^3}\right]\{(1+{\ell }_i^2)-2{\ell }_i^2\}+\frac{2}{m_3}\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{1+{\ell }_i^2}\right]\frac{m_3^2{\ell }_i^2}{(1+{\ell }_i^2)}[1+(1-m_3{)}^2{\ell }_i^2{]}^{-1}\{-{\ell }_i^{-2}[1+(1-m_3){\ell }_i^2]+[2(1-m_3)\left]\right\}$ $=$ $-(1+{\ell }_i^2{)}^{-2}[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)\left]\right\{-2{\ell }_i[1+(1-m_3{)}^2{\ell }_i^2]+(1+{\ell }_i^2)[2{\ell }_i(1-m_3{)}^2]\}$ $\Rightarrow 2m_3\left[\frac{1+(1-m_3{)}^2{\ell }_i^2}{(1+{\ell }_i^2)}\right]\{(1+{\ell }_i^2)-2{\ell }_i^2\}+2m_3\{-[1+(1-m_3){\ell }_i^2]+[2(1-m_3){\ell }_i^2\left]\right\}$ $=$ $-[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\{-2{\ell }_i[1+(1-m_3{)}^2{\ell }_i^2]+(1+{\ell }_i^2)[2{\ell }_i(1-m_3{)}^2\left]\right\}$ $\Rightarrow 2m_3[1+(1-m_3{)}^2{\ell }_i^2\left]\right[\frac{(1-{\ell }_i^2)}{(1+{\ell }_i^2)}]+2m_3[-1+(1-m_3){\ell }_i^2]$ $=$ $-[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\left\{\right[-2{\ell }_i-2(1-m_3{)}^2{\ell }_i^3]+[2{\ell }_i(1-m_3{)}^2]+[2(1-m_3{)}^2{\ell }_i^3]\}$ $\Rightarrow \frac{2m_3}{(1+{\ell }_i^2)}\left\{\right[1+(1-m_3{)}^2{\ell }_i^2][(1-{\ell }_i^2)]+(1+{\ell }_i^2)[-1+(1-m_3){\ell }_i^2]\}$ $=$ $[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]\{1-(1-m_3{)}^2\}2{\ell }_i$ $\Rightarrow \frac{2m_3}{(1+{\ell }_i^2)}\left\{\right[1+(1-m_3{)}^2{\ell }_i^2]-{\ell }_i^2[1+(1-m_3{)}^2{\ell }_i^2]+[-1+(1-m_3){\ell }_i^2]+{\ell }_i^2[-1+(1-m_3){\ell }_i^2\left]\right\}$ $=$ $[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)](2-m_3)2m_3{\ell }_i$ $\Rightarrow (1+{\ell }_i^2{)}^{-1}\{1+(1-m_3{)}^2{\ell }_i^2-{\ell }_i^2-(1-m_3{)}^2{\ell }_i^4-1+(1-m_3){\ell }_i^2-{\ell }_i^2+(1-m_3){\ell }_i^4\}$ $=$ $[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)](2-m_3){\ell }_i$ $\Rightarrow \frac{1}{(1+{\ell }_i^2)(2-m_3){\ell }_i}\left\{\right[m_3-3]m_3{\ell }_i^2+[1-m_3\left]m_3{\ell }_i^4\right\}$ $=$ $[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]$ $\Rightarrow \frac{m_3{\ell }_i}{(1+{\ell }_i^2)(2-m_3)}[(m_3-3)+(1-m_3){\ell }_i^2]$ $=$ $[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)]$ For ${\mu }_e/{\mu }_c=1.00$ the solution to this expression is ${\xi }_i=1.668462981$.

---

file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = SequenceMuRatio100 Example data values drawn from worksheet "SequenceMuRatio100" … $\mathrm{\Delta }\xi =(9.01499598-0.05)/99=0.0905551$ $n_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}$ ${\xi }_i=0.05+(n_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}-1)\cdot \mathrm{\Delta }\xi$ ${\theta }_i$ $A$ ${\eta }_s$ $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ $\log 10{\rho }_0$ $R$ 1 0.05 0.9995836 1.00124818 3.141592582 2.510 0.0009044 1.253 2 0.140555 0.9967235 1.0097655 3.141580334 2.531 0.0071264 1.253 18 1.5894375 0.7367887 1.539943947 2.821678456 3.467 0.663285301 1.126 19 1.6799927 0.7178117 1.566601145 2.775921455 3.470 0.719947375 1.107 20 1.7705478 0.6992927 1.591530391 2.728957898 3.465 0.7767049 1.089 100 9.0149598 0.1886798 1.973119305 0.841461698 1.325 3.62137 0.336

Equilibrium Sequences of $(n_c,n_e)=(5,1)$ BiPolytropes Having ${\mu }_e/{\mu }_c=1.0$ (viewed from several different astrophysical perspectives)
(Interface Pressure${)}^{1/3}$ vs. Radius (Fixed Total Mass)	Mass vs. Radius (Fixed Interface Pressure)	Mass vs. Central Density (Fixed Interface Pressure)
Pressure vs Volume
$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-4}\right(\frac{2}{\pi })A^2{\eta }_s^2$ vs. $\left(\frac{{\mu }_e}{{\mu }_c}{)}^3\right(\frac{\pi }{2^3}{)}^{1/2}\frac{1}{A^2{\eta }_s}$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}A{\eta }_s$ vs. $\frac{{\eta }_s}{\sqrt{2\pi }}$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}A{\eta }_s$ vs. ${\log }_{\mathrm{1}\mathrm{0}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}]$
NOTE:   In all three diagrams, the dashed vertical line identifies the value of the abscissa when it is evaluated for the interface location, ${\xi }_i=1.668462981$. In each case, this vertical line intersects a key turning point along the model sequence.

file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = MuRatio100Fund Data values drawn from worksheet "MuRatio100Fund" … Properties of the Marginally Unstable Model ${\xi }_i$ ${\theta }_i$ $A$ ${\eta }_s$ $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ $\log 10{\rho }_0$ $R$ 1.6639103 0.7211498 1.561995126 2.784147185 3.4698598 0.709872477 1.1107140

Temporary Excel Interpolations

HERE

Properties of Turning-Points Along Sequences Having Various  ${\mu }_e/{\mu }_c$
$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	$P_i$	$R$	$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	${\log }_{10}({\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}})$
		(Fixed $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$)		(Fixed $P_i$)
1.000	1.6684629814	12.03999149	0.092175036	3.46986909	0.712724159
0.9	1.4459132276	13.67957562	0.091291571	3.50879154	0.688526899
0.8	1.0482530437	17.09391244	0.086279818	3.69798999	0.58112284
0.75	0.7170001608	20.48027265	0.079651055	3.91920968	0.484075667
0.74	0.6365283705	21.40307774	0.0777495	3.97973335	0.464464039

Fixed Total Mass

Equilibrium Sequence Expressions

Again, drawing from previous Examples in which ${\rho }_0$ — as well as $K_c$ and $G$ — is held fixed, equilibrium models obey the relations,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]=\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i};$
$R$	$=$	$R^{*}\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right]=\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right](\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\eta }_s}{\sqrt{2\pi }{\theta }_i^2};$
$P_i$	$=$	$P_i^{*}\left[K_c{\rho }_0^{6/5}\right]=\left[K_c{\rho }_0^{6/5}\right]{\theta }_i^6.$

Let's invert the first expression in order to construct equilibrium sequences in which the total mass — rather than ${\rho }_0$ — is held fixed. We find that,

${\rho }_0^{1/5}$	$=$	$\left[K_c^{3/2}{G}^{-3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-1}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}$
$\Rightarrow R$	$=$	$\left[K_c^{1/2}{G}^{-1/2}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\eta }_s}{\sqrt{2\pi }{\theta }_i^2}\right\{\left[K_c^{3/2}{G}^{-3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-1}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}{\}}^{-2}$
	$=$	$\left[K_c^{1/2}{G}^{-1/2}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\eta }_s}{\sqrt{2\pi }{\theta }_i^2}\right[K_c^{3/2}{G}^{-3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-1}{]}^{-2}\left(\frac{{\mu }_e}{{\mu }_c}{)}^4\right(\frac{\pi }{2})\frac{{\theta }_i^2}{A^2{\eta }_s^2}$
	$=$	$\left[K_c^{-5/2}{G}^{5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^3\right(\frac{\pi }{2^3}{)}^{1/2}\frac{1}{A^2{\eta }_s}.$

And,

$P_i$	$=$	$K_c{\theta }_i^6\left\{\right[K_c^{3/2}{G}^{-3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-1}\left]\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}{\}}^6$
	$=$	$K_c\left\{\right[K_c^{3/2}{G}^{-3/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-1}{]}^6\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}\right(\frac{2}{\pi }{)}^3{A}^6{\eta }_s^6\}$
	$=$	$\left[K_c^{10}{G}^{-9}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-6}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}\right(\frac{2}{\pi }{)}^3{A}^6{\eta }_s^6.$

Note as well that,

$P_i\left(\frac{4\pi }{3}{R}^3\right)$	$=$	$\frac{4\pi }{3}\left[K_c^{10}{G}^{-9}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-6}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}\right(\frac{2}{\pi }{)}^3{A}^6{\eta }_s^6\left\{\right[K_c^{-5/2}{G}^{5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2\left]\right(\frac{{\mu }_e}{{\mu }_c}{)}^3(\frac{\pi }{2^3}{)}^{1/2}\frac{1}{A^2{\eta }_s}{\}}^3$
	$=$	$\frac{4\pi }{3}\left(\frac{2}{\pi }{)}^3\right(\frac{\pi }{2^3}{)}^{3/2}\left[K_c^{10}{G}^{-9}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-6}\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}{\eta }_s^3\right\{\left[K_c^{-15/2}{G}^{15/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^6\right]\left(\frac{{\mu }_e}{{\mu }_c}{)}^9\right\}$
	$=$	$\frac{4\pi }{3}\left(\frac{2}{\pi }{)}^3\right(\frac{\pi }{2^3}{)}^{3/2}\left[K_c^{5/2}{G}^{-3/2}\right](\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\eta }_s^3$

Sequence Plots

A plot of $P_i\left[K_c^{-10}{G}^9{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^6\right]$ versus $R^3[K_c^{5/2}{G}^{-5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-2}{]}^3$ at fixed total mass will be generated via the relations,

Ordinate		Abscissa
$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}\right(\frac{2}{\pi }{)}^3{A}^6{\eta }_s^6$	vs	$\left\{\right(\frac{{\mu }_e}{{\mu }_c}{)}^3(\frac{\pi }{2^3}{)}^{1/2}\frac{1}{A^2{\eta }_s}{\}}^3$

Pressure vs Volume

nu vs q

Following the Lead of Yabushita75

Here in the context of $(n_c,n_e)=(5,1)$ bipolytropes, we want to construct an interface-pressure versus volume plot; and mass-versus-central density plots like the ones displayed for truncated isothermal spheres in Figure 1 of an accompanying discussion, and as displayed for a $(n_c,n_e)=(\mathrm{\infty },3/2)$ bipolytrope in Figure 1 (p. 445) of 📚 S. Yabushita (1975, MNRAS, Vol. 172, pp. 441 - 453).

In our accompanying chapter that presents example models of $(n_c,n_e)=(5,1)$ bipolytropes, we have adopted the following normalizations:

${\rho }^{*}$	$\equiv$	$\frac{\rho }{{\rho }_0}$	;	$r^{*}$	$\equiv$	$\frac{r}{[K_c^{1/2}/(G^{1/2}{\rho }_0^{2/5})]}$
$P^{*}$	$\equiv$	$\frac{P}{K_c{\rho }_0^{6/5}}$	;	$M_r^{*}$	$\equiv$	$\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_0^{1/5})]}$

Also, from the relevant interface conditions, we find,

$\left(\frac{K_e}{K_c}\right)$

$=$

${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}.$

Inverting this last expression gives,

${\rho }_0^{4/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}\right(\frac{K_e}{K_c}{)}^{-1}$
$\Rightarrow {\rho }_0^{1/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1/2}{\theta }_i^{-1}\right(\frac{K_e}{K_c}{)}^{-1/4}.$

Hence, we can rewrite the "normalized" expressions as follows:

$r$	$=$	$r^{*}\left[K_c^{1/2}{G}^{-1/2}{\rho }_0^{-2/5}\right]$
	$=$	$r^{*}\left\{K_c^{1/2}{G}^{-1/2}\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1/2}{\theta }_i^{-1}\right(\frac{K_e}{K_c}{)}^{-1/4}{]}^{-2}\}$
	$=$	$r^{*}\left\{K_c^{1/2}{G}^{-1/2}\right[\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2\left(\frac{K_e}{K_c}{)}^{1/2}\right]\}$
	$=$	$r^{*}\left[K_e^{1/2}{G}^{-1/2}\right]\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2.$

Fixed Interface Pressure

Start with the model relation,

$P_i$	$=$	$\left[K_c{\rho }_0^{6/5}\right]P_i^{*}$
	$=$	$\left[K_c{\rho }_0^{6/5}\right](1+\frac{1}{3}{\xi }_i^2{)}^{-3}$

Now, given that,

${\rho }_0^{1/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1/2}{\theta }_i^{-1}\right(\frac{K_e}{K_c}{)}^{-1/4}.$
$\Rightarrow {\rho }_0^{6/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\theta }_i^{-6}\right(\frac{K_e}{K_c}{)}^{-3/2}.$

Fixed Total Mass

Also, from the relevant interface conditions, we find,

$\left(\frac{K_e}{K_c}\right)$

$=$

${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}.$

Inverting this last expression gives,

${\rho }_0^{4/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}\right(\frac{K_e}{K_c}{)}^{-1}$
$\Rightarrow {\rho }_0^{1/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1/2}{\theta }_i^{-1}\right(\frac{K_e}{K_c}{)}^{-1/4}.$

Hence, for a given specification of the interface location, ${\xi }_i$ — test values shown (in parentheses) assuming ${\mu }_e/{\mu }_c=1.0$ and ${\xi }_i=0.5$ — the desired expression for the central density is,

${\rho }_0$

$=$

$\left[K_e^{-5}{K}_c^5{]}^{1/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5};$

and, drawing the expression for the normalized total mass from our accompanying table of parameter values, namely,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}$

$=$

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

we find,

$M_r$	$=$	$M_r^{*}\left[K_c^{3/2}{G}^{-3/2}{\rho }_0^{-1/5}\right]$
	$=$	$M_r^{*}\left[K_c^{3/2}{G}^{-3/2}\right]\left\{\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1/2}{\theta }_i^{-1}(\frac{K_e}{K_c}{)}^{-1/4}{\}}^{-1}$
	$=$	$M_r^{*}\left[K_c^{3/2}{G}^{-3/2}\right(\frac{K_e}{K_c}{)}^{1/4}\left]\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i$
$\Rightarrow M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$\left[K_c^{3/2}{G}^{-3/2}\right(\frac{K_e}{K_c}{)}^{1/4}\left]\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}$
	$=$	$\left[K_e{K}_c^{-5}{G}^{-6}{]}^{1/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}(\frac{2}{\pi }{)}^{1/2}A{\eta }_s,$

where — again, from our accompanying table of parameter values —

${\theta }_i$	$=$	$(1+\frac{1}{3}{\xi }_i^2{)}^{-1/2};$		(0.96077)
${\eta }_i$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)\sqrt{3}{\theta }_i^2{\xi }_i;$		(0.79941)
${\mathrm{\Lambda }}_i$	$=$	$\frac{1}{{\eta }_i}-\frac{{\xi }_i}{\sqrt{3}};$		(0.96225)
$A$	$=$	${\eta }_i(1+{\mathrm{\Lambda }}_i^2{)}^{1/2};$		(1.10940)
${\eta }_s$	$=$	${\eta }_i+\frac{\pi }{2}+{\tan }^{-1}({\mathrm{\Lambda }}_i);$		(3.13637)
$\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{[K_e{K}_c^{-5}{G}^{-6}{]}^{1/4}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}A{\eta }_s;$		(2.77623)
$\frac{{\rho }_0}{[K_e^{-5}{K}_c^5{]}^{1/4}}$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}.$		(1.22153)

Building on Earlier Eigenfunction Details

In the heading of Figure 6 from our accompanying presentation of the properties of marginally unstable oscillation modes in $(n_c,n_e)=(5,1)$ bipolytropes, we point to the (Excel spreadsheet) "Data File" that contains most of the relevant model details. See specifically,

file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble Figure 6: Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation in Marginally Unstable Models having Various ${\mu }_e/{\mu }_c$

Relevant Instabilities

Abstract

The analysis presented by 📚 Eggleton, Faulkner, & Cannon (1998) is essentially an analysis of the $q-v$ diagram. We can determine analytically at what value of ${\xi }_i$ the core-to-total mass ratio reaches a maximum (${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$) for various values of ${\mu }_e/{\mu }_c\le 1/3$. For example, $({\mu }_e/{\mu }_c,{\xi }_i,{\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}})=(\frac{1}{4},4.9379256,0.139270157)$. Our LAWE analysis shows that none of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis does identify a marginally unstable equilibrium configuration along every sequence; even sequences with $\frac{1}{3}\le {\mu }_e/{\mu }_c\le 1$. For example, $({\mu }_e/{\mu }_c,{\xi }_i,\nu )=(1,1.6686460157,0.497747626)$.

Truncated n = 5 Polytrope

In Figure 3 of an accompanying discussion, we show where various turning points lie along the equilibrium sequence of truncated $n=5$ polytropes.

Figure 3:   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres (viewed from several different astrophysical perspectives)
●	${\xi }_e$	†External Pressure vs. Volume (Fixed Mass)	Mass vs. Radius (Fixed External Pressure)	‡Mass vs. Central Density (Fixed External Pressure)	Mass vs. Central Density (Fixed Radius)
●	√3	(a) Pressure-Truncated Isothermal Equilibrium Sequence	(b) Pressure-Truncated Isothermal Equilibrium Sequence	(c) Pressure-Truncated Isothermal Equilibrium Sequence	(d) Pressure-Truncated Isothermal Equilibrium Sequence
●	3
●	√15
●	9.01
		$\left(\frac{2\cdot 3}{\pi }{)}^3\right[{\xi }^{18}(1+\frac{{\xi }^2}{3}{)}^{-12}{]}_{\tilde{\xi }}$ vs. $\left(\frac{\pi }{2\cdot 3}{)}^{5/2}\right[{\xi }^{-15}(1+\frac{{\xi }^2}{3}{)}^9{]}_{\tilde{\xi }}$	$\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-2}{]}_{\tilde{\xi }}$ vs. $\left(\frac{3}{2\pi }{)}^{1/2}\right[\xi (1+\frac{{\xi }^2}{3}{)}^{-1}{]}_{\tilde{\xi }}$	$\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{{\xi }^2}{3}{)}^{-2}{]}_{\tilde{\xi }}$ vs. $\left[\right(1+\frac{{\xi }^2}{3}{)}^{5/2}{]}_{\tilde{\xi }}$	$\left[\frac{2^3\cdot 3}{\pi }{]}^{1/4}\right[{\xi }^{5/2}(1+\frac{{\xi }^2}{3}{)}^{-3/2}{]}_{\tilde{\xi }}$ vs. $[\frac{3}{2\pi }{]}^{5/4}{\tilde{\xi }}^{5/2}$

• KEY RESULT:
  • The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at $\tilde{\xi }=3$.
  • It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.
  • For all $3<n<\mathrm{\infty }$, the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.

  Bipolytropes with (n_c, n_e) = (5, 1)

  q - ν Sequence Plots

  In Figure 1 of an accompanying discussion, we show — via a plot in the $(q,\nu )$ diagram — how the $(n_c,n_e)=(5,1)$ bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, $\frac{1}{4}\le ({\mu }_e/{\mu }_c)\le 1$.

  
  

  Figure 1: Analytically determined plot of fractional core mass ($\nu$) versus fractional core radius ($q$) for $(n_c,n_e)=(5,1)$ bipolytrope model sequences having six different values of ${\mu }_e/{\mu }_c$: 1 (blue diamonds), ½ (red squares), 0.345 (dark purple crosses), ⅓ (pink triangles), 0.309 (light green dashes), and ¼ (purple asterisks). Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, ${\xi }_i$, has one of three values: 0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.

  According to our accompanying discussion, in terms of the parameters,

  ${\ell }_i\equiv \frac{{\xi }_i}{\sqrt{3}};$       and       $m_3\equiv 3\left(\frac{{\mu }_e}{{\mu }_c}\right),$

  the parameter, $\nu$, varies with $\xi$ as,

  $\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}=(m_3^2{\ell }_i^3)(1+{\ell }_i^2{)}^{-1/2}[1+(1-m_3{)}^2{\ell }_i^2{]}^{-1/2}[m_3{\ell }_i+(1+{\ell }_i^2)(\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i){]}^{-1}.$

  KEY RESULT:  Over the range, $\frac{1}{4}\le ({\mu }_e/{\mu }_c)\le \frac{1}{3}$, there is a value of $\nu$ above which no equilibrium configurations exist. We have determined the location of this "turning point" by setting, $d\nu /d\xi =0$; our derived result is,

  $\underset{\mathrm{L}\mathrm{H}\mathrm{S}}{\underbrace{(\frac{\pi }{2}+{\tan }^{-1}{\mathrm{\Lambda }}_i)(1+{\ell }_i^2)[3+(1-m_3{)}^2(2-{\ell }_i^2){\ell }_i^2]}}$

  $=$

  $\underset{\mathrm{R}\mathrm{H}\mathrm{S}}{\underbrace{m_3{\ell }_i[(1-m_3){\ell }_i^4-(m_3^2-m_3+2){\ell }_i^2-3]}}.$

  Maximum Fractional Core Mass, $\nu =M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ (solid green circular markers) for Equilibrium Sequences having Various Values of ${\mu }_e/{\mu }_c$
  $\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	${\theta }_i$	${\eta }_i$	${\mathrm{\Lambda }}_i$	$A$	${\eta }_s$	LHS	RHS	$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$
  $\frac{1}{3}$	$\mathrm{\infty }$	---	---	---	---	---	---	---	0.0	$\frac{2}{\pi }$
  0.33	24.00496	0.0719668	0.0710624	0.2128753	0.0726547	1.8516032	-223.8157	-223.8159	0.038378833	0.52024552
  0.316943	10.744571	0.1591479	0.1493938	0.4903393	0.1663869	2.1760793	-31.55254	-31.55254	0.068652714	0.382383875
  0.3090	8.8301772	0.1924833	0.1750954	0.6130669	0.2053811	2.2958639	-18.47809	-18.47808	0.076265588	0.331475715
  $\frac{1}{4}$	4.9379256	0.3309933	0.2342522	1.4179907	0.4064595	2.761622	-2.601255	-2.601257	0.084824137	0.139370157
  Recall that, ${\ell }_i\equiv \frac{{\xi }_i}{\sqrt{3}};$       and       $m_3\equiv 3\left(\frac{{\mu }_e}{{\mu }_c}\right).$

  The EFC98 Sequence Plot

  📚 Eggleton, Faulkner, & Cannon (1998) also analytically determined the structure of models along various $(n_c,n_e)=(5,1)$ sequences; their Figure 1 displays the behavior of $\nu$ vs. ${\log }_{10}({\rho }_c/{\rho }_i)$ for a range of $\alpha \equiv ({\mu }_e/{\mu }_c{)}^{-1}$. Note that,

  $\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^2\sqrt{3}\right(\frac{{\xi }_i^3{\theta }_i^4}{A{\eta }_s});$
  ${\log }_{10}({\rho }_c/{\rho }_i)$	$=$	${\log }_{10}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}(1+\frac{{\xi }^2}{3}{)}^{5/2}].$

  $\frac{{\mu }_e}{{\mu }_c}={\alpha }^{-1}$	${\xi }_i$	$\nu$	${\log }_{10}\left(\frac{{\rho }_c}{{\rho }_i}\right)$
  $\frac{1}{4}$	$4.9379256$	$0.139370157$	$3.002964$

  KEY RESULT (to be done):  From our original derivation, we have generated a plot intended to replicate Figure 1 from EFC98; then we have marked on each sequence the location of the mass-extremum (i.e., when $d\nu /d\xi =0$) as determined by our above analytically derived result.

  Yabushita75 Guidance

  Alternatively, as derived above, setting $d{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}/d{\ell }_i=0$ leads to the expression,

  $\frac{m_3{\ell }_i}{(1+{\ell }_i^2)(2-m_3)}[(m_3-3)+(1-m_3){\ell }_i^2]$

  $=$

  $[\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)].$

  For ${\mu }_e/{\mu }_c=1.00$ the solution to this expression is ${\xi }_i=1.668462981$. For other parameter choices, see here, for example.

  $\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$
  	LAWE Sol'n	Max. Mtot
  1	1.6686460157	1.668462981
  $\frac{1}{2}$	2.2792811317	n/a

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