SSC/Stability/Yabushita75

Stability of a BiPolytrope with an Isothermal Core

This analysis pulls largely from 📚 S. Yabushita (1975, MNRAS, Vol. 172, pp. 441 - 453); the focus is on bipolytropes having $(n_c,n_e)=(\mathrm{\infty },\frac{3}{2})$. In an accompanying discussion, we summarize the steps that Yabushita took — from 1968 and 1974, to 1975 — that led up to his discovery of an analytic description of the isothermal displacement function.

Equilibrium Structure

We will follow the accompanying formal recipe for building a bipolytropic model, using the step-by-step construction of Milne's (1930) configurations as a guide.

Step 1

The 📚 Yabushita (1975) bipolytrope has an isothermal core $(n_c=\mathrm{\infty })$ and an $n_e=\frac{3}{2}$ polytropic envelope.

Steps 2 & 3

Throughout the core, the properties of this bipolytrope can be expressed in terms of the Lane-Emden function, $\psi (\chi )$, which derives from a solution of the 2^nd-order ODE,

Isothermal Lane-Emden Equation

$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d\psi }{d\xi }\right)=e^{-\psi }$

subject to the boundary conditions,

$\psi =1$       and       $\frac{d\psi }{d\xi }=0$       at       $\xi =0$.

As has been demonstrated in our accompanying mathematics appendix, a series expansion about the center of the isothermal configuration gives,

$\psi (\xi )$

$=$

$\frac{{\xi }^2}{6}-\frac{{\xi }^4}{120}+\frac{{\xi }^6}{1890}-\frac{61{\xi }^8}{1,632,960}+\cdots .$

The solution, $\psi (\xi )$, extends to infinity, so the interface between the core and the envelope can be positioned anywhere within the range, $0<{\xi }_i<\mathrm{\infty }$.

Step4: Throughout the core

Specify: $c_s^2$ and ${\rho }_0\Rightarrow$
$r$	$=$	$[\frac{c_s^2}{4\pi G{\rho }_0}{]}^{1/2}\xi$	(2.2)
$\rho$	$=$	${\rho }_0{e}^{-\psi }$	(2.2)
$P$	$=$	$c_s^2{\rho }_0{e}^{-\psi }$	(2.1)
$M_r$	$=$	$\left[\frac{c_s^6}{4\pi G^3{\rho }_0}{]}^{1/2}\right({\xi }^2\frac{d\psi }{d\xi })$	(2.3)
📚 Yabushita (1975), § 2, pp. 442-443

After adopting the substitute notation, $c_s^2\to K_1$ and ${\rho }_0\to {\lambda }_c$, it is clear that these last four parameter-profile expressions are identical to the ones that appear, respectively, as equations (2.2), (2.1) and (2.3) of 📚 Yabushita (1975).

Step 5: Interface Conditions

$\left(\frac{{\rho }_0}{{\mu }_c}\right)e^{-{\psi }_i}$	$=$	$\left(\frac{{\rho }_e}{{\mu }_e}\right){\theta }_i^{n_e}$	$=$	$\left(\frac{{\rho }_e}{{\mu }_e}\right){\theta }_i^{3/2}$
$c_s^2{\rho }_0{e}^{-{\psi }_i}$	$=$	$K_e{\rho }_e^{1+1/n_e}{\theta }_i^{n_e+1}$	$=$	$K_e{\rho }_e^{5/3}{\theta }_i^{5/2}$
$[\frac{c_s^2}{4\pi G{\rho }_0}{]}^{1/2}{\xi }_i$	$=$	$[\frac{(n_e+1)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{(1-n_e)/(2n_e)}{\eta }_i$	$=$	$[\frac{(5/2)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{(-1/6)}{\eta }_i$
$\left[\frac{c_s^6}{4\pi G^3{\rho }_0}{]}^{1/2}\right({\xi }^2\frac{d\psi }{d\xi }{)}_i$	$=$	$4\pi \left[\frac{(n_e+1)K_e}{4\pi G}{]}^{3/2}{\rho }_e^{(3-n_e)/(2n_e)}\right(-{\eta }^2\frac{d\theta }{d\eta }{)}_i$	$=$	$4\pi \left[\frac{(5/2)K_e}{4\pi G}{]}^{3/2}{\rho }_e^{1/2}\right(-{\eta }^2\frac{d\theta }{d\eta }{)}_i$

This means that,

$\frac{{\rho }_e}{{\rho }_0}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)e^{-{\psi }_i}{\theta }_i^{-3/2};$
$K_e\left[{\rho }_0\right(\frac{{\mu }_e}{{\mu }_c})e^{-{\psi }_i}{\theta }_i^{-3/2}{]}^{5/3}{\theta }_i^{5/2}$	$=$	$c_s^2{\rho }_0{e}^{-{\psi }_i}$
$\Rightarrow \frac{K_e{\rho }_0^{2/3}}{c_s^2}$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/3}{e}^{+2{\psi }_i/3}$
$\Rightarrow \frac{c_s^2}{K_e}$	$=$	$\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{5/2}{\rho }_0{e}^{-{\psi }_i}{]}^{2/3};$
$\frac{{\eta }_i}{{\xi }_i}$	$=$	$\left[\frac{c_s^2}{4\pi G{\rho }_0}{]}^{1/2}\right[\frac{4\pi G}{(5/2)K_e}{]}^{1/2}[{\rho }_e{]}^{1/6}$
	$=$	$\left(\frac{2}{5}{)}^{1/2}\right[\frac{c_s^2}{K_e}{]}^{1/2}\left[{\rho }_0\right(\frac{{\mu }_e}{{\mu }_c})e^{-{\psi }_i}{\theta }_i^{-3/2}{]}^{1/6}{\rho }_0^{-1/2}$
	$=$	$\left(\frac{2}{5}{)}^{1/2}\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{5/2}{\rho }_0{e}^{-{\psi }_i}{]}^{1/3}\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{1/6}{e}^{-{\psi }_i/6}{\theta }_i^{-1/4}\right]{\rho }_0^{-1/3}$
	$=$	$\left(\frac{2}{5}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c})e^{-{\psi }_i/2}{\theta }_i^{-1/4}.$

And, finally,

$(-{\eta }^2\frac{d\theta }{d\eta }{)}_i$	$=$	$\frac{1}{4\pi }\left[\frac{c_s^6}{4\pi G^3{\rho }_0}{]}^{1/2}\right[\frac{4\pi G}{(5/2)K_e}{]}^{3/2}\left[{\rho }_e{]}^{-1/2}\right({\xi }^2\frac{d\psi }{d\xi }{)}_i$
	$=$	$\left(\frac{2}{5}{)}^{3/2}{\rho }_0^{-1/2}\right[\frac{c_s^2}{K_e}{]}^{3/2}\left[{\rho }_0\right(\frac{{\mu }_e}{{\mu }_c}\left)e^{-{\psi }_i}{\theta }_i^{-3/2}{]}^{-1/2}\right({\xi }^2\frac{d\psi }{d\xi }{)}_i$
	$=$	$\left(\frac{2}{5}{)}^{3/2}{\rho }_0^{-1}\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{5/2}{\rho }_0{e}^{-{\psi }_i}\right]\left[\right(\frac{{\mu }_e}{{\mu }_c}\left)e^{-{\psi }_i}{\theta }_i^{-3/2}{]}^{-1/2}\right({\xi }^2\frac{d\psi }{d\xi }{)}_i$
	$=$	$\left(\frac{2}{5}{)}^{3/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^2{e}^{-{\psi }_i/2}{\theta }_i^{3/4}({\xi }^2\frac{d\psi }{d\xi }{)}_i$
$\Rightarrow (-\frac{d\theta }{d\eta }{)}_i$	$=$	$\left(\frac{2}{5}{)}^{3/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^2{e}^{-{\psi }_i/2}{\theta }_i^{3/4}\left(\frac{d\psi }{d\xi }{)}_i\right[\left(\frac{2}{5}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c})e^{-{\psi }_i/2}{\theta }_i^{-1/4}{]}^{-2}$
	$=$	$\left(\frac{2}{5}{)}^{1/2}{e}^{+{\psi }_i/2}{\theta }_i^{5/4}\right(\frac{d\psi }{d\xi }{)}_i.$

Summary Interface Relations
Our Derivations	After setting ${\rho }_e={\rho }_0$	Presented by 📚 Yabushita (1975) (after setting ${\mu }_e/{\mu }_c=1$)
$\frac{{\rho }_e}{{\rho }_0}$ $=$ $\left(\frac{{\mu }_e}{{\mu }_c}\right)e^{-{\psi }_i}{\theta }_i^{-3/2};$ $\frac{c_s^2}{K_e}$ $=$ $\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{5/2}{\rho }_0{e}^{-{\psi }_i}{]}^{2/3};$ $\frac{{\eta }_i}{{\xi }_i}$ $=$ $\left(\frac{2}{5}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c})e^{-{\psi }_i/2}{\theta }_i^{-1/4};$ $(-\frac{d\theta }{d\eta }{)}_i$ $=$ $\left(\frac{2}{5}{)}^{1/2}{e}^{+{\psi }_i/2}{\theta }_i^{5/4}\right(\frac{d\psi }{d\xi }{)}_i.$	${\theta }_i^{3/2}$ $=$ $\left(\frac{{\mu }_e}{{\mu }_c}\right)e^{-{\psi }_i};$ $[\frac{c_s^2}{K_e}{]}^3$ $=$ $\left(\frac{{\mu }_e}{{\mu }_c}{)}^5\right[{\rho }_0{e}^{-{\psi }_i}{]}^2;$ $\frac{{\eta }_i}{{\xi }_i}$ $=$ $\left(\frac{2}{5}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}\left)e^{-{\psi }_i/2}\right[\left(\frac{{\mu }_e}{{\mu }_c}\right)e^{-{\psi }_i}{]}^{-1/6}=\left(\frac{2}{5}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{5/6}{e}^{-{\psi }_i/3};$ $(-\frac{d\theta }{d\eta }{)}_i$ $=$ $\left(\frac{2}{5}{)}^{1/2}\right(\frac{d\psi }{d\xi }{)}_i{e}^{+{\psi }_i/2}\left[\right(\frac{{\mu }_e}{{\mu }_c})e^{-{\psi }_i}{]}^{5/6}=(\frac{2}{5}{)}^{1/2}\left(\frac{d\psi }{d\xi }{)}_i{e}^{-{\psi }_i/3}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{5/6}.$	${\theta }_i^{3/2}$ $=$ $e^{-{\psi }_i};$ (2.9) $[\frac{c_s^2}{K_e}{]}^{3/2}$ $=$ ${\lambda }_c{e}^{-{\psi }_i};$ (2.11) $\frac{{\eta }_i}{{\xi }_i}$ $=$ $(\frac{2}{5}{)}^{1/2}{e}^{-{\psi }_i/3};$ (2.12) $(\frac{d\theta }{d\eta }{)}_i$ $=$ $-\left(\frac{2}{5}{)}^{1/2}\right(\frac{d\psi }{d\xi }{)}_i{e}^{-{\psi }_i/3}.$ (2.13)

Step 8: Throughout the Envelope

Throughout the envelope, we seek the solution, $\theta (\eta )$, of the following Lane-Emden equation:

$\frac{1}{{\eta }^2}\frac{d}{d\eta }\left({\eta }^2\frac{d\theta }{d\eta }\right)=-{\theta }^{3/2}.$

For the envelope, the associated key parameter relations are:

$\rho$	$=$	${\rho }_e{\theta }^{3/2}$
$P$	$=$	$K_e{\rho }_e^{5/3}{\theta }^{5/2}$
$r$	$=$	$[\frac{(5/2)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{-1/6}\eta$
$M_r$	$=$	$4\pi \left[\frac{(5/2)K_e}{4\pi G}{]}^{3/2}{\rho }_e^{1/2}\right(-{\eta }^2\frac{d\theta }{d\eta })$

The surface occurs where the polytropic Lane-Emden function, $\theta$, first goes to zero. We will denote the radius at which this occurs as ${\eta }_s$ — 📚 Yabushita (1975) denotes the same radial location by ${\eta }_1$ — and the slope of the function at the surface will be denoted as $(d\theta /d\eta {)}_s$.

Normalizations

Following 📚 Yabushita (1975), along our model sequence we will hold $c_s^2$ and $K_e$ fixed. From the summary interface relations, then, we find that,

$[\frac{c_s^2}{K_e}{]}^3$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^5\right[{\rho }_0{e}^{-{\psi }_i}{]}^2$
$\Rightarrow {\rho }_0$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}\right[\frac{c_s^2}{K_e}{]}^{3/2}{e}^{{\psi }_i}.$

This is identical to Eq. (3.1) of 📚 Yabushita (1975) if we assume (as did Yabushita75) that ${\mu }_e/{\mu }_c=1$. Given that, following Yabushita75, we have set ${\rho }_e={\rho }_c$, the radius of the equilibrium configuration is given by the expression,

$R^2$	$=$	$\left[\frac{5K_e}{8\pi G}\right]{\rho }_e^{-1/3}{\eta }_s^2$
	$=$	$\frac{5K_e}{8\pi G}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}(\frac{c_s^2}{K_e}{)}^{3/2}{e}^{{\psi }_i}{]}^{-1/3}{\eta }_s^2$
	$=$	$\frac{5K_e}{8\pi G}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{5/6}\left(\frac{c_s^2}{K_e}{)}^{-1/2}{e}^{-{\psi }_i/3}\right]{\eta }_s^2;$

and the configuration's total mass is,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$4\pi \left[\frac{5K_e}{8\pi G}{]}^{3/2}\right[{\rho }_e{]}^{1/2}(-{\eta }^2\frac{d\theta }{d\eta }{)}_s$
	$=$	$4\pi \left[\frac{5K_e}{8\pi G}{]}^{3/2}\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}\right(\frac{c_s^2}{K_e}{)}^{3/2}{e}^{{\psi }_i}{]}^{1/2}(-{\eta }^2\frac{d\theta }{d\eta }{)}_s.$

If we again set ${\mu }_e/{\mu }_c=1$, this expression is identical to Eq. (3.2) of 📚 Yabushita (1975). The configuration's mean density is,

$\overline{\rho }\equiv \frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi R^3}$	$=$	$3\left[\frac{5K_e}{8\pi G}{]}^{3/2}\right[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}\right(\frac{c_s^2}{K_e}{)}^{3/2}{e}^{{\psi }_i}{]}^{1/2}(-{\eta }^2\frac{d\theta }{d\eta }{)}_s\{\frac{5K_e}{8\pi G}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{5/6}\left(\frac{c_s^2}{K_e}{)}^{-1/2}{e}^{-{\psi }_i/3}\right]{\eta }_s^2{\}}^{-3/2}$
	$=$	$3\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/4}\left(\frac{c_s^2}{K_e}{)}^{3/4}{e}^{+{\psi }_i/2}\right](-\frac{1}{\eta }\cdot \frac{d\theta }{d\eta }{)}_s[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/4}\right(\frac{c_s^2}{K_e}{)}^{3/4}{e}^{+{\psi }_i/2}]$
	$=$	$3\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}\left(\frac{c_s^2}{K_e}{)}^{3/2}{e}^{+{\psi }_i}\right](-\frac{1}{\eta }\cdot \frac{d\theta }{d\eta }{)}_s$
$\Rightarrow \frac{\overline{\rho }}{{\rho }_0}$	$=$	$3(-\frac{1}{\eta }\cdot \frac{d\theta }{d\eta }{)}_s.$

See Also

• M. Gabriel & M. L. Roth (1974, A&A, Vol. 32, p. 309) … On the Secular Stability of Models with an Isothermal Core
• M. Gabriel & P. Ledoux (1967, Annales d'Astrophysique, Vol. 30, p. 975) … Sur la Stabilité Séculaire des Modeéles a Noyaux Isothermes
  

  In § 1 (p. 442) of 📚 Yabushita (1975) we find the following reference:   "A somewhat similar problem has been investigated by Gabriel & Ledoux (1967). Gaseous configurations with an isothermal core and polytropic envelope of index 3 were studied by 📚 Henrich & Chandrasekhar (1941) and by 📚 Schönberg & Chandrasekhar (1942). As is well known there is an upper limit (Schönberg-Chandrasekhar limit) to the mass of the core for the configurations to be in hydrostatic equilibria. Gabriel & Ledoux have investigated the stability of these configurations and have shown that secular stability is lost at the configuration that corresponds to the Schönberg-Chandrasekhar limit."