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       $\xi =0$.

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).

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."