SSC/Stability/GammaVariation

How Does Stability Change with γ_g?

Isolated Uniform-Density Configuration

From our separate discussion, the relevant LAWE is,

$\frac{1}{(1-{\chi }_0^2)}\{(1-{\chi }_0^2)\frac{d^{2}x}{d{\chi }_0^2}+\frac{4}{{\chi }_0}[1-\frac{3}{2}{\chi }_0^2]\frac{dx}{d{\chi }_0}+\mathfrak{𝔉}x\}$

$=$

$0,$

where, ${\chi }_0\equiv r_0/R$, $\alpha \equiv (3-4/{\gamma }_{\mathrm{g}})$, and

$\mathfrak{𝔉}$

$\equiv$

$[\frac{3{\omega }^2}{2\pi {\gamma }_{g}G\overline{\rho }}-2(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\right].$

Also, the two relevant boundary conditions are,

$\frac{dx}{d{\chi }_0}=0$        at         ${\chi }_0=0;$

and,

$\frac{d\ln x}{d{\chi }_0}$

$=$

$\frac{1}{{\gamma }_g}(4-3{\gamma }_g+\frac{{\omega }^2}{4\pi G\overline{\rho }})$        at         ${\chi }_0=1.$

Alternatively, this last expression may be written as,

$\frac{d\ln x}{d{\chi }_0}{|}_{{\chi }_0=1}$

$=$

$\frac{1}{6}(\mathfrak{𝔉}-4\alpha ).$

From the general solution derived by 📚 T. E. Sterne (1937, MNRAS, Vol. 97, pp. 582 - 593), we have …

Sterne (1937)

Sterne (1937)

The first few solutions are displayed in the following boxed-in image that has been extracted directly from §2 (p. 587) of 📚 Sterne (1937); to the right of his table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of the solutions presented by Sterne37.

Based on exact eigenvector expressions extracted from §2 (p. 587) of … T. E. Sterne (1937) Models of Radial Oscillation Monthly Notices of the Royal Astronomical Society, Vol. 97, pp. 582 - 593	$\frac{{\omega }^2}{4\pi G\overline{\rho }}$
$j=0;$      $\mathfrak{𝔉}=0;$       $x=1$	$\gamma -4/3$
$j=1;$      $\mathfrak{𝔉}=14;$      $x=1-(7/5){\chi }_0^2$	$2(5\gamma -2)/3$
$j=2;$      $\mathfrak{𝔉}=36;$      $x=1-(18/5){\chi }_0^2+(99/35){\chi }_0^4$	$7\gamma -4/3$
$j=3;$      $\mathfrak{𝔉}=66;$      $x=1-(33/5){\chi }_0^2+(429/35){\chi }_0^4-(143/21){\chi }_0^6$	$12\gamma -4/3$