SSC/Stability/GammaVariation

How Does Stability Change with γ_g?

Isolated Uniform-Density Configuration

Our Setup

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

$\mathfrak{𝔉}$

$\equiv$

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

Also, the two relevant boundary conditions are,

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

and,

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

$=$

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

Alternatively, this last expression may be written as,

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

$=$

$\frac{\mathfrak{𝔉}}{2}.$

The Sterne37 Solution

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	$\frac{{\omega }^2}{4\pi G\overline{\rho }}$
$j=0;$      $\mathfrak{𝔉}=0;$       $x=1$	$\gamma -4/3$
$j=1;$      $\mathfrak{𝔉}=14;$      $x=1-(7/5){\chi }_0^2$	$2(5\gamma -2)/3$
$j=2;$      $\mathfrak{𝔉}=36;$      $x=1-(18/5){\chi }_0^2+(99/35){\chi }_0^4$	$7\gamma -4/3$
$j=3;$      $\mathfrak{𝔉}=66;$      $x=1-(33/5){\chi }_0^2+(429/35){\chi }_0^4-(143/21){\chi }_0^6$	$12\gamma -4/3$

Cross-Check

Check j = 0:    The eigenvector is $x=1$, that is, homologous contraction/expansion, in which case both the first and the second derivative of $x$ are zero. Hence, this eigenvector is a solution to the LAWE only if $\mathfrak{𝔉}=0$. What about the two boundary conditions? Well, the central boundary condition is immediately satisfied; for the outer boundary condition, we see that the logarithmic derivative of $x$ is supposed to be zero, which it is because it equals $\mathfrak{𝔉}/2$. Finally, since $\mathfrak{𝔉}=0$, we see that the oscillation frequency is given by the expression,

$\frac{{\omega }^2}{4\pi G\overline{\rho }}={\gamma }_{\mathrm{g}}-4/3.$

Check j = 1:    The eigenvector is $x=1-\frac{7}{5}{\chi }_0^2$, hence, $dx/d{\chi }_0=-\frac{14}{5}{\chi }_0$, and, $d^{2}x/d{\chi }_0^2=-\frac{14}{5}.$ This means that,

LAWE	$=$	$-\frac{14}{5}(1-{\chi }_0^2)-[1-\frac{3}{2}{\chi }_0^2]\frac{56}{5}+\mathfrak{𝔉}[1-\frac{7}{5}{\chi }_0^2]$
	$=$	${\chi }_0^2[\frac{14}{5}+\frac{168}{10}-\frac{7}{5}\mathfrak{𝔉}]+[-\frac{14}{5}-\frac{56}{5}+\mathfrak{𝔉}]$
	$=$	$\frac{7}{5}{\chi }_0^2[14-\mathfrak{𝔉}]+[\mathfrak{𝔉}-14],$

which goes to zero if $\mathfrak{𝔉}=14$, in which case,

$\frac{{\omega }^2}{4\pi G\overline{\rho }}$

$=$

$\frac{1}{3}[\frac{{\gamma }_{\mathrm{g}}\mathfrak{𝔉}}{2}-4+3{\gamma }_{\mathrm{g}}]=\frac{2}{3}[5{\gamma }_{\mathrm{g}}-2].$

Is the surface boundary condition satisfied? Well …

$\frac{d\ln x}{d{\chi }_0}{|}_{{\chi }_0=1}=[\frac{1}{x}\cdot \frac{dx}{d{\chi }_0}{]}_{{\chi }_0=1}$

$=$

$\left[\right(1-\frac{7}{5}{\chi }_0^2{)}^{-1}(-\frac{14}{5}){\chi }_0{]}_{{\chi }_0=1}=\left[\right(-\frac{2}{5}{)}^{-1}(-\frac{14}{5})]=+7,$

which matches the desired logarithmic slope, $\mathfrak{𝔉}/2$.