= Solution
\b[Radial <epicyclic motion>.] At fixed <specific angular momentum>, use $\Phi_{\rm eff}=\Phi+\ell^2/(2r^2)$ and $\ddot r=-\Phi_{\rm eff}'$. A <circular orbit> at $r_c$ obeys $\Phi'(r_c)=\ell^2/r_c^3=r_c\Omega^2$. Expanding $r=r_c+\xi$ gives $\ddot\xi+\kappa^2\xi=0$, where
$$
\boxed{\kappa^2=\Phi''+\frac{3\ell^2}{r_c^4}
=4\Omega^2+2r_c\Omega\Omega'
=4\Omega^2\left(1+\frac{r_c\Omega'}{2\Omega}\right)}.
$$
For $\kappa^2>0$ the <radial epicyclic frequency> describes <simple harmonic motion>. The angular correction follows from <conservation of angular momentum>, $\dot\phi\simeq\Omega(1-2\xi/r_c)$, and describes an <epicyclic motion> around the guiding <circular orbit>.
\b[Stabilizing terms.] In the given local fluid <dispersion relation>, $\kappa^2$ is rotational support, important at long <wavelength>; $a_0^2k^2$ is random-motion or <pressure> support, important at short <wavelength>. The term $-2\pi G\Sigma|k|$ is destabilizing <self-gravity>. Although the wording refers to stellar dispersion, this algebra uses the supplied fluid approximation; a collisionless stellar treatment has a reduction factor and a different numerical normalization.<On the Gravitational Stability of a Disk of Stars by Alar Toomre>[Abstract and Sections II–V: collisionless stability threshold and random-motion support]{ref}
\b[Axisymmetric threshold.] For $n=0$, set $q=|k|\geq0$. Completing the square gives
$$
\omega^2=a_0^2(q-q_*)^2+\kappa^2(1-Q^{-2}),
\qquad q_*=\frac{\pi G\Sigma}{a_0^2},\quad
Q=\frac{\kappa a_0}{\pi G\Sigma}.
$$
Thus \b[$Q>1$ gives local axisymmetric oscillatory stability; $Q<1$ gives a growing band], with
$$
q_-<q<q_+,\qquad
q_\pm=\frac{\pi G\Sigma}{a_0^2}\left(1\pm\sqrt{1-Q^2}\right).
$$
At $Q=1$, the minimizing mode has zero <frequency> and is marginal, not a strictly restoring oscillation. Local axisymmetric stability does not establish stability to every global nonaxisymmetric mode.
\b[<long-wave and short-wave branches at marginal Toomre stability>.] For $Q=1$ and $n\geq1$,
$$
n^2(\Omega_P-\Omega)^2=(a_0q-\kappa)^2,
\qquad
q=\frac{\kappa\pm n|\Omega_P-\Omega|}{a_0}.
$$
The long-wavelength branch, using the minus sign, requires
$$
\boxed{\Omega-\frac\kappa n\leq\Omega_P\leq\Omega+\frac\kappa n}.
$$
These endpoints are the inner and outer <Lindblad resonance> conditions. \b[The stated <dispersion relation> does not restrict every wave solution to that interval.] Its plus-sign short-wavelength branch has $q>0$ even outside it: for example, $\Omega_P=\Omega+2\kappa/n$ gives $q=3\kappa/a_0$ and satisfies the equation. Additional branch selection or resonant absorption physics would be needed for an absolute confinement claim. No such condition is supplied.
\b[<Toomre stability radius of a flat-rotation disk>.] With $\Omega=v_0/r$, $\kappa=\sqrt2v_0/r$. Using $G=4.3009\times10^{-3}\ \mathrm{pc}\,(\mathrm{km\,s^{-1}})^2M_\odot^{-1}$,
$$
\boxed{Q(r)=\frac{\sqrt2v_0a_0}{\pi G\Sigma r}
\simeq\frac{4.19\ \mathrm{kpc}}r}.
$$
The supplied local model therefore predicts stable radii below about \b[$4.19\ \mathrm{kpc}$], marginality at that radius, and axisymmetric instability farther out while its assumed constant <surface density> and thin-disk approximation remain applicable. An infinite constant-density, flat-rotation disk is an idealization, not a global self-consistent galaxy model.
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2014/paper-1-disk-branches.png]
{title=Original marginal-disk dispersion branches and radial Toomre parameter, showing the long-wave restriction and the surviving short-wave branch}
Back to article page
