Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 57 / 2 / d

The branch originating at $s=0$ has $s=ϵs_1+O({ϵ}^2)$. Keeping first-order terms in the cubic dispersion relation gives ${\omega }^2{s}_1+\Omega ({\omega }^2-{\Omega }^2)=0$. Hence the weak-drag secular gravitational growth rate isCarrying forward the positive-${\omega }^2$ regime of the preceding part, this mode grows when $\boxed{0<{\omega }^2<{\Omega }^2}$. This specifies the scope of the source's abbreviated condition. If ${\omega }^2<0$, the small root written here is instead damped, while the separate dynamical branch already grows; the dust is still unstable, but growth is not identified with this particular small-root continuation. The expansion also fails at ${\omega }^2=0$.

At sufficiently small nonzero $|k|$, ${\omega }^2$ is positive and just below ${\Omega }^2$, sinceTherefore an infinite dust layer with nonzero self-gravity and nonzero drag has a long-wavelength secular gravitational instability of an astrophysical disk, even when $Q>1$. Its small-$k$ growth rate tends to $2\pi ϵG{\sigma }_0|k|/\Omega$, and vanishes at $k=0$; the uniform-density mode itself is neutral. Drag transfers perturbation angular momentum to the gas, weakening rotational support while self-gravity exceeds pressure on long wavelengths.

The instability is not limited to weak drag as an existence statement. For any $ϵ>0$, the cubic has $P(0)=ϵ\Omega ({\omega }^2-{\Omega }^2)<0$ and $P(s)\to +\infty$ on the positive real axis. Thus it has a positive real root whenever ${\omega }^2<{\Omega }^2$. This exact positive-root criterion for dust self-gravity with drag distinguishes the growth criterion from the approximation used to compute its slow rate.