Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 1

In the razor-thin disk approximation, the three-dimensional Poisson equation isFor a horizontal Fourier mode with wavevector $\mathbf{k}$ and $k=|\mathbf{k}|>0$, the Fourier transform of this equation isThe solution that decays away from the disk isConsequently the required razor-thin disk Poisson kernel in the midplane isThe spatially uniform $k=0$ background is excluded from this local perturbation formula.

For a Keplerian shearing sheet, the shear rate is $S=3\Omega /2$ and the shearing-sheet tidal potential isA uniform steady solution isThe Coriolis acceleration of this linear shear flow balances the radial tidal acceleration. The only nonzero background component of the viscous stress tensor that matters is $T_{xy}=-\nu ({\Sigma }_0){\Sigma }_{0}S$, which is spatially constant. Hence $\nabla \cdot \mathbf{T}=0$: a local uniform patch has no stress gradient or torque divergence to drive an accretion flow.

Write the perturbations as $(\sigma ,u,v)e^{i{k}_{x}x+\lambda t}$, set $k=|k_x|$, and evaluate all unmarked background quantities at ${\Sigma }_0$. DefineThe linearized mass conservation equation isThe radial and azimuthal momentum equations arewhere the razor-thin disk Poisson kernel gives $\varphi =-2\pi G\sigma /k$. The term proportional to $\beta$ comes from perturbing the density-dependent background shear stress $T_{xy}=-\nu \Sigma S$.

Eliminating $\sigma ,u,v$ and using $S=3\Omega /2$, so that the radial epicyclic frequency obeys ${\kappa }_r^2=2\Omega (2\Omega -S)={\Omega }^2$, gives the dispersion relation

Setting $\nu ={\nu }_b=0$, the dispersion relation reduces toAn axisymmetric density mode grows whenThe left-hand side is a concave quadratic function of $k$, with maximum $(\pi G{\Sigma }_0{)}^2/v_s^2$ at $k=\pi G{\Sigma }_0/v_s^2$. Growth is therefore possible precisely when the Toomre stability criterion is violated:

The constant term of the real cubic dispersion relation isA negative $c$ guarantees a positive real root because the cubic tends to $+\infty$ as $\lambda \to +\infty$.

If $\beta <0$, then $c<0$ at sufficiently small positive $k$, producing a viscous instability of an accretion disk. This is the local form of the global thin-disk diffusion equationSince $d(\nu \Sigma )/d\Sigma <0$, the effective diffusion reverses sign and amplifies surface-density variations.

If $\beta >0$, instability is still possible when the quadratic expression in parentheses is negative. Its minimum occurs at $k=\pi G{\Sigma }_0/v_s^2$, so the condition isorThis is a secular gravitational instability of an astrophysical disk: viscosity allows self-gravity to overcome rotational support even in part of the range that is stable by the inviscid $Q<1$ criterion.