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

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: