We have studied the $f$-mode oscillations of differentially rotating polytropes by making use of the linear stability analysis. We found that the critical values of $T/|W|$ where the dynamical instability against the $m = 2$ $f$-mode oscillations sets in decrease down to $T/|W| \sim 0.20$ as the degree of differential rotation becomes higher. Here $m$ is an azimuthal mode number and $T$ and $W$ are the rotational energy and the gravitational potential energy, respectively. This tendency is almost independent of the compressibility of the polytropes. These are the { \it first exact results} of the linear stability analysis for the occurrence of the dynamical instability against the $m = 2$ $f$-modes.
Categories: astro-ph gr-qc