  If the dark matter is composed of axions, then axion stars are expected to be
abundant in the Universe. We demonstrate in fully non-linear (3+1) numerical
relativity the instability of compact axion stars due to the electromagnetic
Chern-Simons term. We show that above the critical coupling constant
$g_{a\gamma}^\mathrm{crit} \propto M_s^{-1.35}$, compact axion stars of mass
$M_s$ are unstable. The instability is caused by parametric resonance between
the axion and the electromagnetic field. The existence of stable compact axion
stars requires approximately Planck-suppressed couplings to photons. If the
coupling exceeds the critical value, then all stable axion stars are
necessarily non-compact. Unstable axion stars decay leaving behind a less
massive, less compact, remnant. The emitted radiation peaks at frequency
$\omega \sim 1/R_s$, where $R_s$ is the axion star radius.
