Stability and maximal compactness of relativistic boson stars

We present on progress investigating stability and maximal compactness of relativistic boson stars with non-linear potential. It has been shown that there exist relativistic boson stars so compact that they possess bound circular null geodesics, however a conjecture by Keir suggests that such stars are dynamically unstable in the spherically symmetric case. We test the conjecture and investigate the stability bounds of the boson stars.