Abstract: If the dark matter consists of axions, some of it will condense into axion stars, which are gravitationally bound Bose-Einstein condensates of axions. If an axion star accretes enough axions to exceed a critical mass, it will collapse. The fate of a collapsing axion star has not been established. One possibility for the remnant is a much denser axion star in which the attractive forces of gravity and axion pair interactions are balanced by the mean-field pressure of the axion Bose-Einstein condensate. A nonrelativistic effective field theory for axions can provide insights into the fate of axion stars. The collapsing process of axion stars and the possible denser remnant produce unique monochromatic radio-frequency signals at odd-integer harmonics of the fundamental frequency set by the axion mass.