  The number of nonrelativistic axions can be changed by inelastic reactions
that produce photons or relativistic axions. Any odd number of axions can
annihilate into two photons. Any even number of nonrelativistic axions can
scatter into two relativistic axions. We calculate the rate at which axions are
lost from axion stars from these inelastic reactions. In dilute systems of
axions, the dominant inelastic reaction is axion decay into two photons. In
sufficiently dense systems of axions, the dominant inelastic reaction is the
scattering of four nonrelativistic axions into two relativistic axions. The
scattering of odd numbers of axions into two photons produces monochromatic
radio-frequency signals at odd-integer harmonics of the fundamental frequency
set by the axion mass. This provides a unique signature for dense systems of
axions, such as a dense axion star or a collapsing dilute axion star.
