  Long secondary periods (LSPs), observed in a third of pulsating red giant and
supergiant stars, are the only unexplained type of large-amplitude stellar
variability known at this time. Numerous authors have explored various
scenarios for the origin of LSPs, but were unable to give a final solution to
this problem. We present known properties of LSP variables and show new results
proving that the physical mechanism responsible for LSPs is binarity. Namely,
the LSP light changes are due to the presence of a dusty cloud orbiting the red
giant together with a brown-dwarf companion and obscuring the star once per
orbit. In this scenario, the low-mass companion is a former planet that
accreted a significant amount of mass from the envelope of its host star and
grew into a brown dwarf.
