Published online by Cambridge University Press: 12 April 2016
About 25% of variable asymptotic giant branch stars in the LMC have light curves that are modulated by a long secondary period of length ~ 1–4 yr. Solar-vicinity analogs of these stars show radial velocity curves that are asymmetric and of small amplitude (~3–5km s−1). Possible explanations for the photometric and radial velocity variations include: eccentric motion of an orbiting companion of mass ~0.1 M⊙; radial and nonradial pulsation; rotation of an ellipsoidal-shaped red giant; episodic dust ejection; and star spot cycles. We find that there are severe difficulties with each of these models. Thus, the long secondary periods are the only unexplained type of large-amplitude stellar variability known at this time.