We present a study of a sample of LMC red giants exhibiting Long Secondary Periods (LSPs). We use radial velocities obtained from VLT spectral observations and MACHO and OGLE light curves to examine properties of the stars and to evaluate models for the cause of LSPs. This sample is much larger than the combined previous studies of Hinkle et al. (2002) and Wood, Olivier & Kawaler (2004). Binary and pulsation models have enjoyed much support in recent years. Assuming stellar pulsation, we calculate from the velocity curves that the typical fractional radius change over an LSP cycle is greater than 30 per cent. This should lead to large changes in Teff that are not observed. Also, the small light amplitude of these stars seems inconsistent with the radius amplitude. We conclude that pulsation is not a likely explanation for the LSPs. The main alternative, physical movement of the star -- binary motion -- also has severe problems. If the velocity variations are due to binary motion, the distribution of the angle of periastron in our large sample of stars has a probability of 1.4e-3 that it comes from randomly aligned binary orbits. In addition, we calculate a typical companion mass of 0.09 Msun. Less than 1 per cent of low mass main sequence stars have companions near this mass (0.06 to 0.12 Msun) whereas ~25 to 50 per cent of low mass red giants end up with LSPs. We are unable to find a suitable model for the LSPs and conclude by listing their known properties.
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