  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.
