  As part of the long-term radial velocity monitoring of known transiting
planets -- designed to measure orbital eccentricities, spin-orbit alignments
and further planetary companions -- we have acquired radial velocity data for
the two transiting systems WASP-12 and WASP-14, each harbouring gas giants on
close orbits (orbital period of 1.09 and 2.24 days respectively). In both
cases, the initial orbital solution suggested a significant orbital
eccentricity, 0.049+-0.015 for WASP-12 and 0.091+-0.003 for WASP-14. Since
then, measurements of the secondary eclipse of WASP-12 in the infrared have
indicated that one projection of the eccentricity (e cos w) was very close to
zero, casting doubt on the eccentricity from the initial radial velocity orbit.
Our measurements confirm that the initial eccentricity detection could be
spurious, and show that the radial velocity data is compatible with a circular
orbit. A MCMC analysis taking into account the presence of correlated
systematic noise in both the radial velocity and photometric data gives e=0.017
(+0.015-0.011). By contrast, we confirm the orbital eccentricity of WASP-14,
and refine its value to e=0.088+-0.003. WASP-14 is thus the closest presently
known planet with a confirmed eccentric orbit.
