VLT multi-object spectroscopy of 33 eclipsing binaries in the Small Magellanic Cloud New distance and depth of the SMC, and a record-breaking apsidal motion

Aims. Our purpose is to provide reliable stellar parameters for a significant sample of eclipsing binaries, which are representative of a whole dwarf and metal-poor galaxy. We also aim at providing a new estimate of the mean distance to the SMC and of its depth along the line of sight for the observed field of view. Methods. We use radial velocity curves obtained with the ESO FLAMES facility at the VLT and light curves from the OGLE-II photometric survey. The radial velocities were obtained by least-squares fits of the observed spectra to synthetic ones, excluding the hydrogen Balmer lines. Results. Our sample contains 23 detached, 9 semi-detached and 1 overcontact systems. Most detached systems have properties consistent with stellar evolution calculations from single-star models at the standard SMC metalicity Z = 0.004, though they tend to be slightly overluminous. The few exceptions are probably due to third-light contribution or insufficient signal-to-noise ratio. The mass ratios are consistent with a flat distribution, both for detached and semi-detached/contact binaries. A mass-luminosity relation valid from similar to 4 to similar to 18 M-circle dot is derived. The uncertainties are in the +/- 2 to +/- 11% range for the masses, in the +/- 2 to +/- 5% range for the radii and in the +/- 1 to +/- 6% range for the effective temperatures. The average distance modulus is 19.11 +/- 0.03 (66.4 +/- 0.9 kpc). The moduli derived from the V and from the I data are consistent within 0.01 mag. The 2 sigma depth of the SMC is, for our field, of 0.25 mag or 7.6 kpc under the assumption of a Gaussian distribution of stars along the line of sight. Three systems show significant apsidal motion, one of them with an apsidal period of 7.6 years, the shortest known to date for a detached system with main sequence stars.