  We present new high-resolution, high signal-to-noise optical spectra of HD
77581, the optical counterpart of the X-ray source Vela X-1, and determine
radial velocities from these spectra, as well as from high-resolution IUE
spectra and from digitized photographic spectra. The measured velocities show
strong deviations from a pure Keplerian radial-velocity curve, which are
autocorrelated within one night, but not from one night to another. Since lines
of different ions exhibit very similar changes in profile, these deviations
most likely reflect large-scale motions of the stellar surface akin to
non-radial pulsations. A possible cause could be that the varying tidal force
exerted by the neutron star in its eccentric orbit excites high-order pulsation
modes in the optical star which interfere constructively for short time
intervals. The effect of such velocity excursions on the orbital solution is
estimated by means of a Monte-Carlo simulation technique. We investigate
sources of systematic error, due to, e.g., the tidal deformation of the star,
and find, in particular, evidence for a systematic perturbation of the radial
velocity near the time of velocity minimum. This possible distortion severely
compromises the accuracy of the radial-velocity amplitude, leading to a 95\%
confidence range of 18.0--28.2\,\kms. The corresponding 95\% confidence limits
of the masses are given by $\MX=1.9_{-0.5}^{+0.7}\,\Msun$ and
$\Mopt=23.5_{-1.5}^{+2.2}\,\Msun$.
