  The early acceleration of stellar winds in massive stars is poorly
constrained. The scattering of hard X-ray photons emitted by the pulsar in the
high-mass X-ray binary Vela X-1 can be used to probe the stellar wind velocity
and density profile close to the surface of its supergiant companion HD 77581.
  We built a high signal-to-noise and high resolution hard X-ray lightcurve of
Vela X-1 measured by Swift/BAT over 300 orbital periods of the system and
compared it with the predictions of a grid of hydrodynamic simulations.
  We obtain a very good agreement between observations and simulations for a
narrow set of parameters, implying that the wind velocity close to the stellar
surface is twice larger than usually assumed with the standard beta law.
Locally a velocity gradient of $\beta\sim0.5$ is favoured. Even if still
incomplete, hydrodynamic simulations are successfully reproducing several
observational properties of Vela X-1.
