Spin parameter a_* (see the color bar at the top) as a function of the orbital period and masses of the He-rich stars. The metallicity of all He-rich stars is 0.01 Z_⊙. The gray symbols indicate the initial conditions and the color symbols indicate the final ones. The colored lines linking these two symbols show the evolution of the binary. The color along the line gives a_* along the evolution. Black solid lines refer to the lower limit of the orbital period. At that limit, the He-rich star fills its Roche lobe at the beginning of the core He burning phase. Square: models for which rotation increases the mass lost by 10% with respect to the mass lost by non-rotating models; pentagon: Darwin instability; star: lGRB; Diamond: He-rich star starts to fill its Roche lobe. The three columns correspond to different initial velocities of the He-rich stars and the three rows correspond to different masses for the companion. All the He-rich stars have a metallicity equal to 0.01 Z_⊙.