Numerical setup. The initial magnetic field lines (black solid lines) are dipolar, the magnetic moments are here identical in direction and norm. The angular velocity vectors of the two stars Ω₁ and Ω₂ are either parallel or anti-parallel, and have the same magnitude. The particles are launched from the shell R_⋆ ≤ r_i ≤ R_⋆ + δr with corotation velocity. The size of the injecting shell is exaggerated in the figure. Positrons and electrons are denoted by blue and orange disks. The dashed line marks the radius of the common light cylinder R_LC = c/Ω. The striped zone has a resistivity profile λ(r) (see Eq. (2)). The pair-creation process is pictured on the right, a new pair being created at the expense of the energy of the primary particle.