Baryonic pollution in gamma-ray bursts: The case of a magnetically driven wind emitted from a disk orbiting a stellar mass black hole

Most models for the central engine of gamma-ray bursts involve a stellar mass black hole surrounded by a thick disk formed after the merging of a system of compact objects or the collapse of a massive star. Energy released from the accretion of disk material by the black hole or from the rotation of the hole itself extracted by the Blandford-Znajek mechanism powers a relativistic wind along the system axis. Lorentz factors of several hundreds are needed to solve the compactness problem in the wind which implies the injection of a tremendous power into a very small amount of matter. The Blandford-Znajek mechanism, where the follows magnetic field lines anchored to the black hole is probably the best way to prevent baryonic pollution and can even initially produce a purely leptonic wind. In this paper we rather study the wind emitted from the inner part of the disk where the risk of baryonic pollution is much larger since the outflow originates from high density regions. We show that the baryonic load of this wind sensitively depends on the disk temperature and magnetic field geometry and that the outflow can become ultra-relativistic (Lorentz factor Γ > 100) under quite restrictive conditions only. Conversely, if Γ remains of the order of unity the dense wind emitted from the inner disk could help to confine the central jet but also represent a source of baryon contamination for the Blandford-Znajek mechanism.