Magnetized winds are believed to be present in many astrophysical objects. They were first put forward in the context of the solar wind. More recently they were discovered to probably play a major role in many situations where a relativistic flow is powered by a central rapidly rotating compact object: wind of pulsars, jets in radio galaxies, quasars, Seyfert galaxies and BL Lac objects, microquasars and even possibly gamma-ray bursts.
The first quantitative model of a magnetic stellar wind was developed by Weber & Davis (1967). They were considering the equations of a stationary, axisymmetric, polytropic flow near the equatorial plane in classical MHD. They found that such a wind can carry off most of the angular momentum of the star and are very efficient to accelerate particles up to very high velocities. An important feature of magnetic winds is the existence of three "critical points'' where the velocity of the flow equals the wave velocity of the three MHD wave modes (the slow, Alfvén and fast modes) whereas in comparison non-magnetic winds (Parker 1958) have only one "critical point'' where the velocity of the flow equals the sound speed (sonic waves being the only present wave mode).
The first extension of this theory to relativistic winds is due to Michel (1969) in the context of radio pulsars. This work was considering cold outflows driven by rapidly rotating highly magnetized neutron stars. The main conclusion was that the efficiency of the magnetic to kinetic energy conversion was extremely low compared to the classical case. Goldreich & Julian (1970) studied cool isothermal relativistic winds. Kennel et al. (1983) extended Michel's model to finite temperatures and relativistic injection speeds. All these works were always limited to the equatorial plane and either completely neglected the effect of gravity or adopted an approximative treatment for it. Okamoto (1978) first included an exact general relativistic description of the gravity field. His work was also not limited to the equatorial plane, applying for that the powerful concept of flux tubes. However he restricted his study to pressureless flows only. In a series of papers Camenzind (1986a,b, 1987 derived a complete set of equations describing a stationary axisymmetric relativistic magnetic wind in an arbitrary metric. He then solved these equations in some particular cases (cold flows, jet geometries).
The goal of this paper is to present a formulation of the equations governing a stationary axisymmetric MHD flow in the equatorial plane including an exact treatment of all effects (thermal pressure, gravity and arbitrary shapes of flux tubes) which allows a direct comparison with the classical model of Weber & Davis (1967), so that the relativistic effects can be easily identified. This is done in Sects. 2 and 3, where we worked by analogy with the formulation of the classical case by Sakurai (1985). Then we study in details the efficiency of the magnetic to energy conversion (Sect. 4), in particular the influence of the flux tubes geometry and of the gravity. We confirm and extend the results of Begelman & Li (1994) and we show that a large variety of situations is expected from very inefficient winds like those considered by Michel (1969) to highly efficient cases. Because our model assumes axisymmetry and focus on the equatorial plane, it fully applies only to simple astrophysical objects like isolated neutron stars. On the other hand it can also describe the outer parts of more complex objects, e.g. compact objects with accretion disks or complex magnetospheres, as long as the magnetic field can be approximated as monopole like at these distances from the source. In the particular case of gamma-ray bursts the possibility of Poynting-flux dominated fireballs is briefly discussed in Sect. 5. This is summarized in Sect. 6.
Copyright ESO 2002