Self-Similar Force-Free Wind From an Accretion Disk

We consider a self-similar force-free wind flowing out of an infinitely thin disk located in the equatorial plane. On the disk plane, we assume that the magnetic stream function P scales as P ∝ R , where R is the cylindrical radius. We also assume that the azimuthal velocity in the disk is constant: vφ = Mc, where M < 1 is a constant. For each choice of the parameters ν and M , we find an infinite number of solutions that are physically well-behaved and have fluid velocity 6 c throughout the domain of interest. Among these solutions, we show via physical arguments and time-dependent numerical simulations that the minimum-torque solution, i.e., the solution with the smallest amount of toroidal field, is the one picked by a real system. For ν > 1, the Lorentz factor of the outflow increases along a field line as γ ≈ M(z/Rfp) (2−ν)/2 ≈ R/RA, where Rfp is the radius of the foot-point of the field line on the disk and RA = Rfp/M is the cylindrical radius at which the field line crosses the Alfven surface or the light cylinder. For ν < 1, the Lorentz factor follows the same scaling for z/Rfp < M , but at larger distances it grows more slowly: γ ≈ (z/Rfp) . For either regime of ν, the dependence of γ on M shows that the rotation of the disk plays a strong role in jet acceleration. On the other hand, the poloidal shape of a field line is given by z/Rfp ≈ (R/Rfp) 2/(2−ν) and is independent of M . Thus rotation has neither a collimating nor a decollimating effect on field lines, suggesting that relativistic astrophysical jets are not collimated by the rotational winding up of the magnetic field.