Self-Similar Force-Free Wind From an Accretion Disk

Abstract

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 $P\propto R^ν$, where $R$ is the cylindrical radius. We also assume that the azimuthal velocity in the disk is constant: $v_ϕ= Mc$, 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 $\leq c$ 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 $ν\geq 1$, the Lorentz factor of the outflow increases along a field line as $γ\approx M(z/\Rfp)^{(2-ν)/2} \approx R/R_{\rm A}$, where $\Rfp$ is the radius of the foot-point of the field line on the disk and $R_{\rm A}=\Rfp/M$ 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 $z/\Rfp < M^{-1/(1-ν)}$, but at larger distances it grows more slowly: $γ\approx (z/\Rfp)^{ν/2}$. 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 $z/\Rfp \approx (R/\Rfp)^{2/(2-ν)}$ 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.