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 ^ { \nu } , where R is the cylindrical radius .
We also assume that the azimuthal velocity in the disk is constant : v _ { \phi } = Mc , where M < 1 is a constant .
For each choice of the parameters \nu 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 \nu \geq 1 , the Lorentz factor of the outflow increases along a field line as \gamma \approx M ( z / R _ { fp } ) ^ { ( 2 - \nu ) / 2 } \approx R / R _ { A } , where R _ { fp } is the radius of the foot-point of the field line on the disk and R _ { A } = R _ { fp } / M is the cylindrical radius at which the field line crosses the Alfven surface or the light cylinder .
For \nu < 1 , the Lorentz factor follows the same scaling for z / R _ { fp } < M ^ { -1 / ( 1 - \nu ) } , but at larger distances it grows more slowly : \gamma \approx ( z / R _ { fp } ) ^ { \nu / 2 } .
For either regime of \nu , the dependence of \gamma 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 / R _ { fp } \approx ( R / R _ { fp } ) ^ { 2 / ( 2 - \nu ) } 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 .