On collimation of the outflows in force-free magnetospheres

It is pointed out that ''an indeterminacy'' claimed to exist in the pulsar equation (Sulkanen & Lovelace 1990) or a ''free'' function B-t<(omega)over bar> = beta (Appl & Camenzind 1993a,b) can be uniquely related to the stream function at the boundary of the force-free region to the load region at infinity (Michel 1973b: Okamoto 1974). The form of beta is given by beta = -(alpha/c)(B-p<(omega)over bar>(2))(infinity) from the plasma conditions at infinity, where ct is the angular velocity of field lines. This relation can be interpreted as indicating Ohm's law for the return current from the equatorial region to the polar region on the 'sphere at infinity' with finite resistivity R-L = 4 pi/c = 377 Ohm, thereby ensuring the current closure. The field structure tends to be asymptotically conical, toward the 'sphere at infinity' and fields fill all the space there. It is argued that force-free models for purely cylindrical jets cannot self-consistently conserve electric current. The role of the plasma conditions in the black hole equation is mentioned as well.