ABSTRACT

This paper develops numerical models of stationary, axisymmetric, force-free black-hole magnetospheres, based on the theory origi- nally developed by Blandford & Znajek ru'1.d reformulated and extended by Macdonald

&

Thorne. The, structure of such a magne- tosphere is determined by a single scalar "stream function" satis- fying a nonlinear, second-order partial differential "stream equa- tion" on a region bounded by the black-hole horizon, the accretion disk, and an outer boundary beyond which the force-free condition breaks down. The stream equation is solved numerically, using an iterative relaxation method, for three different poloidal magnetic field configurations: ( 1) (roughly) radial magnetic field; (2) (roughly) uniform magnetic field; and (3) (roughly) paraboloidal magnetic field. The second and third cases also include a force- free gap between the inner edge of the disk and the horizon, with which the horizon may exchange magnetic flux. For the chosen boundary conditions, it is found that the poloidal field structure

*Supported in part by the National Science Foundation [AST82-14126)

galactic nuclei has been a subject of lively speculation since their discovery twenty years ago. The idea that black holes may play a role was recognized within months of the discovery of the first quasar, but it was not until relatively recently that realistic models were proposed for the direct extraction of energy from black holes (Blandford 1976, Lovelace 1976, Harrison 1976, Blandford &

Znajek 1977). These models have in common the assumption of a supermassive ("" 10

⁸

M

_{0 ),}

stationary, axisymmetric black hole, surrounded by an accretion disk that holds a strong ("" 10

⁴

Gauss) magnetic field on the hole. A toroidal com- ponent of the field extracts the rotational energy of the hole and disk and transfers it to accelerated charged particles, which form a jet carrying the energy to the observed double lobed structures. In some variants of these models, the direction of the jet is determined by the spin axis of the black hole , whose large inertia is responsible for the long-term stability of the linear struc- ture of the jet.

In the Blandford-Znajek model, it is assumed that there exists a region near

the horizon where the magnetic field is sufficiently strong, and the plasma

sufficiently tenuous, that the plasma exerts no force on the magnetic field . The

only role of the plasma in the magnetosphere is to provide the charge and

current sources for the field. The field entirely dominates the dynamics, drag-

ging the plasma whither it will. The mathematical expression of this force-free

approximation is the vanishing of the Lorentz force density:

p₀

E + jxB = 0. On

the other hand, in the non-force-free disk the plasma has so much inertia that it

determines the dynamics of the field lines which thread it and are locked into it

by its very high electrical conductivity. This point is crucial to the entire model,

since it is the disk, not the hole's gravity, that holds the magnetic field on the

Znajek (1978) and Damour (1978) have shown that, in its interactions with the electromagnetic field, the horizon behaves as

if

it were an ordinary body with a surface conductivity of

RH

= 4rr = 377 ohms. The field lines thus may slip through the horizon; but as they do, it exerts a torque on them. The angular momentum and mechanical energy thereby extracted from the hole are transmitted along the field lines without loss until they re a ch a region where the force-free approximation breaks down. In this non-force-free "acceleration"

region, the energy and angular momentum presumably are transferred to charged particles . The possible mechanisms operating in this acceleration region have been investigated by Blandford (1976), Lovelace (1976), Lovelace et

al.

(1979) and Phinney (1983).

The Blandford-Znajek theory of

black-ho~e

magnetospheres was recast in a 3 + 1 language by Macdonald & Thorne (1982) (Paper TI in this seri es) using a for- malism developed by Thorne & Macdonald (1982) (Paper I) . This paper (Paper III) extends that work by constructing numeric al models of st a ti onary, axisym- metric, force-free black-hole magnetospheres. Section 2 .1 pre sents without proof the equations of black-hole magnetospheres derived in Macdonald

^&

Thorne (1982) and describes the prescription to be used in generating the numerical models. The procedure used consists of taking known, static, vacuum solutions of Maxwell 's equations in Schwarzschild spacetime and "spinning up"

both the hole and the field to obtain the desired force-free solutions in Kerr spacetime . Section 2.2 briefly describes the numerical methods used in con- structing the models.

Section 3 describes the details of three specific models. Section 3 .1

presents a problem in which the magnetic field lines all thread the horizon and

in

which the initial field, before spinup, is precisely r adial. Section 3.2 considers

and the horizon, with which the horizon may exchange magnetic flux. Section 3.3 describes a model in which the initial field structure is paraboloidal. and also includes a force-free gap.

Throughout this paper, units are used in which the speed of light c and

Newton's gravitational constant G are equal to unity. Electromagnetic quanti-

ties are expressed in cgs units .