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Force-Free Magnetosphere of an Accretion Disk -- Black Hole System. II. Kerr Geometry


Force-Free Magnetosphere of an Accretion Disk — Black Hole System. II. Kerr Geometry
Dmitri A. Uzdensky
1

Kavli Institute for Theoretical Physics, University of California

arXiv:astro-ph/0410715v1 28 Oct 2004

Santa Barbara, CA 93106 uzdensky@kitp.ucsb.edu October 27, 2004 ABSTRACT We consider a stationary axisymmetric force-free degenerate magnetosphere of a rotating Kerr black hole surrounded by a thin Keplerian in?nitely-conducting accretion disk. We focus on the closed-?eld geometry characterize by a direct magnetic coupling between the disk and the hole’s event horizon. We ?rst argue that the hole’s rotation necessarily limits the radial extent of the force-free link on the disk surface: the faster the hole rotates, the smaller the magneticallyconnected inner part of the disk has to be. We then show that this is indeed the case by solving numerically the Grad–Shafranov equation—the main di?erential equation describing the structure of the magnetosphere. An important element in our approach is the use of the regularity condition at the inner light cylinder to ?x the poloidal current as a function of the poloidal magnetic ?ux. As an outcome of our computations, we are able to chart out the maximum allowable size of the portion of the disk that is magnetically connected to the hole as a function of the black hole spin. We also calculate the angular momentum and energy transfer between the hole and the disk that takes place via the direct magnetic link. We ?nd that both of these quantities grow rapidly and that their deposition becomes highly concentrated near the inner edge of the disk as the black hole spin is increased. Subject headings: black hole physics — MHD — accretion, accretion disks — magnetic ?elds — galaxies: active

1

Currently at Princeton University.

–2– 1. Introduction

This paper is devoted to the subject of magnetic interaction between a rotating black hole and an accretion disk around it—a topic that has enjoyed a lot of attention among researchers in recent years. Magnetic ?elds are believed to play an important role in the dynamics of accreting black hole systems (e.g., Begelman, Blandford, & Rees 1984; Krolik 1999b; Punsly 2001). In particular, they can be very e?ective in transporting angular momentum and the associated rotational energy of either the hole or the disk. Where and how this transport takes place and to what observational consequences it can lead, is partly determined by the global geometry of the magnetic ?eld lines. Conceptually, one can think of two basic types of geometry. The ?rst type is the open-?eld con?guration shown schematically in Figure 1. The main topological feature here is that there is no direct magnetic link between the hole and the disk. Instead, all the ?eld lines are open and extend to in?nity. Historically, this con?guration was the ?rst to have been considered, and it has been studied very extensively during the past three decades (see, e.g., Lovelace 1976; Blandford 1976; Blandford & Znajek 1977, hereafter BZ77; MacDonald & Thorne 1982, hereafter MT82; Phinney 1983; Macdonald 1984; Thorne, Price, & Macdonald 1986; Punsly 1989, 2001, 2003, 2004; Punsly & Coroniti 1990; Beskin & Par’ev 1993; Beskin 1997; Ghosh & Abramowicz 1997; Beskin & Kuznetsova 2000; Komissarov 2001, 2002b, 2004a). The reason for this popularity is that this con?guration is related to the famous Blandford– Znajek mechanism (BZ77) now widely regarded as the primary process powering jets in active galactic nuclei (AGN) and micro-quasars. As Blandford and Znajek showed, a largescale, ordered open magnetic ?eld can extract the rotational energy from a spinning black hole and transport it to large distances via Poynting ?ux (a similar process works along the ?eld lines connected to the disk). The second type of magnetic ?eld geometry is the closed-?eld con?guration, shown in Figure 2. Although it has been occasionally discussed in the literature before the last decade (e.g., Zeldovich & Schwartzman, quoted in Thorne 1974; MT82; Thorne et al. 1986; Nitta, Takahashi, & Tomimatsu 1991; Hirotani 1999), it is only in the last ?ve years that it has attracted serious scienti?c attention (e.g., Blandford 1999, 2000, 2002; Gruzinov 1999; van Putten 1999; van Putten & Levinson 2003; Li 2000, 2001, 2002a, 2002b, 2004; Wang, Xiao, & Lei 2002; Wang, Lei, & Ma 2003a; Wang et al. 2003b, 2004). The basic topological structure of magnetic ?eld in this con?guration is very di?erent from that of the open-?eld con?guration. The ?eld lines are closed and directly connect the black hole to the disk. In this so-called Magnetically-Coupled con?guration (Wang et al. 2002), the energy and angular momentum are not taken away to in?nity, but instead are exchanged between the hole and the disk by the magnetic ?eld. Therefore, magnetic coupling, together with the accretion process,

–3– controls the spin evolution and the spin equilibrium of the black hole (Wang et al. 2002, 2003a). In addition, the rotational energy of the hole can be magnetically extracted (just like in the Blandford–Znajek process) and deposited onto the disk leading to a change in the disk energy-dissipation pro?le and hence its observable spectral characteristics (Gammie 1999; Li 2000, 2001, 2002a, 2002b, 2004; Wang et al. 2003a,b). Finally, if the rotating ?eld lines are strongly twisted and become unstable to a non-axisymmetric kink-like instability, a strong variability of the energy release may result, which would be a possible explanation for quasi-periodic oscillations (QPOs) in micro-quasar systems (e.g., Gruzinov 1999; Wang et al. 2004). All these phenomena make the closed-?eld con?guration astrophysically very interesting. Most of the work that has been done on studying the magnetic ?eld structure around an accreting Kerr black hole, including the seminal paper by Blandford & Znajek (BZ77), has been performed under the assumption that the magnetosphere above the thin disk is ideally conducting and force-free. Then, if one also assumes that the system is stationary and axisymmetric, the magnetic ?eld is governed by the general-relativistic version of the forcefree Grad–Shafranov equation (e.g., MT82; for the full-MHD generalization of this equation see Nitta et al. 1991; Beskin & Par’ev 1993; Beskin 1997). Since this is a rather nontrivial nonlinear partial di?erential equation (PDE) with singular surfaces and free functions, it is generally not tractable analytically, except in some special simple cases, such as the slowrotation limit (BZ77). However, over the past 20 years, a number of force-free solutions for the magnetosphere have been obtained numerically, either by solving the Grad–Shafranov equation itself (MacDonald 1984; Fendt 1997) or as an asymptotic steady state of force-free degenerate electrodynamics (FFDE) evolution (Komissarov 2001, 2002b, 2004a). Until now, most of these studies have been done in the context of the open-?eld con?guration, primarily because of its relevance to the jet problem. In contrast, most of the work on closed-?eld con?gurations has been limited to analytic and semi-analytic studies of the e?ects that magnetic link has on the disk radiation pro?le and on the spin evolution of the black hole. The structure of the magnetosphere has not in fact been computed self-consistently. These studies have just assumed the existence of the link and made some simpli?ed assumptions about the ?eld distribution on the horizon. The only exception to this de?ciency is the recent work by Uzdensky (2004) where a force-free magnetosphere linking a Keplerian disk to a Schwarzschild black hole has been numerically computed for the ?rst time. In the present paper, we make the next logical step by extending this previous work to the more general case of a rapidly rotating Kerr black hole. This is indeed the most important

–4– case, not only because real astrophysical black holes are believed to be rotating, but also because the nonlinear terms in the Grad–Shafranov equation, especially the toroidal ?eld pressure, become large in this case. As a result, even the existence of closed-?eld solutions is not guaranteed. And indeed, one of the main goals of our present study is to determine the conditions for existence of such solutions in Kerr geometry. In other words, we aim at determining the limitations that the rotation of the black hole imposes on the direct magnetic link between the hole and the disk. In addition, by computing the global magnetic ?eld structure, we will be able to study the e?ect of the black hole rotation on the magnetic ?eld distribution on the horizon, the poloidal electric current as a function of poloidal magnetic ?ux, and the location of the inner light cylinder, as well as such astrophysically-important processes as angular momentum and energy transfer between the hole and the disk. In order to achieve the goal of obtaining numerical solutions of the force-free Grad–Shafranov equation in Kerr goemetry, we ?rst analize the mathematical structure of this equation. In particular, we pay special attention to its singular surfaces (the event horizon and the light cylinder) and the corresponding regularity conditions. Thus, we use the light-cylinder regularity condition to determine the poloidal current as a function of poloidal magnetic ?ux, similar to way it was done by Contopoulos, Kazanas, & Fendt (1999) for the case of the pulsar magnetosphere (see also Beskin & Kuznetsova 2000; Uzdensky 2003; Uzdensky 2004). The event-horizon regularity condition, also known as Znajek’s (1977) horizon boundary condition, is then used to determine the poloidal ?ux distribution on the horizon. Thus, one does not have the freedom to arbitrarily specify any extra boundary conditions at the horizon, and hence there is no problem with causality, in line with the reasoning presented by Beskin & Kuznetsova (2000) and by Komissarov (2002b, 2004a) (see also Levinson 2004). Finally, although in this paper we deal exclusively with large-scale, ordered magnetic ?elds, we acknowledge the di?culty in justifying the existence of such ?elds around accreting black holes (e.g., Livio, Ogilvie, & Pringle 1999), especially in the closed-?eld con?guration. Also, as recent numerical simulations (e.g., Hawley & Krolik 2001; Hirose et al. 2004), there may be a signi?cant deposition of energy and angular momentum at the inner edge of the disk due to small-scale, intermittent magnetic ?elds connecting the disk to the plunging region (see also Krolik 1999a; Agol & Krolik 2000). The paper is organized as follows. § 2 describes the mathematical formalism of force-free axisymmetric stationary magnetospheres in Kerr geometry. In particular, in § 2.1 we introduce the Kerr metric tensor in Boyer–Lindquist coordinates and list several general geometric relationships for the future use. In § 2.2 we consider steady-state, axisymmetric, degenerate electro-magnetic ?elds and then discuss the force-free condition and the Grad–Shafranov equation. In § 2.3 we consider the black hole’s event horizon as a singular surface of this

–5– equation and discuss the associated regularity condition, which is also known as Znajek’s horizon boundary condition. In § 3 we present a simple but robust physical argument that demonstrates that a force-free magnetic link between a rotating black hole and the disk cannot extend to arbitrarily large distances on the disk, we also argue that the maximal radial extent of the magnetic link should scale inversely with the black hole’s rotation rate in the slow-rotation limit. We con?rm these propositions in § 4, where we present our numerical solutions of the grad–Shafranov equation. Then, in § 5 we discuss the magnetically-mediated angular-momentum and energy exchange between the hole and the disk. We then close by summarizing our ?ndings in § 6.

2.

Axisymmetric force-free magnetosphere in Kerr geometry — basic equations 2.1. Kerr geometry — mathematical preliminaries

In this paper we employ Boyer–Lindquist coordinates (t, r, θ, φ) in Kerr geometry. The metric of the four-dimensional space-time can be written in these coordinates as ds2 = (gφφ ω 2 ? α2 )dt2 ? 2ωgφφ dφdt + grr dr 2 + gθθ dθ2 + gφφ dφ2 , with the components of the metric tensor given by ρ√ α = ?, Σ 2aMr ω = , Σ2 ρ2 , gθθ = ρ2 , grr = ? where ρ2 ≡ r 2 + a2 cos2 θ , (5) (6) (7) (8) (1)

(2) (3) gφφ = ? 2 , (4)

Σ2 ≡ (r 2 + a2 )2 ? a2 ? sin2 θ , Σ ? ≡ sin θ . ρ

? ≡ r 2 + a2 ? 2Mr ,

Here, M and a ∈ [0; M] are the mass and the spin parameter (speci?c angular momentum) of the central black hole, respectively. (Throughout this paper we use geometric units, i.e., we set both the gravitational constant G and the speed of light c to 1).

–6– In order to describe the electromagnetic processes around a black hole, we use the 3+1 split of the laws of electrodynamics introduced by MT82 (see also Thorne et al. 1986). In this formalism, one splits the four-dimensional spacetime into the global time t and the absolute three-dimensional curved space, the geometry of which is described by a three-dimensional (3D) metric tensor with components given by equation (4). The electromagnetic ?eld is represented by the electric and magnetic ?eld 3-vectors E and B measured by local zeroangular-momentum observers (ZAMOs; see Thorne et al. 1986). In order to describe these vectors, we will use both the coordinate basis {?i } = {ei } and the orthonormal basis {e?} i [where the Roman index i runs through the three spatial coordinates (r, θ, φ)]. Because the spatial 3D metric tensor gij is diagonal, these two bases are related via ei = √ gii e? , i i = r, θ, φ (9)

(note: there is no summation over i in this expression!). In particular, in the Boyer–Lindquist coordinates in Kerr geometry, we have ρ eθ = ρeθ , eφ = ? eφ . (10) er = √ er , ? ? ? ? We shall also need the following mathematical expressions: the 3-gradient of a scalar function f (x) = f (r, θ, φ) in the Boyer–Lindquist coordinates is ?f = = and its square is |?f |2 = =
?1 gii (?i f )2 i

gii √
i

?1/2

(?i f )e? i (11)

1 ? 1 (?r f ) er + (?θ f ) eθ + (?φ f ) eφ , ? ? ? ρ ρ ?

? 1 1 (?r f )2 + 2 (?θ f )2 + 2 (?φ f )2 . 2 ρ ρ ?

(12)

Finally, the 3-divergence of a 3-vector A can be written as ? · A = Ai = Ai + Ai (ln ;i ,i |g|),i = 1 |g| |g|Ai
,i

,

(13)

where g is the determinant of the 3-D metric tensor: ρΣ sin θ ρ2 ? . |g| = √ = √ ? ? (14)

–7– 2.2. Stationary axisymmetric ideal force-free magnetosphere in Kerr geometry

As mentioned above, in the 3+1 split formalism of MT82 a magnetosphere of a rotating Kerr black hole is described in terms of two spatial vector ?elds, E and B. Under the assumptions that the magnetosphere is: (1) stationary (?t = 0), (2) axisymmetric (?φ = 0), and (3) ideally-conducting, or degenerate (E · B = 0), these two vector ?elds can be expressed in terms of three scalar functions, Ψ(r, θ), ?F (r, θ), and I(r, θ): B(r, θ) = Bpol + Btor , where Bpol Btor and E(r, θ) = Epol = ? where δ? ≡ ?F ? ω . (19) √ 1 ? = ?Ψ × ?φ = Ψθ er ? Ψr eθ , ? ? ?ρ ?ρ I = Bφ eφ = e? , ? ? α? φ δ? ?Ψ , α Eφ = 0 , (16) (17) (15)

(18)

Here, Ψ(r, θ) is the poloidal magnetic ?ux function, ?F = ?F (Ψ) is the angular velocity of the magnetic ?eld lines, and I(r, θ) is (2/c) times the poloidal electric current ?owing through the circular loop r = const, θ = const. [Note that our de?nitions of Ψ and I di?er from the ones adopted by MT82: Ψ = ψMT82 /2π, I = ?(2/c)IMT82 .] Next, in this work we are interested in the case of a force-free magnetosphere, i.e., a magnetosphere that is so tenuous that the electromagnetic forces completely dominate over all others, including gravitational, pressure, and inertial forces. Even though this framework has been widely accepted as a primary tool in describing magnetospheres of black holes and radio-pulsars, its usefulness and validity near the event horizon has been seriously challenged by Punsly (2001, 2003). However, according to the recent MHD simulations by Komissarov (2004b), these worries seem to be unfounded. Therefore, we shall still employ the force-free approach in this paper. Correspondingly, we shall write the force-balance equation (in the ZAMO reference frame) as ρe E + j × B = 0 , (20) where the ZAMO-measured electric charge density ρe and electric current density j are related to E and B via Maxwell’s equations (see MT82).

–8– The toroidal component of the force-free equation immediately leads to I(r, θ) = I(Ψ) , i.e., the poloidal electric current does not cross poloidal ?ux surfaces. The poloidal component of equation (20), upon using expressions (15)–(18), yields the socalled generally-relativistic force-free Grad–Shafranov equation — the main equation that governs the system. In this paper we shall use as a starting point the form of this equation given in MT82 (i.e., eq. [6.4] of MT82 slightly modi?ed to account for the change in the de?nition of Ψ): α δ?2 ? 2 [1 ? ]?Ψ + ?2 α2 δ? d?F 1 (?Ψ)2 + II ′ (Ψ) = 0 . α dΨ α? 2 ?· (21)

(22)

This is a nonlinear 2nd-order elliptic partial di?erential equation (PDE); it determines Ψ(r, θ) provided that ?F (Ψ) and I(Ψ) are known. We can rewrite this equation as follows: LHS ≡ α? 2? · 1 (α2 ? δ?2 ? 2 )?Ψ = RHS ≡ ?II ′ (Ψ) ? δ??′F (Ψ) ? 2(?Ψ)2 , (23) α? 2

where a prime denotes the derivative with respect to Ψ, e.g., I ′ (Ψ) = dI/dΨ. Upon introducing the quantities D ≡ α2 ? δ?2 ? 2, and Q(r, θ) ≡ (24)

|g| ρΣ ρ2 = = , (25) α? 2 ?? ? sin θ and upon using identity (13), the left-hand side (LHS) of this equation can be written in a compact and convenient form LHS = Q?1 [QD(?Ψ)i ],i = [D(?Ψ)i ],i + D(?Ψ)i?i ln Q . Using expression (11), we get the Grad–Shafranov equation in the following ?nal form: LHS = ?r D? D D Ψr ??r Q + Ψθ ?θ Q Ψr + ?θ 2 Ψθ + 2 ρ2 ρ ρQ = RHS ≡ ?II ′ (Ψ) ? δ??′F (Ψ) ? 2(?Ψ)2 . (26)

(27)

–9– 2.3. Regularity condition at the event horizon

From the Grad–Shafranov equation in the form (27) it is easy to see that, in general, this equation has two types of singular surfaces. One of them is the so-called light cylinder (often called in the literature the velocity-of-light surface or simply the light surface) de?ned as a surface where D = 0. We shall discuss it in more detail later (see § 4.2). There is also another singular surface of the Grad–Shafranov equation: the event horizon de?ned as the surface where ? = 0 = α. (28) This surface will be the main focus of this section. As can be seen from equation (6), the event horizon is a constant-r surface, √ r(θ) = rH = M + M 2 ? a2 = const .

(29)

In addition, the frame-dragging frequency ω de?ned by equation (3) is also constant on the horizon, a ω(r = rH , θ) = ?H = = const . (30) 2MrH This constant is what is conventionally called the rotation rate of the Kerr black hole. Because the horizon is surface of constant r, one can immediately see that it is a singular surface of equation (27). This is because the coe?cient in front of the 2nd-order derivative in the direction normal to this surface (in this case, radial) vanishes, even though the coe?cient in front of the 2nd derivative in the θ-direction does not. The fact that the event horizon is just a singular surface of the Grad–Shafranov equation is extremely important. It means that one cannot impose an independent boundary condition for the function Ψ(r, θ) at the horizon. One can only impose a regularity condition there (Beskin 1997; Komissarov 2002b, 2004a; Uzdensky 2004). Mathematically, this condition means that there should be no logarithmic terms in the asymptotic expansion of Ψ(r, θ) near r = rH (see MT82). Physically, the regularity condition originates from the requirement that freely-falling observers measure ?nite electric and magnetic ?elds near the horizon (see Thorne et al. 1986). Alternatively, the event horizon regularity condition can be obtained from the fast-magnetosonic critical condition in the limit in which plasma density goes to zero and hence the inner fast magnetosonic surface approaches the horizon (Beskin 1997; Beskin & Kuznetsova 2000; Komissarov 2004a). In the present paper, we will not repeat the rigorous derivation of this condition (we refer the reader to MT82 or Thorne et al. 1986). Instead, we just note that as a result of the regularity requirement, one expects both the 1st

– 10 – and 2nd radial derivatives of Ψ to remain ?nite at the horizon. Therefore, when applying the Grad–Shafranov equation (27) at r = rH , one can just simply set ? = 0. Then, after some algebra, one gets: I 2 [Ψ0 (θ)] = δ? where Ψ0 (θ) ≡ Ψ(r = rH , θ) . (32) ? dΨ0 ρ dθ
2

+ const ,

r = rH ,

(31)

In the absence of a ?nite line-current along the axis θ = 0, i.e., when I(θ = 0) = 0, the integration constant is zero and hence I = ±δ? ? dΨ0 , ρ dθ r = rH . (33)

As for the choice of sign in this expression, it can be shown that the correct sign must be plus [remember that MT82 have minus sign because we de?ne I(Ψ) with an opposite sign]; this comes from the requirement that Poynting ?ux measured by a ZAMO in the vicinity of the horizon is directed towards the black hole (e.g., Znajek 1977, 1978; BZ77; MT82). Thus, we have ? dΨ0 2MrH sin θ dΨ0 I[Ψ0 (θ)] = δ? = δ? , r = rH . (34) 2 ρ dθ ρ dθ Equation (34) was ?rst derived by Znajek (1977) and is frequently referred to as the ”Znajek’s horizon boundary condition”. We stress, however, that, because the event horizon is a singular surface of the Grad–Shafranov equation, one cannot really impose a boundary condition there; expression (34) actually follows from the Grad–Shafranov equation itself under the condition that the solution be regular at r = rH . It is interesting to note that, because not only the 2nd-, but also the 1st-order radial derivatives of Ψ drop out of the Grad–Shafranov equation when ? is set to zero, this equation becomes an ordinary (as opposed to a partial) di?erential equation at the horizon! This implies that the horizon poloidal magnetic ?ux distribution, Ψ0 (θ), is connected to the magnetosphere outside the horizon only through the functions I(Ψ) and ?F (Ψ) and not through any radial derivatives. From the practical point of view, this fact means that equation (34) can be viewed as a Dirichlet-type boundary condition that determines the function Ψ0 (θ) once both I(Ψ) and ?F (Ψ) are given. It is important to emphasize that we really have only one relationship on the horizon— equation (34) — between three functions [Ψ0 (θ), I(Ψ), and ?F (Ψ)] and hence one needs to ?nd some other conditions, set somewhere else, to ?x I(Ψ) and ?F (Ψ) if one wants to use (34) to calculate Ψ0 (θ). We shall return to this important point in § 4.1.

– 11 – 3. Disruption of the hole–disk magnetic link by the black hole rotation

The main topic of this paper is a force-free magnetic link between a Kerr black hole and a thin, in?nitely conducting Keplerian accretion disk around it. Thus, we are primarily interested in the closed-?eld con?guration depicted schematically in Fig. 2. In contrast to the open-?eld con?guration, in which all the ?eld lines piercing the event horizon extend to in?nity, in the closed-?eld con?guration, magnetic ?eld lines connect the black hole to the disk, forming a nested structure of toroidal ?ux surfaces. In this section we will examine the conditions under which this con?guration can exist and, in particular, will discuss the limitations that the rotation of the black hole imposes on the radial extent of the force-free magnetic link between the disk and the hole. First, we would like to point out that a magnetically-linked black hole–disk system is dramatically di?erent from a magnetically-linked star–disk system in at least one important aspect. Indeed, let us examine the system’s evolution on the shortest relevant, i.e., rotation, timescale. In the case where the central object is a highly-conducting star, such as a neutron star or a young star, it turns out that no steady state con?guration with the topology similar to that presented in Figure 2 is possible. This is because both the disk and the star can be regarded (on this short timescale) as perfect conductors, so that the footpoints of the ?eld lines that link the two are frozen into their surfaces. Hence, the disk footpoint of a given ?eld line rotates with its corresponding Keplerian rotation rate, ?K (r), whereas the footpoint of the same ?eld line on the star’s surface rotates with the stellar angular velocity ?? . Therefore, each ?eld line connecting the star to the disk [with the exception of a single line connecting to the disk at the corotation radius rco where ?(rco ) = ?? ] is subject to a continuous twisting. This twisting results in the generation of toroidal magnetic ?ux out of the poloidal ?ux, which tends to in?ate and even open the magnetospheric ?ux surfaces after only a fraction of one di?erential star–disk rotation period (e.g., van Ballegooijen 1994; Uzdensky et al. 2002; Uzdensky 2002a,b). On the other hand, in the case of a black hole being the central object the situation is di?erent. The key di?erence is that, unlike stars, black holes do not have a conducting surface. On the contrary, they are actually e?ectively quite resistive, in the language of the Membrane Paradigm (see Znajek 1978; Damour 1978; MacDonald & Suen 1985; Thorne et al. 1986). The rather large e?ective resistivity makes it in principle possible for the ?eld lines frozen into a rotating conducting disk to slip through the event horizon. This fact makes a quest for a stationary closed-?eld con?guration in the black-hole case a reasonable scienti?c task, since it is at least conceivable that such con?gurations may in principle exist. In our previous paper (Uzdensky 2004) we studied exactly this question for the case of a

– 12 – Schwarzschild black hole. We found that a stationary force-free con?guration of the type depicted in Figure 2 indeed exists in this case. At the same time, however, there is of course no guarantee that a similar con?guration will exist in the Kerr case. This is because the nonlinear terms in the Grad–Shafranov equation that correspond to ?eld-line rotation and toroidal ?eld pressure are no longer small in the Kerr case, whereas in the Schwarzschild case these terms, although formally ?nite, were only at a few per cent level. In fact, we can make an even stronger statement: even for a slowly-rotating Kerr black hole, a force-free con?guration in which magnetic ?eld connects the polar region of the horizon to arbitrarily large distances on the disk (which is precisely the geometry depicted in Fig. 2) does not exist! We shall now present the basic physical argument for why this must be the case. Let us suppose that a force-free con?guration of Figure 2, where all the ?eld lines attached to the disk at all radii thread the event horizon, does indeed exist. First, let us consider the polar region of the black hole, r = rH , θ → 0. Suppose that near the rotation axis the ?ux Ψ0 (θ) behaves as a power law: Ψ0 ? θγ (the most natural behavior corresponding to a constant poloidal ?eld being Ψ0 ? θ2 ). Then note that in a con?guration under consideration, the ?eld lines threading this polar region connect to the disk at some very large radius r0 (Ψ) ? rH . Since the ?eld lines rotate with the Keplerian angular velocity of their footpoints on the ?3/2 disk, ?F (Ψ) ? r0 (Ψ) → 0 as Ψ → 0, one ?nds that, for su?ciently small Ψ [and hence su?ciently large r0 (Ψ)], ?F (Ψ) becomes much smaller than the black hole rotation rate ?H = a/2rH . Now let us look at the event horizon regularity condition (34). For the ?eld lines under consideration, we ?nd that sin θdΨ0 /dθ ? θθγ?1 ? Ψ and δ? = ?F (Ψ) ? ?H ? ??H = const = 0, Ψ → 0. Thus, I(Ψ) ? ??H Ψ ? ?aΨ, and, correspondingly, II ′ (Ψ → 0) ? a2 Ψ . (36) as Ψ → 0 , (35)

Now, let us look at the force-free balance on the same ?eld lines but far away from the black hole, at radial distances of the order of r ? r0 ? rH . At these large distances α ≈ 1 and δ?? ? c, so that the electric terms in the Grad–Shafranov equation are small and the coe?cient D is close to 1. Then the LHS of the Grad–Shafranov equation (27) is essentially a linear di?usion-like operator and can be estimated as being of the order of Ψ/r 2 . We see that both the LHS and the RHS given by equation (27) scale linearly with Ψ but the LHS has an additional factor ? r ?2 . Thus we conclude that at su?ciently large distances this term becomes negligible when compared with the II ′ (Ψ)-term (36). In other words, the toroidal ?eld, produced in the polar region of the horizon by the black hole dragging the ?eld lines along, turns out to be too strong to be con?ned by the poloidal ?eld tension

– 13 – at large distances. In fact, this argument suggests that the maximal radial extent rmax of the region on the disk connected to the polar region of a Kerr black hole should scale as rmax ? rH /a in the limit a → 0. One should note that, in the Schwarzschild limit a → 0, this maximal distance goes to in?nity and hence a fully-closed force-free con?guration can exist at arbitrarily large distances, in agreement with the conclusions of our paper I. [Also note that if one tries to perform a similar analysis for the Schwarzschild case, then from the ?3/2 horizon regularity condition one ?nds that I(Ψ) = ?K (Ψ) sin θ(dΨ/dθ)|r=rH ? Ψ · r0 (Ψ). Then, assuming that r0 (Ψ) is a power law at large distances, r ? r0 (Ψ), the toroidal-?eld ?3 pressure term can be estimated as II ′ (Ψ) ? Ψ · r0 (Ψ). We can thus see that at large distances this term becomes negligible compared with the LHS (? Ψr ?2 ), so no limitation on the radial extent of the magnetic link can be derived.] We also would like to remark that this ?nding is not really surprising in view of some important properties axisymmetric force-free magnetospheres, known from the general theory of the (non-relativistic) Grad–Shafranov equation (see, e.g., van Ballegooijen 1994; Uzdensky 2002b). This analogy is so important that we would like to make a digression to describe it here. Let us consider a closed simply-connected (i.e., without magnetic islands) axisymmetric con?guration like the one shown in Figure 2. Then start to increase gradually the overall magnitude (which we shall call λ) of the nonlinear source term II ′ (Ψ) — the so-called generating function — starting from zero. As we are doing this, let’s keep the functional shape of I(Ψ), as well as the boundary conditions for Ψ, ?xed. Then one ?nds the following interesting behavior: there is a certain maximal value λmax (whose exact value depends on the details of the functional shape of I(Ψ) and the boundary conditions), such that one ?nds no solutions of the Grad–Shafranov equation for λ > λmax . For λ < λmax , one actually ?nds two solutions and these two solutions correspond to two di?erent values of the ?eld-line twist angles ?Φ(Ψ). In the limit λ ? λmax the two solutions are remarkably di?erent. One of them corresponds to ?Φ ? λ/λmax ? 1; it is very close to the purely potential closed?eld con?guration and can be obtained as a perturbation from the potential solution. The other solution corresponds to some ?nite distribution ?Φc (Ψ), in general of order 1 radian, and is characterized by very strongly in?ated poloidal ?eld lines. This con?guration in fact approaches the open-?eld geometry in the limit λ → 0. Now, as one increases λ, the difference between the two solutions decreases and they in fact merge into one single solution at λ = λmax . The corresponding con?guration shows some modest in?ation of the poloidal ?eld and corresponds to the ?eld line twist angles that are ?nite (i.e., of order 1 radian) but less than ?Φc (Ψ). Most importantly, as we mentioned above, no solutions with the required simple topology (i.e., without magnetic islands) exist for λ > λmax . Clearly, this is exactly what happens in the Kerr black hole case. Indeed, in this case the regularity condition (34) requires that the generating function II ′ (Ψ) be of the order of

– 14 – a2 Ψ for small Ψ. In a certain sense, the spin parameter a2 e?ectively plays the role of the parameter λ from our example above. If one considers a con?guration in which the magnetic link extends to a radius rmax on the disk and ?xes the disk boundary conditions, it turns out that there is a critical maximum value a2 beyond which no solution can be found. From max the argument presented in the beginning of this section we expect that amax scale inversely with rmax ; in particular, for an in?nitely extended link (rmax → ∞), one ?nds amax → 0 and no solution is found for any a > 0! To sum up, even though the ?eld lines can, to a certain degree, slip through the horizon because the latter is essentially resistive, in some situations the horizon is not resistive enough to ensure the existence of a steady force-free con?guration! Indeed, the ?eld lines are ”dragged” by the rotating black hole to such a degree that, in order for them to slip through the horizon steadily, they must have a certain rather large toroidal component. When, for ?xed disk boundary conditions, the black-hole spin parameter a is increased beyond a certain limit amax (rmax ), this toroidal ?eld becomes so large that the poloidal ?eld tension is no longer able to contain its pressure at large distances. Finally, the argument put forward in this section proves that it may be not only interesting but also in fact necessary to consider hybrid con?gurations in which at least a portion of the ?eld lines are open and magnetic disk-hole coupling plays a more limited role.

4.

Numerical Simulations

In order to verify the proposition put forward in the preceding section and to study the magnetically-coupled disk–hole magnetosphere, we have performed a series of numerical calculations. We obtained the solutions of the force-free Grad–Shafranov equation corresponding to various values of two parameters: the black-hole spin parameter a and the radial extent Rs of the magnetic link. In this section we shall describe the actual computational set-up of the problem, including the boundary conditions and the numerical procedure; we shall also present the main results of our calculations.

4.1.

Problem formulation and boundary conditions

We start by describing the basic problem set-up and the boundary conditions. The simplest axisymmetric closed-?eld con?guration one could consider is that shown in Figure 2. In this con?guration, all magnetic ?eld lines connect the disk and the hole. Fur-

– 15 – thermore, the entire event horizon and the entire disk surface participate in this magnetic linkage; in particular, the ?eld lines threading the horizon very close to the axis θ = 0 are anchored at some very large radial distances in the disk: Ψ0 (θ → 0) ≡ Ψ(r = rH , θ → 0) = Ψdisk(r → ∞) ≡ Ψ(r → ∞, θ = π/2) (37)

However, as follows from the arguments presented in § 3, a steady-state force-free con?guration of this type can only exist in the case of a Schwarzschild black hole; in the case of a Kerr black hole, even a slowly-rotating one (a ? M), such a con?guration is not possible. And indeed, in complete agreement with this point of view, in our simulations, we were not able to obtain a convergent solution even for a Kerr black hole with the spin parameter as small as a = 0.05. Also in §3 we proposed a conjecture that, for a given value of a, the magnetic link between the polar region of the black hole and the disk cannot, generically, extend to distances on the disk larger than a certain rmax (a). The exact value of rmax depends on the details of the problem, such as the exact ?ux distribution Ψd (r) on the surface of the disk, etc. However, we proposed that rmax is a monotonically decreasing function of a, and, more speci?cally, in the limit a → 0, rmax is inversely proportional to a. For a ?nite ratio a/M = O(1), we expect that the magnetic link can only be sustained over a ?nite range of radii not much larger than the radius of the Innermost Stable Circular Orbit rISCO . In order to test these propositions, we set up a series of numerical calculations aimed at solving the Grad–Shafranov equation for various values of two parameters: the black-hole spin parameter a and the radial extent of the magnetic coupling on the disk surface Rs . Correspondingly, in order to investigate the dependence on the radial extent of magnetic coupling, we modi?ed the basic geometry of the con?guration by allowing for two topologicallydistinct regions: region of closed ?eld lines connecting the black hole to the inner part of the disk r < Rs , and the region of open ?eld lines extending from the outer part of the disk all the way to in?nity.2 This con?guration is shown in Figure 3. We count the poloidal ?ux on the disk from the radial in?nity inward, so that Ψd (r = ∞) = 0, and Ψd (r = rH ) = Ψtot . The disk ?ux distribution may still be the same as in the con?guration of Figure 2; however, now there is a critical ?eld line Ψs ≡ Ψd (Rs ) < Ψtot that acts as a separatrix between open
In general, open ?eld lines originating from the disk may carry a magnetocentrifugal wind (Blandford & Payne 1982) and the resulting mass-loading may make a full-MHD treatment necessary for these ?eld lines. Here, however, we shall ignore this complication and will assume the force-free approach to be valid in this part of the magnetosphere as well.
2

– 16 – ?eld lines (Ψ < Ψs ) and closed ?eld lines (Ψs < Ψ < Ψtot ) connecting to the black hole. Correspondingly, the poloidal ?ux on the black hole surface varies from Ψ = Ψs at the pole θ = 0 to Ψ = Ψtot at the equator θ = π/2. It is worth noting that a more general con?guration would also have some open ?eld lines connecting the polar region of the black hole to in?nity. In fact, such a con?guration would be more physically interesting because these open ?eld lines would enable an additional extraction of the black hole’s rotational energy via the Blandford–Znajek mechanism (BZ77). We shall call this a hybrid con?guration because the disk–hole magnetic coupling and the Blandford–Znajek mechanism operate simultaneously. In the present paper, however, we shall assume that no such hole–in?nity open ?eld lines. We make this choice not because of any physical reasons but simply because of technical convenience: we want to isolate the e?ect tof disk-hole coupling. In addition, as we discuss in more detail in § 6, a proper treatment of these open ?eld lines would require a more complicated numerical procedure than that needed for the ?eld lines that connect to the conducting disk. In addition to boundary conditions, one has to specify the angular velocity ?F (Ψ) of the magnetic ?eld lines. Since we assume that the disk is a perfect conductor, and since in our ?eld con?guration all the ?eld lines go through the disk, this angular velocity is equal to that of the matter in the disk. Now let us consider the open ?eld lines Ψ < Ψs . In principle, since they are attached to a rotating Keplerian disk, these lines rotate di?erentially with the angular velocity ?F (Ψ) = ?K [r0 (Ψ)]. Correspondingly, just as the closed ?eld lines going into the black hole or the open ?eld lines in a pulsar magnetosphere, they have to cross a light cylinder and therefore have to carry poloidal current I(Ψ), whose value must be consistent with, and indeed determined by, the regularity condition at the light cylinder. Because this outer light cylinder is very distinct from the inner light cylinder that is crossed by the closed ?eld lines entering the event horizon, we in general would expect the function I(Ψ < Ψs ) be very di?erent from open the function I(Ψ > Ψs ). In particular, we would expect a discontinuous behavior, Is ≡ closed ≡ lim I(Ψ > Ψs ), even though the ?eld-line angular velocity lim I(Ψ < Ψs ) = Is
Ψ→Ψs Ψ→Ψs

?F = ?K [r0 (Ψ)] remains perfectly continuous and smooth at Ψ = Ψs .

Dealing with such a discontinuity in I(Ψ) across the separatrix Ψ = Ψs presents certain numerical di?culties, especially taking into account that the location of the separatrix rs (θ) = r(Ψ = Ψs , θ) is not known a priori. Therefore, in the present study we decided to simplify the problem by introducing the following modi?cations: we require that the outer part of the disk, r > Rs , be nonrotating: ?F (Ψ < Ψs ) ≡ 0.

– 17 – Correspondingly, the open ?eld lines do not cross an outer light cylinder, and so I(Ψ < Ψs ) ≡ 0. To put it in other words, we just take the open-?eld outer part of the disk magnetosphere to be potential. Next, in order to avoid the numerically-challenging discontinuities in ?F (Ψ) and I(Ψ) at Ψ = Ψs , we slightly modify the disk rotation law just inside of Rs by taking ?F smoothly to zero over a small (compared with the total amount of closed ?ux) poloidal ?ux range. In particular, we used the following prescription: ?F (Ψ) = 0 , Ψ < Ψs , Ψ ? Ψs ), ?Ψ Ψ > Ψs , (38)

?F (Ψ) = ?K [r0 (Ψ)] · tan2 (

where ?Ψ = 0.2(Ψtot ? Ψs ) and (see equation [5.72] of Krolik 1999, p. 117) √ M √ . ?K (r) = r 3/2 + a M

(39)

These modi?cations enabled us to focus on examining how black hole rotation (i.e., the spin parameter a) limits the radial extent Rs of the force-free magnetic coupling, while at the same time avoiding certain numerical di?culties resulting from the discontinuous behavior of poloidal current I(Ψ). We believe that these modi?cations do not lead to any signi?cant qualitative change in our conclusions, especially in the case of small a and large Rs . Nevertheless, we intend in the future to enhance our numerical procedure so that it become fully capable of treating this discontinuity. Let us now describe the computaional domain and the boundary conditions. First, because of the assumed axial symmetry and the symmetry with respect to the equatorial plane, we performed our computations only in one quadrant, described by θ ∈ [0, π/2] and r ∈ [rH , ∞]. Thus, we have four natural boundaries of the domain: the axis θ = 0, the in?nity r = ∞, the equator θ = π/2, and the horizon r = rH . Of these, the axis and the equator require boundary conditions for Ψ, whereas the horizon and the in?nity are actually regular singular surfaces and so we only impose regularity conditions on them. The boundary condition on the rotation axis is particularly simple: Ψ(r, θ = 0) = Ψs = const . (40)

The equatorial boundary, θ = π/2, actually consists of two parts: the disk (considered to be in?nitesimally thin) and the plunging region between the disk and the black hole. The border between them, i.e., the inner edge of the disk, is assumed to be very sharp and to lie

– 18 – at the ISCO: rin = rISCO (a); rISCO varies between rISCO = 6M for a Schwarzschild black hole (a = 0) and rISCO = M for a maximally rotating Kerr black hole (a → 1). Let us ?rst discuss the boundary conditions at the disk surface, r > rin . Depending on the resistive properties of the disk, and on the timescale under consideration, one can choose between two possibilities, both of which appear to be physically sensible: 1) If one is interested in time-scales much longer than the characteristic rotation timescale but much shorter than both the accretion and the magnetic di?usion timescales, then it is reasonable to regard the poloidal ?ux distribution across the disk to be a ?xed prescribed function, which must be speci?ed explicitly. Thus, in this case one adopts a Dirichlet-type disk boundary condition: Ψ(r > rin , θ = π/2) = Ψd (r) (41) The function Ψd (r) is arbitrary; the only requirement that must be imposed in accordance with the discussion above is the convention that Ψd (r = ∞) = 0 and Ψd (rin ) = Ψtot . Since we don’t have any good physical reasons to favor one choice of Ψd (r) over any other, we in this paper just choose it arbitrarily to be a power-law with the exponent equal to ?1: Ψd (r) = Ψtot rin . r (42)

2) If one looks for a con?guration that is stationary on timescales much longer than the e?ective magnetic di?usion time (while perhaps still much shorter than the accretion time scale), then one should regard the disk as e?ectively very resistive for the purposes of specifying the disk boundary condition. This situation may arise in the case of a turbulent disk; for such a disk, the e?ective magnetic di?usivity η can probably be estimated as ηturb = αSS cs h, in the spirit of the α-prescription for the e?ective viscosity in the SS73 model. Then, the characteristic radial velocity of the magnetic footpoints across the disk is roughly vfp ? αSS cs (Br /Bz )d . For the ratio (Br /Bz )d of order 1, this velocity is much greater (by a factor of r/h) than the characteristic accretion velocity. Therefore, the only way one can have a steady-state con?guration on the di?usion time-scale (which, according to the above estimate is of the order of the disk sound crossing time r/cs ) is for the poloidal ?eld to be nearly perpendicular to the disk, Br ? Bz . This requirement translates into a simple von-Neumann boundary condition for Ψ(r, θ) at the disk surface: π ?Ψ (r, θ = ) = 0 . ?θ 2 (43)

In our present paper, however, we chose the Dirichlet-type boundary condition represented by equations (41)–(42) and set Ψtot = 1 throughout the paper.

– 19 – In the plunging region (rH ≤ r ≤ rin , θ = π/2) we have chose Ψ(rH ≤ r ≤ rin , θ = π/2) = Ψtot ≡ Ψd (rin ) = const . (44)

This choice appears to be physically appropriate for an accreting (and not just rotating) disk. The reason for this is that the matter in this region falls rapidly onto the black hole and thereby stretches the magnetic loops in the radial as well as the azimuthal directions, greatly reducing the strength of the vertical ?eld component. The horizontal magnetic ?eld then reverses across the plunging region, which is thus described as an in?nitesimally thin non-force-free current sheet lying along the equator. In essense, this situation is directly analogous to the case of a force-free pulsar magnetosphere, where all the ?eld lines crossing the outer light cylinder have to be open and extend out to in?nity, thus forming an equtorial current sheet (Beskin 2003; van Putten & Levinson 2003). In the black-hole case, one could still consider an alternative picture of the plunging region with some ?eld lines crossing the equator inside rin . However, in this case one would still have to have a non-FFDE equatorial current sheet inside the inner light cylinder, as was shown by Komissarov (2002b, 2004a). Finally, as we have discussed in § 2.3, the event horizon is a regular singular surface of the Grad–Shafranov equation. Correspondingly, one cannot and need not impose an additional arbitrary boundary condition here (e.g., Beskin & Kuznetsova 2000; Komissarov 2002b, 2004a). Instead, one imposes the regularity condition (34); this condition has the form of an ordinary di?erential equation (ODE) that determines the function Ψ0 (θ) provided that both ?F (Ψ) and I(Ψ) are given. Thus, from the procedural point of view, this condition can be used as a Dirichlet boundary condition on the horizon. It is important to acknowledge, however, that one does not have the freedom of specifying an arbitrary function Ψ(θ) and then studying how the information contained in this function propagates outward and a?ects the solution away from the horizon. The function Ψ0 (θ) is uniquely determined once ?F (Ψ) and I(Ψ) are given and thus there is no causality violation here. Similarly, the spatial in?nity r = ∞ is also a regular singular surface of the Grad–Shafranov equation and thus can also be described by a regularity condition. In this sense, the horizon and the in?nity are equivalent (e.g., Punsly & Coroniti 1990). Note that, in our particular problem set-up, the sitiuation at in?nity is greatly simpli?ed because we have set ?F (Ψ) = 0 on the open ?eld lines extending from the disk. Because of this, there is no outer light cylinder for these lines to cross and thus one can also set I(Ψ < Ψs ) = 0. Then, at very large distances (r ? rH ), the Grad–Shafranov equation (27) becomes a very simple linear equation: Ψrr + r ?2 sin θ?θ (Ψθ / sin θ) = 0, and the asymptotic solution that corresponds to the open ?eld geometry with ?nite magnetic ?ux is just Ψ(r = ∞, θ) = Ψs cos θ . (45)

– 20 – 4.2. Light-Cylinder Regularity Condition

At this point the problem is almost completely determined. The only thing we still have to specify is the poloidal current I(Ψ). Unlike ?F (Ψ), which was determined from the frozen-in condition on the disk surface, the function I(Ψ) cannot be explicitly prescribed as an arbitrary function on any given surface. Instead, it must be somehow determined self-consistently together with the solution Ψ(r, θ) itself. This means that there must be one more condition that we have not yet used. And indeed, this additional condition is readily found — it is the (inner) light-cylinder regularity condition. Let us look at it more closely. As can be easily seen from the Grad–Shafranov equation (27), the light cylinder, de?ned as the surface where D = 0 ? α = αLC = |δ?|? , (46) is a singular surface, because the coe?cients in front of both the r- and θ- second-order derivatives of Ψ vanish there. Physically speaking, the light cylinder is the surface where the locally-measured rotational velocity of the magnetic ?eld lines with respect to the ZAMOs is equal to the speed of light, vB,φ = c, and where E = Bpol in the ZAMO frame. In general relativity there are two light cylinders, the inner one and the outer one. The outer light cylinder is just a direct analog of the pulsar light cylinder; it is crossed by rotating ?eld lines that are open and extend to in?nity. In our problem, we are interested in the closed ?eld lines, i.e., those reaching the event horizon. These ?eld lines cross the so-called inner light cylinder, whose existence is a purely general-relativistic e?ect, ?rst noticed by Znajek (1977) and by BZ77. Because the inner light cylinder is a singular surface of equation (27), in general this equation admits solutions that are not continuous or continuously di?erentiable at the light cylinder. Such solutions, while admissible mathematically, are not physically possible. Thus, we supplement our mathematical problem by an additional physical requirement that the solution be continuous and smooth across the light cylinder surface. In particular, this means that the 1st and 2nd derivatives of Ψ must be ?nite there. Correspondingly, one can just drop all the terms proportional to D when applying equation (27) at the light cylinder and keep only the terms involving the derivatives of D. The result can be formulated as an expression that determines the function I(Ψ), namely: ?II ′ (Ψ) = ? 1 Ψ (? D)|LC + 2 Ψθ (?θ D)|LC + δ??′F ? 2 (?Ψ)2 |LC , 2 r r ρ ρ (47)

where Ψ, r, and θ are taken at the light cylinder: Ψ = ΨLC (θ) = Ψ[rLC (θ), θ] , (48)

– 21 – and the function rLC (θ) —the shape of the light cylinder surface — is determined implicitly by equation (46). This approach was ?rst used successfully at the outer light cylinder by Contopoulos et al. (1999) in the context of pulsar magnetspheres. In the black hole problem, it was ?rst used by Uzdensky (2004) for the Schwarzschild case. Let us now discuss how one can use the light cylinder regularity condition (47) to determine I(Ψ) in practice. Conceptually, one can think of this condition as follows. Suppose one starts by ?xing all the other boundary and regularity conditions in the problem [including the choice of ?F (Ψ)]. Then, for an arbitrarily chosen function I(Ψ), one can regard the condition (47) as a mixed-type, Dirichlet-Neumann boundary condition because it can be viewed as a quadratic algebraic equation for, say, the ?rst radial derivative. Thus, if I(Ψ) is given, one can express Ψr |LC in terms of ΨLC and Ψθ |LC . Next, one applies this condition separately on each side of the light cylinder and gets a complete, well-de?ned problem in each of the two regions separated by the light cylinder. Then, one can obtain a solution in each of these regions. Because of the use of the regularity condition (47), each of the two solutions is going to be regular near the light cylinder. In general, however, these solutions are not going to match each other at r = rLC (θ) and the mismatch ?ΨLC (θ) will depend on the original choice of the function I(Ψ). This observation suggests a method for selecting a unique function I(Ψ): one can devise a procedure in which one iterates with respect to I(Ψ) until ?ΨLC becomes zero. The corresponding function I(Ψ) is then declared the correct one: only with this choice of I(Ψ) the solution Ψ(r, θ) passes smoothly through the light cylinder. The above method for determining I(Ψ) is conceptually illuminating and can be easily implemented in simple cases. For example, in the case of a uniformly-rotating pulsar magnetosphere, two important simpli?cations take place. First, the location of the light cylinder is known a priori, rLC (θ) = c/? = const, and hence one can choose a computational grid that is most suitable for dealing with the light cylinder (e.g., cylindrical polar coordinates with some gridpoints lying on the cylinder). Second, because ?F = const, the terms quadratic in the derivatives of Ψ disappear, and the task of resolving equation (47) with respect to the derivative normal to the light cylinder becomes trivial. These simpli?cations make the procedure described above very practical and it was in fact used successfully by Contopoulos et al. (1999) (and repeated later by Ogura & Kojima 2003) to obtain a unique solution for an axisymmetric pulsar magnetosphere that was smooth across the outer light cylinder. In the problem considered in this paper, however, the situation is much more complicated. In particular, the light cylinder’s position and shape are not known a priori; instead, they need to be determined self-consistently as part of the solution. Also, equation (47) is, in general, quadratic with respect to ?r Ψ, and hence one has to deal with the problem of the existence of its solutions and with the task of selecting only one of them. Because of this overall

– 22 – complexity, we decided against using this procedure in our calculations. Instead, we chose a much simpler and more straight-forward method: we used equation (47) to determine I(Ψ) [or, rather, the combination II ′ (Ψ) that is actually needed for further computations] directly, by explicitly interpolating all the terms on the right-hand-side of equation (47). We will describe this in more detail in the next section.

4.3.

Numerical procedure

We performed our calculations in the domain {r ∈ [rH , ∞], θ ∈ [0, π/2]} on a grid that was uniform in θ and in the variable x ≡ rH /r (which enabled us to extend the computational domain to in?nity). The highest resolution used was 60 gridzones in the θ-direction and 200 gridzones in the radial (x) direction. To solve the elliptic Grad–Shafranov equation (27), we employed a relaxation procedure similar to the one employed by Uzdensky et al. (2002). In this procedure, we introduced arti?cial time variable t and evolved the ?ux function according to the parabolic equation ?Ψ = ±f (r, θ)(LHS ? RHS) , ?t (49)

where LHS and RHS are the left- and right-hand sides of the Grad–Shafranov equation (27), respectively, and the factor f (r, θ) was an arti?cial multiplier introduced in order to accelerate convergence in regions where the di?usion coe?cients in the x and θ directions are small (e.g., very far away or very close to the horizon). The sign in front of f (r, θ) was chosen according to the sign of the di?usion coe?cient in equation (27): it was plus outside the light cylinder (where D > 0) and minus inside (where D < 0). It is clear that any steadystate con?guration achieved as a result of this evolution is a solution of the Grad-Shafranov equation (27). Here we would like to draw attention to the following non-trivial problem. During the relaxational evolution described by equation (49), the light cylinder generally moves across the grid and, from time to time inevitably gets close to some of the gridpoints. This leads to the danger, ?rst noted by Macdonald (1984), that some gridpoints will oscillate between the two sides of the light cylinder. Indeed, suppose that a given gridpoint P is initially on the outer side of the light cylinder (DP > 0), and so ?ΨP /?t is determined by equation (49) with the plus sign. Let us suppose that the resulting evolution of ΨP is such that DP decreases. The, after some time one may ?nd that DP has become negative; correspondingly, at the next timestep one uses equation (49) with the minus sign and so ΨP starts to evolve in the opposite direction. Because the value of D at a ?xed spatial point P is, locally, a smooth monotonic function of ΨP , it now starts to increase and may become positive again in one or two

– 23 – timesteps. This leads to rapid small-amplitude oscillations of the light cylinder around some gridpoints, instead of a smooth large-scale motion associated with the iteration process. As a result, the light cylinder gets ”stuck” on these gridpoints and the function rLC (θ) becomes a series of steps and plateaus instead of a smooth curve. A simple and e?cient way to avoid this problem turned out to be to update the function D(r, θ) not at every timestep but rather very infrequently. Although it caused some delay in the convergence of the relaxation process, this modi?cation has worked very well in practice, enabling the light cylinder to move freely across the grid and to achieve its ultimate smooth shape. To implement our relaxation procedure numerically, we used an explicit ?nite-di?erence scheme with 1st-order accurate time derivative and centered 2nd-order accurate spatial derivatives. It is also worth mentioning that writing out the full-derivative terms such as ?r (Ψr D?/ρ2 ) as Ψr ?r (D?/ρ2 ) + (D?/ρ2 )Ψrr , etc., and then evaluating them on the grid actually worked better than evaluating these full derivatives directly as they are. The initial condition—the starting point of our relaxation process—was prescribed explicitly as Ψ(t = 0, r, θ) = Ψs + [Ψd (r) ? Ψs ] (1 ? cos θ) . (50)

Also, we found it useful to use cubic-spline interpolation of functions I(Ψ) and ?F (Ψ) to avoid some small-scale rapid oscillations of the solution. Finally, let us describe the particular numerical implementation of the procedure that was used to determine the poloidal current I(Ψ) in our code. As we mentioned at the end of the previous section, we used equation (47) explicitly to determine II ′ (Ψ) by interpolating all the terms on the right-hand side of that equation at the light cylinder. Because the light cylinder surface is roughly spherical, it was convenient to represent I(Ψ) by a tabular function speci?ed on a one-dimensional array {Ψj } of the values of ΨLC at the radial rays LC θ = θj = jhθ , where hθ is the grid-spacing in θ. Along each of these rays, we ?rst had to locate the pair of radially-adjacent gridpoints between which the light cylinder lay. Then we used an interpolation of D(r, θ) to determine the position r = rLC (θj ) of the light cylinder more precisely and to obtain Ψj = ΨLC (θj ), as well as the values of the derivatives Ψr , LC Ψθ , Dr , and Dθ at the light cylinder for each of the rays. Finally, we used (47) to compute the value of II ′ (Ψ) at each Ψj . This is actually not as trivial as it may seem, because the LC condition (47) in such an approach was enforced at all times during the relaxation procedure that determined Ψ(r, θ), whereas the Grad–Shafranov equation itself was satis?ed only after convergence had been reached. Therefore, one had to exercise extra care, for example, in deciding how often I(Ψ) needs to be updated. We found that it was necessary to update I(Ψ) only fairly infrequently during our relaxation procedure.

– 24 – 4.4. Results

The single most important result of the present study is presented in Figure 4. This ?gure shows where in the two-dimensional (a, Ψs ) parameter space force-free solutions exist and where they do not. Filled circles on this plot represent the runs in which convergence was achieved (allowed region), whereas open circles correspond to the runs that failed to converge to a suitable solution (forbidden region). The boundary amax (Ψs ) between the allowed and forbidden regions is located somewhere inside the narrow hatched band that runs from the lower left to the upper right of the Figure [the ?nite width of the band represents the uncertainty in amax (Ψs ) due to a limited number of runs]. As we can see, amax (Ψs ) is a monotonically increasing function. In particular, in the limit Ψs → 0, amax indeed scales linearly with Ψs and hence is inversely proportional to Rs = rin Ψtot /Ψs , in full agreement with our expectations presented in § 3. However, this linear dependence no longer holds for ?nite values of Ψs (and hence of amax ). In order to study the e?ect that black hole spin has on the solutions, we concentrate on several values of a for a ?xed value of Ψs . In particular, we choose Ψs = 0.5 [that corresponds to Rs = 2rin (a)] and considered four values of‘a: a = 0, a = 0.25, a = 0.5, and a = 0.7. Thus, Figure 5 shows the contour plots of the poloidal magnetic ?ux for these four cases. We see that the ?ux surfaces in?ate somewhat with increased a, but this expansion is not very dramatic, even in the case a = 0.7, which is very close to the critical value amax (Ψs = 0.5) that corresponds to a sudden loss of equilibrium. We note that this ?nding is completely in line with our discussion in § 3. The next three Figures present the plots of three important functions that characterize the solutions. In each Figure there are four curves corresponding to our selected values a = 0, 0.25, 0.5, and 0.7 for Ψs = 0.5. Figure 6 shows the event-horizon ?ux distribution Ψ0 (θ); Figure 7 shows the poloidal-current function I(Ψ); and Figure 8 shows the position of the inner light cylinder described in terms of the lapse function αLC (θ). In Figure 6 we also plot Ψ0 (θ) corresponding to the simple split-monopole solution with uniform radial ?eld at the horizon. Note that on the horizon we have ? = 0 and hence 2 Σ = rH + a2 = 2MrH ; therefore Br = ? 1 1 1 1 Ψθ = Ψθ = Ψθ , ?ρ Σ sin θ 2MrH sin θ
(0) (0)

r = rH .

(51)

Thus, Br (θ) = const corresponds to Ψ0 (θ) = Ψs + (Ψtot ? Ψs )(1 ? cos θ), independent of a. ? This function is plotted in Figure 6 (the dashed line) for comparison with the actual solutions. (0) We see that the deviation from Ψ0 (θ) becomes noticeable only when a approaches 1.

– 25 – Figure 8 shows αLC (θ). An interesting feature here is that the light cylinder reaches the event horizon at some intermediate angle 0 < θco < π/2 for small values of a. This is because, when a < 0.359.. M, the inner edge of a Keplerian disk rotates faster than the black hole; correspondingly, somewhere in the disk there exists a corotation point rco > rin such that ?K (rco ) = ?H . The ?eld line Ψco threading the disk at this point corotates with the black hole. Therefore, at the point θco where this line intersects the horizon, we have δ? = 0, and so this point (r = rH , θ = θco ) has to lie on the light cylinder. The location θco of this point moves towards the equator when a is increased and reaches it at a = 0.359.. M. For larger values of a, the entire disk outside of the ISCO rotates slower than the black hole and the light cylinder touches the horizon only at the pole θ = 0. Finally, we also computed all the electric and magnetic ?eld components and checked that E 2 < B 2 everywhere outside the horizon.

5.

Astrophysical Implications/Consequences

In this section we’ll discuss the exchange of energy and angular momentum between the black hole and the disk. Apart from the question of existence of solutions, this issue is one of the most important for actual astrophysical applications. Fortunately, once a particular solution describing the magnetosphere is obtained, computing the energy and angular momentum transported by the magnetic ?eld becomes very simple. Indeed, according to MT82, angular momentum and red-shifted energy (i.e., ”energy at in?nity”) are transported along the poloidal ?eld lines through the force-free magnetosphere without losses. Thus, the amount of angular momentum ?L transported out in a unit of global time t through a region between two neighboring poloidal ?ux surfaces, Ψ and Ψ+?Ψ, as given by equation (7.6) of MT82 (modi?ed to suit our choice of notation), is d?L 1 = ? I?Ψ , dt 2 (52)

and the red-shifted power—?ux of redshifted energy per unit global time t— is expressed as 1 ?P = ? ?F I ?Ψ , 2 (see eq. [7.8] of MT82). Then, taking into account the contributions from both hemispheres and both sides of the disk, we can compute the total magnetic torque exerted by the hole onto the disk per unit t (53)

– 26 – as dL =? dt

Ψtot

I(Ψ)dΨ ,
Ψs

(54)

and, correspondingly, the total red-shifted power transferred from the hole onto the disk via Poynting ?ux is
Ψtot

P =?
Ψs

?F (Ψ)I(Ψ)dΨ .

(55)

Next, since in our problem we have an explicit mapping (42) between Ψ and the radial coordinate r on the disk, we can immediately write down expressions for the radial distributions of angular momentum and red-shifted energy deposited on the disk per unit global time: dΨd d?L(r) = ?I[Ψd (r)] dr , dt dr and ?P (r) = ??K (r) I[Ψd(r)] dΨd dr . dr (56) (57)

Figures 9 and 10 show these distributions for our selected cases a = 0.25, 0.5, and 0.7 for ?xed Ψs = 0.5. We see that in the case a = 0.25 there is a corotation point rco on the disk such that ?disk > ?H inside rco and ?disk < ?H outside rco . Correspondingly, both angular memontum and red-shifted energy ?ow from the inner (r < rco ) part of the disk to the black hole and from the hole to the outer (r > rco ) part of the disk. At larger values of a, however, the Keplerian angular velocity at r = rin is smaller than the black hole’s rotation rate and there is no corotation point; correspondingly, both angular momentum and redshifted energy ?ow from the hole to the disk. Also, as can be seen in Figures 9 and 10, the deposition of these quantities becomes strongly concentrated near the disk’s edge, especially at higher values of a. Next, Figures 11 and 12 demonstrate the dependence of the total integrated angular momentum and red-shifted energy ?uxes (54)–(55) on the black hole spin a for a ?xed value of Ψs = 0.5. We see that both quantities are negative at small values of a (meaning a transfer from the disk to the hole,) but then increase and become positive at larger a. The angular momentum transfer rate depends roughly linearly on a, whereas the red-shifted power P (a) grows even faster, especially at large values of a. It is also interesting to note that the two quantities go through zero at slightly di?erent values of the spin parameter: dL/dt becomes zero at a ≈ 0.23, while P = 0 at a ≈ 0.26. This means that it possible to have the total angular momentum ?ow from the hole to the disk and the total power ?ow from the disk to the hole at the same time.

– 27 – 6. Conclusions

In this paper we investigated the structure and the conditions for the existence of a force-free magnetosphere linking a rotating Kerr black hole to its accretion disk. We assumed that the magnetosphere is stationary, axisymmetric, and degenerate and that the disk is thin, ideally conducting, and Keplerian and that it is truncated at the Innermost Stable Circular Orbit. Our main goal was to determine under which conditions a force-free magnetic ?eld can connect the hole directly to the disk and how the black hole rotation limits the radial extent of such a link on the disk surface. We ?rst introduced (in § 3) a very simple but robust physical argument that shows that, generally, magnetic ?eld lines connecting the polar region of a spinning black hole to arbitrarily remote regions of the disk cannot be in a force-free equilibrium. The basic reason for this can be described as follows. Since the ?eld lines threading the horizon have to ?rst cross the inner light cylinder, and since they generally rotate at a rate that is di?erent from the rotation rate of the black hole, these ?eld lines have to be bent somewhat. In other words, they develop a toroidal magnetic ?eld component, just like the open ?eld lines crossing the outer light cylinder in a pulsar magnetosphere. In the language of the Membrane Paradigm (see Thorne et al. 1986), this toroidal ?eld is needed so that the ?eld lines could slip resistively across the stretched event horizon. The next step in our argument is to look at those ?eld lines that connect the polar region of the horizon to the disk somewhere far away from the black hole. In a force-free magnetosphere, toroidal ?ux spreads along ?eld lines to keep the poloidal current I ? Bφ α? constant along the ?eld. Then one can show that the ? outward pressure of the toroidal ?eld generated due to the black hole rotation turns out to be so large that it cannot be con?ned by the poloidal ?eld tension at large enough distances. In other words, the ?eld lines under consideration cannot be in a force-free equilibrium. Furthermore, one can generalize this argument to the case of closed magnetospheres of ?nite size and derive a conjecture that the maximal radial extent Rmax of the magnetically-coupled region on the disk surface should scale inversely with the black hole spin parameter a in the limit a → 0. In order to verify this hypothesis and to study the detailed structure of magnetically-coupled disk–hole con?gurations, we have obtained numerical solutions of the general-relativistic force-free Grad–Shafranov equation corresponding to partially-closed ?eld con?gurations (shown in Fig. 3). This is an nonlinear 2nd-order partial di?erential equation for the poloidal ?ux function Ψ(r, θ) and it is the main equation governing the system’s behavior. An additional complication in this problem arises from the need to specify two free functions that enter the force-free Grad–Shafranov equation; these are the ?eld-line angular velocity

– 28 – ?F (Ψ) and the poloidal current I(Ψ). Because all the ?eld lines are assumed to be frozen into the disk, the ?rst of this functions is determined in a fairly straightforward way. Namely, for any given ?eld line Ψ, ?F (Ψ) is just the Keplerian angular velocity at this line’s footpoint on the disk. Specifying the poloidal current, on the other hand, is a much more di?cult and nontrivial task. The reason for this is that it cannot be just prescribed explicitly on any given surface and one should look more thoroughly into the mathematical nature of the Grad–Shafranov equation itself to determine I(Ψ). In particular, the most important feature of the Grad–Shafranov equation in this regard is that it becomes singular on two surfaces, the event horizon and the inner light cylinder. This observation is very useful because one can impose a physically-motivated regularity condition at each of these surfaces. One of the most important ideas in our analysis is that one can use the light-cylinder regularity condition to determine, using an iterative procedure, the poloidal current I(Ψ), similar to the way it was done by CKF99 in the context of pulsar magnetospheres. As for the singularity at the event horizon, it is also very important. Basically, it tells us that it is not possible to prescribe an arbitrary boundary condition at the horizon; instead, one can only impose a certain physical condition of regularity there. When combined with the Grad–Shafranov equation itself, this regularity requirement results in a single relationship (historically known as the horizon boundary condition, ?rst derived by Znajek 1977) between three functions: the horizon ?ux distribution Ψ0 (θ), and the two free functions, ?F (Ψ) and I(Ψ) (e.g., Beskin 1997; Beskin & Kuznetsova 2000). What’s important is that there are no other independent relationships that can be speci?ed on this surface. In practical terms, this means that this condition should be used to determine the function Ψ0 (θ) in terms of ?F (Ψ) and I(Ψ), which therefore must be determined outside the horizon. This fact helps to alleviate some of the causality issues raised by Punsly (1989, 2001, 2003) and by Punsly & Coroniti (1990). Since one of the goals of this work was to study the dependence Rmax (a), we performed a series of computations corresponding to various values of two parameters: the black hole spin parameter a and the the magnetic link’s radial extent Rs on the disk surface (the ?eld lines anchored to the disk beyond Rs were taken to be open and non-rotating). At the same time, the disk boundary conditions were kept the same in all these runs, namely, Ψd (r) = Ψtot rin (a)/r. Therefore, varying the value of Rs for ?xed a was equivalent to varying the amount Ψs of open magnetic ?ux threading the disk. Whereas for some pairs of values of a and Ψs we were able to achieve a convergent force-free solution, for others we were not. Thus, as one of the main results of our computations, we were able to chart out the allowed and the forbidden domains in the two-parameter space (a, Ψs ). The boundary between these two domains is a curve amax (Ψs ), which can be

– 29 – easily remapped into the curve Rmax (a). As can be seen in Figure 4, this is a monotonically rising curve with the asymptotic behavior amax ∝ Ψs as Ψs → 0, which is in line with our predictions. We also computed the total angular momentum and red-shifted energy exchanged in a unit of global time t between the hole and the disk through magnetic coupling. We studied the dependence of these quantities on the black hole spin parameter a and found that the angular momentum transfer rate rises roughly linearly with a; it is negative for small a (meaning the angular momentum transfer to the hole) and reverses sign around a ≈ 0.23 (for Ψs = 0.5Ψtot ). The total energy transfer increases with a at an accelerated (i.e., faster than linear) rate, especially at larger values of a; it is also negative at small a, but becomes positive around a = 0.26. This means that there is a narrow range 0.23.. < a < 0.26.. where the integrated angular momentum ?ows from the hole to the disk, whereas the integrated red-shifted energy ?ows in the opposite direction. Finally, we note that, in the case of open or partially-open ?eld con?guration responsible for the Blandford–Znajek process, one has to consider magnetic ?eld lines that extend from the event horizon out to in?nity. Since these ?eld lines are not attached to a heavy in?nitely conducting disk, their angular velocity ?F (Ψ) cannot be explicitly prescribed; it becomes just as undetermined as the poloidal current I(Ψ) they carry. Fortunately, however, these ?eld lines now have to cross two light cylinders (the inner one and the outer one). Since each of these is a singular surface of the Grad–Shafranov equation, one can impose corresponding regularity conditions on these two surfaces. Thus, we propose that one should be able to devise an iterative scheme that uses the two light-cylinder regularity conditions in a coordinated manner to determine the two free functions ?(Ψ) and I(Ψ) simultaneously, as a part of the overall solution process. At the same time, the regularity conditions at the event horizon and at in?nity could be used to obtain the asymptotic poloidal ?ux distributions at r = rH and at r → ∞, respectively. We realise of course that iterating with respect to two functions simultaneously may be a very di?cult task. This purely technical obstacle (in addition to having to deal with the separatrix between the open- and closed-?eld regions) is the primary reason why, in this paper, we have restricted ourselves to a con?guration which has no open ?eld lines extending from the black hole to in?nity. We leave this problem as a topic for future research. It is possible that, instead of solving the Grad–Shafranov equation itself, the easiest and most practical way to achieve a stationary solution will be to use a time-dependent relativistic force-free code, such as one of those being developed now (Komissarov 2001, 2002a, 2004a; MacFadyen & Blandford 2003; Spitkovsky 2004; Krasnopolsky 2004, private communication).

– 30 – I am very grateful to V. Beskin, O. Blaes, R. Blandford, S. Boldyrev, A. K¨nigl, B. C. Low, o M. Lyutikov, A. MacFadyen, V. Pariev, B. Punsly, and A. Spitkovsky for many fruitful and stimulating discussions. I also would like to express my gratitude to the referee of this paper (Serguei Komissarov) for his very useful comments and suggestions that helped improve the paper. This research was supported by the National Science Foundation under Grant No. PHY99-07949.

7.

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– 34 –

B

rin disk black hole
Fig. 1.— Schematic drawing of an open black hole – disk magnetosphere, commonly associated with the Blandford–Znajek (BZ77) process. Rotational energy and angular momentum are extracted from both the black hole and the disk and are transported away by the magnetic ?eld.

– 35 –

B

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

rin disk

black hole

Fig. 2.— Schematic drawing of a fully-closed black hole – disk magnetosphere. Energy and angular momentum are exchanged between the hole and the disk through a direct magnetic link.

– 36 –

sep

arat

open field lines

rix

closed field lines

rin

Rs

Fig. 3.— Schematic drawing of a black hole – disk magnetosphere with a radially-limited magnetic connection. Here, only the inner part of the disk is coupled magnetically to the hole (closed ?eld region), whereas the ?eld lines attached to the outer part of the disk are open and extend to in?nity.

– 37 –

1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0.8 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 forbidden a 0.6 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 0.4 0000000000000000000000000000000000 1111111111111111111111111111111111 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 allowed 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 0.2 0000000000000000000000000000000000 1111111111111111111111111111111111 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 0 0000000000000000000000000000000000 1111111111111111111111111111111111 0 0.1 0.2 0.3 0.4 0.5 0.6
1

0.7

Ψs = Ψ rin / R s tot
Fig. 4.— The 2-D parameter space (a, Ψs ). Filled (black) circles correspond to the runs in which a stationary force-free solution has been obtained, while open (white) circles correspond to the runs that failed to converge to a stationary solution. The shaded band running diagonally across the plot represents the function amax (Rs ), the maximal value of a for which a force-free magnetic link can extend up to a given radial distance Rs on the disk.

– 38 –

a=0.0

a=0.25

a=0.5

a=0.7

Ψopen 0.5 Ψ = total
Fig. 5.— Contour plots of the magnetic ?ux function Ψ(r, θ) for four values of the black hole speci?c angular momentum: a = 0.0, 0.25, 0.5, and 0.7; the amount of open poloidal ?ux in all cases is Ψs = 0.5Ψtot (corresponding to Rs = 2rin).

– 39 –

1

Ψ0(θ)
0.9

Ψs = 0.5

0.8

a=0.7
0.7

a=0.5
0.6

a=0.25 a=0.0
0 0.2 0.4 0.6 0.8 1 1.2

0.5

θ, rad

1.4

1.6

Fig. 6.— The horizon poloidal ?ux distribution Ψ0 (θ) for several values of a and a single value Ψs = 0.5Ψtot of the open magnetic ?ux (corresponding to Rs = 2 rin(a).

– 40 –

0.04

I(Ψ)

0.03

Ψs =0.5
a=0.0 a=0.25
Ψco (a=0.25)

0.02

0.01

0

-0.01

-0.02

a=0.5

-0.03

-0.04

a=0.7
0.5 0.6 0.7 0.8 0.9

-0.05

Ψ

1

Fig. 7.— Poloidal current I as a function of poloidal magnetic ?ux Ψ for several values of a and a single value Ψs = 0.5Ψtot .

– 41 –

0.18

0.16

Ψs =0.5
a=0.5

a=0.7

α LC (θ)

0.14

0.12

0.1

0.08

0.06

a=0.25

a=0.0
θco(a=0.25)

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4

θ, rad

1.6

Fig. 8.— The position of the inner light cylinder represented by the lapse function αLC (θ) for several values of a. The light cylinder touches the horizon at the pole θ = 0 and, for a < 0.36, at the point θ = θco where the corotation ?eld line Ψco ≡ Ψd (rco ) intersects the horizon.

– 42 –

0.012

Ψs = 0.5 a=0.7

dL drdt

2

0.01

0.008

0.006

0.004

0.002

a=0.5
rin rin rin

a=0.25
r co (a=0.25)
6 8 10

0

-0.002 0 2 4

r

12

Fig. 9.— Radial distribution d2 L/drdt of the magnetic torque per unit radius r on the disk surface for a = 0.25, 0.5, and 0.7.

– 43 –

0.0016

Ψs = 0.5 a=0.7

0.0014

dP dr

0.0012

0.001

0.0008

0.0006

0.0004

0.0002

a=0.5
rin rin
0 2

a=0.25
r co (a=0.25)
6 8 10 12

0

rin
4

-0.0002

r

Fig. 10.— Radial distribution dP/dr of the red-shifted power per unit radius r on the disk surface for a = 0.25, 0.5, and 0.7.

– 44 –

0.018 0.016

Ψs = 0.5

L

0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 -0.002 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

a
Fig. 11.— The dependence of the total magnetic torque between the black hole and the disk on the hole’s spin parameter a for ?xed Rs = 2rin .

– 45 –

0.0018 0.0016

Ψs = 0.5

P
0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0 -0.0002 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

a

0.8

Fig. 12.— The dependence of the total red-shifted power exchanged magnetically between the black hole and the disk on the hole’s spin parameter a for ?xed Rs = 2rin .



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