High-Energy Gamma-Ray Emission from Galactic Kerr-Newman Black

THE ASTROPHYSICAL JOURNAL, 498 : 640È659, 1998 May 10 ( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

HIGH-ENERGY GAMMA-RAY EMISSION FROM GALACTIC KERR-NEWMAN BLACK HOLES. I. THE CENTRAL ENGINE BRIAN PUNSLY 4014 Emerald Street No. 116 Torrance CA 90503 Received 1996 September 30 ; accepted 1997 December 9

ABSTRACT A model of the central engine of the unidentiÐed high-latitude galactic hard c-ray (EGRET) sources based on black hole electrodynamics is presented. The c-ray emission is produced in a bipolar outÑow from a charged, rotating black hole (a Kerr-Newman black hole) in a low-density region of the Galaxy, the details of which are provided in a companion paper. The model proposed in this article and its companion is a synthesis of pair creation scenarios for pulsars, the theory of black hole magnetospheres and synchrotron self-Compton large-scale jets. This article describes the physics of the putative central engine. Kerr-Newman black holes are plausible endpoints of the catastrophic gravitational collapse of the most massive magnetized rotating stars. In the following, the ability of a Kerr-Newman black hole to drive a magnetically dominated plasma wind in the charge-starved limit is explored for the Ðrst time. Although there are important analogies to magnetohydrodynamic (MHD) wind theory of Kerr (uncharged rotating) black holes, there are also enormous distinctions. However, previous experience with the MHD Kerr case is exploited to render this more complicated problem tractable. The most important parameter for quantifying the wind energy is the magnetic Ðeld line angular velocity, ) (and F by unfortunately the most difficult to calculate). The determination of ) is tied directly to the process F which a pair plasma is created on large-scale magnetic Ðeld lines through high-energy quantum electrodynamic processes typical of pulsar magnetospheres. It is argued in principle how ) is determined by the plasma injection mechanism. Furthermore, the most signiÐcant result of this e†ortF is the calculation of ) for a model that is determined by a plausible set of astronomical parameters. It is essential to F a distinction from some charge-starved pulsar models : the energy source for the wind is quantirealize Ðed by the Ðeld line rotation rate (the cross-Ðeld potential), ultimately powered by the rotation of the hole through dissipative gravitomagnetic processes and not the volage drop across the vacuum pair creation gap. Subject headings : black hole physics È gamma rays : theory È MHD È radiation mechanisms : nonthermal 1.

INTRODUCTION

Firstly, there is a relevant controversy in the literature as to whether rotating black holes in an astrophysical environment are charged (Kerr-Newman black holes) or uncharged (Kerr black holes). It was originally believed that a charged black hole could not exist for long in a typical astrophysical environment Ðlled with plasma as selective accretion of charge could neutralize the black hole and therefore kill o† the magnetic Ðeld as well (Ru†ini 1973). However, this scenario has become the topic of debate in the literature since we have learned to solve MaxwellÏs equation for simple sources on the spacetime background outside a rotating hole. A series of articles, Wald (1974), Petterson (1975), Chitre & Visheshwara (1975), and King, Lasota, & Kundt (1975), indicates that the minimum energy conÐguration of a rotating hole immersed in the electromagnetic Ðeld of an axisymmetric source term in MaxwellÏs equations is achieved if the hole attains a net charge (i.e., Kerr-Newman black hole). The conÐguration of a simple axisymmetric magnetosphere about a rotating black hole attains a minimum energy conÐguration when the hole and the magnetosphere possess equal and opposite charge with a magnitude that is a function of geometry and the amount of large-scale magnetic Ñux that threads the event horizon (Ru†ini 1977). This is a consequence of the dragging of inertial frames around a rotating black hole ““ mixing ÏÏ the magnetic Ðeld components with the electric Ðeld components of the Maxwell tensor. For a rapidly rotating hole, near the horizon, the electrostatic energy of a particle is

One of the most fascinating and unexpected results of the high-energy c-ray observations of EGRET is the large fraction of sources which cannot be identiÐed with known active sources in other frequency bands. Ozel & Thompson (1996) estimate that possibly half or more of the highlatitude (in galactic coordinates, declination o b o [ 10¡) EGRET detections which are unidentiÐed in other frequency bands are galactic in origin. It is suggested here that a new class of object is responsible for these unidentiÐed sources of c-ray emission, rapidly rotating, charged, galactic black holes of a few solar masses : Kerr-Newman black holes. If such black holes form as a result of stellar collapse, a large proper motion is expected, as for pulsars, and one would therefore also expect a sizable population at high galactic latitude. Furthermore, the model proposed here and in the companion paper (Punsly 1998a) can produce the observed high-energy c-ray luminosity consistently within the bounds of radio and soft X-ray luminosity set by these objects not appearing in surveys at those two frequency bands. The proposed relativistic bipolar outÑow emanating from the black hole magnetosphere is also capable of providing the extreme variability at c-ray wavelengths as it does for models of c-ray loud quasars and BL Lacs (Maraschi, Ghisellini, & Celotti 1992 and Sambruna et al. 1995). There is some related background material concerning Kerr-Newman black holes that the reader should note. 640

GAMMA-RAY EMISSION FROM BLACK HOLES. I. greatly a†ected by the magnetic Ðeld and this energy is actually minimized by separating charge in the presence of the magnetic Ðeld (Petterson 1975). The charge distribution in the magnetosphere can be time stationary as a tenuous plasma can be supported against electrostatic attraction to the hole by centrifugal forces in the rotating environment. In order to avoid subsequent confusion, the reader should review the magnetospheres presented in Hanni (1977), Ru†ini (1977), or the magnetosphere constructed in the Appendix (Fig. 9 below delineates the various elements). Consequently, there is a school of thought that there are no uncharged rotating black holes in the astrophysical environment resulting from the gravitational collapse of a magnetized star, only charged black holes with oppositely charged magnetospheres (as is the case for neutron star models of pulsars ; see Michel 1982 for a discussion). There seem to be at least two types of scenarios that can result in a signiÐcant charge on a galactic black hole. First, when a star of sufficient mass (probably greater than 10 M ) experiences catastrophic gravitational collapse, it is _ commonly believed that a black hole remnant is left behind by the supernova (Shapiro & Teukolsky 1983). If the star is rotating and magnetized, one expects, in analogy to the formation of magnetized white dwarfs and neutron stars, that the magnetic Ðeld is frozen into the collapsing progenitor stellar material. In the case of rotating white dwarfs and neutron stars the Ðnal state of the frozen-in Ñux requires an interior charge density (to remain frozen into the stellar interior) and a surface charge density (to transition to the external Ðeld geometry) of comparable magnitude but opposite total charge to the interior. The black hole has no surface, only the interior of the event horizon. The collapsing progenitor star contracts beyond the event horizon with a frozen-in value of electric charge that is totally analogous to the interior charge of a neutron star in a pulsar. There being no surface in the black hole case requires that the analog of the surface charge be supported by a charge density outside the horizon in the near Ðeld magnetosphere. This charge density acts as a transition to the GoldreichÈ Julian magnetosphere in a similar manner to the surface charge density on the neutron star. However, it is not equivalent ; this is a purely relativistic e†ect that is signiÐcant only for rapidly rotating black holes. The various analyses in the literature previously cited agree that the minimum energy conÐguration of a maximally rotating KerrNewman black hole (those considered here)/axially symmetric magnetospere requires roughly equal and opposite charges in the black hole and magnetosphere. The resulting charge separation is how the fully relativistic Kerr-Newman geometry a†ects the solution of the coupled system of MaxwellÏs equations and the momentum equations governing the gravitational collapse of magnetized frozen-inmatter. Most of the magnetic Ñux is supported by the near Ðeld magnetosphere. (Note that this is not the case for the subset of this Ñux which threads the event horizon. This is primarily sourced within the hole for rapid rotators). Such a scenario for a 7 M black hole considered here requires _ speciÐc charge of the neutron star in about one-quarter the the Crab pulsar. The near Ðeld magnetospheric charge can be supported in a time stationary orbiting disk. The main emphasis of this paper does not rely on the plausibility of a Kerr-Newman black hole existing in total isolation, but on the plausibility of the fully relativistic system of the black hole and the magnetosphere. Such a system can be stable

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only in an isolated environment, as intense accretion onto the central object would disrupt the orbiting charged disk and its Ðelds. The physical picture is similar to charged-starved pulsar models. The Kerr-Newman vacuum electromagnetic Ðeld is such that *FklF \ / 0, where F is the Maxwell tenor (or in kl Kerr-Newman black hole pulsar languageklE Æ B \ / 0). The is charged just like the neutron star in pulsar models (Ruderman & Sutherland 1975). The source of plasma results from particle acceleration along magnetic Ðeld lines produced by the parallel component of the electric Ðeld, E , A as in spark gap models of pulsars (Ruderman & Sutherland 1975 and Arons 1983).1 Above the polar particle acceleration gap, a perfect magnetohydrodynamic (MHD) plasma is established and a relativistic outgoing MHD wind is driven outward along polar Ðeld lines (see Kennel, Fujimura, & Pellat 1979 and Fig. 1). The wind is powered by the gravitomagnetic coupling of the hole to the pair plasma in the black hole ergosphere as discussed in Punsly (1991, 1996). The extracted rotational energy of the hole is manifested predominantly as a magnetically dominated wind as opposed to the energy Ñux due to pair production in the vacuum gap. The wind in the proposed model carries 3 ] 1032 ergs s~1 in Poynting Ñux and the gap injects about 5 ] 1031 ergs s~1 of mechanical energy Ñux into the wind. The Poynting Ñux in the pair plasma MHD wind is converted into mechanical and thermal energy by plasma instabilities and shocks in the asymptotic wind zone : the jet. The detailed structure of the power source considered here is given in Figure 1. The most important result obtained in this article is the determination of ) in principle and for a particular model in ° 4. The analysisF draws on particle creation physics in strong Ðelds, familiar to pulsar physics, as well as the theory of black hole driven MHD winds (gravitomagnetic dynamos). However, the process of incorporating these ideas self consistently in the Kerr-Newman geometry is unique as well as far from trivial. The ability of a KerrNewman black hole to drive a magnetically dominated plasma wind in the charge-starved limit has never been explored previously. The need to create plasma using quantum electrodynamic interactions with the magnetic Ñux in the wind is a distinction from previous treatments of MHD winds in a Kerr magnetosphere. Similarly, the lack of a highly conductive unipolar inductor (the neutron star) is a tremendous distinction from pulsar theory. The resulting object is a hybrid of a pulsar and an MHD Kerr black hole driven wind. The analysis is very nonlinear, as the pair creation mechanism a†ects the Ðeld line angular velocity which in turn a†ects the geometry of interaction of primary c-rays which a†ects the pair creation mechanism, etc. With all of these feedback e†ects, a parametric analysis of how the 1 As of 1996, plasma injection and global Ñow dynamics in pulsar magnetospheres is not a completely solved problem (Daugherty & Harding 1996), and this e†ort by no means claims to resolve these issues. The charge-starved black hole has at least as many ambiguities as the chargestarved pulsar. We merely transcribe the most commonly proposed ideas from pulsar theory, and my choice of polar gap model (Ruderman & Sutherland 1975) does not imply that it is the ““ best ÏÏ theory, just that it is the theory that is the easiest to adapt to black hole magnetospheres. It is left to future work by pulsar experts to clarify inconsistencies in neutron star magnetospheres and the newly introduced charge-starved black hole magnetospheres.

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FIG. 1.ÈKerr-Newman black hole model of galactic hard c-ray sources. At left is a wide-angle view of the twin SSC c-ray emitting jets driven by a compact object. At the bottom center is an exploded view of the compact object. A black hole (indicated by the solid disk) supports a magnetic Ðeld that is primarily dipolar, except at the poles where an MHD wind emanates from the magnetosphere. The hole rotates with an angular velocity, ) , along an axis H aligned with both the dipole moment and the MHD wind bulk velocity. At top right is an expanded view of a black hole magnetosphere. A gravitomagnetic dynamo exists just above the event horizon that powers the outgoing MHD wind. Paired (ingoing and outgoing) MHD winds emerge from a ““ particle creation zone.ÏÏ The Ñow divides within the particle creation zone in a gap containing a strong radial vacuum electric Ðeld.

emissivity changes as the magnetic Ðeld and mass of the hole vary is beyond the scope of the present paper. The purpose of this article is merely to present the plausibility of a Kerr-Newman black hole as a central engine for a c-ray loud bipolar outÑow. The model problem in ° 4 was solved

by an iterative method. A set of parameters were chosen and the model calculated and inconsistencies were noted. This led to a second set of parameters and so on until it appeared that convergence to a solution was imminent. Then the solution was perturbed until the pair creation region was as

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far from the hole as possible for an MHD bipolar outÑow to be established (the reason for this is based on minimum energy requirements as discussed in °° 3 and 4). This solution was then parameterized and calculated to give the results in ° 4. A possible source of confusion is the distinction between the voltage drop across the particle acceleration gap in the pair creation zone and the total energy of the black hole driven wind. The voltage drop across the vacuum gap can relate directly to the energy in the electrodynamic wind driven by some charge-starved pulsar models. In this analysis, the voltage drop across the vacuum gap relates only to the amount of inertia injected into the wind. The energy generated from the central engine is almost purely Poynting Ñux in nature and is generated by a gravitomagnetic dynamo inside the ergosphere of the black hole. For magnetically dominated winds, the amount of Poynting Ñux generated depends only slightly on the injected plasma inertia in the black hole models considered here. The next section is a review of Kerr-Newman electrodynamics. The plausibility of maximally rotating black holes in a galactic environment is established. The charge on the hole is miniscule by gravitational standards and typical by astronomical standards (i.e., on the order of the charge in a neutron star in a pulsar). There is also a review of Kerr black hole driven winds that establishes the formalism and principles used in the present analysis. The new result presented in ° 2 is that any polar particle acceleration gap integral to the production of plasma in the chargestarved limit must hover a ““ few ÏÏ gravitational radii from the event horizon. This is in contrast to the polar gap in a Ðeld aligned pulsar that is bounded by the neutron star surface. Thus the solution of the black hole radiator has a added complication that is not present in the pulsar, the extra degree of freedom of the gap location. Section 3 describes the injection of pair plasma onto the wind carrying polar magnetic Ðeld lines. As much contact is made with the model of Ruderman Sutherland (1975) as possible. However, there are various distinctions with the pulsar that are listed at the beginning of the section. Copious high-energy c-rays are produced in a particle acceleration gap as in the pulsar, eventually producing an MHD pair plasma by quantum electrodynamically scattering o† the strong magnetic Ðeld as in a pulsar. The primary c-rays that seed the gap come from the acceleration of electrons from the interstellar medium, this is a unique concept to this black hole radiator. In this section the zone of closed dead Ðeld lines that protects the black hole from electrical discharge is introduced. The main e†ort of this paper is ° 4 where the Ðeld line angular velocity is determined with the aid of the background set forth in the previous two sections. Principles that were used to deÐne ) for MHD winds in the Kerr geometry in Punsly (1991) inF a qualitative sense are used to quantitatively compute ) . The basic idea is that ) adjusts F stresses associated with F to minimize the magnetic the plasma injection process. This is the Ðrst time that this principle has been used in a detailed calculation of ) in any F black hole wind problem. The second principle that was used in deÐning MHD wind solutions on a Kerr background in Punsly (1998b) is that the physical wind solution is the one providing the minimum torque on (or equivalently energy extraction from) the black hole. This determines the gap location when implemented simultaneously

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with the determination of ) described above. The gap F exists as far from the hole as possible within the constraint that the Ðelds are still strong enough to produce an MHD plasma both inward and outward. Most of the section is the detailed calculation of the physical parameters derived from the aforementioned gap location and value of ) ; from the primary c-rays seeding the gap to the location atF which the pair plasma deposition becomes frozen into the large scale magnetic Ðeld. It is shown that the Poynting Ñux radiated by the black hole gravitomagnetic dynamo far exceeds the high energy radiation Ñux and inertial energy Ñux produced in the acceleration gap. Finally, in the Appendix a detailed model of a black hole magnetosphere is produced. The model is not unique, but it demonstrates the following postulates implicit in the text : 1. An electrodynamic structure can exist in the magnetosphere that allows the black hole to maintain its charge. 2. The magnetosphere possesses electromagnetic sources that produce only second order e†ects to the pair creation scenario near the pole of the black hole as posited in the article for simplicity. 3. The zone of closed dead Ðeld lines is comprised mostly by a rotating distribution of GoldreichÈJulian charge that is time stationary. 2.

KERR-NEWMAN ELECTRODYNAMICS

The Kerr-Newman spacetime is the unique geometry of spacetime outside of the event horizon of a charged, rotating black hole (Misner, Thorne, & Wheeler 1973). The metric in Boyer-Lindquist coordinates is given by the line element

A

B

2Mr dt2 ] o2 dh2 ds2 4 g dxk dxl \ [ 1 [ kl o2 ]

AB C

4Mra o2 dr2 [ sin2 h d/ dt o2 *

] (r2 ] a2) ]

D

2Mra2 sin2 h sin2 h d/2 , (2.1a) o2

where o2 \ r2 ] a2 cos2 h ,

(2.1b)

* \ r2 [ 2Mr ] a2 ] Q2 ,

(2.1c)

and M, a, and Q are the mass, speciÐc angular momentum, and charge of the hole in geometerized units, respectively. The Boyer-Lindquist coordinates reduce to spherical Minkowski coordinates as r ] ]O. There are two event horizons given by the roots of the equation * \ 0, * 4 (r [ r )(r [ r ) \ 0 , (2.2a) ` ~ r \ M ^ JM2 [ a2 [ Q2 , (2.2b) B where r and r are the outer (of relevance here) and inner ` ~respectively. The stationary limit surface is event horizons, given by r \ M ] JM2 [ a2 cos2 h . (2.3) s The region between r and the horizon, r , is the ergos ` sphere. This is the region of spacetime where the gravitomagnetic dynamo that powers the MHD wind operates (Punsly 1991, 1996).

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This paper considers rapidly rotating black holes like those that are commonly believed to reside at the center of active galactic nuclei (AGNs ; see Begelman, Blandford, & Rees 1984). However, this assumption need not be valid for a galactic black hole in isolation. Yet, if a large angular momentum transfer to the central object occurs during the gravitational collapse or the black hole is part of an accreting binary system, the hole can be spun up to near its maximum allowed value of speciÐc angular momentum, a \ 0.998M (Misner et al. 1973). Consider the amount of extractable rotational energy (reducible mass) of a black hole, v. Then (Christodoulou 1970) v B 0.29Mc2 .

(2.4)

For a 7 M black hole v B 4 ] 1053 ergs. _ The bipolar wind described in this paper carries on the order of 7 ] 1032 ergs s~1 from the black hole. Thus, the wind will radiate away less than 1% of the reducible mass in 1011 yr. If a galactic black hole is prepared during formation to have a [ M, then it will be able to support a 7 ] 1032 ergs s~1 wind for over 1011 yr and maintain a [ M. Thus, the rapidly rotating approximation is worthy of pursuit. The main e†ect of a B / M is that the radial coordinate of the event horizon increases. We assume, without loss of generality, that M Z a ? Q [ 0. Thus the radial coordinate of the horizon is found from equation (2.2b) to be r Z M. ` 2.1. T he Kerr-Newman Electromagnetic Field The Maxwell tensor in Boyer-Lindquist coordinates of the vacuum Kerr-Newman electromagnetic Ðeld is (Misner et al. 1973) Qa sin2 h(r2 [ a2 cos2 h) , F \ Õr o4

(2.5a)

2Qar(r2 ] a2) cos h sin h , o4

(2.5b)

Q(r2 [ a2 cos2 h) , o4

(2.5c)

[2Qa2r cos h sin h . o4

(2.5d)

F \ hÕ

F \ rt F \ ht

Note that for r [ 4M, the expression for the electromagnetic Ðeld in Boyer-Lindquist coordinates is the same as that of an electric monopole and a magnetic dipole in spherical coordinates up to a small correction which is 0[(a/ 4M)2]. Boyer-Lindquist coordinates are not orthonormal or even orthogonal. So, in order to give a meaningful value to the radial magnetic Ðeld at the horizon, the Maxwell tensor can be evaluated in an orthonormal frame near the horizon. One can normalize the angular coordinates in terms of the metric coefficients of (2.1), g , using kl eü \ g~1@2 L/L/ , (2.6a) Õ ÕÕ eü \ g~1@2 L/Lh . (2.6b) h hh These two vector Ðelds are two legs of the orthonormal frame of the zero angular momentum observers (ZAMOs), which are physical (noninertial) frames deÐned at Ðxed values of the ““ r ÏÏ and ““ h ÏÏ coordinates. The other two basis vectors of the orthonormal tetrad carried by the ZAMOs

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are (Lightman et al. 1979) *1@2 L , eü \ r o Lr

C

(2.6c)

D

L ) L ] , eü \ a~1 0 Lt c L/

(2.6d)

where a is the lapse function *1@2 sin h , 0¹a¹1 , (2.7) g1@2 ÕÕ and a ] 0 at the event horizon. Strong gravity corresponds to small values of a. The lapse function represents the gravitational redshift between the ZAMOs and the stationary frames at asymptotic inÐnity (deÐned by four velocity, L/Lt). The four velocity of the ZAMO frames is the unit vector Ðeld eü in (2.6d). The angular velocity of the ZAMOs as viewed0 from asymptotic inÐnity is ) a\

g ) \ [ Õt . (2.8) g ÕÕ Note that the radial magnetic Ðeld in the ZAMO basis is well-behaved near the horizon Br \

2Qar(r2 ] a2) cos h , o4[(r2 ] a2)2 [ *a2 sin2 h]1@2

(2.9a)

2Qar cos h ` . (2.9b) lim Br 4 Br \ ` [r2 ] a2 cos2 h]2 ` r?r` The surface radial magnetic Ðeld which is the analog of the polar cap magnetic Ðeld in a pulsar is 2Qar ` . (2.9c) lim Br 4 B \ S [r2 ] a2]2 ` r?r` h?0 Consider the model chosen in this paper, a 7 M black hole _ (if M \ 7 with a B M, then by equation (2.2b) r B 106 cm ` M ). Choose the surface magnetic Ðeld B to be B \ 1010 S or in G._Then, as a consequence of (2.9c), Q B 2S] 1022 esu, geometerized units Q B 5.75 ] 10~3 cm > r . This is ` interior roughly two-thirds of the charge of the neutron star in the Crab pulsar (Michel 1982 ; Ruderman & Sutherland 1975). Note that the expressions above are approximations and carrying out the calculations to two decimal places is not of any signiÐcance. In the following, the standard scientiÐc style of three signiÐcant digits is used merely to keep roundo† errors from accumulating through a long series of calculations. The radial electric Ðeld is also well-behaved in the ZAMO frames near the horizon Er \

Qg1@2 ÕÕ (r2 [ a2 cos2 h)(1 [ )a sin2 h) , o5 sin h

(2.10a)

Q(r2 [ a2) ` . (2.10b) lim Er 4 E \ S [r2 ] a2]2 ` r?r` h?0 Notice that the surface electric Ðeld is virtually zero at the pole. Thus, the particle acceleration gap cannot exist just above the horizon for the rapid rotators under consideration. The particle acceleration gap must hover some distance above the horizon. Furthermore, except at very small values of the lapse function of (2.7), electromagnetic forces

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are 1013 times as strong as gravitational forces. The dominant role of electromagnetism provides an essential commonality with pulsar magnetospheres. Only near the horizon, where a ] 0, does gravity dominate the dynamics. The most important aspect of the Maxwell tensor in equation (2.6) for an isolated black hole is that E Æ B \ / 0, as in a vacuum neutron star electromagnetic Ðeld. This allows for the possibility of pair creation in a strong magnetic Ðeld as in a pulsar spark gap (Ruderman & Sutherland 1975). In complete isolation, a black hole will spontaneously lose its charge (Gibbons 1975). However, in the presence of a tenuous interstellar medium (ISM), seed particles and c-rays are available to initiate pair cascades yielding substantial plasma densities and magnetohydrodynamical (MHD) Ñows (see next two sections). 2.2. Paired MHD W ind T heory The nature of the vacuum electric Ðeld revealed through equation (2.10) implies that a strong component of the electric Ðeld parallel to the magnetic Ðeld, E , can exist in the A polar regions at r D a few times M, even though it vanishes at the horizon. Thus, if a particle acceleration gap exists then it would hover on the order of a few Schwarzschild radii from the hole. Such a gap Ðts nicely into previous models of paired MHD winds in a Kerr (uncharged, rotating hole) magnetosphere with an externally imposed magnetic Ðeld (i.e., there is an azimuthal current source in the accretion disk). The paired wind model was Ðrst described in Phinney (1983) and discussed in Punsly & Coroniti (1990b) and Punsly (1991). In PhinneyÏs model, a particle creation zone hovers well above the hole in each Ñux tube. Gamma-ray collisions within the particle creation zone produce a sufficient pair plasma density so that the Ñows emanating from the region satisfy the perfect MHD relations. The ingoing MHD wind is a accretion wind and the outgoing MHD wind is likened to the pulsar winds of Kennel et al. (1983). The outgoing wind is considered a possible energy supply for the jets in extragalactic radio sources. The central engine that powers the paired winds is not located within the particle creation zone. The concept of the particle creation zone used previously in the AGN context Ðts quite naturally into this analysis of charge-starved Kerr-Newman magnetospheres. Pair creation from a particle acceleration gap can play the role of PhinneyÏs particle creation zone. It will be shown that as in a pulsar model, enough plasma can be created as a consequence of particle acceleration in the gap to short out the vacuum electric Ðelds and paired MHD winds will Ñow from the gap. The problem at hand splits into two separate issues (which must be synthesized in a consistent fashion) : (1) the dynamics of the pair creation from the hovering gap (°° 3 and 4), and (2) the dynamics of the paired MHD wind system. The outgoing and ingoing winds have many similarities. In the models considered here, both are magnetically dominated : over 85% of the energy Ñux is in electromagnetic form (Poynting Ñux). Within the horizon magnetosphere, the dominance of the magnetic Ðeld can be expressed in terms of the pure Alfven speed, U : A U2 (BP)2 A\ , (2.11a) c2 4nn kc2 p where BP is the poloidal magnetic Ðeld in the ZAMO frames, n is the proper number density, and k is the speciÐc P

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enthalpy of the MHD plasma. Furthermore, near the event horizon, (U /c) B 1010. As UP ] U , the gravitational A A forces overwhelm the electromagnetic forces. For particles emerging from the particle creation zone with a typical Lorentz factor c [ 106 and at a lapse function a D 0.7 (as in the models considered here) since UP D a~1 (Punsly & Coroniti 1990b), the plasma will become inertially dominated (UP [ U ), at a lapse function a \ 7 ] 10~5. Thus, A I for lapse function values of a [ 10~3, the ingoing wind will be magnetically dominated as is the case for the outgoing wind. Both of these magnetically dominated MHD winds have a light cylinder. The outer light cylinder is similar to that of standard relativistic wind theory (Kennel, et al. 1983). The inner light cylinder results from the dragging of inertial frames and exists within the ergosphere. For the winds considered here ) D 10~2) , where ) is the angular velocity F H H of the horizon as viewed from asymptotic inÐnity, so that the inner light cylinder exists just within the stationary limit (Punsly 1998b). The winds considered here have an inner light cylinder at a B 5 ] 10~2. Thus, using the value of a above, the winds approach the inner light cylinder as mag-I netically dominated Ñows. This scenario has been analyzed extensively in Punsly (1991 ; 1996 ; 1998b) and is summarized below. Both winds are injected from the particle creation zone into the black hole magnetosphere at a relativistic speed that would certainly exceed the slow magnetosonic speed. The ingoing wind must pass through the Alfven and fast magnetosonic critical points before reaching the horizon (Phinney 1983). The outgoing wind must also pass through the Alfven critical point, and it was argued in Punsly (1998b) that the physical solution will probably pass through the fast critical point as well. The ingoing wind couples to the hole gravitomagnetically and powers the outgoing wind. The ingoing wind is magnetically dominated near the inner light cylinder, yet the light cylinder rotates at minus the speed of light in any physical frame located at this point. Thus, the plasma cannot either corotate with or bend the Ðeld lines signiÐcantly (small inertia). Hence, the plasma pulls o† the Ðeld lines and dissipates near the inner light cylinder (Punsly 1991, 1998b). This dissipation is associated with the dynamo that creates toroidal magnetic Ñux and therefore the Poynting Ñux carried by the outgoing wind (Punsly 1991). It is the ““ dragging of inertial frames ÏÏ and not radial gravity that powers the paired wind system. The total redshifted electromagnetic energy Ñux emerging through a Ñux tube is the integral of the poloidal redshifted Poynting Ñux, SP, over the cross-sectional area, dA, of the Ñux tube (Macdonald & Thorne 1982)

P

aSP dA \

P

) F BTBP dA , 4n

(2.12)

where BT is the toroidal magnetic Ðeld density deÐned in BoyerÈLindquist coordinate as BT 4 o2 sin hFrh .

(2.13)

The energy Ñux depends strongly on the Ðeld line angular velocity ) . F 2.3. T he Field L ine Angular V elocity Determining the Ðeld line angular velocity is crucial to quantifying the energy Ñux generated from the black hole

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magnetosphere (see eq. [2.12]). This is a far more difficult issue in the black hole case than for the pulsar. In a pulsar, the magnetic Ðeld lines are frozen into the highly conductive neutron star. The event horizon, by contrast, is a nonconductor. It is a vacuum spacetime inÐnity for electromagnetic waves and plasma Ñows, and, as such, it will passively accept any externally imposed value of ) (Punsly & F Coroniti 1989, 1990a). Thus any value of ) is allowed, and F it is not associated with the angular velocity of the horizon, ) . Once plasma is threaded onto the Ðeld lines, the Ðeld H lines become essentially free-Ñoating and their rotation rate depends entirely on the plasma injection mechanism (Takahashi et al. 1990 and Punsly 1991). This subsection is intended as an introduction to the physics involved in determining ) . F There are two main issues : 1. Where is the location of the hovering acceleration gap ? 2. Once this gap is located, what are the details of the plasma injection process that determines ) ? F These two questions are not independent and must be answered simultaneously. As a guide as to how to tackle these problems we follow Punsly (1991). The following guiding physical principles are posited. First, the Ðeld line angular velocity adjusts to minimize the magnetic stresses associated with the plasma injection process, within the constraints of MHD causality (principle I). Second, as postulated in (Punsly 1998b) the Ðeld line angular velocity and the resulting MHD wind solution will minimize both the torque on the hole (principle II) ; and third, the entropy generation in the global solution (principle III). Principles II and III were motivated in Punsly (1998b). Principle III is a standard rule governing the evolution of nonequilibrium systems (see chap. 35 of Kittel 1958). Principle II is used to single out the physical solution in relativistic stellar wind theory (Kennel et al. 1983). Principle I is both a second order correction to principle II as well as a logical consequence of the plasma injection mechanism determining ) . The freezing of the plasma onto the magF plasma injection zone is equivalent to the netic Ðeld in the Ðeld being brought into corotation with the injected plasma in an average sense. To elaborate, the injected plasma does not have a single angular velocity but a distribution that is primarily a function of its distance from the event horizon. Consequently, the frozen-in Ðeld cannot corotate with all of the plasma throughout the injection zone. Principle I is equivalent to the Ðeld coming into ““ corotation ÏÏ with the injected plasma as much as possible. In this paper we delineate the solution that satisÐes principles IÈIII above, and thus it is claimed that this is the physical mechanism for the electromagnetic discharge of the Kerr-Newman black hole in the ISM. In order to incorporate principle I, we need to know more about the plasma injection mechanism (see ° 4.1 below). However, principles II and III suggest immediately that the outgoing wind is the Michel (1969) ““ minimum torque solution ÏÏ with the minimum permissible value of ) (chosen consistently F with principle I in mind). 2.3.1. T he Minimum T orque and Minimum Dissipation Solution

It is posited in this paper that the global energy extracting Ñow for a Kerr-Newman black hole will be the

Vol. 498

minimum torque and minimum dissipation solution (principles II and III above). This requires that (1) the minimum value of ) that is consistent with a pair producF ing acceleration gap for a given magnetic Ðeld strength, and that (2), for this value of ) , the outoing wind solution is F that which extracts the minimum amount of angular momentum Ñux (and therefore energy Ñux in a Poynting Ñux dominated wind) from the hole. In a relativistic sense the entropy generation is concentrated in the event horizon. This solution, by the second law of black hole thermodynamics, generates the minimum entropy in the horizon (Thorne, Price, & MacDonald 1986). Thus, the global solution that minimizes torque is also the solution that minimizes entropy generation in a fully relativistic sense (Punsly 1998b). The minimum torque solution (principle II above) was introduced by Michel (1969) and analyzed in detail by Kennel et al. (1983). In the magnetically dominated limit for an asymptotic wind zone that is cylindrically symmetric and homogeneous, the toroidal magnetic Ðeld of (2.13) in each Ñux tube, satisÐes (Punsly & Coroniti 1990b ; Camenzind 1986) BT B =

[) ' F B constant , nc

(2.14)

where ' is the poloidal magnetic Ñux enclosed by a magnetic Ñux tube and the subscript ““ O ÏÏ means evaluated as r ] ]O. Performing the integral in (2.12), for the total poloidal Poynting Ñux in the asymptotic wind, using (2.14) yields

P

)2 '2 aSP dA \ F . (2.15) 8n2c wind Thus, the total energy Ñux is very sensitive to the value of ) . The minimum torque solution is assumed for the outF wind throughout the following. going 3.

THE INJECTION OF PAIR PLASMA

Just as in pulsar models, the source of plasma is the crucial issue for understanding the emission mechanism for isolated, charged, rotating black holes. As with the pulsar, the vacuum electric Ðeld can in principle accelerate charges to ultrarelativistic energies. Curvature radiation in the form of high-energy c-rays can subsequently create electronpositron pairs by scattering o† the strong magnetic Ðeld (Erber 1966). However, there are some key di†erences with the pulsar. First, there are no charges that can be pulled from the surface of the compact object as was originally conjectured by Goldreich & Julian (1969). Second, there is only one moment of the vacuum electromagnetic Ðeld (l \ 0, m \ 0 moment in the relativistic PoissonÏs equation ; see Bicak & Dvorak 1976). All Kerr-Newman black holes have their rotation axis and magnetic axis aligned ; they cannot pulse. Third, a zone of closed dead Ðeld lines is assumed to exist in analogy to pulsar models (Kennel et al. 1979). As the black hole is not a conductor, the closed dead Ðeld lines do not necessarily rotate in consort with the wind zone (see Appendix). The Ðeld line angular velocity is determined by the details of the plasma injection mechanism in this region. Finally, most of the black hole vacuum magnetosphere is electric as opposed to magnetic (FklF \ 0), in contrast to kl star. the vacuum magnetosphere of a neutron

No. 2, 1998

GAMMA-RAY EMISSION FROM BLACK HOLES. I.

With these facts in mind we choose the following pair creation scenario. There is a wind zone with a particle acceleration gap on polar Ðeld lines. The acceleration gap is a Ñow division region within the particle creation zone. The new ingredient is the existence of a mechanism that injects seed pairs into the gap (see Figs. 1 and 2). 3.1. Origin of Seed Photons It is envisioned here that seed pairs arise in the gap indirectly as a consequence of the interaction of the strong vacuum electromagnetic Ðeld and the tenuous interstellar medium (ISM). The average free electron density of the ISM is 3 ] 10~2 cm~3 (Spitzer 1978). We assume that the black hole is located in an isolated region of space, so we choose an electron density of the ISM one order of magnitude less than this (n ) \ 3 ] 10~3 cm~3. We also choose a posie ISM tively charged black hole so that electrons will be accelerated toward the horizon electrically. The structure of the magnetosphere is assumed to consist of the following (see Fig. 1) : 1. A thin polar wind zone with a hovering particle acceleration gap, 2. A zone of closed dead magnetic Ðeld lines, 3. A low density ““ vacuum ÏÏ electrodynamic sheath separating the two regions. In the context of this subsection, region (3) is of interest. This region has no wind within it or pair creation. Electrons from the ISM are accelerated along the magnetic Ðeld lines by the electric Ðeld in the vacuum electrodynamic sheath. As the electrons are accelerated along the vacuum magnetic Ðeld they radiate curvature photons. When r \ 20M, a strong Ñux of curvature photons irradiates the polar wind zone. A fairly intense beam of high energy c-rays illuminates the putative acceleration gap, creating electronÈpositron

FIG. 2.ÈEntrained electrons from the ISM are accelerated toward the hole in the vacuum electrodynamic sheath. As they propagate along the curved magnetic Ðeld lines they radiate primary curvature c-rays. Details in the upper right quadrant of the magnetosphere are illustrated above. The c-rays irradiate the wind zones and the particle creation zone, and pair create in the strong black hole magnetic Ðeld (at small enough values of radial coordinate). Some of the c-rays strike the particle acceleration gap. The c-rays convert to particles as the scatter o† the gap magnetic Ðeld creating the primary ““ seed ÏÏ pairs.

647

seed pairs by scattering o† the magnetic Ðeld within the gap (see Fig. 2). In order to justify and deÐne the hypothesis that charges can be sucked into the central engine through an electrodynamic sheath, the following three conditions must be satisÐed. First, a well deÐned opening to the interstellar medium exists through which charges can enter the inner regions of the black hole magnetosphere, the throat. Second, a small residual electric Ðeld parallel to the magnetic Ðeld exists in the sheath that can accelerate ISM electrons toward the hole. Finally, electron trajectories in the inner magnetospheres have a guiding center along the magnetic Ðeld direction. The throat of the sheath is provided by the magnetic Ðeld topology illustrated in Figure 1. The wind zone Ðeld lines have an inÑection in curvature far from the hole causing a separation from the zone of closed dead Ðeld lines. Thus, condition 1 is a natural consequence of a wind that is restricted to only polar Ðeld lines. Condition (2) is more hypothetical and is not rigorously justiÐable. The source of the sheath electric Ðeld is likely to be a consequence of the global current system of the black hole driven wind. Within the wind zone proper, current Ñows poloidally toward the hole, and a return or closure current must Ñow outside of the wind zone away from the hole in a sheath bounding the wind zone (Punsly 1991, 1996). This return current is driven by a small proper electric Ðeld which is positive. This closure current sheath forms the upper bounding surface of the vacuum electrodynamic sheath. Consider the source of the electric Ðeld in the adjacent vacuum electrodynamic sheath and throat to be essentially an unshielded (in the low density semivacuum) and therefore somewhat stronger version of the same electric Ðeld that drives the closure current bounding the wind zone, E .The magnitude E is likely to be much larger than throat the local magnetic Ðeld throat strength of the ISM near the throat opening. Thus, the dynamics of particle motion are determined by the electric Ðeld at the throat opening independent of the magnetic Ðeld strength of the ISM. This small semivacuum electric Ðeld, E initiates motion along the strong magnetic Ðeld lines throat bounding the electrodynamic sheath. Even a small electric Ðeld will quickly accelerate the electrons to their radiation reaction limited value. As the hole is approached, the electric Ðeld in the sheath begins to resemble the powerful electric Ðeld in the semi-vacuum region of the inner magnetosphere (see Fig. 9 below and related discussions in the Appendix). This is where the third concern above arises since the electric Ðeld becomes stronger than the magnetic Ðeld. At this point, it is necessary to clarify the single particle dynamics as the inner magnetosphere is predominantly electric instead of magnetic. On polar Ðeld lines near the horizon (r \ 20M), o E Â B o \ o B o2 in the ZAMO frames because E and B are almost parallel. Thus, the equations of motion yield a trajectory described by electric Ðeld acceleration parallel to the magnetic Ðeld until the radiation reaction limit is reached (by curvature radiation losses) combined with an E Â B azimuthal drift (Batygin & Toptygin 1964). This concept of ““ seed pairs ÏÏ was chosen because it is plausible, but more importantly it allows one to quantify the value of the injected pair density consistently with the determination of ) in the next section. It might be anotherÏs preference Fto make all of the pairs in the gap

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Vol. 498

proper from a few seed galactic c-rays as is sometimes the case for pulsrs (Kennel et al. 1979). For the model of seed pairs presented here to work the hole needs to be positively charged (to attract ISM electrons). However, it might be that negatively charged holes will radiate away their rotational energy with only a spark gap to produce pairs. This is beyond the scope of this paper. In this paper, we choose one plausible scenario so that we can concentrate on the bigger picture that a Kerr-Newman black hole in the ISM will radiate away its rotational inertia. Within the vacuum electrodynamic sheath, the entrained ISM electrons are accelerated to their radiation reaction limited value, expressed in terms of the maximum attainable Lorentz c-factor (Cheng, Ho, & Ruderman 1986), c

'

\

A

B

E Æ BŒ 1@4 s2 e

(3.1)

where s is the radius of the curvature of the magnetic Ðeld lines. The curvature radiation spectrum from the radiation reaction limiting photons is characterized by the critical frequency, u (Cheng et al. 1986) c E Æ BŒ 3@4 +Jsc . (3.2) +u \ c e

A B

In the vacuum electrodynamic sheath, the Lorentz c factor is always c becuse the particles are reaccelerated ' by the vacuum electric Ðeld after the almost instantaneously emission of each c-ray. Thus, the c-ray luminosity from the vacuum electrodynamic sheath, L , illuminating the particp cle acceleration gap is

P

e(n ) E Æ dx , (3.3) e ISM & where & is the region of the vacuum electrodynamic sheath that illuminates the gap (i.e., mathematically it is the preimage of the gap in the electrodynamic sheath under the mapping induced by ray tracing along the c-ray propagation vectors). The capture cross section for ISM electrons, p , is roughly the area of the ““ throat ÏÏ of the gap between cap dead Ðeld lines and the wind zone, where electromagthe netic collimation of the wind causes a deviation from the dipolar Ðeld conÐguration (see Fig. 1). In our model p is cap on the order of 1016 cm2. L

cp

\ cp

cap

3.2. T he Spark Gap The primary curvature photons from the vacuum electrodynamic sheath scatter o† the strong magnetic Ðeld in the spark gap producing the seed electron-positron pairs. The electron-positron pairs are accelerated by the strong gap electric Ðeld, producing more curvature c-rays that in turn scatter o† the magnetic Ðeld producing additional pairs (see Fig. 3). The gap is bounded from above and below by regions where E Æ B B 0 (small E ) that transition into two perfect A from the gap. The gap height, MHD regions farther away L , is much less than the scattering length for pair creation bygapc-rays in the magnetic Ðeld. This is a thin gap model ; thus, most of the pairs are created in the bounding perfect MHD and near perfect MHD (*F Fkl > o F Fkl o) regions (see Fig. 3). Consequently, there is kl little or no kl electric Ðeld to accelerate these pairs and there is no cascade e†ect in the gap. This is a large distinction from Ruderman & Sutherland (1975) spark gap, where the pair creation ““ oscillates ÏÏ

FIG. 3.ÈDynamics of the particle creation zone. Primary curvature c-rays illuminate the particle acceleration gap. Primary c-rays create electron-positron pairs as they scatter o† the gap magnetic Ðeld. The pairs are accelerated by the strong parallel electric Ðeld, E , within the gap to A ultrarelativistic energies. These pairs radiate high-energy curvature (secondary) c-rays parallel to the local magnetic Ðeld direction. The scattering length for pair creation in the magnetic Ðeld is much larger than the gap height, L . Consequently, pair creation occurs predominantly in gap is very small. regions where E A

in the gap as the pairs are immediately accelerated by the electric Ðeld after creation to reach c , producing more curvature photons and more pairs and'so on. The particle creation mechanism envisioned here is more of a steady discharge than a sparking gap. Again, it is not claimed that the pair creation method described in this paper is any more or less plausible than the spark gap of Ruderman & Sutherland (1975) or the slot gap of Arons (1983). It is merely an expedience to help Ðx the value of ) . F 3.3. T he Particle Creation Zone We now proceed to detail the particle creation zone in Figure 4. The thin vacuum gap model requires the gap height L , satisÐes gap L > a~1(u, r, h) , (3.4) gap where a~1(u, r, h), is the scattering length for pair creation in the magnetic Ðeld of c-rays of energy, +u, as a function of the distance from the point of c-ray emission in the gap (expressed in r and h coordinates). The particle creation zone is not symmetric, the scattering length for inward directed c-rays is di†erent than for outward directed c-rays because the magnetic Ðeld is not homogeneous. Condition (3.4) ensures that plasma is created primarily in the region of small E as in Figure 3. A The number of pairs that are created, n , in terms of pairs Ðeld, n the number of c-rays illuminating the magnetic , photon

No. 2, 1998

GAMMA-RAY EMISSION FROM BLACK HOLES. I.

FIG. 4.ÈDue to the nonzero scattering length for pair creation by curvature c-rays in the magnetic Ðeld, plasma injection occurs primarily in two bands. The upper band is the plasma source for the outgoing wind and the lower band is the plasma source for the ingoing wind.

is given by (Erber 1966) as B [1 [ e~a(u,r,h)z]n , (3.5a) pairs photon where z is the proper distance that the photons have propagated through the strong magnetic Ðeld, and n

A B

B (3.5b) a~1(u, r, h) \ 2j a~1 cr T ~1[s(u)] , c e B M where j is the Compton wavelength of an electron and a is e the Ðnec structure constant. The critical magnetic Ðeld strength is B \ 4.4 ] 1013 G, and B is the magnetic Ðeld cr M strength measured orthogonal to the direction of c-ray propagation. T is a function of the variable s(u, r, h)

C

D

1 +u B (r, h) M , B 2 m c2 cr e and in the parameter range of interest s(u, r, h) 4

AB

(3.5c)

4 . (3.5d) lim T ~1(s) \ 2.17 exp 3s s?0 Equation (3.5) shows us that the absorption coefficient a(u, r, h) is a function of frequency and position. Immediately after c-ray emission in the gap, the c-ray is propagating parallel to the magnetic Ðeld, so B B 0 and the scattering length in (3.5b) is huge ; there isM no pair production. For outgoing c-rays there are two competing processes at work as the c-ray propagates away from the acceleration gap. As a consequence of magnetic Ðeld line curvature, the angle of propagation relative to the Ðeld lines increases (lowers the scattering length by increasing B ) M and the dipolar magnetic Ðeld decreases (raises the scattering length by decreasing B ). Thus, the scattering length, M a~1(u, r, h), will achieve a minimum and then increase again for each value of u. The details of this functional dependence determine how much plasma and of what energy is injected as a function of r and h in the particle creation zone (see Fig. 4). The ingoing c-ray case is much simpler as all

649

e†ects of the inhomogeneous Ðeld tend to decrease the scattering length as the photons propagate towards the hole. Considering the variability of a(u, r, h) described above, we adopt the basic structure of the particle creation zone in Figure 4. A very thin vacuum gap accelerates the seed pairs to high-energy producing curvature c-ray emission. The plasma is injected primarily into two Ðnite bands as a consequence of the nonzero scattering length for pair creation by c-rays in the magnetic Ðeld, one below and one above the gap. The disjoint union of these bands is deÐned as the plasma injection region. The bandlike nature of the plasma injection region is a manifestation of the functionality of the absorption coefficient, a(u, r, h). In practice we Ðnd, roughly speaking, that when a~1 [ 0.1M, the plasma is deposited within a distance Z0.2M upstream of this point. At a certain point, the injected pair density exceeds the GoldreichÈJulian charge density, o (Goldreich & Julian GhJ Ñow occurs at this 1969), and we conclude that an MHD point and beyond. Between the point where the injected plasma density exceeds o and the vacuum gap it is not clear whether a GhJsmall E is present as in Scharlemann, Arons, & distributed Fawley (1978) or if Acharges are pumped to establish perfect MHD. We simply assume in this transition region that E is A negligibly small (possibly zero) and ignore this complication in the dynamics. More speciÐcally if E corresponds to a A value c in (3.1) less than the Lorentz factor of the injected ' pairs then it can be ignored in dynamical considerations. In the next section, we solve for the gap height, location of the gap and the plasma injection region and we Ðnd the Ðeld line angular velocity. 3.4. T he Zone of Closed Dead Field L ines A zone of closed dead Ðeld lines, as in a pulsar (Kennel et al. 1979), is assumed to exist in this treatment. Yet their dynamics are not addressed here and are by no means trivial. The vacuum Kerr-Newman magnetosphere is electric (FklF \ 0) everywhere except just outside the poles of kl (see eq. [2.10a]), and one might expect the hole the horizon to neutralize by selective accretion through the putative zone of closed Ðeld lines. It is argued elsewhere that this is not likely to be the case for Kerr-Newman magnetospheres (Hanni 1977 ; Ru†ini 1977 ; Damour et al. 1978). We assume, in line with the above reference, that the dynamics of the magnetospheric plasma acts to balance the radial force of the electric Ðeld. This concept is referred to by the authors above as a ““ plasma horizon,ÏÏ a boundary where the inÑow of charge is halted by magnetospheric forces (see the Appendix for details). The region of GoldreichÈJulian charge in the zone of closed dead Ðeld lines is e†ectively a shield that insulates the hole from discharge by particles in the ISM. The only place where ISM electrons can reach the hole is through the electrodynamic sheath. However, the discharge dynamics are swamped by the pair creation process and the resulting current closure in the MHD wind system. The electrons accreted through the electrodynamic sheath essentially provide a portion of the return current for the global MHD wind system and actually only help in maintaining the electric charge on the hole and not discharging it. Most of the charge circulation is actually provided by the global current system of the black hole (gravitomagnetic) dynamo just as the neutron star dynamo determines the charge circulation in a pulsar.

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PAIR CREATION DYNAMICS AND )

F

In this section we determine ) self consistently with the F fundamental physical principles introduced in ° 2.3 and the pair creation model described in the last section. 4.1. Establishing the Field L ine Angular V elocity We implement principle I of ° 2.3 to Ðnd the Ðeld line angular velocity, ) . Consider the angular momentum of F the pairs created in the particle creation zone of Figure 4. The component of mechanical angular momentum along the rotation axis of the hole a particle as seen in the stationary frames at asymptotic inÐnity (deÐned by a four velocity, L/Lt) is m (Thorne et al. 1986) m \ L/L/ Æ P ,

m \ g1@2 kuÕ . (4.1b) ÕÕ The ISM electrons entrained by the hole (see Fig. 2) are injected into the black hole magnetosphere axisymmetrically yet they might transport a net component of angular momentum along the symmetry axis of the hole from the surrounding ISM. The number of pairs produced in the particle creation zone greatly exceeds (by 7 orders of magnitude), the number of entrained ISM electrons. Thus, the injected speciÐc angular momentum is likely to be diluted by a factor of 10~7 after pair creation is accomplished. We can then approximate the initial speciÐc angular momentum, m , in the plasma injection bands of int the angular velocity of the pairs Figure 4 by m B 0. Thus, int ) , is initially equal to the ZAMO angular velocity, ) (by p deÐnition), in the plasma injection bands () ) \ ) . (4.2) P int There is a magnetic stress induced as the freshly created pairs become threaded onto the magnetic Ðeld lines. This is the quantity that needs to be minimized (principle I) for the conÐguration of plasma injection indicated by Figure 4. Consider the frozen-in condition for the toroidal magnetic Ðeld (Punsly & Coroniti 1990b) () [ ) ) F P BPg , ÕÕ cbP

(4.3a)

where BP is the ZAMO evaluated poloidal Ðeld strength and bP is the bulk plasma three velocity in that frame. From equation (2.14), for the minimum torque solution with BT B constant for the magnetically dominated wind (Camenzind 1986) ) ' BT B [ F . nc

netic stress when it gets frozen onto the magnetic Ðeld lines. Consider the angular momentum imparted to the plasma injection region above the gap, *L , by the magnetic Ðeld ` (4.3) : as a consequence of equations (4.2) and *L

`

\

(4.3b)

Thus, equation (4.3) implies that the frozen-in plasma in the outgoing wind (noting that bP B ]1), has an angular velocity as viewed from asymptotic inÐnity that is given by () )` B 0 and for the ingoing wind (noting that bP B [1), ()P)frz ~ B 2) . PComparing frz F the initial values of the angular velocity of the injected plasma in equation (4.2) with the Ðnal frozen-in values above, it is clear that the plasma is torqued by mag-

P

V n [[)1 g ] ` ÕÕ , n *m dV B ` ` ` ` ` a

(4.4)

where V is the volume of the plasma injection band, n is ` ` the plasma number density in the ZAMO frame, *m is the ` change in speciÐc angular momentum during the freezing-in process, and )1 is the ““ average ÏÏ ZAMO angular velocity in the plasma` injection band. Similarly, for the lower plasma injection band the induced magnetic stress is approximated by

(4.1a)

where Pk is th four momentum of the particle. In terms of the ZAMO evaluated four velocity, uk, and speciÐc enthalpy, k,

BT \

Vol. 498

*L

~

\

P

V n [2) [ )1]g F ÕÕ , n *m dV B ~ ~ ~ ~ ~ a

(4.5)

where )1 is the average ZAMO angular velocity in the ~ injection band. lower plasma In order to implement principle I, we minimize the magnitude of the total magnetic stress of the plasma injection process, o *L o ] o *L o, by varying ) . Note that equation ` F is solely a conse(4.4) is independent of~ ) , so our result F quence of minimizing (4.5) with respect to ) : F 1 ) B )1 . (4.6) F 2 By choosing the gap as far from the hole as possible, the stresses in equation (4.4) will be minimized as ()g /a) is a monotonically decreasing function. As stated in °ÕÕ2.3, this also minimizes the torque on the hole, satisfying principles II and III as well. In summary, the vacuum gap is situated as far from the hole as possible so that the pair creation mechanism is efficient enough to produce a number density at least a few times o , so that MHD can be established. The functional GhJ pair creation scattering length in equation (3.5b) can be used to Ðnd the lower pair creation band in Figure 4, at which point equation (4.6) is used to determine ) consisF tently wth principle I. At this point ) , is not determined completely since it F depends on )1 , which is a function that depends on the ~ location of the gap. Again, the gap is placed as far from the hole as possible in order to minimize the stresses of plasma injection (see eq. [4.4]), the torque on the hole, and the global entropy generation. If the gap is placed too far from the hole, the electromagnetic Ðeld will be too weak to produce enough pairs to short out the electric Ðeld (establish o ). The problem is very nonlinear, and we estiGhJcenter of the gap (for the parameters chosen in mate that the ° 2, a B M \ 7 M , B \ 1010 G) is near r \ 5.7M. This _ S optimized due to the incredible has not been completely complexity of the feedback e†ects, but it is a close approximation. In the next three subsections we construct the selfconsistent details of a gap centered at r \ 5.7M. We Ðnd that most of the plasma inertia of the ingoing wind is injected in a band between r \ 5.0M and r \ 5.3M. Then Then equation (4.6) yields ) B 2.08 ] 102 s~1 \ 1.39 ] 10~2) . (4.7) F H If the open Ñux tubes comprising the wind zone have a range of polar angles at the event horizon of h to h ' then by equation (2.5b), the total magnetic Ñux in&the wind

No. 2, 1998 zone, '

TOT

, is \

GAMMA-RAY EMISSION FROM BLACK HOLES. I.

651

P

F dh d/ hÕ polar cap \ n/2(r2 ] a2)B [sin2 h [ sin2 h ] . (4.8) ` S ' & The total Poynting Ñux in the wind zone, L , is then wind found from equations (2.15), (4.7), and the parameters of the model introduced in ° 2.1 to be '

L

wind

TOT

4

P

aSP dA wind

A

B 3.5 ] 1032 ergs s

B

sin2 h [ sin2 h 2 ' & . (4.9) 4.41 ] 10~3

4.2. T he Electric Field in the Gap The strength of the electric Ðeld in the gap is a crucial ingredient for quantifying the pair creation rate as a consequence of equations (3.1) and (3.2) that describe the radiation reaction limiting curvature radiation. We show in this section that the electric Ðeld contribution from the GoldreichÈJulian charge density above and below the gap represents only a small correction to the Kerr-Newman Ðeld for a thin gap at r \ 5.7M, with gap height L > M. gap charge First, we should quantify the GoldreichÈJulian density starting with MaxwellÏs equation in BoyerÈ Lindquist coordinates L MJ[gFhtN B 4no8 (e) , h B 0 , (4.10a) GhJ J[g Lh 1

where the square root of the determinant of the metric satisÐes J[g \ o2 sin h ,

(4.10b)

and o8 is the GoldreichÈJulian charge density in BoyerÈ GhJ coordinates. Using the frozen-in condition in Lindquist Boyer-Lindquist coordinates ) (F ) \ F (F ) ht MHD hÕ c

(4.11)

(F ia assumed to be the Kerr-Newman magnetic Ðeld), hÕ with equations (2.1) and (2.5) near h B 0, one Ðnds in along equation (4.10a) Br o8 B [) [ ) ]a~2[1 ] O(sin2 h)] , h B 0 , GhJ 2nec F (4.12a) with Br given by (2.9a). Note that the sign o8 depends on GhJ rate of the whether ) is faster or slower than the rotation F local geometry, ). Far from the hole, ) ] 0 and the standard pulsar result is attained. In the ZAMO frames the charge density is o \ ao8 . GhJ GhJ At the base of the gap, from (2.9a) and (4.7), L o (e) B 7 ] 10~2 esu cm~3 , r \ 5.7M [ gap . GhJ 2

(4.12b)

(4.13)

The plasma above and below the gap in Figure 5 is considered to have a small E by the deÐnition of the gap, A

FIG. 5.ÈElectrodynamic structure that is associated with a particle acceleration gap that has a signiÐcant electric Ðeld contribution from the GoldreichÈJulian charge densities in the adjacent wind zones. Charge densities just above and below the gap, o, are approximately equal to the GoldreichÈJulian charge density, o . The electric Ðeld transforms from GhJ magnetic Ðeld below the gap to being purely orthogonal to the poloidal having a signiÐcant parallel component within the gap in a transition layer of width, d. It is argued that this situation does not occur. The electric Ðeld from o is too small to noticeably a†ect the Kerr-Newman vacuum Ðeld GhJ in the gap.

namely, *F Fkl > o F Fkl o , kl kl

L o r [ 5.7M o Z gap . 2

(4.14)

This implies that the charge density, o, above and below the gap satisÐes oBo

GhJ

[0 ,

L o r [ 5.7M o Z gap . 2

(4.15)

The contribution to the gap electric Ðeld from the charge density below the gap arises due to the absence of the GoldreichÈJulian charge density directly above (the gap). Thus, the electric Ðeld changes from being predominantly in the h direction in the nearly perfect MHD zone to having a signiÐcant radial component in a transition layer of thickness d in Figure 5. Note that just above the gap in Figure 5, from equation (4.12a), o [ 0 as well and higher up it changes sign due to GhJ nature of ). This means that just above the the functional gap, it is of the wrong sign to terminate the gap electric Ðeld. Thus, the superposition of contributions to the gap electric Ðeld from the hole and the GoldreichÈJulian charge density below the gap induces a negative surface charge layer, p , ` bounding the outgoing wind zone from below. Now, we estimate the electric Ðeld in the gap produced by the GoldreichÈJulian charge density at r [ 5.7M \ [L /2. If the gap is thin as assumed, one can use the plane gap approximation to Ðnd the contribution to the gap parallel electric Ðeld, *E , gap *E B 4no (e)d , (4.16) gap GhJ where half of the electric Ðeld results from the charge density induced at the base of the outgoing wind zone through a contribution from p . Applying equation (4.13) `

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to express o in (4.16), GhJ *E

gap

Vol. 498

A B

B 0.9

d statvolt cm~1 . 1 cm

From equation (2.10a), the Kerr-Newman radial electric Ðeld at r \ 5.7M is Er B 5.6 ] 108 statvolt cm~1. Thus, K.N. *E > Er if d \ 100M, which is an unreasonably large gap K.N. value for d. We conclude that the GoldreichÈJulian charge density in the wind is insigniÐcant in the determination of the gap electric Ðeld. Now, it is possible to create surface charge densities bounding the wind zones, p and p , that could conceiv` ~ ably create any electric Ðeld strength in the gap. However, there is no obvious physical reason why this would happen. The vacuum Kerr-Newman Ðeld and the GoldreichÈJulian charge density are the only necessary dynamical components for the model. Thus, it is conjectured here that the Kerr-Newman electric Ðeld is essentially equal the gap electric Ðeld at r \ 5.7M. Note that we have ignored any possible contributions from charges in the zone of closed dead Ðeld lines so this might not be a valid assumption, however, a model magnetosphere is constructed in the Appendix that is consistent with this assumption to within 10%. Consider the outer regions of the particle acceleration gap deÐned by a magnetic Ðeld line that emanates from the horizon at an angular coordinate h \ 4¡, then by (3.1) FOOT the gap electric Ðeld above can accelerate electrons and positrons to the radiation reaction limited value of c \ ' 3.58 ] 108, where the radius of curvature, s, was approximated by its dipolar value at r \ 5.7M in accord with the discussion of (2.5), so that s \ 1.17 ] 108 cm. The corresponding critical frequency deÐned in (3.2) is +u \ 12.1 c ergs. 4.3. T he Plasma Injection Mechanism We are Ðnally in position to quantify the amount of inertia produced in the particle acceleration gap. We begin with the production of seed pairs as described in ° 3.1 and Figures 1 and 2. The parameters incorporated into the model so far are : M \ 7 M , B \ 1010 G, ) \ 2.08 S ] 102 s~1, and [h ] \_ 4¡, where [h ] F is the FOOT ' FOOT ' angular coordinate at the base of the Ðeld line (the horizon) marking the edge of the polar cap wind zone. 4.3.1. Seed Pair Injection

It is conjectured here that seed c-rays, primary c-rays, are produced in a vacuum electrodynamic sheath as the vacuum Kerr-Newman electric Ðeld accelerates entrained ISM electrons toward the hole (see ° 3). These high-energy curvature c-rays illuminate the gap and produce seed pairs by scattering o† the magnetic Ðeld. The details are described below for a vacuum gap at r \ 5.7M, assuming a vacuum electrodynamic sheath the domain of which is the set of Ðeld lines with angular coordinates at their base 4¡ \ [h ] \ 4.4¡. This choice of maximum angular coorFOOT issheath dinate justiÐed later in the subsection. The appropriate geometry for the sheath conjectured above is illustrated in Figure 6. The gap is primarily illuminated by photons produced in the sheath at h [ 4¡.4 FOOT and r B 11M. Primary curvature (sheath) photons radiated above r B 15M reach the wind zone too far from the hole to produce substantial pair densities (Br is too weak). This is built into the model by the minimum energy and entropy generation principles (II and III) as discussed in ° 2.3. The gap is as far from the hole as possible for pair creation in the

FIG. 6.ÈGeometry of the scattering of primary c-rays with the magnetic Ðeld within the gap.

magnetic Ðeld to be efficient. Sheath curvature photons radiated below r \ 10M primarily irradiate the ingoing wind zone causing additional pair creation and added inertia. Consider a primary curvature photon emitted near r \ 11M and h \ 4¡.4. The angle between the electric FOOT is found by Ðrst evaluating the angular and magnetic Ðelds coordinate of the magnetic Ðeld line at r \ 11M, h . This is found in the dipolar approximation by the implicitB relation (Daugherty & Harding 1996), M sin2 h , (4.17) B sin2 h FOOT where h is the angular coordinate of the Ðeld line at the horizon.FOOT The angle that the Ðeld line makes with the vertical (symmetry axis of the hole) in the dipolar approximation is given by r\

3 cos h sin h B B. (4.18) 3 cos2 h [ 1 B Using equations (4.17) and (4.18), the primary curvature photon is created at r \ 11M and h \ 14¡.74 and propagates at an angle to the symmetry Baxis of the hole ( \ 22¡.24. The radius of curvature of the Ðeld line at r \ 11M, h \ 4¡.4 is s \ 9.68 ] 107 cm. Then, using the scattering FOOT geometry above and (2.10a) inserted into (3.1), the maximum Lorentz factor of the entrained ISM electrons at r \ 11M is c \ 2.36 ] 108 and the critical frequency of ' the resulting curvature radiation +u \ 4.30 ergs. c to produce pairs in a The scattering length, l, for a c-ray tan ( \

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GAMMA-RAY EMISSION FROM BLACK HOLES. I.

magnetic Ðeld is given by Erber (1966)

A BA B +c e2

magnetic Ðeld can be deÐned using (2.13) and (2.6),

CD

+ B 4 cr exp , (4.19) m c B 3s e M where B and s are deÐned in (3.5c) and (3.5d), respectively, cr l is deÐned by (3.5) , l B 4.4

l \ a~1(u, r, h) ,

(4.20)

m is the mass of the electron, B is the component of the e M magnetic Ðeld perpendicular to the propagation vector of the c-ray in the particle acceleration gap. The primary c-ray from r \ 11M strikes the center of the gap at r \ 5.7M at the Ðeld line deÐned by the horizon coordinate h \ 3¡.5 and propagates at an angle of 9¡.67 FOOT to the Ðeld. Considering the geometry described above and in Figure 6 and using the Ðeld strength values in the gap from (2.9a), B \ 3.41 ] 107 G. Choose l to be the maximum photon M through the gap and wind zone. If r is the crosspath sectional (cylindrical) radius of the gap Mfor geometry described above l\

2r M B 5 ] 106 cm . sin (

(4.21)

Inserting this value of l and B into (4.19) yields M for the threshold s~1 \ 14.39. This allows one to solve c-ray energy for pair creation for a sheath photon that traverses the wind zone and pair creates within the gap, +u \ 0.144 ergs. Any photon with an energy less threshold than this from the sheath has a a very low probability of creating pairs in the wind zone or gap proper. Comparing this result to the critical frequency for sheath generated c-rays, (u /u ) \ 0.033. By thethreshold explicit cnature of the curvature radiation spectrum, this result implies that 41% of the primary sheath c-rays are of sufficient energy to produce pairs in the gap (and 94% of the sheath c-ray luminosity is in the form of pair producing c-rays). Again, using the spectrum of curvature radiation (Jackson 1975) and the geometry described above, the number of seed pairs created in the gap is approximately given by L

L cp n \ 1.28 ] 109 cm~3 , gap seed 1027 ergs s

(4.22)

where L is the c-ray luminosity from the sheath in the cp form of curvature c-rays produced in the region & deÐned in the integral of equation (3.3). For a vacuum gap of proper length L , we crudely approximate this region of the elecgap sheath as r such that [(L /2) ] 11M \ r \ trodynamic gap (L /2) ] 11M and h B 4¡.4. gap FOOT 4.3.2. T he Capture Cross Section for ISM Electrons

To compute L in equation (3.3), we need to know the cp for ISM electrons, p . This is intercapture cross section cap toroidal magpreted through the collimation e†ects of the netic Ðeld in MHD wind theory (Sakurai 1985). The hoop stresses associated with the toroidal Ðeld, BÕ, can overwhelm pressure gradients associated with the plasma and gradients in the vertical magnetic Ðeld, Bz. In the ZAMO frames Bz is the component of the magnetic Ðeld along the symmetry axis of the hole Bz \ BP cos ( ,

653

(4.23a)

where ( is deÐned in equation (4.18). The ZAMO toroidal

BÕ \ a~1g~1@2 BT . (4.23b) ÕÕ For a dipolar magnetic Ðeld, Bz decreases with r very rapidly for h \ 4¡, r B 100M. Using (2.14), (4.7), (4.8), FOOT and (4.23b) in our model at the outer edge of the wind zone

A

BÕ B 2.13 ] 103 G

r LC r sin h

B

(4.23c) B where r is the light cylinder radius associated with ) , LC F r \ 1.44 ] 108 cm. For the purely dipolar Ðeld conÐguLC ration, at the outer edge of the wind zone, we have o BÕ o [ Bz, if r [ 118M. Thus, we anticipate signiÐcant hoop stress collimation before the Ñow reaches r \ 118M. To really know the cross-Ðeld structure of the wind one needs to solve the Grad-Shafronov equation (Nitta, Takahashi, & Tomimatsu 1991) ; however, this is beyond the scope of the present e†ort. We assume that at r B 113M, the wind zone magnetic Ðeld topology is deviating signiÐcantly from the dipolar Ðeld zone as indicated in Figure 1. This naturally produces the ““ throat ÏÏ leading into the electrodynamic sheath as depicted in Figure 1. We can justify the choice of h \ 4¡.4 in ° 4.3.1 as the Ðeld line that forms the outerFOOT boundary of the electrodynamic sheath. It is conjectured that the outer regions of the magnetosphere of closed dead Ðeld lines are static as in the standard pulsar model. Michel (1982) argues that a static magnetosphere is charge separated. For electrons to be accelerated from the ISM toward the hole along a Ñux tube, one would expect a negative charge density along its entire length (Michel 1982). Consider the standard expression for the GoldreichÈJulian charge density (i.e., the asymptotic form of equation [4.12] generalized to all angles), as given for example by equation (1) of Ruderman & Sutherland (1975). Only Ñux tubes for which Bz [ 0 over their entire length have a negative charge density everywhere and are therefore capable of attracting ISM electrons to the black hole. The outermost Ðeld line for which this can occur is the one that is horizontal (B \ 0) at the opening of z the throat of the electrodynamic sheath. At r \ 113M, the Ðeld line with an angular coordinate at the horizon, h \ 4¡.40 realizes this concept. Consequently, the FOOT cross section in the integral is capture p B cap

P

h/45]

2nR2 sin h dh

K

. (4.24) h/54.73] R/113M This implies in the model that p B 1.04 ] 1016 cm2. cap equations (3.1) and Evaluating L in equation (3.3) using cp (4.24) yields L

cp

\ 7.37 ] 1028

L gap ergs s~1 . M

(4.25)

In deriving (4.25), the contribution to the electric Ðeld at r \ 11M from charges in the zone of dead Ðeld lines was ignored ; again this might not be a valid assumption. The seed pair density in the vacuum gap can be found by substituting (4.25) into (4.22) n \ 9.43 ] 104 cm~3. seedbased on the geometry of a single This result was derived c-ray emitted at r \ 11M, h \ 4¡.4. It is believed that FOOTsubsection is typical of the the scattering described in this c-rays that illuminate the gap. However, it should be noted that in this sense that the value of n above is only an approximate result. It would be better seed to perform a detailed

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modeling at a later time, but it is anticipated that the result would be similar to that above. 4.3.3. Creation of the MHD W ind Plasma

When the seed pairs of ° 4.3.1 are created within the gap, they are immediately accelerated by the gap electric Ðeld, to their radiation reaction limited value of c , producing sec' ondary (curvature) c-ray emission (see Figure 3). Consider the opacity to pair creation of these secondary curvature c-rays in the magnetic Ðeld as described in equation (3.5). We analyze the local scattering geometry in Figure 7 in order to quantify the functionality of B in the expression M for the variable absorption coefficient, a, in equation (3.5). Due to inhomogeneous Ðeld line curvature, most of the inertia is created near the outer edge of the gap (Daugherty & Harding 1996), so we consider a Ðeld line with angular coordinate at the horizon h \ 4¡ in Figure 7. FOOT 4.3.3.1 The Ingoing Wind

The curvature photons are emitted at an angle ( \ E 14¡.38 to the vertical. Consider the ingoing c-rays, in particular the absorption coefficient for pair production of curvature c-rays in the magnetic Ðeld equation (3.5). In Figure 7, we have B \ sin o ( [ ( o BP , (4.26) M F E where ( is the angle that the magnetic Ðeld makes with the vertical. FAs the c-ray propagates toward the hole both sin o ( [ ( o and BP increase and one expects virtually a comF conversion E plete of c-ray energy Ñux from the gap into ingoing pair inertia. The scattering length a~1(u, r, h) is a function of photon energy. Explicit calculations of a~1(u, r, h) show that near r [ 5.2M, the scattering length for over half of the ingoing gap c-ray luminosity is on the order of 0.2M or less. Thus, over half the inertia is deposited in the ingoing wind in the narrow band 4.95M \ r \ 5.3M (see Fig. 8). This result was applied to equation (4.6) to determine ) in ° 4.1.1. The injected plasma number density exceeds F the GoldreichÈ

FIG. 7.ÈGeometry of pair creation in the large-scale magnetic Ðeld by c-rays radiated from within the gap. Note that the curvature of the Ðeld lines is greatly exaggerated.

Vol. 498

Julian charge density at r \ 5.4M. Thus most of the plasma is injected where E B 0 as anticipated in the model that A was developed through Figures 3 and 4. The injected plasma is not accelerated appreciably by E , thus the A injected inertia is a good estimate of the mechanical energy Ñux of the wind [there are some synchrotron losses, but they are O(sin o ( [ ( o) \ 3% of the total energy]. F E 4.3.3.2 The Outgoing Wind

Now consider the outgoing wind. We do not expect a complete conversion of the gap c-ray energy to pair inertia in the scattering process as a consequence of the functional nature of the opacity given by (3.5). Notice that in (4.26), BP-decreases outward of the gap and (sin o ( [ ( o) F E increases. Consequently, the geometry of Figure 7 and equation (4.26) implies that B has a broad maximum when M r B 7.0MÈ8.5M, with (B ) Z 3 ] 106 G. The threshold M' energy for c-ray conversion to pairs in the outgoing wind zone is found using (3.5) to be (+u /+u ) \ 0.18. From thresholdradiation c the explicit nature of the curvature spectrum (Jackson 1975), 60% of the power from the gap is converted into pairs, we call this power efficiency v \ 0.6. Note that p c-rays emitted this corresponds to only 18% of the total from the gap as a consequence of the curvature radiation spectrum, we denote as the number efficiency of the process v \ 0.18. n 4.3.3.3. The Establishment of an MHD Wind

The total power emitted outward from the gap in the form of curvature c-rays is

C DC

P B [n ng (r \ 5.7M)] curv seed ÕÕ

L

gap 2

D

2 e2c c4 3 s2 '

(4.27a)

B 7.64 ] 1026[L

] ergs s~1 . (4.27b) gap Choose the height of the gap, L in (4.27), so that the gaplarger than o injected plasma density is sufficiently when GhJ injecthe plasma gets frozen-in (at the center of the plasma tion band in Figure 8). This allows the currents to become decoupled from the bulk motion of the GoldreichÈJulian

FIG. 8.ÈDetailed structure of the particle creation zone for the model described in the text. Tick marks on the right-hand side of the wind zone indicate the radial coordinate in units of the black hole mass, M.

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GAMMA-RAY EMISSION FROM BLACK HOLES. I.

charge density. A density of plasma n [ o 10o o can GhJ support the particle drifts necessary to produce the MHD wind currents in the plasma injection band. Introducing a density larger than this would add extra (unnecessary) magnetic stresses to the plasma injection process as detailed in equation (4.4) and would violate principle I of ° 2.3. In order to quantify the injected number density we must recall the expression for the GoldreichÈJulian charge density in the ZAMO frames, equation (4.12). From (3.5) and Figure 8 most of the plasma inertia is injected in the range 6.4 \ r \ 7.0M. The GoldreichÈJulian charge density at the peak plasma inertia injection region, r B 6.6M, is found from (4.12) to be o \ 1.58 ] 107 cm~3 at GhJ r \ 6.6M. Compare the GoldreichÈJulian charge density to the number density of pairs created in the region r B 6.6M. First, the number density of curvature c-rays from the gap is given in terms of the luminosities of these secondaries, P , curv numerically by the approximate expression 17.9P curv n B , (4.28a) c +ucn(r2 ] a2)c sin2 h B where h is the maximum angular coordinate of the wind zone at rB\ 6.6M and P is evaluated in (4.27), curv L gap . (4.28b) n (r \ 6.6M) B 8.09 ] 108 c 105 cm

C

D

Evaluating the absorption coefficient in (3.5) yields the number efficiency, described in ° 4.3.3.2, at r \ 6.6M as v (r \ 6.6M) B 0.10. The number density of created plasma isn (by the deÐnition of v ) given in terms of the number of gap produced c-rays, n inn (4.28) as c n \ 2v n . (4.29) n c By (4.28a) if L \ 9.16 ] 104 cm, then using (4.29) yields n \ 1.48 ] 108gap cm~3 \ 9.38 o o o. This should be a sufficient charge density to short GhJ out the electric Ðelds and decouple the MHD currents from the bulk Ñow of the GoldreichÈJulian charge density. Yet, this is not an excess of plasma that would require undesirable excess magnetic stresses associated with the plasma injection process (principle I). Note that L > M in the model. Also, we gap stress again that we only considered the scattering geometry in the outer regions of the gap, h [ 4¡. The results of FOOT this subsection are approximated and a full numerical model of the gap would certainly be superior. 4.4. Energy Flux from the V acuum Gap The process of seed pair acceleration in the vacuum gap injects roughly equal energy Ñuxes into the ingoing and outgoing winds. For the ingoing wind, virtually all of the energy Ñux ends up in inertial form. Using the gap height, L , found above implies an ingoing inertial energy Ñux, gap is established with a magnitude given by (4.27a) of S~, I S~ \ 1031 ergs s~1. I For the outgoing wind, the model predicts that 40% of the energy Ñux is in the form of high-energy c-rays that do not convert to inertia in the magnetic Ðeld, S`, S` \ 2.8 c c ] 1031 ergs s~1. Similarly, pair creation in the plasma injection region accounts for an outgoing inertial energy Ñux S` \ 4.2 I ] 1031 ergs s~1. There is another component of outgoing inertia from the

655

seed pairs accelerated within the gap. They are decelerated from c , due to curvature losses and produce an energy ' Ñux (S )` Z 2 ] 1029 ergs s~1. I gap The created pairs have a component of momentum orthogonal to the magnetic Ðeld as well that is almost instantaneously released as c-ray synchrotron emission. The perpendicular component of the momentum is sin [( [ ( ] of the total momentum (see Fig. 7). The F E synchrotron energy Ñux is found to be S` B 6 ] 1029 synch ergs s~1. The other component of the energy Ñux is the Poynting Ñux in the wind. The total extractable rotational energy Ñux from the hole by the paired wind system is given by (4.9). Everything has been previously determined except for h . Near the pole, the Ðeld lines are not curved enough (B &is M too small to produce short scattering lengths, a~1(u, r, h), in [3.5]) to generate the pair densities required to carry the MHD currents of the wind. It is estimated that h \ 1¡.2. & (4.9) Taking h \ 4¡ as throughout the text, equation ' yields a wind luminosity L \ 3.54 ] 1032 ergs s~1. Although we have beenwind neglecting inertial and radiative e†ects in the wind dynamics so far, we now incorporate them into the energy budget, as some of the energy Ñux from the hole was dissipated in the gap. Thus, the outgoing Poynting Ñux just above the particle creation zone is S` \ 2.84 ] 1032 ergs s~1. The magnetic dominance of theEM outgoing wind leaving the particle creation zone is quantiÐed by the ratio (S`/S` ) \ 0.148. I EMto remember that the Poynting Ñux is It is important generated in the gravitomagnetic dynamo near the inner light cylinder at a lapse function of a B 5 ] 10~2 as discussed in ° 2.2 (Punsly 1991). The total extractable energy from the black hole is given by equations (2.15) and (4.9). The result above shows that the gravitomagnetic dynamo introduces one order of magnitude more energy than that which is injected in the particle acceleration gap and as a consequence the wind is magnetically dominated. This also has the important theoretical consequence of decoupling the voltage drop across the acceleration gap from the energy Ñux driven by the hole. 5.

DISCUSSION

This paper demonstrates a method by which an isolated Kerr-Newman black hole/axially symmetric magnetospheric system can radiate twin MHD winds or jets. By adopting conventional pair creation scenarios from pulsar theory to the Kerr-Newman geometry an MHD Ñow is established which, in turn, allows us to implement previously known MHD gravitomagnetic dynamo theory around rotating black holes. The power source becomes identical to what has been previously found for Kerrr (uncharged, rotating) black holes in the MHD case (since the amount of charge considered here is negligible to the spacetime metric) for an imposed value of ) > ) . The F extraction H fortuitous circumstance of using a known energy mechanism renders this imposing nonlinear problem tractable. Three new issues were addressed in the text in a self consistent manner. 1. The determination of ) , F 2. The location of the ““ hovering ÏÏ polar acceleration gap (a distinction from pulsar polar gaps), and 3. The generation of primary c-rays that seed the acceleration gap.

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The most important of these is the determination of ) F consistent with points (2) and (3) above. A fourth issue is addressed less rigorously in the Appendix, the enveloping axially symmetric magnetosphere which prevents spontaneous discharge from the hole. We Ðnd for a near maximally rotating black hole a BM \ 7 M and a 1010 G surface polar magnetic Ðeld _ that ) B 208 s~1. The center of the particle acceleration F gap lies about 5.7M from the hole. An outgoing MHD wind emerges in each hemisphere frozen into the magnetic Ðeld at 6.3M from the hole and transports an intrinsic luminosity, L B 3.54 ] 1032 ergs s~1 about 85% of which is in elecwind tromagnetic form (Poynting Ñux) and 10% in inertial energy Ñux and 5% in stray c-ray Ñux. It will be discussed in a companion paper that the jet driven by the physical process of this paper can produce the broad band spectrum of the high latitude, unidentiÐed galactic hard c-ray sources.

Vol. 498

In summary, in spite of some remaining technical issues, this article is a preliminary attempt at understanding the black hole analog of the charge-starved pulsar. Just as in the original pulsar models of Goldreich & Julian (1969) and Rudeman & Sutherland (1975), this e†ort has introduced many subjects requiring future intensive study. It is hoped that charged black hole magnetosophere/wind systems can provide the same level of intellectual interest as pulsar theory. This paper was the end result of many intense exchanges with Nancy Parks. Without her boundless encouragement and support, this article would have never reached completion. I would also like to thank Frank Ibbott for the typing of the manuscript and the excellent composition of the Ðgures.

APPENDIX A A MODEL OF THE ZONE OF CLOSED DEAD FIELD LINES This appendix constructs a model of the zone of closed dead Ðeld lines that protects the charged Kerr-Newman black hole from spontaneous electric discharge. The zone of closed dead Ðeld lines presented here is consistent with the model of the black hole wind/magnetosphere developed in the text. In particular, as described in the introduction ° 1 and ° 3.4 the electromagnetic sources in the zone of closed dead Ðeld lines have a small e†ect on the dynamics responsible for plasma injection and germination of the MHD wind (i.e., the magnetic Ñux through the hole is altered by less than 1% and the electric Ðeld in the particle acceleration gap is decreased by approximately 10%). The zone of closed dead Ðeld lines is illustrated in Figure 9. The zone consists of the following discrete elements : A charged current ring at a radius r \ 10M from the hole, A plasma horizon, A semivacuum region between the black hole and the plasma horizon that envelopes the charged current ring, A large zone of closed dead Ðeld lines Ðlled with GoldreichÈJulian charge density o . GhJ We describe these regions in order in the next four subsections.

1. 2. 3. 4.

A.1.

THE CHARGED CURRENT RING

It has been shown that the minimum energy conÐguration of simple rotating black hole/magnetosphere systems requires the magnetosphere and black hole to have equal and opposite electrical charge (see Ru†ini 1977, and references therein). Consequently from the model in ° 2.1 we pick the charge of the magnetosphere, Q , to be M Q \ [2 ] 1022 esu \ [Q . (A.1) M Furthermore, we assume that this charge is concentrated in the current ring at r \ 10M \ 107 cm, depicted in Figure 9. The current ring is chosen to be protonic as opposed to positronic in nature. It might be thought of as representative of a small inert fossil disk resulting from the original gravitational collapse. If the disk is large, one might think of the current ring as the inner edge. If one were to consider positronic matter, the issues of annihilation and pair creation would have to be analyzed within the context of our desire to produce a time stationary magnetosphere, and this would be very complicated. The current ring is chosen to have a large number density N [ 5 ] 1020 cm~3 ,

(A.2)

and the dynamics are dominated by MHD. The charge Q of equation (A.1) resides in a surface charge distribution and the M than 1 part in 1019). Consequently, the mass of the ring is larger plasma is very close to charge neutral (i.e., it is charged to less than 5 ] 1017 g, and gravitational attraction of the ring to the hole exceeds the electrostatic attraction. The ring maintains its equilibrium by balancing the gravitational force with a centrifugal force. The ring plasma satisÐes perfect MHD so the Ðeld line angular velocity of the magnetic Ñux threading the ring () ) equals the mechanical angular velocity of the ring FR material ) R ) \ () ) B 103 s~1 . (A.3) R FR Note that this Ðeld line angular velocity di†ers from that of the MHD wind zone in ° 4.3. This is allowable since the magnetic Ñux is not frozen into the horizon and any value of ) is physically acceptable and it can vary from Ðeld line to Ðeld line F (Punsly & Coroniti 1990a).

No. 2, 1998

GAMMA-RAY EMISSION FROM BLACK HOLES. I.

657

FIG. 9.È““ Eastern ÏÏ hemisphere of the zone of closed dead Ðeld lines is shown in cross section. The plasma horizon separates a large static GoldreichÈ Julian charge density from a semi-vacuum region, virtually devoid of plasma. The null lines deÐned by zero GoldreichÈJulian charge density, o \ 0, GhJ The separates the regions of positive and negative charge. A charged current ring orbits the hole at the Keplerian velocity at a radial coordinate r \ 10M. ring has a charge, [Q, equal and opposite of the hole and an azimuthal current, IÕ, into the page (i.e., in the same sense as the holeÏs rotation).

The ring also is chosen to carry an azimuthal current that is decoupled from the bulk Ñow of charge since this is an MHD plasma. Current drifts produce a current opposite to and exceeding that of the bulk Ñow of charge. The total current is IÕ \ 4 ] 1025 esu s~1 \ [2) Q \ 2) Q . R M R The magnetic moment of the ring, m , is therefore R nIR2 m \ B 21Qa \ 21m , R H c

(A.4)

(A.5a)

where R is the radius of the ring, R \ 107 cm ,

(A.5b)

and m is the magnetic moment of the hole H m \ Qa \ 2 ] 1028 esuÈcm . (A.5c) H For a Kerr-Newman black hole with a/M [ 1, as posited in the text, the charged current ring alters the magnetic Ñux through a hemisphere of the event horizon by less than 1% and the resulting electric Ðeld in the particle acceleration gap is roughly 87% of the value given in ° 4.2. It should be noted that farther from the hole (i.e., in the particle creation zone), the magnetic Ðeld of the ring is important. It does not alter the Ñux in the wind zone because the ingoing and outgoing winds are frozen-into the Ñux emanating from the hole and this is therefore conserved. The Ñux from the ring would provide a collimating magnetic pressure. However, the e†ects of this on the pair creation scattering geometry were ignored in the text for simplicity. A.2.

THE PLASMA HORIZON

A necessary consequence of the pairing of a charged black hole with an equal but oppositely charged current ring is the existence of a plasma horizon. The basic idea is that the quadrupole moment of the electric Ðeld becomes dominant at radii

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Vol. 498

FIG. 10.ÈField line angular velocity, )1 , as a function of latitude, h, on the plasma horizon F

larger than that of the ring, r [ R, while the magnetic Ðeld is dipolar. At a large enough radius, the particles can exist in E Â B drift trajectories and not be sucked into the hole. We are however, looking for a very particular type of plasma horizon, a boundary beyond which a tenuous charged separated magnetosphere can exist in time stationary form. Inside the boundary there exist unbalanced electrostatic forces. For this boundary to be stable, one needs no net electromagnetic force orthogonal to the plasma horizon

C

D

VÕ (E ) \ [ F eü xB Æ eü , M M vac c Õ

(A.6)

where (E ) is the component of the electric Ðeld from the hole and ring that is orthogonal to the plasma horizon. The unit vac eü are in the azimuthal direction and orthogonal to the plasma horizon, respectively. The rotational velocity of vectors eü M and Õ M the Ðeld is V Õ \ )1 r sin h , (A.7) F F where )1 is the angular velocity of the Ðeld lines in the zone of closed and dead Ðeld lines. The Ðeld line angular velocity is a functionFthat can vary from Ðeld line to Ðeld line since the magnetic Ðeld is not frozen into the black hole. Relation (A.6) is equivalent to the vanishing of the orthogonal component of the proper electric Ðeld, E@ . The vanishing of M the parallel component, E , yields A (E ) \ [()1 r sin /eü ] B) Æ eü , (A.8) A vac F Õ A where expression (A.8) is evaluated at the plasma horizon and eü is the poloidal unit vector in the plasma horizon. ConseA within the GoldreichÈJulian magnetosphere on the other quently, at the interface (E ) is sourced signiÐcantly by charges A vac side of the plasma horizon. We chose the plasma horizon to be deÐned by r \ ph

2.5 ] 107 cm , (sin h)1@2

27¡.7 \ h \ 152¡.3 .

(A.9)

The range of angular coordinate is set by the intersection of the plasma horizon with the boundary of the zone of closed dead Ðeld lines (see Fig. 9). The Ðeld line angular velocity along the plasma horizon is determined from (A.6) and (A.9) is is plotted in Figure 10. Notice that the angular velocity changes sign near h \ 60¡. This is a reÑection of the fact that the quadrupole moment of the electric Ðeld is very prominent. The orthogonal electric Ðeld, (E ) , changes sign and the charges move to cancel the proper electric M vac Ðeld. The resulting azimuthal motion and the frozen-in condition are the dynamics that determine )1 on the Ðeld lines. F Consequently, as the electric Ðeld changes sign, so must )1 . F A.3.

SEMIVACUUM REGION

Inside the plasma horizon is a region in which the dynamics are dominated by vacuum electromagnetic Ðelds. The region is considered to be relatively devoid of plasma and nontime stationary. There is no obvious reason for plasma to enter or be created in this region, and any plasma that is introduced will either enter the hole or join the current ring.

No. 2, 1998

GAMMA-RAY EMISSION FROM BLACK HOLES. I. A.4.

659

GOLDREICHÈJULIAN CHARGE ZONE

The major component of the zone of closed dead Ðeld lines is a time stationary zone of GoldreichÈJulian charge. We pick the plasma to be charge separated, but there is nothing precluding a tenuous MHD plasma supporting the GoldreichÈJulian charge density. Notice from Figures 9 and 10 that at no point do the Ðeld lines rotate at or beyond the speed of light, so a static charge density is possible and there are no open Ðeld lines transporting charges out of the zone of closed dead Ðeld lines. The GoldreichÈJulian charge density is more difficult to compute than for a pulsar since )1 is not a constant. Using F equation (A.5) and the dipolar approximation

C

D

r sin2 h L L 22 Qa (3 cos2 h [ 1))1 [ ()1 ) ] cos h sin h ()1 ) . o \[ F GhJ 2 Lr F Lh F 2ne r3 The frozen-in condition means that )1 is constant along each Ðeld line F L 1 L 2 cos h ()1 ) ] sin h ()1 ) \ 0 , Lr F r Lh F

(A.10)

(A.11a)

A B

sin h , )1 \ )1 F r1@2

(A.11b)

where )1 is a function of the variable sin h/r1@2. Applying (A.11) to (A.10), we obtain

C

DA B

sin h ) BZ 22Qa sin h 3 cos2 h ] 1 )@ , o \[ F [ GhJ r1@2 r1@2 2ne 2ner3 4

(A.12)

where the Ðrst term is what is found for pulsar magnetospheres and the second term is a consequence of the gradient in )1 . F Using the values of )1 on the plasma horizon, (A.12) can be solved numerically. One has the approximate analytic F expressions ) Bz o B F , GhJ 2ne

22Qa o B GhJ 2ner3

G

27¡.7 \ h \ 40¡ ,

[ ) (3 cos2 h [ 1) ] F

87¡ \ h \ 93¡ ,

140¡ \ h \ 152¡.3 ;

H

(3 cos2 h ] 1) (R /r)1@2 sin h 0 [426 s~1] , 4 [1 [ (R /r) sin2 h]1@2 0 95¡ \ h \ 135¡ ;

(A.13a)

45¡ \ h \ 85¡ , (A.13b)

R 4 2.5 ] 107 cm . (A.13c) 0 Note that in equation (A.13b) the gradient in )1 is the dominant term. Thus o [ 0 even when )1 \ 0, as indicated in F is a very small region near the GhJequator just beyond F the plasma horizon Figure 9. Also, equation (A.13a) implies that there where o \ 0. This region is too small to be resolved in Figure 9. GhJ REFERENCES Arons, J. 1983, ApJ, 266, 215 Misner, C., Thorne, K., & Wheeler, J. 1973, Gravitation (San Francisco : Batygin, V. V. & Toptygin, I. N. 1964, Problems in Electrodynamics (New Freeman) York : Academic Press, Inc.) Nitta, S., Takahashi, M., & Tomimatsu, A. 1991, Phys. Rev. D, 44, 2295 Begelman, M., Blandford, D., & Rees, M. 1984, Rev. Mod. Phys., 56, 265 Ozel, M. E., & Thompson, D. J. 1996, ApJ, 463, 105 Bicak, J. & Dvorak, L. 1976, Gen. Relativ. & Gravitation, 7, 959 Petterson, J. A. 1975, Phys. Rev. D, 12, 8 Camenzind, M. 1986, A&A, 156, 137 Phinney, E. S. 1983, Ph.D. thesis, Univ. Cambridge Cheng, K. S., Ho, C., & Ruderman, M. 1986, ApJ, 300, 522 Punsly, B. 1991, ApJ, 372, 424 Chitre, D. M., & Visheshwara, C. V. 1975, Phys. Rev. D, 12, 1538 ÈÈÈ. 1996, ApJ, 467, 105 Christodoulou, D. 1970, Phys, Rev. Lett., 25, 1596 ÈÈÈ. 1998a, ApJ, 498, 660 (Paper II) Darmour, T., Hanni, R. S., Ruffini, R., & Wilson, J. R. 1978, Phys. Rev. D, ÈÈÈ. 1998b, ApJ, submitted 17, 1518 Punsly, B., & Coroniti F. 1989, Phys. Rev. D, 40, 3834 Daugherty, J. K., & Harding, A. K. 1996, ApJ, 458, 278 ÈÈÈ. 1990a, ApJ, 350, 318 Erber, T. 1966, Rev. Mod. Phys., 38, 626 ÈÈÈ. 1990b, ApJ, 354, 583 Gibbons, G. W. 1975, Commun. Math. Phys., 44, 245 Ruderman, R. A., & Sutherland, P. G. 1975, ApJ, 196, 51 Goldreich, P., & Julian, W. H. 1969, ApJ, 157, 869 Ru†ini, R. 1973, in Black holes, ed. B. Dewitt & C. Dewitt (New York : Hanni, R. S. 1977, in Proc. 1st Marcel Grossman Meeting on General Gordon & Breach), 525 Relativity, ed. R. Ruffini (Amsterdam : North-Holland), 429 ÈÈÈ. 1977, in Proc. 1st Marcel Grossman Meeting on General RelaJackson, J. D. 1975, Classical Electrodynamics (New York : Wiley) tivity, ed. R. Ruffini (Amsterdam : North-Holland), 349 Kennel, C., Fujimura, F., & Okamoto, I. 1983, Geophys. Astrophys. Fluid Sakurai, T. 1985, A&A, 152, 121 Dyn., 26, 147 Sambruna, R. M., et al. 1995, ApJ, 449, 567 Kennel, C., Fujimura, F., & Pellat, R. 1979, Space Sci. Rev. 24, 407 Scharlemann, E. T., Arons, J., & Fawley, W. M. 1978, ApJ, 222, 297 King, A. R., Lasota, J. P., & Kundt, W. 1975, Phys. Rev. D, 12, 3037 Shapiro, S. L. & Teukolsky, S. A. 1983, Black Holes, White Dwarfs, and Kittel, C. 1958, Elementary Statistical Physics (New York : Wiley) Neutron Stars (New York : Wiley) Lightman, A. P., et al. 1979, Problem Book in Relativity and Gravitation Spitzer, L. 1978, Physical Processes in the Interstellar Medium (New York : (Princeton, NJ : Princeton Univ. Press) Wiley) Macdonald, D., & Thorne, K. 1982, MNRAS, 198, 345 Takahashi, M., et al. 1990, ApJ, 363, 206 Maraschi, L., Ghisellini, G., & Celotti, A. 1992, ApJ, 397, L5 Thorne, K. S., Price, R., & MacDonald, D. ed. 1986, Black Holes : The Michel, F. C. 1969, ApJ, 157, 1183 Membrane Paradigm (New Haven : Yale Univ. Press) ÈÈÈ. 1982, Rev. Mod. Phys., 54, 1 Wald, R. M. 1974, Phys. Rev. D, 10, 1860