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05WEB LINK FOR HIGH RESOLUTION DOCUMENT IN FOOTNOTEPreprint typeset using LATEX style emulateapj v. 11/26/04

JET FORMATION IN BLACK HOLE ACCRETION SYSTEMS II: NUMERICALMODELS

JONATHAN C. MCK INNEY1

June 16, 2005

ABSTRACTIn a companion theory paper, we presented a unified model of jet formation. We suggested that primarily two

types of relativistic jets form near accreting black holes:a potentially ultrarelativistic Poynting-dominated jetand a Poynting-baryon jet. We showed that, for the collapsarmodel, the neutrino-driven enthalpy flux (classicfireball model) is probably dominated by the Blandford-Znajek energy flux, which predicts a jet Lorentz factorof Γ∼ 100− 1000. We showed that radiatively inefficient AGN, such as M87, are synchrotron-cooling limitedto Γ∼ 2− 10. Radiatively efficient x-ray binaries, such as GRS1915+105, are Compton-drag limited toΓ. 2,but the jet may be destroyed by Compton drag. However, the Poynting-baryon jet is a collimated outflow withΓ ∼ 1− 3. Here we present general relativistic hydromagnetic simulations of black hole accretion with paircreation used to simulate jet formation in GRBs, AGN, and x-ray binaries. Our collapsar model shows thedevelopment of a patchy “magnetic fireball” with typicallyΓ∼ 100−1000 and a Gaussian structure. Temporalvariability of the jet is dominated by toroidal field instabilities for& 102 gravitational radii. A broader Poynting-baryon jet withΓ∼ 1.5 could contribute to a supernova.

Subject headings: accretion disks, black hole physics, galaxies: jets, gammarays: bursts, X-rays : bursts,supernovae: general, neutrinos

1. INTRODUCTION

General relativistic magnetohydrodynamics (GRMHD) isthe black hole mass-invariant (nonradiative) physics com-monly used to describe black hole accretion systems. Suchsystems often exhibit jets. However, the observed jet prop-erties, such as collimation and speed, are not uniform be-tween systems. In McKinney (2005b), our goal was to deter-mine the unifying, or minimum number of, pieces of physicsthat would explain jet formation associated with gamma-raybursts (GRBs), active galactic nuclei (AGN), and x-ray bi-naries. Similar ideas have been explored by other authors(Ghisellini & Celotti 2002; Ghisellini 2003; Meier 2003).The goal of McKinney (2005b) was to understand jetforma-tion and explain the origin of the energy, composition, colli-mation, and Lorentz factor.

This paper implements some of the theoretical ideas ofMcKinney (2005b) into a GRMHD numerical model. In par-ticular, we study the collapsar model with pair creation andan effective model for Fick diffusion of neutrons that con-taminate the jet with an electron-proton plasma. The genericmodel parameters also allow the numerical model to be inter-preted as a model for AGN systems such as M87. We alsocomment on the applicability of the model to x-ray binarysystems.

§ 2 summarizes the proposed unified model to explain jetformation in all black hole accretion systems, where more de-tails are provided in McKinney (2005b).

§ 3 presents the results of a fiducial numerical model corre-sponding to the collapsar model and describes the jet structureand formation. The relevance of the presented model to AGNand x-ray binaries is discussed. The jet is found to have somepiece-wise self-similar features, and curve fits are given foruse by other modellers. Acceleration of the GRB jet is foundto occur over a large range in radii rather than occurring close

1 Institute for Theory and Computation, Harvard-Smithsonian Center forAstrophysics, 60 Garden Street, MS 51, Cambridge, MA 02138,USAHigh Res. Figures:http://rainman.astro.uiuc.edu/~jon/jet2.pdfElectronic address: jmckinney@cfa.harvard.edu

to the black hole. The introductory estimates of the Lorentzfactor in McKinney (2005b) are refined based upon these nu-merical models. The jet characteristic structure, such as theformation of a double transonic (transfast) flow, is discussed.

§ 4 discusses the results and their possible implications.The results are compared to similar investigations, and thelimitations of the models presented are discussed.

§ 5 summarizes the key points.In appendix A, we give the GRMHD equations of motion

solved when accounting for pair creation. See also McKinney(2005b) for relevant discussions. These are the equations nu-merically solved that give the results discussed in section3.These are straightforward and how one handles the detailsof the pair creation ends up not making any qualitative (ormuch quantitative) difference in the results, hence why thematerial is in the appendix. That is, the energetics of thePoynting-dominated jet are dominated by the electromagneticfield. Appendix B summarizes the well-known characteristicsof the GRMHD equations and other surfaces of physical in-terest used in the discussion in section 3.

2. GRMHD PAIR INJECTION MODEL OF JET FORMATION

In McKinney (2005b), we showed that one can identify asmall subset of physics that can explain the jet energy, com-position, collimation, and Lorentz factor for all black hole ac-cretion systems.

One key idea of that paper is that the terminal Lorentz fac-tor is determined by the toroidal magnetic energy per unit pairmass density energy near the location where pairs can escapeto infinity (beyond the so-called “stagnation surface”). ForGRBs, neutron diffusion is crucial to explain (and limit) theLorentz factor and explain the baryon-pollution or baryon-contamination problem. For AGN and x-ray binaries, sincea negligible number of baryons cross the field lines, pair-loading from radiative annihilation is crucial to determine theLorentz factor of the Poynting-dominated jet since this deter-mines the rest-mass flux and density in the jet.

Figure 1 shows the basic picture for GRB systems, whilefigure 2 shows the basic picture for AGN and x-ray binary

2 McKinney, J.C.

e+e−

νe _νe

EM JET

DISK

FAR−FIELD JET

FIREBALLMAGNETIC

100−1000 RgSHOCKSTRANSONIC

CORONA

NEUTRONDIFFUSION

STAGNATIONSURFACE

CORONALOUTFLOW

BH

ENVELOPE

FIG. 1.— Schematic of pair-production model and subsequent magneticfireball formation for GRB disks. Fireball is extremely optically thick. Belowa stagnation surface, pairs are accreted by the black hole and so do not loadthe jet. HereRg = GM/c2.

e+e−

γ softγhard

EM JET

CORONA

DISK

FAR−FIELD JET

PATCHY, COLDJET

SURFACESTAGNATION

TRANSONICSHOCKS100−1000 Rg

CORONALOUTFLOW

BH

ISM/ETC

FIG. 2.— Schematic of pair-production model and subsequent shock-heating and emission. AGN jet is optically thin and emits nonthermal andthermal synchrotron, while x-ray binary jet can be marginally optically thickand emit via self-absorbed synchrotron and by severe Compton drag. SevereCompton drag can lead to destruction of the Poynting-dominated jet.

systems. An accreting, spinning black hole creates a mag-netically dominated funnel region around the polar axis. Therotating black hole drives a Poynting flux into the funnel re-gion, where the Poynting flux is associated with the coiling ofpoloidal magnetic field lines into toroidal magnetic field lines.The ideal MHD approximation holds very well, and so onlyneutral particles such as photons, neutrinos, and neutronscancross the field lines and load the Poynting-dominated jet thatemerges.

The accretion disk emits neutrinos in a GRB model (γ-rayand many soft photons for AGN and x-ray binaries) that an-nihilate and pair-load the funnel region within some “injec-tion region.” For GRB systems, neutrons Fick-diffuse acrossthe field lines and collisionally decay into an electron-protonplasma.

Many pairs (any type) are swallowed by the black hole, butsome escape if beyond some “stagnation surface,” where thetime-averaged poloidal velocity is zero and positive beyond.

Pairs beyond the stagnation surface are then accelerated bythe Poynting flux in a self-consistently generated collimatedoutflow. In the electromagnetic (EM) jet, the acceleration pro-cess corresponds to a gradual uncoiling of the magnetic fieldand a release of the stored magnetic energy that originatedfrom the spin energy of the black hole.

One key result of this paper is that the release of magneticenergy need not be gradual once the toroidal field dominatesthe poloidal field, in which case pinch (and perhaps kink) in-stabilities can occur and lead to a nonlinear coupling (e.g.ashock) that converts Poynting flux into enthalpy flux (Eichler1993; Begelman 1998). In the proposed GRB model, this con-version reaches equipartition and the jet becomes a “magneticfireball,” where the toroidal field instabilities drive large vari-ations in the jet Lorentz factor and jet luminosity.

In McKinney (2005b), we showed that in AGN systems,nonthermal synchrotron from shock-accelerated electronsandsome thermal synchrotron emission releases the shock en-ergy until the synchrotron cooling times are longer than thejet propagation time. For AGN, jet acceleration is negligi-ble beyond theextended shock zone, as suggested for blazarsbeyond the “blazar zone” (Sikora et al. 2005). In x-ray binarysystems, the shock is not as hot and also unlike in the AGN (atleast those like M87) case the jet can be optically thick. Thusthese x-ray binary systems self-absorbed synchrotron emitifthey survive Compton drag.

For all these systems, at large radii patches of energy fluxand variations in the Lorentz factor develop due to toroidalinstabilities. These patches in the jet could drive internalshocks and at large radii they drive external shocks with thesurrounding medium. The EM jet is also surrounded by amildly relativistic matter coronal outflow/jet/wind, which isa material extension of the corona surrounding the disk. ThisPoynting-baryon, coronal outflow collimates the outer edgeofthe Poynting-dominated jet, which otherwise internally col-limates by hoop stresses. The luminosity of the Poynting-baryon jet is determined, like the Poynting-dominated jet,bythe mass accretion rate, disk thickness, and black hole spin.

In this paper, this unified model is studied numericallyusing axisymmetric, nonradiative, GRMHD simulations tostudy the self-consistent process of jet formation from blackhole accretion systems. These simulations extend the workof McKinney & Gammie (2004) by including pair creation(and an effective neutron diffusion for GRB-type systems) toself-consistently treat the creation of jet matter, investigatinga larger dynamic range in radius, and presenting a more de-tailed analysis of the Poynting-dominated jet structure.

2.1. Units and Notation

The units in this paper haveGM = c = 1, which sets thescale of length (rg ≡ GM/c2) and time (tg ≡ GM/c3). Themass scale is determined by setting the (model-dependent)observed (or inferred for GRB-type systems) mass accretionrate (M0[g s−1]) equal to the accretion rate through the blackhole horizon as measured in a simulation. So the mass isscaled by the mass accretion rate (M0) at the horizon (r =rH ≡ rL(1+

√

1− j2)), such thatρ0,disk ≡ M0[r = rH ]tg/r3g and

the mass scale is then justm ≡ ρ0,diskr3g = M0[r = rH ]tg. Un-

less explicitly stated, the magnetic field strength is giveninHeaviside-Lorentz units, where the Gaussian unit value is ob-tained by multiplying the Heaviside-Lorentz value by

√4π.

The value ofρ0,disk can be determined for different sys-tems. For example, a collapsar model withM = 0.1M⊙s−1 and

Black Hole Jet Formation: Numerical Models 3

M ≈ 3M⊙, thenρ0,disk ≈ 3.4×1010gcm−3. M87 has a massaccretion rate ofM ∼ 10−2M⊙ yr−1 and a black hole massof M ≈ 3× 109M⊙ (Ho 1999; Reynolds et al. 1996) givingρ0,disk ∼ 10−16gcm−3. GRS 1915+105 has a mass accretionrate of M ∼ 7× 10−7M⊙ yr−1 (Mirabel & Rodriguez 1994;Mirabel & Rodríguez 1999; Fender & Belloni 2004) with amass ofM ∼ 14M⊙ (Greiner et al. 2001), but see Kaiser et al.(2004). This givesρ0,disk ∼ 3×10−4gcm−3. This disk densityscales many of the results of the paper. The disk height (H) toradius (R) ratio is written asH/R.

The GRMHD notation follows MTW. For example, the4-velocity components areuµ (contravariant) oruµ (covari-ant). For a black hole with angular momentumJ = jGM2/c,j = a/M is the dimensionless Kerr parameter with−1≤ j ≤ 1.The rest-mass density is given asρ0, internal energy density

as u, magnetic field 3-vector asBi ≡ ∗

Fit, where

∗

F is thedual of the Faraday tensor. The contravariant metric com-ponents aregµν and covariant components aregµν , wherebeyond the frame-dragging of the black hole the metric inBoyer-Lindquist coordinates is approximately diagonal suchthat an orthonormal contravariant component isuµ =

√gµµuµ

for any spatial component ofuµ. The comoving energy den-sity is b2/2, wherebµ is the comoving magnetic field. SeeMcKinney & Gammie (2004) for details.

The Lorentz factor of the jet can be measured either as thecurrent time-dependent value, or, using information abouttheGRMHD system of equations, one can estimate the Lorentzfactor at large radii from fluid quantities at small radii. TheLorentz factor as measured by a static observer at infinity is

Γ≡ ut = ut√−gtt (1)

in Boyer-Lindquist coordinates, where no static observersexist inside the ergosphere. This is as opposed toW ≡ut√

−1/gtt, which is the Lorentz factor as measured by thenormal observer as used by most numerical relativists.

In McKinney (2005b) we show that the Lorentz factor andφ-velocity at large distances are

Γ∞ =E = −hut +ΦΩFBφ (2)

uφ∞ =L = huφ +ΦBφ, (3)

whereh = (ρ0+ug + p)/ρ0 is the specific enthalpy,Φ is the con-served magnetic flux per unit rest-mass flux,ΩF is the con-served field rotation frequency,Bφ is the covariant toroidal

magnetic field, anduφ∞ = uφ,∞. HereE andL simply rep-

resent the conserved energy and angular momentum flux perunit rest-mass flux. Notice the matter and electromagneticpieces are separable, such that

Γ∞ =Γ(MA)∞ +Γ

(EM)∞ (4)

uφ,∞ =u(MA)φ,∞ + u(MA)

φ,∞, (5)

See McKinney (2005b) for more details.

3. NUMERICAL EXPERIMENTS

This section presents the results of GRMHD numeri-cal models with pair creation to self-consistently describethe Poynting-dominated and Poynting-baryon jet formationprocess. Our numerical scheme is HARM (Gammie et al.2003a), a conservative, shock-capturing scheme for evolv-ing the equations of general relativistic MHD. Compared tothe original HARM, the inversion of conserved quantities toprimitive variables is performed by using a new faster and

more robust two-dimensional non-linear solver (Noble et al.2005). The new HARM also uses a parabolic interpola-tion scheme (Colella 1984) rather than a linear interpolationscheme. The new HARM also uses an optimal TVD thirdorder Runge-Kutta time stepping (Shu 1997) rather than themid-point method. For the problems under consideration, theparabolic interpolation and third order time stepping methodreduce the truncation error significantly, including magneti-cally dominated regions withb2/ρ0 ≫ 1.

Notice that no explicit reconnection model is included.However, HARM checks the effective resistivity by measur-ing the rest-mass flux across field lines. For the boundary be-tween the coronal wind and the Poynting-dominated jet, thetotal mass flux across the field line is negligible compared tothe rest-mass flux along the field line and the pair creationrate. An unresolved model would load field lines with rest-mass that crosses field lines and not properly represent anyphysical resistivity (McKinney 2004).

3.1. Computational Domain

The computational domain isaxisymmetric, with a grid thattypically extends fromrin = 0.98rH , whererH is the horizon,to rout = 104rg, and fromθ = 0 to θ = π. Full 3D calcula-tions with this dynamic range are not possible with today’scomputers. Our numerical integrations are carried out on auniform grid in so-called “modified KS” (MKS) coordinates:x0,x1,x2,x3, wherex0 = t[KS], x3 = φ[KS]. The radial coordi-nate is chosen to be

r = R0 + exnr1 , (6)

whereR0 is chosen to concentrate the grid zones toward theevent horizon (asR0 is increased from 0 torH ) andnr is con-trols the enhancement of inner to outer radial regions. Forstudies where the disk and jet interaction is of primary inter-est,R0 = 0 andnr = 1 are chosen. For studies where in additionthe far-field jet is of interest,R0 = −3 andnr = 10 are chosen.Theθ coordinate is chosen to be

θ = πx2 +12

(1− h(r))sin(2πx2), (7)

whereh(r) is used to concentrate grid zones toward the equa-tor (ash is decreased from 1 to 0) or pole (ash is increasedfrom 1 to 2). The jet at large radii is resolved together withthe disk at small radii using

h(r) = 2− Q j(r/r0 j)−n jg j (8)

with the parameters ofQ j = 1.3 − 1.8, r0 j = 2.8, n j = 0.3,r1 j = 20, r2 j = 80, andg j = g j(r) = 1

2 + 1π

atan( r−r2 j

r1 j) is used to

control the transition between inner and outer radial regions.To perform convergence tests,Q j and the overall radial andθ resolution are varied. An alternative to this fixed refine-ment of the jet and disk is an adaptive refinement (see, e.g.,Zhang & MacFadyen 2005).

3.2. Initial Conditions and Problem Setup

All the experiments evolve a weakly magnetized torusaround a Kerr black hole in axisymmetry. The focus of ournumerical investigation is to study a high resolution modelofthe collapsar model. Other simulations were performed, andthe fiducial model’s relevance to AGN and x-ray binary sys-tems is discussed at the end of the section. A generic modelis used so the results mostly can be applied to any black holesystem. A black hole spin ofj = 0.9375 is chosen, but this

4 McKinney, J.C.

produces similar results to models with 0.5. j . 0.97, whichincludes most black hole accretion systems.

The initial conditions consist of an equilibriumtorus which is a “donut” of plasma with a blackhole at the center (Fishbone & Moncrief 1976;Abramowicz, Jaroszinski, & Sikora 1978). The donut issupported against gravity by centrifugal and pressure forces.The solutions of Fishbone & Moncrief (1976) correspondingto utuφ = const. are used. The initial inner edge of the torusis set atredge = 6. The equation of state is a gamma-lawideal gas withγ = 4/3, but otherγ lead to similar results(McKinney & Gammie 2004). This donut is a solution to theaxisymmetric stationary equations, but the donut is unstableto global nonaxisymmetric modes (Papaloizou & Pringle1983). However, when the donut is embedded in a weakmagnetic field, the magnetorotational instability (MRI)dominates those hydrodynamic modes. Small perturbationsare introduced in the velocity field to seed the instability.Themodels haveutuφ = 4.281, the pressure maximum is locatedat rmax = 12, the inner edge at (r,θ) = (6,π/2), and the outeredge at (r,θ) = (42,π/2).

This is consistent with the accretion disk that is expected toform in the collapsar model (MacFadyen & Woosley 1999).A GRB in the collapsar model is formed after the collapseof a massive star as a black hole withMBH ∼ 3M⊙ and spinj ∼ 0.75−0.94 accretes atM0 ∼ 0.1M⊙/s from a torus of mat-ter within r < 200km that has a height (H) to radius (R) ra-tio (“thickness”) ofH/R ∼ 0.2− 0.4. MacFadyen & Woosley(1999) find that an energetic outflow develops in the polar re-gion whereρ ∼ 107gcm−3 near the horizon,ρ ∼ 105gcm−3

at r ∼ 200km, andρ ∼ 103−4gcm−3 at r ∼ 2000km (seemodel 14A in MacFadyen & Woosley 1999). In the par-lance of collapsar models, ourutuφ model corresponds to aj16 ≡ j/(1016cm2s−1) of j16 ∼ 5, within the range suggestedby MacFadyen & Woosley (1999) to produce a GRB. Also,theH/R∼ 0.2−0.4 as suggested that forms according to neu-trino emission models (Kohri & Mineshige 2002; Kohri et al.2005).

For radiatively inefficient AGN and x-ray binaries this issimply a reasonable approximate model for the inner-radialaccretion flow. While a slightly thicker disk (H/R∼ 0.9) maybe more appropriate for radiatively inefficient flows, this isnot expected to significantly affect these results. A study ofthe effect of disk thickness, especially thin disks, is leftforfuture work.

The orbital period at the pressure maximum 2π(a +(

rmax/rg

)3/2)tg ≃ 267tg, as measured by an observer at infin-

ity. The model is run for∆t = 1.4× 104tg, which is about52 orbital periods at the pressure maximum and about 1150orbital periods at the black hole horizon. For the collapsarmodel this is only∼0.2 seconds, and at the spin chosen wouldcorrespond to the late phase of accretion when the Poyntingflux reaches its largest magnitude. For the AGN M87 this cor-responds to∼ 7 years. For the x-ray binary GRS 1915+105this corresponds to∼ 1 second.

Notice that since the model is axisymmetric, disk turbu-lence is not sustained after aboutt ∼ 3000tg ; that is, the anti-dynamo theorem prevails (Cowling 1934). However, whilethis affects the disk accretion, this does not affect the evolu-tion of the Poynting-dominated jet. That is, from the time ofturbulent accretion to “laminar” accretion, the funnel regionis mostly unchanged. Indeed, the far-field jet that has alreadyformed is causally disconnected from the region where the

accretion disk would still be turbulent.For the collapsar model, at the quasi-steady state mass ac-

cretion rate of 0.1M⊙/s the disk will last fort . 0.42s. Thus,this model requires a more extended disk or a fresh supplyof plasma into the disk from the surrounding stellar enve-lope through an “accretion shock” to generate a long dura-tion GRBs (MacFadyen & Woosley 1999). However, sinceany such system modelled is in quasi-steady state, the resultshere do not strongly depend on the mass of the disk as long asthe disk is not too massive, which would require taking intoaccount the self-gravity of the disk.

The numerical resolution of these models is 512×256 com-pared to 4562 in McKinney & Gammie (2004). However, dueto the enhancedθ grid, the resolution in the far-field jet regionis∼ 10 times larger. Also, with the use of a parabolic interpo-lation scheme, the overall resolution is additionally enhanced.Compared to our previous model this gives us an effectiveθresolution of≈ 9000.

As with McKinney & Gammie (2004), into the initial torusis put a purely poloidal magnetic field. The field can be de-scribed using a vector potential with a single nonzero com-ponentAφ ∝ MAX( ρ0/ρ0,max − 0.2,0) The field is thereforerestricted to regions withρ0/ρ0,max > 0.2. The field is nor-malized so that the minimum ratio of gas to magnetic pres-sure is 100. The equilibrium is therefore only weakly per-turbed by the magnetic field. It is, however, no longer sta-ble (Balbus & Hawley 1991; Gammie 2004). Hirose et al.(2004); McKinney & Gammie (2004) show that no initiallarge-scale net vertical field is necessary, since a large-scalepoloidal field is self-consistently generated. The field con-nects the black hole horizon (as observed by a physical ob-server) to large distances. Apart from the existence of an over-all poloidal sign, the results are insensitive to the details of theinitial magnetic field geometry for any physically motivatedgeometry (McKinney & Gammie 2004).

3.3. Floor vs. Pair Creation Model

Numerical models often “model” the injection physics inthe Poynting-dominated jet by employing a so-called floormodel. This model forces a minimum on the rest-mass den-sity (ρ f l) and internal energy density (u f l), which are usuallyset to several orders of magnitude lower than the disk density.Actually, however, floor models have always been treated asa purely numerical invention in order to avoid numerical ar-tifacts associated with the inability to numerically solvetheequations of motion when the density is low. Indeed, the ideawas one should convergence test by gradually loweringρ f landu f l. This is perhaps a reasonable for numerical study ofstars, but accretion flows are so hot (baryons withT & 1010Kin the thick disk state near the black hole) that pairs are cre-ated when neutrinos annihilate (or when photons annihilatefor x-ray binary and AGN systems).

In previous numerical models of Poynting jets, the funnelregion is always completely evacuated, and floor-models nec-essarily generate mass at least where the poloidal velocityup = 0 at the stagnation surface. That is, matter inside the stag-nation surface falls into the black hole, while matter outsideit is ejected as part of the jet. Indeed, arbitrary floor-modelsviolate the ideal-MHD condition far away from the black holewhere no pairs should be produced. For example, if a numer-ical model usesρ f l ∼ Const., then this leads to a completelyunphysical model of pair creation and the ideal-MHD approx-imation is violated for the entire length of the jet in the funnel.

While pair creation has so far been ignored by those doing

Black Hole Jet Formation: Numerical Models 5

numerical models of jets from black hole accretion systems,ithas been shown that the properties of the Poynting-dominatedjet are almost completely determined by the pair creationphysics, which thus cannot be ignored (Phinney 1983; Punsly1991; Levinson 2005).

Our collapsar model accounts for pair creation and Fick dif-fusion of neutrons. In this paper, there is an injected rest-mass, enthalpy, and momentum. For the fiducial collapsar-like model, rest-mass, enthalpy, and momentum are injectedwith energy at infinity fractions of, respectively,fρ = 0.05,fh = 0.45, and fm = 0.5 as described in McKinney (2005b).The total energy at infinity injected is determined by neutrinoannihilation rates and Fick diffusion rates. The energy injec-tion rate from neutrino annihilation is based upon viscous diskmodels (Popham et al. 1999; MacFadyen & Woosley 1999),where the rate of injection is determined by choosing a vis-cosity coefficientα = 0.01 and mass accretion rateM0 =0.1M⊙s−1 for the collapsar model. As discussed in that pa-per, the rest-mass is dominated by electron-positron pairsonlyvery close to the black hole.

The value ofα and M0 determine the energy injected inproportion to the disk density per unit light crossing time (tg).Thus any system with comparable normalized energy injec-tion rate would follow similar results. For example, the M87model is similar, where there the electron-positron pairs donot annihilate and represent the true density in the jet. In theM87 model there is no Fick diffusion, yet the dimensionlessmodel parameters are approximately the same.

The momentum direction is chosen by settinguφin j = uφ and

uθin j = 0, so thatfm determinesur

in j. It turns out that the choiceof fh, fm, and the injected momentum direction very weaklydetermine the jet structure forr & 6rg. This is because a largefraction of the pairs are lost to the black hole and magneticforces, rather than the injected momentum, dominate the mo-tion of the plasma near the black hole.

Beyond this injection region, the rest-mass is dominatedby electron-protons created in collision events from neutronsthat Fick diffuse across the field lines into the jet region. Forthe collapsar model this coincidentally corresponds to thein-jected rest-mass injected in this model. Since the electron-positron pair annihilation only gives an additional∼ 10% ofinternal energy, then pair annihilation need not be considered,and the rest-mass injected well-models the rest-mass injectedas an electron-proton plasma from Fick-diffused neutrons thatcollisionally decay.

This physical model of the injected particles is augmentedwhen the rest-mass or internal energy density reaches verylow values. If the density drops below some “floor” value,then the density is returned to the floor value. The floor modelchosen hasρ f l = 10−7r−2.7ρ0,disk and u f l = 10−9r−2.7ρ0,disk.HARM keeps track of how often the density goes below thefloors and how this modifies the conservation of mass, en-ergy, and angular momentum. The floor model contributionis negligible compared to the physical injection model at allspatial locations and at all times. The coefficient of the floorwas chosen to be close to the resulting density near the blackhole horizon. It was not chosen arbitrarily small in order toavoid numerical difficulties in integrating the equations when,rarely, the floor is activated in regions of large magnetic en-ergy density per unit rest-mass density.

3.4. Boundary Conditions

Two different models are chosen for the outer boundarycondition. One model is called an “outflow” boundary con-dition, for which all primitive variables are projected into theghost zones while forbidding inflow. The other model is toinject matter at some specified rate at the outer boundary, un-less there is outflow. This would correspond to a Bondi-likeinfall for AGN and x-ray binaries, or would correspond tothe collapsing envelope of a massive star for collapsar mod-els. In particular, unless there is outflow, the outer bound-ary is set to inject mass at the free-fall rate, with a densityof ρ0 = 10−4r−3/2ρ0,disk andu = 10−6r−5/2ρ0,disk and no angularmomentum. Presupernova models suggest that the infallingmatter is at about 30% the free-fall rate we have chosen above(MacFadyen & Woosley 1999), but this difference is unlikelyto significantly affect the jet formation. Also, for their collap-sar model, the density structure in the equatorial plane variesbetweenρ0 ∝ r−3/2 to ρ0 ∝ r−2 and the internal energy isu ∝ r−5/2 to u ∝ r−2.7 (MacFadyen & Woosley 1999). This issimilar to our simplified model, which happens to also modela Bondi accretion for AGN and x-ray binaries. Thus the re-sults are indicative of GRBs, AGN, and x-ray binaries. Theouter grid radius corresponds to about 20 presupernova coreradii (Woosley & Weaver 1995) or∼ 1010km or about 1/10ththe entire star’s radius.

3.5. Fiducial Model

The overall character of the accretion flow is unchangedcompared to the descriptions given in McKinney & Gammie(2004). The disk enters a long, quasi-steady phase in whichthe accretion rates of rest-mass, angular momentum, and en-ergy onto the black hole fluctuate around a well-defined mean.Meanwhile, as in McKinney & Gammie (2004), a Poynting-dominated jet and Poynting-baryon jet (coronal outflow) haveformed. The Poynting-dominated jet forms once the ram pres-sure of the funnel material is lower than the toroidal mag-netic pressure. Afterwards the baryons are spread apart in thelaunch of a magnetic tube filled with pairs. This occurs withint . 500tg.

The Poynting-dominated jet forms as the differential rota-tion of the disk and the frame-dragging of the black hole in-duce a significant toroidal field that launches material awayfrom the black hole by the same force described in McKinney(2005b).

A coronal outflow is also generated between the disk andPoynting-dominated jet. In this model the coronal outflowhasΓ∞ ∼ 1.5. The coronal-funnel boundary contains shockswith a sonic Mach number ofMs ∼ 100. The inner-radialinterface between the disk and corona is a site of vigorousreconnection due to the magnetic buoyancy and convectiveinstabilities present there. These two parts of the corona areabout 100 times hotter than the bulk of the disk. Thus thesecoronal components are a likely sites for Comptonization andnonthermal particle acceleration.

Figure 3 and figure 4 show the final time (t = 1.4×104tg) log of rest-mass density and magnetic field projectedon the Cartesian z vs. x plane. For the purposes ofproperly visualizing the accretion flow and jet, we followMacFadyen & Woosley (1999) and show both the negativeand positivex-region by duplicating the axisymmetric re-sult across the vertical axis. Color represents log(ρ0/ρ0,disk)with dark red highest and dark blue lowest. The final statehas a density maximum ofρ0 ≈ 2ρ0,disk and a minimum ofρ0 ∼ 10−13ρ0,disk at large radii. Grid zones are not smoothed

6 McKinney, J.C.

FIG. 3.— Jet has pummelled its way through presupernova core andthrough1/10th of entire star. Time ist = 1.4×104tg. Panel (A) shows final distribu-tion of logρ0 on the Cartesian plane. Black hole is located at center. Red ishighest density and black is lowest. Panel (B) shows magnetic field overlayedon top of log of density. Outer scale isr = 104GM/c2.

to show grid structure. Outer radial zones are large, but outerθ zones are below the resolution of the figure.

Clearly the jet has pummelled its way through the surround-ing medium, which corresponds to the stellar envelope in thecollapsar model. By the end of the simulation, the field hasbeen self-consistently launched in to the funnel region andhas a regular geometry there. In the disk and at the surfaceof the disk the field is curved on the scale of the disk scaleheight. Withinr . 102rg the funnel field is ordered and sta-ble due to the poloidal field dominance. However, beyond

FIG. 4.— Strongest magnetic field near black hole in X-configuration dueto Blandford-Znajek effect and collimation of disk+coronal outflow. As infigure 3, but outer scale isr = 102GM/c2. Time ist = 1.4×104tg. Black holeis black circle at center. Color scale is same as in figure 3.

r ∼ 10− 102rg the poloidal field is relatively weak comparedto the toroidal field and the field lines bend and oscillate er-ratically due to pinch instabilities. The radial scale of theoscillations is 102rg (but up to 103rg and as small as 10rg),wherer ∼ 10rg is the radius where poloidal and toroidal fieldstrengths are equal. By the end of the simulation, the jethas only fully evolved to a state independent of the initialconditions atr ≈ 5× 103rg, beyond which the jet featuresare a result of the tail-end of the initial launch of the field.The head of the jet has passed beyond the outer boundary ofr = 104rg. Notice that the magnetic field near the black hole

Black Hole Jet Formation: Numerical Models 7

FIG. 5.— Contours fort ≈ 1500tg and an outer scale ofr ≈ 102rg, wherethe disk-corona boundary is a cyan contour whereβ ≈ u/b2 = 3, the corona-wind boundary is a magenta contour whereβ = 1, and the jet-wind boundaryis a red contour whereb2/(ρ0c2) = 1. Black contour denotes boundary be-yond which material is unbound and outbound (wind + jet). Beyondr ≈ 10rgthe jet undergoes poloidal oscillations due to a toroidal pinch instability.

is in an X-configuration. This is due to the BZ-effect havingpowerPjet ∝ sin2θ, which vanishes at the polar axis. The X-configuration is also related to the fact that the disk+corona iscollimating the Poynting-dominated jet. The field is mostlymonopolar near the black hole, and such field geometriesdecollimate for rapidly rotating black holes in force-free elec-trodynamics (Krasnopolsky et al. 2005).

Figures 5 and 6 show the energy structure of the disk,corona, and jet. This figure is comparable to the left panel offigure 2 in McKinney & Gammie (2004). Figure 5 shows con-tours fort ≈ 1500tg and an outer scale ofr = 102rg, while fig-ure 6 shows contours fort ≈ 1.4×104tg and an outer scale ofr = 103rg. The disk-coronal boundary is represented as a cyancontour whereβ ≡ pg/pb ≡ 2(γ − 1)u/b2 = 3, the coronal-wind boundary as a magenta contour whereβ = 1, and thejet-wind boundary as a red contour whereb2/(ρ0c2) = 1. Theblack contour denotes the boundary beyond where material isunbound and outbound (wind or jet). Beyondr ≈ 10rg the jetundergoes poloidal oscillations due to toroidal pinch instabil-ities, which subside byr ≈ 102 − 103rg where the jet is ther-mally supported. At large scales, the cyan and magenta con-tours closer to the equatorial plane are not expected to cleanlydistinguish any particular structure.

Figure 7 shows the disk, corona, and jet magnetic fieldstructure during the turbulent phase of accretion att ≈ 1500tg.Compared to figure 4, this shows the turbulence in the disk,but is otherwise similar. The jet, disk, and coronal structuresremains mostly unchanged at late times despite the decay ofdisk turbulence. That is, the current sheets in the disk donot decay and continue to support the field around the blackhole that leads to the Blandford-Znajek effect. The coronathickness and radial extent do not require the disk turbulence,which only adds to the time-dependence of the coronal out-flow.

Figure 8 shows a color plot ofΓ∞, where red is highest and

FIG. 6.— Same as figure 5 but fort ≈ 1.4×104tg and an outer scale ofr ≈103rg. Disk wind leads to broad coronal outflow. Jet remains collimated to afixed opening angle. Byr &100rg, pinch instabilities subside once magneticenergy converted into thermal energy and supports jet. Residual slow denseand fast diffuse patches from the instability are present inthe jet, such as theslower and cooler dense blob shown at top left corner of figure.

FIG. 7.— Field geometry near black hole fort ≈ 1500tg during phase ofstrong disk turbulence.

reaches up toΓ∞ ∼ 103 −104 and yellow hasΓ∞ ∼ 102 −103.Panel (A) shows outer scale ofr = 104rg, while panel (B)shows outer scale ofr = 103rg with same color scale. Theinner-radial region is not shown sinceΓ∞ is divergent nearinjection region where ideal-MHD breaks down. Different re-alizations (random seed of perturbations in disk) lead to upto aboutΓ∞ ∼ 104 as shown for the lower pole in the colorfigures. This particular model was chosen for presentation forits diversity between the two polar axes. The upper polar axisis fairly well-structured, while the lower polar axis has under-

8 McKinney, J.C.

FIG. 8.— Jet becomes conical at large radii with a core half-opening angleθ j ≈ 5. Plot of logΓ∞ with red highest (Γ∞ ∼ 104) and blue lowest.Yellow is Γ∞ ∼ 102 − 103. Panel (A) has outer scale ofr = 104GM/c2.Panel (B) outer scaler = 103GM/c2. Jet is independent of initial conditionsby r ≈ 5×103rg.

gone an atypically strong magnetic pinch instability. Variousrealizations show that the upper polar axis behavior is typi-cal, so this is studied in detail below. The strong hollow-conestructure of the lower jet is due to the strongest field being lo-cated at the interface between the jet and envelope, and thisisrelated to the fact that the BZ-flux is∝ sin2θ, which vanishesidentically along the polar axis. It is only the disk+coronathathas truncated the energy extracted, otherwise the peak powerwould be at the equator.

Figure 9 shows the velocity structure of the Poynting-

FIG. 9.— A radial cross section along a mid-level field of the jet showingthe velocity structure. Top panel shows formation of magnetic fireball wherematter (MA) and electromagnetic (EM) energy flux per unit rest-mass fluxhasΓ(MA)

∞ ∼ Γ(EM)∞ .

dominated jet along a mid-level field line. The top panelshows the late-time time-averaged values ofΓ

(EM)∞ (solid line),

Γ(MA)∞ (dotted line), andΓ = ut [BL − coords] (dashed line) as a

function of radius along a mid-level field line. For 2. r . 10,the value ofΓ(EM)

∞ is highly oscillatory. This shows the lo-cation of the stagnation surface, where the poloidal veloc-ity up = 0, is temporally variable. The stagnation surfacevaries between 2. r . 10, within the range studied in priorsections. This is where ideal-MHD approximation is break-ing and thus at any moment the value ofΓ

(EM)∞ nearly di-

verges. Notice that while atr . 103rg the Poynting flux dom-inates, the Poynting flux energy is slowly converted into en-thalpy flux due to nonlinear time-dependent shock heating.Shocks are expected beyond the magneto-fast surface (see,e.g., Bogovalov & Tsinganos 2005) and here are due to pinchinstabilities (Eichler 1993; Begelman 1998). Forr & 103rg,the enthalpy flux and Poynting flux are in equipartition.

Thus a “magnetic fireball” has developed and the terminalLorentz factor is of orderΓ∞ ∼ 400 for this choice of flowline. The Lorentz factor byr ∼ 104 is still only 5. Γ . 10,with smaller patches withΓ ∼ 25. This implies there will bean extended acceleration region where the magnetic fireballloses energy during adiabatic expansion. These results shouldalso help motivate the boundary conditions used in jet simula-tions (Aloy et al. 2000; Zhang, Woosley, & MacFadyen 2003;Zhang W. et al. 2004), which sometimes start out withΓ∼ 50andu/ρ0 ∼ 150 near the black hole. Such a high enthalpyonly occurs by 103rg in our simulations. Their simulations aresomewhat indicative of the Lorentz factor growth beyond oursimulated time and radial range. This suggests thatΓ ∼ 100should occur byr ∼ 106rg and thatΓ ∼ 103 should occur byr ∼ 107rg. This is sufficient to avoid the compactness prob-lem. A simulation of the acceleration region is left for futurework.

The next lower panel of figure 9 shows the electromag-netic contribution to the specific angular momentum at infin-ity (L = ΦBφ) (solid line), the matter contribution to the spe-

Black Hole Jet Formation: Numerical Models 9

FIG. 10.— A θ cross-section of the jet atr = 5× 103GM/c2 showingthe velocity, density, energy, and magnetic structure in Gauss. Velocity andenergy structure show Gaussian fits as dashed lines. Vertical dashed linesmarks outer and core jet.

cific angular momentum at infinity (huφ) (dotted line), and thespecific angular momentumuφ (dashed line).

The bottom panel of figure 9 shows the orthonormal ra-dial (ur =

√grrur) (solid line) andφ (uφ =

√gφφuφ) (dotted

line) 4-velocities in Boyer-Lindquist coordinates. The di-rected motion becomes relativistic, while theφ-componentof the 4-velocity becomes sub-relativistic. However, the ro-tation is still large enough that it may stabilize the jet againstm = 1 kink instabilities, as shown in the nonrelativistic case(Nakamura & Meier 2004).

Figure 10 shows theθ cross-section of the jet atr = 5×103rg for the upper polar axis according to figures 3, 4, and 8.This is a location by which the jet has stabilized in time, wherefarther regions are still dependent on the initial conditions.The hot fast “core” of the jet includesθ . 5 at this radiusand is marked by the left dashed vertical line. Also, it is usefulto notice thatθ ≈ 0.14 corresponds to the radius obtained forthe “mid-level” field line shown for radial cross-sections infigures 9 and 11.

The region withinθ. 27 is an expanded cold slow portionof the jet. It is possible that the gamma-ray burst photons aredue to Compton drag from soft photons emitted by this jet“sheath” dumping into the faster spine (Begelman & Sikora1987; Ghisellini et al. 2000; Lazzati et al. 2004). Based uponthe data fits described below forr & 120rg and a radiation-dominated plasma, the outer sheath’s (θ ≈ 0.2) seed photontemperature as a function of radius is

Tγ,seed ∼ 50keV

(

r5×103rg

)−1/3

(9)

These seed photons can be upscattered by the jet and producea GRB and high energy components. A self-consistent studyof a Compton dragged jet that determinesΓ and the emissionenergy is left for future work.

The right dashed vertical line marks the boundary betweenthe last field line that connects to the black hole. Beyond thisregion is the surrounding infalling medium. The upper-left

panel showsΓ∞ (solid line),Γ (dotted line) and Gaussian fitsto these two as dashed and long-dashed overlapping lines. Thelower left panel shows the angular energy structure of the jetwhereǫ(θ) ≡ −r2T r

t ≈ r2ρ0urΓ∞ with an overlapping Gaus-

sian fit as a dotted line. The right panels show the density andmagnetic structure of the jet. The upper right panel shows thedensity (solid line) and internal energy density (dotted line).The lower right panel shows the orthonormal Boyer-Lindquistcoordinate toroidal (solid line) and radial (dotted line) fieldcomponents.

This jet structure is weakly due to an interaction with thesurrounding medium, and primarily due to internal evolutionof the jet. Clearly the slower jet envelope is non-negligible,giving credence to the universal structured jet (USJ) model,but see Lamb et al. (2004) and see Zhang B. et al. (2004);Lloyd-Ronning et al. (2004). Gaussian jets have been usedto explain a universal connection between GRBs and x-rayflashes (Zhang B. et al. 2004). In most of our simulatedmodels the jet structure is Gaussian withǫ(θ) = ǫ0e−θ2/2θ2

0 ,whereǫ0 ≈ 0.18 andθ0 ≈ 8, within the range that they sug-gest fits observations. The total luminosity per pole isL j ≈0.023M0c2, where 10% of that is in the “core” peak Lorentzfactor region of the jet within a half-opening angle of 5.Also,Γ∞ is approximately Gaussian withΓ∞,0 ≈ 3×103 andθ0 ≈ 4.3. Also,Γ is approximately Gaussian withΓ0 ≈ 5 andθ0 ≈ 11. This can be folded into various models, such as theprobability of observing polarised emission in Compton dragemission models (Ghisellini et al. 2000; Lazzati et al. 2004).

For the collapsar model the above givesL j ≈ 4 ×1051erg s−1 for this j = 0.9375 black hole model. This isapproximately equal to the luminosity emitted by the blackhole plus the energy in pairs produced by annihilating neutri-nos. Not all of this energy can be observed inγ-rays to ob-tain E j ≈ 2× 1051erg for a 30 second event (Zhang B. et al.2004), unless the black hole has spinj ≈ 0.6 for most of theburst, which would suggest little spin evolution that requiresa much thicker disk (Gammie, Shapiro, & McKinney 2004)and so less neutrino emission and annihilation efficiency. InMcKinney (2005b), we found thatΓ∞ is weakly dependenton the black hole spin fora & 0.6, so the Lorentz factor islikely unaffected by such changes.

It is more probable that the emission inγ-rays is ineffi-cient or only a fraction of the total energy is observed (i.e.this requires. 2% for each if sole effect) ; or the true blackhole mass accretion rate is lower than predicted by currentcollapsar models (i.e.M0 ∼ 0.002M⊙s−1 rather thanM0 ∼0.1M⊙s−1 if sole effect). For the standard collapsar model, ifone only observed the core of the jet, which has only 10% ofthe luminosity, then an efficiency of converting toγ-rays of≈ 20% is required to fit the model of Zhang B. et al. (2004).

Notice that the disk thickness (and so mass accretion rate)may strongly determine the collimation angle. In highermass accretion rate systems, the disk is thicker than describedhere, and the final opening angle may be smaller. Also,if the mass accretion rate is much lower, the disk is alsothicker (Kohri et al. 2005). The collapsar type model givesthe thinnest disk near the black hole and small changes inthe mass accretion rate lead to a small change in the diskthickness (Kohri et al. 2005). Also, for the relevant blackhole spins, the spin only weakly determines the final colli-mation angle (McKinney & Gammie 2004). The universal jetmodel requires quite large opening angles that would not besupported by invoking stellar (and so black hole) spin depen-

10 McKinney, J.C.

FIG. 11.— A radial cross-section of both poles of (solid and dotted lines)jet along a mid-level field line in the jet showing the jet collimation, density,

and magnetic structure (Bφ,r in Gauss). Overlapping dashed lines are fits.

dence. It would also not be supported by invoking mass ac-cretion rate dependence, since a system with a higher or loweraccretion rate actually has a thicker disk (Kohri et al. 2005)and so stronger collimation.

There is a weak dependence on the density of thestellar envelope compared to that seen in relativis-tic hydrodynamic simulations of jets (Aloy et al. 2000;Zhang, Woosley, & MacFadyen 2003; Zhang W. et al. 2004),since here the confinement is mostly magnetic. During thisenergetic phase of the jet, the medium plays little role in shap-ing the core of the jet, as has been suggested for other reasons(Lazzati et al. 2005).

The peak Lorentz factors are obtained in a narrow regionwithin a core with half-opening angle ofθ ≈ 5. Noticeas well that the very inner core of the jet is denser andslower, a feature predicted in the theoretical discussion ofMcKinney (2005b). This feature is simply a result that theBlandford-Znajek flux is∝ sin2θ. Most of the Poynting en-ergy is initially at the interface between the jet and surround-ing medium, until the jet internally evolves to collimate intoa narrower inner core. Notice that in McKinney (2005b), wepredictedΓ∞ ∼ 1000 for j = 0.9, where the simulated resultsshow a peakΓ∞ ≈ 3×103 and typical 102 . Γ∞ . 103. Thisis in reasonable agreement with analytic estimates given thejet structure was not taken into account.

Figure 11 shows the collimation angle along a mid-levelfield line in the Poynting-dominated jet, along the same fieldline as in figure 9, and for both poles of the simulation. Gen-erally the two poles are similar but not necessarily identical.Many simulations that were performed show at least one polarjet develop some pinch instabilities. In the model shown, oneside develops very strong instabilities by the end of the simu-lation. The development of the pinch instability mostly affectsthe far-radial Lorentz factor (Γ andΓ∞ — and so densities).The value ofΓ is up to 5 times larger in pinched regions thannon-pinched regions, whileΓ∞ can be up to a factor 10 timeslarger in pinched regions compared to non-pinched regions.

In figure 11, for 7rg . r . 100rg there is a region of col-

limation slightly faster than logarithmic. For the field lineclosest to the coronal wind (not shown) starting atθ j ≈ 1.0,collimation is logarithmic with

θ j ≈ 1

(

r2.8rg

)−1/3

. (10)

Closer to the polar axis the collimation is faster. For the fieldline starting atθ j ≈ 0.3, approximately

θ j ∝ r−2/5 (11)

up to r ∼ 102rg. The inner-radial collimation is due to con-finement by the disk+corona, while in the outer-radial rangethe coronal outflow collimates the Poynting flow. The coronais likely required for the Poynting-dominated jet collima-tion, since without a disk, a monopole-like field near theblack hole actually decollimates at higher black hole spin(Krasnopolsky et al. 2005). Far from the coronal outflow, theinternal jet is collimated by internal hoop stresses. Note thatthe classical hoop-stress paradigm that jets can self-collimateis not fully tested here, but these results suggest that collima-tion is in large part due to, or at least requires, the coronaloutflow.

Notice from figure 9 that there is little acceleration be-yond r ∼ 102rg while the magnetic fireball is forming. It iswell known that magnetic acceleration requires field lines tocollimate (Begelman & Li 1994). Once the flow is pseudo-conical the flow is unable to transfer magnetic energy intokinetic energy and proceeds to convert it into enthalpy flux.After r & 103rg the flow oscillates around a pseudo-conicalasymptote with a mean half-opening angle ofθ j ∼ 5. Theopening angle has been found to be weakly dependent on theblack hole spin (McKinney & Gammie 2004) and, as foundin this paper, the details of the injection model. This resultingpseudo-conical asymptote results once the magnetic energyno longer dominates and the coronal wind no longer helpscollimate the flow. This opening angle is in line with expecta-tions built around afterglow achromatic break measurementsand estimates of the opening angle based upon energy of theburst. During the evolution of the burst, the jet may continueto collimate due to a more extended coronal outflow. Thiswould be observed as an overall hard-to-soft evolution ofγ-rays since one is gradually exposed to the slower edges of thejet since one likely does not observe exactly emission alongthe core center. The additional loading of neutrons from thecoronal outflow may also lead to a hard-to-soft evolution ofthe jet as the overall Lorentz factor decreases over the dura-tion of the burst.

Figure 11 shows the toroidal field and pitch angle, whichshows that eventually the toroidal field completely domi-nates the poloidal field, and the toroidal field remains or-dered. Thus, large polarizations up to 60% are possible(Granot & Königl 2003). The inner radial toroidal field iswell fit by

Bφ

√

ρ0,diskc2[Gauss] = 0.0023

(

r390rg

)−0.7

. (12)

The outer radial field is well fit by

Bφ

√

ρ0,diskc2[Gauss] = 0.0023

(

r390rg

)−1.5

. (13)

The transition radius isr ≈ 390rg. Notice that this is alongone mid-level field line, and the coefficients and power laws

Black Hole Jet Formation: Numerical Models 11

are slightly different for each field line. The pitch angle showsthat the field becomes toroidally dominated and at large ra-dius the magnetic loops have a pitch angle of. 1 by 103rg.For regions not on the axis there is no possibility of recon-nection unless the flow is kink unstable (Drenkhahn 2002;Drenkhahn & Spruit 2002). Given the regularity of the flowfield and the lack of randomized field, reconnection isnot nec-essary or likely. Here, the pinch instability drives a nonlinearcoupling such as shocks that converts the Poynting flux to en-thalpy flux. This mechanism should be investigated in futurework.

The toroidal field dominance leads tom = 0 pinch mag-netic instabilities. Them = 1 kink mode may also operate, butstudying the 3D jet structure is left for future work. As suchoscillations develop, this leads to arbitrarily sized patchesmoving at arbitrary relative velocities as in the internal shockmodel. The typical size of a patch is∼ 100rg to∼ 103rg in thelab frame along the length of the jet. For long-duration GRBslasting about 30 seconds, the number of pulses should be nolarger than the ratio of the scale of the instability to the lengthof the event (106rg), or about 100− 1000 pulses. Differentrealizations of the same simulation show that the likelihoodof an instability growing is random, which could lead to thelarge variations in the number of observed pulses and explainthe diversity of observed pulses (Nakar & Piran 2002). Thatis, some jets may have very few patches and so pulses. Thistype of instability is likely required to produce the diversityof GRBs. It is uncertain whether toroidal field instability in-duced variability dominates pair creation loading variabilitydue to disk structure variability, which can be addressed byfuture work that self-consistently treats the neutrino emissionfrom the disk.

Figure 11 also shows the rest-mass density per disk density,internal energy density per rest-mass density, and the ratio of(twice) the comoving magnetic energy density per unit rest-mass density. Notice that the density is close to that estimatedin McKinney (2005b), so those calculations are likely reason-able approximations to the simulation results. The inner radialrest-mass density is well fit by

ρ0

ρ0,disk= 1.5×10−9

(

r120rg

)−0.9

. (inner) (14)

The outer radial rest-mass density is well fit by

ρ0

ρ0,disk= 1.5×10−9

(

r120rg

)−2.2

. (outer) (15)

The transition radius isr ≈ 120rg. Likewise, the inner radialinternal energy density is moderately fit by

uρ0,diskc2

= 4.5×10−9

(

r120rg

)−1.8

. (inner) (16)

The outer radial internal energy density is moderately fit by

uρ0,diskc2

= 4.5×10−9

(

r120rg

)−1.3

. (outer) (17)

The transition radius isr ≈ 120rg and is due to the presenceof the thick disk at small radii. Notice that this is along onemid-level field line, and the coefficients and power laws areslightly different for each field line. One can compare this tothe “envelope” density model used with

ρ0

ρ0,disk= 8×10−8

(

r120rg

)−1.5

. (envelope) (18)

FIG. 12.— Jet becomes superfast and toroidal field dominates at largeradii. Stagnation surface time-dependent but stable. Poynting-dominated jetcharacteristic (and other interesting) surfaces. Red lines are field lines. Fromr = 0 outwards: Blue#1: horizon + ingoing-fast ; Cyan#1: ingoing-Alfvén;

Black#1:Bφ = Br ; Green#1: ergosphere ; Purple#1: ingoing-slow ; Black#2:stagnation surface where poloidal velocityup = 0 ; Purple#2: outgoing-slow ;

Cyan#2: outgoing-Alfvén; Black#3:Bφ = Br again ; Black#4: light cylinder; Blue#2: outgoing-fast.

The envelope has little impact on the jet structure.Figure 12 shows the characteristic structure of the jet. The

figure shows a log-log plot of one hemisphere of the time-averaged flow at late time. In such a log-log plot, 45 linescorrespond to lines of constantθ. Lines of constant sphericalpolarr are horizontal near thez-axis and vertical near theR-axis. The field lines are shown as red lines. The disk andcoronal regions have been truncated with a power-law cutofffor r . 100rg and a conical cutoff for larger radii. The inner-most blue line is the event horizon. Clearly the field lines arenearly logarithmic untilr ∼ 100rg. After this point the fieldlines stretch out and oscillate around a conical asymptote.

Blue lines in figure 12 show the ingoing and outgoing fastsurfaces. Clearly the transition to a supercritical (superfast)flow has occurred. Withinr . 10rg the plotting cutoff createsthe appearance that the fast surface and other lines terminatealong the cutoff.

The inner magenta line is the ingoing slow surface, whilethe outer magenta line is the outgoing slow surface. The in-ner cyan line is the ingoing Alfvén surface. Energy can beextracted from the black hole if and only if the Alfvén surfacelies inside the ergosphere (Takahashi et al. 1990). The ergo-sphere is shown by the green line, which is indeed outside theingoing Alfvén surface. The outer cyan line is the outgoingAlfvén surface. The inner black line (next to the ergosphere)is the surface whereBφ = Br in Boyer-Lindquist coordinates.At larger radii there are two overlapping black lines. Onecorresponds to, again, where the toroidal field is equal to thepoloidal field. The other corresponds to the light “cylinder”surface.

Other small artifacts in the plot are a result of unsteady na-ture of the flow. However, the despite the unsteady nature ofthe flow the characteristic structure is quite simple and rela-

12 McKinney, J.C.

tively smooth in appearance. This indicates that the flow ismostly stationary. The stagnation surface is fairly unsteady,but stable.

All the results discussed in this paper are robust to increasesor decreases in numerical resolution. Convergence testinghasbeen performed by choosing different models of theθ grid,including differentQ jet parameters that controls whether thedisk or jet is more resolved at small radii. Another resolu-tion of 256×256 instead of 512×256 was also chosen. Noqualitative differences were found. The interpolation schemewas also changed from parabolic to linear and no qualitativedifferences for the far-field jet region were found.

We find that the pair injection model weakly determines thebulk structure of the flow at large radii. Other injection mod-els were simulated by varying the total energy injectedein j,fraction of rest-mass createdfρ, fraction of internal energycreatedfh, fraction given to momentum energyfm, and theφ-velocity of the injected particles. Onlyein j and fρ stronglydetermine the bulk jet flow.

Most of these results are insensitive to the two differentmodels of the surrounding medium. One model is a surround-ing infall of material and the other is an “evacuated” exteriorregion. The jet structure is negligibly broader or narrowerinthe evacuated case due to magnetic confinement and collima-tion by the coronal outflow.

3.6. Simulation as Applied to AGN and X-ray Binaries

Notice that, quantitatively, all of these results can be simplyrescaled by density in the jet in order to apply to other GRBmodels, AGN, and even approximately x-ray binary systems.This is because by the outer regions simulation, the jet hasonly reached aboutΓ∼ 5− 10 andΓ∞ ∝ 1/ρ0, jet.

In particular, numerical models for M87 give the same re-sults with 2. Γ . 10 patches byr ∼ 103rg. The key dif-ference is that while the collapsar jet is very optically thick,the M87 jet is not so most of the shock-heated energy is lostto synchrotron radiation. The same structured jet is formed,and is useful to explain TeV BL Lac objects and radio galax-ies (Ghisellini et al. 2005) and may help explain observationsof blazars (Blandford & Levinson 1995; Ghisellini & Madau1996; Chiaberge et al. 2000). The full opening angle of the jetcore of∼ 10 also agrees with the observations of the far-fieldjet in M87 (Junor et al. 1999). As discussed previously, theshocks are synchrotron cooled and the heat generated does notcontribute to larger scale acceleration. While self-consistentsynchrotron cooling should be included, these results suggestthe shock zone (or “blazar zone” for blazars) should be quiteextended betweenr ∼ 102rg and up to aboutr ∼ 104rg.

Similarly, simulations were performed using an injectionmodel for GRS 1915+105 giving patches with 1.5 . Γ . 5by r ∼ 102rg. Due to the efficient pair-loading there is lit-tle extra Poynting flux to convert to enthalpy flux or kineticenergy flux by the time shocks occur atr ∼ 102rg. As dis-cussed in McKinney (2005b), the Poynting-lepton jet in theGRS 1915+105 model is possibly destroyed by Compton dragby disk photons and synchrotron self-Compton and limited toat mostΓ . 2, but the jet could be completely destroyed byCompton drag.

3.7. Summary of Fits

A summary of the fits along a fiducial field line is given.Near the black hole the half-opening angle of the fullPoynting-dominated jet isθ j ∼ 1.0, while byr ∼ 120rg, θ j ∼

0.1. This can be roughly fit by

θ j ∼(

rrg

)−0.4

(inner) (19)

for r < 120rg andθ j ∼ 0.14 beyond. The core of the jet fol-lows a slightly stronger collimation with

θ j ∝ r−2/5 (20)

up tor < 120rg andθ j ∼ 0.09 beyond. Also, roughly for M87and the collapsar model, the core of the jet has

Γbulk ∼(

r5rg

)0.44

(inner) (21)

for 5 < r . 103rg and constant beyond for the M87 modelif including synchrotron radiation, while the collapsar modelshould continue accelerating and the power law will trun-cate when most of the internal and Poynting energy is lostto kinetic energy and the jet becomes optically thin at aboutr ∼ 109rg or internal shocks take the energy away. If theacceleration is purely thermal without any magnetic effect,thenΓ ∝ r (Mészáros & Rees 1997). However, it is not clearhow the equipartition magnetic field affects the acceleration.Roughly for GRS 1915+105 the core of the jet has

Γbulk ∼(

r5rg

)0.14

(inner) (22)

for 5 < r ∼ 103rg and constant beyond, with no account forCompton drag or pair annihilation. Also, for any jet systemthe base of the jet hasρ0 ∝ r−0.9 (inner) for r . 120rg andρ0 ∝ r−2.2 (outer) beyond. For the collapsar and M87 models

ρ0

ρ0,disk∼ 1.5×10−9

(

r120rg

)−0.9

(inner) (23)

andρ0

ρ0,disk∼ 1.5×10−9

(

r120rg

)−2.2

(outer), (24)

while for GRS 1915+105 the inner-radial coefficient is 10−5

and outer is 6×10−3. For the collapsar model, the inner radialinternal energy density is moderately fit by

uρ0,diskc2

= 4.5×10−9

(

r120rg

)−1.8

(inner). (25)

The outer radial internal energy density is moderately fit by

uρ0,diskc2

= 4.5×10−9

(

r120rg

)−1.3

(outer). (26)

The transition radius isr ≈ 120rg. For M87 the internal en-ergy is near the rest-mass density timesc2 until r ∼ 120rgwhen the dependence is as for the collapsar case. ForGRS1915+105 the internal energy is near the rest-mass den-sity timesc2 until r ∼ 120rg and then rises to about 2.5 timesthe rest-mass density timesc2. The inner radial toroidal labfield is well fit by

Bφ

√

ρ0,diskc2[Gauss] = 0.0023

(

r390rg

)−0.7

(inner) (27)

for 5< r < 390rg. The outer radial toroidal lab field is well fitby

Bφ

√

ρ0,diskc2[Gauss] = 0.0023

(

r390rg

)−1.5

(outer) (28)

Black Hole Jet Formation: Numerical Models 13

for r > 390rg.For the typical jet with no atypical pinch instabilities, the

energy and velocity structure of the jet follow

ǫ(θ) = ǫ0e−θ2/2θ20 , (29)

whereǫ0 ≈ 0.18 andθ0 ≈ 8. The total luminosity per poleis L j ≈ 0.023M0c2, where 10% of that is in the “core” peakLorentz factor region of the jet within a half-opening angleof5. Also,Γ∞ is approximately Gaussian

Γ∞(θ) = Γ∞,0e−θ2/2θ20 , (30)

whereΓ∞,0 ≈ 3×103 andθ0 ≈ 4.3. Also,Γ is approximatelyGaussian

Γ(θ) = Γ0e−θ2/2θ20 , (31)

whereΓ0 ≈ 5 andθ0 ≈ 11. The outer sheath’s (θ≈ 0.2) seedphoton temperature as a function of radius is

Tγ,seed ∼ 50keV

(

r5×103rg

)−1/3

. (32)

4. DISCUSSION

A collapsar-type GRMHD simulation with a neutrino an-nihilation model and Fick diffusion model has been studied.A self-consistent Poynting-dominated jet is produced and theLorentz factor at large distances isΓ∞ ∼ 103 in the core ofthe jet, but an equally important structured component existswith Γ∞ ∼ 102. The Lorentz factor of the jet is determined bythe electron-proton loading by Fick diffusion of neutrons.No-tice that estimates of the baryonic mass-loading from the op-tical flash of GRB990123 suggest a baryon loading that givesΓ∼ 103 (Soderberg & Ramirez-Ruiz 2003), which is in basicagreement with the findings here. The half opening angle ofthe core of the jet isθ j ∼ 5, while there exists a significantstructured component with a half opening angle ofθ j . 25.It is likely that this jet component survives traversing there-maining portion of the stellar envelope (r ∼ 105rg, anotherdecade in radius).

The jet at large distances is unstable to, at least, anm = 0pinch instability due to the dominance of the toroidal fieldto the poloidal field (see, e.g. Begelman 1998). Here it isfound that the energy of the jet is patchy. Typical realizationsof the same model have a random number of patches with100. Γ∞ . 103. This likely results in a pulsed sequenceof magnetic fireballs that move at large relative Lorentz fac-tors, as required by the internal shock model (Mészáros 2002;Ghirlanda et al. 2003; Piran 2005). The acceleration regionand internal shock region was not simulated since the dynam-ical range required is another∼ 6 orders of magnitude in ra-dius.

Some models assume the Poynting energy to be con-verted into radiation far from the collapsing star by internaldissipation (see, e.g., Thompson 1994; Mészáros & Rees1997; Spruit, Daigne, & Drenkhahn 2001; Drenkhahn2002; Drenkhahn & Spruit 2002; Sikora et al. 2003;Lyutikov, Pariev, & Blandford 2003) such as magneticreconnection. However, in our model, significant internaldissipation occurs already by 103rg.

A very efficient model that reduces the compactness prob-lem invokes Compton drag (bulk Comptonization) to gen-erate the emission, where the stellar envelope or presuper-nova region provides soft photons (Ghisellini et al. 2000;Lazzati et al. 2004). Our numerical model shows a jet struc-ture with a slow cold “sheath” necessary for supplying the

seed photons, so this process might explain the GRB emis-sion process.

As in McKinney & Gammie (2004), a mildly relativisticPoynting-baryon jet is launched from the inner edge of thedisk with a half-opening angle of roughly 16 to 45, how-ever long-time study of this jet component depends on sus-taining long-time disk turbulence, which cannot be simulatedin axisymmetry. For GRBs, this hot baryon-loaded jet mightplay a significant role in the subsequent supernova (by pro-ducing, e.g.,56Ni) and be an interesting site for the r-process(Kohri et al. 2005). The mass ejection rate is found to beM0,coronal ≈ 0.03M0, which for a 30 second event gives 0.1M⊙of baryonic mass. This may be sufficient to power super-novae. The Lorentz factor is aboutΓ ∼ 1.5− 3, such as ob-served in a component of SN1998bw (Kulkarni et al. 1998).

For GRBs, the picture that emerges is that Poynting fluxis emitted from the black hole atr ∼ rg and is loaded withelectron-positron pairs withinr . 20rg. A similar mass-fraction of neutrons Fick-diffuses into the jet, and they dom-inate the rest-mass once the temperature decreases toT .6× 109K by r ∼ 10rg. Magnetic acceleration occurs overr . 102rg up to Γ ∼ 10. Notice that this is in contrastto Mészáros & Rees (1997) who assumed the initial bulkLorentz factor isΓ[r = rH ] ∼ Γ

(EM)∞ , which assumed instan-

taneous magnetic acceleration.Once the toroidal field dominates the poloidal field, mag-

netic instabilities develops aroundr ∼ 10− 102rg which turnsthe jet into an equipartition “magnetic fireball” byr ∼ 103rg.The spatial structure of the magnetic fireball energy flux ispatchy on the instability scale ofr ∼ 102rg. Simulations thatdemonstrate how these features would later interact requiremuch more radial dynamic range. Between 104rg . r . 108rgthe flow accelerates due to adiabatic expansion with a radialdependence up toΓ ∝ r (Mészáros & Rees 1997), where thePoynting flux provides a reservoir of magnetic energy that iscontinuously shock-converted into thermal energy.

After passing the stellar surface atr ∼ 105rg, the plasmabecomes transparent toγ-rays at 106rg . r . 107rg, wherepossibly Compton drag of the sheath seed photons producesthe GRB emission (Ghisellini et al. 2000; Lazzati et al. 2004).By r ∼ 107, patches of varyingΓ∼ 100− 1000 have reachedtheir terminal velocity. Between 108rg . r . 1010rg the fire-ball is optically thin to Compton scattering, radiation escapes,and pairs no longer annihilate but can continue to be magnet-ically accelerated (Mészáros & Rees 1997). Within this sameradial range, internal shocks proceed to convert the remain-ing kinetic energy to nonthermalγ-rays. Beyond 1010rg a re-verse shock may occur and external shocks with the interstel-lar medium (ISM) occur.

Simulations of AGN were performed, which show similarΓ vs. radius as the collapsar case. That is, there is an earlytransfer of Poynting flux to kinetic energy flux leading up toaboutΓ∼ 5−10 by aboutr ∼ 103rg for an M87 model. This isconsistent with the lack of observed Comptonization featuresin blazars (see, e.g., Sikora et al. 2005), although this is alsoconsistent with the fact that the optical depth to Comptoniza-tion is low in such systems.

While prior work assumingΓbulk ∼ 10 andθ j ∼ 15 foundthat the BZ power is insufficient to account for Blazar emis-sion (Maraschi & Tavecchio 2003), as shown here the struc-ture of the Poynting-dominated jet is nontrivial. The re-gion with Γbulk ∼ 10 is narrower withθ j ∼ 5 and jet emis-sion is likely dominated by shock accelerated electrons with

14 McKinney, J.C.

Γe & 100 with an extended high energy tail. This lowers thenecessary energy budget of the jet to be consistent with BZpower driving the jet.

4.1. Theoretical Contemporaneous Comparisons

Theoretical studies of ideal MHD jets focused on coldjets and the conversion of Poynting energy directly to ki-netic energy (see, e.g., Li et al. 1992; Begelman & Li 1994;Daigne & Drenkhahn 2002). Indeed, much of the workhas focused on self-similar models, of which the only self-consistent solution found is suggested to be R-self-similarmodels (Vlahakis 2004). Unfortunately, such models resultin only cylindrical asymptotic solutions, which is apparentlynot what is observed in AGN jets nor present in the simula-tions discussed in this paper. The previous section showedthat some aspects of the jet are nearly self-similar and thatthere is some classic ideal-MHD acceleration occurring inthis region. However, once the toroidal field dominates thepoloidal field, the Poynting flux is converted into internal en-ergy until they reach equipartition. The conversion is due toa time-dependent nonlinear coupling, such as shocks, due tothe magnetic pinch instability. Cold ideal-MHD jet modelswould have no way of addressing this. This region also in-volves relatively rapid variations in the flow, so stationary jetmodels would have difficulty modelling this region.

Notice that GRMHD numerical models show that the mag-netic field corresponding to the Blandford-Znajek effect dom-inates all other magnetic field geometries (Hirose et al. 2004;McKinney 2005a). For example, there are no dynamically sta-ble or important field lines that tie the black hole to the outeraccretion disk as in Uzdensky (2005). All black hole fieldlines tie to large distances or tie to the horizon-crossing accre-tion disk. Also, there are no field lines that connect the inner-radial accretion disk and large distances. The Blandford-Znajek associated magnetic field is completely dominant, asshown in, e.g., figure 3 and figure 4.

We have not directly included those effects of neutron dif-fusion that induce a jet structure (Levinson & Eichler 2003).However, there is probably strong mixing within the jet andthe large-scale jet is probably not affected by the details ofthe interface where neutron-diffusion occurs. Alternatively,the Gaussian structure we find may dominate the power lawstructure they find since the Gaussian structure is due to astrong internal electromagnetic evolution of the jet and matterplays little role in setting the jet structure.

4.2. Numerical Contemporaneous Comparisons

MacFadyen & Woosley (1999) evolve a presupernova corewith a black hole inserted to replace part of the core andevolve the nonrelativistic viscous hydrodynamic equations ofmotion for various values of viscosity coefficient (α), massaccretion rate, nuclear burning, stellar angular momentum,and some models have an injected energy at the poles at somefixed energy rate. For models in which they inject energyat some specified rate, they find the baryon-contaminationproblem is somewhat alleviated by forming a hot bubble. Asshown in their figure 28, they find the jet hasu/(ρ0c2) ∼ 10and they suggest that this corresponds toΓ∞ ∼ 10, insuffi-cient to avoid the compactness problem. Theirα = 0.1 modelshows a significant disk outflow, which over the duration ofthe GRB would yieldM ∼ M⊙ in mass that could power asupernova. In theirα = 0.01 model there are insignificant out-flows.

Our MHD results are most comparable to theirα = 0.01model. We find a disk outflow yieldingM ∼ 0.1M⊙, whichmay still be sufficient to produce a supernova component. Wehave self-consistently evolved the jet formation and includedan approximate model for the pair creation physics. In ourcase the baryon-contamination problem is self-consistentlyavoided by magnetic confinement of the jet against baryons,where only a small number of neutrons Fick-diffuse into thejet and self-consistently determine the Lorentz factor at largedistances. We find that neutrino annihilation energy is proba-bly dominated by energy from the black hole. Notice that wefind Γ∞ ∼ 100− 1000, which is much larger than they find.The difference between their and our model is the presenceof a magnetic field and a rotating black hole, which drive theBZ-effect and a stronger evacuation of the polar jet region.

Proga (2005) have performed nonrelativistic MHD simula-tions of collapsars with a realistic equation of state. Theysug-gest MHD accretion is able to launch, collimate, and sustainastrong Poynting outflow, although they measure a jet velocityof v ∼ 0.2c. They find the jet is Poynting flux dominated suchthatΓ∞ . 10, insufficient to avoid the compactness problem.The rotation of the black hole is crucial to generate a suffi-ciently Poynting-dominated jet. Similar fully relativistic sim-ulations by Mizuno et al. (2004) have performed with a jetvelocity of onlyv ∼ 0.3c, which is due to their choice of the“floor” model in the polar regions and the short time of inte-gration.

Zhang, Woosley, & MacFadyen (2003); Zhang W. et al.(2004) use a relativistic hydrodynamic model to simulatejet propagation through the stellar envelope and subsequentbreakout through the stellar surface. They assume a highlyrelativistic jet is formed early near the black hole and theyinject the matter with a large enthalpy per baryon of aboutu/ρ0c2 ∼ 150. They also tune the injected energy to comparewith observations (Frail et al. 2001). In their view, variabilityis due to hydrodynamic instabilities between the cocoon andcore of the jet. They find the stellar envelope can collimatethe flow.

We have self-consistently evolved theformation of the jetalong with the disk without having to assume a jet struc-ture. We suggest that the injected energy per baryon is onlyu/ρ0c2 . 20 unless super-efficient neutrino mechanisms areinvoked. We also find that the self-consistent energy releasedis larger than observed, suggesting an fairly inefficient genera-tion ofγ-rays and a dominant lowerΓ jet component that mayexplain x-ray flashes. We suggest magnetic instabilities dueto pinch or kink modes dominate the variability rather thanhydrodynamic instabilities, which are known to be quenchedby magnetic confinement. We find that the jet structure isnegligibly broader for models with no stellar envelope due tomagnetic confinement and we find that the outer portion of thePoynting-dominated jet is collimated by the coronal outflow.

Many hydrodynamic jet propagation simulations have beenstudied (for a review see Scheck et al. 2002). Scheck et al.(2002) evolve hydrodynamic relativistic hadronic and lep-tonic jets using a realistic equation of state. They find thattheresulting morphology of the jets are similar, despite the differ-ent composition. This suggests jet gross morphology is not auseful tool to differentiate jet composition. In our model,themost relativistic jet component is lepton-dominated in AGNand x-ray binaries and baryon-dominated in GRB systems.Notice that magnetic confinement quenches most hydrody-namic instabilities.

De Villiers et al. (2005b) evolve a fully relativistic, black

Black Hole Jet Formation: Numerical Models 15

hole mass-invariant model, showing a jet with hot blobs mov-ing with Γ ∼ 50. A primary result in agreement with theresults here is that a patchy or pulsed “magnetic fireball” isproduced. This is gratifying since it suggests that the devel-opment of a “magnetic fireball” is not an artifact of the nu-merical implementation but is a likely result of shock heating.We also agree in finding that the core of the jet is hot and fastand is surrounded by a cold slow flow. One difference is thatthey say they seem to find the flow is cylindrically collimatedby r & 300rg, while we find nearly logarithmic collimationuntil r ∼ 102rg and an oscillatory conical asymptote beyond.Another difference is that they suggest temporal variability isdue to injection events near the black hole, while we suggestit is due to pinch (or perhaps kink) instabilities atr & 102rg.

Another difference is that they find a larger value ofΓ atsmaller radius than we find. Indeed, one should wonder whytheir black hole mass-invariant model applies to only GRBsand not AGN or x-ray binaries. This is a result of theirmuchlower “floor” density ofρ0 ∼ 10−12ρ0,disk, which is not con-sistent with self-consistent pair-loading or neutron diffusionloading. Also, because they use a constant density floor, thejet region at large radius must be additionally loaded with anarbitrary amount of rest-mass. The typical “floor” model addsrest-mass into the comoving frame, but this artificially loadsthe jet with extra mass moving at highΓ. We suggest thatrather than the acceleration occurring withinr . 700rg of theirsimulation, that acceleration occurs much farther away beforethe emitting region atr . 109rg. In their Γ ∼ 50 knots thegas has 103 . u/ρ0c2 . 106. This implies that their actualterminal Lorentz factor is on the order of 105 . Γ∞ . 107,which would imply that the external shocks occur before in-ternal shocks could occur.

Komissarov (2005) study the BZ process and MHD Pen-rose process. They found that the prior “jet” results of(Koide, Shibata, Kudoh, & Meier 2002), who evolve only fort ∼ 100tg, are only a transient phenomena associated withthe initial conditions. They also suggest that there are seri-ous problems for the MHD-driven model of jets since theydo not find jets in their models except for contrived field ge-ometries. However, this is potentially due to three effects.First, their outer radius isr ∼ 50rg, while here we study out tor ∼ 104rg. The magnetic acceleration only leads to a logarith-mic increase in the velocity, so a large radial range is requiredto observe relativistic motion. Second, magnetic accelerationrequires collimating field lines (Begelman & Li 1994), and adisk or disk wind is probably required to collimate the mag-netic outflow (Okamoto 1999, 2000). The “disk” that formsin their models is relatively thin. Third, to avoid numericalerrors, they limit their solution tob2/(ρ0c2) . 100. Here wefind that a self-consistent source of jet matter from pair cre-ation and neutron diffusion leads tob2/(ρ0c2) ∼ 105 near theblack hole. These three effects are probably why they find norelativistic jets.

4.3. Limitations

As has been pointed out by Komissarov (2005), noncon-servative GRMHD schemes often overestimate the amountof thermal energy produced. All GRMHD numerical modelssuffer from some numerical error. Shock-conversion of mag-netic energy to thermal energy is modelled by HARM in theperfect magnetic fluid approximation with total energy con-served exactly. However, our numerical model may overesti-mate the amount of magnetic dissipation in shocks, and so the

shocks may more slowly convert magnetic energy to thermalenergy. If this dissipation was delayed until theγ-ray photo-sphere atr ∼ 107 − 108rg, then those shocks could be directlyresponsible for theγ-ray emission from GRBs. Clarificationof this issue is left for future work. However, we expect thattoroidal field instabilities drive efficient magnetic dissipationas shown in the numerical results presented here.

In order to evolve for a longer time than simulated in thispaper, other physics must be included. For long-term evolu-tion of a GRB model, one must include disk neutrino cool-ing, photodisintegration of nuclei, and a realistic equation ofstate. If one wishes to track nuclear species evolution, a nu-clear burning reactions network is required. For the neutrinooptically thick region of the disk, radiative transport shouldbe included. The self-gravity of the star should be includedto evolve the core-collapse. This includes a numerical rel-ativity study of the collapse of a rotating magnetized mas-sive star into a black hole (see review by Stergioulas 2003,§4.3). The jet should be followed through the entire star andbeyond penetration of the stellar surface (Aloy et al. 2000;Zhang, Woosley, & MacFadyen 2003; Zhang W. et al. 2004).

As applied to all black hole accretion systems, someother limitations of the numerical models presented includethe assumption of axisymmetry, ideal MHD, and a non-radiative gas. The assumption of axisymmetry is likelynot important for the inner jet region since our earlier re-sults (McKinney & Gammie 2004) find quantitative agree-ment with 3D results (De Villiers, Hawley, & Krolik 2003).The primary observed limitation of axisymmetry appears tobe the decay of turbulence (Cowling 1934), which we attemptto avoid by requiring a resolution that gives quasi-steady tur-bulence for much of the simulation. Also, the jet at large dis-tances has already formed by the time turbulence decays, andby that time the jet at large radius is not in causal contact withthe disk.

When the toroidal field dominates the poloidal field, even-tually m = 1 kink instabilities and higher modes may appear(see, e.g., Nakamura et al. 2001; Nakamura & Meier 2004).Thus our models may underestimate the amount of oscillationin the flow and the conversion of Poynting flux to enthalpyflux.

The limitation of axisymmetry may also limit the efficiencyof the Blandford-Znajek process. The magnetic arrested disk(MAD) model suggests that any accretion flow likely accumu-lates a large amount of magnetic flux near the black hole. Thismeans the efficiency of extracting energy would be higher(Igumenshchev et al. 2003; Narayan et al. 2003).

We have neglected low-energy jet and disk radiative pro-cesses in the numerical simulations. Future work should in-clude a model of synchrotron radiation for an AGN model inorder to simulate the “jet” emission to verify the claims madein McKinney (2005b) that the broad “jet” emission observedby Junor et al. (1999) is actually the coronal outflow ratherthan the Poynting-dominated jet.

Also for AGN, a self-consistent simulation with syn-chrotron emission would likely show the continuous loss ofPoynting flux until the synchrotron cooling timescale is longerthan the jet propagation timescale, which still suggests the jetPoynting flux is finally in equipartition with the enthalpy flux.This would suggest that the shock-zone, and so emission re-gion, is more extended (r ∼ 102 − 104rg) than the simulatedshock-zone (r ∼ 102 − 103rg).

For AGN and x-ray binaries, the radiatively inefficient disk

16 McKinney, J.C.

approximation, which assumes electrons couple weakly toions, may not hold. If the electrons and ions eventually couplenear the black hole, then the disk might collapse into an un-stable magnetically dominated accretion disk (MDAF) (Meier2005). Like the MAD model, this might drastically alter theresults here, although it is uncertain whether jets are actuallyproduced under the conditions specified by the MDAF model.

The single-fluid, ideal MHD approximation breaks downunder various conditions, such as during the quiescent out-put of x-ray black hole binaries, where a two-temperatureplasma likely forms near the black hole as ions and electronsdecouple (see, e.g., Narayan & Yi 1995). Resistivity plays arole in current sheets where reconnection events may gener-ate flares as on the sun, such as possibly observed in Sgr A(Genzel et al. 2003). Finally, radiative effects may introducedynamically important instabilities in the accretion disk(e.g.Gammie 1998; Blaes & Socrates 2003).

5. CONCLUSIONS

Primarily two types of relativistic jets form in black hole(and perhaps neutron star) systems. The Poynting-dominatedjet region is composed of field lines that connect the rotatingblack hole to large distances. Since the ideal MHD approxi-mation holds very well, the only matter that can cross the fieldlines are neutral particles, such as neutrinos, photons, and freeneutrons.

In McKinney (2005b), we showed that the primary differ-ences between GRBs, AGN, and black hole x-ray binariesis the pair-loading of the Poynting-dominated jet, a similarmass-loading by free neutrons in GRB-type systems, the opti-cal depth of the jet, and the synchrotron cooling timescale ofthe jet.

In McKinney (2005b), we showed that for GRB-type sys-tems the neutron diffusion flux is sufficiently large to be dy-namically important, but small enough to allowΓ ∼ 100−1000. Beyondr ∼ 10rg many of the electron-positron pairsannihilate, so the Poynting-dominated jet is dominated inmass by electron-proton pairs from collision-induced neutrondecay. Most of the energy is provided by the BZ effect insteadof neutrino-annihilation.

In McKinney (2005b), we showed that for AGN and x-ray binaries, the density of electron-positron pairs establishednear the black hole primarily determines the Lorentz factoratlarge distances. Radiatively inefficient AGN, such as M87,achieve 2. Γ∞ . 10 and are synchrotron cooling limited.The lower theγ-ray radiative efficiency of the disk, the moreenergy per particle is available in the shock-zone. Radia-tively efficient systems such as GRS1915+105 likely have noPoynting-lepton jet due to strong pair-loading and destructionby Comptonization by the plentiful soft photons for x-ray bi-naries with optically thick jets. However, all these systemshave a mildly relativistic, baryon-loaded jet when in the hard-low state when the disk is geometrically thick, which can ex-plain jets in most x-ray binary systems.

A GRMHD code, HARM, with pair creation physics wasused to evolve many black hole accretion disk models. Thebasic theoretical predictions made in McKinney (2005b) thatdetermine the Lorentz factor of the jet were numerically con-firmed. However, Poynting flux is not necessarily directlyconverted into kinetic energy, but rather Poynting flux is firstconverted into enthalpy flux into a “magnetic fireball” due toshock heating. Thus, at large distances the acceleration ispri-marily thermal, but most of that thermal energy is providedby shock-conversion of magnetic energy. In GRB systems

this magnetic fireball leads to thermal acceleration over anextended radial range. The jets in AGN and x-ray binariesrelease this energy as synchrotron and inverse Compton emis-sion and so the jet undergoes negligible thermal accelerationbeyondr ∼ 102 − 103rg.

Based upon prior theoretical (McKinney 2005b) and thisnumerical work, basic conclusions for collapsars include:

1. Black hole energy, not neutrino energy, typically pow-ers GRBs.

2. Poynting-dominated jets are mostly loaded bye−e+

pairs close to the black hole, and bye− p pairs forr & 10rg.

3. BZ-power and neutron diffusion primarily determinesLorentz factor.

4. Variability is due to toroidal field instabilities.

5. Poynting flux is converted into enthalpy flux and leadsto the formation of a “magnetic fireball.”

6. Patchy jet develops 102 . Γ∞ . 103, as required byinternal shock model.

7. Random number of patches (< 1000 for 30 secondburst) and so random number of pulses.

8. Energy structure of jet is Gaussian withθ0 ≈ 8.

9. Core of jet withθ j ≈ 5 can explain GRBs.

10. Extended slower jet component withθ j ≈ 25 can ex-plain x-ray flashes.

11. Coronal outflows withΓ ∼ 1.5 may power supernovae(by producing, e.g.,56Ni) with M ∼ 0.1M⊙ processedby corona.

Based upon prior theoretical (McKinney 2005b) and thisnumerical work, basic conclusions for AGN or x-ray binariesinclude:

1. Poynting-dominated jetse−e+ pair-loaded unless advectcomplicated field.

2. γ-ray radiative efficiency, and so pair-loading, deter-mines maximum possible Lorentz factor.

3. Poynting-lepton jet is collimated withθ j ≈ 5.

4. Extended slow jet component withθ j . 25.

5. For fixed accretion rate, variability is due to toroidalfield instabilities.

6. Poynting flux is shock-converted into enthalpy flux.

7. In some AGN, shock heat in transonic transition lostto synchrotron emission and limits achievable Lorentzfactor to 2. Γ. 10 (e.g. in M87).

8. Coronal outflows produce broad inner-radial jet fea-tures in AGN together with well-collimated jet compo-nent (e.g. in M87).

9. In some x-ray binaries, Compton drag loads Poynting-lepton jets and limits Poynting-lepton jet toΓ. 2 or jetdestroyed.

Black Hole Jet Formation: Numerical Models 17

10. In some x-ray binaries, Poynting-lepton jet opticallythick and emits self-absorbed synchrotron.

11. Coronal outflows have collimated edge withΓ. 1.5.

12. Coronal outflows may explain all mildly relativistic andnonrelativistic jets in radiatively efficient systems (mostx-ray binaries).

For AGN and X-ray binaries, the coronal outflow collimationangle is strongly determined by the disk thickness. The aboveconclusions regarding the collimation angle assumedH/R ∼0.2 near the black hole andH/R∼ 0.6 far from the black hole,while H/R ∼ 0.9 (ADAF-like) is perhaps more appropriate

for some systems. The sensitivity of these results toH/R isleft for future work.

ACKNOWLEDGMENTS

I thank Avery Broderick for an uncountable number of in-spiring conversations. I also thank Charles Gammie, BrianPunsly, Amir Levinson, and Ramesh Narayan, with whomeach I have had inspiring conversations. I thank Scott No-ble for providing his highly efficient and accurate primitivevariable solver. I thank Xiaoyue Guan for providing her im-plementation of parabolic interpolation. This research wassupported by NASA-ATP grant NAG-10780 and an ITC fel-lowship.

APPENDIX

A. GRMHD EQUATIONS OF MOTION WITH PAIR CREATION

A single-component GRMHD approximation that accounts for baryon conservation is summarized. Leptons are assumed tobe conserved in the equation of state (EOS) and by accountingfor free-streaming neutrinos. We assume an ideal gas EOSof relativistic particles and neglect radiative transportsince we assume all particles either stream freely or are trapped in thefluid. Alternatively, we assume the initial conditions well-model the steady-state radiative equilibrium. We model the streamingphoton/neutrino-annihilation into pairs by injecting rest-mass and energy-momentum at an appropriate fraction of the baryondensity.

The black hole has a Kerr metric written in Kerr-Schild coordinates, where the Kerr metric in Kerr-Schild coordinates andthe Jacobian transformation to Boyer-Lindquist coordinates is given in McKinney & Gammie (2004). We use Kerr-Schild ratherthan Boyer-Lindquist because in Kerr-Schild coordinates the inner-radial boundary can be placed inside the horizon and so outof causal contact with the flow. It is difficult to avoid interactions between the numerical inner boundary and the jet whenusingBoyer-Lindquist coordinates. This interaction leads to excessive variability in the jet since the ingoing superfast transition is noton the grid and then the details of the boundary condition significantly impact the jet. Numerical models of viscous flows havehistorically had similar issues (see, e.g., McKinney & Gammie 2002).

A single-component MHD approximation is assumed such that baryon number is conserved up to a source term due to paircreation due to either radiative annihilation or neutron diffusion, such that

(ρ0uµ);µ = Sρ, (A1)

whereρ0 ≡ mbnb, mb ≈ mn the neutron mass,nb = nn + np, andmp is the proton mass. One may choose an arbitrary mass weightto defineρ0 in the single component fluid approximation.

For a magnetized plasma the conservation of energy-momentum equations are

Tµν;ν =

(

TµνMA + Tµν

EM

)

;ν= Sµ

T . (A2)

whereTµν is the stress-energy tensor, which can be split into a matter(MA) and electromagnetic (EM) part. In the fluidapproximation

TµνMA = (ρ0 + ug)u

µuν + pgPµν , (A3)

with a relativistic ideal gas pressurepg = (γ − 1)ug, γ = 4/3, andPµν = gµν + uµuν is the projection tensor. Eitherγ = 5/3 orγ = 4/3 for either the baryon-dominated component or lepton-dominated component does not change the results described insection 3.

A.1. Injection Physics

A detailed pair creation model would self-consistently determine the distribution of rest-mass and energy-momentum injected,which is left for future work. In this paper, the rough results of Popham et al. (1999); MacFadyen & Woosley (1999) are usedtoapproximate the neutrino annihilation and pair creation. They determine the energy density creation rate in pairs as measured byan observer at infinity (ein j). A monoenergetic, monomomentum injection of rest-mass and energy-momentum is assumed. Thatis, for a Lorentz invariant particle distribution functionof

f =dN

dx1dx2dx3du1du2du3, (A4)

the fluid equations are derived from the Boltzmann equationd fdτ

=dnin j

dτδ[uµ − uµ

in j] (A5)

for a particle creation density ratednin j/dτ in the comoving frame of the existing fluid. There is no collisional term in the idealcase. After mass-weighting and taking 4-velocity moments of the Boltzmann equation one hasSρ = (ρ0,in ju

µin j);µ, whereρ0,in j is

the injected rest-mass density anduµin j is the 4-velocity of the injected particles. Also,Sµ

T = Tµνin j ;ν

. Here,

dnin j

dτ≡ me

mb

dne−e+,in j

dτ=

1mb

dρ0,e−e+,in j

dτ, (A6)

18 McKinney, J.C.

so the fluid is still treated as a single component with a single mass weightmb. As discussed in McKinney (2005b), this is agood approximation since the late-time Poynting-dominated jet region is dominated by lepton mass while the remaining regionis dominated by baryon mass. The simultaneous advected flux of injected energy is neglected, so thenST t =

√−gein j is valid inBoyer-Lindquist, Kerr-Schild, or modified Kerr-Schild coordinates. This two-step approach (similar to operator splitting), whereparticles are injected andthen advected with the normal fluid, is a good approximation for typical numerical integrations that usea multi-step timestep approach. This is also a good approximation because the injection region has a small 4-velocity beyond thestagnation surface. Also, the error in the injection approximations being made is larger than the error in neglecting the advectionterm. Thus

Sρ = ∂/∂t(√

−gρ0,in jutin j), (A7)

andSTµ = ∂/∂t

(√−g

(

q0,in jutin ju

in jµ + pin jδ

tµ

))

. (A8)

whereq0,in j ≡ ρ0,in j + uin j + pin j. This represents the source of energy-momentum, such as photon and neutrino losses emitted andabsorbed and the pairs created by annihilation of photons orneutrinos. This accounts for, in the guise of the MHD approximation,“non-local” transport of energy and momentum. The injectedparticles are a relativistic gas withγ = 4/3 such thatpin j = (γ−1)uin j.

Two approximate injection methods are used to treat the injected momentum. One assumes that the momentum injected ismostly aligned with the axis withuθ

in j = 0 and eithervφin j = 0 or vφin j = Ω[R = rstag] for some typical origin of the neutrinosrstag.For the other, the particles are injected in the comoving frame of the existing fluid (i.e.uµ

in j = uµ). It turns out that these differentapproaches lead to qualitatively similar results for the solution beyond the region where injection is important (r & 6rL in thejet). The total energy density injected (ein j as defined by an observer at infinity) is partitioned between rest-mass, enthalpy, andmomentum energy. The fraction of rest-mass injected is defined asfρ and so

∂/∂t(√

−gρ0,in jutin j) = fρ

√−gein j. (A9)

The fraction of internal energy related injected energy is defined asfh, and so

∂/∂t(√

−g((uin j + pin j)utin ju

in jt + pin j)) = fh

√−gein j. (A10)

The fraction of rest-mass plus momentum energy injected is defined asfρ + fm, and so

∂/∂t(√

−gρ0,in jutin ju

in jt ) = ( fρ + fm)

√−gein j. (A11)

Here fρ + fh + fm = 1.McKinney (2005b) definesein j, which along with the injection fractions anduµuµ = −1 completely define the injection process

by allowing one to solve forρin j, uin j, utin j, anduin j

t for a fixed time intervaldt. Radiative cooling effects are important for studyinghundreds of dynamical times, but otherwise can be neglectedif the initial model is consistent with the time-averaged radiativeproperties. Future work will refine the treatment of the paircreation process, considering the creation effects in the locally flatcomoving frame, once a self-consistent model is available.

A.2. Pair Plasma Annihilation

The electron-positron pair plasma that forms may annihilate itself into a fireball if the pair annihilation rate is faster thanthe typical rate of the jet (c3/GM) near the black hole. Also, if the pair annihilation timescale is shorter than the dynamicaltime, then pair annihilation would give a collisional term in the Boltzmann equation. From the pair annihilation rate givenin McKinney (2005b), one finds thattpa ≫ GM/c3 for AGN and marginally so for x-ray binaries. Thus, pairs mostly do notannihilate, and so formally the pair plasma that forms in thelow-density funnel region is collisionless so that the Boltzmannequation should be solved directly. Plasma instabilities and relativistic collisionless shocks are implicitly assumed to keep thepairs in thermal equilibrium so the fluid approximation remains mostly valid, as is a good approximation for the solar wind (see,e.g. Feldman & Marsch 1997; Usmanov et al. 2000). This same approximation has to be invoked for the thick disk state in AGNand x-ray binaries, such as for the ADAF model (McKinney 2004). For regions that pair produce slower than the jet dynamicaltime, each pair-filled fluid element has a temperature distribution that gives an equation of state withP = ρ0,e−e+ kbTe/me ratherthanP = (11/12)aT4, wherea is the radiation constant. So most of the particles have a Lorentz factor ofΓe ∼ u/(ρ0,e−e+ c2)and little of the internal energy injected is put into radiation. This also allow the use of a single-component approximation. Aself-consistent Boltzmann transport solution is left for future work.

On the contrary for GRB systems, due to the relatively high density of pairs, the time scale for pair annihilation istpa ≪ GM/c3

along the entire length of the jet. Thus a pair fireball forms and the appropriate equation of state is that of an electron-positron-radiation fireball. Thus, formally the pair fireball rest-mass density is not independent of the pair fireball internal energy density.However, because the pairs are well-coupled to the radiation until a much larger radius ofr ∼ 108 −1010rL, the radiation providesan inertial drag on the remaining pair plasma. That is, the relativistic fluid energy-momentum equation is still accurate. So theeffective rest-mass density is∼ ρ0 + u (u the total internal energy of the fireball), and so the effective rest-mass is independent ofthe cooling of the fireball until the fireball is optically thin (see, e.g., Mészáros & Rees 1997).

For GRB systems, the mass conservation equation is reasonably accurate. Even though the electron-positron pairs annihilate,the rest-mass of pairs injected is approximately that of thepairs that are injected due to Fick-diffusion of neutrons (see appendix Aof McKinney 2005b). The annihilation energy from electron-positron pairs contributes a negligible additional amountof internalenergy, so can be neglected, especially compared to the Poynting energy flux that emerges from the black hole. Thus, the rest-mass can always be assumed to be due to baryons rather than theelectron-positron pairs. This also suggests that the neutrinoannihilation is a negligible effect if the BZ power is largerthan the neutrino annihilation power.

In summary, the rest-mass evolution discussed in section 3 is accurate for GRB, AGN, and marginally so for x-ray binaries.This is despite the lack of Boltzmann transport for the collisional system, or a collisional term due to pair annihilation.

Black Hole Jet Formation: Numerical Models 19

A.3. Electromagnetic Terms

In terms ofFµν , the Faraday (or electromagnetic field) tensor,

TµνEM = FµγFν

γ −14

gµνFαβFαβ , (A12)

where a factor of√

4π is absorbed into the definition ofFµν . The induction equation is given by the space components of∗

Fµν

;ν = 0, where∗

F is the dual of the Faraday, and the time component gives the no-monopoles constraint. The other Maxwellequations,Jµ = Fµν

;ν , define the current densityJµ. The comoving electric field is defined as

eν ≡ uµFµν =12ǫµνkλuν

∗

Fλk = η jν , (A13)

whereη corresponds to a scalar resistivity for a comoving current density jµ = JνPνµ, wherePνµ ≡ gνµ + uνuµ projects any4-vector into the comoving frame (i.e.Pνµuµ = 0). The classical ideal MHD approximation thatη = eµ = 0 is assumed. Thecomoving magnetic field is defined as

bν ≡ uµ

∗

Fµν

=12ǫµνkλuν

∗

Fλk, (A14)

and so the stress-energy tensor can be written as

TµνEM =

b2

2(uµuν + Pµν) − bµbν, (A15)

and∗

Fµν

= bµuν − bνuµ (A16)

and soFµν = ǫµνσǫuσbǫ (A17)

since∗

Fµν

= 12ǫ

µνκλFκλ. Hereǫ is the Levi-Civita tensor. Following the notation of MTW,ǫµνλδ = − 1√−g [µνλδ], where [µνλδ]is the completely antisymmetric symbol and = 0,1, or −1. Notice thateνuν = bνuν = 0, so they each have only 3 independentcomponents and are space-like 4-vectors.

With Bi ≡ ∗

Fit

andE i ≡ F it , the no-monopoles constraint becomes

(√

−gBi),i = 0, (A18)

and the magnetic induction equation becomes

(√

−gBi),t = −(√

−g(biu j − b jui)), j. (A19)

The ideal MHD approximation assumes thateµ = 0, and so the invarianteµbµ = 0. Since the Lorentz acceleration on a particle isfµl = qeµ, then this implies that the Lorentz force vanishes on aparticle in the ideal MHD approximation. Equation A14 impliesbt = Biui andbi = (Bi + uibt)/ut, so the magnetic induction equation becomes

(√

−gBi),t =−(√

−g(Biv j − B jvi), j

=−(√

−g(ǫi jkεk)), j, (A20)

wherevi = ui/ut, εi = −ǫi jkv jBk = −v×B is the EMF, andǫi jk is the spatial permutation tensor. A more complete account of therelativistic MHD equations can be found in Anile (1989).

B. CHARACTERISTIC (AND OTHER) SURFACES

The ideal MHD dispersion relation is given, e.g., in Gammie et al. (2003a) (there is a sign typo there), and summarized here.In the comoving frame, the dispersion relation is

D(kµ) = 0ω(

ω2 − (k ·vA)2)

×(

ω4 −ω2(

K2c2ms + c2

s (k ·vA)2/c2)

+ K2c2s (k ·vA)2

)

,(B1)

wherec2ms = (v2

A + c2s (1− v2

A/c2)) is the magnetosonic speed,c2s = (∂(ρ+ u)/∂p)−1

s is the relativistic sound speed,vA = B/√E is the

relativistic Alfvén velocity,E = b2 + w, andw ≡ ρ+ u + p. Herec is the (temporarily reintroduced) speed of light. The invariantscalars defining the comoving dispersion relation areω = kµuµ, K2 = KµKµ = kµkµ +ω2, whereKµ = Pµνkν = kµ +ωuµ is theprojected wave vector normal to the fluid 4-velocity,v2

A = bµbµ/E , and (k ·vA) = kµbµ/√E . The terms in the dispersion relation

correspond to, respectively from left to right, the zero frequency entropy mode, the left and right going Alfvén modes, and theleft and right going fast and slow modes. The eighth mode is eliminated by the no-monopoles constraint.

The dispersion relation gives the ingoing and outgoing slow, Alfvén, and fast surfaces. Energy can be extracted from theblack hole if and only if the Alfvén point lies inside the ergosphere (Takahashi et al. 1990). Optimal acceleration of theflow byconversion of Poynting flux to kinetic energy flux occurs beyond the outer fast surface (Begelman & Li 1994). Other surfaces

20 McKinney, J.C.

include: the horizon atrH ≡ 1+√

1− j2 ; the ergosphere atr ≡ 1+√

1− ( jcosθ)2 ; the coordinate basis light surface in where√gφφ = c/ΩF , where asymptotically

√gφφ = rsin(θ) is the Minkowski cylindrical radius ; the surface in Boyer-Lindquist coor-

dinates where the toroidal field equals the poloidal field where Bφ = Br. Finally, there is a stagnation surface where the poloidalvelocity up = 0. In a field confined jet where no matter can cross field lines into the jet, and if the jet has inflow near the blackhole and outflow far from the black hole, then this necessarily marks at least one location where rest-mass must be createdeitherby charge starving the magnetosphere till the Goldreich-Julian charge density is reached, or pair production rates sustains therest-mass density.

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