The Force-Free Magnetosphere of a Rotating Black Hole

0.3 MB·English

by  Ioannis Contopoulos

Draftversion January23,2013
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE FORCE-FREE MAGNETOSPHERE OF A ROTATING BLACK HOLE
Ioannis Contopoulos
ResearchCenterforAstronomy,AcademyofAthens,Athens 11527,Greece; [email protected]
Demosthenes Kazanas
NASA/GSFC,Code663,Greenbelt,MD20771, USA;[email protected]
3
1 Demetrios B. Papadopoulos
0 DepartmentofPhysics,AristotleUniversityofThessaloniki,Thessaloniki54124, Greece;[email protected]
2 Draft version January 23, 2013

n
ABSTRACT
a
J We revisitthe Blandford&Znajek(1977)processandsolvethe fundamental equationthatgoverns
2 the structure of the steady-state force-free magnetosphere around a Kerr black hole. The solution
2 depends onthe distributions ofthe magneticfield angularvelocityω andthe poloidalelectric current
I. Thesearenotarbitrary. Theyaredeterminedself-consistentlybyrequiringthatmagneticfieldlines

] cross smoothly the two singular surfaces of the problem, the inner ‘light surface’ located inside the
E ergosphere, and the outer ‘light surface’ which is the generalization of the pulsar light cylinder. We
H find the solution for the simplest possible magnetic field configuration,the split monopole, through a
. numericaliterativerelaxationmethodanalogoustotheonethatyieldsthestructureofthesteady-state
h axisymmetric force-free pulsar magnetosphere (Contopoulos, Kazanas & Fendt 1999). We obtain the
p
rate of electromagnetic extraction of energy and confirm the results of Blandford and Znajek and of
-
o previous time dependent simulations. Furthermore, we discuss the physical applicability of magnetic
r field configurations that do not cross both ‘light surfaces’.
st Subject headings: Physical Data and Processes: Accretion, accretion disks — Black hole physics —
a Magnetohydrodynamics (MHD)
[

2
1. INTRODUCTION black hole is still lacking. Nevertheless, the broader
v
astrophysical interest in the Blandford-Znajek process,
0 The electromagnetic extraction of energy from a spin-
has prompted in the past decade several researchers to
2 ning black hole can be one of the most powerful and ef-
seek and obtain the steady-state structure of the spin-
3 ficient engines in the Universe. It was Blandford & Zna-
ningblackholemagnetospheresthroughtime-dependent
0 jek(1977)(hereafterBZ77)whofirstarguedthataspin-
. ningblackholethreadedbyasufficientlystrongmagnetic general relativistic force-free numerical simulations (e.g.
2 Komissarov & McKinney 2007, Palenzuela et al. 2011,
fieldandpermeatedby anelectron-positronplasmagen-
1
Lyutikov & McKinney 2011). These works established
2 erated from pair cascades, establishes a force-free mag-
the universalapplicabilityofthe electromagneticextrac-
1 netosphere, in direct analogy to the theory of the ax-
tion of the rotation energy of a Kerr black hole in pow-
: isymmetric pulsar magnetosphere developed a few years
v ering active galactic nuclei, jets in X-ray binaries, and
earlier by Goldreich & Julian (1969). They derived the
i even gamma-raybursts.
X fundamental equation that governs the structure of this
The presenceofthis extrasourceofenergyinaddition
magnetosphere, the general relativistic force-free Grad-
r to that of the accretion disk may account for some of
a Shafranovequation,andobtainedasecondorderpertur-
the phenomenology. One should bear in mind, however,
bative solution for small values of the Kerr parameter
thatthesituationismorecomplicated. Forinstance: (1)
a and a scaling for the electromagnetic power extracted
Almost 90% of extragalactic sources are radio quiet, i.e.
from the rotating black hole.
theydonotproduceelectromagneticoutflowsintheform
The structure, singular surfaces and general prop-
of radio jets, and yet they are believed to contain both
erties of this equation for magnetic fields threading
constituents of the Blandford-Znajek process, namely a
Schwarzschild and Kerr black holes were discussed re-
spinning black hole and a strong magnetic field thread-
cently by Uzdensky (2004, 2005). He emphasized the
ingit (e.g. Kukula 2003);(2)The formationanddisrup-
need for a self-consistent determination of the distri-
tion of radio jets in black hole X-ray binaries over short
butions of the magnetic field angular velocity and the
time scales is at odds with this general notion (e.g. Bel-
poloidalelectriccurrent,somethingthatearlyrelaxation
loni2010);(3) There seems to existstrong observational
methods (Macdonald 1984) and later higher order per-
evidence that the power in the radio jet is not directly
turbative solutions (Tanabe & Nagataki 2008) did not
relatedto black hole spin, contraryto the main result of
take into account. He also noted that this may be a
BZ77 (Fender, Gallo & Russell 2010; however, see also
very difficult task, and indeed, to the best of our knowl-
Narayan & McClintock 2012). Notice, though, that this
edge a solution of this equation in the general case of
may be due to an erroneous determination of the black
an open magnetic field configuration threading a Kerr
hole spin.
2 Contopoulos et al.
The above justify an independent re-evaluation of the electromagnetic field tensor Fµν is related to the elec-
Blanford-Znajekprocessthroughthesolutionofthegen- tric and magnetic fields Eµ,Bµ measured by ZAMOs
eralrelativisticforce-freeGrad-Shafranovequation. The through Fµν = UµEν UνEµ + ǫµνλρB U (ǫ
λ ρ µνλρ
special relativistic form of that equation, the so called [µνλρ]det(g )−1/2 is−the 4-dimensional Levi-Civit≡a
µν
pulsarequation,wasonlysolvedin1999byContopoulos, tensor)|. Under|these conditions, the fundamental equa-
Kazanas and Fendt who showed that the magnetic field tionthat governsthe steady-state structure of the force-
structure can only be obtained together with the deter- free magnetosphere around a Kerr black hole becomes
mination of the unique electric current distribution that
allows for a smooth and continuous structure to exist. ρeE˜+J˜ B˜ =0 . (3)
×
In that respect, steady-state solutions offer a deeper in-
ρ andJ˜arethe electric chargeandcurrentdensities re-
sightin the physics ofthe problemthan numericaltime- e
spectively. Eq. (3) is supplemented by Maxwell’s equa-
dependent ones. It should be noted that, while BZ77
provided an estimate of the energy extraction efficiency tions of electrodynamics
of this process, they did not obtain an exact solution of ˜ B˜=0
the structure of the black hole magnetosphere. Also, at ∇·
that time, the critical role of the singular surfaces and ˜ E˜=4πρe
∇·
the distributions of the magnetospheric electric current ˜ (αB˜)=4παJ˜
and field line angular velocity had not been appreciated. ∇×
(αE˜)=0 . (4)
Today,werevisitthisproblemwithalltheknowledgewe
∇×
carryfromour13-yearlonginvestigationoftheforce-free Here,
pulsar magnetosphere. ˜ A˜ Aj , A˜ B˜ gijA B ,
∇· ≡ ;j · ≡ i j
2. THEGENERALRELATIVISTIC PULSAREQUATION (˜ A˜)i [ijk]det(g )−1/2A ,
lm k;j
In order to derive the fundamental equation that gov- ∇× ≡ | |
ernsthesteady-statestructureoftheforce-freemagneto- (A˜ B˜)i [ijk]det(g )−1/2A B .
lm j k
× ≡ | |
spherearoundaKerrblackholewefollowcloselythe3+1
Forseveralapplicationsinastrophysics,perfect(infinite)
formulationofThorne&Macdonald(1982)usedbymost
conductivity is a valid approximation. In this case,
researchersin the astrophysicalcommunity (e.g. Uzden-
sky 2005). We restrict our analysis to steady-state and E˜ B˜ =0 , (5)
axisymmetric space times where (...) =(...) =0. In ·
,t ,φ
thatcase,thegeneral4-dimensionalspace-timegeometry and the electric and magnetic vector fields can be ex-
maybe written inBoyer-Lindquistsphericalcoordinates pressedin terms of three scalar functions, Ψ(r,θ), ω(Ψ),
xµ (t,r,θ,φ) as and I(Ψ) as
≡
ds2=g dxµdxν 1 2I√Σ
µν B˜(r,θ)= Ψ , √∆Ψ , (6)
,θ ,r
Asin2θ √Asinθ ( − α )
= α2dt2+ (dφ Ωdt)2
− Σ −
Ω ω
Σ E˜(r,θ)= − √∆Ψ ,Ψ ,0 . (7)
+ dr2+Σdθ2 . (1) α√Σ ,r ,θ
∆ n o
ω is the angular velocity of the magnetic field lines, I is
Here, α (∆Σ/A)1/2 and Ω 2aMr/A are the lapse
the poloidalelectric currentflowing through the circular
≡ ≡
function and angular velocity of ‘zero-angular momen- loopr =const.,θ =const.,andΨisequalto(2π)−1 times
tum’ fiducial observers (ZAMOs) respectively,
thetotalmagneticfluxenclosedinthatloop. Noticethat
the electric field changes sign close to the horizon with
∆ r2 2Mr+a2 , Σ r2+a2cos2θ ,
≡ − ≡ respect to its sign at large distances. As explained in
A (r2+a2)2 a2∆sin2θ , BZ77, a rotating observer (ZAMO) will in general see
≡ − a Poynting flux of energy entering the horizon, but he
M is the black hole mass, and a its angular momentum will also see a sufficiently strong flux of angular momen-
(0 a M). Throughout this paper we adopt geo- tum leaving the horizon. That ensures that energy is
≤ ≤
metric units where G = c = 1. Semicolon stands for extracted from the black hole. The poloidal component
covariantderivative,comma forpartialderivative. Latin of Eq. (3) then yields the general relativistic force-free
indices denote spatialcomponents (1 3), Greek indices Grad-Shafranov equation
−
denote space-time components (0 3), and ‘ ’ denotes
− ∼ 1 A Σ Ψ cosθ
thespatialpartofvectors. ZAMOsmovewith4-velocity ,r ,r ,θ
Ψ + Ψ +Ψ
Uµ = (1/α ,0 ,0 ,Ω/α) orthogonal to hypersurfaces of ,rr ∆ ,θθ ,r A − Σ − ∆ sinθ
(cid:26) (cid:18) (cid:19) (cid:27)
constant t. The force-free magnetosphere of a spinning
ω2Asin2θ 4Mαωrsin2θ 2Mr
blackholeischaracterizedbytheelectromagneticenergy-
1 +
momentum tensor · − Σ Σ − Σ
(cid:20) (cid:21)
1 1
Tµν = (FµFνα F Fαβgµν) , (2) A,r Σ,r cosθ A,θ Σ,θ
4π α − 4 αβ − A − Σ Ψ,r− 2sinθ − A + Σ
(cid:18) (cid:19) (cid:18) (cid:19)
and the condition Tµν = 0. Here, the rest mass
;ν Ψ
and pressure contribution have been neglected. The (ω2Asin2θ 4Mαωrsin2θ+2Mr) ,θ
· − ∆Σ
Blandford-Znajek revisited 3
2Mr A,r 1 4ωMαrsin2θ is self-consistently determined through an iterative nu-
+ Ψ +
Σ A − r ,r Σ merical technique that implements a smooth crossing of
(cid:18) (cid:19)
the relativistic Alfv`en surface, which is the usual light
1 A,r Ψ,θ A,θ cylinder where rsinθ = c/ω (Contopoulos, Kazanas &
Ψ
· ,r r − A − ∆ A Fendt 1999, Timokhin 2006). In the case of a spinning
(cid:26) (cid:18) (cid:19) (cid:27)
blackhole,thesituationisqualitativelysimilarbutmore
ω′sin2θ(ωA 2αMr) Ψ2 + 1Ψ2 complicated. Contrary to a neutron star, the black hole
− Σ − ,r ∆ ,θ doesnothaveasolidsurface,andthereforeitcannotim-
(cid:18) (cid:19)
pose any restriction on ω other than the ‘radiation con-
4Σ
′ dition’ Eq. (11). The only naturalrestriction that is im-
= II (8)
− ∆ posedbythephysicalproblemisthatmagneticfieldlines
must be smooth and continuous everywhere. ω(Ψ) must
(Eq. (3.14) of BZ77 re-written in our notation). Hence-
therefore be determined together with I(Ψ) through the
forth, primes will denote differentiation with respect to
condition of smooth crossing of both LS of the prob-
Ψ. One sees directly that if we set α =0 and M =0 in
lem, the inner one inside the ergosphere, and the outer
eq. (2) we obtain
one at large distances. In the next section, we will see
1 2Ψ 1 cosθ that this canbe achievedthroughaniterative numerical
Ψ + Ψ + ,r Ψ [1 ω2r2sin2θ]
,rr r2 ,θθ r − r2 sinθ ,θ · − technique analogous to the one employed in the pulsar
(cid:18) (cid:19) magnetosphere. As remarked previously, iterating with
2Ψ,r 2ω2cosθsinθΨ respect to two functions simultaneously is indeed a very
− r − ,θ difficulttask. ThisiswhyUzdensky(2005)optedtocon-
sider only field lines that connect the black hole horizon
1
ωω′r2sin2θ Ψ2 + Ψ2 = 4II′ , (9) to the surrounding accretion disk where ω(Ψ) is deter-
− ,r r2 ,θ − mined by the Keplerian angular velocity of the disk.
(cid:18) (cid:19)
There exists one more interesting complication.
whichisthewellknownpulsarequation(Scharlemann&
Eq. (11) applies only to open field lines that cross the
Wagoner1973). The zeroingofthe termmultiplying the
eventhorizon. In the case of a star, field lines that cross
second order derivatives in eq. (2),
thestellarsurfacerotatewiththe angularvelocityofthe
ω2Asin2θ 4Mαωrsin2θ 2Mr star, but they do not cross an event horizon (either be-
1 + =0 , (10) cause no event horizon forms, or because they simply
− Σ Σ − Σ
avoid the horizon), and the above restriction does not
yields the singular surfaces of the problem. We will apply. In the present case, we are interested in studying
henceforth call the singular surfaces ‘light surfaces’ the fundamentals of the Blandford-Znajek process, and
(LS)1. When we set M = α = 0, it yields the stan- therefore,wewillconsiderablackhole‘ascleanaspossi-
dard pulsar light cylinder rsinθ = c/ω. In general, the ble’. Obviously,amagnetizedastrophysicalblack hole is
shape of the outer LS is only asymptotically cylindrical notisolated. Inthatrespect,wewouldlikeheretostudy
as θ 0 (see figure 2 below), and an inner LS appears anastrophysicalblackholewiththeminimumnumberof
→
inside the ergosphere. We will henceforth use the no- extraelements,namelyasurroundingthindiskofmatter
tation ‘light cylinder’ (LC) and ‘outer LS’ interchange- to hold the magnetic field throughthe necessaryelectric
ably. Itis interestingto note that the outer boundaryof currents and charges. Charges and currents will be pro-
the ergosphere corresponds to the solution of the singu- duced through pair production in the black hole magne-
larity condition for ω = 0, whereas the inner boundary tosphere. We will ignoremagnetic field lines that do not
(the event horizon at r = r M +√M2 a2) cor- cross both the inner and outer LS because we will have
BH
≡ −
responds to the solution of the singularity condition for no physical way to determine their ω and the poloidal
ω = Ω a/(r2 +a2), where Ω is the angular ve- electric current I contained inside them. We will also
BH ≡ BH BH
locityoftheblackhole. Itisalsointerestingtonotethat ignore magnetic field lines that cross the equator out-
thenatural‘radiationcondition’atinfinity(energymust side the black hole eventhorizonbecause such field lines
flow outwards along all field lines) requires that need a source of poloidal electric current on the equa-
tor (if they rotate, they will need to cross at least one
0 ω Ω (11)
≤ ≤ BH LSwheretheconditionofsmoothcrossingwillingeneral
(BZ77),andthereforeindeed the inner LSlies inside the requireanonzeroelectriccurrenttoflowalongthatline).
ergosphere. Therefore,inthepresentworkwewillonlystudyblack
BothEqs. (2) and(9) containthe two functions, ω(Ψ) hole magnetosphereswhere allmagnetic field lines reach
and I(Ψ), which must be determined by the physics the event horizon, and all magnetic field lines are open.
of the problem. In the case of an axisymmetric spin- As argued in Lyutikov (2012), such configuration is the
ning neutron star, ω is usually taken to be equal to the most natural end stage of stellar collapse. Notice once
neutron star angular velocity Ω 2. In pulsars, I(Ψ) againthatitis sustainedbyelectric currentsinanequa-
NS
torialcurrentsheet. Theeffectofasurroundingdiskofa
1 These are none other than the ‘velocity-of-light surfaces’ of certain height possibly threaded by the return magnetic
Macdonald (1984) where the speed of a particle moving purely field (as in the scenario predicted by the Cosmic Bat-
toroidallywithangularvelocityω equalsc.
2 Notice that in the presence of particle acceleration magne- tery; Contopoulos & Kazanas 1998) will be considered
in a future work.
tospheric ‘gaps’, this is not 100% exact (Ruderman & Suther-
land 1975, Contopoulos 2005). In particular, in old pulsars near
theirdeath lineω≪ΩNS.
4 Contopoulos et al.
3. THENUMERICALITERATIVESCHEME 0.1[Ψ(r+,θ) Ψ(r−,θ)] , (15)
− −
We will here describe the technical numerical details whereas at the outer LS we implement
that led to the solution of Eq. (2). Firstly, we change
the radial variable from r to R(r) r/(r +M). Our Inew(Ψnew)=Iold(Ψnew)
numerical integration extends from R≡min R(rBH) (the +0.1[Ψ(r+,θ) Ψ(r−,θ)]
event horizon) to R = 1 (radial infinit≡y). The θ co- −
max ω (Ψ )=ω (Ψ )
ordinate extends from θ = 0 (the axis of symmetry) new new old new
to θ = π/2 (the equamtionrial plane). We implemented 0.1[Ψ(r+,θ) Ψ(r−,θ)] , (16)
max − −
a 256 64 numerical grid uniform in R and θ. Notice
× where
that this grid has a very high resolution in r around the Ψ 0.5[Ψ(r+,θ)+Ψ(r−,θ)] (17)
blackholeeventhorizonwherethe innerLSlies,butnot new ≡
as high around the outer LS, and in particular at low θ at each LS. The reasoning here is that we impose
and a values. weighted corrections on ω(Ψ) and I(Ψ) based on the
Secondly,wespecifyboundaryconditionsontheaxisof non-smoothness of the Ψ(r,θ) distribution along all grid
symmetry,thehorizon,theequatorialplane,andinfinity. points inside and outside the two LS, and not through
Ψ(θ =0)=0. Ψ(r ,θ)isdeterminedbytheintegration the regularization conditions as we did in Contopoulos,
BH
over θ of the so called Znajek regularizationcondition Kazanas & Fendt (1999). This is a very general proce-
dure that may be applied to any similar singular equa-
Mr sinθ
I(Ψ)= BH (Ω ω)Ψ (12) tion.
−r2 +a2cos2θ BH− ,θ Our numerical scheme does not converge easily, and
BH
its parameters need to be empirically adjusted by fol-
(Znajek 1977). Notice that this condition is updated as
lowing the progress of the iteration. The signs of the
the distribution I(Ψ) is updated along with the numeri-
weighting coefficients are determined by trial and er-
cal solution for Ψ(r,θ). The equatorial boundary condi-
ror: when we choose the wrong signs, the mismatches
tion is interesting. As we discussed in the previous sec- Ψ(r+,θ) Ψ(r−,θ) increaseandtheΨ(r,θ)distribution
tion,theequatorialregionmaycontainanaccretiondisk, | − |
is very quickly destroyed. Their magnitudes are also ob-
withorwithoutitsownmagneticfield. Beforeinvestigat-
tained empirically: too large values (around unity) lead
ingsuchacomplexastrophysicalsystem,wewouldliketo
to over-corrections and instability, too small values (be-
understand first the simplest case, that of a magnetized
low 0.01) do not correct fast enough for the iteration
blackholewheretheelectriccurrentssupportingitsmag-
to converge. Furthermore, in order to facilitate conver-
netic field are distributed on a thin equatorial current
gence, at the inner LS we update only ω, whereas at
sheet. In that case, the equatorial boundary condition
the outer LS we update both I and ω every other relax-
becomes Ψ(r,θ = π/2) = Ψ . The outer boundary
max ation iteration for Ψ. Also, in order to avoid numerical
conditionisalsonotimportantsincetheouterboundary
instabilities, we smooth out the distributions of I and
has practically been moved to infinity. In practice, we
ω every 50 relaxation iterations for Ψ. We found that
choseΨ (R=1)=0,butanyotherboundarycondition
,r morefrequent smoothing inhibits convergence. Our goal
yields the same results. We initialize our numerical grid
is to minimize the residuals in the discretized form of
with a split monopole field configuration
eq. (2) together with the non-smoothness in Ψ(r,θ), but
Ψ(r,θ)=Ψ (r,θ) Ψ (1 cosθ) , (13) our method fails for poor grid resolutions at the inner
SM max
≡ − and/or the outer LS (see Discussion section).
with The results of our numerical integration are shown in
ω(Ψ)=0.5ΩBH and figure 1. We show here the distributions of ω(Ψ) (top
panel),2I(Ψ) (centerpanel),andPoyntingfluxatlarge
I(Ψ)=ISM(Ψ)≡−0.25ΩBHΨ[2−(Ψ/Ψmax)] . (14) distances|2I(Ψ| )ω(Ψ) (bottom panel) after 6000 relax-
| |
The reader can directly check that Eqs. (13) & (14) are ation steps for Ψ(R,θ), for six values of the black hole
anexactanalyticalsolutionofthepulsarequation(Eq.9; spin parameter a. As is the case in pulsars, the mag-
Michel 1982). Notice the negative sign in Eq. (14). For netospheric electric current is non-zero at Ψ = Ψ ,
max
the particular magnetic field configuration where Br on and the global electric circuit closes through an equa-
the axis is aligned with the black hole spin direction, torial current sheet. Notice that we have no way to
thiscorrespondstoanoutflowofelectrons. Differentini- update ω along the axis, and therefore we have cho-
tial configurations yield the same final solution. We dis- sen ω(Ψ = 0) = 0.5Ω .3 On the equator we imple-
BH
′ ′
cretize all physicalquantities on our 256 64 (R,θ) grid mented ω (Ψ=Ψ )=I (Ψ=Ψ )=0. The reader
max max
×
and we update Ψ(R,θ) through simultaneous overrelax- should check the qualitative agreement with the results
ationwithChebyshevacceleration(subroutineSORfrom of the time-dependent force-free general relativistic sim-
Numerical Recipes; Press, Flannery & Teukolsky 1986). ulations performed by Tchekhovskoy, Narayan & McK-
Thirdly,weupdate the distributionsofω(Ψ)andI(Ψ) inney (2010) (their figure 4).
as follows: At each latitude θ, we check where the sin- In figure 2 we show the characteristic magnetic field
gularity condition (Eq.10) is satisfied in r. At each such configurationfora/M =0.9999nearthe blackholehori-
radial position, we extrapolate Ψ inwards from larger r zon (left panel) and at larger distances (right panel).
(Ψ(r+,θ)) and outwards from smaller r (Ψ(r−,θ)). In We also plot the inner LS inside the ergosphere, and
general,Ψ(r+,θ) and Ψ(r−,θ) differ. Then, at the inner
LS we implement 3 We have also run our simulations with ω′(Ψ = 0) = 0 and
thesolutionsweresimilar,onlytheconvergenceofωbecamemore
ω (Ψ )=ω (Ψ ) unstable.
new new old new
Blandford-Znajek revisited 5
4
0.54
3.5
3
0.52
2.5
mega 0.5 r/M 2
o
1.5
0.48 1
0.5
0.46
0
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.2 0.4 0.6 0.8 1 r/M
Psi/Psi_max
0.6 14
0.5 12
10
0.4
8
M
2|I| 0.3 r/ 6
0.2 4
2
0.1
0
0 2 4 6 8 10 12 14
0
0 0.2 0.4 0.6 0.8 1 r/M
Psi/Psi_max
0.3 Figure 2. Poloidal magnetic field lines (lines of constant Ψ in
20 equal subdivisions of Ψmax) for a/M = 0.9999 close to the
event horizon (top panel) and further out (bottom panel). The
0.25 thicklinesaretheeventhorizon,theinnerLS,theouterboundary
of the ergosphere, and the outer LS (from the center outwards
respectively).
0.2
configurations of figure 1 of Tchekhovskoy, Narayan &
mega 0.15 McKinney (2010) because the scale is not adequate, but
2|I|o is implied in their figure 7. Notice that ω(Ψ) is equal
to 0.5Ω to within 10%, and that I(Ψ) is very close to
BH
0.1 thesplitmonopoleexpressionI (Ψ)alsotowithin10%
SM
(Eqs.13&14). Therefore,theZnajekcondition(Eq.12)
can be rewritten using Eq. (6) as
0.05
Ψ Σ
Br max
0 0 0.2 0.4 0.6 0.8 1 |r=rBH ≈ 2MrBH√A
Psi/Psi_max
a2Ft|iIgl(auΨrrg)e|en1do.irsmtDaanilscitzeresidb2ut|otIi(oΩΨnB)s|HωoΨf(Ψmω)(aΨxn)o(rcnmeonratmleirzaelpdizaentdoelt)Ωo,2BaΩHnBdΨHPmo(atyxonpt(ibpnoagtntfleoulm)x, = (Ψ2Mma)x2 "1+ [1+ (1a/−M(a)2/M)2]2 cos2θ# . (18)
panel) for various values of the black holespin parameter a/M =
Finally, we calculatepthe total rate of electromag-
0.7(solid),0.8(shortdashed),0.9(dotted),0.99(smalldots),0.999
(dash dotted), 0.9999 (short dashdotted). Allthree distributions neticenergyextraction(frombothhemispheres)asmea-
are close to the analytical split monopole expressions of Eq. (14) sured at infinity through Eq. (4.11) of Blanford & Zna-
(thickdots).
jek (1977) as
the outer LS which becomes asymptotically cylindrical Ψmax
= 2 2π 2I(Ψ)ω(Ψ)dΨ
as θ 0. Notice that magnetic field lines are monopo- E − ×
lar at→large distances (beyond a few times the radius of ZΨ=0
2π
the event horizon), and they bend upwards towards the =k k Ω2 Ψ2 . (19)
ESM ≡ 3 BH max
axis as they approach the horizon. This effect is also
seen in the numerical simulations of Lyutikov & McKin- We have here normalized our results to the split
ney (2011) for a=0.99. It is not discernable in the field monopole value with ω = 0.5Ω . k 0.8 for
SM BH
≈
6 Contopoulos et al.
a/M = 0.9999 and is practically indistinguishable from forced to implement our inner boundary condition at a
unity for a/M <0.9 (see bottom panel of figure 1), thus radial distance more than a hundred times larger than
confirming the∼main result of BZ77, namely that, in an the actual neutron star radius (Spitkovsky 2006, McK-
open magnetic field configuration with an infinitely thin inney2006,Contopoulos&Kalapotharakos2010). This,
surrounding equatorial disk, electromagnetic energy is however, posed no problem since the angular velocity of
extractedat a rateproportionalto both Ω2 andΨ2 . the field lines is set by the rotation of the pulsar. The
BH max
blackholeproblemismorecomplicatedbecausetheblack
4. DISCUSSION hole surface is not infinitely conducting, and the numer-
We believe that the study of the steady-state open icalcode must adequately resolveboth the inner bound-
force-free black hole magnetosphere offers a deeper in- ary and the light cylinder. Notice that our numerical
sight in the physics of the problem than time-dependent method works best as a/M approaches unity since in
numerical simulations. In analogy to the axisymmetric that case the inner LS is well separated from the event
force-free pulsar magnetosphere, we argued that steady- horizon,andtheouterLSisatitsclosestapproachtothe
state solutions can only be obtained through a self- originofthe numericalgridwhere its radialresolutionis
consistentdeterminationof the distributions of both the the highest. In that case, a numerical resolution of 256
magnetic field angular velocity and the magnetospheric radialgridpointscorrespondsto34gridcellsbetweenthe
electric current ω(Ψ) and I(Ψ) respectively. These can horizon and the inner LS, and 94 grid cells between the
only be determined when open field lines cross both sin- two LS along the equator. When a/M >0.9, our results
gular LS, the inner LS inside the ergosphere outside the are rather resolution independent. On∼the other hand,
black hole horizon, and the outer LS which is a deformed asa/M approacheszero,theergosphereshrinksinwidth
light cylinder. We developed an iterative numerical pro- and the inner LS approaches so close to the event hori-
cedureanalogoustotheoneweimplementedinthestudy zonthat our numericalgrid does not have enoughradial
of the pulsar magnetosphere. Iterating with respect to resolution for the iterative procedure that yields ω(Ψ)
two functions simultaneously is a very difficult task and to properly work4. At the same time, as a/M decreases
our method may certainly be improved in the future. so does Ω , pushing the outer LS so far out that our
BH
Yet, we are confident enough to suggest that the dis- radiallyexpandingnumericalgridhasverylowradialres-
tributions are unique for the particular boundary con- olution at the position of the outer LS for the iterative
ditions of the problem (all magnetic field lines thread procedure to properly converge. Similarly, the resolu-
theblackholehorizon,allareopen,andthesurrounding tion of our radially expanding grid on the light cylinder
diskisinfinitelythin)sincedifferentinitialconditionsfor becomes worse as θ 0, and this too complicates the
Ψ, ω and I evolve towards the same final solution (for convergence of our it→erative method. Other numerical
each value of a that is). Our solutions may serve as test approaches certainly face similar complications at short
cases for time-dependent numerical codes. We plan to and/oratlargedistances. Ourmethodmaybeimproved
applyourmethodinthestudy ofvariousastrophysically byareformulationinradiallyexpandingcylindricalcoor-
interesting situations where the surrounding disk is not dinates,inKerr-Schildcoordinates,orinstandardspher-
infinitely thin but expands vertically with distance. icalcoordinateswithanumericalintegrationouterradial
All field lines that cross the black hole horizon are boundary.
boundtocrosstheinnerLS.Thisisnotthecase,though, In summary,the force-free spinning black hole magne-
for the outer LS. In fact, several numerical simulations tosphereis notverydifferent fromthe force-freeaxisym-
existintheliteraturewheretheblackholeisthreadedby metricpulsar magnetosphere. Infact, beyondadistance
a vertical uniform magnetic field, and yet they converge ontheorderofafewtimestheblackholeradius,thetwo
to certain distributions of ω(Ψ) and I(Ψ) (e.g. Komis- problemsarealmostidentical. Animportantcorollaryof
sarov & McKinney 2007, Palenzuela et al. 2011). We thissimilarityisthataspinningblackhole surroundedby
expect that in that case, the solution is mathematically a thin equatorial plasma disk does not generate a colli-
indeterminate, and the solutions shown depend on the mated outflow, only an uncollimated wind (as in Con-
particularchoiceofouternumericalboundaryconditions. topoulos, Kazanas & Fendt 1999). Thus, the problem
For example, the lateral boundary conditions in Palen- of the black hole jet formation remains open. A possi-
zuelaet al. (2011)arenotperfectly absorbingbutreflect ble resolution must involve a surrounding thick disk, or
outgoing waves as is manifested in the oscillations seen a surrounding magnetic disk wind that gradually colli-
in their figure 2. We plan to study such magnetic field mate the initially monopolar black hole wind into a col-
configurationsinafutureworkandcheckwhetherwecan limated jet outflow, as seenin the numericalsimulations
indeed freely specify one of the two free functions, ω(Ψ) of Tchekhovskoy, Narayan& McKinney (2010). It is in-
orI(Ψ). Wewillthuscheckwhetherthe resultsofprevi- teresting to note here, again similarly to pulsars, that
ous time-dependent numerical simulations are unique or beyond the light cylinder the Lorentz factor Γ in the
not. monopolar wind increases linearly with cylindrical dis-
We would like to caution the reader about certain nu-
merical complications that arise in all numerical simu-
lations of the spinning black hole magnetosphere, time 4 Thesolutionsfora/M =0.7&0.8wereobtainedwitharadial
dependent andsteady state. Numericalsimulations that grid resolution of 1024. For even smaller values of a, our numer-
do not adequately resolve both the inner and outer LS ical integration fails because the inner LS is practically indistin-
should not be trusted. We faced a similar problem in guishablefromtheevent horizon. Wecanbypass thisproblemby
choosing the inner boundary condition some distance outside the
our study of the pulsar magnetosphere. In that case,
blackholehorizon,set ω≈0.5ΩBH, anditerateonlywithrespect
there exists no inner LS, and in order to study the tran- tothedistributionofpoloidalelectriccurrentI(Ψ)thatallowsfor
sition through the outer LS (the light cylinder) we were asmoothcrossingoftheouterLS.
Blandford-Znajek revisited 7
tance as Contopoulos,I.&Kazanas,D.1998, ApJ,508,859
rsinθ rΩBHsinθ Contopoulos,I.&Kazanas,D.2002, ApJ,566,336
Γ (20)
∝ c/ω ≈ 2c Contopoulos,I.,Kazanas, D.&Fendt, C.1999,ApJ,511,351
Contopoulos,I.&Kalapotharakos, C.2010,MNRAS,404,767
(Contopoulos & Kazanas 2002), possibly up to the fast Fender,R.P.,Gallo,E.&Russell,D.2010,MNRAS,406,1425
Goldreich,P.&Julian,W.H.1969,ApJ,157,869
magnetosonic point of the outflow. Therefore, accord-
Komissarov,S.S.&McKinney,J.C.2007, MNRAS,377,L49
ing to Eq. (20), collimated field/flow lines that do not Kukula,M.J.2003,NewAstro.Rev.,47,215
extend as far from the axis as uncollimated ones are ex- Lyutikov,M.2012,arXiv:1209.3785
pected to reach lower Lorentz factors. Of course, the Lyutikov,M.&McKinney,J.C.2011, Phys.Rev.D.,84,4019
final Lorentz factor of such outflows will also depend on Macdonald,D.A.1984,MNRAS,211,313
McKinney,J.C.2006,MNRAS,368,L30
themassloadingofagivenmagneticfieldlines,butsuch
Michel,F.C.1982,Rev.Mod.Phys.,54,1
considerationsarebeyondthe scopeofthe presentwork. Narayan,R.&McClintock,J.E.2012, MNRAS,419,69
Weplantocontinueourinvestigationoftheinterrelation Palenzuela,C.,Bona,C.,Lehner,L.&Reula,O.2011,Class.
between the black hole and disk outflows in the future. Quant.Grav.,28,4007
Press,W.H.,Flannery,B.P.,Teukolsky,S.A.1986,‘Numerical
recipes.Theartofscientificcomputing’,CambridheUniv.Press
We would like to thank Prs. George Contopoulos and Ruderman,M.A.&Sutherland, P.G.1975, ApJ,196,51
Scharlemann,E.T.&Wagoner, R.V.1973,ApJ,182,951
Maxim Lyutikov for interesting discussions. This work
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was supported by the General Secretariat for Research KipS.Thorne,K.S.&Macdonald,D.1982,MNRAS,198,339
andTechnologyofGreeceandtheEuropeanSocialFund Tanabe,K.&Nagataki, S.2008,Phys.Rev.D,78,024004
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