Dynamical behaviour of the `beads' along the magnetic field lines near a rotating black hole

0.07 MB·English

by  Xinwu Cao

Mon.Not.R.Astron.Soc.000,1–??(1997) Printed19January2010 (MNLATEXstylefilev2.2)
Dynamical behaviour of the ‘beads’ along the magnetic field lines
near a rotating black hole
⋆
Xinwu Cao
ShanghaiAstronomicalObservatory,ChineseAcademyofSciences,80NandanRD,Shanghai200030China
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0 Accepted1997June5.February1.Received1997April4;
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ABSTRACT
J
Theelementsofthecoldmagneticdrivenflowsbehavelikebeadsonthemagneticfieldline.
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Theinclinationofthefieldlinesatthesurfaceofthediscplaysacrucialroleonthenatureof
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the magneticallydrivenoutflow.For the non-relativisticcase, a centrifugallydrivenoutflow

] of matter from the disc is possible, if the poloidal component of the magnetic field makes
E an angleof less thana critical60◦ with the disc surface.The collimatedflowsejected from
H
activegalacticnucleimayprobablystartfromtheregionneartheblackhole.Weinvestigate
. the dynamical behavior of the ’beads’ on the magnetic field line start from the disc near a
h
p black hole. It is foundthat the critical angle becomeslarger than 60◦ for the rotating black
- holecase,whichmayimplythattheflowsareeasyacceleratedintheinneredgeofthedisk
o
surroundingarotatingblackhole.
r
t
s
Keywords: Accretiondisc,jets,blackhole.
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1 1 INTRODUCTION
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2 Fortenyears,production of magnetohydrodynamic jetsbyanaccretiondischasbeenconsidered asapromisingparadigmtoexplainthe
3 collimatedflowsejectedfromactivegalacticnucleiandmorerecentlybyyoungstellarobjects(BlandfordandPayne1982;Lovelace1976
. andCamezind1987).Inthesemodelsthediscissupposedtobethreadedbyaverticalmagneticfieldthatextendsthroughadensecorona,
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0 themagnetosphere aboveitbecomingfilledwithplasmawhichcanbeflungoutcentrifugallybythemagneticfieldlines.Inmagnetically
0 drivenflows,thestrengthanddistributionofthemagneticfluxatthediscsurfaceplaysanimportantroleasaboundary conditionforthe
1 problem.Theangularmomentumlossrateoftheflowdependsmostlyonthestrengthofthefield.Theinclinationofthefieldlinesatthe
:
v surfacestronglyinfluencesthewayinwhichtheflowisaccelerated.Theelementsofrelativelytenuousgasonthemagneticfieldlinesbehave
i likebeadsonarigidwire.BlandfordandPayne(1982)pointoutthatthebeadscanbelaunchedcentrifugallyfromaKeplerianaccretiondisc
X
ifthepoloidalfielddirectionisinclinedatananglelessthan60◦totheradialdirection.CaoandSpruit(1994)investigateinsomedetailhow
r
a theflowpropertieschangewiththefieldinclinationatthediscsurface.Atlowinclination,theyfindamagneticallydrivencirculationalong
thediscsurfaceratherthanahighdensityflow.Thejetsobservedinsomeactivegalacticnucleiarebelievedtobeacceleratedclosetothe
blackhole.Inthispaperweanalyzethedynamicalfeaturesofthebeadsalongthemagneticfieldlineneararotatingblackholeingeneral
relativisticframe.ThebasicequationsfortheproblemarelistedinSect.2.Sect.3containsabriefdiscussion.
2 BASICEQUATIONS
Thebehaviour offlowsflungoutcentrifugallybythemagneticfieldsislikebeadsonrigidwires.Ifthe’beads’startfromrestatthedisc
surfaceand arecarriedwithconstant angular velocitybythe’wire’,thentheeffectivepotential for abead onawire,corotatingwiththe
Keplerianangularvelocityataradiusr ,is
d
GM 1 r 2 r
Ψ (r,z)=− + d , (1)
eff rd (cid:20)2(cid:16)rd(cid:17) (r2+z2)1/2(cid:21)
⋆ E-mail:[email protected]
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2 X. Cao
wheretheNewtoniangravitationalpotentialisusedinthederivation.Aftersomemathematicalderivations,onefindsthattheequilibriumat
r=r isunstableiftheprojectionofthewireonthemeridionalplanemakesanangleoflessthan60◦withtheequatorialplane(Blandford
d
andPayne1982).Thecriticalangleof60◦ playsanimportantroleonthejetformation.Thismechanicalanalogydoesexhibittheessential
dynamicalfeatureofcentrifugallydrivenflowsnotveryclosetoacompactcentralobject.Hereweinvestigatethedynamicalpropertiesof
beadsonthemagneticfieldlineneararotatingblackhole.
Thespace-timegeneratedbyarotatingblackholecanberepresentedbyKerrmetricinsphericalcoordinates
2 △ 2 2 sin2θ 2 2 2 ρ2 2 2 2
ds =− [dt−asin θdφ] + [(r +a )dφ−adt] + dr +ρ dθ , (2)
ρ2 ρ2 △
where
2 2
△≡r −2Mr+a ,
2 2 2 2
ρ ≡r +a cos θ,
aistheangularmomentumperunitmassoftheblackhole,M isthemassoftheblackhole(settingG=c=1).
TheradialmotionofafreeparticleclosetoaKerrblackholeisgovernedbytheequation:
d2r + 1grrg′ dr 2− 1grrg′ dθ 2− 1grrg′ dφ 2− 1grrg′ dt 2−grrg′ dφ dt =0, (3)
dτ2 2 rr dτ 2 θθ dτ 2 φφ dτ 2 tt dτ tφdτ dτ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
wherethemetriccoefficientsgrr,g ,aregivenbyEq.(2),andg′ = ∂gµν.
µν µν ∂r
Sinceweonlyintendtoinvestigatetheinstabilityconditionof theparticlerotatingwiththeKeplerianangular velocitynearthedisc
surface, the term−1grrg′ dθ 2 in Eq. (3) could be omitted, similar to the non-relativistic approach. Combining Eqs. (2) and (3), we
2 θθ dτ
obtaintheequationdescribing(cid:0)the(cid:1)radialmotionofthebeads:
dur2 + gr′rur2+ gφ′φΩ2+2gt′φΩ+gt′tur2− grr(gφ′φΩ2+2gt′φΩ+gt′t) =0, (4)
dr g g Ω2+2g Ω+g g Ω2+2g Ω+g
rr φφ tφ tt φφ tφ tt
whereur2 = dr,Ω= dφ.
dτ dt
ThebeadsmovealongarigidmagneticfieldlinehavetheconstantangularvelocitymeasuredbyadistantobserverΩ = dφ.Forthe
dt
Keplerianaccretiondisc,Ω=Ω ,givenby
K
uφ
Ω = , (5)
K ut
whereuφ,utarevelocitiesofafreeparticlemovinginacircularKeplerianorbit.Usingthefour-velocitypresentedinBardeenetal.(1972),
wefinallyobtainKeplerianangularvelocitymeasuredbyadistantobserver
M1/2
Ω =± , (6)
K r3/2±aM1/2
d
wheretheuppersignisforthedirectorbitwhilethelowersigntheretrogradeorbit,r istheradialpositionofthefootpointofthefieldline.
d
Theradiusofminimumstablecircularorbitr surroundingarotatingblackis
ms
rms=M{3+Z2∓[(3−Z1)(3+Z1+2Z2)]1/2}, (7)
where
Z1 ≡1+(cid:18)1− Ma22(cid:19)1/3h(1+ Ma )1/3+(1− Ma )1/3i,
3a2 2 1/2
Z2 ≡(cid:18)M2 +Z1(cid:19) ,
theuppersignforthedirectorbit.
IntegratingEq.(4),wecangettheeffectivepotentialforabeadonamagneticfieldlinecorotatingwiththeKeplerianangularvelocity
Ω :
K
1 C
Ψ =− − , (8)
eff g g (g Ω2 +2g Ω +g )
rr rr φφ K tφ K tt
wheretheintegralconstantCisdeterminedbytheboundarycondition
π
Ψ =0, at r=r , θ= . (9)
eff d 2
Now,theinstabilityconditioncouldthenbederivedfromthesecondderivativeoftheeffectivepotential.Aftersomemathematicaldeductions,
wegetthecriticalangleα ofthemagneticlineatthesurfaceofthediscasfollows
crit
r2[(2r3+4Ma2)Ω2 −8MaΩ +4M] 1/2
α =arctan d d K K . (10)
crit (cid:26)(2rd5+2a2rd3+8Ma2rd2+4Ma4)Ω2K−8Ma(rd2+a2)ΩK+4Ma2(cid:27)
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Magneticfieldsneara rotatingblackhole 3
Figure1.Thecriticalangleθcritvs.radialpositionofthemagneticfieldfootpointatthediscsurfacewithrespecttothedifferentrotatingblackholeangular
momentum,a=0.5,0.9,1.0M.
Inthenon-rotatingblackholecase,a = 0,wehaveα = 60◦,whichissameasthatobtainedbyBlandfordandPayne(1982)forthe
crit
non-relativisticcase.InFig.1wedepicttherelationsbetweenthecriticalangleandradialpositionofthefootpointofthemagneticlinefor
therotatingblackholewithdifferentangularmomentuma.
3 DISCUSSION
Theconfigurationandstrengthofthemagneticfieldofthediscplaysacrucialroleonthedynamicalpropertiesofthecentrifugallydriven
magnetohydrodynamic flow.Assumingthatthefieldconfigurationabovethediscisclosetoapotentialfield,theshapeofthediscfieldis
determinedentirelybythemagneticfieldB atthediscsurface.Theelementsofthecoldmagneticdrivenflowsbehavelikebeadsonthe
z
magneticfieldlines.Whetherthematteratthediscsurfacecanbeflungoutmainlydependsontheinclinationangleofthefieldlineatthe
discsurface(BlandfordandPayne1982).BlandfordandPayne(1982)pointoutthattheflowcanbecentrifugallydrivenifthefieldlines
inclinedlessthanthecriticalangleof60◦ withrespecttotheradialdirection.Thespaceneartheinnerregionofthediscisa’deadzone’,
wherethemagneticfieldlineistoosteep,theflownotbeingabletoovercomeabarrierofgravitationalpotential.
WeinvestigatethedynamicalpropertiesofthebeadonthemagneticfieldsthreadingtherotatingdiscaroundaKerrblackhole.Wehave
thesamecriticalangle60◦ forthenon-rotatingblackhole(a = 0)asNewtoniancase.EvenifintheregionveryclosetoaSchwarzschild
black hole, the flow could not be centrifugally flung out if the field lines inclined greater than the critical angle 60◦ with respect to the
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4 X. Cao
radialdirection.Abramowicz(1990) point outthatthecentrifugalforcereversessignatr = 3M.Forthefreeparticleincircularmotion,
theminimumradiusis4M (unstable).Theresultsobtainedherecouldnotbeextendedtotheregionr = 3M.Cao(1995)studythesame
problembyusingapseudo-Newtonian potentialtosimulategeneral relativisticeffects.Theyfindthatthecriticalanglebecomeslargerin
theregionclosetoanon-rotatingblackhole.Thereasonisthattheangularvelocityderivedfromthepseudo-Newtonianpotentialsimulates
dφ,not dφ.Ingeneralrelativisticframe, dφ and dφ havedifferentvalues.Thebeadsmovealongarigidmagneticfieldlinehaveaconstant
dτ dt dτ dt
angularvelocity dφ measuredbyadistantobserver.
dt
TheresultsareshowninFig.1fortherotatingblackholes.Thecriticalangleθ increaseswiththereductionof theradiusof the
crit
magneticfieldlinefootpoint.ForthebeadsattheminimumstableradiusofanextremeKerrblackhole,thecriticalangleθ couldbeas
crit
largeas90◦,whichmayimplythattheflowscouldbecentrifugallyacceleratedtohighvelocitiesevenbythemagneticfieldlineswithlow
inclinationanglesveryclosetotheblackhole.Wenotethatthecriticalanglefallsrapidlyinthecasesofrelativeslowlyrotatingblackhole,
forexample,θ ∼70◦attheinneredgeofthediscfora=0.9.Theresultsobtainedherestronglyimplythattherotatingblackholewill
crit
behelpfulonthecentrifugallyaccelerationofflowsveryclosetotheblackhole.The’deadzone’foracceleratingoverthetheinnerregion
ofthediscwilldisappearfortherapidlyrotatingblackholewithanappropriatediscmagneticfieldconfiguration.Thefurtherinvestigation
onthisproblemusingthefullrelativisticMHDapproachisnecessary.
Wealsostudythecasethatthematterinthediscisintheretrogradeorbitaround theblackhole,thoughitisnotsurewhethersuch
discexistsinreality.Theresultsshowthatthecriticalangleisslightlyinfluencedbytherotatingblackhole,thecriticalangleisalwaysless
than,butverycloseto60◦.ThecriticalangleattheminimumstableradiusfortheextremeKerrblackholeisabout57◦,almostsameasthe
non-rotatingcase.
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