A magnetic reconnection model for quasi-periodic oscillations in black hole systems

0.25 MB·English

by  Chang-Yin Huang

ResearchinAstron.&Astrophys.Vol.0(200x)No.0,000–000
Researchin
(http://www.raa-journal.org)
Astronomyand
Astrophysics
A magnetic reconnection model for quasi-periodic oscillations in
black hole systems
3
1
0 Chang-YinHuang1,2,Ding-XiongWang1∗,Jiu-ZhouWang1 andZhi-YunWang3
2
n 1 SchoolofPhysics,HuazhongUniversityofScienceandTechnology,Wuhan430074,China;
a 2 SchoolofMathematicsandStatistics,HuazhongUniversityofScienceandTechnology,Wuhan430074,
J China;

2 3 SchoolofPhysicsandElectronicEngineering,HubeiUniversityofArtsandScience,Xiangyang
441053,China

] ∗[email protected]
E
H
.
h Abstract The quasi-periodic oscillations (QPOs) in black hole (BH) systems of different
p scalesareinterpretedbasedonthemagneticreconnectionofthelarge-scalemagneticfields
- generatedbythetoroidalelectriccurrentsflowingintheinnerregionofaccretiondisk,where
o
thecurrentdensityisassumedtobeproportionaltothemassdensityoftheaccretingplasma.
r
t The magnetic connection (MC) is taken into account in resolving the dynamic equations
s
a of the accretion disk, in which the MC between the inner and outer disk regions, the MC
[ betweentheplungingregionandthedisk,andtheMCbetweentheBHhorizonandthedisk

are involved.It turnsoutthatthe single QPO frequencyof severalBH systems of different
1
v scalescanbefittedbyinvokingthemagneticreconnectionduetotheMCbetweentheinner
2 andouterregionsofthedisk,wheretheBH binariesXTEJ1859+226,XTEJ1650-500and
6 GRS1915+105andthemassiveBHsinNGC5408X-1andREJ1034+396areincluded.In
1 addition,theX-rayspectracorrespondingtotheQPOsare fittedforthesesourcesbasedon
0
thetypicaldisk-coronamodel.
.
1
0 Key words: accretion, accretion disks — black hole physics — magnetic fields — stars:
3 individual(XTEJ1859+226,XTEJ1650-500,GRS1915+105)—galaxies:individual(NGC
1
5408,REJ1034+396)
:
v
i
X
1 INTRODUCTION
r
a
Asiswellknown,X-rayquasi-periodicoscillation(QPO)isacommonphenomenonintheradiationfrom
black-hole binaries (BHBs). The high-frequencyQPOs (HFQPOs) have been observedin several BHBs,
some of which show interesting 3:2 frequency pairs (GRS 1915+105, GRO J1655-40, XTE J1550-564,
H1743-322), and others display single QPO (XTE J1859+226, XTE J1650-500). QPOs have also been
observed in ultraluminousX-ray sources (ULXs), e.g., a 54 mHz QPO in M82 X-1 and a 20 mHz QPO
inNGC5408X-1werediscoveredrespectivelybyStrohmayer&Mushotsky(2003)andStrohmayeretal.
(2007) withXMM-Newton.Strohmayer& Mushotsky(2009) discoveredanotherstrong10mHzQPO in
NGC5408X-1,andfoundthatthecorrelationoftimingandspectralpropertiesofthissourceissimilarto
thoseofGalacticBHBs.
ThefirstconvincingQPOofactivegalacticnuclei(AGNs)wasdiscoveredbyGierlinskietal.(2008)
in narrow line Seyfert 1 RE J1034+396, which opened a window for the comparative timing studies of
stellar-mass and supermassive black holes (BHs). The 1-hourX-ray QPO observedin RE J1034+396is
analogousto the 67 Hz QPO in BHB GRS 1915+105(Middelton& Done 2010). In earlier years, quasi-
periodic signals like QPOs were discovered in some supermassive BHs. For example, the power density
2 C.Y.Huang,D.X.Wang,J.Z.Wang&Z.Y.Wang
spectraoftwoX-rayflaresofSgrA*observedin2000and2002showfivedistinctpeaksatperiodsof 100,
219,700,1150,2250seconds(Aschenbachetal.2004)andaquasi-periodicfluxmodulationwithaperiod
of22.2minuteswasdiscoveredintheX-raydataoftheSgrA*flarein2004(Blangeretal.2006).
A number of models have been proposed to interpret HFQPOs in BHBs. However, none of them
canfullyexplainthecharacteristicsofQPOs, especiallythecorrelationsofspectralandtimingproperties
(Remillard&McClintock2006;Maitra&Miller2010).Asiswellknown,HFQPOsinBHBsarestrongly
correlatedtosteeppowerlaw(SPL)state.AsuccessfulmodelofQPOsshouldlinkthecorrespondingspec-
tral state, allowing for a highly dynamicalinterplay between thermal and nonthermalprocesses with the
mechanismsoperatingoverawiderangeofluminosity(Remillard&McClintock2006).QPOsobservedin
massiveBHsareprobablyrelatedtothebrightstatewhichissimilartotheSPLstateofBHBs(Strohmayer
& Mushotsky2003, 2009; Middeltonetal. 2009), and they mayhave the same originas the HFQPOs in
BHBs(Gierlinskietal.2008;Bian&Huang2010).
InverseComptonscatteringisgenerallythoughttobethepromisingradiationmechanismofSPLstate
(Zdziarski2000;McClintock&Remillard2006,hereafterMR06),anddisk-coronamodelissuccessfulin
fitting the spectrum of SPL state (Gan et al. 2009, hereafter G09; Huang et al. 2010). Zhao et al. (2009)
(hereafterZ09)interpretedHFQPOsinBHBsasthemagneticreconnectionoflarge-scalemagneticfields
generated by toroidal electric currents flowing in the disk without considering the spectral state. In this
paper,weimprovethemodelofZ09basedonthedisk-coronamodelwiththemagneticconnection(MC)
effects.We fitboththe QPO frequenciesandthecorrespondingX-rayspectra offourBH systemsofdif-
ferentscales,inwhichtwoBHBs(XTEJ1859+226,XTEJ1650-500),oneULX(NGC5408X-1)andone
supermassiveBH(REJ1034+396)areincluded.
This paper is organized as follows. A description of the model is given in Section 2. The QPO fre-
quenciesandX-rayspectraofthefoursourcesarefittedbasedonthedisk-coronamodelwithMCeffects
inSection3.AndsomeissuesrelatedtoourmodelarediscussedinSection4.Throughoutthispaper,the
geometricunitsG=c=1areused.
2 MODELDESCRIPTION
2.1 OriginofmagneticfieldsinBHsystems
Large-scalemagnetic fields play veryimportantroles in high energyastrophysicalphenomena.The rela-
tivisticjetsfromAGNsandBHBsarelaunchedandcollimatedbyinvokingtheopenlarge-scalemagnetic
fields relatedto the BZ and BP processes(Blandford& Znajek 1977; Blandford& Payne 1982), and the
broadironlinesobservedinBHsystemscouldbeinterpretedbytransferringenergyandangularmomentum
froma spinningBH to itssurroundingaccretiondisk byinvokingthemagneticconnectionvia theclosed
large-scalemagneticfieldsconnectingthehorizonwiththeinnerdisk(Wilmsetal.2001;Milleretal.2002;
Li 2002a; Wang et al. 2002, hereafterW02).However,a consensuson the origin of large-scalemagnetic
fieldsinBHsystemshasnotbeenreached.
Unlike neutron stars, magnetic fields cannotexist on the horizonof an isolated BH, and they should
be maintainedby the surroundingenvironment,such as an accretiondisk. For BHBs, the magneticfields
probablycomefromtheplasmaofthecompanion.Oneofthemostpromisingoriginsoflarge-scalemag-
neticfieldsinBHsystemsisaccretiondiskaroundtheBHs,andsomeauthorscalculatedthemagneticfield
configurationsbyassumingthetoroidalelectriccurrentinthedisk(Li2002b;Z09).However,theoriginof
thecurrentsremainstobeclarified.
Inthispaper,weintendtointerprettheoriginoftheelectriccurrentbasedontheassumptionthatthe
accretingplasmadeviatessomehowfromelectricneutralityasitflowsthroughtheLagrangepointintothe
Roche lobe of the BHBs, resulting in toroidalelectric currentsflowing in the accretion disk. In addition,
weresolvethedynamicalequationsoftheaccretiondiskbytakingtheMCeffectsintoaccountandfigure
outthemassdensityandcurrentdensityinthedisk,andthendeterminetheconfigurationofthelarge-scale
magnetic fields by consideringthe interaction between the electric currentand the disk with the iterative
algorithm.ItturnsoutthatwecanfittheassociationofQPOswithspectralstatesinBHsystemsofdifferent
scalesbasedonourmodel.
Amagneticreconnectionmodelforquasi-periodicoscillationsinblackholesystems 3
8
6
MCHD
W
H
M
(cid:144)r4
MCPD
2
BH MCDD AccretionDisk
Disk
00 2 rrin 4 rr0 6 8 rin r0
r(cid:144)M BlackHole
Fig.1 Theconfigurationoflarge-scalemagnetic Fig.2 Aschematicdrawingofmagneticrecon-
fields generated by the continuous distribution nectioninMCDD,wheretheredlinerepresents
of toroidal electric current flowing in the disk. thefieldlineconnectingradiirin andr0.
Themagneticfieldlinesareplottedinthinlines
andtheboundariesbetweenthreetypesofmag-
neticconnection(MCHD,MCPD,andMCDD)
areshownbythethicklines.Theredlinerepre-
sentsthemagneticfieldlineconnectingtheradii
rinandr0.ThefigureisplottedwithmBH =10,
a = 0.8,m˙ = 0.1,α = 0.3,η = 10−13, and
∗
n = 5, where m˙ and α are the mass accretion
rate in unit of the Eddingtonaccretionrate and
theviscosityparameter,respectively.
Thedeviationfromelectricneutralityisdescribedbydefiningaparameterηasfollows,
η n n /n , (1)
e p p
≡| − |
wheren andn arethenumberdensitiesofelectronsandprotons,respectively.Thechargedensitycanbe
e p
expressedas
ρ =ηen , (2)
e p
wheree=4.8 10−10e.s.uistheelectroncharge.
×
Thelarge-scalemagneticfieldsgeneratedbythetoroidalelectriccurrentmayinturnaffectthecurrent.
For example, the field lines may pipe hot electrons into the corona above the disk therefore change the
charge density in the disk. This effect should be strongest in the inner disk because the magnetic field
intensity is strongest near the inner edge of the disk and decreases rapidly outwards. For simplicity, we
assumethiseffectdecreaseswith theincreasingdiskradiusasa powerlawandthesurfacedensityofthe
currentinthediskcanbeexpressedas
ηe
j =ρ 2h ν r−n =ηen 2h Ω r1−n = ρ hΩ r1−n, (3)
e K p K m K
· · · · · µm
p
whereh,ρ ,andm arethehalfheightofthedisk,themassdensityandtheprotonmassrespectively.The
m p
powerlawindexnisafreeparameterforfittingtheQPOfrequency,andµ = 0.615istheweight-average
molecularweightof the gas. The toroidalelectric currentis assumed to distribute fromthe inner edgeof
the disk to the outer boundary of the disk-corona system, the radius of which is set at r = 100M in
out
calculations,sincetheradiationoftheaccretiondiskmainlycomesfromtheinnerregion.
4 C.Y.Huang,D.X.Wang,J.Z.Wang&Z.Y.Wang
13
12 190Hz
11
n
10
9
8
10-16 10-15 10-14 10-13 10-12
Η
Fig.3 Thecontourof190HzQPOofXTEJ1859+226inη nparameterspacewithm =12,
BH
−
a =0.96,m˙ =0.38,andα=0.3.
∗
FollowingtheworkofZnajek(1978),Linet(1979),Li(2002b)andZ09,wecancalculatethetoroidal
componentoftheelectricvectorpotentialdeterminedbythecurrentgivenbyEq.(3)withagivenmassden-
sityofthediskmatter.Andtheboundariesofinner-outerregionsrelatedtothemagneticfieldconfiguration
can be determined.As shown by Fig.1, there are three types of magnetic field configurationgenerated—
MCoftheBH with thedisk(MCHD),MCoftheplungingregionwiththedisk(MCPD),andMCofthe
innerandouterdiskregions(MCDD).IntheMCDDregion,themagneticfieldlinesarefrozeninthedisk
attheinnerandouterfootpoints.Thefieldlineswilltwistthemselvessincetheinnerandouterfootpoints
havedifferentangularvelocities.InFig.1,theinnerandouterfootpointsofthefieldlineinredlocateatr
in
(theinneredgeofthedisk)andr ,respectively.Thevalueofthe angularvelocitydifferencebetweenthe
0
footpointsofthisfieldlineismaximalamongallthelinesinMCDDregion,sothislinewilltwistitselffirst
andtriggerthemagneticreconnection,asshowninFig.2,whichreleasesmagneticenergyperiodicallyto
generateflareswith the frequencyofthe differencebetweenKeplerianfrequenciesofthe innerand outer
footpoints.WeinterpretthisfrequencyastheQPOfrequency,anditreads:
ν = ΩK(rin)−ΩK(r0) =ν (r3/2M−3/2+a )−1 (r3/2M−3/2+a )−1 , (4)
QPO 2π 0h in ∗ − 0 ∗ i
where ν0 ≡ m−BH13.23×10−4Hz and mBH ≡ M/M⊙ is the black hole mass in unit of the solar mass.
a a/M isthedimensionlessBHspin.r isinitializedattheinnermoststablecircularorbit(ISCO).
∗ in
≡
As shown by Eq. (3), parametersn and η determine the currentdensity and thereforemagnetic con-
figurationandQPOfrequencyoncethesurfacedensityofdiskmatterisgiven.Thevalueofη determines
thecurrentintensityandthereforemagneticfieldstrength,whilethatofndeterminestheconcentricityof
thecurrentsandmagneticfields.AlargernleadstoasmallerMCDDregion,asmallerdistancebtweenrin
andr0,andhencealowerQPOfrequency,whileasmallernleadstotheoppositeresults.Asanexample,
acontourof190HzQPOofXTEJ1859+226isplottedinη nparameterspaceasshowninFig.3.Itis
foundthatamaximumη 10−12isrequiredtoavoidatoost−rongmagneticfieldforastablesolutionwith
∼
theMCeffect.
2.2 ThedynamicalequationsofdiskandcoronawithMCeffects
ThedynamicalequationsoftheaccretionflowaremodifiedbyconsideringtheMCeffectsonthetransfer
ofenergyandangularmomentum(Li2002a;G09),
d
(M˙ L† g)=4πr(QL† H ), (5)
D MC
dr − −
d
(M˙ E† gΩ )=4πr(QE† H Ω ), (6)
dr D − i − MC i
Amagneticreconnectionmodelforquasi-periodicoscillationsinblackholesystems 5
d¶Di+1 dZDi+1 d¶D dZD d¶D dZD
dZC dZC dZC
I I I
d¶Di dZDi d¶P dZP d¶H dZH
Fig.4 AnequivalentcircuitforMCDD,MCPDandMCHDgivenfromlefttoright.
where Q and g are respectivelythe viscously dissipated energyper unitdisk surface and interiorviscous
torqueofthedisk,andQconsistsoftwoparts,
Q=Q +Q , (7)
d cor
whereQ isradiatedfromdiskasblackbody,
d
Q =σT4, (8)
d eff
andQ istransportedintothecoronatoheatitbymagneticreconnectionoftangledsmall-scalemagnetic
cor
fields,anditreads(Liuetal.2002)
B2 B3
Q = DV = D , (9)
cor A
4π 4π√4πρ
whereB ,V andρaretheintensityofthesmall-scalemagneticfields,theAlfvenspeed,andmassdensity,
D A
respectively.
ThequantityH inEqs.(5)and(6)isthefluxofangularmomentumanditreads
MC
2
H 1 dTMC = 1 dΨ Ωi−Ωi+1, (10)
MC
≡ 4πr dr 2πrdr (cid:18)2π(cid:19) dZ
i
where dT and dΨ are the torque exerted on to an infinitesimal annulus of the disk and magnetic flux
MC
threadingtheinfinitesimalannulus,respectively.
WecancalculateavarietyofmagnetictransferofenergyinBHaccretiondiskbyusinganequivalent
circuitinananalogouswaytoMacdonald&Thorne(1982)andW02asshowninFig.4.ThreetypesofMC
(MCDD,MCPDandMCHD)indicatedinFig.1aregiveninFig.4,inwhichaseriesofloopscorrespondto
theadjacentmagneticsurfacesarisingfromtherotationoftheclosedfieldlines.
ThequantitiesdZ ,dZ ,dZ anddZ aretheresistancescorrespondingtothetwoadjacentmagnetic
D P H C
surfacesinthedisk,theplungingregion,theBH horizonandthecorona,respectively.Thequantitiesdε
D
anddε arerespectivelytheelectromotiveforcesgeneratedbytherotationofthediskandtheBHhorizon
H
asgivenbyW02,anddε istheelectromotiveforceduetotherotationoftheplungingregion.
P
ThequantitiesΩi,Ωi+1 anddZi inEqs.(6)and(10)areexplainedforthethreedifferentdiskregions
asfollows.
(1) MCDD region: Ωi and Ωi+1 are respectively the angular velocities of the inner footpointi and
outerfootpointi+1oftheclosedfieldlineonthedisk,andtheresistancedZ isdefinedby
i
R dr
dZ =dZ = cor , (11)
i cor
2πr
whereR istheaveragearealresistivityofthediskandcorona,whichisrelatedtoR ,thesurfaceresis-
cor H
tivityoftheBHhorizonasfollows,
R =η R
cor R H (12)
(cid:26)RH =4π =377ohm,
6 C.Y.Huang,D.X.Wang,J.Z.Wang&Z.Y.Wang
whereη isaparametertoadjustthevalueofR .Forsimplicity,wesetη =0.1incalculations.
R cor R
(2)MCHDregion:Ωi =ΩHandΩi+1 =ΩDaretheangularvelocitiesoftheBHhorizonandthedisk
respectively.Theresistance
dZ =dZ +dZ , (13)
i H cor
where
dZ =2ρ dθ/̟. (14)
H H
TheparametersinEq.(14)aregivenby
̟ =(Σ /ρ )sinθ, ρ2 r2 +a2cos2θ,
H H H ≡ H
Σ 2Mr , r =M(1+q), q = 1 a2. (15)
H ≡ H H − ∗
p
(3)MCPD region:Ωi = ΩP andΩi+1 = ΩD are respectivelythe angularvelocitiesof the plunging
regionandthedisk,andΩ isgivenasfollows(Shapiro&Teukolsky1983;Wang2000),
P
(r 2M)L†+2aME†
Ω = − . (16)
P (r3+a2r+2Ma2)E† 2aML†
−
TheresistanceisthesameasthatinMCDDexpressedbyEq.(11).
We adopt the same assumption as given by G09 about the corona: the optical depth of the corona
τ =1andtheheightofthecoronal=10r .
cor ms
3 FITTINGTHEQPOFREQUENCIESANDX-RAYSPECTRA
In our model, the toroidalelectric currentinteractswith the dynamicsof the accretion disk. To solve the
dynamicEqs.(5)and(6),wemustknowtheconfigurationofthelarge-scalemagneticfieldsgeneratedby
thetoroidalelectriccurrent,whosedistributionisrelatedtothesurfacedensityofthediskmatterbyEq.(3),
and ρ is in turnfiguredoutby solvingthe dynamicalequations.For simplicity,we assume thatinitially
m
thereisnoelectriccurrentinthedisk,andsolvetheEqs.(5)and(6)withH =0togettheglobalsolution
MC
ofthedisk-coronasystem.Theemergedspectrumofthedisk-coronasystemisthensimulatedwithMonte
Carlo method based on the code developed by G09. The free parameters of the disk-corona model, e.g.,
BHspina andmassaccretionratem˙ ,canbedeterminedbyfittingtheobservedspectrum.Sothesurface
∗
density of the disk matter is obtained. Then we consider the interaction between the electric currentand
disk-coronawiththeiterativealgorithmwhichconsistsofthefollowingsteps:
(i)AssumingavalueofthepowerlawindexninEq.(3),e.g.n=5.
(ii)Calculatingthesurfacedensityofthetoroidalelectriccurrentinthediskandtheconfigurationof
thelarge-scalemagneticfields.
(iii)Resolvingthedisk-coronasystembytakingtheMCeffectsintoaccount.
(iv)Repeatingsteps(ii)and(iii)untilthesurfacedensityofdiskmatterremainsstable.
(v)Calculatingthefrequencyν byEq.(4).
QPO
(vi)Repeatingsteps(i)-(v)untiltheQPOfrequencyisinaccordancewiththeobservations.
Steps(ii)-(iii)shouldberepeatedseveraltimesbeforethesurfacedensityofdiskmatterbecomesstable.
Finally,wesimulatetheemergedspectrumagainuntilitisunchanged.
To avoid the negative radiation flux from the inner disk due to the transfer of energy and angular
momentumtotheouterdiskbytheMCDDasarguedbyGanetal.(2007),weadjusttheradiusoftheinner
boundaryofthedisktodeviateoutwardsfromISCO.Althoughthedeviationislessthan10%oftheradius
ofISCO,itresultsin 30%decreaseofQPOfrequency.
∼
3.1 FittingtheHFQPOsinXTEJ1859+226andXTEJ1650 500withSPLstate
−
ObservationsshowthatHFQPOsinBHBsareassociatedwiththeSPLstatewhichischaracterizedbyhigh
luminosity (therefore high accretion rate), strong power-law component and the steep power-law index
(Γ > 2.4).WefitthesingleHFQPOsandthecorrespondingX-rayspectraoftwoBHBsXTEJ1859+226
and XTE J1650-500. The comparisons between the simulated spectra and the observed ones are shown
Amagneticreconnectionmodelforquasi-periodicoscillationsinblackholesystems 7
211---PhotonscmskeV0.00.000.1111 HaL+++++++++++++++++++++++++++++++++++++++++++++++++++X+H+T1+9+E9+9+J++O1mmaΑ+ +*8cB==+=t+H50+0+10=9..+363.+1+9+-8++26+1282L6 211---PhotonscmskeV0.00.000.1111 HbL+++++++++++++++++++++++++++++++++++++++++++++++++++X+H+T1+9+E9+9+J++O1mmaΑ+ +*8cB==+=t+H50+0+10=9..+362.+1+9+-7++29+12882L6 211---PhotonscmskeV0.00.000.1111 HcL+++++++++++++++++++++++++++++++++++++++++++++++++++X+H+T1+9+E9+9+J++O1mamΑ+ +*8c==+B=t+5H0+0+10=9..+363.++97+-1++9.+612882L6
10-4 10-4 10-4
2 5 10 20 50 100 2 5 10 20 50 100 2 5 10 20 50 100
EnergyHkeVL EnergyHkeVL EnergyHkeVL
Fig.5 The simulated spectra of XTE J1859+226 with different parameters: (a) m = 12,
BH
a = 0.96, m˙ = 0.38 and α = 0.3; (b) m = 12, a = 0.998, m˙ = 0.27 and α = 0.3;
∗ BH ∗
(c) m = 7.6, a = 0.998, m˙ = 0.31, and α = 0.3. The total spectrum and its thermal
BH ∗
and Comptonized components are plotted in solid, dotted and dashed lines, respectively. The
observationdataaretakenfromMR06.Thesourcedistanceissetat11kpc(Zuritaetal.2002;
MR06)andtheinclinationi=70◦isassumed.
for XTE J1859+226 and XTE J1650 500in Figs. 5 and 6, respectively.The total spectra consist of the
−
thermalcomponentemittedfromthediskandthepower-lawcomponentgeneratedbytheinverseCompton
scatteringofthesoftphotonsbytherelativisticelectronsinthecorona.
ThefittingparametersarelistedinTable1,inwhichm˙ andαarerespectivelythemassaccretionratein
termsoftheEddingtonaccretionrate(1.4 1018m gs−1)andtheviscosityparameter.TheBHmassof
BH
×
XTEJ1859+226wasestimatedintherange7.6 12M (MR06).We fitthespectrumofXTEJ1859+226
⊙
−
observedinOctober16 18of1999(MR06)withtheBHmassoftheupperlimit12M andofthelower
⊙
−
limit7.6M . Sincethe BH massof XTEJ1650 500hasnotbeenwell estimated,we take a largermass
⊙
−
7.3M andasmallerone4M ,whichareestimatedbyOroszetal.(2004)tofitthespectrum.Verylarge
⊙ ⊙
BH spin is needed to fit the spectra for the lower limit to the BH mass, and the lower limit to BH spin
correspondingto each BH mass is given in Table 1. The spectrum of XTE J1650 500is fitted with the
−
lowerandupperlimitstospinforeachBHmass,whilethespectrumofXTEJ1859+226canbefittedonly
foranextremeKerrBHwiththelowerlimittotheBHmass7.6M .Thevalueoftheviscosityparameter
⊙
αisselectedproperlyintherange0.1 0.3tofitthespectrumofeachsource.Thedifferenceofthevalue
−
arisesprobablyfromthedifferenceofthestrengthofthesmall-scalemagneticfieldinthediskiftheviscous
processisdominatedbythetangledsmall-scalemagneticfield.Thevaluesofthefittingparametersnand
η for QPO frequencies are listed in the last two column in Table 1. We fix η at a certain value for each
scaleofBHmass.BHsystemswithsmallermasshavestrongermagneticfieldallowingoflargervalueof
η. A larger value of n is needed with a smaller BH mass, implying that the toroidalelectric currentsare
concentratedveryclosetoISCO.
3.2 QPOsinNGC5408X-1andREJ1034+396
OneofattractivefeaturesofourmodelisthatitisapplicabletofitQPOswithcorrespondingX-rayspectra
observedintheBHsystemsofdifferentscales.Strohmayeretal.(2007)discoveredastrong20mHzQPO
intheULXNGC5408X-1,andtheX-raytimingandspectralpropertiesareanalogoustothoseexhibited
byGalacticstellar-massBHsinthe‘veryhigh’orSPLstate.ThefittingparametersfortheQPOandX-ray
spectrumofNGC5408X-1arelistedalsoinTable1.Thesimulatedspectrafordifferentparametersandthe
comparisonstotheobservedenergyspectrumareshowninFig.7.However,atpresent,theBHmassesin
theULXsareconstrainedtoawiderangefrom102M to105M (Milleretal.2003;Cropperetal.2004;
⊙ ⊙
Robertsetal.2005;Wu&Gu2008).WetakeaverylargeBHmass105M tofitthedataofNGC5408X-1
⊙
sinceboththeX-rayspectrumandQPOfrequencyarefittedbetterforlargerBHmass.TheQPOfrequency
andspectrumarebothtakenfromStrohmayeretal.(2007)withthesameHydrogencolumndensity,which
arefittedinourmodelwiththelowerandupperlimitstotheBHspinasshowninTable1.
8 C.Y.Huang,D.X.Wang,J.Z.Wang&Z.Y.Wang
XTEJ1650-500 XTEJ1650-500
211---PhotonscmskeV01.000.00-114 ++Ha+L++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++H++2++0+0++1mamΑ+ *+S=B=+=eH0+0p0=+..2.1+278+01.7+3L+ + 211---PhotonscmskeV01.000.00-114 ++Hb+L++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++H++2++0+0++1mmΑa+ +*S=B=+=eH0+0p0=+..20.+279+31.9+3L8+ +
10-5 ++ 10-5 ++
15 20 30 50 70 100 150 200 15 20 30 50 70 100 150 200
EnergyHkeVL EnergyHkeVL
XTEJ1650-500 XTEJ1650-500
211---PhotonscmskeV01.000.00-114 ++Hc+L++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++H++2++0+0++1mΑm+a +S*==B+=eH0+0p0+.=.21+.249+711+L+ + 211---PhotonscmskeV01.000.00-114 ++Hd+L++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++H++2++0+0++1mΑma+ +*S==B+=eH0+0p0+.=.20.+294+619+L8+ +
10-5 ++ 10-5 ++
15 20 30 50 70 100 150 200 15 20 30 50 70 100 150 200
EnergyHkeVL EnergyHkeVL
Fig.6 The simulated spectra of XTE J1650 500 with different parameters: (a) m = 7.3,
BH
−
a = 0.87,m˙ = 0.10andα = 0.2;(b)m = 7.3,a = 0.998,m˙ = 0.03andα = 0.2;(c)
∗ BH ∗
m = 4, a = 0.91, m˙ = 0.17, and α = 0.2; (d) m = 4, a = 0.998, m˙ = 0.06, and
BH ∗ BH ∗
α=0.2.TheobservationdataaretakenfromMiniuttietal.(2004).Thesourcedistanceissetat
2.6kpc(Homanetal.2006)andtheinclinationi=70◦isassumed.
The first convincing QPO of AGNs was reported by Gierlinski et al. (2008) in RE J1034+396.
Middelton&Done(2010)suggeststhattheQPOdiscoveredinREJ1034+396hasananalogytothe67Hz
QPOseenintheBHBGRS1915+105duetotheirsimilar‘hotdiskdominated’energyspectra.Unlikeother
HFQPOs, the 67 Hz QPO in GRS 1915+105is an exceptionalcase, which appears in thermal-dominant
(TD) state (MR06).In thissubsection,we fit boththe 67 Hz QPO in GRS 1915+105andthe 0.00027Hz
QPO in RE J1034+396 as a comparison. The spectrum of GRS 1915+105 showing the 67 Hz QPO is
takenfromMiddleton&Done(2010).AndthespectrumofREJ1034+396showingthe0.00027HzQPOis
takenfromMiddletonetal.(2009)withthesameminimumgalacticabsorptionfixedat1.31 1020cm−2.
×
The‘hotdiskdominated’spectraof GRS 1915+105and RE J1034+396are bothfitted with almostmax-
imum BH spinsas shown in Table 1, and their comparisonswith the observedspectra are shown in Figs
8 and 9, respectively.The disparities between the simulated spectra and the observed ones in the energy
bands10 30keVforGRS1915+105and0.5 1keVforRE J1034+396indicateprobablythata second
− −
ComptonizationprocessisneededtogeneratetheTDspectrumasshowninFigs10and11ofMiddelton&
Done(2010),wherealowtemperature,opticallythickthermalComptonizationisaddedtofitthespectra.
ThisComptonizationmaybegeneratedfromthetransitionlayerbetweenthediskandthecorona.
InspectingTable 1 and Figs 5 9,we find that the QPOs and the correspondingX-ray spectra of BH
−
systemsofdifferentscalescanbefittedwiththemagneticreconnectionofthelarge-scalemagneticfields
basedonthedisk-coronamodel.QPOsinmassiveBHshavesimilarfeatureswiththoseofBHBs,e.g.very
centralizeddistributionoftheelectriccurrentsorlargevalueofn,anditsinverseproportiontoBH mass.
Theyare eitherrelated to the SPL state, like XTE J1859+226and XTE J1650 500,or related to the TD
−
state,likeGRS1915+105.ThesesimilarcharacteristicshintprobablythatQPOsinBHsystemsofdifferent
scalesmayhavethesameoriginandareassociatedwiththesamespectralstate.
Amagneticreconnectionmodelforquasi-periodicoscillationsinblackholesystems 9
Table1 ThefittingparametersforQPOfrequenciesandX-rayspectra.
Source νQPO(Hz) mBH a∗ m˙ α n η
XTEJ1859+226 190a 12b 0.96 0.38 0.3 8.4
0.998 0.27 0.3 8.9
7.6b 0.998 0.31 0.3 14.6
XTEJ1650−500 250c 7.3d 0.87 0.10 0.2 3.1
0.998 0.03 0.2 9.1 10−12
4d 0.91 0.17 0.2 17.7
0.998 0.06 0.2 23.1
GRS1915+105 67e 18b 0.998 0.25 0.14 16.9
10b 0.998 0.40 0.16 27.0
NGC5408X-1 0.02f 1.0×105 0.95 0.012 0.1 15.1 10−16
0.998 0.005 0.1 15.5
REJ1034+396 0.00027g 7.0×106h 0.99 0.15 0.3 16.4 10−17
0.998 0.14 0.3 16.4
2.0×106h 0.998 0.13 0.3 39.7 10−15
aCui et al. (2000); bMR06; cHoman et al. (2003); dOrosz et al. (2004); eMorgan
etal.(1997);fStrohmayeretal.(2007);gGierlinskietal.(2008);hZhouetal.(2010)
0.01 0.01
NGC5408X-1 NGC5408X-1
211---onscmskeV011.0000--154 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++H+2+0+0amΑm +6*=B==HJ000a=..1n.09115502L5 211---onscmskeV011.0000--154 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++H+2+0+0maΑm +6*=B==HJ00a0=..1n.01905095L58
hot + + hot + +
P 10-6 HaL + + P 10-6 HbL + +
+ +
0.5 1.0 2.0 5.0 10.0 0.5 1.0 2.0 5.0 10.0
EnergyHkeVL EnergyHkeVL
Fig.7 The simulated spectra of NGC 5408 X-1 with different parameters: (a) m = 105,
BH
a =0.95,m˙ =0.012andα=0.1;(b)m =105,a =0.998,m˙ =0.005andα=0.1.The
∗ BH ∗
plotstyleisthesameasFigure5.TheobservationdataaretakenfromStrohmayeretal.(2007).
Thesourcedistanceissetat4.8Mpc(Karachentsevetal.2002)andtheinclinationi = 75◦ is
assumed.ThetotalHydrogencolumndensityissetasnH =13 1020cm−2(Strohmayeretal.
×
2007).
4 DISCUSSION
Inthispaper,atoymodelfortheQPOsinBHsystemsofdifferentscalesisproposedbasedonthemagnetic
reconnectionoflarge-scalemagneticfieldsgeneratedbythetoroidalelectriccurrentsinthedisk.Thedy-
namicalequationsofaccretiondiskareresolvedbasedontheinteractionbetweentheelectriccurrentswith
thedisk-coronasystembyusingtheiterativealgorithm.The190Hzand250HzsingleHFQPOsinBHBs
XTEJ1859+226andXTEJ1650 500associatedwithSPLstatesarewellfittedbasedonthedisk-corona
−
modelwithelectriccurrentsflowingintheinnerdisk.AndthesimilarQPOsobservedinULXNGC5408
X-1andSeyfert1AGNREJ1034+396andthecorrespondingX-rayspectraarealsofitted.Thespectrum
ofNGC5408X-1isfittedwithstrongpower-lowcomponentandsteeppower-lawindexsuggestingthatthe
QPOissimilartotheHFQPOsinBHBsXTEJ1859+226andXTEJ1650-500andareprobablyassociated
withthesamespectralstate—SPLstate.WhiletheQPOinREJ1034+396isanalogoustothe67HzQPO
inGRS1915+105whichisassociatedwithTDstate.
10 C.Y.Huang,D.X.Wang,J.Z.Wang&Z.Y.Wang
100 100
1-L GRS1915+105 1-L GRS1915+105
21--cmskeV 101 ++ +++++++++++++++++++++++++++++++++ H1996Αamm A*==B=p0H00r.=.1.229415998L8 21--cmskeV 101 ++ +++++++++++++++++++++++++++++++++ H1996Αamm A*==B=p0H00r.=.1.42961990L8
V + V +
ke ++ ke ++
HFkeVE 0.1 ++++++++++++ ++++++++ HFkeVE 0.1 HbL ++++++++++++ ++++++++
E +++ + E +++ +
0.01 +++ +++++ + ++ 0.01 +++ +++++ + ++
1 2 5 10 20 50 100 200 1 2 5 10 20 50 100 200
EnergyHkeVL EnergyHkeVL
Fig.8 ThesimulatedspectraofGRS1915+105withdifferentparameters:(a)m = 18,a =
BH ∗
0.998,m˙ = 0.25andα = 0.14;(b)m = 10,a = 0.998,m˙ = 0.4andα = 0.16.Theplot
BH ∗
style is the same as Figure 5. The observationdata are taken from Middleton & Done (2010).
Thesourcedistanceissetat11kpc(McClintocketal.2006)andtheinclinationi=66◦(Fender
etal.1999)isused.
0.020 0.020 0.020
211---HLFkeVkeVcmskeVE25´´000011....0000000001--125044 HaL +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R+++++++E+H++2++m+0maΑ+J +0+*B=1=+7+=H+000+=0M+..3+31.7+954a+´9y+++13+03169L+6 211---HLFkeVkeVcmskeVE25´´000011....0000000001--125044 HbL +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R+++++++E+H++2++m+0maΑ+J +0*+B=1=+=7+H+000+0=M+..3+31.79+44a+´9y++8+13+03169L+6 211---HLFkeVkeVcmskeVE25´´000011....0000000001--125044 HcL +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R+++++++E+H++2++m+Α0ma+J +0*=+B1=+=7+H0+00+0=.M+.33+1.29+34a+´9y++8+13+03169L+6
E1´10-4 + E1´10-4 + E1´10-4 +
0.2 0.5 1.0 2.0 5.0 10.0 0.2 0.5 1.0 2.0 5.0 10.0 0.2 0.5 1.0 2.0 5.0 10.0
EnergyHkeVL EnergyHkeVL EnergyHkeVL
Fig.9 ThesimulatedspectraofREJ1034+396withdifferentparameters:(a)m = 7 106,
BH
a =0.99,m˙ =0.15andα=0.3;(b)m =7 106,a =0.998,m˙ =0.14andα=0×.3;(c)
∗ BH ∗
m =2 106,a =0.998,m˙ =0.13,andα=×0.3.TheplotstyleisthesameasFigure5.The
BH ∗
×
observationdataaretakenfromMiddletonetal.(2009).Thesourcedistanceissetat125.9Mpc
(z=0.042,H0=100km s−1 Mpc−1) andthe inclinationi = 40◦ is assumed.We only consider
theminimumgalacticabsorptionfixedat1.31 1020cm−2(Middletonetal.2009).
×
SimilaritiesoftheQPOsinBHsystemsofdifferentscalesenableustoestimatesomephysicalquan-
tities of the massive BHs which have not been well constrained at present. As suggested by MR06, the
frequenciesofHFQPOsinBHBsshowingthe3:2frequencypairsareinverseproportionaltotheBHmass.
Abramowiczetal.(2007)usedthe1/M scalingtoexpectQPOfrequenciesforBHsofdifferentscalesand
neutronstars.Similarly,wecanusethisrelationtoestimatetheBHmassinULXNGC5408X-1.Including
thethreeBHBsshowingthesingleHFQPO,wehavetherelationship,m = 1460ν−1 ,betweentheBH
BH QPO
massandtheQPOfrequencyasshowninFig.10.TheBHmassofNGC5408X-1isthenestimatedabout
7.3 104M withthe0.02HzQPOasshownbytheblackdotinFig.10.
⊙
×
Thequasi-periodicsignalssimilartoQPOswerealsodiscoveredintheX-rayfluxofSgrA*.Ifthese
signalsare indeedQPOs andaretriggeredby themagneticreconnectiondescribedin this paper,thenthe
lowerlimitoftheBHspincanbeconstrainedbecausetheouterfootpointsofthemagneticfieldlinescannot
extendtotheinfinitedistance.WeestimatethelowerlimitoftheBHspinofSgrA*as0.448byfittingthe
22.2minutessignalsdiscoveredintheX-rayflareon2004August31(Blangeretal.2006)usingthemass
4.4 106M (Genzeletal.2010),whichisveryclosetothevaluea 0.44 0.08estimatedbyKatoet
⊙ ∗
× ≈ ±
al.(2010)usingtheQPOmethodinthecontextofdisk-seismology.