A model of rotating hotspots for 3:2 frequency ratio of HFQPOs in black hole X-ray binaries

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by  Ding-Xiong Wang

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
A model of rotating hotspots for 3:2 frequency ratio of
HFQPOs in black hole X-ray binaries
Ding-Xiong Wang1,3, Yong-Chun Ye1, Guo-Zheng Yao1,2 and Ren-Yi Ma1
1 Department of Physics, Huazhong Universityof Science and Technology, Wuhan,430074,China
5 2 Department of Physics, Beijing Normal University, Beijing100875, China
0 3 Send offprint requests to: D.-X. Wang ([email protected])
0
2

n 2February2008
a
J

1 ABSTRACT
2
We propose a model to explain a puzzling 3:2 frequency ratio of high frequency
1
quasi-periodic oscillations(HFQPOs) in black hole (BH) X-ray binaries,GRO J1655-
v
40, GRS 1915+105 and XTE J1550-564. In our model a non-axisymmetric magnetic
6
coupling (MC) of a rotating black hole (BH) with its surrounding accretion disc co-
5
exists with the Blandford-Znajek (BZ) process. The upper frequency is fitted by a
4
rotating hotspot near the inner edge of the disc, which is produced by the energy
1
0 transferred from the BH to the disc, and the lower frequency is fitted by another ro-
5 tating hotspotsomewhere awayfromthe inner edge ofthe disc, whicharisesfromthe
0 screw instability of the magnetic field on the disc. It turns out that the 3:2 frequency
/ ratioofHFQPOsintheseX-raybinariescouldbewellfittedtotheobservationaldata
h
with a much narrower range of the BH spin. In addition, the spectral properties of
p
- HFQPOs are discussed. The correlation of HFQPOs with jets from microquasars is
o contained naturally in our model.
r
t Key words:
s
a accretion, accretion discs — black hole physics — magnetic field — instabilities —
: stars:individual (GRO J1655-40)— stars:individual(GRS 1915+105)— stars:indi-
v
i vidual (XTE J1550-564)— stars: oscillations — X-rays: stars
X
r
a
1 INTRODUCTION ries.Strohmayer(S01a;2001b)investigatedcombinationsof
the azimuthal and radial coordinate frequencies in general
Quasi-periodic oscillations in X-ray binaries havebecome a relativity to explain the HFQPO pairs in GRO J1655-40
very active research field since the launch of the Rossi X- and GRS 1915+105. Wagoner et al. (2001) regarded the
Ray Timing Explorer (RXTE; Bradt, Rothschild & Swank HFQPOpairsasfundamentalg-modeandc-modediscoseis-
1993). A key feature in these sources is that some of high mic oscillations in a relativistic accretion disc. Abramowicz
frequency quasi-periodic oscillations (HFQPOs) appear in & Kluzniak (2001) explained thepairs in GRO J1655-40 as
pairs. Fiveblack hole (BH) X-raybinaries exhibittransient a resonance between orbital and epicyclic motion of accret-
HFQPOs, of which three sources have pairs occurring in ing matter. It seems that more than one physical model is
GRO J1655-40 (450, 300Hz; Strohmayer 2001a, hereafter required to fit all of theHFQPO observations.
S01a; Remillard et al 1999), GRS 1915+105 (168, 113Hz;
As is well known, the Blandford-Znajek (BZ) process
McClintock & Remillard 2003, hereafter MR03), and XTE
isan effectivemechanism for powering jets from AGNsand
J1550-564 (276, 184Hz; Miller 2001; Remillard et al 2002)
quasars, and it is also applicable to jet production in mi-
withapuzzling3:2ratiooftheupperfrequencytothelower
croquasars(Blandford&Znajek1977,Mirabel&Rodrigues
frequency.Veryrecently,the3:2frequencyratio(henceforth
1999). Recently, the magnetic coupling (MC) of a rotating
3:2 ratio) was discovered in the X-ray outburst of H1743-
BHwithitssurroundingdischasbeeninvestigatedbysome
322(240, 160Hz;Homan etal.2004; Remillard etal. 2004).
authors(Blandford1999;Li2000a,2002a;Wangetal.2002,
These discoveries give the exciting prospect of determining
hereafterW02),whichcanberegardedasoneofthevariants
BHmassandspin,aswellastestinggeneralrelativityinthe
oftheBZprocess.TheMCprocesscanbeusedtoexplaina
strong-field regime.
verysteepemissivityintheinnerregionofthedisc,whichis
A numberof different mechanisms have been proposed foundbytherecentXMM-Newton observationofthenearby
to explain the origin of HFQPO pairs in BH X-ray bina- brightSeyfert1galaxyMCG-6-30-15(Wilmsetal.2001;Li
2
2002b; Wanget al. 2003a, hereafter W03a). Inaddition, we
explained the HFQPOsin X-ray binaries based on a model
AstrophysicalLoad
of a rotating hotspot due to the MC of a rotating BH with
the inner region of the disc (Wang et al. 2003b, hereafter
W03b). However, the 3:2 ratio of HFQPOs has not been
discussed by virtueof the MC process.
Very recently, we discussed the condition for the coex- JetProduction
istence of the BZ and MC processes, and found that this
coexistence always accompanies the screw instability of the
magnetic field in BH magnetosphere, provided that theBH
spin and the power-law index for the variation of the mag- i
netic field on the disc are greater than some critical values θS i+1
(tWivealnyg).et al.2003c, 2004, hereafterW03cand W04,respec- θL PQ R S
In this paper we propose a model to explain the 3:2
BlackHole AccretionDisc
ratio of HFQPOs in BH X-ray binaries by virtue of non- rms InnerHotspot OuterHotspot
axisymmetric MC of a rotating BH with its surrounding rS
accretion disc. The upper frequency arises from a rotating
Figure1.PoloidalmagneticfieldconnectingarotatingBHwith
hotspot near the inner edge of the disc, and the lower fre-
aremoteastrophysicalloadandthesurroundingdisc.
quencyisproducedbythescrewinstabilitysomewhereaway
from theinneredge.Itturnsoutthatthe3:2ratioofHFQ-
POsinGROJ1655-40,GRS1915+105 andXTEJ1550-564
non-axisymmetric as assumed in W03b, varying with the
iswellfittedbyenergytransferredintheMCprocess,where
azimuthal angle φ as follows (see Fig.2 in W03b),
six parameters are used to describe the BH mass, spin and
distribution of a non-axisymmetric magnetic field. The 3:2 1+δ, 0<φ<∆φ,
Bp (φ)= (Bp)2 f(φ), f(φ) (1)
ratioobtainedinourmodelprovidesamuchnarrowerrange H q H ≡(cid:26) 1, ∆φ6φ62π,
(cid:10) (cid:11)
of the BH spin compared with the other models. Further-
more,thejetsfromtheseBHbinariesareexplainednaturally where (Bp)2 is root-mean-square of the poloidal mag-
H
by theBZ process coexisting with the MC process. q
neticfield(cid:10)overth(cid:11)eangularcoordinatefromθ=0toθ .The
L
This paperis organized as follows. In section 2 we give
parameter δ is used to describe the strength of the bulging
a brief description of our model. In section 3 we fit the up-
magnetic field in the azimuthal region 0<φ<∆φ.
perandlowerfrequenciesofHFQPOsbytheinnerandouter
(3)Themagnetosphereisassumedtobeforce-freeout-
hotspotsco-rotatingwiththedisc.Thediscussionisfocused
sidetheBHandthedisc,andtheclosedmagneticfieldlines
at the issues of the outer hotspot. A scenario of how the
are frozen in the disc. The disc is thin and perfectly con-
presence of the screw instability leads to the outer hotspot
ducting,liesintheequatorialplaneoftheBHwiththeinner
is given. In addition, the time-scale of the screw instabil-
boundarybeing at ISCO.
ity is estimated by using an equivalent R-Lcircuit, and the
(4) The poloidal magnetic field varies as a power law
spectral properties of HFQPOs are discussed by assuming
with theradial coordinate of thedisc as follows,
the existence of corona above the disc. In section 4 we dis-
cusssomeissuesofastrophysicalmeaningsrelatedtothe3:2 Bp ξ−n, (2)
D ∝
ratio.ThroughoutthispaperthegeometricunitsG=c=1 whereBp representsthepoloidalmagneticfieldonthedisc,
are used. D
andtheparameternisthepower-lawindex,andthedimen-
sionlessradialcoordinateisdefinedasξ r/r .Theradius
ms
≡
r isrelatedtotheBHmassM andthedimensionlessspin
ms
parameter a∗ a/M as given in NT73.
≡
2 DESCRIPTION OF OUR MODEL (5) The magnetic flux connecting the BH with its sur-
roundingdisctakesprecedenceoverthatconnectingtheBH
Weintendtofitthe3:2ratioofHFQPOsbyvirtueofanro- with the remote load. As argued in W03c and W04, as-
tatingBHsurroundedbyamagnetizedaccretiondiscbased sumption 5is crucialfor thecoexistence of theBZ andMC
on thefollowing assumptions. processes.
(1) The configuration of the poloidal magnetic field is ThemagnetosphereanchoredinaKerrBHanditssur-
shown in Fig. 1. The radius rms indicates theinner edge of rounding disc is described in Boyer-Lindquist coordinates,
the disc, being the innermost stable circular orbit (ISCO) inwhichthefollowing parametersareinvolved(MacDonald
(Novikov & Thorne 1973, hereafter NT73), and rS is the and Thorne 1982, hereafter MT82).
radiuswherethescrewinstabilityofthemagneticfieldmight
occur,whichisrelatedtotheangleθ bythesamemapping Σ2 = r2+a2 2 a2∆sin2θ, ρ2=r2+a2cos2θ,
S −
rbeoluatnidonargyivbeentwineeWn0t4h.eAospsehnowanndincFloisge.d1,fiθeSldislitnheesaonngutlhaer  ∆α==ρr√(cid:0)2+∆aΣ2.−(cid:1)2Mr, ̟=(Σ/ρ)sinθ, (3)
horizon, and θL is the lower boundary angle for the closed  In W03b(cid:14)the BZ and MC powers in non-axisymmetic
fieldlinesconnectingtheBHwiththedisc.Throughoutthis
case are related to those in axisymmetic case by
paper θ =0.45π is assumed in calculations.
L
(2) The poloidal magnetic field on the BH horizon is PNA=λPA , (4)
BZ BZ
A model of rotating hotspots for 3:2 frequency ratio of HFQPOs in black hole X-ray binaries 3
PMNCA =λPMAC, (5) only depends on the parameters a∗, and n, i.e., it is inde-
pendent of the parameters m , B , δ and ε. Considering
where PNA and PNA are the powers in non-axisymmetic BH 4
BZ MC that the hotspot is frozen at the disc, we have the upper
case,andPBAZ andPMAC arethepowersinaxisymmeticcase, frequency νupper by substituting ξ into the Keplerian
respectively.Theparameterλisexpressedintermsofδand QPO max
angular velocity as follows,
ε ∆φ/2π by
λ≡=[(1+δ)ε+(1 ε)]2 =(1+δε)2. (6) νQPO =ν0(ξ3/2χ3ms+a∗)−1, (15)
− whereν (m )−1 3.23 104Hz.Itturnsoutthatνupper
Similarly, the BZ and MC torques in non-axisymmetic depends0o≡n theBHthree×param×eters, mBH, a∗ and n. QPO
case are related to those in axisymmetic case by The lower frequency of HFQPOs is fitted by an outer
TNA=λTA , (7) hotspot rotating with the Keplerian angular velocity ex-
BZ BZ pressed byequation (15),and a scenario for theproduction
TNA =λTA , (8) of theouter hotspot is given as follows.
MC MC
It is well known that the magnetic field configurations
where TNA and TNA are the torques in non-axisymmetic
BZ MC withbothpoloidalandtoroidalcomponentscanbescrewin-
case,andTA andTA arethetorquesinaxisymmeticcase,
BZ MC stable (Kadomtsev 1966; Bateman 1978). According to the
respectively. The powers PA and PA , and the torques
BZ MC Kruskal-Shafranov criterion (Kadomtsev 1966), the screw
TA and TA are given in W02 as follows.
BZ MC instability will occur,if thetoroidal magnetic fieldbecomes
θ k(1 k)sin3θdθ so strong that the magnetic field line turns around itself
PBAZ P0 =2a2∗Z 2 (−1 q)sin2θ, 0<θ<θS, (9) about once. Recently, some authors discussed the screw in-
(cid:14) 0 − − stability in BH magnetosphere (Gruzinov 1999a; Li 2000b;
θ β(1 β)sin3θdθ Tomimatsu et el. 2001). In W04 we suggested that the cri-
PMAC P0 =2a2∗Z 2 (−1 q)sin2θ, θS <θ<θL. (10) terion for the screw instability in the MC process could be
(cid:14) θS − − expressed by
TBAZ(cid:14)4Ta∗0(=1+q) 0θ 2(1−−(1β−)sqi)ns3inθ2dθθ, 0<θ<θS, (11) wFShcerreewL(ai∗s;ξt,hne)p−olLoi/d(a2lπ̟lenDg)th6o0f,the closed field line c(o1n6)-
R
TA T = nectingtheBHandthedisc,̟D isthecylindricalradiuson
MC4(cid:14)a∗0(1+q) θθS 2(1−−(1β−)sqi)ns3inθ2dθθ, θS <θ<θL. (12) thedSiesctt,ianngdeqFuSacrlietwy(ian∗;eξq,una)tiiosna(f1u6n)c,tiwoenoobftaa∗in, ξthanedcrnit.i-
R cal radial coordinate ξ r /r for the screw instability,
Inequations(9)—(12),kandβ aretheratiosoftheangular S ≡ S ms
velocities of the magnetic field lines to the angular velocity which dependson theparameters a∗ and n.
It is believed that a disc is probably surrounded by
oftheBHhorizonintheBZandMCprocesses,respectively.
a high-temperature corona analogous to the solar corona
Thequantityq≡ 1−a2∗ isafunctionoftheBHspin,and (Liang & Price 1977; Haardt 1991; Zhang et al 2000). Very
P0 and T0 are defipnedas recently, some authors argued that the coronal heating in
P (Bp)2 M2 B2m2 6.59 1028erg s−1, some starsincluding theSunis probably related to dissipa-
0≡ H ≈ 4 BH× × · (13)
(cid:26) T0≡ (cid:10)(BHp)2(cid:11)M3 ≈B42m3BH ×3.26×1023g·cm2·s−2, tviaorniaotifocnurorfemntasg,naentdicvfiereyldsstr(oGnaglsXga-raaryde&mPisasironneslla2r0is0e4;frPome-
(cid:10) (cid:11)
where B4 is the strength of the poloidal magnetic field on teret al. 2004).
the horizon in units of 104gauss, and mBH is defined as Analogously, if the corona exists above the disc in our
mBH ≡M/M⊙. model,weexpectthatthecoronaaboveξS mightbeheated
Sofarsixparametersareinvolvedinourmodel,inwhich by the induced current due to the screw instability of the
mBH anda∗areusedfordescribingthemassandspinofthe non-axisymmetric magnetic field. Therefore a very strong
Kerr BH, and the three parameters, B4, δ and ε are used X-rayemission wouldbeproducedabovetherotatingouter
for describing the non-axisymmetric magnetic field on the hotspot,andthelowerfrequencyνlower isobtaineddirectly
QPO
horizon, the power-law index n is for the variation of the bysubstituting ξ intoequation (15).
S
magnetic field with theradial coordinate of thedisc. Thetime-scaleofthescrewinstabilitycanbeestimated
byusinganequivalentcircuitasshowninFig.2,inwhichthe
segmentsLM andKN representthetwoadjacentmagnetic
surfacesconnectingtheBHhorizonandtheouterhotspotas
3 INNER AND OUTER HOTSPOTS FOR 3:2
shown in Fig. 2a. Considering the existence of the toroidal
FREQUENCY RATIO
magneticfieldthreadingeachloop,weintroduceaninductor
As argued in W03b, the upper frequency of HFQPOs is into the circuit, which is represented by the symbol ∆L in
modulated by an inner hotspot, which corresponds to the Fig. 2b.
maximum of function F as follows,
QPO ∆ε =(∆Ψp/2π)Ω , ∆ε = (∆Ψp/2π)Ω , (17)
H H D D
F r2F r2 F =ξ2(F +F )/F , (14) −
QPO ≡ ms 0 DA MC 0 where ∆Ψp is the flux of the poloidal magnetic field sand-
where F is(cid:14)the radiation flux due to disc accretion, and wiched by the magnetic surfaces LM and KN. The resis-
DA
F is radiation flux arising from the energy transferred tance at the BH horizon is given by
MC
fromtherotatingBHintotheinnerdiscintheMCprocess. ∆l 2ρ ∆θ
Wefindthattheradialcoordinateoftheinnerhotspot,ξmax, ∆RH =σH2π̟ = H̟ H, (18)
H H
4
where σ = 4π = 377ohm is the surface resistivity of the
H
BH horizon (MT82), and ∆θ is the angle on the horizon
H
spannedbythetwosurfaces.Thequantitiesρ and̟ are
H H
the Kerrmetric parameters at the horizon. The inductance
∆L in the R-Lcircuit is definedby
∆L=∆ΨT Ip, (19)
(cid:14)
where Ip and ∆ΨT are the poloidal current flowing in the
circuit and theflux of thetoroidal magnetic field threading
the circuit, respectively. The flux ∆ΨT can be integrated
overthe region KLMN as follows,
θ K
S
L
∆ΨT =I BT√grrgθθdrdθ, (20) M N
loop
BlackHole AccretionDisc
where the toroidal magnetic field measured by “zero-
angular-momentum observers” (ZAMOs) is given as rms
rS
BT =2Ip/(α̟), (21) (a)
where α is thelapse function definedin equation (3). ∆ε ∆L
D
Unfortunately, the geometric shapes of the magnetic
N M
surfaces LM and KN are unknown. As an approximate
estimation we assume that the surfaces LM and KN are
formedbyrotatingthetwoadjacentcirclesaroundthesym-
metricaxis,and∆ΨT canbecalculated byintegratingover
the region KLMN as shown in Fig. 3. Finally, we calcu-
late thequantityτ ∆L/∆R byincorporating equations IP
H
≡
(18)—(21), which is independentof IP.
Asasimpleanalysiswecandivideoneeventofthescrew
instability into two stages. In stage 1 the instability starts,
and the magnetic energy is released rapidly, the poloidal
currentandthusthetoroidalmagneticfielddecreasetozero
in very short time. In stage 2 the toroidal magnetic field
recoversgraduallywiththeincreasingpoloidalcurrentIP in K L
∆ε ∆R
theR-Lcircuit,untilthenexteventoftheinstabilityoccurs H H
accordingtothecriterion(16).Itseemsreasonablethatthe (b)
duration in stage 1 is much less than that in stage 2, and
Figure2.(a)TwoadjacentmagneticsurfacesconnectingtheBH
thetime-scaleofthescrewinstabilitydependsmainlyonthe
horizonandtheouter hotspot, (b)oneloopofequivalent circuit
duration of stage 2, which is regarded as the relax time of
forscrewinstability.
the poloidal current IP in the R-L circuit. The current IP
is governed by thefollowing equation,
∆LddItP +∆RHIP =∆εH+∆εL. (22) tScrew >trelax>1 νQupPpOer >1 νQloPwOer. (26)
(cid:14) (cid:14)
As pointed out in MR03, the spectral properties of ac-
Settingthe initial condition, IP =0, we havethe solution,
cretingblackholesinstateswhichshowHFQPOsareusually
IP(t)=Ip 1 e−t/τ , (23) dominated by a steep power-law (SPL) component. How-
steady −
whereIp is(cid:0)thesteady(cid:1)currentintheR-Lcircuit,andit ever, the origin of X-ray power-law remains controversial.
steady Most of models for the SPL state invoke inverse Compton
reads
scattering as the operant radiation mechanism, and the ra-
Ip =(∆ε +∆ε )/∆R . (24) diation spectrum formed through Comptonization of low-
steady H L H
frequency photons in a hot thermal plasma cloud may be
Fromequation(23)weobtainthatthepoloidalcurrent
described by a power law (Pozdnyakov, Sobol & Syunyaev
attains 99.3% of Ip in the relax time t = 5τ, im-
steady relax 1983). It is noted that the origin of the Comptonizing elec-
plying therecovery of thetoroidal magnetic field. Thus the
trons involves magnetic instabilities in the accretion disc
time-scale of the screw instability can be estimated as
(Poutanen&Fabian1999),withwhichthescrewinstability
t >t =5τ. (25) in our model is consistent.
Screw relax
Suppose that the spectral properties of the inner and
Summarizing theaboveresults, we havetheupperand
outer hotspots arise from blackbody spectra, the effective
lower frequencies of HFQPOs, and the corresponding val-
radiation temperature (T ) is expressed by
ues of ξ , ξ and t as shown in Table 1. It is easy to HS eff
max S relax
check from Table 1 that the time-scale of the screw insta- T =T [f +f ]1/4, (27)
eff 0 DA MC
bilityisgenerally greater thanthecorrespondingperiods of
HFQPOs,i.e., T =(F /σ)1/4 4.8 105B1/2(K), (28)
0 0 ≈ × 4
A model of rotating hotspots for 3:2 frequency ratio of HFQPOs in black hole X-ray binaries 5
Table 1. The 3:2ratioof HFQPOsproduced by the inner and outer hotspots withδ=0.5and ε=0.2. Henceforth BHB-I,II and III
representGROJ1655-40, GRS1915+105andXTEJ1550-564,respectively.
InnerHotspot OuterHotspot
BHB a∗ n mBH trelax (sec) ξmax EHupSper B41/2 νQupPpOer ξSC EHloSwer B41/2 νQloPwOer
(kev) . (Hz) (kev) . (Hz)
0.777 6.22 6.8 7.61×10−3 1.277 7.23×10−3 450 2.015 4.96×10−3 300
I
0.730 6.27 5.8 6.57×10−3 1.281 6.72×10−3 450 2.097 4.46×10−3 300
0.773 6.23 18 2.02×10−2 1.277 7.19×10−3 168 2.021 4.92×10−3 113
II
0.603 6.17 10 1.11×10−2 1.297 5.55×10−3 168 2.428 3.27×10−3 113
0.788 6.19 11.5 1.28×10−2 1.276 7.37×10−3 276 2.000 5.09×10−3 184
III
0.696 6.29 8.5 9.72×10−3 1.284 6.38×10−3 276 2.159 4.14×10−3 184
Note: The value ranges of the BH mass corresponding to GRO J1655-40, GRS 1915+105 and XTE J1550-564 are adopted from
Greeneetal.(2001), MR03andOroszetal.(2002), respectively.
4 DISCUSSION
Inspecting Table 1, we find that the 3:2 ratios of HFQPOs
in GRO J1655-40, GRS 1915+105 and XTE J1550-564 are
wellfittedbyadjustingthetwoparameters,a∗andn,forthe
given value ranges of m . In this section we discuss some
BH
issues of astrophysical meanings related to the 3:2 ratio of
HFQPOsin the BH binaries.
K
L
4.1 3:2 ratio and magnetic field on the BH
M N
horizon
A B
AccretionDisc
It is pointed out that the twin HFQPOs are not always
detected together in GRO J1655-40, i.e., 300 Hz QPO is
BlackHole observed sometimes without 450 Hz detection, while 450
Hz QPO is observed sometimes without 300 Hz detection
(S01a). In our model the upper and lower frequencies of
HFQPOs are produced by the inner and outer hotspots,
which are located at different sites of the disc. In addition,
ourmodelisproposedbasedonthenon-axisymmetricmag-
neticfieldonthehorizon,andtwodifferentphysicalmecha-
nismsareinvolved,i.e.,theMCprocessfortheinnerhotspot
Figure 3. Twoadjacent magnetic surfaces produced bytwo ro- and the screw instability for the outer hotspot. Since the
tating circles, where the arrows represent the current flowing in non-axisymmetric magnetic field cannot be stationary on
theR-Lcircuit. the horizon, we can understand that the twin HFQPOs do
not haveto occur simultaneously in some cases.
In Table 1, both Eupper and Elower reach the energy
HS HS
where σ is the Stefan-Boltzmann constant. The energy of levelofemitting X-ray,providedthat theroot-mean square
theinner and outer hotspots can beestimated by of the magnetic field on the horizon is strong enough to be
109gauss (B 105). Considering that a system cannot
E k T =E [f +f ]1/4, (29) ∼ 4 ∼
HS ≡ B eff 0 DA MC radiate a given flux at less than the blackbody tempera-
E =k (F /σ)1/4 4.14 10−2B1/2(kev), (30) ture (Frank et al. 1992), the strength of magnetic field of
0 B 0 ≈ × 4 109gauss might be regarded as a upper limit for energy
∼
wherek istheBoltzmannconstant.Substitutingξ and level of emitting X-ray in BH binaries.
B max
ξ into equation (29), we have the energy of the inner and Itis pointedout in S01athat theenergy spectraof the
S
outer hotspots Eupper and Elower as shown in Table 1. 300 HzQPO appear to besignificantly different from those
HS HS
Inspecting equation (29) and the expressions for f of the 450 Hz QPO in GRO J1655-40: The former has a
DA
and f given in W03b, we find that both Eupper and typical amplitude in the 2–12 keV band, while the latter is
MC HS
EHloSwer depend not only on the parameters, a∗ and n, but detected in the hard band, 13–27 keV. It is also shown in
also on the parameters for the non-axisymmetric magnetic Fig.4.16ofMR03thatthepowerdensitiescorrespondingto
field, B1/2, δ and ε. It is worthy to point out that both theupperfrequencyaregreaterthanthosecorrespondingto
4
Eupper and Elower are independent of the BH mass of the thelowerfrequencyofHFQPOsinsomeBHX-raybinaries,
HS HS
X-ray binaries. such as GRO J1655-40 (450, 300Hz) and XTE J1550-564
6
(276, 184Hz). These observations are consistent with the
results obtained in our model, i.e., Eupper is greater than
HS 0.12
Elower for each BH binary as shown in Table 1.
HS 0.998
0.95
0.1
4.2 BH spin and mass constrained by 3:2 ratio
0.9
0.08
It is widely believed that HFQPOs in BH binaries might
beauniquetimingsignatureconstrainingtheBHmassand P0
/0.06
spinviaamodelrootedin generalrelativity.Thusaprecise AZ 0.8
NB
P
measurement of the 3:2 ratio of HFQPOs becomes an ap-
0.04
proach to estimate the BH mass and spin. Combining the 0.7
fitting results in Table 1 with the observations of the three
0.02 0.6
BH binaries, we discuss the issues related to the BH mass
0.5
and spin as follows.
0
The BH spin in GRO J1655-40 has been estimated by
4 5 6 7 8 9 10
someauthors.InterpretingtheX-rayspectra,Zhang,Cui&
n
Chen (1997) suggested that the most likely value is 0.7 <
a∗ <1,whileSobczaketal.(1999)giveanupperlimita∗ < (a)
0.7. Assuming that a bright spot appears near the radius 10
of the maximal proper radiation flux from a disc around a 0.12 9
rotatingBH,Gruzinov(1999b)inferredanupperlimita∗ < 8
0.6 for the BH in GRO J1655-40. Abramowicz & Kluzniak 0.1 7
(2001) suggested that the 3:2 ratio in GRO J1655-40 arises
from a resonance between orbital and epicyclic motion of P00.08 6
accreting matter, and they estimated that the value range /
of the BH spin is 0.2 < a∗ < 0.65. Compared with the NAPBZ0.06 5
aboveestimationranges,weprovideamuchnarrowerrange
0.04
for the BH spin in GRO J1655-40, 0.730 < a∗ < 0.777, as
4
showninTable1,wheretheBHspinsinGRS1915+105and
0.02
XTEJ1550-564arealsoconstrainedinrathernarrowranges,
0.603<a∗ <0.773 and 0.696<a∗ <0.788, respectively. 0
0.5 0.6 0.7 0.8 0.9 1
a
*
4.3 3:2 ratio and jets from microquasars
(b)
TheBHX-raybinaries,GROJ1655-40,GRS1915+105and
Figure4.TheBZpowerPNA(a)varyingwithnwiththegiven
XTE J1550-564, are also regarded as microquasars, from BZ
valuesofa∗,(b)varyingwitha∗ withthegivenvaluesofn.
which relativistic jets are observed (Mirabel & Rodrigues
1998, 1999 and references therein; see also Table 4.2 of
MR03).Thiscorrelation canbeexplainedverynaturallyby
equations (4) and (9) we find that the BZ power increases
virtueofthemagneticfieldconfigurationasshowninFig.1.
monotonically withthepower-lawindexnforthegivenBH
The jet power is assumed to be provided by the BZ mech-
anism and expressed byequations (4)and (9).Substituting spin a∗, while it varies non-monotonically with a∗ for the
given n as shown in Fig. 4.
a∗, n, mBH and ξS into these equations, we have the jet
powers as shown in Table 2. In calculations the parameter
k=0.5 is taken for theoptimal BZ power (MT82). 4.4 3:2 ratio and electric current on disc
Mirabel & Rodriguez (1999) pointed out that GRS
1915+105 may have a short-term jet power of 1039erg/s AsshowninTable1,averylargepower-lawindexnvarying
initsveryhighstate,whichisalargefractionoft∼heobserved from 6.17 to6.29 isrequiredbythe3:2ratioofHFQPOsin
accretion power. From Table 2 we have the angular bound- the BH binaries. It implies that the poloidal magnetic field
ary and the corresponding BZ power for GRS 1915+105 as increases inward in a very steep way governed by equation
follows. (2). It is worthy to point out that this result is consistent
with that in W03a, where the power-law index n varying
PBNZA =4.87×1029B42(erg/s), 0<θ<50.50, (31) from 6 to 8 is required to explain a very steep emissivity
PNA =4.68 1028B2(erg/s), 0<θ <42.40. (32) (4-5) observed in Seyfert 1 galaxy MCG-6-30-15.
BZ × 4 S The poloidal magnetic field arises most probably from
Equations (31) and (32) imply that a large fraction of the toroidal electric current on the disc, and the current
the rotating energy is extracted from the BH contained in density j is related to Bp byAmpere’s law as follows.
ϕ D
GRS 1915+105 for the jet power, and PNA can reaches
1039erg/s, provided that the magnetic fiBelZd on the horizo∼n j = 1 dBDp = 1 dBDp = n(BDp)msξ−(n+1). (33)
is strong enough to attain 109gauss (B 105). This ϕ 4π dr 4πrms dξ − 4πrms
4
∼ ∼
strength of the magnetic field is the same as that required Inspecting equation (33), we obtain two features of the
bytheinnerandouterhotspotsforemittingX-ray.Byusing toroidal electric current on thedisc.
A model of rotating hotspots for 3:2 frequency ratio of HFQPOs in black hole X-ray binaries 7
Table 2.Thejetpowersaccompanying 3:2ratioofHFQPOswithδ=0.5andε=0.2.
BHB a∗ n mBH θS PBNZA B42(erg/s)
(cid:14)
0.777 6.22 6.8 50.70 7.09×1028
I 0.730 6.27 5.8 48.90 3.98×1028
0.773 6.23 18 50.50 4.87×1029
II
0.603 6.17 10 42.40 4.68×1028
0.788 6.19 11.5 50.90 2.14×1029
III
0.696 6.29 8.5 47.70 7.01×1028
(i)Theminussigninequation(33)showsthatthecur- the 3:2 ratio, which play very important roles in fitting the
rent flows in the opposite direction to the disc matter; (ii) 3:2 ratio of HFQPOs.
thepower-law indexn+1in equation (33) impliesthat the Inthispaperanon-axisymmetricmagneticfieldonthe
profile of the current density in the central disc is steeper BHhorizon isassumedinanadhocway.Unfortunately,we
thanthatofthemagneticfield.Wegiveaprimaryexplana- cannot give a good explanation for its existence at present.
tion for these features as follows. Probably the non-axisymmetric magnetic field on the BH
Probably the direction of the current arises from the horizon arises from non-axisymmetric accretion disc, since
effect of the closed magnetic field lines on the charged par- the magnetic field is brought and held by the surrounding
ticles of the disc matter. In our model the angular velocity magnetized disc. We hope to approach this difficult task in
of the BH is required to be greater than that of the disc to future.
producetheinnerandouterhotspotsonthedisc.Therefore
thechargedparticles ofthediscmatterwill bedragged for- Acknowledgments. We thank the anonymous referee for
ward by theclosed field lines. Considering that the mass of his/her many constructive suggestions. This work is sup-
thepositivechargedparticle,suchasproton,ismuchgreater portedbytheNationalNaturalScienceFoundationofChina
than that of electron, we think that the bulk flow of elec- underGrant Numbers10173004, 10373006 and 10121503.
tronsmightbealittleaheadofthatofprotons.Probablythe
minute difference in the two kinds of the bulk flows results
in a current in theopposite direction to thedisc.
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