Effects of Magnetic Coupling on Radiation from Accretion Disc around a Kerr Black Hole

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by  Zhao-Ming Gan

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Effects of Magnetic Coupling on Radiation from Accretion
Disc around a Kerr Black Hole
Zhao-Ming Gan1, Ding-Xiong Wang1,2 and Yang Li1
7 1 Department of Physics, Huazhong Universityof Science and Technology, Wuhan 430074, China
0 2 Send offprint requests to: D.-X. Wang([email protected])
0
2

n 5February2008
a
J

8 ABSTRACT
1 The effects of magnetic coupling (MC) process on the inner edge of the disc are dis-
cussedindetail.Itisshownthattheinneredgecandeviatefromthe innermoststable
1
circular orbit (ISCO) due to the magnetic transfer of energy and angular momentum
v
between a Kerr black hole (BH) and its surrounding accretion disc. It turns out that
2
3 the inner edge could move inward and outward for the BH spin a∗ being greater and
lessthan0.3594,respectively.TheMCeffectsondiscradiationarediscussedbasedon
5
the displaced inner edge. A very steep emissivity canbe providedby the MC process,
1
0 which is consistent with the observation of MCG-6-30-15. In addition, the BH spins
7 of GRO J1655-40 and GRS 1915+105are detected by X-ray continuum fitting based
0 on this model.
/
h Key words: accretion, accretion disc— black hole physics — magnetic field
p
-
o
r
t
s 1 INTRODUCTION between the BH and the disc, just like the energy trans-
a
portation between a dynamo and a motor (Macdonald &
: As is well shown, disc accretion is a very efficient energy
v Thorne 1982, Thorne, Price & MacDonald 1986). In other
source in astrophysics, which is widely used to explain the
i words,theBHexertsatorqueatthediscintheMCprocess,
X high-energyradiationofAGNs,X-raybinariesandsoon.In
resulting in the transfer of energy and angular momentum.
r the model of standard accretion disc (SAD), the motion of The transfer direction is determined by the difference be-
a theaccretingmatterisassumedtobeKeplerianwithasmall
tweentherotationoftheBHandthatofthedisc,i.e.,ifthe
radial velocity, and theinner edge of the disc lies at thein-
BHrotatesfasterthanthedisc,energyandangularmomen-
nermoststablecircularorbit(ISCO)ofradiusr ,within
ISCO tumare extracted from theBH and transferred to thedisc,
whichthematterplungesveryfastontotheblackhole(BH).
otherwise thedirection is reversed.
Inthiscase,therewouldbenosignificanttorquesexertedat
the inner edge, and the “no torque boundary condition” is Recently, Li (2004, hereafter L04) studied the MC be-
averygood approximation(Bardeen 1970, Shakura&Sun- tween a BH and its surrounding disc. It is argued that the
yaev 1973, Novikov& Thorne 1973, Page & Thorne 1974). inner edge of the disc moves out to a radius where the an-
gular velocity of the disc is equal to that of the BH for
Not long ago, some authors discussed the possibility
that magnetic fieldsmay inducea nonzero torqueat ISCO, a∗ < 0.3594. However, the inner edge remains at ISCO for
whichinvolvesthemagneticconnectionofthematterinthe theBH spin with a∗ >0.3594. This result seems somewhat
inconsistent. Why is the inner edge of the disc affected by
plunging region inside ISCO with the matter in the disc
outside ISCO (Krolik 1999; Gammie 1999; Agol & Krolik theMC for a∗ <0.3594, but not for a∗ >0.3594 ?
2000). On the other hand, some authors argued that the In this paper, we intend to discuss the MC effects on
torqueatISCOofathindiskisveryweak,sinceitdepends the inner edge of the disc. In our model the inner edge of
on the vertical thickness of the disk (Paczynski 2000; Li the disc is not assumed to be ISCO in advance. The MC
2002a, hereafter L02; Afshordi & Paczynski 2003). effectsonthediscradiationlieinthefollowingaspects.The
Recently,muchattentionhasbeenpaidtothemagnetic profileoftheradiationfluxandtheinteriorviscoustorqueat
coupling(MC) betweenaBH anditssurroundingaccretion thedisccanbechangedduetothemagnetictorqueexerted
disc (Blandford 1999, Li 2002a, hereafter L02, Wang et al. at the disc. It is shown that if the BH rotates faster than
2002), which can be regarded as one of the variants of the the disc, the disc radiation would be more concentrated at
Blandford-Znajek(BZ)process(Blandford &Znajek1977). the inner disc in the MC process. This feature can be used
By virtue of the magnetic field connecting the rotating BH to explain thevery steep emissivity index in the innerdisc,
andthedisc,energyandangularmomentumaretransferred which is consistent with the XMM-Newton observation of
2
thenearbybright Seyfert 1 galaxy MCG-6-30-15 (Wilmset
al. 2001; Li 2002b; Wang et al. 2003 hereafter W03).
The disc radiation is derived by considering the mag-
1
netictransferoftheenergyandangularmomentumbetween
a rotating BH and a relativistic thin disc (Page & Thorne
1974; L02), and a criterion of determining the inner edge 0.98
of the disc is proposed based on a reasonable constraint,
i.e., a reasonable disc radiation should not be negative. It
+
is shown that the value a∗ = 0.3594 can be regarded as a E0.96
critical BH spin, which corresponds to the angular veloc-
ity of ISCO equal to that of the BH. As the BH spin a∗
is less than 0.3594, i.e., the energy and angular momentum 0.94
are transferred from the inner disc to the BH, and the bal-
anceofenergyandangularmomentumonthediscwouldbe
0.92
disrupted in the MC process. Thus the inner disc becomes
0 5 10 15 20 25 30
unstable, as the disc cannot suffer a negative interior vis- r(cid:144)M
cous torque. It turns out that the MC effects on the disc
radiation are so strong that the inner edge of the disc has
5.5
tomoveoutwardandinward fromISCOfortheBHspina∗
less and greater than 0.3594, respectively.
5
In order to facilitate the discussion of the MC effects
ontheinneredgeofthediscwemakethefollowingassump-
tions: M4.5
(cid:144)
1.Thelargescalemagneticfieldremainsconstantatthe +
L
BH horizon, while it varies as a power-law with the disc 4
radius as given in W03 ratherthan concentrated at ISCO.
2.Thediscisthin,perfectlyconductingandKeplerian, 3.5
lying in the equatorial plane of a KerrBH;
3.Consideringthatthe“notorqueboundarycondition”
3
remains controversial, we assume that no torque is exerted
0 5 10 15 20 25 30
at the inner edge of a thin disc in the MC with a central r(cid:144)M
BH.
This paperis organized as follows. In 2 theproperties Figure1.Specificenergyandangularmomentumoftheaccret-
of the inner edge of SAD and the radial p§rofiles of specific ingmatterofSADversusthediscradiuswitha∗=0.5.Theblack
pointsindicatethepositionradiusofISCO.
energy and angular momentum are discussed in detail. In
3 a criterion for the inner edge of a disc is proposed by
§
consideringtheMCeffects.In 4someofthecharacteristics
§
and applications of the MC process are presented, includ- where M˙ is the accretion rate. The quantities F and
D DA
ing fitting the steep emissivity index of MCG-6-30-15 and g are the radiation flux and interior viscous torque of
DA
theBHspinsofGROJ1655-40andGRS1915+105.Finally, thedisc,respectively.Itisemphasized thatthesequantities
in 5, we summarize our main results. Throughout this pa- canneverbenegativeinanyphysicalcases. E†,L† andΩ
§ D
per Boyer-Lindquist coordinates (t,r,θ,ϕ) and the geomet- arerespectivelythespecificenergy,specificangularmomen-
ric unitsG=c=1 are used. tumand angular velocity of thetest particles movingalong
geodesiccircularorbitsintheequatorialplaneofaKerrBH
(Bardeen, Press & Teukolsky 1972, hereafter B72).
2 INNER EDGE OF SAD WITH NO TORQUE Thequantity isthetotalluminosity,andg isthe
DA in
L
BOUNDARY CONDITION exterior torqueexerted at theinnerboundary with angular
velocity Ω = Ω (r ). Generally, the “no torque bound-
Asiswellknown,thefulldescriptionofaKerrBHneedsonly in D in
arycondition”isassumedinthetheoryofSAD,i.e.,g =0
two parameters, i.e., mass M and spin a∗ ≡ a/M ≡ J/M2 (Page & Thorne 1974). From equation (1)-(3) we findinthat
(−1<a∗ <1). the radiation flux and interior viscous torque could be sig-
Based on Page&Thorne(1974) andL02wederivethe
nificantlydifferentinthecaseofg =0,bywhichanextra
expressionsfortheradiationflux,interiorviscoustorqueand in 6
energy is provided.
total luminosity of a relativistic thin disc as follows.
The specific energy and angular momentum vary non-
F = 1 −dΩD/dr [ r (E† Ω L†)M˙ dL†dr monotonically with the disc radius, attaining their minima
DA 4πr(E†−ΩDL†)2 Rrin − D D dr (1) at ISCO indicated by a black dot as shown in Figure 1.
+g (E† Ω L† )] The radial profile of E† and L† is very different from the
in in− in in monotonous profile of these quantities obtained in Newto-
g = E†−ΩDL†4πr F (2) nian mechanics. It is the radial feature of E† and L† in
DA −dΩD/dr · DA generalrelativitythatresultsintwoconstraintstotheinner
∞
=2 E†F 2πr dr=M˙ (1 E† )+g Ω (3) edge of SAD: (1) it is always located at or outside ISCO
LDA Z DA· · D − in in· in withoutextraenergyandangularmomentumtransferredto
rin
Effects of Magnetic Coupling on Radiation from Accretion Disc around a Kerr Black Hole 3
0
12
0
-25
10
HaL HbL -50 HcL
8 -10
F0 F0 F0 -75
(cid:144) (cid:144) (cid:144)
FDA 6 FDA-20 FDA
-100
4
-125
-30
2
-150
0
1 1.5 2 r(cid:144)2.r5in 3 3.5 4 1 1.5 2 r(cid:144)2.r5in 3 3.5 4 1 1.5 2 r(cid:144)2.r5in 3 3.5 4
Figure 2.RadiationfluxfromSADwithdifferentinneredges versusthediscradiusfordifferentBHspins,wherea∗=0.8,0.6and0.3
correspond to dotted, dashed and solid lines, respectively. The parameter λ=1.0, 0.90 and 0.85 correspond to panels (a), (b) and (c),
respectively.
the accreting matter, and (2) it could move inward within
ISCO,providedthatsomeextraenergyandangularmomen-
AstrophysicalLoad
tum are transferred to the accreting matter. In 3 we shall
§
elaboratethispointindetail,andproposeacriterionforthe
inneredge of a thin disc by considering the MC effects.
Thelower limit r totheintegral inequations(1) and
in
(3)istheradiusoftheinneredgeofthedisc,whichisusually
assumed to lie at ISCO in SAD. Abramowitz et al. (1978)
argued that if the disc is geometrically thick,its inner edge
locatesbetweenISCOandtheinnermostboundcircularor-
bit.
Aparameterλisintroducedtolabelthepositionofthe i
inneredge, which is definedas
r λ2 r , for λ> r /r , (4) ΘM i+1
in ISCO H ISCO
≡
p
where r is theradius of theBH horizon (B72).
H
Combining equations (1) with (4), we have the curves BlackHole AccretionDisk
of FDA/F0 versus r/rin for gin = 0 as shown in Figure 2, rin
where thequantityF is defined as F =10−4 M˙ /r2 .
0 0 · D in
Asshown inFigure2a,theradiation fluxbecomeszero
at r with λ = 1. However, an unphysical disc region Figure3.Thepoloidalmagneticfieldconfigurationinourmodel.
ISCO
with negative radiation flux appears for λ less than unity
asshown inFigures2band2c,implyingthattheradiusr
in
cannot besmaller than rISCO. This unphysicalcase can be ones. In addition, as argued in the next section, the inner
understood based on the radial profile of E†, L† given in edgeof an accretion disccan deviatefrom ISCO in theMC
Figure 1. process due to the magnetic transfer of energy and angu-
In SAD the accreting matter outside rISCO losses its lar momentum between a rotating BH and its surrounding
energy and angular momentum dueto thedifferential rota- accretion disc.
tion, and thus enters into the smaller circular orbit. Since
the radius r corresponds to the minimum specific en-
ISCO
ergy and specific angular momentum as shown in Figure 1,
3 MC EFFECTS ON INNER EDGE OF DISC
the accreting matter has to absorb rather than release en-
ergy and angular momentum to get into the smaller orbit. The magnetic field configuration of our model is shown in
TherealpictureinsiderISCO isthattheaccretingmatteris Figure3.Theparameternisthepower-lawindexindicating
plungedintotheBHwithahugeradialvelocityduetolack the variation of the poloidal magnetic field with the disc
of enough angular momentum. radius, BP r−n. The mapping relation between between
D ∝
Thus we conclude that SAD with an inner edge within the angular coordinate on the BH horizon and the radial
rISCO is not stable, while any radius larger than rISCO is coordinateonthedisccanbederivedfromtheconservation
possible. Itisworth notingthat rISCO,theradiusof ISCO, of themagnetic flux(Wang et al. 2002, W03) and it reads
idsisnco,totnhlyeubneiinqguethcheosimcealfloerstthraediniunsearmedognegoafllaosftathnedaprodsstihbilne cosθ= rrinG(a∗;r′,n) dr′+C0, (5)
R
4
4
8 HaL HbL
2
6
0
F0 F0
(cid:144) (cid:144)
F4 F-2
2 -4
-6
0
-8
1 1.5 r(cid:144)2rin 2.5 3 1 1.5 2 r(cid:144)2.r5in 3 3.5 4
Figure4. Theradiationfluxesversusr/rinwithFtotal/F0,FDA/F0andFMC/F0plottedinsolid,dottedanddashedlines,respectively.
Theparametersn=6andκm=1aretakenwithλ=0.974,a∗=0.45andλ=1.000,a∗=0.10,inpanels(a)and(b),respectively.
G(a∗;r,n)=(r/rin)1−n√1+a2∗M2r−2+2a2∗M3r−3 (6) Ftotal= 4π1r(E−†d−ΩΩDD/Ld†r)2Rrrin(E†−ΩDL†)× (11)
2 (1+a2∗M2ri−n2+2a2∗M3ri−n3)(1−2Mr−1+a2∗M2r−2) (M˙D dL†/dr+4πrH)dr FDA+FMC
· ≡
p
tflwhuhixse,rpeBaCpP0eari)sn.daAntchicneotrepdgorilnaoglidctaoolnmstthaaegnnctoe(ntwisceerfiavdealodtpiootnnCot0fh=ethhceoosrmi0za.o4gn5nπeBtiinc gMC = E−†d−ΩDΩ/DdLr†4πr·Ftotal (12)
D H ∞
are related by (Wanget al. 2002, W03) =M˙ (1 E† )+4π HΩ r dr (13)
Ltotal D − in Z D ·
dψ= B 2π(r2 +a2)sinθ dθ rin
=−BPH2·π rH4+r2a2+2a2Mr· dr, (7) As the magnetic field on the BH is supported by the
D· q r2+a2−2Mr · surrounding disc, there is some relation between BH and
M˙ . As a matter of fact the relation might be rather com-
where ψ = ψ(r,θ) is the magnetic flux through a surface D
plicated,and would beverydifferentin differentsituations.
bounded bya circle with r=constant and r=constant.
One possibility has been suggested by Moderski, Sikora &
Lasota (1997) and is based upon the balance between the
Based onMacdonald&Thorne(1982)andW03weex-
pressure of the magnetic field on the horizon and the ram
press theMC torqueand power as follows,
pressure of theinnermost parts of an accretion flow, i.e.,
 dTMC =2(cid:0)d2ψπ(cid:1)2 (ΩHdZ−HΩD), (8) BH2 (8π)=Pram∼ρc2 ∼M˙D 4πrH2 , (14)
(cid:14) (cid:14)(cid:0) (cid:1)
 dP =2 dψ 2 ΩD·(ΩH−ΩD). Considering thatequation(14)isnotacertain relation
MC 2π dZH between BH and M˙D, we rewrite it as follows,
 (cid:0) (cid:1)
awnhdere ΩH = a∗/(2 rH) is the angular velocity of the BH BH ≡q2κmM˙D/rH2 , (15)
dZ = RH rH2 +a2cos2θ dθ dr (9) whereκm isaparameter indicatingtherelativeimportance
H 2π (r2 +a2)sinθ dr of the MC process with respect to the disc accretion. The
(cid:16) (cid:17) H (cid:16) (cid:17)
MC process dominates over disc accretion for κ 1, and
m
In equations (8) and (9) Z is the resistance of the BH ≫
H itisdominatedbythelatterforκ 1.Weemphasizethat
m
horizon with the surface resistivity given by R = 4π/c ≪
H ≈ equation (15) is independentof (14),one can always fixκm
377 ohm. with any given relation between B and M˙ .
H D
Incorporating the MC effects with the conservation of
AsshowninFigure4a,thetotalradiationfluxfromthe
energy and angular momentum, we havethefollowing rela-
disc with the inner edge within r (λ = 0.974) can be
ISCO
tions:
positive, though more energy and angular momentum are
d (M˙ L† g )=4πr(F L† H), neededtokeeptheKeplerian orbitsof theaccreting matter
 dr D − MC total − (10) within rISCO. The reason is that the energy and angular
 ddr(M˙DE†−gMC·ΩD)=4πr(FtotalE†−HΩD) 0m.3o5m9e4n(tBumlanadrefotrrdan1s9f9e9rr)e,danfrdomthethpeoBsiHtivteoctohnetdriibscutfoiornad∗u>e

where H (1/4πr) dT /dr is the angular momentum to the MC effects exceeds the negative one due to the disc
MC
≡ ·
flux due to the MC effects, and gMC is the interior viscous accretion.TheBHspina∗ =0.3594correspondstoΩH equal
torqueof thedisc in theMC process. to Ω at ISCO.
D
Thus we can derive theradiation fluxand thetotal lu- On the other hand, as shown in Figure 4b, the total
minosity of a thin disc by resolving equation (10). radiationfluxfromtheinnerdisccouldbenegativefora∗ <
Effects of Magnetic Coupling on Radiation from Accretion Disc around a Kerr Black Hole 5
1.5
10
1
8
M 6 (cid:144)LF0 0.5
(cid:144) F
r Hg
o
l
0
4
2 -0.5
0
0 0.2 0.4 a* 0.6 0.8 1 1 1.5 2 r(cid:144)2.r5in 3 3.5 4
Figure 5. Curves of the radius of the inner edge (solid line), Figure 6. The curves of FDA/F0(dotted line) and FMC/F0
theradiusofISCO(dottedline)andtheradiusoftheinnermost (solidline)versusr/rin fora∗=0.75,n=6andκm=1.0.
bound orbit (dashed line) versus the BH spin with n = 6 and
κm=1.
Incorporating equations (11) and (16), we have the ra-
dius of the inner edge of the disc, r /M, varying with the
in
0.3594,givingrisetoanunphysicalradiantregion,although BHspina∗asshowninFigure5.InspectingFigure5,wefind
theinneredgeislocatedatISCOwithλ=1.000.Thisresult thatthedeviationofrinfromrISCOdependsontheBHspin
arises from the magnetic transfer direction of the energy asfollows: (1)rin>rISCO fora∗ <0.3594, (2)rin<rISCO
and angular momentum from the inner disc to the BH for for a∗ >0.3594 and (3) rin =rISCO at a∗ =0.3594.
a∗ <0.3594. Based on the criterion (16), we can evaluate the
FromtheabovediscussionwefindthattheMCprocess influence of the parameter κm on the deviation of rin from
is a very efficient mechanism in transferring energy and rISCO: (1) the turning point of the deviation a∗ = 0.3594
angular momentum between the BH and its surrounding is independent of the value of κm; (2) the greater κm
disc. The inner region of the disc would be eventually corresponds to the greater MC effects, and thus to the
disrupted for a∗ < 0.3594, because too much energy and greater deviation, and (3) we haverin =rISCO for κm =0,
angular momentum are transferred to the BH via the whichcorrespondstoSADwithoutMCeffectsaccordingto
MC process. Thus we infer that the inner edge will move equation (15).
outward beyond ISCO. On the other hand, the inner edge
can be extended within ISCO for a∗ >0.3594, because the Inspecting equations (11) and (12), we find that the
excess energy and angular momentum can be transferred MCtorquealwaysvanishesattheinneredge,implyingthat
magnetically from therotatingBH totheinnerdiscviathe the MC of the BH with a thin disc does not disrupt the no
MC process. torque boundary condition. However, there is an exception
that the closed field lines concentrate at the inner edge of
Fromequation(12),weinferthatgMChasthesamesign thedisc,wheretheMCtorquerelatedtogMC atrin canbe
as the total radiation flux. Combining the MC effects with written in the form
thefactthatanaccretiondisccannotsustainanegativevis- dT /dr 4πrH =(g ) δ(r r ),
cboeucsotnosrtqrauien,ewdebsyuaggpeosstittihvaetraadrieaatsioonnaflbulex.inJunsetrleikdegethsehocuaslde (cid:26) (gMMCC)in =≡τ0·sign(ΩHM−CΩiinn·). − in (17)
This situation has been discussed in L04, where the MC
of SAD, one can always find some reasonable radii for the
inner edge of the disc, as long as they are far away enough torquedoesnotvanishattheinneredgeofthedisc.Fora∗ <
0.3594, the criterion for the inner edge can be equivalently
from the BH, to keep the radiation flux positive through-
written as
out the whole disc. And what we have to do is to find the
smallest radius as the inner edge. Following Wang (1995), ΩH(a∗)=ΩD(rin,a∗). (18)
we consider that the inner edge of the disc is the position
wherethetransportation ofenergyandangularmomentum Equation (18) implies that the inner edge is located
onthediscjustbeginstoexceedtheadjustablerangeofthe at the radius where the disc matter has the same angular
interior viscous torque. However, this criterion can hardly velocity as the BH, which is consistent with the conclusion
in L04.
beformulatedaccurately.Alternatively,wesuggestthatthe
criterion for the inner edge of a thin disc with a smoothly
distributed exterior torquecan be written as
4 MC EFFECTS ON DISC RADIATION
dF d2F
total =0, total >0 (16) TheMCprocesshasasignificanteffectonthediscradiation.
(cid:16) dr (cid:17)ri+n (cid:18) dr2 (cid:19)ri+n Compared with SAD, the radiation flux is much more con-
6
closertotheinneredge.Incorporatingequations(19)—(21)
withthecriterion(16),wehavetheBHspinsofGROJ1655-
40 and GRS 1915+105 as shown in Table 1.
5
In Z97, the factor Q(a∗) is set to unity, and it is ar-
guedthattheapproximationisaccuratetowithin10%fora
Α 4 widerange ofthemodelparameter space.In thispaper,we
x
e check the relation in equation (21) carefully and find that
d
in theapproximationmadeinZ97workswellonlyforthenon-
3
y relativistic discs. Oncethe relativistic effects are taken into
t
i
iv account,thecorrection fromQ(a∗)woulddeviateunitysig-
ss2 nificantly by a factor of several orders. On the other hand,
i
m the correction above in our paper is also partially from the
e
MCeffects.Weemphasizethattheresultsinourpapercon-
1
tainsfullyrelativisticeffects,whichisdifferentfromtheones
in Z97. And one can also find that the derived spins of the
0 black holes in Z97 would be much smaller if the correction
1.2 1.4 1.6 1.8 2 2.2 2.4
r(cid:144)rin Q(a∗) is considered seriously.
Accordingtotheobservation,jetsappearbothinGRO
Figure 7. The curves of α ≡ −dlnFtotal/dlnr versus r/rin J1655-40 and GRS 1915+105. As shown in Figure 3, two
fora∗ =0.998 and κm =10. The dotted, dashed and solidlines
kinds of magnetic field configurations are contained in our
correspondton=6,7and8,respectively.Theemissivityindexof
theSeyfert 1galaxy MCG-6-30-15inferredfromtheobservation model. The closed field lines connecting the BH and the
ofXMM-Newtonisshownbytheshadedregion. disccorrespondtotheMCprocess.Theopenfieldlinescon-
nectingtheBHtotheremoteastrophysicalloadcorrespond
to BZ process, which is widely used to interpret the jet-
production in X-ray binaries, AGNs and GRBs (Blandford
centratedattheinnerdisc.Byusingequations(1)and(11)
& Znajek 1977, Blandford 1999). Based on the argument
wehavethecurvesofF /F andF /F versusr/r as
DA 0 MC 0 in
given in W03 we find that the BZ process can coexist with
shown in Figure 6.
the MC process for the parameters in Table 1. Thus the
From equation (11) we obtain a very steep emissivity
jet-productioninGROJ1655-40andGRS1915+105canbe
indexupto4.3 5.0asshowninFigure7,wheretheemissiv-
∼ interpreted naturally based on this model.
ity index is defined as α dlnF /dlnr. This result is
total
consistentwiththeXMM≡-N−ewton observationofthenearby
brightSeyfert1galaxyMCG-6-30-15 (WilmsJ.etal. 2001;
Branduardi-Raymontet al. 2001). 5 CONCLUSION AND DISCUSSION
Zhang et al. (1997, hereafter Z97) firstly invented an
approachtomeasuretheBHspinbydeterminingtheradius In this paper the MC effects on the radiation from a rela-
ofISCOinfittingthespectrumoftheX-raycontinuumfrom tivistic thin disc are discussed, and some issues related to
athinrelativisticdisc.Consideringthedeviationoftheinner thistoy model are addressed as follows.
edgefrom ISCOduetotheMCeffects, wemodify equation (1)AsarguedbyUzdensky(2005),theMCofarotating
(3) in Z97 as follows, BH with its surrounding disc can be regarded as a stable
magnetic connection, which is dramatically different from
1/2 2
r =ηD Fearth fcolfGR(θ,a∗) (19) the magnetic connection of a rotating neutron star with a
in (cid:20)2σg(θ,a∗) Q(a∗)(cid:21) (cid:20) Tcol (cid:21) disc. Since a BH does not have a conducting surface, the
·
magnetic field lines frozen into a rotating conducting disk
whereg(θ,a∗) and fGR(θ,a∗) havethesame meaning given can slip on the horizon. Compared with the BH, a neutron
inZ97.Weignoretheeffectofmagneticfieldonthegeodesic stars is a highly-conducting star. Therefore, each field line
ofphotons,sotheyareonlydeterminedbythemetricaround connecting the star to the disk is subject to a continuous
the BH. In this paper, we use the values of g(θ,a∗) and twisting, and no steady magnetic connection exists in the
fGR(θ,a∗) from Table 1 in Z97. η rin/rpeak. And the case of a rotating neutron star with a disc (Ghosh, Lamb
≡
other quantitiesare defined as follows: & Pethick 1977, Ghosh & Lamb 1979a, b; Eksi, Herquist &
Fearth=g(θ,a∗)Ltotal/2πD2, (20) Nara(y2a)nT2h0e05M).C discussed in this paper involves a large-
Tcol=fGR(θ,a∗)fcol Tpeak
· scale magnetic field connecting a rotation BH with a thin
Compared with Z97, the MC effects are taken into ac- disc. As argued in L02, this type of MC is very different
countinourmodel.Inaddition,weintroduceafactorQ(a∗) fromthoseinvolvingasmall-scalemagneticfieldconnecting
tomodify theapproximate relation between thebolometric the matter inside ISCO with the matter outside ISCO
luminosity of the disc and the peak emission region (Mak- (Krolik 1999; Gammie 1999; Agol & Krolik 2000), and the
ishima et al. 1986) as follows, “no torque boundary condition” is assumed based on the
Ltotal≡4πσrp2eakTp4eak·Q(a∗) (21) consideration of this MC with a thin disc.
The MC effects on the data fitting behave at least two ThecontributionofthispaperliesintheMCeffectson
aspects:(1) theinneredgedeviatesfrom ISCO,and(2)the discradiationbasedonthedeviationoftheinneredgefrom
radius r of the peak value of the radiation flux is much ISCO,and themain results are summarized as follows.
peak
Effects of Magnetic Coupling on Radiation from Accretion Disc around a Kerr Black Hole 7
Table1.MeasuringBHspinsofGROJ1655-40andGRS1915+105basedonX-raycontinuumwiththeMCeffects(n=6.0andκm=1.0).
Observedparameters
Source BHspin
Fearth(erg·cm−2·s−1) Tcol(k/Kev) θ(deg) M(M⊙)
GROJ1655-40 3.3×10−8 1.36 70 6.0∼6.6 0.932∼0.960
GRS1915+105 4.4×10−8 2.27 70 10∼18 0.927∼0.996
ItisshownthattheinneredgeofSADshouldbelocated Ghosh,P.,Lamb,F.K.,&Pethick, G.J.,1977,ApJ,217,578
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of SAD. ISCO is the smallest one among all the possible —.1979b,ApJ,234,296
stable circular orbits in SAD, although it is not compelling Krolik,J.H.,1999,ApJ,515,L73
Li,L.-X.2002a,ApJ,567,463(L02)
to taken ISCO as the inneredge of thedisc.
—.2002b, A&A,392,469
Considering the magnetic transfer of energy and angu-
—.2004, PASJ,56,685(L04)
lar momentum between a rotating BH and its surrounding
Macdonald,D.,&Thorne,K.S.1982,MNRAS,198,345
accretion disc,weproposeacriterion fortheposition ofthe
Makishima,T.,etal.1986, ApJ,308,635
inneredgebasedonareasonableconstrainttotheradiation Misner, C.W., Thorne, K.S., Wheeler, J.A. 1973, Gravitation
fluxfromthedisc.Itturnsoutthatthedeviationoftheinner (Freeman,SanFrancisco)
edge from ISCO depends on the direction of the magnetic Moderaki, R., Sikora, M., Lasota, J. P. 1997, in Ostrowski M.,
transfer,whichiseventuallydeterminedbytheBHspin:the Sikora M., Madejski, G., Belgelman M., eds, Proc. Interna-
radiusrin couldbegreaterandlessthanrISCO fora∗ being tionalConf.,RelativisticJets inAGNs.Krakow,p.110
less and greater than 0.3594, respectively. Novikov,I.D.,&Thorne,K.S.1973,inBlackHoles,ed.Dewitt
C,(GordonandBreach,NewYork)
In addition, the MC effects on the boundary condition
Paczynski,B.,2000,preprint(astro-ph/0004129)
of the disc are discussed. It is argued that the “no torque
Page,D.N.,&Thorne,K.S.1974,ApJ,191,499
boundary condition” is not affected at the presence of the
Shakura,N.I.&Sunyaev, R.A.1973, Astron.Astrophys,24,337
MC process except that the magnetic field is concentrated
Thorne,K.S.1974,ApJ,191,507
completely at theinneredge. Only in thisextreme case the Thorne,K.S.,Price,R.H.&Macdonald,D.A.1986,Blackholes:
MC effectscan betreated effectively as a“non-zero bound- TheMembraneParadigm(New Haven:YaleUniv.Press)
ary torque”. Uzdensky,D.A.,2005,ApJ,620,889
It has been argued that no stable geodesic circular Wang,D.-X.,Xiao,K.2002, MNRAS,335,655
orbits exist within ISCO (B72). The conclusion is derived Wang,D.-X.,Ma,R.-Y.,Lei,W.-H.andYao,G.-Z.2003,ApJ,
based on geodesic of atest particle in theKerrmetric. It is 595,109(W03)
Wang,Y.M.1995,ApJ,449,L153
worth notingthattheorbitsof theaccretingmattercannot
WilmsJ.etal.2001, MNRAS,328,L27
be regarded as geodesics because of the interior viscous
Zhang,S.N.,Cui,W.&Chen,W.1997,ApJ,482,L155(Z97)
torque and the presence of a strong magnetic field related
to the MC process, and the result that r is less than
in
r for a disc around a fast-rotating BH seems not in
ISCO
conflict with theconclusion given in B72.
Acknowledgments. This work is supported by the Na-
tional Natural Science Foundation of China under Grant
Numbers10573006and10121503.Theanonymousrefereeis
thankedfor his (her) helpful comments and suggestions.
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