The Dependence of Spectral State Transition and Disk Truncation on Viscosity Parameter $α

0.13 MB·English

by  Erlin Qiao

PASJ:Publ.Astron.Soc.Japan,1–??,hpublicationdatei
(cid:13)c 2009.AstronomicalSocietyofJapan.
The Dependence of Spectral State Transition and Disk Truncation on Viscosity
Parameter α
ErlinQiao1,2andB.F.Liu1
1NationalAstronomicalObservatories/YunnanObservatory,ChineseAcademyofSciences,P.O.Box110,Kunming650011,P.R.China
2GraduateSchoolofChineseAcademyofSciences,Beijing100049,P.R.China
[email protected]
bfl[email protected]
9
0 (Receivedhreceptiondatei;acceptedhacceptiondatei)
0
Abstract
2

n AwealthofGalacticaccretingX-raybinarieshavebeenobservedbothinlow/hardstateandhigh/softstate. The
a transition between these two states was often detected. Observationshows that the transition luminositybetween
J these two states is differentfor differentsources, ranging from 1% to 4% of the Eddingtonluminosity. Even for

5 the same source the transition luminosity at different outbursts is also different. The transition can occur from
0.0069 to 0.15 Eddington luminosity. To investigate the underlying physics, we study the influence of viscosity
]
E parameter α on the transition luminosity on the basis of the disk-coronamodel for black holes. We calculate the
massevaporationrateforawiderangeofviscosityparameter,0.1≤α≤0.9. Byfittingthenumericalresults,we
H
obtainfittingformulaeforboththetransitionaccretionrateandthecorrespondingradiusasafunctionofα. Wefind
.
h thatthetransitionluminosityisverysensitivetothevalueofα,L/L ∝α2.34. For0.1≤α≤0.6,thetransition
Edd
p luminosityvariesbytwoordersofmagnitude,from0.001to0.2Eddingtonluminosity.Comparingwithobservations
-
wefindthatthetransitionluminositycanbefittedbyadjustingthevalueofα,andthemodeldeterminedvaluesof
o
r αaremostlyintherangeofobservationallyinferredvalue. Meanwhileweinvestigatethetruncationofthediskin
st thelow/hardstateforsomeluminoussources.Ourresultsareroughlyinagreementwiththeobservations.
a Keywords:accretion,accretiondisks—X-rays:individual(GX339-4,GS2000+251)—X-rays:stars
[

1
1. Introduction (Narayan &Yi 1995b; Abramowicz et al. 1995) and thus is
v
derivedfromtheassumptionthatalloftheaccretionenergyis
5
7 It is well known that there are two basic spectral states, a transferred to electrons and eventually radiates away (q+ ∼
vis
4 high/softstateandalow/hardstateinaccretingX-raybinaries. q ). The disk-corona evaporation model (Meyer & Meyer-
ie
0 At the high/soft state, the accretion is dominantly via a stan- Hofmeister 1994; Meyer, Liu, Meyer-Hofmeister 2000a; b;
1. dard disk (Shakura & Sunyaev 1973); while at the low/hard Liu et al. 2002) can naturally explain both the transition of
0 state, the accretion is dominantlyvia an advectiondominated spectral state and truncation of the thin disk in the low/hard
9 accretionflow(ADAF)oraradiativeinefficientaccretionflow state. Theinteractionbetweenthediskandthecoronaleadsto
0 (RIAF) (Narayan & Yi 1994, 1995a,b; Abramowicz et al. massevaporatingfromthedisktothecorona.Theevaporation
:
v 1995; for reviews see Narayan 2005 and Kato et al. 2008). ratereachesamaximumvalueatafewhundredSchwarzschild
i Similaraccretionmodelsareproposedinactivegalacticnuclei radii.Whentheaccretionratefeedingtheaccretionintheouter
X
(Narayan & Yi 1994, Yuan et al. 2003). Detailed fits to the disk is higherthanthe maximalevaporationrate, the disk ex-
r observationalspectra show thatin the soft/highstate of black tendsdownto the innermoststable orbit. Whenthe accretion
a
hole the thin disk extends down to the innermost stable cir- rateislowerthanthemaximalevaporationrate,thediskisfirst
cularorbit(ISCO);while in the hard/lowstates, the accretion truncatedatthemostefficientevaporationregionandthenthe
flowconsistsoftworadialzones,i.e.,aninneradvectiondom- innerdiskcouldbecompletelydepletediftheaccretionrateis
inatedaccretionflow(ADAF)thatextendsfromISCO(oreven too low. The final steady state at high accretion rate is then
insidetheISCO,seeWatarai&Mineshige2003)tosometrun- thesoftstatedominatedbydiskaccretionandatlowaccretion
cationradiusand anouterthin accretiondisk, abovewhichis ratethehardstatedominatedbytheADAF/RIAF.Themaximal
thecoronawhichprovideshotmassaccretionflowfortheinner evaporationratecorrespondstothetransitionaccretionrate.
ADAF(Esinetal. 1997). Investigationofthesoft-to-hardstatetransitionforX-raybi-
The mechanisms to facilitate the state transition have been naries with goodmeasurementsof flux, distance and mass of
investigatedby Honma (1996), Manmoto & Kato (2000) and the compact object shows that the transition luminosity is at
Lu et al. (2004), which are based on a radial conductive en- about 1-4% of the Eddington luminosity, with a mean of 2%
ergytransportprocess,andbyMeyeretal. (2000b),Ro`z˙an`ska (Maccarone 2003). This is in good agreementwith the theo-
& Czerny(2000),Spruit& Deufel(2002),and Dullemond& reticalpredictionof2%bythediskcoronaevaporationmodel
Spruit (2005), which are based on a vertical evaporativepro- (Meyeretal2000a).Nevertheless,thetransitionluminosityof
cess. In strong ADAF principle, the transition luminosity is somesourcesdeviatesthisvaluelargely.Forinstance,thetran-
defined as the critical accretion rate for an ADAF to exist sition luminosity observed for GX 339-4 reaches 0.15 L ,
Edd
2 ErlinQiao&B.F.Liu [Vol.,
whileitisonly0.0069L forGS2000+251. Studiesofin- is thenconducteddownintolower, cooleranddensercorona.
Edd
dividualobject,GX339-4,showthatevenforthesameobject If the density in this layer is sufficiently high, the conductive
thetransitionoccurredatdifferentluminosityduringdifferent flux is radiated away. If the density is too low to efficiently
outbursts (Zdziarski et al. 2004, Belloni et al. 2006). Even radiate the energy, cool matter is heated up and evaporation
inthesameoutburst,thetransitionluminositycandifferinthe into the corona takes place. The mass evaporation goes on
riseanddecayphases(Miyamotoetal. 1995). Ifitisthedisk untilan equilibriumdensity is established. The gasevaporat-
evaporationthatcausesthestatetransitioninabovecases,there ing into the corona still retains angular momentum and will
shouldbesomethingchangedinthecorona. differentiallyrotate aroundthe centralobject. By friction the
Oneofthepotentialfactorsleadingtodifferenttransitionlu- gas looses angular momentum and drifts inward thus contin-
minosityisthechangeofviscosityparameter,whichisfixedto uously drains mass from the corona towards the central ob-
0.3 in previousstudies (Meyer et al. 2000a;b). Although the ject. This is compensated by a steady mass evaporationflow
effective temperature profile and thus spectra do not depend fromtheunderlyingdisk. Theprocessisdrivenbythegravita-
onthe αvalueatallin thestandarddisk, theemissivity inan tionalpotentialenergyreleasedbyfrictionintheformofheat
optically thin corona depends on the density and the density in the corona. Therefore, mass is accreted to the central ob-
is roughly inversely proportional to α. Therefore, the radia- jectpartiallythroughthecorona(evaporatedpart)andpartially
tive luminosity from an optically thin corona/ADAFdepends through the disk (the left part of the supplying mass). In the
on the value of viscosity parameter. On the other hand, there inner region the evaporation becomes so efficient that all the
is a long research history of the α value throughthe study of matteraccretingthroughthediskisheatedupandaccretedto-
outbursts of dwarf novae and X-ray novae (Meyer & Meyer- wardstheblackholeviathecorona. Onlywhentheaccretion
Hofmeister 1983; Smak 1984; Lin et al. 1985; Mineshige & rate is higher than the evaporation rate, can the disk survive.
Wheeler1989). Themostrecentstudyforblackholes(King, Therefore,thediskistruncatedattheradiuswheretheevapo-
Pringle&Livio2007)yields0.1<α<0.6(iftheviscosityis rationrateequalstotheaccretionrate. Inthecaseofaccretion
definedasν= 2αV H). ratehigherthanthemaximalevaporationrate,thediskextends
3 s
For the range of 0.1≤ α≤0.3, the influence of viscosity downtotheISCO.Inthiswork,weinvestigatehowthemaxi-
has been studied by Meyer-Hofmeister & Meyer (2001) for malevaporationrateandthecorrespondingradiusvarieswith
a few number of α value. In their calculations, the temper- viscosityparameterα. We usethemodelofLiuetal. (2002)
ature of electrons is assumed equal to that of ions, that is a andincludethemodificationontheenergyequation. Forclar-
one-temperaturemodel. In reality,the accretionflowsaround ity, we list the equations describing the physics of corona as
a black hole can reach a temperatureof ∼109K or higher, at follows:
thissituationelectronsandionsdecouple. Therefore,wetake Equationofstate
the two-temperaturedisk coronamodel (Liu et al. 2002) and
ℜρ
carry out more detailed calculations for a series of viscosity P = (T +T ), (1)
i e
2µ
parameters in the range of 0.1≤α≤0.9 , which covers the
recent observationalestimates of the viscosity range (King et where µ=0.62 is the molecular weight assuming a standard
al. 2007). chemicalcomposition(X=0.75,Y =0.25)forthecorona.For
In this work, fitting formulae of the numerical results for convenience, we assume the numberdensity of ion n equals
i
both the maximal mass accretion rate and the corresponding to that of electron ne, which is strictly true only for a pure
radius dependingon the viscosity parameterα are given. On hydrogenplasma.
the basis of this, we compare our theoretical results with ob- Equationofcontinuity
servationsforindividualX-raybinaryandAGNs. InSect.2we
d 2 2z
briefly describe the physics of the corona abovethe thin disk dz(ρvz)=ηMRρvR−R2+z2ρvz. (2)
andlistequations.InSect.3weshowournumericalresultsand
inSect.4wecomparethemodelpredictionswithobservations. Equationofthez-componentofmomentum
OurdiscussionsandconclusionaregiveninSect.5andSect.6. dv dP GMz
ρv z =− −ρ . (3)
z dz dz (R2+z2)3/2
2. Themodel
Theenergyequationofions
The disk corona model for accreting black holes is estab-
lishedindetailbyMeyeretal. (2000a)andlaterthismodelis ddz(cid:26)ρivz(cid:20)v22 +γ−γ1Pρii −(R2G+Mz2)21(cid:21)(cid:27)
extendedclosetothecentralblackholebytakingintoaccount = 3αPΩ−q
the decoupling of ions and electrons (Liu et al. 2002). It is 2 ie
furtherdevelopedbyincludingtheinflowandoutflowofmass, +ηER2ρivR(cid:20)v22 +γ−γ1Pρii −(R2G+Mz2)12(cid:21) (4)
energyandangularmomentumfromandtowardsneighboring
zones(Meyer-Hofmeister& Meyer 2003). The basic physics − 2z ρ v v2 + γ Pi − GM ,
ofthemodelisbrieflydescribedbelow. R2+z2(cid:26) i z(cid:20) 2 γ−1ρi (R2+z2)12(cid:21)(cid:27)
We consider a hot corona above a geometrically thin stan- whereη isthemassadvectionmodificationtermandη is
M E
dard disk around a central black hole. In the corona, viscous the energy modification term. We take η = 1 for the case
M
dissipation leads to ion heating, which is partially transferred without consideration of the effect of mass inflow and out-
totheelectronsbymeansofCoulombcollisions. Thisenergy flow from and towads neighboring zones in the corona, and
No.] TheDependenceofSpectralStateTransitionandDiskTruncationonViscosityParameterα 3
η =η +0.5isamodificationtopreviousenergyequations 3. Numericalresults
E M
(for details see Meyer-Hofmeister& Meyer 2003). q is the
ie
exchangerateofenergybetweenelectronsandions, InourcalculationwefixthemassofblackholeasM=6M
⊙
andvarytheviscousparameterα intherangeof0.1-0.9. For
1 1
q = 2 23melnΛσ cn n (κT −κT )1+T∗2 (5) everygivenα,theevaporationrateiscalculatedasm˙ 0≡ρvz at
ie (cid:18)π(cid:19) 2mp T e i i e T∗32 thelowerboundaryandthenisintegratedintheradialone-zone
region,M˙ ≈2πR2m˙ . Thisintegratedvaluerepresentsthe
with evap 0
evaporationratein agivenregionarounddistanceRandthus
T = κTe 1+me Ti , (6) canbecomparedwiththeaccretionrate.
∗ mec2(cid:18) mi Te(cid:19) Fig.1showshowtheevaporationratevarieswiththedistance
fromtheblackhole. Heretheevaporationrateisscaledbythe
where m and m is the proton and electron mass, κ is
p e Eddingtonrate,M˙ =L /ηc2=1.39×1018M/M gs−1
Boltmannconstant,σistheThomasscatteringcrossandlnΛ= Edd Edd ⊙
(whereη=0.1istheenergyconversionefficiency)andthera-
20istheCoulomblogarithm.
diusisscaled bythe Schwarzschildradius, R =2GM/c2=
Theenergyequationforboththeionsandelectrons S
2.95×105M/M cm. It can be seen that for every given α
⊙
d ρv v2 + γ P − GM +F the evaporationrate reaches a maximum at a certain distance
dz z 2 γ−1 ρ (R2+z2)1/2 c
n h i o and decreases very quickly outside the maximal evaporation
= 3αPΩ−n n L(T)
+η2 2ρv v2e+i γ P − GM (7) region. Theoccurrenceofmaximalevaporationrateisaresult
ER R 2 γ−1 ρ (R2+z2]1/2 of energybalance in the disk corona system. In the outer re-
h i
− 2z ρv v2 + γ P − GM +F , gion, the evaporationrate increases toward centralblack hole
R2+z2n zh 2 γ−1 ρ (R2+z2)1/2i co since the released accretionenergyincreases(∝R−1), which
wheren n L(T)isthebremsstrahlungcoolingrateandF isthesourceofenergyforevaporation.However,withincrease
e i c
thethermalconduction(Spitzer1962), of numberdensity in the corona, the radiation becomesmore
andmoreefficient(sincethebremsstrahlungisproportionalto
dT
F =−κ T5/2 e (8) the square of number density) and only a very little (or even
c 0 e dz
no)gasisheatedupandevaporates.Thus,theevaporationrate
with κ =10−6ergs−1cm−1K−7/2 for fully ionized plasma. reachesamaximumvalueanddecreasesagainintheinnerre-
0
Allotherparametersinaboveequationsareunderstandarddef- gion.
inition, and are in cgs units. The five differential equations, Fig.1alsoshowsthatthevalueofevaporationratesincreases
Eqs.(2),(3),(4),(7)and(8),whichcontainfivevariablesP(z), withthevalueofviscosityparameter,whichissignificantinthe
T ,T ,F ,andm˙(z)(≡ρv ),canbesolvedwithfiveboundary innerregionbutnotobviousintheouterregion. Theincrease
i e c z
conditions. of evaporationrate with viscosity is understoodas a result of
Atthelowerboundaryz (theinterfaceofdiskandcorona), efficient heating and unimportant advection. In the inner re-
0
thetemperatureofthegasshouldbetheeffectivetemperature gion, the advectivecooling(related to α) is not the dominant
of the accretion disk. Previousinvestigations(Liu, Meyer, & cooling process, as shown by our numerical data. For larger
Meyer-Hofmeister1995)showthatthecoronaltemperaturein- value of α, the viscousheating is more efficient, which leads
creases from effective temperature to 106.5K in a very thin to more energy that is conducted down to the chromosphere
layer and thus the lower boundary conditions can be reason- above the disk. There the conducted energy is partially radi-
ablyapproximated(Meyeretal. 2000a)as, ated away andpartiallyused to evaporatethe coolgas. Thus,
the evaporation rate enhances for larger α. This effect is not
T =T =106.5K, andF =−2.73×106P atz=z . (9)
i e c 0 significantinregionsfarawayfromtheblackholesincemost
Atinfinity,thereisnopressureandnoheatflux. Thisrequires of the viscous heating is transferred to the internal energy of
soundtransitionatsomeheightz=z . Wethenconstrainthe coronagas,onlyaverysmallpartisconducteddownandrele-
1
upperboundaryas, vanttoevaporation.
The maximalevaporationrateis verysensitiveto the value
ℜ
Fc=0andvz2=Vs2≡P/ρ= 2µ(Ti+Te)atz=z1.(10) of α. The values of the maximal evaporation rate for α=0.1
and α=0.9 are 0.001 and 0.2 Eddingtonrate, respectively, in-
With such boundary conditions, we assume a set of lower creasingbyafactorofabout200fromα=0.1toα=0.9. The
boundaryvaluesforP andm˙ to start the integrationalongz. locationofthemaximalevaporationratesforα=0.1andα=0.9
OnlywhenthetrialvaluesforP andm˙ fulfilltheupperbound- are ∼1880R and 25R , decreasing by a factor of about 75
S S
ary conditions, can the presumed P and m˙ be taken as true fromα=0.1toα=0.9. Thedetailedresultsassociatedwith
solutionsofthedifferentialequations. themaximalevaporationratearelistedTable1.
Herewewishtopointoutthattheviscosityparameterαwe Plotting the data in Fig.2, we find that the dependence of
are studying in this work only appears in the viscous heating both the maximal evaporation rate and the corresponding ra-
rate, 23αPΩ,andradialdriftspeed,vR∝α,whichiseventually dius on the viscous parametercan be expressed in power-law
associatedwiththeheatingandcoolingrates. forms.Thebestfitstothedataaregivenas
(M˙ /M˙ ) ≈0.38α2.34, (11)
evap Edd max
and
4 ErlinQiao&B.F.Liu [Vol.,
downtotheinnermoststableorbit. Therefore,theaccretionis
dominatedbythethindisk,andthecoronaisveryweak. The
spectrum is dominated by multi-color blackbody. Therefore,
the value of maximal evaporation rate represents the critical
mass accretion rate for the transition from hard/low state to
high/softstate. The correspondingradiusindicateshowclose
totheblackholethethindiskisbeforethestatetransition.
Previous study (Meyer & Meyer-Hofmeister 2000a, Liu et
al. 2002) reveals that the properties of evaporation are inde-
pendent on the black hole mass. To clarify this for the case
of differentviscous parameters, we also perform calculations
withablackholemassof108M . Wefindthatforanygiven
⊙
αintherangeofα=0.1to0.9,theresultsaremassindepen-
dent as the evaporation rate is scaled in Eddington accretion
rateandthedistanceinSchwarzschildradius. Asanexample,
welistthevalueofα=0.3,0.5,onecanseefromTable1.This
indicates that our modeland predictionsare notonly valid in
stellarmassblackhole,butalsoareapplicabletosupermassive
Fig.1.Thedistributionofevaporationratealongdistancesforaseries blackholeslikeAGNs. Themostdirectimplicationisthatthe
ofviscosityparameters. Frombottomtotopαis0.1,0.15,0.2,0.3, disk truncation and state transition should occur at the same
0.5,0.9respectively. Itshowsthattheevaporationrateincreaseswith accretionrateforboththeblackholeX-raybinariesandAGNs
increasingα. Forlargerαthemaximalevaporationrateishigherand
ifthevaluesofαarethesame.
thecorrespondingradiusissmaller.
4. Comparisonwithobservation
Table1.Theevaporationfeaturefordifferentvalueofα
α R /R M˙ /M˙ Our calculations show that for standard viscosity, α=0.3,
max s max Edd the maximal evaporation rate is ∼2% of the Eddington rate,
M =6M 0.10 1883.65 1.19×10−3
⊙ which implies the transition from hard to soft state occurs at
0.15 891.25 4.58×10−3
an accretionrate ∼2%of the Eddingtonrate. Thisis in good
0.20 446.68 1.01×10−2
agreementwithMaccarone(2003)’sobservationalstudyonthe
0.25 281.84 1.78×10−2
X-raybinaries,whereitisfoundtheaveragetransitionrateis
0.30 223.87 2.73×10−2
also2%oftheEddingtonrate.
0.40 112.20 5.16×10−2
Thechangeofviscosityprovidesapossibleexplanationfor
0.50 70.79 8.11×10−2
the deviationof the transition accretionrate from the average
0.70 37.58 1.49×10−1
value. For X-ray binaries, King et al. (2007) found a range
0.90 25.12 2.24×10−1
from 0.2 to 0.4, which is 0.3 to 0.6 in our definition of α. In
M =108M 0.3 223.87 2.73×10−2
⊙ thisrangethe disk evaporationmodelpredictsa rangeforthe
0.5 70.79 8.11×10−2
transition accretion rate of 2% to 11%. The outer disk can
Note:M˙maxandRmax representthemaximalevaporationrate reach down to 52RS before transitions to the soft state. For
andthecorrespondingradiusfordifferentα. a lower viscousparameterthe transition accretionrate can be
even smaller. For instance, in the case of α = 0.2, the pre-
(R/R ) ≈18.80α−2.00. (12) dictedaccretionrateis1%.Thestrongdependenceofmaximal
S M˙=M˙max
evaporationrateontheviscousparameterindicatesthatasmall
Asithasbeenpointedoutbefore(Meyeretal. 2000a;b),the
fluctuationofviscositycouldleadtotheobservationalinferred
existenceofamaximalevaporationratealongradialdirection
variationinthetransitionrateandmayexplainwhythespectral
hasveryimportantconsequenceonthespectralstatetransition.
statetransitsfromhardtosoftstateatdifferentaccretionrates
When the accretion rate supplying to the outer disk is lower
for different objects, or for the same object at different out-
thanthisvalue,thediskiscompletelydepletedfirstatthemost
bursts. Thedependenceofthetruncationradiusatstatetransi-
efficientevaporationregion,i.e. theregionwherethemaximal
tionontheviscousparametersalsoprovidesapossiblemech-
evaporationrate reaches. The disk is thus truncated and then
anismforthechangeofobservationalinferreddisktruncation
recedesoutwardsuntiltheevaporationistooweaktocompete
extending down to ∼ 100R at hard state before transition .
withthemassflowingrateinthedisk. Atthefinalsteadystate S
The numerical fitting formulae given in this work are useful
thediskistruncatedatadistancewherethesupplyingaccretion
todeterminethevalueoftheviscosityparameterfromtheob-
rateisequaltotheevaporationrate. Intheinnerregionapure
servedtransitionluminosityandthendeterminethetruncating
corona/ADAFfillsin. Therefore,theaccretionisviaaninner
radius for individual objects. On the other hand, if viscosity
ADAF and an outer thin disk, and the radiation from the ac-
can be determined by some other way, Eqs.(11) and (12) can
cretionflowsformsahard-statespectrum. Whentheaccretion
predicttheaccretionrateatspectraltransitionandspectralen-
rate is higher than the maximal evaporation rate, evaporation
ergydistributioncontributedbythediskandthecorona.
cannotdeplete the disk at anydistances, and the disk extends
No.] TheDependenceofSpectralStateTransitionandDiskTruncationonViscosityParameterα 5
Fig.2.Left:Themaximalevaporationratevs. viscosityparameter. Right:Theradiuscorrespondingmaximalevaporationratevs. viscosityparameter.
Thefigureshowsthatthemaximalevaporationrateincreasesandthecorrespondingradiusdecreaseswithincreaseofviscousparameterα
4.1. Fitstothetransitionluminosity beingabletocheckwhetherthetransitionluminosityiscaused
bythechangeoftheviscosity.
Fromobservationsoneobtainstheluminosityatthespectral
transition. Adopting η =0.1 for the energy conversioncoef- 4.2. Fitstothetruncationradiusinthelow/hardstate
ficient,wehaveL /L =M˙ /M˙ . Thus,accordingto
tran Edd Edd Inthelow/hardstateofblackholeaccretingsystems,theac-
ourevaporationmodelthetransitionluminosityisafunctionof
cretionrateissmallerthanthemaximalevaporationrate,thus
viscousparameterα,thatis,
the disk is truncatedat a certain distance where the accretion
L /L ≈0.38α2.34. (13) rate is equal to the evaporation rate. For a standard viscos-
tran Edd
ityparameter,α=0.3,theevaporationmodelpredictsthatthe
Tofitthetransitionluminosity,wecollectdatafromliteratures
truncationradiusislargerthan224R ,aslistedinTable1.This
forthesourcesthatthetransitionluminosityiswellmeasured. S
value is larger than the values inferred from observations for
The detailed results are listed in Table 2. If the variation of
someobjects. Previousinvestigationontheeffectofmagnetic
transitionluminosityiscausedby the changeof viscosity, we
fieldsandconduction(Qianetal. 2007;Meyer-Hofmeister&
can calculate the value of α from equation (13). The derived
Meyer2006)showthatthetruncationcanshiftto∼100R if
valuesare also listed in Table 2. One can see that thoughthe S
the magnetic field and reduced conduction are taken into ac-
transitionluminosityvarieslargelyfromabout0.0069L in
Edd count.However,forsomeobjects,thistruncationradiusisstill
GS 2000+251to 0.15L in GX 339-4,the valuesof α are
Edd toolargetoaccountfortheobservation.Thiscontradictioncan
in a relatively small range from 0.18 to 0.67. Here we must
bealleviatedbytakingintoaccountthefluctuationofthevis-
note the different definition of α in different literatures. We
cousparameterα. Sincethetruncationradiusinsystemswith
takeν= 2αV2Ω(Shakura&Sunyaev1973)forthekinematic
3 s accretionratelessthan∼2-3percentofEddingtonratecanbe
viscosity, while King et al. (2007) adopt ν = αV2Ω (Frank
s wellexplainedbychangingthestrengthofmagneticfieldand
et al. 2002). Thus, the value of α in our work differs by a
thermalconductionforstandardviscousparameter(Qianetal.
factor of 3/2 from that of King et al. (2007). Translating the
2007), here we only collect observationaldata with the mass
above values of viscous parameters to the definition of King
accretionratelargerthanthisvalue.
etal. (2007),thefittedvalueofαisintherangeofα∼0.12
NGC4593isknownasaSeyfertgalaxy. Byfittingthecon-
- 0.45. These values are in agreement with the observational
tinuumspectrumof optical-UVand Fe K line profile, Lu&
values of 0.1- 0.4 inferred by King et al. (2007). Here we α
Wang (2000) find the observational evidence supporting that
wtraisnhsittioonpoluinmtionuotsitthyatLther/eLare undcueerttaointthieesuinncethrteaiinntfieersreind thin disk truncates at >∼ 30Rs. The corresponding mass ac-
tran Edd cretion rate in the outer thin disk is about 0.055M˙ . This
measuringthemassofblackholeanditsdistance. Thisleads Edd
requires that a value of α ≥ 0.44 from equation (11), with
toasmalldeviationinthevalueofviscosityparameterinfitting
whichthemaximaltruncationradiusisabout98R according
thetransitionluminosity. Nevertheless,weexpectthevalueof S
to equation (12). If a large α is associated with strong mag-
αisstillintherangeofKingetal. (2007).
neticfields,thestrongmagneticfieldsinthisobjectwillresult
With the value of viscous parameter α, we can calculate
inatruncationradiussmallerthan98R . Thediscrepancyof
thetruncationradiusofthediskatthespectraltransitionfrom S
a factor of 3 between the observation and theoretical predic-
equation(12). Ifobservationscandeterminethetransitionra-
tioncouldbedecreasedbytakingboththemagneticfieldand
dius,say,bymeansofemissionlines,onecancomparetheob-
thermalconductionintoaccount(Qianetal. 2007).Therefore,
servationallyinferredvaluewiththemodelprediction,thereby
6 ErlinQiao&B.F.Liu [Vol.,
weexpectthattheverysmalltruncationofthethindiskinthe 5.2. Thecombinedeffectsofmagneticfields,conductionand
low/hard state of NGC 4593 is a combined effect of viscos- viscosity
ity,magneticfieldandconduction. Notethattheviscosityand
As it has beenshown in previousstudy (Qian et al. 2007),
conductionaremutuallyrelatedtothemagneticfields.
themagneticfieldsandconductionhardlyaffectthetransition
GX 339-4 is of well known black hole X-ray binaries.
luminosity, but lead to a small truncation radius at the state
Zdziarskietal.(2004)studythe15outburstsofGX339-4from
transition. Here it is shown that increase of viscosity results
1987to2004. Wheninthelow/hardstatethehighestmassac-
innotonlya decreasedtransitionradiusbutalsoanenhanced
cretionratecanreach∼0.26M˙ ,andalsoatdifferenttime
Edd transition luminosity. Combinationof these effects can make
theaccretionratecanbe∼0.14,0.17,0.07,0.02M˙ inthe
Edd our modelmore flexible in fits to the observationallyinferred
low/hardstate(Zdziarskietal. (2004).Fromequation(11)this
transition luminosityand and the truncationradius. Here one
requiresthevalueofαlargerthan0.85,0.65,0.71,0.49,0.28
shouldnotethephysicallinkbetweenmagneticfieldstrengths
respectively. Forthese valueofα, themaximaltruncationra-
and viscosity parameter. In fact, the viscosity is mostly pro-
diusis derivedfrom equation(12)as 26, 44, 37, 80, 231 R .
S videdbymagneticstress,thoughwearenotveryclearthede-
Theseresultsareinagreementwiththeobservationalvaluesof
taileddependenceofαonmagneticstress.
R ∼20-200R ,althoughthediscrepancyremains.
in S
5.3. Thevalueofviscousparameterα
5. Discussion
Collecting and analyzing observational results from dwarf
novae,softX-raytransientsandAGNs, Kingetal.(2007)find
The disk-corona evaporation model provides a physical
the range of viscous parameter is 0.1≤α≤0.4. Fits to the
mechanism for the spectral state transition between low/hard
observationaltransition luminosity by our modelgive similar
stateandhigh/softstateandfortruncationofthethindiskinthe
rangefor the viscousparameter. However, the valueof α de-
lowstate. Nevertheless,itshouldbepointedoutthatthereare
rivedbymanyMHDsimulationsislessthan0.02.Forinstance,
stilluncertaintiesinquantitativefitstoobservations.Thevalue
Stoneetal. (1996)deriveα<0.01;Brandenburgetal.(1995)
ofthemassevaporationrateinourmodelisaffectedbyseveral
get0.001< α <0.005,Hirose, Krolik& Stone(2006)obtain
factors,suchasthemagneticfield(Qianetal. 2007),thether-
α≃0.016,Hawley&Krolik(2001)getα∼0.1. Theanalysis
mal conduction (Meyer-Hofmeister & Meyer 2006), and the
ofdetailedobservationaldata(Kingetal. 2007)indicatesthat
inverseComptonscattering(Liu etal. 2002). Italsodepends
mostMHDsimulationsunderestimatethevalueofα.
sensitively on the viscosity parameter α, as shown from our
Cautionshouldbetakentothefactthattheαvalueobtained
calculations. Therefore,oneshouldtakeintoaccountallthese
fromthenumericalsimulationsdependshowtocalculatethem,
factorsinquantitativefitstoobservations.
(e.g., Machida, Hayashi & Matsumoto 2000, Machida &
5.1. TheeffectofComptoncooling Matsumoto 2003, Machida, Nakamura & Matsumoto 2004).
Thisisbecausethemagnetizedaccretionflowshowshighlyin-
Comptonupscatteringof photonsfromthe disk can be im-
homogeneousstructureandsignificanttimefluctuations.Thus,
portantincoolingofthecorona(Liuetal. (2002). Athighac-
thederivedalphavaluesdependhowtomaketimeandspatial
cretionrates,thethindiskextendsdowntotheISCO,Compton
averages of physical quantities. Generally speaking, the ra-
cooling becomes efficient and hence less heat is conducted
tio ofmagneticenergyto gasenergyandthusthealphavalue
downwards,whichleadstoa decreasedevaporationrate. The
increaseswith increaseof height; the alphavaluessometimes
maximal evaporation rate also decreases, indicating that the
even exceed unity in corona (see, e.g., Machida et al. 2000).
transitionluminosityfromsofttoharddecreasesasaresultof
Hence,ifwemakethespatialaveragewithweightofmagnetic
Comptoncooling. Ontheotherhand,atlowaccretionratethe
energyinsteadofgasdensityorgasenergy,thederivedalpha
diskistruncated.Comptonupscatteringofdiskradiationisnot
valuestendtobehigher.
efficient,therebyhardlyaffectstheevaporationrate.Therefore,
ourmodelinthisworkisvalidforthetransitionfromlow/hard 5.4. Thedependenceofwindlossonα
statetohigh/state.Forthetransitionfromhigh/softtolow/hard
The fraction of mass carried away by the wind sensitively
state, the transition luminosity is a few times lower than that
dependsonthevalueofα. Forα=0.3,thefractionincreases
fromlow/hardtohigh/softtransition(Meyer-Hofmeisteretal.
from∼7%atthemaximalevaporationregion(R=102.35R )
2005; Liu et al. 2005), producingthe observedhysteresis. It S
to∼9.8%atR=104.25R . Forα=0.1,thefractionincreases
isinterestingtonotethatdifferentalphavaluesmayalsocon- S
from∼1.4%atthemaximalevaporationregion(R=103.27R )
tributetothehysteresis.Thismakesourpredictiononhystere- S
to∼2.7%atR=104.25R . Forα=0.9,thefractionincreases
sis closer to the observations for objects showing very large S
from∼16%at R=25R to 19%atR=104.25R . Onecan
hysteresis(e.g.Miyamotoetal1995). S S
see the fraction of mass carried away by the wind increases
In the case of a weak inner disk existing at low/hardstates
systemically with an increase of the value of α. This can be
(Liuetal. 2007),theComptoneffectcanalsoleadtoa lower
understoodasthehighertemperatureandpressureinthelarger
evaporationrate,butitcannotchangethemaximalevaporation
αcase.
ratesignificantly.
5.5. Amechanismforthe“strongADAF”principle
Ourmodelindicatesthattheaccretionflowchangesfroma
corona/ADAF to a thin disk when the accretion rate exceeds
No.] TheDependenceofSpectralStateTransitionandDiskTruncationonViscosityParameterα 7
Table2.TransitionluminosityandfittingvalueoftheviscosityparameterinaccretingX-raybina-
ries
source L /L α references
tran Edd
XTEJ1859+226 0.12 0.61 (Gierlin´ski&Newton2006)
XTEJ1739-278 0.13 0.63 (Gierlin´ski&Newton2006)
XTEJ1650-500 0.02 0.28 (Gierlin´ski&Newton2006)
4U1543-47 0.07 0.49 (Gierlin´ski&Newton2006)
NovaMus91 0.031∗ 0.34 (Maccarone2003)
GS2000+251 0.0069∗ 0.18 (Maccarone2003)
CygX-1 0.028∗ 0.33 (Maccarone2003)
GROJ1655-40 0.0095∗ 0.21 (Maccarone2003)
LMCX-3 0.014∗ 0.24 (Maccarone2003)
AqlX-1 0.019∗ 0.28 (Maccarone2003)
4U1608-52 0.042∗ 0.39 (Maccarone2003)
4U1728-34 0.050∗ 0.42 (Maccarone2003)
XTEJ1550-564 0.13 0.63 (Gierlin´ski&Newton2006)
XTEJ1550-564 0.03 0.34 (Gierlin´ski&Newton2006)
XTEJ1550-564 0.034∗ 0.36 (Maccarone2003)
GX339-4 0.15 0.67 (Gierlin´ski&Newton2006)
GX339-4 0.06 0.45 (Gierlin´ski&Newton2006)
Note: (1) Aql X-1, 4U 1608-52, 4U 1728-34 are neutron stars, the others are
block holes or black hole candidates. (2) The luminosity band in Gierlin´ski &
Newton (2006) is 1.5-12 kev. (3) Asterisk denotes the transition luminosity from
high/soft state to low/hard state, for which the fitting value of α should be slightly
higher if the Compton cooling by disk photons is included in our model. (4)
DifferenttransitionluminositiesforXTEJ1550-564andGX339-4arefromdifferent
outbursts.
themaximalevaporationrate. Thus,themaximalevaporation magnitude.Annumericalfittingformulaeofthemaximalmass
raterepresentsthecriticalaccretionratefortheaccretiontak- evaporation rate as a function of viscosity is given by fitting
ingtheformofacorona/ADAF.Ourdetailedcalculationsshow thenumericaldata. Thecorrespondingradiusisalso givenas
that the maximalevaporationrate dependson the viscouspa- afunctionofα. Sincethemaximalevaporationraterepresents
rameterapproximatelyintheformofm˙ ∝α2.34. Thisresultis the critical accretion rate at the transition from low/hard to
similartothecriticalaccretionrateforanADAFtoexist,that high/softstate, wecompareourmodelwithobservations. We
is,m˙ ∝α2 (Narayan&Yi1995b;Abramowiczetal. 1995; find that the transition luminosity can be fitted by adjusting
crit
Mahadevan1997). Thesimilardependentlawsontheviscos- the value of α, and the model determined values of α are
ityareeasytounderstandsinceboththemaximalevaporation mostlyin the rangeofobservationallyinferredvalue(Kinget
rateandcriticalaccretionrateforanADAFtoexistarederived al. 2007). Meanwhileweinvestigatethetruncationofthedisk
from the assumption that all of the viscous heating is trans- in the low/hard state for some luminous sources. Our results
ferredtoelectronsandeventuallyradiatedaway(q+ ∼q ,for areroughlyinagreementwiththeobservations.
vis ie
details see Mahadevan 1997). Now that the evaporation as a
consequenceofdiskcoronainteractioncancompletelydeplete We thank F. Meyer and E. Meyer-Hofmeister for their
the inner disk and turn it into a pure corona/ADAFwhen the comments. This work is supported by the the National
accretion rate is less than the maximal evaporation rate, the Natural Science Foundation of China (grants 10533050 and
disk evaporation model provides a naturally physical mecha- 10773028).
nism for the “strong ADAF” principle, that is, whenever the
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