A disc corona-jet model for the radio/X-ray correlation in black hole X-ray binaries

0.3 MB·English

by  Erlin Qiao

Mon.Not.R.Astron.Soc.000,1–9() Printed16January2015 (MNLATEXstylefilev2.2)
A disc corona-jet model for the radio/X-ray correlation in black hole
X-ray binaries
5 Erlin Qiao 1⋆ and B.F. Liu 1
1 1NationalAstronomicalObservatories,ChineseAcademyofSciences,Beijing100012,China
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2

n
a 16January2015
J

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1 ABSTRACT
The observed tight radio/X-ray correlation in the low spectral state of some black hole X-

] ray binaries implies the strong coupling of the accretion and jet. The correlation of L ∝
E R
L∼0.5−0.7 was well explained by the coupling of a radiatively inefficient accretion flow and
H X
a jet. Recently, however,a growingnumberof sources show more complicatedradio/X-ray
. correlations, e.g., L ∝ L∼1.4 for L /L & 10−3, which is suggested to be explained by
h R X X Edd
p thecouplingofa radiativelyefficientaccretionflowanda jet. In thiswork,we interpretthe
- deviation from the initial radio/X-ray correlation for LX/LEdd & 10−3 with a detailed disc
o corona-jetmodel.Inthismodel,thediscandcoronaareradiativelyanddynamicallycoupled.
tr Assuming a fraction of the matter in the accretionflow, η ≡ M˙jet/M˙, is ejected to form the
s jet, we can calculate the emergentspectrum of the disc corona-jetsystem. We calculate L
a R
andL atdifferentM˙,adjustingηtofittheobservedradio/X-raycorrelationoftheblackhole
[ X
X-raytransientH1743-322forL /L >10−3.ItisfoundthatalwaystheX-rayemissionis
X Edd
1 dominatedbythedisccoronaandthe radioemissionisdominatedbythejet.We notedthat
v thevalueofηforthedeviatedradio/X-raycorrelationfor L /L > 10−3,issystematically
X Edd
5 lessthanthatofthecase for L /L < 10−3, whichisconsistentwith thegeneralidea that
6 X Edd
thejetisoftenrelativelysuppressedatthehighluminosityphaseinblackholeX-raybinaries.
5
3 Keywords: accretion,accretiondiscs—blackholephysics—X-rays:individual:H1743-
0
322—X-rays:binaries—radiocontinuum:stars
.
1
0
5
1
: 1 INTRODUCTION al. 2010). Meanwhile, the low/hard spectral state is often associ-
v
atedwiththeproductionofcollimated,relativisticjets,whichare
i A black hole X-ray binary (BHB) isa gravitationally bound sys-
X quenchedinthehigh/softspectralstate(Fenderetal.2004).
temcomposed ofablackholeandanormalstar.BHBsarelumi-
r nous in X-rays, which is believed to be resulted by accreting the A correlation between the radio luminosity and X-ray lu-
a
matterofthenormalstarontotheblackhole.AccordingtotheX- minosity was found in the low/hard spectral state of BHBs, i.e.,
rayspectralfeaturesandthetimingproperties,twotypicalspectral LR∝LbX,withb∼0.5−0.7(Corbeletal.2003,2008,2013;Gallo
stateswereidentifiedinBHBs,i.e.,thehigh/softspectralstateand etal.2003).Theexistingoftheradio/X-raycorrelationforBHBs
thelow/hardspectralstate(forreviews,seeTanaka&Lewin1995; ismainlyfromtheobservationsoftwosources,i.e.,GX339-4and
Remillard&McClintock2006).Inthehigh/softspectralstate,the V404 Cyg (Corbel et al. 2013). Yuan & Cui (2005) interpreted
spectrumisdominatedbyapeakemissionaround1keV,whichis this radio/X-ray correlation, i.e., LR ∝ LX∼0.7 for LX ≃ 10−6LEdd
believedtobeproduced byacool discextending downtothein- toLX ≃ 10−3LEdd (withLEdd = 1.26×1038M/M⊙ ergs−1 )within
nermost stable circular orbit (ISCO) of a black hole (Shakura & theframeworkofaradiativelyinefficientaccretionflow(RIAF)-jet
Sunyaev1973;Mitsudaetal.1984;Bellonietal.2000).Thespec- model, in which a fraction of the matter, η, in the accretion flow
tralfeaturesofthelow/hardspectralstatearecomplicated, which isassumed tobeejectedtoformajet(withη ≡ M˙jet/M˙).Inthis
aregenerallythoughttobeproducedbyaninnerhotaccretionflow model,theradioemissionwasdominatedbytheself-absorbedsyn-
andanoutertruncatedcooldisc(Reesetal.1982;Esinetal.1997, chrotronemissionofasteady,collimatedcompact jet,andtheX-
2001; McClintock et al. 2001; Yuan et al. 2005; Narayan & Mc- rayemissionwasdominatedbyaRIAFviathermalComptoniza-
Clintock 2008). Recently, some observations indicate that a cool tion process. In the RIAF-jet model, if a constant η is assumed
discmayalsoexistintheregionveryclosetoISCOinthelow/hard fordifferent M˙,theradioluminosityfromthejetcanbescaledas
spectralstate(Milleretal.2006a, b;Tomsicketal.2008; Reiset LR ∝ ηM˙ξ withξ ∼ 1(e.g.,Heinz&Sunyaev2003),andbecause
of the nature of theadvection in RIAF,the X-rayluminosity can
bescaledasL ∝ M˙qwithq∼2.Then,thepredictedradio/X-ray
X
⋆ E-mail:[email protected] correlationisLR∝LξX/qwithξ/q∼0.5,whichisroughlyconsistent
(cid:13)c RAS
2 ErlinQiaoandB.F.Liu
withobservations. TheRIAF-jetmodel cannotonlyinterpretthe agivenradioluminosity,thehigherX-rayluminosityofthetrack
radio/X-raycorrelation,butalsocanexplainthebroadbandspectral withb∼1.4isprobablyresultedbythetransitionoftheaccretion
energydistribution(SED)andmostofthecomplextimingfeatures flowfromaRIAFtoaradiativelyefficientlyaccretionflow,e.g,a
ofsomeblackholeX-raytransients,e.g.,XTEJ1118+480(Yuanet disccoronasystembyHaardt&Maraschi(1991,1993),DiMatteo
al.2005;Malzacetal.2004).WhenBHBsenterthequiescentstate, et al. (1999), Liu et al. (2002a, 2003), Merloni & Fabian(2002),
i.e.,L .10−6L ,Yuan&Cui(2005)predictedasteeperradio/X- Cao(2009) and Huang et al. (2014), or aluminous hot accretion
X Edd
raycorrelationofL ∝L1.23withintheframeworkoftheRIAF-jet flowbyYuan(2001),Xie&Yuan(2012)andMa(2012).Meyer-
R X
model,inwhichtheradioemissionisdominatedbythejet,mean- Hofmeister &Meyer (2014) proposed that recondensation of gas
whiletheX-rayemissionisalsodominatedbythejetratherthanthe fromthecoronaintoaninnerdisccanprovideadditionalsoftpho-
RIAF.Later,byfittingtheSEDsoftheblackholeX-raytransient tonsforComptonization,whichleadstoahigherX-rayluminosity
GROJ1655-40andV404Cyginthequiescentstate,itisfoundthat comparedtotheunchanged radioemission.Intheradioquiethy-
boththeradioemissionandtheX-rayemissioncanindeedbeex- pothesis, a varied η is assumed for different M˙. As suggested by
plained by the jet (Pszotaet al. 2008; Xieet al. 2014). However, Coriatetal.(2011),ifthereisalinerdependence ofηon M˙,i.e.,
duetotheveryfaintemissioninthequiescentstate,thepresenceof η∝ M˙,thepredictedradio/X-raycorrelationshouldbeL ∝L2ξ/q.
R X
theradio/X-raycorrelationofL ∝ L1.23forL .10−6L isstill BecauseforRIAFqis∼ 2,thepredictedslopeoftheradio/X-ray
R X X Edd
controversial(Galloetal.2006). correlationisalso roughly close tothetrack withb ∼ 1.4. How-
We need to keep in mind the complexities of the observed ever,theoreticallythedependenceofηonM˙ isunclear,(e.g.,Pe’er
radio/X-ray correlation. For example, by critically examining the &Casella2009),furtherstudiesarestillneededtoputconstraints
radio/X-ray correlation in a sample of microquasars, Xue & Cui ontherelationbetweenηandM˙.
(2007) found thatthecorrelationvariedsignificantlyamongindi- Observationally, there is evidence of a coupling of the hot
vidualsources,notonlyintermsoftheshapebutalsoofthedegree plasmaandthejetinbothBHBsandactivegalacticnuclei(AGNs),
ofthecorrelation.Recently,astatisticalanalysisofthedataledto i.e.,thecouplingofthedisccoronaandjetathighmassaccretion
the new claim of dual tracks, with some referring as the ‘univer- ratesandthecoupling of the RIAFand jet atlow massaccretion
sal’trackandtheotherasbeing‘outlier’trackforL /L &10−3 rates(Wuetal.2013;Zdziarskietal.2011;Merlonietal.2003;Fal-
X Edd
(Galloetal.2012;Jonkeretal.2012).Weshouldnotethattheob- ckeetal.2004).Meanwhile,theoretically,forM˙ &α2M˙Edd(withα
served radio/X-ray correlation isquitecomplex, and theclaimof theviscosityparameter,M˙Edd=1.39×1018M/M⊙gs−1),theaccre-
the simple two tracks is also fairly dubious (Coriat et al. 2011). tionflowwilltransitfromaRIAFtoadisccoronasystem,which
Anyhow, so far, a growing number of sources have been discov- hasbeencomprehensivelystudiedbymanyauthors,e.g.,Meyeret
ered with a steeper radio/X-ray correlation, namely, L ∝ L∼1.4, al.(2000a,b),Liuetal.(2002b)andQiao&Liu(2009,2010,2013)
R X
during L & 10−3L [e.g., IGR J17497-2821 (Rodriguez et al. withintheframeworkofthediscevaporationmodel,orbyNarayan
X Edd
2007), XTE J1650-500 (Corbel et al. 2004), Swift J1753.5-0127 & Yi (1995b), Abramowicz et al. (1995) and Mahadevan (1997)
(Soleri et al. 2010)]. The study of the radio/X-ray correlation of withintheframeworkoftheRIAFsolution.
L ∝ L∼1.4 for L & 10−3L is the purpose of this work. By Consequently, in this paper, we propose a disc corona-jet
R X X Edd
collectingthearchivequasi-simultaneousradioandX-raydatabe- modeltoexplaintheradio/X-raycorrelationof LR ∝ LX∼1.4 during
tween2003to2010,Coriatetal.(2011)comprehensivelystudied thehighluminosityphase LX/LEdd & 10−3,inwhichafractionof
therelationshipbetweentheradioluminosityandX-rayluminosity thematter,η,inthecoronaisassumedtobeejectedtoformthejet.
oftheblackholeX-raytransientH1743-322.Itisfoundthatduring So far, the theoretical understanding of the jet formation is poor.
ahighluminosityphase,correspondingtoL /L ∼10−3to10−1, Specifically,itisdifficulttoput constraintsonthedependence of
X Edd
H1743-322followstheradio/X-raycorrelationwithasteeperslope ηon M˙ inourmodel,sowesetηasanindependentparameteron
ofb∼1.4.Whileduringalowluminosityphase,correspondingto M˙ to fitthe observations. As an example, by fittingthe observed
L /L . 10−5, H1743-322 follows the radio/X-ray correlation radio/X-raycorrelationduringthehighluminosityphaseofblack
X Edd
witha slope of b ∼ 0.6. For LX/LEdd between 10−5 to 10−3, it is holeX-raytransientH1743-322forLX/LEdd>10−3,wefoundthat
probablycorrespondingtoatransitionregionbetweenthetwocor- η is weakly dependent on M˙, and the mean fitting result of η is
relations. ∼0.57%.Thederivedrelativelysmallerfittingvalueofηsupports
The change of the radio/X-ray correlation from the low lu- thegeneralideathatthejetisoftensuppressedatthehighluminos-
minosity phase to the high luminosity phase in H1743-322 may ityphaseofBHBs.Thedisccorona-jetmodelisbrieflydescribed
implyeither thechange of theproperties of theaccretionflow or insection2.ThenumericalresultsarepresentedinSection3.Some
thechange of thedifferent coupling intheaccretionflowandjet, comparisonswithobservationsareshowninSection4.Discussions
e.g.,thechange ofthedependence of thefractionofthematterη areinSection5,andtheconclusionsareinsection6.
ejectedtoformthejetonthemassaccretionrate M˙.Thesources
withtheradio/X-raycorrelationofL ∝L∼1.4couldbeconsidered
R X
as aX-ray loud hypothesis, i.e.,for a given radio luminosity, the
simultaneousX-rayluminosityofthetrackwithb ∼ 1.4ishigher 2 THEMODEL
thanthatofthetrackwithb ∼ 0.5−0.7,orinturncouldbecon-
2.1 Accretionflows
sideredasaradioquiethypothesis,i.e.,foragivenX-rayluminos-
ity,thesimultaneousradioluminosityofthetrackwithb ∼ 1.4is Theaccretion flowsadopted here isageometrically thindisc en-
lowerthanthatofthetrackwithb∼0.5−0.7.Inbothhypotheses, closedbyageometricallythick,hotcoronaaroundacentralblack
theradioemissionisdominatedbytheself-absorbed synchrotron hole(Liuetal.2002a,2003).ThediscisastandardShakura&Sun-
emissionofthesteady,collimatedcompactjet,andtheX-rayemis- yaev(1973) disc,whichistightlycoupledwiththeplane-parallel
sionisdominatedbytheaccretionflow.Themaindifferenceofthe corona. Magnetic fields are assumed to be generated by dynamo
two hypotheses isthe different dependence of η on M˙. In the X- action.AsaresultofParkerinstability,magneticfluxloopscontin-
rayloudhypothesis,ifaconstantηisassumedfordifferent M˙,for uouslyemergefromthedisctothecoronaandreconnectwithother
(cid:13)c RAS,MNRAS000,1–9
3
loops.Inthisway,theaccretionenergytakenfromthethindiscis Giventhevalues oftheinput parametersm,m˙,α,ℓ,andthe
released in the corona as thermal energy and eventually emitted initialvaluesofλ andλ ,wecansolveequation(9)for f.ThenT
τ u
away mostly in X-ray band via inverse Compton scattering. The andnaresolvedfromequations(7)and(8).Theeffectivetempera-
densityofthecoronaisdeterminedbyanenergybalancebetween tureofthesoftphotosT
R
thedownward heatconduction andmassevaporation inthechro-
σT4 = 3GMM˙φ(1−f) + cUre
mosphericlayer.Adetaileddescriptionofthemodelcanbefound R 8πR3 4 rad (10)
≈ cUre
in Liu et al. (2002a, 2003). The equations describing these pro- 4 rad
cessesinthecoronaarelistedasfollows. can also be calculated by combing equations (3) and (5) (with
B2 4kT albedoalwaysbeingassumedtobezeroinourcalculations).With
V ≈ τ∗cU , (1)
4π A mec2 rad T,nandTR,thespectraofthedisc-coronasystemcanbecalculated
byMonteCarlosimulation.
k T7/2 γ kT 1/2 Inourdetailedcalculations,wealsotakeintoaccountthein-
0 ≈ nkT  , (2) fluencesofλ andλ andthelengthofmagneticloops.Sincethe
ℓ γ−1 mH coronatempeτratureTu,densitynandenergyfraction f donotsen-
whereT isthecoronaltemperature,nisthecoronalnumber den- sitivelydependonλ andλ (seeequations(7)–(9)),andthevalue
τ u
sity, Bisthestrengthofmagneticfield,Urad istheenergydensity ofλτandλushouldbe∼1inorderofmagnitude,thecoronaspec-
ofthesoftphotonfieldforComptonscattering,V = B2 isthe tracannotbesignificantlyaffectedbythechosenvalueofλ and
A q4πρ τ
Alfve´nspeed, ℓisthelengthofmagneticloop, τ∗ istheeffective λu.Nevertheless,werepeattheMonteCarlosimulationbyadjust-
optical depth, defined as τ∗ ≡ λτnσTℓ with λτ ∼ 1. Other con- ing the value of λτ and λu. We find a set of reasonable value for
stants have their standard meanings. From equations (1) and (2), λτ and λu until the downward-scatted luminosity is equal to the
thetemperatureT anddensity ninthecorona canbedetermined seedluminosityandtheupward-scattedluminosityisequaltothe
foragivenmagneticfield BandradiationfieldU ,andthenthe releasedgravitationalenergy.Inthiswaywefindaself-consistent
rad
radiativespectrumcanbecalculatedout. solutionofthedisc-coronasystemandgetthecorrespondingemer-
For a coupled disc and corona system, the magnetic field is gentspectrum.Wetesttheeffectofthelengthofmagneticloopℓ
derived from an equipartition of gas pressure and magnetic pres- ontheemergent spectrum,itisfoundthattheemergentspectrum
sure in the disc, i.e., β ≡ ndisckTdisc = 1. The soft photons (U ) isveryweaklydependentonℓ.Soinourmodel,ℓisnotafreepa-
are composed of intrinsicdisBc2/r8aπdiation Uin and the reprocessraedd rameter,andwesetℓ = 10RS throughoutthecalculations.Forthe
radiationofbackwardComptonemissionorfadthecoronaUre .With detailed description of the Monte Carlo simulation, one can also
rad refertoLiuetal.(2003).
energytransportingtothecoronabythemagneticfield,thediscis
expectedtobegaspressuredominant.Thus,themagneticfieldand
softphotonfieldcanbeexpressedasfunctionsofblackholemass,
2.2 Coupleddisccorona-jetmodel
accretionrate,anddistance,
Thecalculationofjetemissionisbasedontheinternalshocksce-
B=2.86×108α−9/20β−1/2m−9/20[m˙ φ(1− f)]2/r−51/40G, (3)
0.1 1 0.1 10 nario as described in Yuan et al. (2005). In the disc corona-jet
Urinad =aTe4ff = 4c3GMM8˙π(R13− f)φ (4) mM˙ojedte=l,ηaM˙s,miaslalsfsruamcteiodntoofbetheejemctaetdtetroifnortmhethaeccjreett.ioInntflhiosww,oi.rek.,,
=1.14×1014m−1m˙0.1φ(1− f)r1−03ergscm−3, aconicalgeometryisconsidered forthejet,andthehalfopening
Ure =0.4λ U , (5) angleof thejet isfixedat ϕ = 0.1. Changing thevalue of ϕwill
rad u B leadtoachangeofthedensityofthejet,howeverthiseffectcanbe
where m, m˙0.1, α0.1, β1, r10 are the mass of black hole, the ac- absorbedbythemassaccretionrate.ThebulkLorentzfactorofthe
cretion rate, the viscosity parameter, the equipartition factor, and jetisfixedatΓ =1.2,whichiswellconsistentwiththetypically
thedistancerespectivelyscaledby M⊙,0.1M˙Edd,0.1,1and10RS. observedvaluejoetfthejetinthelow/hardspectralstateofBHBs(e.g.
φ ≡ 1−(R∗/R)1/2 andR∗ = 3RS isadoptedastheISCOofanon- Fender2006).Withinthejet,internalshockisproducedduetothe
rotatingblackhole(RS=2.95×105 M/M⊙ cm).λu isafactorin- collisionoftheshellswithdifferentvelocities.Theinternalshocks
troducedfortheevaluationoftheseedfieldinHaardt&Maraschi willaccelerateasmallfractionoftheelectronstobeapower-law
(1991, 1993), which isaround 1 in order of magnitude. Here the energydistributionwithindexp.Wefixp=2.1inthecalculations
energyfractiondissipatedinthecorona, f,isnotafreeparameter assuggested by Zhang et al. (2010) for fittingthe SEDsof three
butcanbeexpressedas, blackholeX-raytransientsJ1753.5-0127,GROJ1655-40andXTE
F B2 3GMM˙φ −1 J1720-318. Thetwoparameters,ǫe andǫB,describingtheratioof
f ≡ Fctootr =4πVA 8πR3  . (6) tfiheeldentoertghyeosfhtohcekaecnceerlgeyratiendtheeleschtroocnksfarnodntthaereamfixpeldifiaetdǫmea=gn0e.t0i4c
Combingequations(1),(2),(3),(5)and(6),wegetasolution and ǫ = 0.02 respectively, which are consistent with the obser-
B
of the corona above agas pressure-dominated disc inthecase of vationsofgamma-raybursts(GRB)afterglows(e.g.,Panaitescu&
Ure >>Uin , Kumar2001,2002).BecausetheComptonscatteringopticaldepth
rad rad
ofthejetissmall,onlysynchrotronemissionisconsideredinthe
T = 3.86×109α−9/80β−1/8λ−1/4λ−1/4m1/80
0.1 1 τ u (7) calculation(Markoffetal.2001).
×[m˙ φ(1− f)]1/10r−51/160ℓ1/8K,
0.1 10 10
n= 1.61×1018α−9/40β−1/4λ−1/2λ−1/2m−39/40
0.1 1 τ u (8)
×[m˙ φ(1− f)]1/5r−51/80ℓ−3/4cm−3, 3 NUMERICALRESULT
0.1 10 10
f = 3.73×104(1− f)11/10α−99/80β−11/8λ1/4λ1/4m11/80 We calculate the emergent spectra of the disc corona-jet model
0.1 1 τ u (9)
×(m˙ φ)1/10r−81/160ℓ3/8. aroundastellar-massblackholewithmassMwhentheparameters
0.1 10 10
(cid:13)c RAS,MNRAS000,1–9
4 ErlinQiaoandB.F.Liu
including M˙,α,andηarespecified.Inthispaper,wefixtheblack differentM˙ toproducetheradio/X-raycorrelation.However,actu-
hole massat M = 10M , assuming a typical viscosity parameter ally the dependence of η on M˙ is unclear, so η will be set as an
⊙
α=0.3,andaconstantηfordifferentM˙. independentparameteron M˙ tofitthedetailedobservationsinthe
Givenη = 0.2%,theemergent spectraareplottedintheleft nextsection.
panel of Figure1fordifferent massaccretionrates,i.e.,fromthe
bottom up M˙ = 0.02,0.05,0.1,0.3 and 0.5 M˙ . The solid lines
Edd
arethetotalemergentspectra,andthedottedlinesaretheemergent
4 APPLICATIONTOBLACKHOLEX-RAYTRANSIENT
spectrafromthejets.FromtheleftpanelofFigure1,wecanclearly
H1743-322
seethatX-rayemissionisdominatedbythedisc-coronasystemand
theradio emissionisdominated by thejet for themass accretion H1743-322isaX-raytransientdiscoveredin1977(Kaluzienski&
ratesfrom M˙ = 0.02−0.5 M˙ .Basedontheemergentspectra, Holt1977;Doxseyetal.1977),thendetectedagainbytheInterna-
Edd
wecalculatetheradioluminosityL andtheX-rayluminosity tional Gamma-ray Astrophysics Laboratory (INTEGRAL)during
8.5GHz
L for different M˙. The best-fitting linear regression for the theburstin2003.H1743-322iswellstudiedinbothradioandX-
2−10keV
correlationbetweenL /L andM˙ canbeexpressedas, rayband. Bycomparingboththespectral featuresandthetiming
8.5GHz Edd
propertiesofH1743-322tothewell-studiedblackholeX-raytran-
L8.5GHz=A(η) M˙ ξ(η), (11) sientXTEJ1550-564,McClintocketal.(2009)arguedthatH1743-
 LEdd  M˙Edd 3is22infiserareadcctoretbieng∼bl1a0cMkho(lee..gT.,hSetebilnaecrkehtoalel.m20a1ss2)o.fCHo1ri7a4t3e-t32a2l.
where A(η)| = 10−8.47 and ξ(η)| =1.42. The best-fitting ⊙
η=0.2% η=0.2% (2011) analyzed all the archive data of H1743-322 observed by
linearregressionforthecorrelationbetween L /L and M˙
2−10keV Edd RXTEbetween2003January1and2010Februay13,meanwhile
canbeexpressedas,
theauthorsconductedquasi-simultaneous(△t.1d)radioobserva-
L M˙ q tionsthroughTelescopeCompact Array(ATCA),CompactArray
 2L−E10dkdeV=10−1.23M˙Edd , (12) BofroVaLdAbanindtBheaclikteenradtu(rCeA. BB),andcollectedtheobservationaldata
whereq = 1.06,whichisroughlyconsistentwitharadiativelyef- Based on the data of H1743-322 from Coriat et al. (2011),
ficientaccretionflowwithLX ∝ M˙ (e.g.Haardt&Maraschi1991, we plot the relation between L3−9keV/LEdd and L8.5GHz/LEdd with
1993).Combingequations11and12,wecanderivethat, theredsign‘•’intherightpanelofFigure2.ForL /L >
3−9keV Edd
ξ(η)/q 10−3,thebest-fittinglinearregression forthecorrelationbetween
L L
 8.5GHz=A(η)101.23ξ(η)/q 2−10keV , (13) L3−9keV/LEddandL8.5GHz/LEddisasfollows,
 LEdd   LEdd  L L 1.39
dwehreivreedξ(sηlo)/pqe|ηo=0f.2r%ad∼io1/X.3-5raayndcoArr(eηl)a1ti0o1n.23fξ(rηo)/mq|ηt=h0e.2%di=sc1c0o−r6o.8n.aT-jheet  L8.5EGddHz=10−5.9 3L−E9dkdeV . (14)
model issteeper than thatof theRIAF-jetmodel, i.e.,for afixed Thisisplottedintherightpanel of Figure2withthedottedline.
X-rayluminosity,theincreaseoftheradioluminosityofthedisc- Fixing M = 10M , α = 0.3, we calculate the radio luminosity
⊙
corona system is more quick than that of the RIAF. One of the L and X-ray luminosity L for different M˙, adjusting η
8.5GHz 3−9keV
reasonsisthat,foragivenradioluminosity,duetothenatureofthe tofitequation 14.It isfound that η isvery weakly dependent on
high radiative efficiency, the simultaneous X-ray luminosity pre- M˙,i.e.,for M˙ = 0.02 M˙ ,η = 0.54%;for M˙ = 0.05 M˙ ,η =
Edd Edd
dicted by the disc-corona system is intrinsically higher than that 0.52%; for M˙ = 0.1 M˙ , η = 0.57%; for M˙ = 0.3 M˙ , η =
Edd Edd
of theRIAF.Moreover, because theX-rayluminosityof RIAFis 0.62%;for M˙ = 0.5 M˙ ,η = 0.62%. Themeanfittingvalueof
Edd
roughlyL ∝M˙2,andtheX-rayluminosityofthedisc-coronasys- η is ∼ 0.57%. The thick solid line in the right panel of Figure 2
X
temisroughlyL ∝M˙1.06,forafixedradioluminosity,theincrease isthemodel line.Thecorresponding emergent spectrawithmass
X
oftheX-rayluminosityoftheRIAFismorequickthanthatofthe accretionratesareplottedintheleftpanelofFigure2.Yuan&Cui
disc-coronasystem,whichinturnisequivalenttothecase,i.e.,for (2005) fitted the radio/X-ray correlation of L ∝ L∼0.7 for L ≃
R X X
afixedX-rayluminosity,theincreaseoftheradioluminosityofthe 10−6L to L ≃ 10−3L withintheframeworkoftheRIAF-jet
Edd X Edd
disc-coronasystemismorequickthanthatoftheRIAF. model. They found that the fitting result was very sensitive to a
Inordertochecktheeffectsofηontheradio/X-raycorrela- parameterδ,whichdenotesthefractionoftheviscositydissipated
tion,weassumeanotherconstantηfordifferentM˙,i.e.,η=0.5%, energy directly heating the electrons in the RIAF. For δ = 0.5,
to calculate the emergent spectra for comparisons with that of theyfoundthatηwashighlydependentonM˙,decreasingfromη∼
η = 0.2%. The emergent spectra are shown in the right panel of 10%toη ∼ 1%withmassaccretionratefrom M˙ ∼ 10−3M˙ to
Edd
Figure1. Thebest-fittinglinear regression for the correlation be- ∼ 10−2M˙ . For a smaller value of δ = 0.01, they found that η
Edd
tween L /L and M˙ can be expressed as, L /L = wasnearlyaconstantwithM˙,i.e.,η∼1%.Indeed,wehavetested
8.5GHz Edd 8.5GHz Edd
A(η)(M˙/M˙ )ξ(η) with A(η)| = 10−7.94 and ξ(η)| = the value of η for H1743-322 during the phase L /L . 10−3
Edd η=0.5% η=0.5% X Edd
1.40, and the best-fitting linear regression for the correlation be- withintheframework of RIAF-jetmodel. Wefound η ∼ 10%. It
tween L /L and M˙ can be expressed as, L /L = isclearthatthefittingvalueofηfromtheRIAF-jetmodelforthe
2−10keV Edd 2−10keV Edd
10−1.25(M˙/M˙ )q with q = 1.06. Then, the predicted radio/X- lowluminosityphaseissystematicallyhigherthanthatofthefitting
Edd
raycorrelationisL /L =A(η)101.25ξ(η)/q(L /L )ξ(η)/q valueofηfromthedisccorona-jetmodel forthehighluminosity
8.5GHz Edd 2−10keV Edd
with A(η)101.25ξ(η)/q| = 10−6.3 and ξ(η)/q| ∼ 1.32. phase,whichisconsistentwiththegeneralideathatthejetisoften
η=0.5% η=0.5%
Roughly,itisclearlyshownthatasystematicincreaseofthefrac- relativelysuppressedatthehighluminosityphaseinBHBs(Fender
tionofthematterejectedtothejetleadstheradiofluxtoasystem- 2004).
aticincrease,howeverdoesnotchangetheslopeoftheradio/X-ray Inthepresentjetmodel,besidesthemassrateinthejet M˙ ,
jet
correlationmuch. therearestillthreeparameters,whichcanaffecttheemissionofthe
Intheabovecalculations,weassumethatηisaconstant for jet,i.e.,thepower-lawenergydistributionindexoftheaccelerated
(cid:13)c RAS,MNRAS000,1–9
5
Figure1.
Emergentspectraofthedisccorona-jetmodelaroundablackholewithM=10M⊙assumingα=0.3.Leftpanel,η=0.2%isassumed.Fromthebottomup,
thesolidlinesarethecombinedemergentspectraofthedisccoronaandjetforM˙ =0.02,0.05,0.1,0.3and0.5M˙Eddrespectively.Thedottedlinesarethe
emergentspectrafromthejets.Rightpanel:η=0.5%isadoptedandthemeaningofthelinestylesarethesameasintheleftpanel.
electrons p,theratiooftheenergyoftheacceleratedelectronsand M˙ onthejetemission,whichisbeyondthestudyofthepresent
jet
theamplifiedmagneticfieldtotheshockenergy,ǫ andǫ .Inthis paper.Inordertocomparewithotherworksforthefractionofthe
e B
work, p = 2.1, ǫ = 0.04 and ǫ = 0.02 are fixed respectively. matterejectedtoformthejetinthelowluminosityphase(e.g,Yuan
e B
By modeling the broadband emission of the afterglow of eight &Cui2005), wetakethesamevalueofǫ = 0.02asadopted in
B
GRBs,Panaitescu&Kumar(2001)derivedthat p = 1.87±0.51, Yuan&Cui(2005).Inaword,althoughtheparametersinthejetare
ǫ = 0.062±0.045 and logǫ = −2.4±1.2. Yuan et al. (2005) uncertain,westillcanconcludethatthejetisrelativelysuppressed
e B
fitted the multiwavelength observations of XTE J1118+480 with atthehighluminosityphaseinBHBs.
RIAF-jet model, in which the best fitting results are p = 2.24,
The mechanism of jet formation is unclear. Narayan & Mc-
ǫ = 0.06and ǫ = 0.02 respectively. BasedonRIAF-jetmodel,
e B Clintock(2012) collectedasmall sample composed of fiveblack
Zhangetal.(2010)fittedthesimultaneousmultiwavelengthobser-
holeX-raytransientswithprecisespinmeasurements.Meanwhile
vations of three black hole X-ray transients J1753.5-0127, GRO
theauthorsestimatedtheballisticjetpowerusingthedataat5GHz
J1655-40, andXTEJ1720-318. Itisfoundthatthebestfittingre-
radioobservations.Itisfoundthattheestimatedjetpoweriscor-
sultsinJ1753.5-0127arep=2.1,ǫ =0.04andǫ =0.02,inGRO
e B relatedwiththesquareofa (witha =cJ/GM2,Jisangularmo-
∗ ∗
J1655-40arep=2.1,ǫ =0.06andǫ =0.02,andinXTEJ1720-
e B mentum of the black hole), which isvery close tothe theoretical
318arep=2.1,ǫ =0.06andǫ =0.08.Ithasbeentestedthatthe
e B scalingderivedbyBlandford&Znajek(1977).However,bysepa-
differentvaluesof phasveryminoreffectsontheradioemission,
ratelyconsideringtheballisticjetandthehardsteadyjet,Fender,
howeverhassignificantlyeffectsontheX-rayemission(O¨zeletal.
Gallo&Russell(2010)foundthatthereisnoevidenceforthecor-
2000;Figure4inYuanetal.2003).DuetotheX-rayemissionisal-
relationbetween the jet power and theblack hole spin. Recently,
waysdominatedbytheemissionofdiscandcorona,thechangeofp
aninterestingpaperarguedthatthejetformationmaybecorrelated
inanobservationallyreasonablerangewillnotchangeourresults.
withthe hot plasma, namely, the jet power is correlated with the
Throughoutthecalculation p = 2.1isfixed.Wetesttheeffectsof
RIAFwhentheEddingtonratioislessthan∼1%andthejetpower
ǫ andǫ onthejetemission.Intheleftpanel ofFigure3,fixing
e B iscorrelatedwiththehotcoronaabovethecooldiscwhentheEd-
M = 10M ,α = 0.3, M˙ = 0.5,η = 0.5%, p = 2.1andǫ = 0.02,
⊙ B dingtonratioisgreaterthan∼1%(Wuetal.2013).Ourstudymay
weplottheemergentspectraofjetforǫ =0.01,0.04,0.08,0.1re-
e putsomeconstrainsonthemechanismofjetformation,i.e.,bysug-
spectively.Itisfoundthat,ǫ issensitivetotheX-rayemission,and
e gestingthattherelativestrengthofthejetpowermaybeinversely
insensitivetotheradioemission. However, intheobservationally
correlatedwiththeEddingtonratioinanaccretingblackhole(e.g.,
inferredrange of ǫ ∼ 0.01−0.1,it isclear that theX-rayemis-
e Ko¨rding,Falcke,&Markoff2002;Fender,Gallo,&Jonker2003).
sionisstilldominatedbythediscandcoronainsteadofthejet,so
thechangeofǫ willnotchangeourresults(seetherightpanelof Asacomparison,intherightpanelofFigure2,wealsoplot
e
Figure1forcomparison).Throughoutthecalculationǫe = 0.04is the relation between L3−9keV/LEdd and L8.5GHz/LEdd for GX 339-4
fixed.Intherightpanel of Figure3,fixing M = 10M ,α = 0.3, with the sign of green ‘△’, and V404 Cyg with the sign of or-
⊙
M˙ = 0.5,η = 0.5%, p = 2.1andǫe = 0.04,weplottheemergent ange ‘(cid:3)’. For LX > 10−3LEdd, the radio/X-ray correlation with
spectraofjetforǫB = 0.01,0.02,0.05,0.1respectively.Itisfound L8.5GHz ∝ L3∼−09.6keV is expected to be interpreted within the frame-
that,increasingthevalueofǫ willincreasetheradioemission,and workoftheRIAF-jetmodel,assuggestedfortheblackholeX-ray
B
nearlydoesnotchangetheX-rayemission.AsdiscussedinSection transientXTEJ1118+480.However,howtojustifytheexistenceof
3.1,anincreaseofM˙ willalsoincreasetheradioemission.Sode- theRIAFatthehighluminosityphase?Qiao&Liu(2009)studied
jet
tailedspectralfittingareneededtodisentangletheeffectsofǫ and theeffectoftheviscosityparameterαonthecriticalmassaccretion
B
rateforthetransitionfromaRIAFtoadisccoronasystemwithin
(cid:13)c RAS,MNRAS000,1–9
6 ErlinQiaoandB.F.Liu
theframeworkofthediscevaporationmodel.Theauthorsderived tonsoftheaccretiondiscitself.Achangethevalueofthealbedo
that M˙ ∝ α2.34 M˙ , i.e., a larger value of α can increase the ‘a’,e.g.,anincreasethevalueof‘a’meansthesoftphotonlumi-
crit Edd
criticalmassaccretionratefortheexistenceoftheRIAF.Asimi- nositycausedbyreprocessdecreases, sotheComptonluminosity
larresulttothecriticalmassaccretionratefortheexistenceofthe decreases, meanwhile the reflection luminosity increases. In our
RIAF, i.e., M˙ ∝ α2 M˙ was also derived from the RIAF so- Monte Carlo simulation, for the energy conservation, we always
crit Edd
lution(Narayan&Yi1995b; Mahadevan 1997).Bysummarizing set that the emergent luminosity is equal to the released gravita-
theobservationaldataofdwarfnovaoutbursts,outburstsofX-ray tionalenergy.Soachangethevalueofthealbedo‘a’inareason-
transients,andvariabilityinAGNandsoon,Kingetal.(2007)in- able range will change the shape of the X-ray spectrum (relative
ferredthatthevalueofαisintherangeof∼0.1−0.4.Extremely, strengthoftheComptoncomponentandthereflectioncomponent),
Narayan(1996)tookα=1toexplaintheBHBsystemswithhigh butwillslightlychangetheX-rayluminosity.Sotheeffectsofthe
transitionluminosities.Theoreticalunderstandings fortheviscos- albedo on the emergent spectrum have only very little change to
ityhavebeenstudiedformanyyearssincethepioneeringworkby ourresults.
Shakura&Sunyaev(1973)(Balbus&Hawley1991;Hawleyetal. Asweknow,theobservedhighfrequencyquasi-periodicos-
1995;Hawley&Krolik2001).Theestimatedvalueofαfromnu- cillation(HFQPO),e.g.,150-450Hz,areconsistentwiththeKep-
mericalsimulationsisroughlyoneorderofmagnitudesmallerthan lerianfrequencyneartheISCOofaSchwarzschildblackholewith
thatoftheobservations(Stone1996;Hiroseetal.2006).However, masses 15-5 solar masses. Meanwhile, the observed HFQPOsdo
indeedtherearenumericalsimulationsshowingthatthevalueofα notchangesignificantlydespitethesizablechangeintheX-raylu-
sometimesevencanexceedunityinthecorona(e.g.,Machidaetal. minosity,suggestingtheconnectionsbetweentheHFQPOsandthe
2000). massandspinoftheblackhole.Iftheblackholemassiswellcon-
strained,theHFQPOscanbeusedtomeasurethespinoftheblack
hole.Theobservedpairsoffrequenciesina3:2ratiosuggestthat
the HFQPOs are probably produced by some types of resonance
5 DISCUSSION
mechanism (e.g., Abramowicz & Kluzniak 2001). In the present
Inthis work, weproposed a disccorona-jet model toexplain the disc-coronamodel,forsimplicity,weassumetheaccretiondiscal-
observedradio/X-raycorrelationofL ∝L∼1.4forL /L &10−3 waysextendingdowntotheISCOofaSchwarzschildblackhole,
R X X Edd
inBHBs.Wenoted thatasimilardisccorona-jet model wasalso i.e, ISCO is fixed at 3RS. The incorporation of the effects of the
proposedforexplainingthisradio/X-raycorrelation,inwhichthe spintothedisc-coronamodelisnecessaryinthefuturetomakethe
X-rayemissionisalsodominatedbythediscandcorona,andthe modelmorerealistic,meanwhiletomatchtheobservedHFQPOs.
radioemissionisdominatedthejet(Huangetal.2014).Bothinour Observationally,thereisapositivecorrelationbetweentheEd-
workandHuangetal.(2014),forthedisc-coronamodel,itisas- dington ratio λ and hard X-ray index Γ for λ & 0.01 (Wu & Gu
sumedthatthemagneticfieldisgeneratedbydynamoactioninthe 2008, Qiao & Liu 2013). In the present disc-corona model, the
accretion disc, then due to buoyancy, the magnetic loops emerge hard X-ray photon index is Γ ∼ 2.1, and does not change with
from the accretion disc into the corona and reconnect with other the mass accretion rates. In the present paper, for simplicity, we
loops. In this way the accretion energy is released in the corona onlyconsiderthegas-pressuredominatedcaseforthestructureof
asthermalenergyandeventuallyemittedawayviainverseComp- the accretion disc, i.e., only the hard-state solution is considered
tonscattering.Bystudyingtheenergybalanceofthediscandthe (Liuetal.2002;2003).AsdiscussedinLiuetal.(2003),thegas-
corona, Huang et al. (2014) solved the structure of the cold ac- pressure dominated accretion disc can exist for all the mass ac-
cretiondisc.However,theycannotself-consistentlydeterminethe cretion rates. When the system is accreting at M˙ > 1.2M˙Edd, the
temperature of the corona T and the Compton scattering optical radiation-pressuredominatedaccretiondisccanexistextendingto
depth τ, which aretwo very important quantities for determining 50RS,whichpredictsasoft-statesolution.Whentheaccretionrate
theshapeoftheComptonemergentspectrum.Theoretically,T and is at 0.3M˙Edd . M˙ . 1.2MEdd, the accreting system can be at a
τ should be solved independently for determining the Compton statebetweenthehardstateandthesoftstatewithhardX-rayin-
emergentspectrum.InHuangetal(2014),itisarguedthattheob- dexvaryingwithmassaccretionrates.
servationallyinferredvalueofτisintherangeof∼0.1−0.8,then Meanwhile,inHuangetal.(2014),thejetpowerisestimated
throughtheircalculation,τ=0.5isfixedtosolveT forfittingthis according to the proposed hybrid jet model (Meier 1999, 2000),
radio/X-ray correlation. In our model, we do not fix the value of fromwhichtheradioluminosityisestimatedbasedontheempir-
τ.Weperformedself-consistentMontecarlosimulationtotreatthe ical relation from Cyg X-1 and GRS 1915+105 (Falcke & Bier-
structureofthedisc-coronasystem.Thefractionofthedissipated mann1996;Heinz&Sunyeav2003;Heinz&Grimm2005).Inthe
energy in the corona f, the temperature of the corona T and the presentpaper,basedontheinternalshockscenario,wecalculated
Comptonscatteringopticaldepthτcanbeself-consistentlydeter- the emergent spectrum of the jet, which makes our results more
mined,whichareimportantadvantagesofourmodel. easytocomparewithobservations.
We note that in our model, currently in order to simplify
the calculation of the complex interaction between the disc and
corona,wealwayssetthealbedo,i.e.,‘a=0’,toconducttheMonte
6 CONCLUSION
Carlo simulation, which means that the irradiation photons from
thecoronaarefullyabsorbedbytheaccretiondisc,thenarerepro- In this work, we investigate the radio/X-ray correlation of LR ∝
cessed as the soft photons for the inverse Compton scattering in LX∼1.4forLX/LEdd&10−3withintheframeworkofadisccorona-jet
the corona. Since albedo ‘a=0’ is adopted, we don’t have reflec- model,inwhichafractionofthematter,η,isassumedtobeejected
tioncomponent intheemergent spectra.Weshould keepinmind toformthejet.Wecalculatetheslopeoftheradio/X-raycorrelation
that in our model, in the gas pressure dominated case, the origin byassumingaconstantηfordifferentM˙.Forη=0.2%,itisfound
of soft photons for the inverse Compton scattering is dominated thatL8.5GHz ∝ Lξ2/−q10keV withξ/q ∼ 1.35.Forη = 0.5%,wederive
bythereprocessed softphotons ratherthantheintrinsicsoftpho- thatL ∝ Lξ/q withξ/q∼ 1.32,whichisveryclosetothe
8.5GHz 2−10keV
(cid:13)c RAS,MNRAS000,1–9
7
Figure2.
Leftpanel:Emergentspectraofthedisccorona-jetmodelaroundablackholewithM=10M⊙assumingα=0.3formodelingtheradio/X-raycorrelationof
theblackholeX-raytransientH1743-322forL3−9keV/LEdd>10−3.Fromthebottomup,thesolidlinesarethecombinedemergentspectraofthedisc
corona-jetmodelforM˙ =0.02,0.05,0.1,0.3and0.5M˙Edd,andthecorrespondingdottedlinesaretheemergentspectrafromthejetwithη=0.54%,0.52%,
0.57%,0.62%and0.62%respectively.Rightpanel:L8.5GHz/LEddasafunctionofL3−9keV/LEdd.Thered‘•’aretheobservationsforH1743-322,thegreen
△’aretheobservationsforGX339-4,andorange‘(cid:3)’aretheobservationsforV404Cyg.Thedottedlineisthebest-fittinglinearregressionofH1743-322for
L3−9keV/LEdd>10−3.Thedashedlineisthebest-fittinglinearregressionofGX339-4andV404Cyg.Thethicksolidlineisthemodelline,andthemodel
spectraareshownintheleftpanel.
Figure3.
Emergentspectraofthejetfordifferentparameters.Intheleftpanel,M=10M⊙,M˙ =0.5M˙Edd,η=0.5%,p=2.1andǫB=0.02arefixed,andfromthe
bottomup,ǫeare0.01,0.04,0.08and0.1respectively.Intherightpanel,M=10M⊙,M˙ =0.5M˙Edd,η=0.5%,p=2.1andǫe=0.04arefixed,andfromthe
bottomup,ǫBare0.01,0.02,0.05and0.1respectively.
caseofη=0.2%.Asanexample,fordifferentM˙,bychangingthe ofthejetpowerisrelativelysuppressedduringthehighluminosity
valueofη,wefittheobservedradio/X-raycorrelationofblackhole phaseinBHBs.
X-ray transients H1743-322 for L /L > 10−3. It is found
3−9keV Edd
thatηisweaklydependentonM˙,andthemeanfittingvalueofηis
∼0.57%.Wenoteaninterestingresult,i.e.,themeanfittingresult
ACKNOWLEDGMENTS
ofη ∼ 0.57%fortheradio/X-raycorrelationofL ∝ L∼1.4 during
R X
thehighluminosityphaseissystematicallylessthanthatofthecase E.L.QiaothanksthejetcodefromDr.HuiZhang.Wethanktheref-
forthelowluminosityphase(atleastη&1%),whichmayputsome ereeforhis/herveryusefulsuggestionsandcomments,especially
constraintsonthejetformation,i.e.,bysuggestingthatthestrength the expert view on the observational aspects to the manuscript.
E.L.Qiao appreciates Dr. Francesca Panessa for the Englishcor-
(cid:13)c RAS,MNRAS000,1–9
8 ErlinQiaoandB.F.Liu
rection. We thank the very useful discussions with Prof. Weimin Ma,R.,2012,MNRAS,423,L87
Yuan. This work is supported by the National Natural Science Machida,M.,Hayashi,M.R.,&Matsumoto,R.,2000,ApJ,532,
FoundationofChina(grants11303046,11033007,11173029,and L67
U1231203)andtheStrategicPriorityResearchProgramTheEmer- Malzac,J.,Merloni,A.,&Fabian,A.,2004,MNRAS,351,253
genceofCosmo-logicalStructuresoftheChineseAcademyofSci- Mahadevan,R.,1997,ApJ,477,585
ences(GrantNo.XDB09000000). Markoff,S.,Falcke,H.,&Fender,R.,2001,A&A,372,L25
McClintock,J.E.,Garcia,M.R.,Caldwell,N.,Falco,E.E.,Gar-
navich,P.M.,&Zhao,P.,2001,ApJ,551,147
McClintock,J.E.,RemillardR.A.,RupenM.P.,TorresM.A.P.,
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