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 0 2 n a 16January2015 J 5 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). 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