Wang Jiuzhou，Wang Dingxiong

(School of Physics，Huazhong University of Science and Technology，Wuhan 430074，China)

1 Introduction

Spectral states observed in black hole X-ray binaries(BHXBs)involve a number of unresolved issues in astrophysics，displaying complex variations not only in the luminosities and energy spectra，but also in presence/absence of jets and quasi-periodic oscillations(QPOs).

Belloni(2006，hereafter B06)［1］classified the states of BHXBs as low/hard(LH)state，hard intermediate state(HIM)，soft intermediate(SIM)state and high/soft(HS)state，which display different luminosity and hardness associated with different behavior of QPOs and radio loudness.

McClintock ＆ Remillard (2006，hereafter MR06)［2］used four parameters to define X-ray states based on the very extensive RXTE data archive for BHXBs，in which three states，i.e.，thermal –dominant(TD)state，low/hard(LH)state and steep power law(SPL)state are included.In TD state，the flux is dominated by the thermal radiation from the inner accretion disk，and QPOs are absent or very weak.The spectral profile of LH state is characterized by a hard power-law component at 2～20 keV with a photon index Γ in the range ≈ 1.4～2 and an exponential cut-offatabout 100 keV，which is associated with a quasi-steady radio jet.SPL state has a strong power-law component with Γ～2.5，which is associated withhigh-frequencyQPOs.Although a consensus on classification of spectral states of BHXBs has not been reached，it is widely accepted that these states can be reduced to only two basic states，i.e.，a hard state and a soft one，and jets can be observed in hard states，but cannot be in soft states.

The accretion flow in LH state is usually supposed to be a truncated thin disk with an inner advectiondominated accretion flow(ADAF)in the prevailing scenario(MR06).However，a thermal component has been observed in a number of sources in LH state，and the accretion disks in these sources are consistent with extending close to the innermost stable circular orbit(ISCO).In addition，XMM-Newton observations of GX 339-4 show that a broad iron line together with a dim，hot thermal component was present in its spectra during the hard state.This effect seems to be observed in a few other sources such as Cygnus X-1，and SWIFT J1753.5-0127(Miller，Homan ＆ Miniutti 2006)［3］.Not long ago，Miller et al(2006)pointed out that the XMM-Newton spectra of GX 339-4 in the LH state clearly reveal the presence of a cool accretion disk componentand a relativistically-broadened Fe K emission line.Theresultsoffitsmade to both components strongly suggest that a standard thin disk remains at or near to ISCO，at least in bright phases of the LH state.Recent broad iron line detections and estimates of the disk component strength suggest that a non-recessed disk could still be present in bright hard states.Very recently，Reis，Fabian ＆ Miller(2010)［4］presented an X-ray study of eight black holes in the LH state，and they found that a thermal disk continuum with a color temperature consistent with L∝T4is clearly detected in all eight sources and the detailed fits to the line profiles exclude a truncated disk in each case.On the contrary，the disk-corona model reveals its advantages in modeling the LH state of BHXBs(Merloni ＆ Fabian 2002)［5］.

One of the main characteristics of the LH state of BHXBs is its association with quasi-steady jets，and the most promising mechanisms of powering the jets are the Blandford-Znajek(BZ)process(Blandford ＆ Znajek 1977)［6］and the Blandford-Payne(BP)process(Blandford ＆ Payne 1982)［7］，in which largescale magnetic fields are invoked to extract energy from a spinning black hole and from itssurrounding accretion disk，respectively.Recently，Gan et al(2009，hereafter GWL09)［8］proposed a model of magnetically induced disk-corona for BHXBs，in which the closed magnetic field lines pipe the hot matter evaporated from the disk，and shape it in the corona above the disk to form a magnetically induced diskcorona system.Later，we combined epicyclic resonanceswith the disk-corona modelgiven by GWL09 to interpret the high frequency QPOs with 3:2 pairs observed in SPL states of the three BHXBs，i.e.，GRO J1655-40(450，300Hz)，XTE J1550-564(276，184Hz)and GRS 1915+105(168，113Hz)(Huang et al 2010，hereafter HGW10)［9］.Although some spectral profiles have been fitted in the LH state，a clear association with a quasi-steady jet has not been revealed，and this becomes a great challenge to the present theoretical models.

In this paper，we intend to model the LH state of BHXBs based on a disk-corona model，in which the inner edge of the accretion disk is assumed to extend to ISCO，and the jets are driven by the patched open magnetic field suggested by Spruit ＆ Uzdensky(2005，hereafter SU05)［10］.In addition，we discuss the correlation of state transitions of BHXBs with the evolution of the magnetic field configuration in black hole accretion disk.This paper is organized as follows.In section 2，we present a detailed description of our model，in which the jet is driven by the patched open magnetic field，and the coupling of the jet with accretion process is taken into account based on the conservation of energy and angular momentum.In section 3，the spectral profiles of the LH state of five BHXBs are fitted based on the disk-corona model，and the relation between jet power and X-ray luminosity is fitted by adjusting accretion rate and the outer boundary of the corona over the disk.Finally，in section 4，we discuss some issues related to our model.

Throughout this paper the geometric units G=c=1 are used.

2 Description of our model

In order to interpret the association of the LH state of BHXBs with a quasi-steady jet，we take the scenario of magnetic patches suggested by SU05，i.e.，a large portion of the net vertical magnetic flux threading a disk gets concentrated into patches of strong field due to'flux expulsion'effect of convective turbulence(Zeldovich 1956［11］;Parker 1963［12］).A schematic drawing of disk-corona model with magnetic patches is shown in Fig.1.

The configuration given in Fig.1 hassome advantages on interpreting the LH state of BHXBs.(a)The outward diffusion of trapped fields by turbulent reconnection can be reduced significantly once the magnetic field locally becomes strong enough to avoid being tangled，and the jet can be driven effectively by the open large-scale magnetic field concentrated in the patches，(b)the disk luminosity can be depressed effectively due to part of energy is transferred into the jet，(c)the corona geometry is scarcely affected by the magnetic patches，which distribute dispersedly on the disk and occupy a small fraction of the disk surface，and(d)the large scale patched open magnetic field is apt to collimate the jet from black hole accretion disk.Thus the LH state of BHXBs associated with a quasi-steady jet can be fitted based on disk-corona model with the magnetic patches.

As argued in SU05，the magnetic field strength Bp inside the patches is estimated by

where βMRI=O(1)is a parameter related to magnetorotational instability(MRI)，and Pgasis the gas pressure dominating over the radiation pressure inside the disk.According to Balbus ＆ Hawley(1991)［13］，the interior viscous process is dominated by tangled small-scale magnetic field，BD，and the viscous pressure is comparable to magnetic pressure，and thus we assume

In this paper，we introduce two parameters，λrand λφ，to describe the distribution of magnetic patches in the disk as follows，

In equation(3)the radial coordinate of the patch rpis related to the disk radius r by the radial fraction factor λr，and the toroidal coordinate of the patch φpto the toroidal coordinate φ of the disk by the toroidal fraction factor λφ.Henceforth the subscript'p'represents the quantities related to the magnetic patches.Thus the magnetic torque exerted on the magnetic patches is given by

where dAp=rdrpdφpis the area element of the magnetic patch，and γ is the ratio of the toroidal component|BΦ，p|to Bp.The jet power driven by each magnetic patch is

where Ωpis the average angular velocity of magnetic patches.Since the magnetized patch drifts inward with a speed greater than that of accreting matter，Ωp=kΩkshould be less than the Keplerian angular velocity，and k is a factor less than unity.

Incorporating equations(1)～(5)，we have the jet power within the radius r driven by the open magnetic field of all patches as follows

where four parameters k，γ，λrand λφare incorporated into fp=kγλrλfwith βMRI≈1.

The flux of angular momentum Hptransferred into the jet is related to the flux of energy Spby Hp=Sp/ΩK，and Spcan be calculated by

Incorporating equations(1)，(6)and(7)，we have

Since dPpis the jet power driven by the magnetic patches in the ring of width dr，the quantities Spand Hpare respectively the average fluxes of energy and angular momentum transferred from the ring by the trapped magnetic fields as shown in equations(7)and(8).The coupling of jet and accretion is taken into account by using the conservation equations of energy and angular momentum as follows，

where g is the viscous torque in the disk，the quantities L+and E+are the specific energy and angular momentum given by Bardeen，Press ＆ Teukolsky(1972)［14］.

Incorporating equations(8)～(10)，we have the radiation flux from disk as follows

where QDAis the viscously dissipated energy due to disk accretion，and it reads

Inspecting equation(11)，we find that the total radiation flux F(r)is less than QDAdue to the negative contribution ofthe second term atRHS ofthis equation，and this term represents the jet effect of reducing the total radiation.So we expect that diskcorona model with trapped magnetic fields anchored at the patches can be applicable to the LH state with a quasi-steady jet observed in several BHXBs，and the detailed fits will be given in the next section.

According to typical disk-corona scenario，part of the viscously dissipated energy Q is released asin the disk，emitting eventually as black-body radiation and supplying seed photons for Comptonization of corona.The rest dissipated energy，heats corona and maintains its relativistic temperature via magnetic buoyancy and reconnection(Liu et al 2002)［15］.The quantityis proportional to magnetic energy density and local Aflven speed，and we have

3 Fitting LH state of BHXBs

In this section we fit the LH state and its associated jet in two steps，where the BHXBs，4U 1543-475，GX 339-4，XTE J1550-564，GRO J1655-40 and GRS 1915+105 are included.Firstly，we fit the spectral profiles of the LH state of these sources，and then we fit the relation between jet power and X-ray luminosity based on the first step.

3.1 Fitting Spectra

At the first step the spectra of the LH state are fitted based on disk-corona model，and the code given in GWL09 is modified in two aspects:(i)the MC process with the closed magnetic field in GWL09 is replaced by the jet launching process with the patched open magnetic field as shown in Fig.1，and(ii)the outer boundary of corona taken as that of the closed magnetic field in GWL09 is replaced by the radius routas an adjustable parameter in fitting the LH state in this paper.

The main characteristics of the five BHXBs are given in Tab.1，and the values of the input parameters and those of LJand LXare given in Tab.2，where LJand LXdenote the jet power derived from equation(6)and the X-ray luminosity，respectively.The spectral profiles of the LH state are shown in Fig.2.It is noted that the luminosities and accretion rates are given in terms of Eddington luminosity，1.25 ×1038(M/M⊙)erg/s，and the disk radius is given in unit of Rg=GM/c2.

It is noted that the spectral profiles of the LH states of the five BHXBs given in Fig.2 are in good agreement with the observation data given in Fig.4.11 of MR 06.

3.2 Relation between X-ray luminosity and jet power

At the second step in fitting the LH state we check the relation between the jet power and the X-ray luminosity as follows

This relation was first proposed by Fender，Gallo ＆ Jonker(2003)［22］，and the coefficient Asteadyvaries between 6 ×10－3and 0.3(Fender，Belloni ＆ Gallo 2004［23］，hereafter FBG04;Malzac，Merloni ＆ Fabian 2004)［24］.In our fits LJis regarded as the jet power Pjetgiven by equation(15)，and LXcan be calculated based on the spectral profiles of the LH state as shown in Fig.2.

In addition，we obtain the values of LXfor each BHXB based on the spectral profile of the LH state obtained in the first step，and the coefficient Asteadycan be determined by equation(15)，which is a constant in the fitting for each source.It turns out that the relation between LXand LJcan be satisfied by adjusting accretion rate in and the outer boundary radius routrout of corona.The fitting results are given in Tab.3 and Fig.3.

Inspecting Tab.3 and Fig.3，we find that LJand LXdo obey equation(15)with the coefficient Asteadyranging between 6 ×10－3and 0.3，and both quantities increase monotonically with the increasing accretion rate in as well as the increasing outer boundary routof the corona.These results are consistent with the rising phase of transient outburst of the BHXBs.

Although the origin of the magnetic field in BHXBs remains unclear，it is most probably related to accretion process in the following aspects.

(i)Seed magnetic field is regarded as one of the possibilities of the origin of the magnetic field，and seed magnetic field is brought from a companion in accretion process;

(ii)Seed magnetic field can be amplified via dynamo mechanism，and this mechanism arises from differential rotation of accretion disk;

(iii)Large-scale magnetic field mightbe generated by toroidal electric current，and this kind of current exist probably in the accretion due to total deviation from electric neutrality in accreting plasma coming from its companion.

4 Discussion

In this paper，we fit the spectral profiles of five BHXBs in the LH state associated with the quasi-steady jets by introducing the large-scale patched magnetic fields into disk-corona model.It turns out that a quasisteady jet does associate with the LH state，and the relation between the jet power and the X-ray luminosity does hold based on our model.

It is noted that the coupling of patched magnetic field with accretion disk plays an essential role in the fits.On the one hand，the patched magnetic field reduces the luminosity effectively by extracting energy from the accretion disk to drive jets，and thus it affects disk dynamics.On the other hand，this effect gives rise to a feedback to the patched magnetic field itself，reducing its outward diffusion and increasing its drift inward.The main characteristics of the spectral profiles of the LH state can be retained with a quasi-steady jet driven by the patched magnetic field distributed dispersedly on the disk-corona system.Some issues related to our model are addressed as follows.

(1)The mechanisms of driving jet.

In this paper，the magneto-centrifugal mechanism(e.g.，Bisnovatyi-Kogan ＆ Ruzmaikin 1976［25］;Blandford ＆ Payne 1982)is adopted to drive jet via the patched field instead of the BZ process because of the following reasons.(i)The coupling of the patched magnetic fields with the disk-corona system is realized via the magneto-centrifugal mechanism，and it is helpful to interpret the LH state with a quasi-steady jet.(ii)As shown in Figure 1，the patched fields assumed in different direction dissipate probably in magnetic reconnection asthey driftclose to the innermost region of the disk，and thus the BZ process cannot work due to very few magnetic fields brought to the black hole.In addition，this configuration could provide a possible interpretation for producing episodic jets in the transition from hard state to soft state(FBG04;Yuan et al.2009，hereafter Y09).

(2)Relationbetweenjet powerandX-ray luminosity.

From the fitting results given in Tab.3 and Fig.3 we find that the relation between jet power and X-ray luminosity is fitted numerically in our model.Both jet power and X-ray luminosity increases monotonically with the increasing accretion rate and the outer boundary of the corona，and these fits are consistent with the rising phase of transient outburst of the BHXBs as shown in the X-ray hardness-intensity diagram(HID)given by FBG04.These results can be roughly understood as follows.

On the one hand，both jet power and X-ray luminosity are powered essentially by gravitational energy released in accretion process，so these two quantities increase with the increasing accretion rate.On the otherhand，required bythe unchanged hardness of the LH state，the outer boundary of the corona should increase to contribute more power law component via Comptonization in the corona with the increasing accretion rate.

(3)The uncertainty of the fitting parameters.

The most difficult problem related to our model is how to describe the distribution and the drifting motion of the magnetic patches.As indicated above，four parameters are incorporated into one parameter，fp=kγλrλφ，and fp≈0.0001 is taken in the calculations.Fortunately，the fitting results are insensitive to the value，i.e.，we can obtain the same spectral profiles with a quasi-steady jet power for fpvarying around 0.0001.The maximum values of fpare constrained by the requirementthatthe radiation flux given by equation should be non-negative，and those maximum values are shown in the rightmost column of Tab.3.

The fitting results presented in this paper show that the spectral profiles in the LH state could be well fitted by disk-corona model with the inner edge of disk remains close to ISCO，thus a cool accretion disk componentand a relativistically-broadened Fe K emission line can be naturally explained.Nevertheless，we have to make some assumptions with several parameters for describing the patched magnetic fields due to lack of knowledge on'flux expulsion'effect of convective turbulence，and we hope to improve our model based on further study on convective turbulence in accretion disk.

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