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Chang-Yin Huang, Qingwen Wu, Ding-Xiong Wang, Modelling the ‘outliers’ track of the radio–X-ray correlation in X-ray binaries based on a disc–corona model, Monthly Notices of the Royal Astronomical Society, Volume 440, Issue 2, 11 May 2014, Pages 965–970, https://doi.org/10.1093/mnras/stu364
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Abstract
A universal radio–X-ray correlation (|$F_{\rm R}\propto F_{\rm X}^{b}$|, b ∼ 0.5–0.7) has been found for a sample of black hole X-ray binaries (BHBs) in their low/hard states. This can roughly be explained by a coupled model of a jet and radiatively inefficient advection-dominated accretion flow. However, more and more ‘outliers’ have been found in the last few years, which evidently deviate from the universal radio–X-ray correlation and usually show a much steeper correlation with an index of ∼1.4. Based on simple physical assumptions, it is speculated that radiatively efficient accretion flows occur in these outliers. In this work, we test this hypothesis by modelling the outliers track based on the radiatively efficient disc–corona model and the hybrid jet model. We find that our model predicts a steeper radio–X-ray correlation, with slopes ≳ 1.2 for typical viscosity values of α ∼ 0.05–0.2. In particular, the slope is ∼1.4 for the case α ∼ 0.1, which is consistent with observational results for H1743−322. Our results suggest that the outliers track might be regulated by the disc–corona model.
INTRODUCTION
Most black hole X-ray binaries (BHBs) are transient systems that undergo occasional outbursts that display complex spectral and timing features. Normally, there are two main states in BHBs: the high/soft (HS) state and the low/hard (LH) state (e.g. Zdziarski 2000; McClintock & Remillard 2006). The HS state is characterized by strong thermal emission and a weak power-law component. In contrast, the thermal emission is normally weak in the LH state, and most of the radiation comes from the non-thermal power-law component. Both the soft X-ray bumps observed in the HS state of BHBs and the optical/UV bumps observed in quasars can be naturally interpreted as resulting from the multitemperature blackbody emission from a cold, optically thick, geometrically thin standard accretion disc (SSD; Shakura & Sunyaev 1973). The prevalent accretion model for LH-state BHBs and low-luminosity active galactic nuclei (AGNs) is that of the hot, optically thin, geometrically thick advection-dominated accretion flows (ADAFs; also called radiatively inefficient accretion flows, RIAFs) that have been developed for black holes (BHs) accreting at low mass accretion rates (e.g. Ichimaru 1977; Narayan & Yi 1994, 1995; Abramowicz et al. 1995; see Kato, Fukue & Mineshige 2008; Narayan & McClintock 2008, for recent reviews).
There is a strong association between the radio and the X-ray emission in the LH state of BHBs. The quasi-simultaneous radio and X-ray fluxes roughly follow a universal non-linear correlation (|$F_{\rm R}\propto F_{\rm X}^{b}$|, b ∼ 0.5–0.7, Hannikainen et al. 1998; Corbel et al. 2003; Gallo, Fender & Pooley 2003; Corbel et al. 2013). This non-linear correlation seems to be maintained down to the quiescent state (Corbel et al. 2003; Gallo et al. 2006; Corbel, Koerding & Kaaret 2008). By taking into account the BH mass, the relation was extended to AGNs, and is called the ‘Fundamental Plane’ of BH activity (Merloni, Heinz & Matteo 2003; Falcke, Körding & Markoff 2004; Wang, Wu & Kong 2006; Körding, Falcke & Corbel 2006; Li, Wu & Wang 2008; Yuan, Yu & Ho 2009; Gültekin et al. 2009; Plotkin et al. 2012). In recent years, however, several BHBs have been found to lie well outside the scatter of the original radio–X-ray correlation (e.g. H1743−322, Jonker, Miller-Jones & Homan 2010; Coriat et al. 2011; Swift 1753.5−0127, Cadolle Bel et al. 2007; Soleri et al. 2010; XTE J1650−500, Corbel et al. 2004; XTE J1752−223, Ratti et al. 2012). These ‘outliers’ roughly form a different track (the ‘outliers’ track), following a steeper correlation with an index of b ∼ 1.4, as initially found for H1743−322 (Coriat et al. 2011). Some of these sources (e.g. H1743−322, XTE J1752−223, MAXI J1659−152) jump to the standard universal correlation when they fade towards quiescence (Jonker et al. 2010, 2012; Coriat et al. 2011; Ratti et al. 2012).
Although jets have been observed in various kinds of high-energy objects (e.g. AGNs, BHBs, Gamma-ray bursts, etc.), the detailed physical mechanism for jet formation is still unknown. The popular mechanisms for jet production include the Blandford–Znajek (BZ) process (Blandford & Znajek 1977) and the Blandford–Payne (BP) process (Blandford & Payne 1982). Both mechanisms involve energy extraction via open large-scale magnetic fields, either from a rotating BH in the form of Poynting flux (BZ process) or from the rotating accretion disc in the form of a magnetically driven wind (BP process). The hybrid model proposed by Meier (1999), as a variant of the BZ model, combines the BZ and BP effects through the large-scale magnetic fields threading the accretion disc outside the ergosphere and the rotating plasma within the ergosphere. This model seems to be supported by magnetohydrodynamic (MHD) simulations (e.g. Koide et al. 2000; McKinney & Gammie 2004; Hirose et al. 2004; Hawley & Krolik 2006) and by some recent observations (e.g. Nemmen et al. 2007; Wu & Cao 2008; Wu, Cao & Wang 2011; Li & Cao 2012).
The radiatively inefficient ADAF is expected to produce X-ray emission with |$L_{\rm X}\propto \dot{M}^{q}$| (where |$\dot{M}$| is the accretion rate, q ∼ 2.0; e.g. Narayan, Garcia & McClintock 1997; Merloni, Heinz & Matteo 2003; Yuan & Cui 2005; Wu & Cao 2006), which can roughly explain the universal radio–X-ray correlation (|$L_{\rm R}\propto L_{\rm X}^{0.7}$|) if considering a scaling between the jet luminosity and jet power of |$L_{\rm R}\propto Q_{\rm jet}^{1.4}$| (e.g. Blandford & Königl 1979; Falcke & Biermann 1996; Heinz & Sunyeav 2003) and |$Q_{\rm jet}\propto \dot{M}$| (e.g. Falcke & Biermann 1995; Wu et al. 2011). Detailed calculations based on the ADAF–jet model support this scenario (e.g. Yuan & Cui 2005). From simple physical assumptions, the outliers of BHBs can be understood if they are accreting through a radiatively efficient accretion disc, where LX is roughly proportional to |$\dot{M}$|. The standard disc–corona model (Shakura & Sunyaev 1973; Liu, Mineshige & Shibata 2002) and the luminous hot accretion flow (Yuan 2001) are two possible candidates for the radiatively efficient accretion flow. Recently, observational evidence has pointed to the possible presence of a cool inner disc in the bright hard state of BHBs if the luminosity is higher than ∼0.1 per cent of the Eddington luminosity (e.g. Miller, Homan & Miniutti 2006a; Miller et al. 2006b; Rykoff et al. 2007; Ramadevi & Seetha 2007; Reis, Miller & Fabian 2009; Reis, Fabian & Miller 2010, but see also Done & Trigo 2010; Plant et al. 2013 for a different opinion). From a theoretical point of view, Liu et al. (2007) also found that an inner cool disc may exist for the LH state of BHBs owing to the condensation of matter from the ADAF if the accretion rate is larger than 0.1 per cent of the Eddington rate.
The outliers are normally in the bright hard state, and the steeper radio–X-ray correlation regulated by the radiatively efficient model has been discussed by a number of authors (e.g. Merloni et al. 2003; Falcke et al. 2004; Coriat et al. 2011; Corbel et al. 2013). In this work, we present detailed calculations based on a jet that forms from the disc–corona system, and test whether or not this model can explain the radio–X-ray correlation of the outliers track. In Section 2, we present the disc–corona model and the jet model. The theoretical result and its comparison with observations are presented in Section 3. We discuss our results in Section 4.
THE MODEL
Disc–corona model
By solving equations (1)–(7) numerically, we can obtain self-consistent global solutions of the cold disc. For a given optical depth τ, the electron temperature of the corona at a given radius can be derived from equation (8). Spectral studies of BHBs in the LH state indicate that the value of τ is about ≲ 0.5–2 (e.g. Gierliński et al. 1997; Zdziarski 1999; Torii et al. 2011). Theoretical studies of the disc–corona model also reveal that the value of τ lies in the range 0.1–0.8 (e.g. Liu, Mineshige & Shibata 2002; Liu, Mineshige & Ohsuga 2003; Liu et al. 2012; Yao et al. 2005; Cao 2009; Qiao & Liu 2012; You, Cao & Yuan 2012). In this work, we consider a slab corona with optical depth τ ∼ 0.1–1, height Hc = 20Rg (Liu et al. 2002), inner boundary Rin = Rms and outer boundary Rout = 100Rg, where Rg ≡ GMBH/c2 and Rms are the gravitational radius and the radius of the innermost stable circular orbit (ISCO), respectively.
The radiative spectrum of the disc–corona system was simulated using the Monte Carlo method developed in our earlier work (see Gan, Wang & Lei 2009, for more details). We briefly summarize it here, in four steps: (i) sample a seed photon from the cold disc based on the Planck spectrum for a given Td at R; (ii) select a value for its free path and test whether it can leave the corona; (iii) simulate the interaction of the photon with the electrons in the corona; (iv) repeat steps (ii) and (iii) until the photon leaves the coronal system. The X-ray luminosity can be derived by integrating the spectrum.
Jet model
RESULTS
Because the photon index of the X-ray spectrum is determined by both the coronal electron temperature and the optical depth, we can reproduce the hard X-ray spectrum as that observed in LH-state outliers by adjusting the proper optical depth of the corona for a given accretion rate. For example, we present spectra with the hard X-ray photon index Γ ≃ 1.8 for our disc–corona model in Fig. 1, where different optical depths are chosen for different accretion rates in the cases of a* = 0.5 and MBH = 10 M⊙. We note that the 1–10 keV X-ray luminosity of our model is not very sensitive to the value of optical depth (e.g. 0.1 < τ < 1) when the dissipated energy in the corona is fixed (see equation 3), even there are some changes in the shape of the spectrum. For simplicity, we fix τ = 0.5 when exploring radio–X-ray correlations.
The radio–X-ray relationships for values of the viscosity parameter α = 0.05, 0.1 and 0.2 are presented in the top panel of Fig. 2 for a given typical BH mass of 10 M⊙ and a BH spin of a* = 0.5. It can be seen that the radio and X-ray luminosities are positively correlated even though they do not follow a simple power-law correlation, with the relation becoming slightly steeper at high luminosities for a given α parameter. Furthermore, a larger α value leads to a steeper radio–X-ray correlation. We find that the slope of the radio–X-ray relation is greater than 1.2, even for α = 0.05. For comparison, a thick solid line with slope 1.4 is plotted in the top panel of Fig. 2, which is roughly consistent with the case α = 0.1 for the typical luminosity range of the outliers track with |$4\times 10^{36}\,{\rm erg \, s}^{-1}\lesssim \it L_{\rm X}\lesssim \rm 10^{38}\,{\rm erg \, s}^{-1}$|. In the bottom panel of Fig. 2, we show the radio–X-ray correlation for various BH spins when the viscosity parameter α = 0.1. Larger BH spin parameters lead to a higher radio luminosity (or jet power), because the jet power is more sensitive to the BH spin parameter than the X-ray emission (or disc luminosity).
We compare our theoretical results with the observed radio–X-ray correlation for the outliers track in Fig. 3, which is extracted from Corbel et al. (2013). We find that the observed radio–X-ray correlation of the outliers track can be roughly reproduced with α ∼ 0.1, and the reproduced correlation is not very sensitive to the BH spin parameters. For comparison, we also plot the case of τ = 0.3 and a* = 0.8 (dotted line), which is roughly the same as that for a slightly larger optical depth τ = 0.5 with the same BH spin (thick solid line) . This suggests that our main conclusion will not change even if different optical depths of the corona occur for different luminosities. GRS 1915+105 deviates slightly from our above model predictions. However, it is roughly consistent with model predictions if MBH = 15 M⊙ is adopted (thin solid line in Fig. 3).
DISCUSSION
The universal radio–X-ray correlation of |$F_{\rm R}\propto F_{\rm X}^{0.5-0.7}$| of BHBs in their LH state (e.g. Hannikainen et al. 1998; Corbel et al. 2003; Gallo et al. 2003; Corbel et al. 2013) was quantitatively explained by the coupled ADAF–jet model in Yuan & Cui (2005), where the radio and X-ray emissions are dominated by the radiation from the jet and the ADAF respectively. However, the physical mechanism for the outliers track is still unclear. In this work, we explore the radio–X-ray correlation based on the jet and disc–corona model, and find that this model can roughly reproduce the steeper correlation found in outliers, if it is assumed that the physical mechanisms of the jet launching and radiation are identically in the ‘outliers’ and universal tracks.
Wu & Gu (2008) found that there exist two kinds of hard-state sources, where the hard X-ray photon index is anticorrelated with the luminosity for BHBs in the dim hard state, but positively correlated for the sources in the bright hard state. The anticorrelation and positive correlation are consistent with the predictions of the ADAF model and the disc–corona model, respectively (e.g. Cao 2007; You et al. 2012; Qiao & Liu 2013). The critical Eddington ratio for the positive correlation and anticorrelations is ∼1 per cent, which is also roughly consistent with the prediction for the disc transition (Wu & Gu 2008). The hard X-ray photon index is also positively correlated with the luminosity for these outliers, which is consistent with the prediction of the disc–corona model (Cao, Wu & Dong 2014). Furthermore, the broad Fe K emission lines observed in the bright hard state of some outliers (e.g. H1743−322, McClintock et al. 2009; XTE J1650−500, Rossi et al. 2005; XTE J1550−564, Sobczak et al. 2000) suggest that the line emission region extends very close to the ISCO and that an inner cool disc may exist. GX 339–4 also has a positive correlation between the hard X-ray photon index and the luminosity in its bright hard state (Wu & Gu 2008; Cao et al. 2014), even it was suggested to form the universal correlation (Coriat et al. 2011). It is interesting to note that the radio–X-ray correlation also becomes steeper for GX 339–4 when the X-ray luminosity LX ≳ 1037 erg s− 1 (|$F_{\rm 8\,GHz}\propto F_{\rm 3-9\,keV}^{\sim 1.14}$|; for example figs 8 and 9 in Corbel et al. 2013 and fig. 1 in Cao et al. 2014). Therefore, all BHBs in the bright hard state may follow the so-called outliers track.
We find that the slopes of the radio–X-ray correlation based on the radiatively efficient disc–corona model are much steeper than the traditional slope of 0.7, even though the correlation is not a simple power-law relation. For example, the slope is ≳ 1.2 even for the smallest adopted viscosity parameter α = 0.05. For larger α values, the radio–X-ray correlation becomes steeper (see top panel of Fig. 2), because the magnetic field (or jet power) becomes stronger (see equation 4), while the X-ray emission (disc/corona) is not very sensitive to the viscosity parameter. Even for a given α parameter, the radio–X-ray correlation becomes steeper at high luminosities. The physical reason is that the radio emission (or jet power) is roughly proportional to the accretion rate, while the corona (X-ray emission) roughly saturates at high accretion rates owing to the stronger cooling (e.g. Liu & Taam 2009 Fig. 1). We find that our model prediction with α ∼ 0.1 can roughly reproduce the observed radio–X-ray correlation for this outliers track, where α ∼ 0.1 is a typical viscosity parameter constrained from MHD simulations (e.g. ∼0.05–0.2, Hawley & Balbus 2002). The radio–X-ray correlation of our model is not very sensitive to the BH spin parameter, because both the X-ray emission from the disc and the radio emission from the jet are related to the BH spin. We cannot constrain the BH spin for a given BHB through this comparison owing to the uncertainties of our model and the uncertainties in converting the jet power to radio luminosity. Fortunately, the BH spin parameter does not affect the slope of the radio–X-ray correlation, which will not change our main conclusion (see bottom panel of Fig. 2).
In our disc–corona model, we find that the hard 1–10 keV X-ray emission is roughly proportional to |$\dot{M}^{0.8}$|. This suggests that the corona becomes weak with increasing |$\dot{M}$| (e.g. |$L_{\rm cor}/L_{\rm bol}\propto \dot{M}^{-0.2}$|), which is qualitatively consistent with observations (see also Cao 2009; Wang, Watarai & Mineshige 2004). The magnetic field of the disc obeys the relationship |$B_{\rm d} \propto P_{\rm mag}^{0.5}\propto \dot{M}^{0.4}$| for the case of the adopted magnetic stress tensor. Considering the relationship between the jet power and radio luminosity, |$L_{\rm R}\propto Q_{\rm jet}^{17/12}$|, for the theoretical jet model, we find |$L_{\rm R}\propto L_{\rm X, 1-10\;keV}^{1.4}$| for our model because the jet power |$Q_{\rm jet} \propto B_{\rm d}^{2} \propto \dot{M}^{0.8} \propto L_{\rm X, 1-10\,keV}$|. We note that the exact slope of the radio–X-ray correlation is also affected by the viscosity parameter (see top panel of Fig. 2). However, the slopes of the radio–X-ray correlation based on the jet–disc/corona model are generally steeper than 1.2 for typical values of the viscosity parameter of 0.05–0.2, and are much steeper than those predicted by the ADAF–jet model (e.g. ∼0.7, Yuan & Cui 2005). Therefore, our results with α ∼ 0.1 give a natural explanation for the radio–X-ray correlation of the outliers, and support the idea that this outliers track may be regulated by the disc–corona model, as pointed out by Coriat et al. (2011).
The radio–X-ray correlation based on the disc–corona model and the ADAF model can roughly explain the outliers track and the universal track respectively. The transition track may be regulated by a transition state of accretion flow. The radiative efficiency will increase very rapidly at a critical accretion rate, at which the low radiative efficiency of the ADAF will transit to the high radiative efficiency of the disc–corona. If this is the case, we expect that the X-ray luminosity will increase much faster than the radio luminosity if the accretion rate approaches the critical rate for disc transition (e.g. ∼0.01 Eddington accretion rate), which can roughly explain the flat transition track (see Fig. 3). It should be noted that different values of α will lead to different critical accretion rates for disc transition (e.g. Liu & Taam 2009; Qiao & Liu 2009). For a larger α, the critical accretion rate will be large. This may be the reason why the universal radio–X-ray correlation can extend to higher X-ray luminosities in some BHBs (e.g. GX 339−4).
Yuan (2001) has shown that a hot accretion flow may exist when the accretion rate is larger than a critical rate, and this hot accretion flow may be also radiatively efficient (e.g. Xie & Yuan 2012). If this is the case, the jet launched from the hot accretion flow may also provide a possible explanation for the outliers track of the radio–X-ray correlation. More direct calculations are expected to test this scenario, which is beyond the scope of our work. However, it is still unclear whether a hot accretion flow with very high accretion rates exists or not if considering the possible condensation of matter from the hot plasma to a cool disc, as Liu et al. (2007) found that the presence of a small cool disc is possible for luminosities greater than 0.1 per cent of the Eddington luminosity. Moreover, the broad Fe K emission lines observed in the bright hard state of some outliers do imply the existence of an inner cool disc, which cannot be explained by hot accretion flows. We suggest that high-sensitivity X-ray observations (e.g. XMM–Newton) of the cool disc for these outliers may shed light on this issue.
We thank the anonymous referee for constructive comments, which greatly improved the manuscript. Q.Wu thanks Henk Spruit and the Max Planck Institute for Astrophysics, where part of the work was done, for their hospitality. This work is supported by the National Basic Research Program of China (2009CB824800), New Century Excellent Talents in University (NCET-13-0238) and the NSFC (grants 11103003, 11133005 and 11173011).