Magnetic-reconnection-heated corona in active galactic nuclei: refined disc-corona model and application to broad-band radiation
Huaqing Cheng, B. F. Liu, Jieying Liu, Zhu Liu, Erlin Qiao, Weimin Yuan
aa r X i v : . [ a s t r o - ph . H E ] J un MNRAS , 1–14 (2019) Preprint 5 June 2020 Compiled using MNRAS L A TEX style file v3.0
Magnetic-reconnection-heated corona in active galactic nuclei:refined disc–corona model and application to broad-band radiation
Huaqing Cheng, , ⋆ B. F. Liu, , † Jieying Liu, , , Zhu Liu, Erlin Qiao, , and Weimin Yuan , Key Laboratory of Space Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China National Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences, Kunming 650011, P. R. China Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, P. R. China Center for Astronomical Mega-Science, Chinese Academy of Sciences, Beijing 100012, P. R. China
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
A long-standing question in active galactic nucleus (AGN) research is how the corona isheated up to produce X-ray radiation much stronger than that arising from the viscous heatingwithin the corona. In this paper, we carry out detailed investigations of magnetic-reconnectionheating to the corona, specifically, studying how the disc and corona are self-consistentlycoupled with the magnetic field, and how the emergent spectra depend on the fundamentalparameters of AGN. It is shown that diverse spectral shapes and luminosities over a broadbandpass from optical to X-ray can be produced from the coupled disc and corona within alimited range of the black hole mass, accretion rate and magnetic field strength. The relativestrength of X-ray emission with respect to optical/ultraviolet (UV) depends on the strengthof the magnetic field in the disc, which, together with accretion rate, determines the fractionof accretion energy transported and released in the corona. This refined disc–corona modelis then applied to reproduce the broad-band spectral energy distributions (SEDs) of a sampleof 20 bright local AGNs observed simultaneously in X-ray and optical/UV. We find that,in general, the overall observed broad-band SEDs can be reasonably reproduced, except forrather hard X-ray spectral shapes in some objects. The radiation pressure-dominant region,as previously predicted for the standard accretion disc in AGN, disappears for strong X-raysources, revealing that AGN accretion discs are indeed commonly stable as observed. Ourstudy suggests the disc–corona coupling model involving magnetic fields to be a promisingapproach for understanding the broad-band spectra of bright AGNs.
Key words: accretion, accretion discs – magnetic fields – galaxies: active – galaxies: nuclei.
It is widely believed that the radiation of the active galactic nu-clei (AGNs) is powered by the accretion of matter onto a super-massive black hole. The broad-band spectral energy distributions(SEDs) in high-luminosity AGNs are usually interpreted as radia-tion from a geometrically thin, optically thick accretion disc sand-wiched in a geometrically thick, optically thin corona. Specifically,the optical/UV spectra, which are often referred to as the ‘big bluebump’, are contributed by the thermal emissions from the cold disc(e.g. Shields 1978; Malkan 1983), while the power-law emissionsin hard X-ray band are produced by thermal electrons in the hot ⋆ E-mail: [email protected] † E-mail: bfl[email protected] corona through Comptonization of soft photons from the disc (e.g.Svensson & Zdziarski 1994; Magdziarz et al. 1998).Interaction between the cold disc and the hot coronais essential. Theoretical models elucidating the interactionshave been studied for decades (e.g. Haardt & Maraschi 1991,1993; Nakamura & Osaki 1993; Svensson & Zdziarski 1994;Poutanen & Svensson 1996; Dove et al. 1997; Liu et al. 2002a,2003; Wang et al. 2004; Cao 2009; You et al. 2012, 2016;Liu, Qiao & Liu 2016; Poutanen et al. 2018). Of these investiga-tions the radiation coupling between the disc and corona, that is theComptonization of disc photons in the corona and the coronal irra-diation to the disc, has been well considered. Heat conduction andenthalpy/mass flow between the hot corona and cold disc were alsotaken into account in some of the studies (e.g. Meyer et al. 2000b;Liu et al. 2002b). Detailed studies reveal that the corona is weaker © 2019 The Authors
H. Cheng et al. at higher accretion rates because efficient Compton cooling leads togas condensation to disc, predicting weak X-ray emission at highstates (e.g. Liu et al. 2002b; Meyer-Hofmeister et al. 2012). Even ifthe condensation of hot gas is neglected, viscous heating within thecorona is insufficient to produce the observed X-ray luminosity. Inorder to account for the strong X-ray radiation in luminous AGNs,it is often assumed that the energy released by disc accretion issomehow transported to the corona and radiates in X-rays. Hence,additional heating mechanism has been explored to keep the coronasufficiently strong to produce the observed X-ray luminosity.One of the most promising mechanisms for the coronal heat-ing is through magnetic field (e.g. Tout & Pringle 1992; Di Matteo1998; Miller & Stone 2000; Merloni & Fabian 2001; Liu et al.2002a, 2003; Merloni 2003; Wang et al. 2004; Cao 2009; You et al.2012). The concept of magnetic-reconnection heating is initiatedfrom solar flares (e.g. Shibata & Yokoyama 1999). The magneticfield is generated by the dynamo process in the disc and emergesinto the corona as a result of Parker buoyancy instability. Thestored magnetic energy is then released in the corona via magnetic-reconnection. In this manner, part of the accretion energy liberatedin the disc is transferred to the corona through magnetic field, heat-ing up the corona during reconnection and eventually radiated awayin X-rays.A self-consistent disc–corona model was proposed in theframe of magnetic-reconnection heating and disc evaporation(Liu et al. 2002a, 2003). In this model, the corona is assumed tobe heated by the aforementioned magnetic-reconnection, which isbalanced by inverse Compton (IC) scattering. Heat conducted fromthe hot corona to the disc surface leads to continuous evapora-tion, replenishing gas to the corona during its accretion towardsthe black hole. Specifically, the fraction of the energy liberated inthe corona, f , is physically determined by the coupling of the discand corona involving magnetic fields, which was usually taken asa free parameter in previous studies. In addition, the disc evapo-ration provides a natural and reasonable explanation for the masssupply of the hot corona. Therefore, such a model provides a phys-ical mechanism for both the formation and heating of the corona.Numerical calculations show that a significant fraction of accretionenergy can be transported to the corona, making it possible for thecorona to produce strong X-ray emissions as observed (Liu et al.2003; Qiao & Liu 2015). The model has been further developedto more general cases, and is expected to be applicable to the va-riety of observed spectra in AGNs (Liu, Qiao & Liu 2016). Nev-ertheless, these earlier studies mainly focused on X-ray emissionoriginating from a small region near the black hole, neglecting thecontribution of outer disc emission to the optical/UV band. Radia-tion from the entire accretion flow needs to be taken into accountwhen the overall spectrum is modelled. Also, the self-consistencyof the model needs to be improved to a higher precision. These re-finements are crucial for the model to be realistically applicable tomodelling observational data, and are one of the motivations of thiswork.On the other hand, we have recently conducted a study of thebroad-band SEDs for a well-selected sample of Seyfert 1 galax-ies, by making use of their simultaneous optical, UV and X-rayobservations with the Neil Gehrels
Swift
Observatory (Cheng et al.2019). The physical properties of the objects in the sample arefound to spread over a broad range, for example the black hole mass M BH ≈ – M ⊙ and Eddington ratio λ Edd ≡ L bol / L Edd ≈ . – , which provides a reasonable pre-requisite for the applica-tion of the disc–corona model.The motivation of this work is to understand quantitatively the physical origin of the broad-band SEDs of AGN, in the frame-work of the magnetic-reconnection-heated corona model (Liu et al.2002a, 2003; Liu, Qiao & Liu 2016). First, the model is furtherimproved and refined (see Section 2) to be more realistic and ap-plicable to compare directly with observations. Then, the model isapplied to reproducing the observed broad-band SED of the afore-mentioned AGN sample. We find that for the majority of the sam-ple objects, the observed SEDs can generally be reproduced by thedisc coupled with a strong magnetic reconnection-heated corona,though for some the X-ray spectra are too flat to be accounted for.The paper is organized as follows. In Section 2, the magnetic-reconnection-heated corona model is briefly described and illustra-tive calculations of the spectrum for typical AGNs are presented.In Section 3, the spectral modelling of the broad-band SEDs is de-scribed and main results for individual sources are shown and illus-trated. The discussion and conclusions are given in Section 4 and5. A magnetic-reconnection-heated corona model is proposed inLiu et al. (2002a, 2003) and revisited by Liu, Qiao & Liu (2016).In this model, it is assumed that a standard geometrically thin andoptically thick disc (Shakura & Sunyaev 1973) extends to the in-nermost stable circular orbit, sandwiched by a hot geometricallythick and optically thin, accreting corona. The magnetic field isassumed to be generated by the dynamo process operating in theaccretion disc. Under the influence of Parker instability, the mag-netic loops emerge into the corona and reconnect with other loops.In this manner, some fraction ( f ) of the accretion energy storedin the magnetic field is transferred into the corona and convertedinto thermal energy of the electrons. This energy is eventually ra-diated through the process of IC scattering. The disc is heated par-tially by accretion and partially by illumination from the corona. Inthe chromospheric layer between the disc and corona, thermal con-duction heats up gas, leading to the disc gas evaporating into thecorona and supplying for the steady accretion of coronal flow (e.g.Meyer et al. 2000a,b; Ró˙za´nska & Czerny 2000; Spruit & Deufel2002). The disc and corona are coupled through the magnetic field,the gas evaporation and coronal illumination to the disc. The struc-ture of disc and corona can be self-consistently determined by tak-ing into account this coupling, from which the emergent spectrumcan be calculated by Monte Carlo simulations.Note that the magnetic heating to coronal electrons is muchmore efficient than heating through Coulomb collisions with ions,the latter is neglected in our calculation of electron temperature.The ions in the corona are heated up to a higher temperature byviscously released energy as they accrete to the black hole, lead-ing to a two-temperature corona. Such an approximation is justi-fied by our calculations for bright AGN, and is, in principle, validfor sources with X-ray emission stronger than that from a pureadvection-dominated accretion flow (ADAF, Narayan & Yi 1994).The equations determining the structure of the accretion flowshave been list in detail in Liu, Qiao & Liu (2016), which are sum-marized as follows.Equations determining the coronal density and electron tem-perature are, B π V A ≈ kT e m e c τ cU rad , (1) MNRAS000
Observatory (Cheng et al.2019). The physical properties of the objects in the sample arefound to spread over a broad range, for example the black hole mass M BH ≈ – M ⊙ and Eddington ratio λ Edd ≡ L bol / L Edd ≈ . – , which provides a reasonable pre-requisite for the applica-tion of the disc–corona model.The motivation of this work is to understand quantitatively the physical origin of the broad-band SEDs of AGN, in the frame-work of the magnetic-reconnection-heated corona model (Liu et al.2002a, 2003; Liu, Qiao & Liu 2016). First, the model is furtherimproved and refined (see Section 2) to be more realistic and ap-plicable to compare directly with observations. Then, the model isapplied to reproducing the observed broad-band SED of the afore-mentioned AGN sample. We find that for the majority of the sam-ple objects, the observed SEDs can generally be reproduced by thedisc coupled with a strong magnetic reconnection-heated corona,though for some the X-ray spectra are too flat to be accounted for.The paper is organized as follows. In Section 2, the magnetic-reconnection-heated corona model is briefly described and illustra-tive calculations of the spectrum for typical AGNs are presented.In Section 3, the spectral modelling of the broad-band SEDs is de-scribed and main results for individual sources are shown and illus-trated. The discussion and conclusions are given in Section 4 and5. A magnetic-reconnection-heated corona model is proposed inLiu et al. (2002a, 2003) and revisited by Liu, Qiao & Liu (2016).In this model, it is assumed that a standard geometrically thin andoptically thick disc (Shakura & Sunyaev 1973) extends to the in-nermost stable circular orbit, sandwiched by a hot geometricallythick and optically thin, accreting corona. The magnetic field isassumed to be generated by the dynamo process operating in theaccretion disc. Under the influence of Parker instability, the mag-netic loops emerge into the corona and reconnect with other loops.In this manner, some fraction ( f ) of the accretion energy storedin the magnetic field is transferred into the corona and convertedinto thermal energy of the electrons. This energy is eventually ra-diated through the process of IC scattering. The disc is heated par-tially by accretion and partially by illumination from the corona. Inthe chromospheric layer between the disc and corona, thermal con-duction heats up gas, leading to the disc gas evaporating into thecorona and supplying for the steady accretion of coronal flow (e.g.Meyer et al. 2000a,b; Ró˙za´nska & Czerny 2000; Spruit & Deufel2002). The disc and corona are coupled through the magnetic field,the gas evaporation and coronal illumination to the disc. The struc-ture of disc and corona can be self-consistently determined by tak-ing into account this coupling, from which the emergent spectrumcan be calculated by Monte Carlo simulations.Note that the magnetic heating to coronal electrons is muchmore efficient than heating through Coulomb collisions with ions,the latter is neglected in our calculation of electron temperature.The ions in the corona are heated up to a higher temperature byviscously released energy as they accrete to the black hole, lead-ing to a two-temperature corona. Such an approximation is justi-fied by our calculations for bright AGN, and is, in principle, validfor sources with X-ray emission stronger than that from a pureadvection-dominated accretion flow (ADAF, Narayan & Yi 1994).The equations determining the structure of the accretion flowshave been list in detail in Liu, Qiao & Liu (2016), which are sum-marized as follows.Equations determining the coronal density and electron tem-perature are, B π V A ≈ kT e m e c τ cU rad , (1) MNRAS000 , 1–14 (2019) agnetic-reconnection-heated corona in AGNs k T e ℓ c ≈ γγ − n c kT e (cid:18) kT e µ m H (cid:19) / . (2)Eq.(1), describes the energy balance in the corona. The magneticloops emerge at Alfvén speed V A ≡ p B / πµ m H n c , bringing mag-netic energy flux B π V A into corona and being radiated by IC scat-tering. Here T e , n c , and τ are the coronal temperature, density, andmodified scattering depth, respectively; U rad is the energy densityof soft photons. Eq.(2) describes the energy balance of conduction-induced evaporation, where ℓ c is the length of magnetic loops in thecorona. Eqs.(1) and (2) determine the coronal density and electrontemperature for given soft photon field and magnetic field, whichare coupled with the thin disc.Constants in above equations are the Boltzmann constant k = . × − erg K − , the thermal conduction coefficient k = − erg cm − s − K − / , the ratio of specific heats γ = / ,the molecular weight for pure hydrogen plasma µ = . , the massof hydrogen atom m H , the mass of electron m e and the light speed c . The length of magnetic loops is set to be the vertical scale heightof corona, ℓ c ≈ H c , as the loops emerge from the disc and thenexpand in the corona, where H c ≈ R for an optically thin, two-temperature hot accretion flow. The scattering depth is modified bythe average optical depth for the isotropic incident photons under-going upscattering in a sandwich corona geometry and the multiplescattering of soft photons, τ = λ t n c σ T H c , where λ t is larger than 1.The energy density of soft photons ( U rad ) and the magneticfield ( B ) in Eq.(1) are determined by the coupling between the discand corona. The soft photons are contributed by disc accretion andcoronal illumination, U rad = c ( GM Û M π R " − (cid:18) R S R (cid:19) / − B π V A ) + U re , (3)where the first term on the right-hand side of the equation denotesthe density of net energy from accretion gain and magnetic loss;The second term, U re , denotes energy density originating from lo-cal corona illumination. When the hard photons from the coronahave been reprocessed in the disc and emit into the corona, the en-ergy density of these soft photons is proportional to the energy den-sity of corona emission, with a coefficient relevant to the fractionof downward propagation and reflection albedo ( ∝ ( − a ) giventhat the IC scattering is isotropic). Since the coronal radiation en-ergy is eventually from magnetic energy, U re can be approximatedas U re = . λ u B π , where a = . is adopted, and λ u includesthe deviation from isotropic scattering and the ratio of the speed ofmagnetic energy release (at Alfvén speed) and radiation (at lightspeed). The value of λ u is around unit in order of magnitude, whichis determined from Monte Carlo simulation in our calculations.The strength of magnetic field is parametrized with a magneticequipartition coefficient β , which is defined as the ratio of the gasplus radiation pressure to the magnetic pressure ( P B = B π ), B π = β (cid:18) ρ d kT d µ m p + aT (cid:19) , (4)therefore, the magnetic field can be determined once the disc den-sity and mid-plane temperature ( ρ d , T d ) are derived from the discsolutions for given mass of black hole, accretion rate, and viscos-ity. It should be noted that a fraction ( f ) of the accretion energy is taken away from the disc, f ≡ B π V A3 GM Û M π R (cid:20) − (cid:16) R S R (cid:17) / (cid:21) , (5)where f is an implicit function of M , Û M , α, β as from Eq.(4). Thisleads to corresponding change in the equations of energy conser-vation and angular momentum conservation for a steady accretionflow, that is, GM Û M ( − f ) π R " − (cid:18) R S R (cid:19) / = σ T ( κ es + κ ff ) H d , (6) Û M ( − f ) Ω " − (cid:18) R S R (cid:19) / = π H d α P d , (7)where the coefficients for the electron scattering and absorption are κ es = . g − , κ ff = . × ρ d T − / cm g − ; H d is thedisc thickness, H d = c s / Ω = p P d / ρ d / Ω , and P d is the total pres-sure exerted by gas, radiation and magnetic field, which can be ex-pressed as function of T d , ρ d .Eqs. (1)–(7) are the complete set of equations for seven un-knowns, that is the electron temperature T e and density n c in thecorona, the mid-plane temperature T d and density ρ d in the disc,the magnetic field strength B , the energy fraction released in thecorona f and the soft photon energy density U rad emitted from thedisc. Given the black hole mass M , accretion rate Û M , viscosity pa-rameter α and magnetic coefficient β , the equations from (1) to (7)can be numerically solved for different radius with initial value for λ t = . and λ u = . . With the obtained disc and corona param-eters we calculate the emergent spectra of the disc–corona systemby Monte Carlo simulations. We further check the self-consistencyfrom the Monte Carlo results, that is, (1) whether the luminosity ofdownward-scattered photons is approximately equal to irradiationluminosity assumed in the structure calculation (relevant to λ u ),and (2) whether the luminosity of escaped photons, composed ofnon-scattered, upward-scattered, and reflected photons, is approxi-mately equal to the liberated rate of total gravitational energy. Weadjust the two parameters λ t and λ u in the iterative computation un-til the above two conditions are fulfilled. This iterative calculationmodifies the density of re-processed soft photons and the scatter-ing depth adopted in calculating the disc and corona parameters byMonte Carlo simulations. For a detailed explanation on the iterativeprocedure, see Liu et al. (2003).In this work, significant improvements on our disc–coronamodel have been made. First, the radiation from the the whole sta-ble disc–corona system is taken into account. The outer boundaryof the disc–corona system is set to the self-gravity radius R sg ofseveral hundreds to thousand R S (Laor & Netzer 1989). Whilst inthe previous work only the inner part of R ≤ R S was consid-ered when the study focused on X-ray emission. This is crucialfor a more realistic description of the emergent spectra in the op-tical band, as this portion of the broad-band SED is mainly con-tributed by the thermal emissions at larger distances. Second, thelength of magnetic loop is set as l c ≈ R , which is more reason-able for a geometrically thick, two-temperature hot accretion flows(e.g. Narayan & Yi 1994, 1995a,b; Yuan & Narayan 2014) than aconstant value of R S for all distances adopted in previous stud-ies. Third, the precision of the energy conservation is substantiallyimproved by implementing the iterative procedures at each local- MNRAS , 1–14 (2019)
H. Cheng et al. ized disc region rather than taking the whole disc as one region inprevious studies. Fourth, reflection (for an albedo of 0.2) is takeninto account in the iterative calculation. With these refinements, themodel is now more self-consistent and strictly energy-conservative(both locally and globally), which can be applied to explain the ob-servational data over a wide energy band from the optical/UV toX-rays.We point out that the X-ray emission is dominantly con-tributed by the innermost corona since it is powered by the re-leased gravitational energy ( ∝ / R ), even though the corona ex-tends to large distances. Detailed study of hot corona (Liu et al.2017) shows that the bulk X-ray emissions concentrate in a regionwithin R S − R S . The disc–corona model is not contradictorywith the observed ‘compact’ size measured by X-ray radiation (e.g.Reis & Miller 2013; Fabian et al. 2015; Chartas et al. 2016) . In our model, the viscosity parameter α is fixed to the value of . , as have been done in previous investigations (Liu et al. 2003;Liu, Qiao & Liu 2016). The structure and emergent spectrum ofthe disc–corona depend on the black hole mass m (in units of solarmass M ⊙ ), the mass accretion rate Û m (in units of the Eddingtonrate Û M Edd = . × MM ⊙ g s − for an efficiency η = . ) and themagnetic equipartition coefficient β . The effects of the parameter m , Û m , and β on the disc–corona structure and further the broad-band spectral features are investigated in details in the followingsubsections. The main spectral features and their dependence on m , Û m , and β are summarized in the last two paragraphs of thissubsection. It has been speculated that the spectrum of an object with givenblack hole mass is solely determined by the mass accretion rate.The variation of the spectrum is then caused by a change in theaccretion rate. With magnetic field involved in transporting energyfrom the disc to the corona, an additional parameter, namely theequipartition coefficient β , also plays a role in determining therelative strength of radiations from the disc and the corona. To studythe effect of accretion rate, we take values of m = and β = , and show the structure of accretion flows and radiation spectrain Fig. 1 and Fig. 2.In the panel (a)–(d) of Fig. 1, the radial distributions of ef-fective temperature in the disc, the coronal electron temperature,the coronal modified optical depth, and fraction of the energy re-leased in the corona are plotted for the mass accretion rate Û m = . (red), . (green), and . (blue), respectively. An obvious featureis that the coronal temperature, optical depth, and energy fractionturn over at some distance, with a larger extent at higher accretionrates. This is because the radiation pressure dominates over the gaspressure in the inner disc region. With the increase of the accre-tion rate, the radiation pressure-dominant region becomes large,leading to the turn-over at large distance. As the radiation pres-sure in the disc solution does not depend on the accretion rates, thestrength of magnetic field does not vary with Û m under the equipar-tition assumption. Therefore, the energy fraction ( f ) transferred tothe corona decreases with increase of accretion energy measuredby Û m . Consequently, the disc radiation increases, while the coronaldensity (equivalent to optical depth) and temperature are lower dueto less heating (small f ) and more cooling (caused by strong disc radiation) at higher accretion rates. As shown in panel (b) and (c) ofFig. 1, the coronal electron temperature decreases from ∼ . × to ∼ × K, and the optical depth decreases from ∼ . to ∼ . when the accretion rate increases from . to . . We note that theradial distribution of the effective temperature is not similar to thatof corona. This is because roughly half of the corona radiation goesback to illuminate the disc, no matter how much accretion energy istransferred to the corona. The inclusion of illumination could alsolead to a quantitative deviation from the above analyses but couldnot change the variation trend with Û m .The spectrum is determined by the structure of the two-phaseaccretion flow. With the dependence of the disc–corona structureon the accretion rate, we expect that the ratio of X-ray emission todisc decreases with increase of Û m as a result of decrease in energyfraction f ; The optical/UV spectrum shifts to higher frequenciesdue to more accretion energy released in the disc; The decreaseof coronal temperature and optical depth leads to a large decreaseof Compton y -parameter, thereby significantly softens the X-rayspectrum. The detailed spectral index from IC scattering in opti-cally thin corona is α ≈ ln τ es / ln A , where the amplification factor A is determined by electron temperature, A = + θ e + θ e2 with θ e = kT e m e c . Such predictions are confirmed by our computationalresults, as illustrated in Fig. 2. That is, from Û m = . to . , the X-ray spectrum becomes weaker and softer, and the disc componentbecomes stronger and shifts to higher frequency. Such a feature isin agreement with observations in AGN (e.g. Shemmer et al. 2006;Vasudevan & Fabian 2007; Jin et al. 2012). Nevertheless, the ob-tained spectra as shown in Fig. 2 are still too soft compared to theobservational X-ray spectra in some AGNs. A stronger magneticfield is expected to produce a harder X-ray spectrum for a givenaccretion rate, which is discussed in the following subsection. The magnetic equipartition coefficient, β , is defined as the ratio ofthe sum of the gas and radiation pressure to the magnetic pressure.A large value of β implies a weak magnetic field. For different β values, we calculate the disc–corona structure and then the emer-gent spectrum. The radial distributions of effective temperature inthe disc, the coronal electron temperature, the coronal modified op-tical depth, and the fraction of the energy released in the corona areplotted in Fig. 3 for fixed m = , Û m = . , and different magneticequipartition coefficient β = (blue), (green), and (red). The emergent spectra are shown in Fig. 4. It can be seenthat the increase of the magnetic field (equivalent to a decrease in β ) results in an increase in the coronal temperature, optical depth(density), and hence strong X-ray radiation, whilst the disc emis-sion decreases. This is easy to understand since a stronger mag-netic field means more energy can be transported into the corona,which evaporates more gas to a higher temperature and producesstronger Compton radiations. In this manner, the magnetic fieldcan significantly affect the coronal temperature and density, andthereby changes the spectrum significantly, in particular, the hardX-ray spectral shape. In contrast, the effective temperature of thedisc depends only weakly on β because the energy released in thecorona partially comes back to heat the disc through irradiation.We find that for β = and Û m = . , nearly all the accretionenergy is transferred to the corona by magnetic field ( f ≈ ), andthe disc is heated by the coronal irradiation rather than viscosity,which is referred to a passive disc. Further increase of magneticfield does not change the results. We note that critical value of β MNRAS , 1–14 (2019) agnetic-reconnection-heated corona in AGNs Figure 1.
Effect of mass accretion rate on the radial distribution of the effective temperature in the thin disc (panel a), the electron temperature in the corona(panel b), the modified optical depth in the corona (panel c), and the fraction of energy released in the corona (panel d). The black hole mass and magneticparameter are fixed to m = and β = , while the accretion rate changes from 0.05, 0.1, to 0.5. The coronal temperature ( T e ), optical depth ( τ ),and energy fraction ( f ) decrease dramatically when the disc changes from gas pressure-dominant at large distances to radiation pressure-dominant at smalldistances. Figure 2.
Effect of mass accretion rate on the emergent spectra from the disc–corona system. Parameters are the same as Fig.1. The luminosity has been scaledto the bolometric luminosity. for becoming a passive disc ( f ≈ ) depends on the mass accretionrates. For instance, the passive disc solution ( f ≈ ) is satisfiedwhen β decreases to ∼ at an accretion rate of Û m = . , com-pared to β = at Û m = . ; However, in the case of Û m = . ,this can be achieved for β = .The magnetic field also affects the radial structure of disc andcorona, as shown in the radial distribution of coronal temperatureand optical depth in Fig. 3. In the case of strong magnetic field, the disc can be gas pressure-dominant at almost all radii since a largeamount of energy is transferred to the corona (say, β = ); whilein the case of weak magnetic field ( β = ), only ∼ per centof the accretion energy is released in the corona, and the radiationpressure-dominant region in the disc extends to a distance of R ∼ R S . The size of the radiation pressure-dominant region in thedisc leads to a corresponding change of the coronal temperatureand optical depth along distance, as discussed in above subsection. MNRAS , 1–14 (2019)
H. Cheng et al.
Figure 3.
Effect of magnetic field on the radial distribution of the effective temperature in the thin disc (panel a), the electron temperature in the corona (panelb), the modified optical depth in the corona (panel c), and the fraction of energy released in the corona (panel d). The black hole mass and magnetic parameterare fixed to m = and Û m = . , while the magnetic parameter changes from 100, 200, to 1000. The coronal temperature ( T e ), optical depth ( τ ), and energyfraction ( f ) decrease dramatically when the disc changes from gas pressure-dominant at large distances to radiation pressure-dominant at small distances. Figure 4.
Effect of magnetic field on the emergent spectra from the disc–corona system. Parameters are the same as Fig.3. The luminosity has been scaled tothe bolometric luminosity.
The black hole mass in AGN spans typically from to so-lar mass. This can lead to significant differences in the effectivetemperature of a standard disc ( T eff ∝ m − / ) for individual AGN.We study the influence of black hole mass on both the disc andcorona structure and the emergent spectrum. In the panels (a)–(d)of Fig. 5, we plot the effective temperature in the disc, the elec-tron temperature, the modified optical depth and the energy frac- tion in the corona as a function of radius for the black hole mass m = (red), (green), and (blue), with given Û m = . and β = . Similar to the standard Shakura–Sunyaev disc, the effec-tive temperature of the disc is lower for a higher black hole mass,and the radiation shifts to lower frequencies as shown in Fig. 6. Onthe other hand, the effects of the black hole mass on the distributionof coronal electron temperature T e and modified optical depth τ cannearly be neglected. This is a common nature of optically thin, hotaccretion flows, no matter it is an ADAF, a friction-heated corona, MNRAS , 1–14 (2019) agnetic-reconnection-heated corona in AGNs Figure 5.
Effect of black hole mass on the radial distribution of the effective temperature in the thin disc (panel a), the electron temperature in the corona (panelb), the modified optical depth in the corona (panel c), and the fraction of energy released in the corona (panel d). The accretion rate and magnetic parameterare fixed to Û m = . and β = , while the black hole mass changes from , , to solar mass. The coronal temperature ( T e ), optical depth ( τ ),and energy fraction ( f ) decrease dramatically when the disc changes from gas pressure-dominant at large distances to radiation pressure-dominant at smalldistances. Figure 6.
Effect of black hole mass on the emergent spectra from the disc–corona system. Parameters are the same as Fig.5. The luminosity has been scaledto the bolometric luminosity. or magnetic-reconnection-heated corona. Consequently, the hardX-ray spectral index is independent of the black hole mass. Com-bined with the effect of m on the disc radiation, the SED from discand corona shifts to lower frequency for higher black hole mass, asshown in Fig. 6.To summarize, the theoretical spectrum from the disc andmagnetic-reconnection-heated corona consists of two primary por-tions, that is the optical-to-EUV radiation is dominated by the mul- ticolor blackbody emission from optically thick disc, while the ra-diation from EUV to hard X-ray band is dominated by the Comp-tonization process of the disc photons, forming a power-law spec-trum in the hard X-ray band below ∼ keV or so. The emer-gent spectrum from the disc–corona varies with three physical pa-rameters inherent in the model, that is the black hole mass m , themass accretion rate Û m , and the magnetic equipartition coefficient β . The X-ray spectrum is steeper at higher accretion rate, implying MNRAS , 1–14 (2019)
H. Cheng et al. a stronger disc (relative to the corona) when the source is brighter.Similarly, the X-ray emission increases with the increase of mag-netic field, meanwhile the disc emission decreases. Compared tothe effect of accretion rate, the spectral features in the X-ray bandare more sensitive to the variations in β , since it directly deter-mines the energy released in the corona. The black hole mass onlyaffects the disc component. The higher the black hole mass is, thelower frequency of disc radiation peaks at.Such a model provides a physical mechanism to interpret thevariety of observational AGN spectra with limited free parame-ters. In particular, the magnetic buoyant instability and magnetic-reconnection scenario solves the energy shortage of the corona,making it possible to produce sufficiently strong X-ray emissionas observed in AGN. The energy fraction liberated in the corona, f , is self-consistently determined by the magnetic behavior and thecoupling of the disc and the corona, which was usually taken as afree parameter without a physical mechanism in previous studies.The predicted spectral features are in qualitative agreement withobservational characteristics in AGN. Modelling of observed SEDfor a sample of AGN is performed in the next section. On the basis of above theoretical investigations, we are ready tomodel the observational spectra of AGN from just the two funda-mental parameters – the black hole mass and accretion rate – plusthe magnetic equipartition coefficient. In the following spectral fit-ting, m , Û m , and β are set to be as free parameters, which can beconstrained from the shapes and luminosities of the observed SEDsof individual AGN. In a recent study, we obtained the broad-band SEDs of a sample of local AGNs (Cheng et al. 2019). The sample is adopted for spec-tral modelling in this investigations given the following advantages.First, the optical, UV, and X-ray data are obtained by simultaneousobservations with the UV/Optical telescope (UVOT, Roming et al.2005) and X-ray telescope (XRT, Burrows et al. 2005) onboard theNeil Gehrels Swift
Observatory (Gehrels et al. 2004), which are es-sential for SED studies. Moreover, the sample was selected to belargely free from dust reddening using the Balmer decrement asan indicator. Secondly, photometry of the nuclei was measured inan elaborate way by eliminating the host galaxy starlight. Thirdly,their optical spectral parameters were measured accurately in a ho-mogenous way by taking advantage of the Sloan Digital Sky Sur-vey (SDSS, York et al. 2000) spectroscopic observations of all thesample objects (see Cheng et al. 2019, for details). Therefore, thevirial black hole masses ( M vir ) were reliably measured from thesingle-epoch spectral parameters, and thus the Eddington ratios forthe sample objects.The luminosities of the sample objects in the optical, UV, andX-ray bands are taken from Cheng et al. (2019). It should be notedthat the optical/UV luminosities are derived from the measuredphotometric fluxes for disc emission assuming an inclination an-gle of ◦ of the disc normal with respect to the line of sight (seeCheng et al. 2019, Section 4.1); whereas the X-ray luminosities arederived assuming isotropic emission of the optically thin corona.To facilitate the modelling of the SED, the uncertainties of some of the measurements in Cheng et al. (2019) are estimated or re-fined. First, the errors on the measurement of the luminosities inoptical/UV band are quantitively reevaluated. For the error on themeasurement of the luminosities in the optical ( u, b, v ) band, weinclude the standard measurement error, the systematic error due tothe calibration and an extra error of per cent to take into accountthe uncertainty in performing the AGN–host galaxy decomposition,which is estimated via the comparison of our results with those ofKoss et al. (2011) for the seven same objects in the two samples.For UV ( uvw1, uvm2, and uvw2 ) bands, the standard error fromPoisson noise is combined with the flux calibration error and anadditional error of per cent on the uncertainty associated withthe possible host galaxy contribution, which is estimated based onthe results in Vasudevan et al. (2009). The uncertainty associatedwith the estimation of the dust extinction effect is not consideredas the sample is selected in a way that there is little or no intrinsicdust reddening as indicated by the Balmer decrement of the broademission lines.It should be noted that the soft X-ray data ( . – keV) isnot taken into account in the spectral fitting. This is because thisenergy band is often contributed by the so-called soft X-ray ex-cess, whose origin is yet unknown (e.g. Magdziarz et al. 1998;Done et al. 2012) and whose contribution to the total accretion en-ergy budget is negligible (see Cheng et al. 2019, for details). There-fore, only the X-ray spectra in the – keV band and the opti-cal/UV data are utilized. In some of the objects the 2–10 keV spec-tra are of low quality due to the relatively low effective area of XRTabove 2 keV.Since we consider our disc–corona model to be applicable inthe range of Û m , that is accretion rate in units of the Eddington rate,from . to (see below), three of the sample objects with signif-icantly lower Eddington ratios ( λ Edd << . ) are excluded. Thusthe final sample used in the following data modelling comprisesonly objects, whose basic data are listed in Table 2 and theirbroad-band SEDs are shown in Fig. 7. The fitting procedure is implemented by comparing the observedbroad-band spectra with theoretical disc–corona SEDs. For thispurpose, we construct a grid of models covering wide ranges inthe parameter space of the black hole mass, mass accretion rate andmagnetic equipartition coefficient. Specifically, a grid of
496 400 models is constructed, with the parameter ranges of . – . for log ( M / M ⊙) , . – . for Û m and . – . for / β (see Table1). The range of black hole mass is chosen based on the distributionof the virial masses of the sample considering their typical system-atic uncertainty of . dex (Grier et al. 2017). We note that for eachindividual AGN, the black hole mass is allowed to vary within anuncertainty range of ± . dex of the virial mass only, if the fittedmass value (with its uncertainty taken into account) exceeds sig-nificantly the allowed range. The range of the mass accretion rateof . – . is chosen to validate the thin disc assumption, beyondwhich the thin disc could be replaced by an ADAF (e.g. Ichimaru1977; Rees et al. 1982; Narayan & Yi 1995a,b; Abramowicz et al.1995; Yuan & Narayan 2014) or a slim disc (e.g. Abramowicz et al.1988; Wang et al. 1999; Ohsuga et al. 2002). Here the critical ac-cretion rate for a slim disc to occur is assumed to be (rather than . ) since part of the accreted energy is carried away by the mag-netic field. The range of the magnetic equipartition coefficient β is set in such a way that the energy fraction f varies from to .For each set of the model parameters, the χ statistic is cal- MNRAS000
496 400 models is constructed, with the parameter ranges of . – . for log ( M / M ⊙) , . – . for Û m and . – . for / β (see Table1). The range of black hole mass is chosen based on the distributionof the virial masses of the sample considering their typical system-atic uncertainty of . dex (Grier et al. 2017). We note that for eachindividual AGN, the black hole mass is allowed to vary within anuncertainty range of ± . dex of the virial mass only, if the fittedmass value (with its uncertainty taken into account) exceeds sig-nificantly the allowed range. The range of the mass accretion rateof . – . is chosen to validate the thin disc assumption, beyondwhich the thin disc could be replaced by an ADAF (e.g. Ichimaru1977; Rees et al. 1982; Narayan & Yi 1995a,b; Abramowicz et al.1995; Yuan & Narayan 2014) or a slim disc (e.g. Abramowicz et al.1988; Wang et al. 1999; Ohsuga et al. 2002). Here the critical ac-cretion rate for a slim disc to occur is assumed to be (rather than . ) since part of the accreted energy is carried away by the mag-netic field. The range of the magnetic equipartition coefficient β is set in such a way that the energy fraction f varies from to .For each set of the model parameters, the χ statistic is cal- MNRAS000 , 1–14 (2019) agnetic-reconnection-heated corona in AGNs Table 1.
Ranges and steps of the parameters adopted to construct the gridof models used for SED fittingParameter Min–Max values Step Number of steps log ( M / M ⊙) . − . .
02 146 Û m . − . .
01 1001 / β . − .
034 0 .
001 34 culated using the SED data. The best fit is obtained by searchingfor the global minimum of the resulting χ grid over the entireparameter space specified as in Table 1. The confidence range ofa given parameter at the 90 per cent level for one interesting pa-rameter is derived as the boundaries around the best-fitting valuebeyond which the χ increment ∆ χ ≥ . (with the other pa-rameters freely fitted). To assess the statistical goodness-of-fit, thenull hypothesis probability p of the best fit is also derived from the χ distribution. The best-fitting parameters of the disc–corona model are listed inTable 2 and the fitted SEDs are overplotted in Fig. 7. As can beseen, the models can generally reproduce the overall SEDs acrossthe optical/UV and X-ray bands reasonably well. For the major-ity of the sample, the model fits are acceptable (or marginally) inthe statistical sense ( p ≥ . in 13), although in several objectsthe X-ray spectral quality is too poor to yield meaningful con-straint. In the remaining objects, the fits are poor as the modelapparently fails to account for the spectral slopes in either X-ray(four objects) or optical/UV (three objects). For the overall sample,the modelled 2–10 keV X-ray photon indices are in the range of Γ = . – . (without taking into account the disc-reflection com-ponent). These values, however, appear to be steeper than thoseof many Seyferts which have Γ ∼ . – . as directly observedand . – . for the primary continuum when the reflection com-ponent is accounted for (e.g. Nandra & Pounds 1994; Corral et al.2011; Rivers et al. 2013; Lubi´nski et al. 2016; Ricci et al. 2017), al-though AGNs with relatively higher Eddington ratios tend to showsteeper Γ (e.g. Shemmer et al. 2006, 2008; Brightman et al. 2013).See Section 4.1 for further discussion.The distribution of the fitted mass accretion rates is plottedin Fig. 8. It shows that the accretion rates centre around . ,with a minimum . and a maximum . . In particular, Û m issmaller than . in out of the 20 objects. In only two sourcesrelatively larger Û m are yielded, namely Ton 1388 ( . ) and RXJ1355.2+5612 ( . ). Therefore, the distribution of Û m from thespectral fitting supports the basic assumption of our model, thatthe gravitational energy is efficiently released via the standardShakura–Sunyaev disc ( η ≈ . ).The distribution of β is plotted in Fig. 9, showing a rangefrom ∼ to ∼ with a median of . The relative small val-ues of β suggest a strong magnetic-reconnection-heated coronaresiding in most of the sources. The energy fraction released inthe corona (averaged along radius) exceeds 80 (90) per cent in 15(12) objects, as listed in Table 2. In such cases, the intrinsic discemission is so weak that the reprocessed radiation from the down-ward Compton scattering completely dominates the energy densityof the soft photons. This implies that the radiation pressure in thedisc (mid-plane) is much weaker than that in a standard disc forthe same accretion rate. Specifically, from the spectral fitting we find that the innermost radius of the disc region where the radiationpressure is indeed less than or comparable to the gas pressure cangenerally extend down to R ≤ ∼ R S for f > ∼ . and all the waydown to the last stable orbit for f close to (the exact radius de-pends also on Û m ). An important consequence is that the radiationpressure-induced disc instability is ruled out in these AGN. Thisexplains why there is no observational evidence for the instabilitywhich is predicted to exist in a standard disc around supermassiveblack hole. The magnetic field plays a key role in both producingstrong X-ray emission and stabilizing the accretion flows in AGN.For the remaining five objects, weaker X-ray emission (rela-tive to the optical/UV component) is found and a weaker coronasolution is deduced. Less energy is carried away by the mag-netic field, leading to a decrease in the coronal energy fraction f .Among these, three (RX J1209.8+3217, PG 1307+085, and RXJ1702.5+3247) have f = . – . and thus 20–30 per cent of theaccretion energy is released in the disc; the radiation pressure-dominant region begins to appear in the inner disc. For the othertwo objects, which have the highest accretion rates among the sam-ple, Ton 1388 ( Û m ≈ . and L bol ≈ . × erg s − ) and RXJ1355.2+5612 ( Û m ≈ . and L bol ≈ . × erg s − ), about halfof the accreted energy ( f ∼ . ) is released in the disc. The innerregions of their discs (up to radii R < ∼ R S ) are completelydominated by radiation pressure, although the reduced heating andhence the disc radiation can mitigate somewhat the photon-trappingeffect therein. Compared to other AGNs in the sample, a strong discand relatively weak corona scenario is favoured for them.The black hole mass can, in principle, also be determinedfrom the spectral modeling. It would be interesting to compare thethus inferred masses with those derived independently via the virialmethod from the single-epoch SDSS optical spectra in our previouswork (Cheng et al. 2019), as given in Table 2. For the majority ( )of the sample objects, the fitted mass values are broadly consistentwith M vir within their mutual uncertainties (0.5 dex for M vir andthe per cent error for the fitted mass). A comparison of the fittedmasses with M vir for these 17 objects is shown in Fig.10. Whereasfor the remaining three objects, the mass value just reaches the endof the allowed range (reaching the higher end log M vir + . forall). We note that this bias of fitting the mass in these objects arisesmainly from matching their relatively flat/red optical/UV slopes ob-served with the turnover of the disc emission bump, which requiresa higher black hole mass so as to shift the peak to a lower frequency(see Section 4.1 for discussion). In several objects (particularly SBS 1136+594, Mrk 1310, Mrk290, and Ton 730), the observed X-ray spectra are too flat/hard( Γ ≈ . − . ; Cheng et al. 2019) to be accounted for by themodel. Whereas in several others (e.g. RX J1007.1+2203, Mrk 142,and KUG 1618+410) the X-ray spectral data are of too low qual-ity to constrain strongly the model. For the overall sample, ourmodel generally yields Γ = . – . , which appear to be steeperthan those often observed in typical Seyfert galaxies ( Γ ∼ . – . ; e.g. Nandra & Pounds 1994; Dadina 2007, 2008; Corral et al.2011; Lubi´nski et al. 2016). Here we discuss this discrepancy fromboth observational and theoretical considerations.The observed X-ray spectra of some AGN may show a com-plex shape and deviate significantly from the original form of the
MNRAS , 1–14 (2019) H. Cheng et al.
Figure 7.
Broad-band SED data of the sample objects in optical/UV and X-ray observed simultaneously with
Swift
UVOT and XRT (taken from Cheng et al.2019), overplotted with the best-fitting SEDs of the magnetic-reconnection-heated disc–corona model (solid line). The best-fitting parameters (black holemass, accretion rate, and magnetic equipartition coefficient) are labelled, as well as the derived fraction of energy released in the corona ( f ).MNRAS , 1–14 (2019) agnetic-reconnection-heated corona in AGNs Figure 7 (continued)
Table 2.
Fitting results and the virial black hole masses of sample objectsObject log ( M vir / M ⊙) log ( M / M ⊙) Û m β χ ν p f (1) (2) (3) (4) (5) (6) (7) (8)Mrk 1018 . . + . − . . + . − . + /
16 0 .
97 0 . Mrk 493 . . + . − . . + . + − /
11 0 .
93 0 . Mrk 1392 . . + . − . . + . − . + /
17 0 .
90 0 . Ton 1388 . . + . − . . + . − . + − /
11 0 .
87 0 . CBS 126 . . + . − . . + . − . + / .
78 0 . RX J1007.1+2203 . . + . − . . + . + − / .
70 0 . Mrk 142 . . + . − . . + . − . + / .
70 0 . MCG 04-22-042 . . + . − . . + . − . + /
27 0 .
63 0 . Mrk 705 . . + . − . . + . − . + /
10 0 .
53 0 . PG 1138+222 . . + . − . . + . − . + /
16 0 .
39 0 . PG 1307+085 . . + . − . . + . − . + − /
15 0 .
38 0 . Mrk 771 . . + . − . . + . − . + − /
13 0 .
25 0 . KUG 1618+410 . . + . − . . + . − . + / .
20 0 . Ton 730 . . + . − . . + . − . + /
12 0 .
07 0 . RX J1209.8+3217 . . ( b ) . + . − . + − / .
06 0 . RX J1702.5+3247 . . ( b ) . + . − . + − / .
06 0 . Mrk 1310 . . + . − . . + . − . + / .
02 0 . Mrk 290 . . + . − . . + . − . + /
17 0 .
02 0 . RX J1355.2+5612 . . ( b ) . + . − . + − / .
005 0 . SBS 1136+594 . . + . − . . + . − . + /
33 0 .
004 0 . Notes: Column (1): NED name of sample objects; column (2): virial black hole masses obtained from Cheng et al.(2019); column (3): black hole masses; those reaching the boundary of the allowed range ( log M vir ± . ) are markedwith ’b’; column (4): accretion rate; column (5): magnetic equipartition coefficient; column (6): reduced χ ν ; column(7): null hypothesis probability; and column (8): averaged fraction of energy released in corona.Uncertainties quoted are at the 90 per cent level for one interesting parameter; no uncertainties given mean that theparameters cannot be constrained.MNRAS , 1–14 (2019) H. Cheng et al.
Figure 8.
Distribution of the best-fitting mass accretion rates Û m of the sam-ple objects. Figure 9.
Distribution of the best-fitting magnetic equipartition coefficients β of the sample objects. primary corona emission. First, any absorption in excess of thatis accounted for by models (usually the Galactic absorption) orpartial coverage of the emission region will harden the observedX-ray spectra. Secondly, the X-ray spectra of some AGN are ob-served to be significantly contributed, or even dominated, by a re-flection component from the accretion disc and dusty torus, whichis peaked around – keV and can flatten the observed overallspectra below keV (e.g. George & Fabian 1991; Ross & Fabian2005), and not included in our modeling. Observationally it hasbeen demonstrated that, in many Seyferts with typical Γ = . – . as observed, their primary X-ray continua have actually steeperslopes of Γ = . – . when the reflection component is accountedfor (e.g. Nandra & Pounds 1994; Dadina 2007; Corral et al. 2011;Rivers et al. 2013; Ricci et al. 2017). This is also confirmed by oursimulations that the observed X-ray spectra will flatten by ∆Γ = . – . when an assumed reflection component is included, byusing XSPEC and its disc reflection model
XILLVER (García et al.2013). However, for the relatively steep X-ray slopes of our mod-eling, this effect is not sufficient to explain the rather flat spectralindices of Γ ∼ . – . , unless in some extreme conditions. For ex-ample, this happens when part of the primary X-ray emission fromcorona is hampered to reach the observer by various processes, Figure 10.
Comparison of the fitted black hole masses and the virial masses(from Cheng et al. 2019). Only 17 objects with fitted mass values consistentwith M vir within the mutual uncertainties are plotted; whereas for the oth-ers, the mass values reach the end of the allowed range ( log M vir ± . dex).The error bars are at the 90 per cent confidence level. The solid line repre-sents the one-to-one relation, and the dashed lines denote the uncertainties( ± . dex) of the virial mass. such as obscuration or light bending (e.g. Fabian & Vaughan 2003;Miniutti & Fabian 2004). The current X-ray data quality, as ob-tained with Swift /XRT, is insufficient to disentangle these complexcomponents from the primary coronal emission. We defer more rig-orous modeling of the AGN spectra by taking into account the re-flection component to a future work.Furthermore, it has recently been suggested that the primaryX-ray spectral slopes of AGNs depend on the Eddington ratio λ Edd (e.g. Brightman et al. 2013; Trakhtenbrot et al. 2017; Wang et al.2019). Specifically, AGNs showing flat spectra Γ ≤ have gen-erally lower Eddington ratios λ Edd ≤ . , and the spectra steepenwith the increase of λ Edd . Since some of our sample objects have λ Edd > . , their X-ray spectral indices are expected to be Γ > ,somewhat steeper than . – . of the primary continuum as foundin many Seyferts. In fact, our model naturally predicts the trend ofspectral steepening with increasing λ Edd .However, we expect that the above factors may only allevi-ate somewhat, rather than eliminate, the discrepancy between themodel and observed spectral indices, due to the limitation of thisand other similar models. A theoretical limit to the hardness ofthe spectral slope can be understood as the following. In the disc–corona coupling model, around half of the coronal radiation is in-tercepted by the disc, which is mostly absorbed (or scattered) andeventually re-emits at lower energy bands. This implies that the discemission, contributed by both the irradiation and intrinsic radiation,is usually stronger than that of corona. Only at the extreme case thatall the viscous energy is transported into the corona ( f ≈ ), can thecoronal luminosity be comparable with that of disc. With decreaseddisc emission the Compton cooling rate also decreases, leading toan increase in the coronal temperature and density. Therefore, theCompton- y parameter (defines as y = kT e m e c τ es ∝ n c T e ) reachesits upper limit at f ≈ , corresponding to the hardest possibleX-ray spectral slope. In fact, this has been a long-standing prob-lem for the non-truncated planar disc geometry around black holes(for a detailed review, see Poutanen et al. 2018), and one way to MNRAS000
Comparison of the fitted black hole masses and the virial masses(from Cheng et al. 2019). Only 17 objects with fitted mass values consistentwith M vir within the mutual uncertainties are plotted; whereas for the oth-ers, the mass values reach the end of the allowed range ( log M vir ± . dex).The error bars are at the 90 per cent confidence level. The solid line repre-sents the one-to-one relation, and the dashed lines denote the uncertainties( ± . dex) of the virial mass. such as obscuration or light bending (e.g. Fabian & Vaughan 2003;Miniutti & Fabian 2004). The current X-ray data quality, as ob-tained with Swift /XRT, is insufficient to disentangle these complexcomponents from the primary coronal emission. We defer more rig-orous modeling of the AGN spectra by taking into account the re-flection component to a future work.Furthermore, it has recently been suggested that the primaryX-ray spectral slopes of AGNs depend on the Eddington ratio λ Edd (e.g. Brightman et al. 2013; Trakhtenbrot et al. 2017; Wang et al.2019). Specifically, AGNs showing flat spectra Γ ≤ have gen-erally lower Eddington ratios λ Edd ≤ . , and the spectra steepenwith the increase of λ Edd . Since some of our sample objects have λ Edd > . , their X-ray spectral indices are expected to be Γ > ,somewhat steeper than . – . of the primary continuum as foundin many Seyferts. In fact, our model naturally predicts the trend ofspectral steepening with increasing λ Edd .However, we expect that the above factors may only allevi-ate somewhat, rather than eliminate, the discrepancy between themodel and observed spectral indices, due to the limitation of thisand other similar models. A theoretical limit to the hardness ofthe spectral slope can be understood as the following. In the disc–corona coupling model, around half of the coronal radiation is in-tercepted by the disc, which is mostly absorbed (or scattered) andeventually re-emits at lower energy bands. This implies that the discemission, contributed by both the irradiation and intrinsic radiation,is usually stronger than that of corona. Only at the extreme case thatall the viscous energy is transported into the corona ( f ≈ ), can thecoronal luminosity be comparable with that of disc. With decreaseddisc emission the Compton cooling rate also decreases, leading toan increase in the coronal temperature and density. Therefore, theCompton- y parameter (defines as y = kT e m e c τ es ∝ n c T e ) reachesits upper limit at f ≈ , corresponding to the hardest possibleX-ray spectral slope. In fact, this has been a long-standing prob-lem for the non-truncated planar disc geometry around black holes(for a detailed review, see Poutanen et al. 2018), and one way to MNRAS000 , 1–14 (2019) agnetic-reconnection-heated corona in AGNs resolve this issue is to reduce the the interception fraction of thecoronal radiation. To reach such a situation, several models havebeen proposed beyond the canonical plane-parallel corona config-uration, such as the patchy/clumpy/truncated disc–corona model(e.g. Galeev et al. 1979; Haardt et al. 1994; Stern et al. 1995) andthe dynamically outflowing corona model (e.g. Beloborodov 1999;Malzac et al. 2001).We now consider the spectral deviations in the optical/UVband in some of the objects, e.g. RX J1209.8+3217, RXJ1355.2+5612, and RX J1702.5+3247. In fact, their observed spec-tra are redder than those of the other sample objects and thanthe prediction of the standard Shakura–Sunyaev disc (Cheng et al.2019). This is most likely not caused by dust reddening since thesample objects were selected to be essentially free from dust extinc-tion, as indicated by their Balmer decrement in the optical spec-tra (Cheng et al. 2019). As demonstrated by Cheng et al. (2019),the observed flattening of the optical/UV spectra is likely a conse-quence of the loss of accreted mass from outside inward, by meansof outflows or disc/winds, hence leading to a flattened temperatureprofile from outside inward. Such an effect is not taken into ac-count in the current disc–corona model. To summarize, additionalprocesses or spectral components can practically complicate the ap-pearance of the spectra as observed in AGN. For the majority of the sample objects, a strong corona with anirradiation-heated disc scenario ( f > per cent) is suggested byour disc–corona model. In these objects, the X-ray emission is rela-tively strong with regard to optical/UV emission, arising from rela-tively strong magnetic fields. We suggest that this scenario is likelycommon in typical Seyfert galaxies. Indeed, our sample is biased tolocal X-ray bright AGN which are predominantly Seyfert galaxieswith moderate accretion rates. As for high-luminosity quasars, thesituation is less clear as there are only a few quasars in our sam-ple. Specifically, the two objects with the highest accretion rates,Ton 1388 ( Û m ≈ . , L bol ≈ . × erg s − ; Pâris et al. 2017)and RX J1355.2+5612 ( Û m ≈ . , L bol ≈ . × erg s − ),are of typical quasar luminosities. Their observed SEDs requireonly about half of the accretion energy transported into the corona( f ∼ . ), mainly due to a much higher accretion rate than the oth-ers ( Û m ∼ . and . , respectively), and their observed X-ray lu-minosities are much lower than the optical/UV luminosities. We an-ticipate that this may be a commonplace in many luminous quasarswith high accretion rates, whose SEDs are dominated by a strongoptical/UV bump. However, this has to be tested in the future with alarger sample of quasars. We suggest that in AGNs there may exista wide range of the relative magnetic field strengths, and hence ofthe fractions of energy transported into the corona, leading to thediverse relative strengths of the corona X-ray emission with respectto the optical/UV emission.A small fraction of AGNs are known to have signifi-cantly weak intrinsic X-ray emission compared to their opti-cal/UV emission, referred to as X-ray weak AGN (e.g. Yuan et al.1998; Brandt et al. 2000; Williams et al. 2004; Dong et al. 2012;Luo et al. 2014). They are characterized by much larger ef-fective optical-to-X-ray spectral indices (defined as α ox = − .
384 log L ( )/ L ( Å ) , Tananbaum et al. 1979) than thoseexpected from the α ox – L relation ( L is the monochromaticluminosity at Å). In the framework of our model, these AGNscan be understood if their magnetic fields in the disc are muchweaker (larger β values) than those in normal AGNs. In the ex- treme case when the coronal energy fraction is too low ( f ∼ ),the emergent spectrum will approach that of the standard Shakura–Sunyaev disc. As a simple comparison with observations, we cal-culate theoretical α ox for a given range of L = – erg s − Hz − assuming m = and β = . We find α ox span-ning a range of ∼ . – . , which are in good agreement with thoseof the X-ray weak AGNs in the Dong et al. (2012) sample. Thissuggests that the observed weak X-ray emissions in these objectscan be attributed to a very weak magnetic activity of the underlyingdisc. Aiming at the long-standing problem of how the disc corona isheated so as to sustain strong X-ray emission as observed in brightAGNs, we have studied a thin disc coupled with a corona throughmagnetic field and radiation in detail. The magnetic-reconnection-heating model (Liu et al. 2002a, 2003; Liu, Qiao & Liu 2016) hasbeen refined and applied to modelling the observed SED of a sam-ple of AGNs. Our theoretical investigation shows that the modelcan produce a variety of broad-band SEDs for objects with vari-ous black hole masses, accretion rates and magnetic field strengths.The accretion rate and magnetic field play key roles in determiningthe X-ray spectral slope and the relative radiation strength betweenthe disc and the corona. Specifically, a higher accretion rate leadsto a steeper X-ray spectrum and weaker X-ray emission (relativeto the optical/UV emission), as a consequence of less evaporation;a stronger magnetic field results in a flatter X-ray spectrum andstronger X-ray emission as it transports more accretion energy fromthe disc to the corona; the black hole mass, however, only shifts thepeak wavelength of the optical/UV emission. The magnetic fieldtransports accretion energy into the corona, providing a solution tothe problem of energy shortage in corona as a strong X-ray emitterin AGNs. Meanwhile, this process also helps naturally eliminateor alleviate the unstable radiation pressure-dominant region in thecentral part of accretion discs in AGN.The model is then applied to fitting the broad-band SEDs ofa sample of local AGNs studied extensively in our previouswork (Cheng et al. 2019), which have simultaneous X-ray and op-tical/UV observations and well-measured AGN parameters. It isfound that, in general, the overall observed broad-band spectralshapes can be reasonably reproduced by our disc–corona model,although in some objects the X-ray spectra are too flat to be ac-counted for or too noisy to yield meaningful constraint. In the ma-jority of the sample (15/20), the accretion energy is mostly trans-ported into the corona ( f > per cent) by the magnetic field, sup-porting a strong corona coexisting with an irradiation-heated disc.Meanwhile, the fitting results rule out largely a radiation pressure-dominant region in the central discs. In only a few bright objectsat quasar luminosities (e.g. Ton 1388 and RX J1355.2+5612), thefits lead to only about half of the energy released in the corona( f ∼ . ), given their much higher observed optical/UV luminosi-ties relative to the X-ray ones. The fitted black hole masses aregenerally consistent with the virial masses within their mutual un-certainties for the majority.We note that our model generally predicts relatively steep 2–10 keV photon indices, Γ = . – . . They appear to be steeper thanthe rather flat spectra observed in several sample objects, as well asin many other Seyferts ( Γ < . ), though the discrepancy can bealleviated to some extent by considering some effects (e.g. contri-bution of a disc-reflection component). Further theoretical investi- MNRAS , 1–14 (2019) H. Cheng et al. gation is needed to resolve this issue, which is generally inherent inmodels with similar disc–corona geometry as ours.
ACKNOWLEDGEMENTS
The authors thank the referee for constructive suggestions thathelp to improve the paper. This work is supported by the Na-tional Program on Key Research and Development Project (grantno. 2016YFA0400804), the National Natural Science Foundationof China (grants , , and ), the Strate-gic Priority Research Program of the Chinese Academy of Sci-ences (grant no. XDB23040100), and the Strategic Pioneer Pro-gram on Space Science, Chinese Academy of Sciences (grantsXDA15310300 and XDA15052100). HQ thanks Alice Breeveld,He-Yang Liu, and Bei You for useful comments and suggestions.Part of this work is based on the observations obtained by the NeilGehrels Swift
Observatory. We acknowledge the entire
Swift teamfor providing the data that made this work possible. This researchhas made use of the NASA/IPAC Extragalactic Database (NED),which is funded by the National Aeronautics and Space Adminis-tration and operated by the California Institute of Technology.
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