The physical relation between disc and coronal emission in quasars

We propose a modified version of the observed non-linear relation between the X-ray (2 keV) and the ultraviolet (2500 \AA) emission in quasars (i.e. $L_{\rm X}\propto L_{\rm UV}^{\gamma}$) which involves the full width at half-maximum, FWHM, of the broad emission line, i.e. $L_{\rm X}\propto L_{\rm UV}^{\hat\gamma}~FWHM^{\hat\beta}$. By analysing a sample of 550 optically selected non-jetted quasars in the redshift range of 0.36-2.23 from the Sloan Digital Sky Survey cross matched with the XMM-Newton catalogue 3XMM-DR6, we found that the additional dependence of the observed $L_{\rm X}-L_{\rm UV}$ correlation on the FWHM of the MgII broad emission line is statistically significant. Our statistical analysis leads to a much tighter relation with respect to the one neglecting FWHM, and it does not evolve with redshift. We interpret this new relation within an accretion disc corona scenario where reconnection and magnetic loops above the accretion disc can account for the production of the primary X-ray radiation. For a broad line region size depending on the disc luminosity as $R_{\rm blr}\propto L_{\rm disc}^{0.5}$, we find that $L_{\rm X}\propto L_{\rm UV}^{4/7} ~FWHM^{4/7}$, which is in very good agreement with the observed correlation.


INTRODUCTION
One of the observational evidences for the link between the accretion disc and the X-ray corona in active galactic nuclei (AGN) is given by the observed non-linear correlation between the monochromatic ultraviolet luminosity at 2500 Å (L UV ) and the one in the X-rays at 2 keV (L X ). Such relationship (parameterised as log L X = γ log L UV + β) exhibits a slope, γ, around 0.6, implying that optically bright AGN emit relatively less X-rays than optically faint AGN [1]. The value of the slope in this relation does not depend on the sample selection, the correlation is also very tight (∼ 0.24 dex, Lusso and Risaliti 2), and independent on redshift [3,4,5,6,7,8,9,10,11]. Recently, the L X − L UV relationship (or, more precisely, its version with fluxes) has also been employed as a distance indicator to estimate cosmological parameters such as Ω M and Ω Λ by building a Hubble diagram in a similar way as for Type Ia supernovae [12,13, and references therein], but extending it up to z ∼ 6 [14,15]. Yet, the challenge in interpreting such relation on physical grounds is that the origin of the X-ray emission in quasars is still a matter of debate.
Some attempts at explaining the quasar X-ray spectra were based upon the reprocessing of radiation from a non-thermal electron-positron pair cascade (e.g. Svensson 16,17,Zdziarski et al. 18). Another possibility rests on a two-phase accretion disc model, where a fraction f of gravitational power is dissipated via buoyancy and reconnection of magnetic fields in a uniform, hot (T cor ∼ 100 keV ∼ 10 9 K) plasma close to the cold opaque disc [19,20,21,22,23]. The scenario for this model is the following. Phase 1 is the optically thick "cold" (tens of eV) disc, whilst phase 2 is a hot optically thin plasma located above (and below) the disc. The seed disc photons illuminate the hot tenuous plasma, and a fraction of them is up-scattered to hard X-rays via inverse Compton scattering, providing the main source of cooling of the plasma. About half of these hard X-ray photons is irradiated back to the disc, contributing to its energy balance, whilst the rest escape and are observed. A stable disc-corona system is then in place only if there is a strong coupling between ultraviolet and X-ray photons. This model retrieves the photon index slope of the coronal X-ray spectrum (i.e. Γ X 2), in close agreement with real data. Nonetheless, it also predicts a nearly equal amount of ultraviolet and X-ray radiation (i.e. γ = 1), which is not in agreement with the observed correlation. If the corona is not uniform, but a rather patchy medium, and only a fraction ( f ) of the accretion power is released in the hot phase, the resulting γ value is < 1, but one needs to consider a rather arbitrary number of active blobs [24]. Magnetic field turbulence has been recognised not only as an additional heating mechanism in the formation of the corona (e.g. Galeev et al. 25,Merloni and Fabian 26,Liu et al. 27), but also as an efficient means for the transport of the disc angular momentum (e.g. Balbus 28). Yet, the value of the fraction f of gravitational power dissipated in the hot corona (needed for keeping the plasma at high temperatures), and how the coronal physical state depends on the black hole mass and disc accretion rate still remain a matter of debate.
In Lusso and Risaliti [29] we outlined a simple but physically motivated, ad-hoc model to interpret the observed correlation between the ultraviolet and the X-ray emission in terms of physical parameters such as the black hole mass (M BH , here we considered the normalized value m = M BH /M ), the accretion rate (ṁ =Ṁ/Ṁ Edd , whereṀ Edd is the accretion rate at Eddington), and the distance to the black hole (r = R/R S , where R S = 2GM BH /c 2 is the Schwarzschild radius). Our main aim was to link such relations with observable quantities (i.e. the observed ultraviolet and X-ray luminosities), thus obtaining a relation that can be then compared with the data.
Here, we further analyse the L X − L UV − FWHM relation with the goal of understanding its physical origin in the context of black hole accretion physics.

THE OBSERVED
The quasar sample considered by Lusso and Risaliti [29] is obtained by cross-matching the quasar SDSS catalogue published by Shen et al. [30] with the serendipitous X-ray source catalogue 3XMM-DR6 [31]. Filters are also applied in order to select a clean quasar sample where biases and contaminants are minimised, namely: (i) jetted, broad absorption line quasars and sources with high levels of absorption in the optical (E(B − V) > 0.1) are removed from the sample; (ii) we selected those quasars with a full-width half-maximum, FWHM, for the MgII λ2800 Å higher than 2000 km s −1 ; (iii) only sources with good X-ray data (i.e. S/N>5 in the 0.2-12 keV EPIC band) and low levels of X-ray absorption (i.e. with an X-ray photon index Γ X > 1.6) are considered; and finally (iii) the Eddington bias (see also Risaliti and Lusso 14 and Lusso and Risaliti 2 for further details) is minimised by including only quasars whose minimum detectable X-ray flux is lower than the expected one in each observation. The FWHM values for the MgII emission line as well as the ultraviolet 2500Å luminosities are taken from the Shen et al. [30] catalogue, which have been measured from the continuum/emission line fitting of the SDSS spectra (see their Section 3 for further details). The final clean sample is composed of 550 quasars and it is only ∼25% of the initial sample (∼2,100 quasars with both soft 0.5-2 keV and hard 2-12 keV fluxes), resulting from the stringent filters mentioned above. In the future we will refine the treatment of the systematics by considering additional instrumental systematics, and developing large mock simulations to better understand the impact of the Eddington bias on the results. The main aim is to reduce the rejection fraction, thus maximizing the statistics of the final sample.
with an observed dispersion of 0.22 dex, which is a very tight relation. To fit the data, we adopted emcee [32], which is a pure-Python implementation of Goodman & Weare's affine invariant Markov chain Monte Carlo ensemble sampler.
We then divided the quasar sample into equally spaced narrow redshift intervals in log z with a ∆ log z < 0.1 to minimise the scatter due to the different luminosity distances within each interval. We split the sample into 10 intervals with ∆ log z = 0.08 and, for each redshift interval, we performed a fit of the F X − F UV − FWHM relation, log F X =γ z log F UV +β z FWHM +K, with freeγ z andβ z . The results for the best-fit slopesγ z andβ z as a function redshift are shown in Figure 1. Despite the large scatter, bothγ z andβ z slopes do not show any significant evolution with time and they are consistent with a constant value in the redshift interval 0.3-2.

THE TOY MODEL
Previous works in the literature suggest that the amount of gravitational energy from the accretion disc released in the hot plasma surrounding the disc itself, likely depends on the black hole mass M BH , the mass accretion rateṀ, and the spin of the black hole. In Lusso and Risaliti [29], the L X − L UV correlation was reproduced if the energy transfer from an optically thick, geometrically thin accretion disc to the corona is confined to the gas-pressure dominated region of the disc. In such simple model, the monochromatic ultraviolet and X-ray luminosities show an extra dependence on the FWHM, thus on the black hole mass and accretion rate as L UV ∝ M 4/3 BH (Ṁ/Ṁ Edd ) 2/3 and L X ∝ M 19/21 BH (Ṁ/Ṁ Edd ) 5/21 , respectively. Assuming a broad line region size function of the disc luminosity as R blr ∝ L 0.5 disc we have that where α is the standard disc viscosity parameter, κ is a calibration constant in the broad line region radius-luminosity relation (i.e. R blr = kL 0.5 bol Trippe 33), and J(r) = (1 − √ 3/r).
In this model the bulk of the radiation budget of the X-ray corona is measured at the transition radius (r tr ) where the gas pressure in the accretion disc equates the radiation pressure, and it is defined as where m = M BH M , andṁ =Ṁ M Edd . The transition radius can vary from a few gravitational radii (r g ) to several hundreds r g for a black hole mass m = 10 8−9 , with an accretion rate in the rangeṁ = 0.1 − 0.3, and α = 0.4. This parameter also depends on the value of f (i.e. the higher is f , the smaller is r tr ). Yet, the effective location of the corona may also be placed at small radii of less than tens of r g . From a qualitative perspective, the torque (caused by the disc rotation) of magnetic field lines (originated at r ≥ r tr ) that connects two opposite sides of the accretion disc (e.g. magnetic reconnection, a schematic representation of our model is provided in Figure 2) could cause particle acceleration at the magnetic reconnection site, located closer to the black hole (at r < r tr ), where particles lose their energy radiatively via the inverse Compton process.

CONSTRAINING THE FRACTION OF ACCRETION POWER RELEASED IN THE CORONA
Our simple (but physically motivated) model also provides constraints on the fraction of accretion power released in the hot plasma in the vicinity of the accretion disc, f , as a function the broad line region size, from the observed normalization of L X − L UV − FWHM plane. Figure 3 shows how the normalization of equation (2) changes as a function of α, κ, and f . We fixed the κ factor to 1.3 × 10 −6 , which corresponds to a broad line region size of about 1.3 × 10 17 cm (∼ 50 light days) and a bolometric luminosity of 10 46 erg s −1 , typical of quasars (e.g. Kaspi et al. 34). The viscosity parameter varies from 0.1 to 0.4 [35], whilst f ranges in the interval 0-0.99. Although the normalization of the relation dsiplays a large scatter, which can be a factor of ∼ 2 in logarithm (i.e. orders of magnitude), the data suggest a value of f in the range 0.9-0.95.
In the future, by studying larger sample of quasars with M BH and accretion rate values (which are all related to f ), we will compare our results on f with the expectations from numerical simulations (e.g. Jiang et al. 36) and other theoretical models (e.g. Merloni and Fabian 26, Cao 37,Liu et al. 38), especially those r tr P gas >>P rad P gas =P rad P gas <<P rad X-rays optical-UV Disc corona Figure 2. A schematic representation of the toy model discussed in §3. The optically thick, geometrically thin accretion disc emits the seeds photons that illuminate the hot tenuous (possibly clumpy) plasma. A fraction of them is up-scattered to hard X-rays via inverse Compton scattering. Part of these hard X-ray photons is irradiated back to the disc, contributing to its energy balance, whilst the rest escape and are observed. Stable magnetic loop can be formed only in the radiation pressure dominated part of the disc.

CONCLUSIONS
Our modified L X − L UV relationship in quasars, which takes into account the full-width half-maximum of the quasar emission line (i.e. L X ∝ Lγ UV FWHMβ), has an observed dispersion of ∼0.2 dex over ∼3 orders of magnitude in luminosity and indicates that there is a good "coupling" between the disc, emitting the primary radiation, and the hot-electron corona, emitting X-rays.
We interpreted such relation through a simple (but physically motivated) model based on the ones presented by Svensson and Zdziarski [21] and Merloni and Fabian [26], where a geometrically thin, optically thick accretion disc is coupled with a uniform hot plasma. We assumed that the bulk of the corona emission is mainly powered by the accretion disc and it is located at the transition radius (r tr ) where the gas pressure equates the radiation pressure in the disc. Assuming a broad line region size function of the bolometric luminosity as R blr ∝ L 0.5 bol we have that M BH ∝Ṁ/Ṁ Edd FWHM 4 , which leads to the final relation L X ∝ L 4/7 UV FWHM 4/7 . Such a relation is remarkably consistent with the fit obtained from a sample of 550 optically selected quasars from SDSS DR7 cross matched with the XMM-Newton catalogue 3XMM-DR6. The toy model we presented, although simplistic, is capable of making robust predictions on the X-ray luminosities (at a given ultraviolet emission and FWHM) of unobscured/blue quasars, and puts observational constraints on the fraction of accretion power released in the hot plasma in the vicinity of the accretion disc, f , as a function the broad line region size, from the observed normalization of L X − L UV − FWHM plane. The latter result will provide a new vantage point for the next generation of semi-analytical and magneto-hydrodinamical simulations investigating on the physical link between the accretion disc and the X-ray corona.
The proposed relation L X ∝ L 4/7 UV FWHM 4/7 does not show significant evolution with time in the redshift range covered by our data, z = 0.3 − 2.2, and thus it can be employed as a cosmological indicator to estimate cosmological parameters (e.g. Ω M , Ω Λ ).