corona by e.g. illumination increases (indeed, Ftot−hard correlates with Frel−refl), produc-ing a hardenproduc-ing (Γwc decreases) of the warm corona spectrum.
The main problem with this interpretation is the evolution of the coronal tempera-ture. We would expect the temperature to decrease/increase when the spectrum steep-ens/hardens. The present observations do not show clear evolution of these temperatures.
This suggests that something else may have to change, e.g. the optical depth, to keep the temperature roughly constant.
Note that we have assumed a constancy of the warm corona geometry. Its variation is however plausible and should add another free parameter to explain the observations.
Indeed, variability of the optical-UV-to-X-rays emission of NGC 4593 could result from geometrical variations of the ‘two coronae’ but also of the ‘warm corona’ and outer part of the disc, not covered by the warm corona, and, potentially, also contributing to the optical-UV emission. A detailed analysis of the parameter space, also adopting more sophisticated models (e.g. agnsed,Kubota & Done 2018) self-consistently accounting for the disc contribution in the frame-work of two Comptonising coronae is deferred to a future work.
In turn, we find upper limits for the optical depth of the hot corona. We notice that kThc and τhc anticorrelate (Pcc=-0.85 P(r>)=4%).
• All spectra display a remarkable soft-excess, and a warm Comptonisation model best describes this component. The warm medium is characterized by a variable photon index 2.35 < Γwc < 2.74and a constant kTwc = 0.12±0.01keV. The optical depth of the warm corona is variable 35 ≤ τwc ≤ 47. According to this analysis, most of the accretion power is released in the warm-corona than in the accretion disc.
• For the first time, we observe an anticorrelation between the photon indexes of the hot and the warm corona. The origin of this trend cannot be ascribed to a model degeneracy (see Fig. 4.19). The interpretation of such anticorrelation is not straightforward but can result from the radiative feedback between the two coronae.
• The present test on the two-corona model indicates that it reliably allows for re-producing the AGN broadband emission. Indeed, besides observational differences with other sources that can be explained in terms of different physical properties (e.g. Eddington ratio), notably the two-corona model provides good representa-tions of the data suggesting that it accounts for a common Comptonising mech-anism occurring in AGN. On the other hand, the existence of a warm corona at such temperature and optical depth above the accretion disk is expected to produce emission/absorption features from the ionized matter. The fact that we do not see any of them would imply specific physical properties (e.g. turbulence) that have to be tested with accurate radiative transfer codes.
Estimating the coronal parameters with MoCA
As discussed in Chapter 3, both the photon index and high energy cut-off of the X-ray primary emission depend on the intrinsic properties of the Comptonising medium, namely its temperature, optical depth and geometry.
In this Chapter, we derive and discuss formulae through which it is possible to directly relate the phenomenological quantities Γ-Ec with the corresponding coronal properties kTe-τe for the cases of a slab-like and a spherical corona.
The interplay between coronal parameters and the AGN X-ray spectral shape has been so far object of several investigations, especially when observatories capable of de-tecting hard X-rays have been launched. Perola et al. (2002) using BeppoSAX data collected and studied the photon index and Ecof a sample of AGN, while a similar work was performed on INTEGRAL data (e.g. Malizia et al. 2014). Then, NuSTAR (see Ap-pendix A) greatly helped in studying the exponential cut-offs of the nuclear continuum in several AGN, (see e.g. Fabian et al. 2015,2017; Tortosa et al. 2018).
These space missions allowed for a substantial corpus of high energy cut-off and pho-ton index measurements, and many efforts have been made to find a way for converting such parameters into those describing the coronal properties.
In Petrucci et al. (2000) an approximate relation between the observed high-energy cut-off of NGC 5548 and the thermal energy of the corona is given (Ec∼2-3kTe), while inPetrucci et al. (2001) it was discussed for a larger sample of Seyfert 1s. However, this relation has been proved to be accurate for an extended slab geometry only, but it is often used for coronae of any geometry and size.
On the other hand, Beloborodov (1999) reports a relation between the spectral pho-ton index and the Comppho-ton parameter y. This parameter introduced in Chapter 2 (Eq.
2.35) is the product of the average fractional energy change per scattering and the mean number of scatters that the photon undergoes (Rybicki & Lightman 1979). Then, y is
used to characterise the energy gain of a photon scattering within a finite medium. The Compton parameter encodes the physical conditions of the coronal plasma, and it can be defined as follows (see also Chapter 2 and Chapter 3):
y = 4(θe+ 4 θ2e)τT(τT+ 1) , (5.1) where θ = kTe/mec2, and τT is the Comptonising medium opacity. Beloborodov (1999) found that the photon index is related to y according to Γ=49 y−2/9. In his work, Be-loborodov(1999) studied the dependence of Γ on y for only two values of electron temper-atures (kTe=50 keV and kTe=100 keV) and letting free to vary the opacity of the corona.
Moreover, in his work, the Comptonising corona is infinitely extended and spherical.
In the following, we use MoCA (MonteCarlo code for Comptonisation in Astrophysics) (Tamborra et al. 2018), to derive formulae for directly translating high energy cut-offs and photon indices into the corresponding physical parameters of the hot corona, namely the optical depth and electron temperature. We will derive formulae for a slab-like corona and for the case of a sphere.
5.1 MoCA & simulations setup
To our knowledge, MoCA (see detailed description inTamborra et al. 2018) is the first MonteCarlo code devoted to study accretion of astrophysical sources based on a single photon approach and working in a fully special relativistic scenario and also accounting for polarisation signals. Moreover, compared to different codes in the literature, such as compTT, compPS, nthcomp (Titarchuk 1994; Poutanen & Svensson 1996; Zdziarski et al. 1996;Życki et al. 1999), the energy dependent Klein-Nishina cross section is taken into account (differently from Schnittman & Krolik 2010) and multiple geometries can be tested and investigated (see e.g. Beheshtipour et al. 2017).
MoCA is a flexible code, and it can account for various and different physical conditions of the AGN and of the corona.
Indeed, before running every simulation, MoCA allows for defining the source BH mass and its accretion rate. These two quantities mainly affect the seed photons spec-trum produced by the multi-colour accretion disc. Accretion discs are taken with the typical properties for a Shakura-Sunyaev disc (see Chapter 2 for details and references), and it is possible to set different values for its inner and the outer radii.
The synthetic spectra produced through MoCA also account for the physical con-ditions of the hot corona that is modelled as a thermal distribution of electrons. Two geometries were considered for the corona: a sphere and a very oblate spheroid to which we will refer to as a slab. In this work, we have assumed the Comptonising corona to be as extended as the disc. Moreover, for the slab geometry case, MoCA allows for setting
up the corona height above the accretion disc (set to 10 rg in our simulations). MoCA also allows for tuning the physical conditions of the corona, indeed it is possible to set up the coronal temperature and its opacity.
In the forthcoming analysis, we will use simulations computed assuming the source BH mass and accretion rate to be the same as Ark 120 (e.g. Nardini et al. 2016;Reeves et al. 2016b; Porquet et al. 2018) i.e. MBH = 1.5 × 108M and ˙m = Lbol/LEdd = 0.1. Then disc radii are always set to be Rout = 500 rg and Rin = 6rg, respectively. For both the slab and spherical geometries, we have simulated the Comptonised spectra using a wide range of values for the electrons temperature and their optical depth:
0.5 < τe < 4.5 and 20 <kTe < 120 for the slab geometry, while 0.5 < τe < 7 and 20 <kTe < 120were used for the spherical corona.
In Fig. 5.1 we show examples of spectra obtained produced by MoCA.
Figure 5.1. MoCA simulated spectra assuming τ =1.5 and different electron tempera-tures ranging from 50 keV (black) to 120 keV (orange) and adopting a step of 10 keV.
Spectra assuming a slab-like corona are shown in the left panel, while, in the right panel, a spherical corona is adopted. Colour code is the same for the two panels. It is worth to notice that the same τ -kT couple gives rise to different spectra when the slab-like or the spherical geometries are considered.