Inverse Compton (IC) scattering is thought to be the dominant process responsible for X-ray emission in AGN. When UV seed photons from the accretion disk, with energies h ν , pass through the coronal plasma, energy gets transferred from the hot ( 1 0 9 K) electrons to the photon field by repeated IC scattering. This happens if h ν < 4 k T e , where h is the Planck constant, ν is the photon frequency, k is the Boltzmann constant and T e is the temperature of the electrons.

For a plasma of non-relativistic electrons in thermal equilibrium with energy k T e ≪ m e c 2 , having an optical depth τ es , one can define the Compton y parameter: y = m a x ( τ es , τ es 2 ) ( 4 k T e / m e c 2 ) , where m e is the electron mass and c is the speed of light. For y ≫ 1 , the average photon energy reaches the thermal energy of the electrons and is called ‘saturated inverse Compton scattering’. The case for unsaturated Comptonization ( y > 1 ) is however of most interest in black hole systems, and in such a case, the output spectrum in the X-ray is a power law with a high energy cut-off E C determined by the electron temperature, which is typically approximated to be E C ∼ 2 − 3 k T e (Petrucci et al., 2001; Fabian et al., 2015).

The high densities of electrons around the magnetic field in AGN corona makes it a significant synchrotron emitter predominantly between 5 − 200 G H z (Laor and Behar, 2008; Panessa et al., 2019; Baldi et al., 2022; Kawamuro et al., 2022; Ricci et al., 2023). The fact that we see 1) ubiquitous unresolved mm emission even with high spatial resolution, 2) flat radio slopes, and 3) strong correlation between the radio and X-rays in RQ-AGN are indicators of radio emission from the corona through synchrotron processes (Panessa et al., 2019). For example, recent results (Ricci et al., 2023) point toward a tight correlation between 2 − 10 keV and 100 GHz luminosity for a volume-limited sample of nearby hard X-ray selected RQ-AGN. Similarly, the core radio flux at 5 GHz and the 2 − 10 keV luminosity for nearby radio quiet AGN have been found to show an interesting correlation L R, 5 GHz / L 2 − 10 keV ∼ 1 0 − 5.5 (Laor and Behar, 2008) which is similar to that found in coronally active stars (such as the Sun) and is popularly known as the Gudel-Benz relation (Guedel and Benz, 1993). As a caveat we note here that the coronal magnetic field can be as high as B = 1 0 2 − 1 0 5 Gauss and significant synchrotron self absorption (SSA) effects may limit our detection at lower frequencies (below ∼ 40 GHz).

Direct measurement of magnetic fields in AGN corona has not yet been possible, but we can estimate a typical range from the analogy of RL-AGN. Large B 0 values on event horizon scales are feasible considering that measurements from AGN jets have previously found magnetic field strengths of ∼ 0.1 G on ∼ 1 pc scales from core frequency-shift methods (O’Sullivan and Gabuzda, 2009) and ∼ 10 G on ∼ 0.1 pc scales from Faraday rotation measurements (Martí-Vidal et al., 2015). Such observational values are consistent with B 0 ≳ 1 0 5 G at the base of the jet with a 1 / r decay of the magnetic field, and are thereby consistent with theoretical and numerical predictions for launching relativistic jets (Tchekhovskoy et al., 2011).

Sources which are physically compact and highly luminous, like the AGN corona, are also radiatively compact. This means that the photons and the particles in the plasma are in constant interaction with each other. In such a plasma, photon–photon collisions can lead to e − − e + pair production, when the photons are energetic enough. The resulting e − − e + pair density is proportional to the luminosity ( L ) and electron temperature ( k T e or Θ = k T e / m e c 2 ), and inversely proportional to the source size ( R ) assuming a spherical source. This is typically expressed by the compactness parameter l = L σ T / ( R m e c 3 ) where σ T is the Thomson cross-section. Thus, when the energy content of corona increases, manifested by an increase in both l and k T e , the extra energy goes into creating more pairs, rather than increasing the temperature. Therefore, the process acts as a natural thermostat for the corona (Done and Fabian, 1989; Fabian et al., 2015; 2017). See Figure 2 left panel.

If magnetic reconnection is a dominant form of energy production mechanism in the X-ray corona (Di Matteo, 1998), then one would expect a fraction of non-thermal electrons (Done and Fabian, 1989). The existence of non-thermal particles in the corona would result in a distribution of photon energy that extends into the MeV band. This small number of high energy particles could be highly effective in seeding pair production. Moreover, the cooled non-thermal pairs could share the total available energy, thus reducing the mean energy per particle and therefore decreasing the temperature of the thermal population. Such hybrid coronal plasma, consisting of thermal and non-thermal electron populations, might have been found in a few nearby AGN (See Figure 2 right panel), in which the Comptonizing plasma is found well below the pair production line in the l − k T e plane (Fabian et al., 2015; 2017).

Figure 1 left panel highlights the primary spectral and physical characteristics of X-ray coronae and their typical range of values. The left panel of Figure 3 shows the photon index ( Γ ) distribution for a large sample of AGN (both obscured and unobscured) studied with broad-band X-ray observations (0.3–150 keV), with a median value of 1.78 ± 0.1 (Ricci et al., 2017). Recent broad-band X-ray studies also show that the spectral slope of the 14 − 195 keV emission is steeper than the 0.3 − 10 keV band, suggesting the high energy cut-off is ubiquitous in AGNs (Ricci et al., 2017).

The photon index is dependent on both the plasma temperature and the optical depth, and it can be estimated as Γ ∼ [ 9 4 + m e c 2 k T e τ ( 1 + τ / 3 ) ] 0.5 − 1 2 (Rybicki and Lightman, 1979). By measuring both Γ and E C it is possible to estimate the optical depth of the Comptonizing plasma, assuming a geometry (see, for example, Brenneman et al., 2014). Recent studies of nearby AGN estimate a median value of the optical depth of τ = 0.25 ± 0.06 (Ricci et al., 2017).

The typical coronal luminosity can span a large range L 2 − 10 keV ∼ 1 0 40 − 45 erg s − 1 (Piconcelli et al., 2005; She et al., 2017; Ricci et al., 2017), with the lower limit being only loosely defined by detector sensitivity and increasing contribution of non-AGN process to the X-ray emission. On this low-luminosity end of the distribution are the sources that are either accreting at low Eddington ratios or host intermediate mass black holes ( log ( M BH / M ⊙ ) < 6 , Dong et al., 2012). On the extreme high-luminosity end, are the hyperluminous AGN with L 2 − 10 keV ≥ 1 0 45 erg s − 1 , typically found at z ∼ 2 − 4 encompassing the cosmic peak of quasar activity (Martocchia et al., 2017).

The contribution of the X-rays to the total AGN emission is usually parametrized with the X-ray bolometric correction ( κ 2 - 10 ) : κ 2 - 10 ∼ L bol / L 2 - 10 keV . Studies of nearby AGN have shown that more luminous sources typically have weaker coronal X-ray emission relative to their bolometric luminosity, with κ 2 - 10 ≃ 15 − 25 at λ Edd < 0.1 , and κ ∼ 40 − 70 at λ Edd > 0.1 (Vasudevan and Fabian, 2007).

The high energy cut-off of the power law component is related to the coronal temperature as E C ∼ 2 k T e , for an optically thin plasma, i.e., τ ≲ 1 . On the other hand when the plasma is optically thick, i.e., τ ≫ 1 the relation is E C ∼ 3 k T e , both approximated for a corona of slab geometry (Petrucci et al., 2000; 2001). The right panel of Figure 3 (Kamraj et al., 2022) shows the distribution of high-energy cut-offs inferred from NuSTAR observations of a sample of nearby AGN. The median value of the cut off energy obtained for the sample is ∼ 84 ± 9 keV . A large study of a sample of nearby Swift-BAT detected AGN finds a median cut-off energy in local AGN that is significantly higher ( E C ∼ 200 ± 29 keV ; Ricci et al., 2017). Indirect constraints on the cut-off energy have been obtained by fitting the cosmic X-ray background (CXB), and have shown that the mean cut-off energy is likely below 300 keV (Gilli et al., 2007; Treister et al., 2009; Ueda et al., 2014), in agreement with the observational studies reported above.

Analyzing a sample of ∼ 200 AGN, Ricci et al. (2018) found that, while E cut is not related to the mass of the black hole or the 14 − 150 keV luminosity, it appears to be related to the Eddington ratio ( λ Edd ) . Sources with λ Edd > 0.1 were shown to display significantly lower median cut-off energy ( E cut = 160 ± 41 keV ) than those with λ Edd ≤ 0.1 ( E cut = 370 ± 51 keV ) . This supports the idea that more radiatively compact coronae are cooler, because they tend to avoid the region in the temperature-compactness parameter space where runaway pair production would dominate (See Figure 2). In some extreme cases, coronal temperatures as low as k T ∼ 10 − 20 keV have been measured in a few nearby AGN (Buisson et al., 2018). Interestingly, and in agreement with the results of Ricci et al. (2018), such cool corona are often detected in several high Eddington AGN (Kara et al., 2017; Tortosa et al., 2022). These plasma are either not pair-production dominated, or they are hybrid, as discussed in the Introduction.

Sample studies of AGN in hard X-rays with NuSTAR detected an anti-correlation between k T e and τ (Tortosa et al., 2018; Kamraj et al., 2022; Serafinelli et al., 2024). On average, the lower mass, highly accreting narrow line Seyfert 1 galaxies (NLSy1s) exhibit a steeper photon index ( Γ > 2 ) , suggesting the corona might be cooler or less optically thick compared to other AGN (e.g., Brandt et al., 1997; Gallo, 2018).

Compton scattering induces polarization of the X-ray photons, which is an important tool to study the geometry of the emitting plasma. The polarization of the X-ray photons measured both in degree and position angle, is energy- and geometry dependent. For example, a polarization degree of 4 % or higher can possibly rule out spherical and lamp-post coronal models, because symmetry reduces the polarization degree (Zhang et al., 2019; Ursini et al., 2022a). The same is true for the orientation of the polarization angle, that is model-specific (different models predict different values of polarization angle).

Launched in December 2021, the Imaging X-ray Polarimetry Explorer (IXPE), is the first X-ray spectro-imaging polarimeter satellite sensitive in the 2–8 keV band (Weisskopf et al., 2022). IXPE successfully measured a polarized signature in NGC 4151 (Gianolli et al., 2023) with a polarization degree of 4.9% ± 1.1% at a position angle of 86 ° ± 7 ° east of north at 68% confidence level. The amount of polarization associated with the corona is of the order of 4%–8%, which directly excludes a spherical geometry (Beheshtipour et al., 2017; Ursini et al., 2022b). The polarization angle measured for this source, which is parallel to its radio jet, suggests that the corona could be distributed along the accretion disk or perhaps it’s a part of the inner accretion flow (as in a slab geometry). On the other hand, upper limits (at 99% confidence level) of 3.2% and 6.2% were obtained for the polarization degree in MCG − 5 − 23 − 16 (Marinucci et al., 2022a; Tagliacozzo et al., 2023) and IC 4329A (Ingram et al., 2023), respectively, implying that, for those two objects, we cannot directly rule out a spherical and/or lamppost corona. However, the orientation of the polarization angle in those two AGN seems to be more consistent with an extended corona (along the equatorial plane) rather than with a polar or spherical corona, because their tentatively measured polarization angle (at < 3 σ ) are also parallel to the detected radio structures and polar winds (Tagliacozzo et al., 2023; Ingram et al., 2023).

Here we also mention an interesting result from an X-ray binary, Cygnus X-1, for which the polarization degree could be constrained exceptionally well, at (4.0 ± 0.2)% between 2 − 8 keV, and a polarization angle parallel to the jet axis (Krawczynski et al., 2022). This suggests that, similar to AGNs, the hot X-ray corona is likely spatially extended in a plane perpendicular to the jet axis, parallel to the inner accretion flow, and rules out the commonly used lamp-post model.

Gravitational microlensing of quasar light by a foreground mass (lens) can be used to probe the sizes related to the accretion disc and corona (e.g., Chartas et al., 2009; Morgan et al., 2008; Dai et al., 2010). Using such methods, the size of the X-ray emitting region (the corona) is estimated to be very compact, around 5 − 10   R G (e.g., Dai et al., 2010).

Capturing the transit of the X-ray source by an obscuring cloud is fortuitous, but not rare (see for, e.g., Risaliti et al., 2007; Turner et al., 2018; Gallo et al., 2021; Ricci and Trakhtenbrot, 2022). Such events are important as they can be used to constrain the sizes of the X-ray region based on the duration of the eclipse. This method assumes that the cloud is gravitationally bound to the central SMBH in a Keplerian orbit and the eclipse occurs when the cloud moves across our line of sight to the central engine. In the objects for which eclipses could be used to measure the size of the corona (e.g., Risaliti et al., 2011; Gallo et al., 2021), the results have been consistent with those obtained using other methods.

One of the most common coronal variability pattern is the chaotic total intensity variation, or stochastic variation. Early studies using observations from Ariel-V and EXOSAT show that 40 % of AGN exhibit stochastic variability on a timescale less than 1 day, and 97% of them showed variability on longer timescales (e.g., Grandi et al., 1992; McHardy and Czerny, 1987). More recently, a linear relationship between the rms amplitude of short-term variability and flux variations on longer timescales has been found in AGN X-ray light curves (Gaskell, 2004; Uttley et al., 2005a; Vaughan et al., 2011). This has been dubbed the “rms-flux” relation (Uttley et al., 2005b). This is an important feature of the aperiodic variability of accreting compact objects, including black hole X-ray binaries (Gleissner et al., 2004; Heil et al., 2012).

It is still unclear how the X-ray coronal variability at different-timescales is produced. Popular models predict that inward propagation of random accretion rate fluctuations in the accretion flow could create such stochastic variations in the coronal X-ray emission (Lyubarskii, 1997; Kotov et al., 2001; King et al., 2004; Kelly et al., 2011; Ingram and van der Klis, 2013; Cowperthwaite and Reynolds, 2014). The longer term variability may be produced by accretion rate changes (Mushotzky et al., 1993), but the origin of the short timescale variations (a few ks or less) are still debated. The magnetic field reconnection in accretion disk threading the coronal plasma likely play a role (Di Matteo, 1998) as they do in the solar corona.

An important measurement of the variability is the power density spectrum (van der Klis, 1989; Vaughan et al., 2003a; b), which describes the amount of power (the amplitude squared, i.e., the power of the signal) as a function of temporal frequency. When the X-ray light curve can be described as random displacements around a mean value, then the power density spectrum (PSD) shows a constant value, that is, all frequencies have equal power. This is known as a white noise spectrum. On the other hand, a red noise spectrum is created when the points in the light curve have a random displacement from its adjacent point rather than from the mean. In such a case the variations at lower frequencies have more power. Red noise is the characteristic of several astrophysical systems including the Sun (Lu and Hamilton, 1991) and black hole binaries (Belloni and Hasinger, 1990), and it is closely related to the stochastic nature of such non-linear systems. In AGNs, over the frequency range f = 1 0 − 3 to 1 0 − 5 Hz, the power spectral density of most Seyfert galaxies has a mean slope of α ∼ 2.0 in the 2 − 10 keV band, exhibiting no characteristic timescales (González-Martín and Vaughan, 2012), and indicating that red-noise steeply decreases at higher frequencies (that is shorter time scales). In some AGN there is a break in the PSD slope at f = 2 × 1 0 − 4 Hz, from a much flatter slope of 2 at lower frequencies to a steeper slope of 3 at higher frequencies, and the break is connected with the black hole mass. This is similar to the three slope PSD detected in black hole binaries (BHB): α ∼ 0 for low frequencies ( < 0.2 Hz), α ∼ 1 for intermediate frequencies ( ∼ 0.2 − 3 Hz) and α ∼ 2 at higher frequencies, above 3 Hz (Vaughan et al., 2003b).

The origin of QPOs in AGN are highly debated, they are still very rare and they have mostly been discovered in the 2 − 10 keV or harder X-ray bands. For example, such QPOs have been found at 2.6 × 1 0 − 4 Hz ( ∼ 1 hour) in RE J1034 + 396 (Gierliński et al., 2008; Alston et al., 2014; 2016), at 1.5 × 1 0 − 4 Hz ( ∼ 2 hours) in MS 2254.9–3712 (Alston et al., 2015), at 2.7 × 1 0 − 4 Hz ( ∼ 1 hour) in 1H 0707–495 (Pan et al., 2016). A QPO of a period of ∼ 3.8 hours was detected from an ultra-soft AGN candidate 2XMM J123103.2 + 110648 (Lin et al., 2013). A systematic study of AGN X-ray variability in a sample of 104 sources in search for QPOs detected only two sources with QPOs (González-Martín and Vaughan, 2012). Very recently a recurrent QPO has been discovered in the post-changing-look AGN 1ES 1927 + 654, where the QPO frequency increased from ∼ 0.9 mHz to ∼ 2.3 mHz over a period of 2 years ² (Nature, in press).

X-ray flares with different amplitude at different timescales are common in AGN. Typically, flares can exhibit flux increases of ∼ 5 − 10 times over time spans ranging from hours to days (Gallo et al., 2019; Lawther et al., 2023; Wilkins et al., 2022; Reeves et al., 2021; Ding et al., 2022). During X-ray flares, a spectral softening (softer-when-brighter) and a decreasing reflection fraction have been observed in some AGNs (e.g., Mrk 335 Gallo et al., 2019, 1H 0707–495 Wilkins et al., 2014). In Mrk 335, the decade-long low flux state has been marked by occasional X-ray “flares,” (see Figure 4 left panel) which have sometimes brightened by a factor of 50 within a single day (Grupe et al., 2012; Wilkins and Gallo, 2015a; Gallo et al., 2019). In some highly accreting, low SMBH mass AGN, X-ray flares have been linked to the radial expansion of the corona over the accretion disk: the corona appears brighter when it is more extended outward, while it dims when it is compact and located closer to the SMBH (e.g., Wilkins et al., 2014). Extreme variability and flares in AGN corona suggest that the compact corona must be a dynamic structure since the time scales for heating and cooling processes for the hot electrons are less than the light crossing time of the corona (Fabian et al., 2015), which prevents the system to settle down to an equilibrium. In addition, the plasma properties also change during a flare. For example, Wilkins et al. (2022) found that during a flare, the cutoff energy E C of the primary energy continuum dropped from 14 0 − 20 + 100 keV to 4 5 − 9 + 40 keV . Another example is the Seyfert 1 galaxy I Zwicky 1 that also showed such dramatic changes in the plasma properties, when the corona rapidly cooled from E C ∼ 200 to ∼ 15 keV within 5 days in January 2020, as caught by XMM-Newton and NuSTAR, (Ding et al., 2022).

X-ray flaring events in many astrophysical objects are generally associated with magnetic reconnection (e.g., Petropoulou et al., 2016; Mehlhaff et al., 2020), a fundamental plasma process where magnetic energy is converted into thermal and nonthermal particle energy (e.g., Lyubarsky, 2005; Takahashi et al., 2011). Magnetic reconnection has a short dissipation time, and short flares with durations that generally do not exceed a few times t ∼ R G / c should be preferentially associated with magnetic reconnection (e.g., Petropoulou et al., 2016; Christie et al., 2018). Longer flaring episodes could be associated with magnetic flux accumulation in the corona because of accretion, changing the properties of particle acceleration and thus that of the emitted photons (e.g., Liska et al., 2020; Scepi et al., 2021; Ripperda et al., 2022).