Magnetic-Reconnection-Heated Corona Model: Implication of Hybrid Electrons for Hard X-ray Emission of Luminous Active Galactic Nuclei

It is believed that active galactic nuclei (AGNs) are powered by the gravitational energy liberation of the accretion matter. Luminous AGNs generally have relatively high Eddington ratio of $L_{\rm bol}/L_{\rm Edd}>0.02$, where $L_{\rm bol}$ is the bolometric luminosity, and $L_{\rm Edd}=1.25\times 10^{38}(M_{\rm BH}/M_{\odot})$ erg s^-1 is the Eddington luminosity. Their spectral features are characterized by three components: the so-called “big blue bump” ranging from the optical to the ultraviolet (UV) band, which is explained by a standard thin disc extending into the innermost stable circular orbit (e.g., Shakura & Sunyaev, 1973), the soft X-ray excess, whose origin is under debate (e.g., Done et al., 2007, where various origins were summarized), and the power-law emission in hard X-ray band that is commonly contributed by the inverse Compton scattering of optical/UV photons from the accretion disc. Because the low-energy photons are scattered up to X-ray band through the interaction with hot electrons in the corona (Svensson & Zdziarski, 1994; Magdziarz et al., 1998; Zdziarski et al., 2000; Lubiński et al., 2016), the observational feature of the luminous AGNs in X-ray band is the key point to reveal the relation between the disc and the corona at different accretion rates (Wang et al., 2004; Shemmer et al., 2006; Vasudevan & Fabian, 2007; Zhou & Zhao, 2010; Arcodia et al., 2019; Cheng et al., 2019; Weng et al., 2020).

Heating mechanism for AGN corona against strong inverse Compton cooling is unclear. Magnetic energy is suggested to be a major energy source that is carried into the corona by magnetic field and released as thermal energy through the magnetic reconnection process (e.g., Di Matteo, 1998; Miller & Stone, 2000; Merloni & Fabian, 2001; Liu et al., 2002a, 2003; Merloni et al., 2003; Wang et al., 2004). In particular, Liu et al. (2003) obtained both soft and hard X-ray spectra in different AGN accretion cases by the disc-corona model, where the corona is heated by the magnetic reconnection. Applying the magnetic-reconnection-heated corona model, one can well explain the positive correlation between the hard X-ray bolometric correction factor and the Eddington ratio. Meanwhile, the trend of the X-ray photon index increasing with the Eddington ratio can be also interpreted (Cao, 2009; You et al., 2012; Liu et al., 2016). The coupling of the magnetic-reconnection-heated disc-corona and the jet was used to illustrate the “outlier” relation of $L_{\rm R}\propto L_{\rm X}^{\sim 1.4}$ in the black hole X-ray binaries by Qiao & Liu (2015). They suggested that the X-ray emission of $2$–$10$ keV is dominated by the radiation of the disc-corona system. In order to reproduce the observed data of optical/UV and X-ray band in luminous AGNs, Cheng et al. (2020) further refined the model in Liu et al. (2016). They set the outer boundary of the disc-corona accretion flow as the self-gravity radius and efficiently considered both local and global energy-conservation. They found that the accretion energy transported to the corona depends on both the magnetic field and the accretion rate, which is consistent with the results found by Liu et al. (2016). They successfully applied this model to fit the spectra of 16 luminous AGNs.

We notice that the former magnetic-reconnection-heated corona model reproduces a relatively steep X-ray spectrum with the photon index $\Gamma_{\rm 2-10\,keV}>2.1$ (Liu et al., 2003; Cao, 2009; Liu et al., 2016; Cheng et al., 2020). Such steep spectra are understood as a consequence of the disc-corona coupling. When soft photons from the disc are up-scattered in the corona, it causes cooling of electrons and condensation of coronal gas to the disc. The higher the accretion rate in the disc, the lower the temperature and density in the corona. This leads to a small Compton $y-$parameter ($y=4kT_{\rm c}\tau_{\rm es}/m_{\rm e}c^{2}$ for optical-thin corona, where $T_{\rm c}$ is the temperature of the corona and $\tau_{\rm es}$ is the electron scattering depth), and a steep X-ray spectrum in luminous AGNs. Even all the accretion energy is efficiently transported from the disc to the corona by the magnetic field and subsequently emitted in the X-ray band, approximately half of the isotropic X-ray photons return to the disc and are mostly reprocessed as soft photons to take part in the Compton cooling again in the corona. Thus, the Compton $y-$parameter is anyhow not sufficiently large for producing a flat X-ray spectrum.

Some observational evidences suggest that certain luminous AGNs have much flat X-ray emission, i.e., $\Gamma_{\rm 2-10\,keV}=1.89\pm 0.05$ for radio-quiet quasars and $\Gamma_{\rm 2-10\,keV}\sim 1.7-1.9$ for Seyfert 1 galaxies (e.g., Reeves & Turner, 2000; Lubiński et al., 2016; Akylas & Georgantopoulos, 2021). If these hard X-ray photons are emitted through the inverse Compton scattering of thermal electrons, the Compton $y$-parameter should exceed the theoretically predicted value.

In order to produce a flat spectrum as observed in radio-quiet quasars or Seyfert 1 galaxies, it is necessary to reduce the coronal illumination on the disc, resulting in a decrease in the energy of soft photons. This can be realized by assuming a truncated disc, the cold clumps, or an outflowing corona (Zdziarski et al., 2000; Yuan, 2003; Liu et al., 2014). Nevertheless, the truncated disc is hard to produce a soft-state spectrum, and the cold clumps is to be further studied for the dynamical existence. Even though the radiation pressure-driven outflows could be optically thick with relatively low temperature, sustaining such an optically thick “corona” for steady X-ray emission is still questionable. On the other side, Cheng et al. (2020) have argued that the reflection component flattens the X-ray spectrum with $\Delta\Gamma_{\rm 2-10\,keV}\sim 0.1-0.2$. The large reflection component is suggested to present in the Seyfert population (Ricci et al., 2017; Akylas & Georgantopoulos, 2021). However, there is poor evidence for this component in quasars. Additionally, other accretion flows, such as those where coronal gas condenses into discs, are also suggested to produce relatively flat X-ray spectra (Liu et al., 2015). However, the current version of this model is only limited for the accretion rates below 0.1 times of the Eddington accretion rate.

The electrons in the disc-corona system are commonly assumed to be thermal. However, we notice that some works have suggested that thermal and non-thermal electrons co-exist in the corona. The thermal and non-thermal electrons have a hybrid energy distribution. The investigations on the X-ray emission of hybrid electrons began in 1980s (Guilbert et al., 1983; Kazanas, 1984; Zdziarski & Lightman, 1985; Svensson, 1987; Lightman & Zdziarski, 1987). In general, these works proposed that the $\gamma-\gamma$ process generates electron-positron pairs. This process takes effects on the optical depth of the corona and makes the result in changing X-ray emission. The inverse Compton scattering of coronal hybrid electrons was applied to the X-ray spectrum extending to $\sim 800$ keV in the soft state of Cyg X-1, (e.g., Gierliński et al., 1999). Without making any assumption about the shape of the particle distribution, Belmont et al. (2008) successfully solved the time-dependent kinetic equations for homogeneous, isotropic distributions of photons, electrons, and positrons. The particle heating and the radiation processes were fully considered. When the steady state is reached, non-thermal electrons are formed in the high-energy band. Thus, the emission properties of compact high energy sources can be well reproduced. Poutanen & Vurm (2009) and Vurm & Poutanen (2009) solved the coupled integro-differential kinetic equations for photons and electrons/positrons without any limitations on the photon and lepton energies. They reproduced the X-ray emission of Cyg X-1 in the soft state and suggested that the corona is magnetically dominated. In particular, magnetic reconnection in the context of accreting black-hole systems was considered by Beloborodov (2017) and Sridhar et al. (2021, 2023). Recently, Zhong & Wang (2022) suggested that the non-thermal electrons are accelerated by the first-order Fermi mechanism which is related to the magnetic-reconnection process. Therefore, in the corona, due to the continuous magnetic reconnection, some electrons can be accelerated to be non-thermal (see Kagan et al., 2015, for review). The thermal and non-thermal electrons are expected to be co-existing presented by a hybrid energy distribution in the corona. Consequently, they can modify the radiation spectrum of the thermal electrons. The inverse Compton scattering of these non-thermal electrons in the hot corona can be used to explain the observed hard X-ray spectrum.

Although the radiation process of non-thermal electrons for X-ray emission has been well investigated, the coupling of the disc and the corona containing hybrid electrons needs to be further studied. We perform the magnetic-reconnection-heated corona model in this paper. The model has been well established (Liu et al., 2002a, 2003, 2016; Cheng et al., 2020). In this work, the contribution of the radiation from the non-thermal electrons is included in the model, enabling us to successfully reproduce the flat X-ray spectrum ($\Gamma_{\rm 2-10\,keV}<2.1$). In light of the coexistence between thermal electrons and non-thermal electrons in the magnetic reconnection process, we suppose that a fraction of the magnetic energy ($f_{\rm th}$) in the corona is allocated to thermal electrons, while the rest is allocated to non-thermal electrons. We show the dependence of the structure and spectrum of the disc-corona system on the energy allocated to the thermal electrons. We also investigate the effects of the non-thermal electrons and the magnetic field on the spectrum. Through comparison to the data of five individual luminous AGNs with $\Gamma_{\rm 2-10\,keV}<2.1$ by our modeling results, we suggest that the Comptonization of hybrid electrons can well reproduce the flat X-ray spectra if a large fraction of magnetic energy ($>40\%$) is assigned to the non-thermal electrons.

This paper is organized as follows. In section 2, we introduce the model of disc and corona coupled by the magnetic field, which includes hybrid electrons in the corona. In section 3, we present the computational results on the structure and the spectrum of the disc-corona system in our model. In section 4, we apply the model for some individual objects in the hard X-ray band. The discussion and the conclusion are presented in Sections 5 and 6, respectively.

In this paper we adopt the magnetic-reconnection-heated corona model proposed by Liu et al. (2002a) and developed by Liu et al. (2003, 2016) and Cheng et al. (2020). In the model, the thin disc extending into the innermost stable orbit is sandwiched by the plane-parallel corona. The magnetic fields are generated by the dynamo mechanism in the accretion disc. Because of the magnetic buoyancy, the magnetic flux loops emerge into the corona at the Alfvén speed $V_{\rm A}$ and reconnect with the opposite-direction loops. During the reconnection process, the magnetic energy is released as heating to the corona. The heating flux $Q_{\rm cor}^{+}$ in the corona can be expressed by the magnetic field $B$ and the Alfvén speed $\rm V_{A}$ as

where the strength of the magnetic field $B$ is characterized by the magnetic coefficient $\beta_{\rm 0}$ as

In the equation above, $P_{\rm g,d}=\frac{\rho_{\rm d}kT_{\rm d}}{\mu m_{\rm H}}$ and $P_{\rm r,d}=\frac{aT_{\rm d}^{4}}{3}$ are the gas-pressure and the radiation-pressure in the disc respectively, where $T_{\rm d}$ and $\rho_{\rm d}$ are the temperature and the density in the disc middle plane respectively, $k=1.38\times 10^{-16}\,\rm erg~{}K^{-1}$ is the Boltzmann constant, $m_{\rm H}=1.67\times 10^{-24}\,\rm g$ is the mass of the hydrogen atom, $\mu=0.5$ is the molecular weight for an assumed chemical abundance of pure hydrogen, and $a=7.56\times 10^{-15}\,\rm erg~{}K^{-4}~{}cm^{-3}$ is the radiation constant.

In the new scenario, non-thermal electrons are supposed to coexist with the thermal electrons in the corona, following a power-law distribution in the energy range between $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$ as $n_{\rm e,pl}(\gamma)=C_{\rm 1}N_{\rm e,pl}\gamma^{-p}$, where $N_{\rm e,pl}$ is the number density of the non-thermal electrons and $C_{\rm 1}\,=\,\frac{1-p}{\gamma_{\rm 2}^{1-p}-\gamma_{\rm 1}^{1-p}}$.

The Alfvén speed in Equation (1) is a function of magnetic field and coronal density, $V_{\rm A}=B/\sqrt{4\pi\mu m_{\rm H}(N_{\rm e,th}+N_{\rm e,pl})}\approx B/\sqrt{4\pi\mu m_{\rm H}N_{\rm e,th}}$, where the number density of thermal electrons, $N_{\rm e,th}$, is usually much larger than that of non-thermal electrons, $N_{\rm e,pl}$, as shown in section 5.2.

The ratio of the heating flux $Q_{\rm cor}^{+}$ to the total gravitational energy is defined as

where $R_{\rm s}\equiv 2GM_{\rm BH}/c^{2}$ is the Schwarzschild radius, $Q_{\rm grav}={3GM_{\rm BH}\dot{M}\over 8\pi R^{3}}\left[{1-\left({3R_{\rm S}\over R}\right)^{1/2}}\right]$ is the total accretion energy flux liberated in the accretion disc, $R$ is the radius in the disc, $\dot{M}$ is the accretion rate, $\rm{M_{BH}}$ is the black hole mass, $c$ is the light speed, and $G$ is the gravitational constant.

Since a fraction $f$ of the accretion energy is carried into the corona, the equation for the energy conservation in the thin disc is modified as

where the scattering opacity is $\kappa_{\rm es}=0.4\,\rm~{}{cm^{2}~{}g^{-1}}$, the free-free opacity is $\kappa_{\rm ff}=6.4\times 10^{22}\rho_{\rm d}T_{\rm d}^{-7/2}\,\rm~{}{cm^{2}~{}g^{-1}}$, and the Stefan-Boltzmann constant is $\sigma=5.67\times 10^{-5}\,\rm erg~{}K^{-4}~{}cm^{-2}~{}s^{-1}$. The disc thickness is $H_{\rm d}=\sqrt{P_{\rm t,d}/\rho_{\rm d}}/{\Omega}$, where the total pressure is $P_{\rm t,d}=(1+1/\beta_{0})(P_{\rm g,d}+P_{\rm r,d})$, and the angular velocity is $\Omega=\sqrt{GM_{\rm BH}/R^{3}}$.

The corresponding angular momentum equation in the thin disc is

where $\tau_{\rm r\varphi}=\alpha P_{\rm t,d}$ is the viscosity stress, and $\alpha$ is the viscosity parameter. From Equation (4) and Equation (5), we can find that the temperature and the density in the disc are tightly related with the input parameters of $M_{\rm BH}$, $\dot{M}$, $\alpha$, and $\beta_{\rm 0}$. Moreover, they are also affected by the coronal density through the definition of the energy fraction $f$ in Equation (3).

The magnetic energy transferred from the disc is liberated in the corona via the process of the magnetic reconnection. If the density of the corona is not sufficiently high to radiate the gained energy, strong heat conduction to the chromospheric layer between the disc and the corona makes the gas evaporation into the corona. Once the density is reached a certain value, an equilibrium is established between magnetic heating and Compton cooling. Meanwhile, the evaporating gas steadily supplies for coronal accretion. Therefore, the corona and the disc are coupled in terms of energy and matter. Due to the coexistence of thermal and non-thermal electrons in the corona, a fraction ($f_{\rm th}$) of magnetic energy is suggested to heat the thermal electrons while the rest portion ($1-f_{\rm th}$) is used to accelerate the non-thermal electrons. The corona with certain density will be cooled by the inverse Compton scattering of the hybrid-distributed electrons. Finally, a steady corona with hybrid electrons is formed when the equilibrium is reached between the magnetic-reconnection heating and the Compton cooling. These coupling processes between the disc and the corona can be described using the following equations.

The energy balance equations for thermal and non-thermal electrons are

and

where $G(\gamma_{\rm 1},\gamma_{\rm 2},p)=\frac{1}{3}\int_{\rm\gamma_{\rm 1}}^{\gamma_{\rm 2}}\gamma^{2-p}\beta^{2}d\gamma$.

The thermal conduction-induced evaporation as

where $T_{\rm c}$ is the temperature of the thermal electrons. The constants in the above equations are: the ratio of specific heats $\gamma_{\rm 0}=5/3$ and the thermal conduction coefficient $k_{0}=10^{-6}\,\rm{erg~{}cm^{-1}~{}s^{-1}~{}K^{-7/2}}$. The length of the magnetic loops in the corona $\ell_{\rm c}$ is approximate to $R$ for the optical-thin corona. As the isotropic incident photons are up-scattered in the plane-parallel corona, in Equations (6) and (7), we introduce a coefficient $\lambda_{\rm\tau}$, which is generally larger than 1. The energy density of soft photons, $U_{\rm rad}$, is contributed by the intrinsic disc emission released by the viscous heating and the reprocessing of the coronal irradiation. We present it in the following.

The inverse Compton scattering of the isotropically distributed electrons implies that about half of the Comptonization photons upward-escape from the corona, while the rest photons are backward to the disc. Within only a small fraction undergoing reflection (albedo $a=0.2$), these backward photons are reprocessed as seed photons for Compton cooling in the corona. Therefore, the soft photon flux for coronal inverse Compton scattering is $F_{\rm s}=(1-f)Q_{\rm grav}+\frac{1}{2}(1-a)F_{\rm c}$, where $F_{\rm c}$ is the total Compton emission flux from both the thermal electrons ($F_{\rm th}$) and the non-thermal electrons ($F_{\rm pl}$). The thermal flux $F_{\rm th}$ can be expressed as the sum of the net flux gained during scattering and the original flux brought by the scattered soft photons, i.e., $F_{\rm th}=f_{\rm th}fQ_{\rm grav}+F_{\rm s}(1-e^{-\tau_{c}})$, where $\tau_{c}\equiv\lambda_{\rm\tau}N_{\rm e,th}\sigma_{\rm T}\ell_{\rm c}$ is the effective optical depth and $(1-e^{-\tau_{c}})$ is the scattering probability of a soft photon. The flux $F_{\rm pl}$ is corresponding to the magnetic energy allocated to the non-thermal electrons, i.e., $F_{\rm pl}=(1-f_{\rm th})fQ_{\rm grav}$. Therefore, the flux and the energy density for the soft photons can be expressed as

respectively.

For given values of the black hole mass $M_{\rm BH}$, the accretion rate $\dot{M}$, the viscosity parameter $\alpha$, the magnetic coefficient $\beta_{\rm 0}$, the fraction of magnetic energy to thermal electrons $f_{\rm th}$, and the parameters for the non-thermal electrons including $\gamma_{\rm 1}$, $\gamma_{\rm 2}$ and $p$, we can numerically solve the Equations (4), (5), (6), (7), and (8) by combing Equations (1), (2), (3), and (9) at the grid point of radius R, with an initial value of $\lambda_{\rm\tau}=1.0$. The temperature $T_{\rm d}$ and the density $\rho_{\rm d}$ in the disc, the temperature $T_{\rm c}$ and the number density $N_{\rm e,th}$ of thermal electrons, and the number density $N_{\rm e,pl}$ of non-thermal electrons in the corona are subsequently derived. The values of $f$ and $U_{\rm rad}$ can be also obtained according to their definitions in Equation (3) and Equation (9). Utilizing the structural parameters $T_{\rm c}$, $N_{\rm e,th}$ and $U_{\rm rad}$ of the disc-corona system, we can calculate the X-ray spectrum emitted by thermal electrons through Monte Carlo simulation. We need to check the self-consistency from the Monte Carlo simulation result, i.e., whether the flux of escaped photons in the Monte Carlo simulation is equal to the combined flux from upward Compton emission and directly escaped (non-scattered) soft photons, i.e., ${F_{\rm th}/2}+F_{\rm s}e^{-\tau_{\rm c}}$. In order to fulfill this condition, we fine-tune the parameter $\lambda_{\rm\tau}$ and repeat the calculation for the disc-corona structure and the Monte Carlo simulation. Subsequently, we update the value of $R$ to the adjacent grid point to computer the structure and spectrum of the disc-corona system. Ultimately, we determine the global structural parameters of the disc-corona system and the overall spectrum contributed by thermal electrons.

In order to calculate the radiation spectrum of non-thermal electrons, we use the modified Compton spectrum formalism in the Thomson limit proposed by Zhong & Wang (2013). The presentation is

where the classical electron radius is $r_{\rm 0}=2.82\times 10^{-13}$ cm and $\varsigma(\frac{\epsilon_{\rm 1}}{\epsilon},\gamma)=2\frac{\epsilon_{\rm 1}}{\epsilon}\rm{ln}\frac{\epsilon_{1}}{\epsilon(1+\beta)^{2}\gamma^{2}}+\frac{2\beta}{1+\beta}\frac{\epsilon_{1}}{\epsilon}+(1+\beta)(1+\beta^{2})\gamma^{2}-\left(\frac{\epsilon_{1}}{\epsilon}\right)^{2}\frac{1}{(1+\beta)\gamma^{2}}$. The parameters $\epsilon$ and $\epsilon_{\rm 1}$ represent the energy of the initial photon and the energy of the photon after scattering, respectively. The differential density of the seed photons at the radius $R$ is

where $T_{\rm eff}(R)=cU_{\rm rad}/(2\sigma)$ is the effective temperature at the radius $R$ of the thin disc. Finally, the total radiation rate emitted by the non-thermal electrons per volume is

where the lower-energy limit is $\epsilon_{\rm 0}=\epsilon_{\rm 1}(1-\beta_{\rm 2})/(1+\beta_{\rm 2})$ with $\beta_{\rm 2}=\sqrt{1-1/\gamma_{\rm 2}^{2}}$.

We can calculate the total luminosity of electrons in the corona as

where $R_{\rm in}$ and $R_{\rm out}$ are the inner boundary and outer boundary of the accretion disc-corona system, respectively. As approximately half of the Compton photons are backward to the disc, the radiation luminosity from the coronal upper layer is also half of the total luminosity.

In our previous works (Liu et al., 2016; Cheng et al., 2020), we have shown the influence of the accretion rate and black hole mass on both the structure and spectrum. In this paper, we focus on the effect of the Comptonization of non-thermal electrons on the X-ray spectrum in luminous AGNs. Thus, we fix the black hole mass $M_{\rm BH}=10^{8}M_{\rm\odot}$ and the accretion rate $\dot{M}=0.1\dot{M}_{\rm Edd}$. The viscosity parameter $\alpha$ is 0.3. We change the energy fraction $f_{\rm th}$, the parameters ($\gamma_{\rm 1}$, $\gamma_{\rm 2}$, $p$) for the distribution of non-thermal electrons, and the magnetic coefficient $\beta_{\rm 0}$ in order to investigate their effects on the properties of the structure and the spectrum of the disc-corona system.

The numerical results in the preceding section demonstrate that the X-ray spectrum is significantly affected by the Comptonization of non-thermal electrons. Especially, for a relatively narrow energy range of the non-thermal electrons, the Comptonization in the corona will generate a relatively flat X-ray spectrum with $\Gamma_{\rm 2-10\,keV}<2.1$. Therefore, we can use this model to reproduce the observed X-ray spectrum with $\Gamma_{\rm 2-10\,keV}<2.1$.

Cheng et al. (2020) applied the refined magnetic-reconnection heating corona model to fit the multi-band spectra of 20 luminous AGNs. In their model, only thermal electrons are included in the corona. In general, the modeling fitting is suitable for the majority of AGN spectra. However, there were four objects (Ton 730, Mrk 290, SBS 1136$+$594, and Mrk 1310) that were not well reproduced because their X-ray spectra are excessively flat. The X-ray spectra of these objects can be fitted by a power-law with photon indices $\Gamma_{\rm 2-10\,keV}$ in the range of $1.53$–$1.64$. The accretion rates of these objects are around $\dot{m}\sim 0.04-0.18$. Besides, PG 1138+222 is also included due to its accretion rate $\dot{m}=0.07$ and its X-ray spectrum fitted by a power-law with photon index $\Gamma_{\rm X}=1.92^{+0.07}_{-0.07}$ (Cheng et al., 2019). In this section, we apply our model to explain the observational spectra of these AGNs in the optical/UV (UVOT) and X-ray ($2$–$10$ keV) wavelengths.

We take the numbers of the black hole mass, the accretion rate, and the outer boundary radius for each object as same as those in Cheng et al. (2020), and make sure that the optical/UV spectral data of the AGNs can be well reproduced. Due to the lack of higher-energy data, we only apply our model to produce the hard X-ray emission in the energy band of $2$–$10$ keV , and set the upper limit of the electron Lorentz factor less than $50$. In general, for each object, we initially set $\beta_{\rm 0}$ to be $50$. If the X-ray luminosity generated by the disc-corona system, which includes only the thermal electrons, is significantly larger than the observed X-ray luminosity, we then increase $\beta_{\rm 0}$. Afterwards, we adjust the parameters of $f_{\rm th}$, $\gamma_{\rm 1}$, $\gamma_{\rm 2}$, and $p$ to reproduce the observed X-ray spectrum. As shown in the subsection 3.3, the effect of $p$ on the spectrum is weak when the electron Lorentz factors are in a narrow range. Therefore, we only consider two cases of $p=1.5$ and $p=2.5$ when comparing the observed data.

The modeling result for each object is plotted by the thick black line in Figure 8. The parameters are noted as $m=M_{\rm BH}/M_{\rm\odot}$ and $\dot{m}=\dot{M}/\dot{M}_{\rm Edd}$. For Ton 730, Mrk 290, and PG 1138+222, their observed luminosity ratios $L_{\rm 2-10\,keV}/L_{\rm UVOT}$ are 0.26, 0.45, and 0.34, respectively. The number of $f_{\rm th}$ is around $0.5$–$0.6$. In other words, more than $40\%$ of the magnetic energy in the corona is allocated to the non-thermal electrons. For SBS 1136+594 and Mrk 1310, their observed luminosity ratio $L_{\rm 2-10\,keV}/L_{\rm UVOT}$ are 0.81 and 1.15, respectively. A smaller $f_{\rm th}$ number ($\sim 0.2-0.3$) is adopted to reproduce their flat X-ray spectra. The grey line in each panel of Figure 8 is the spectrum calculated from the model that only includes thermal electrons in the corona. Because the case of pure thermal electrons has the same magnetic field to the case of the hybrid-distributed electrons, the emission reproduced by the pure thermal electrons can be compared to the observational one. However, its photon index is dramatically larger than that obtained from the hybrid-distributed electrons.

In conclusion, the current model containing the hybrid electrons can well reproduce the observed flat X-ray spectra with $\Gamma_{\rm 2-10\,keV}<2.1$ in luminous AGNs.

We develop the magnetic-reconnection-heated corona model in which both thermal and non-thermal electrons are included. In the model, the magnetic energy liberated in the corona is cooled through the inverse Compton scattering by the hybrid-distributed electrons. We have found that the parameter, $f_{\rm th}$, describing the fraction of coronal energy allocated to thermal electrons, takes significant effects on the structure and radiation spectrum of the disc-corona system. Additionally, the spectrum strongly depends on both the non-thermal electron energy distribution and the magnetic field. The X-ray luminosity and the X-ray photon index ($\Gamma_{\rm 2-10\,keV}$) depend strongly on both the energy fraction $f_{\rm th}$ and the energy distribution of non-thermal electrons. We apply our model to explain the observed optical/UV and X-ray data of five luminous AGNs with the flat X-ray spectra. We suggest that the disc-corona system can make a flat X-ray spectrum ($\Gamma_{\rm 2-10\,keV}<2.1$) if a large fraction ($>40\%$) of the magnetic energy is allocated to the non-thermal electrons.

We thank the referee for helpful suggestions and comments. We appreciate Dr. Huaqing Cheng for providing the observed data and Dr. Xiaogu Zhong and Dr. Xiaolin Yang for useful suggestions. This work has the financial support of the National Key R&D Program of China (2023YFE0101200 and 2021YFA1600402). J.Y. is supported by the Strategic Priority Research Program of Chinese Academy of Sciences, grant No. XDB 41000000, the Natural Science Foundation of Yunnan Province (No. 202201AT070158), the Yunnan Revitalization Talent Support Program Young Talent Project (Notice: The publicity has been completed, but the official document has not yet been issued), the National Natural Science Foundation of China (grants 12133011, 12288102), and the International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001). J.M. is supported by the National Natural Science Foundation of China 11673062, CSST grant CMS-CSST-2021-A06, and the the Yunnan Revitalization Talent Support Program (YunLing Scholar Project). B.F. is supported by the National Natural Science Foundation of China 12073037 and 12333004.

The data underlying this article will be shared on reasonable request to the corresponding author.