A MAGNETIC RECONNECTION ORIGIN FOR THE SOFT X-RAY EXCESS IN AN ACTIVE GALACTIC NUCLEUS

Soft X-ray excess is a major component of many Type 1 active galactic nuclei (AGNs), especially Narrow Line Seyfert 1 (NLS1) galaxies (e.g., Boller et al. 1996; Laor et al. 1997). Below 1 keV in the X-ray band, the X-ray data are above the low energy extrapolation of the best-fitting 2–10 keV power law characterized by a power law index of Γ ∼ 2.0 (Gondhalekar et al. 1997); this is so-called soft X-ray excess. Historically, the soft X-ray excess has often been fit using a blackbody model, yielding best-fit temperatures in the range of 0.1–0.2 keV (Czerny et al. 2003; Gierliski & Done 2004). The standard disk model (Shakura & Sunyaev 1973) gives a maximum effective temperature of the accreting material of $kT\sim 10({\dot{m}}/{m_8})^{1/4}$ eV, where $\dot{m}$ is the accretion rate in units of the Eddington accretion rate (i.e., $\dot{m}=\dot{M}/\dot{M_E}$) and m₈ = M/10⁸ M_☉. However, according to this model, assuming a high accretion rate and low-mass black holes, the peak of the accretion disk blackbody spectrum is just 0.1 keV. Current observations show, however, that the black hole mass of a source with a soft X-ray excess exceeds 10⁶ M_☉ (Crummy et al. 2006). Because different NLS1 galaxies have different masses and accretion rates, it is difficult to maintain a stable temperature within a range of 0.1–0.2 keV for the blackbody spectrum of the accretion disk. For the standard disk model, it is difficult to explain these properties (Turner & Pounds 1989; Done et al. 2012).

From the analysis of the soft X-ray data, it is found that strong emission and absorption features appear in the soft X-ray spectra of most AGNs. As discussed in Gierliski & Done (2004), the imprint of spectral features such as the strong jump in opacity at ∼0.7 keV from O vii, O viii, and the Fe M-shell can lead to an apparent soft excess below this energy. However, the problem is that some parts of the soft excess are often smooth and certain predicted features are not evident. Schurch & Done (2007, 2008) computed the X-ray spectra of outflowing gas with density and velocity, demonstrating that a high outflow velocity would be required if absorption in outflows solely explains the soft excesses observed in AGN spectra. A high velocity of 0.9c even exceeds the speeds of the relativistic wind components detected through energy-shifted absorption lines. Schurch et al. (2009) found that the models of line-driven accretion-disk winds do not attain sufficiently high velocities.

The origin of the soft X-ray excess in AGNs is as yet unknown. We try to explore a new radiation mechanism by considering the role of the magnetic field in accretion disks. We present a new scenario as follows: magnetic reconnection could happen in a thin layer on the surface of the accretion disk as in a solar active region (Kurtz 1984). Electrons are accelerated by the shock wave and turbulence is triggered by magnetic reconnection (Kirk et al. 1994). These electrons then emit X-rays through Compton scattering with photons from the accretion disk. In Section 2, we estimate the magnetic field and the energy released by magnetic reconnection in accretion disks to show whether the released magnetic energy is sufficient to produce the soft X-ray excess. In Section 3, we depict the radiative processes and obtain the Compton spectrum. In Section 4, we apply a new model to AGNs with strong soft X-ray excesses. Finally, Section 5 summarizes our main results and presents our conclusions.

We assume that magnetic reconnection emerges frequently in a thin layer on the surface of the standard accretion disk. In previous works, the magnetic field B was calculated adopting the equipartition of gas energy and magnetic energy in the disk given by Miller & Stone (2000): ξ ≡ n_diskkT_disk/(B²/8π). This result is not a useful guide, however, because the authors ignore radiation, assume an isothermal equation of state, and do not show how the magnetic field scales in the radiation-dominated regime. In fact, it is unclear why the magnetic energy density should scale as a constant fixed ratio of the midplane gas pressure. The best numerical simulations have been done by Hirose et al. (2009a, 2009b) and Blaes et al. (2011). Their simulations show that the vertically averaged rms of the magnetic field scales as the square root of the total (gas + radiation) pressure, i.e.,

$$\begin{eqnarray} &&B=\sqrt{8\pi \xi ^{-1}\left(nkT+\frac{aT^4}{3}\right)}, \end{eqnarray} \tag{ 1 }$$

where a = 7.564 × 10⁻¹⁵ erg cm⁻³ K⁻⁴. Figure 1 shows the distribution of the magnetic field versus the radius of the accretion disk in units of R_S with the parameters α = 0.1, ξ = 1, and 10⁸ M_☉. It is shown that the magnetic field rapidly declines along the radius of the accretion disk inside about 20 R_S, but is relatively flat outside 20 R_S. The different curves are calculated at accretion rates of $\dot{m}$ = 0.01, 0.1, and 1, respectively ($\dot{m}$ has units of the Eddington accretion rate). The magnetic field in the inner disk is stronger than that in the outer disk, implying that the magnetic energy is mainly deposited in the inner disk. We calculate the energy released by fast magnetic reconnection on the accretion disk, in which the magnetic dissipation scales as B²V_A, where V_A is the Alfvén speed. But it has an upper limit near the disk surface, which is limited by the Poynting flux (see Figure 2) as (Hirose et al. 2006)

$$\begin{eqnarray} &&F_p=fc_s\frac{B^2}{8\pi }, \end{eqnarray} \tag{ 2 }$$

where the factor f represents the departure from hydrostatic balance that permits the upward motion and c_s is the sound velocity. From Figure 2, we know that the Poynting flux rapidly decreases from the innermost steady radius to 20 R_S. Because the gas pressure is far less than the radiation pressure inside 20 R_S, the Poynting flux is approximately given by

$$\begin{eqnarray} F_p&\approx& 4.98\times 10^{18}\left(\frac{4}{3}+\frac{2}{\xi }\right)^{\frac{1}{2}} \alpha ^{-1}m_8^{-1}\dot{m} {\left(\frac{R}{R_S}\right)}^{-3}\nonumber\\ &&\times f\xi ^{-1}\ {\rm erg\ cm^{-2}\ s^{-1}}, \end{eqnarray} \tag{ 3 }$$

where R_S is the Schwarzschild radius. Assuming that almost all the energy released by magnetic reconnection is converted to soft X-ray radiation energy, we can estimate the maximum soft X-ray luminosity by integrating the Poynting flux from the inner radius to 20 R_S as

$$\begin{eqnarray} &&L_{{\rm Soft}}\approx 2.81\times 10^{46}\left(\frac{4}{3}+\frac{2}{\xi }\right)^{\frac{1}{2}} \alpha ^{-1}m_8^{2}\dot{m}f\xi ^{-1}\ {\rm erg\ s^{-1}}. \nonumber\\ \end{eqnarray} \tag{ 4 }$$

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3.1. The Electron Spectrum

Considering that electrons are accelerated by shock waves and turbulence in magnetic reconnection, we can obtain the electron energy spectrum as (Droege & Schlickeiser 1986)

$$\begin{eqnarray} &&F(E)=AE^{-q(E)}, \end{eqnarray} \tag{ 5 }$$

where the electron spectrum index is

$$\begin{eqnarray} &&q(E)=-\frac{\eta +a+1}{2}+\frac{3+a-\eta }{2}\!\left[\!\!1+ \frac{4\lambda {\sqrt{2m_e}}^{\eta +b}}{{(3+a-\eta)}^2}E^{(\eta +b)/2}\!\right]^{1/2},\nonumber\\ \end{eqnarray} \tag{ 6 }$$

and the parameters η, a, λ, and b are determined by the microcosmic structure of the shock wave and turbulence expressed as follows (Bogdan & Schlickeiser 1985; Droege & Schlickeiser 1986):

$$\begin{eqnarray*} T(p)&=&T_0p^{-b},\nonumber \\ K(p)&=&\delta \upsilon (p)p^{2-q}=\kappa p^{\eta },\nonumber \\ D(p)&=&\alpha _2\frac{V_A^{2}}{K}p^{2},\nonumber \\ \dot{p_G}&\approx& \alpha _1\frac{V_S^{2}}{K}p^{2}, \nonumber \end{eqnarray*}$$

where a = α₁V_S²/α₂V_A², λ = κ/α₂V_A²T₀. When the electrons are not accelerated to high energies for fast magnetic reconnection, the electron spectrum will become a power law with constant index (Droege & Schlickeiser 1986), given by

$$\begin{equation} F(E)=AE^{-q}, E_1\le E\le E_2; \end{equation} \tag{ 7 }$$

E₁ and E₂ are the cutoffs in the spectrum. When E₂ ≫ E₁ and q > 1, the factor A is given by $A=N_e(q-1)E_1^{q-1}$, in which N_e is the number density of electrons. The kinetic energy E of non-relativistic electrons is expressed as E = (γ − 1)m_ec², in which $\gamma =1/\sqrt{1-\beta ^2}$ and βc are the Lorentz factor and velocity of the electron, respectively. When E ≪ m_ec², $\beta =\sqrt{2E/m_ec^2}$.

3.2. Comparison of Electron Energy Losses through Bremsstrahlung and Compton Radiation

The electrons in the magnetic field produce Bremsstrahlung, inverse Compton scattering, and synchrotron emission. Because the electrons are non-relativistic, we can ignore synchrotron radiation. Next, we will compare the electron energy losses through Bremsstrahlung and Compton radiation.

The energy loss rate due to Compton radiation of an electron with energy E is given by (Blumenthal & Gould 1970):

$$\begin{eqnarray} &&\frac{{dE}}{dt}=\sigma _Tc\gamma ^2\int (1-\beta \cos \theta)^2\epsilon dn=\sigma _Tc\gamma ^2\left(1+\frac{1}{3}\beta ^2\right)U_{{\rm ph}},\nonumber\\ \end{eqnarray} \tag{ 8 }$$

where σ_T is the Thompson cross section and U_ph is the energy density of soft photons. The total energy loss rate of the non-thermal electrons is given by

$$\begin{eqnarray} P_c&=&\sigma _Tc U_{{\rm ph}}\int ^{E_2}_{E_1}A\gamma ^2\left(1+\frac{1}{3}\beta ^2\right)E^{-q}{dE},\nonumber \\ &=&\sigma _Tc U_{{\rm ph}}N(q-1)\left(\frac{E_1}{m_ec^2}\right)^{q-1}\nonumber\\ &&\times \int ^{\beta _2}_{\beta _1}\gamma ^5\beta \left(1+\frac{1}{3}\beta ^2\right)(\gamma -1)^{-q}d\beta. \end{eqnarray} \tag{ 9 }$$

The energy loss rate due to Bremsstrahlung radiation of an electron with energy E is given by (Gould 1980)

$$\begin{equation} \frac{{dE}}{dt}=\frac{2}{\pi }\sigma _T n_Z Z^2 \alpha _{f}m_ec^3\beta \left(1+\frac{5}{12}\beta ^2\right), \end{equation} \tag{ 10 }$$

α_f = 1/137 is the fine structure constant, and n_Z is the number density of the ion. The total energy loss rate is

$$\begin{eqnarray} P_b&=&\sum _Zn_Z Z^2\frac{2}{\pi }\sigma _T \alpha _{f}m_ec^3A\int ^{E_2}_{E_1}\beta \left(1+\frac{5}{12}\beta ^2\right)E^{-q}dE,\nonumber \\ &=&\sum _Zn_Z Z^2\frac{2}{\pi }\sigma _T \alpha _{f}m_ec^3N(q-1)\left(\frac{E_1}{m_ec^2}\right)^{q-1}\nonumber\\ &&\times \int ^{\beta _2}_{\beta _1}\gamma ^3\beta ^2\left(1+\frac{5}{12}\beta ^2\right)(\gamma -1)^{-q}d\beta. \end{eqnarray} \tag{ 11 }$$

Then, we have

$$\begin{eqnarray} &&\frac{P_c}{P_b}=\frac{\pi U_{ph}}{2\alpha _f n_p m_ec^2}\frac{H_1(\beta _1,\beta _2)}{H_2(\beta _1,\beta _2)}, \end{eqnarray} \tag{ 12 }$$

where n_p = ∑n_ZZ²,

$$\begin{equation} H_1(\beta _1,\beta _2)=\int ^{\beta _2}_{\beta _1}\gamma ^5\beta \left(1+\frac{1}{3}\beta ^2\right)(\gamma -1)^{-q}d\beta, \end{equation} \tag{ 13 }$$

and

$$\begin{eqnarray} &&H_2(\beta _1,\beta _2)=\int ^{\beta _2}_{\beta _1}\gamma ^3\beta ^2\left(1+\frac{5}{12}\beta ^2\right)(\gamma -1)^{-q}d\beta. \end{eqnarray} \tag{ 14 }$$

In the accretion disk, where $U_{ph}=\sigma T_{{\rm eff}}^4/c$, T_eff is the effective temperature of the disk, σ is the radiation constant, and n_p is the number density of the proton, we obtain

$$\begin{eqnarray} \frac{P_c}{P_b}&=&\frac{\pi \sigma T_{{\rm eff}}^4}{2\alpha _f n_p m_ec^3}\frac{H_1(\beta _1,\beta _2)}{H_2(\beta _1,\beta _2)}\nonumber\\ &=&1.78\times 10^7 \alpha \dot{m}^3 r^{-\frac{9}{2}}(1-r^{-\frac{1}{2}})^2 \frac{H_1(\beta _1,\beta _2)}{H_2(\beta _1,\beta _2)}, \end{eqnarray} \tag{ 15 }$$

where r = R/3R_s, $\dot{m}=\dot{M}/\dot{M}_{{\rm Edd}}$, and $\dot{M}_{{\rm Edd}}$ is the Eddington accretion rate. Because H₁(β₁, β₂) > H₂(β₁, β₂), P_c ≫ P_b in the inner region of the accretion disk for the larger accretion rate, indicating that the Compton radiation of electrons dominates the Bremsstrahlung radiation.

3.3. Compton Radiation

Because the magnetic field decreases along the radius of the accretion disk, the parameters of accelerated electrons will change with radius. Assuming that E₁, E₂, and q are independent of the magnetic field, we can estimate the number density of electrons along the radius. Assuming that the energy loss rate of electrons equals the magnetic dissipation, i.e., P_c = B²V_A/8πh_r, in which h_r is the characteristic scale of reconnection, we obtain

$$\begin{equation} N(R)(q-1)(\frac{E_1}{m_ec^2})^{q-1}=\frac{B^2 V_A}{8\pi h_r \sigma _Tc U_{ph}}H_1^{-1}(\beta _1,\beta _2). \end{equation} \tag{ 16 }$$

When an electron is not accelerated to a high energy, the Compton spectrum in the Thomson limit given by Blumenthal & Gould (1970) for a relativistic electron should be changed as

$$\begin{eqnarray} \frac{dN_{\gamma,\epsilon }}{dtd\epsilon _1}&=&\frac{\pi r_0^2 c}{2\gamma ^4 \beta ^4}\frac{n(\epsilon)d\epsilon }{\epsilon ^2} \left\lbrace 2\epsilon _1 ln\frac{\epsilon _1}{(1+\beta)^2 \gamma ^2 \epsilon }+\frac{2\beta }{1+\beta }\epsilon _1\right.\nonumber\\ &&\left.+(1+\beta)(1+\beta ^2)\gamma ^2\epsilon -\frac{\epsilon _1^2}{(1+\beta)\gamma ^2\epsilon }\right\rbrace, \end{eqnarray} \tag{ 17 }$$

where n()d represents the differential photon density, ₁ is the energy of the scattered photon, and the electron has an energy E = (γ − 1)m_ec² and a velocity βc. When β → 1, the above formula returns to the Compton spectrum produced by a high-energy electron in the Thomson limit (Blumenthal & Gould 1970). We note that the above distribution function equals zero at β_* = (₁/ − 1)(₁/ + 1)⁻¹. Since the differential number of electrons is dN = AE^−qdE, the total Compton spectrum is given by

$$\begin{eqnarray} \frac{{dN}_{\epsilon }}{dt d\epsilon _1}&=&A\int E^{-q}dE \frac{dN_{\gamma,\epsilon }}{dtd\epsilon _1}=N_e(R)(q-1)\left(\frac{E_1}{m_ec^2}\right)^{q-1}\nonumber\\ &&\times \int _{\beta _{{\rm min}}}^{\beta _{{\rm max}}} (\gamma -1)^{-q}\gamma ^3\beta d\beta \frac{dN_{\gamma,\epsilon }}{dtd\epsilon _1}, \end{eqnarray} \tag{ 18 }$$

where dγ = γ³βdβ, β_min = max [β_*, β₁], β_max = β₂, β₁, and β₂ correspond to the energy cutoffs E₁ and E₂ in the electron spectrum. For the soft photons in the accretion disk, the differential photon density n(, R)d in a radius R is given by

$$\begin{eqnarray} &&n(\epsilon,R)d\epsilon =\frac{8\pi \epsilon ^2}{h^3c^3}(e^{\frac{\epsilon }{kT(R)}}-1)^{-1}d\epsilon, \end{eqnarray} \tag{ 19 }$$

where T(R) is the temperature of the disk surface at a radius R. We then obtain the Compton spectrum along the radius

$$\begin{eqnarray} \frac{{dN}}{dtd\epsilon _1}(R)&=&N_e(R)(q-1)\left(\frac{E_1}{m_ec^2}\right)^{q-1}\frac{\pi r_0^2 c}{2}\nonumber\\ &&\times \int _{\epsilon _0}^{\epsilon _1}\frac{n(\epsilon,R)}{\epsilon }H\left(\frac{\epsilon _1}{\epsilon },q\right)d\epsilon \nonumber \\ &=&\frac{3}{16}\frac{B^2 V_A}{8\pi h_r U_{ph}}H_1^{-1}(\beta _1,\beta _2)\nonumber\\ &&\times \int _{\epsilon _0}^{\epsilon _1}\frac{n(\epsilon,R)}{\epsilon }H\left(\frac{\epsilon _1}{\epsilon },q\right)d\epsilon, \end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray} &&H\left(\frac{\epsilon _1}{\epsilon },q\right)=\int _{\beta _{{\rm min}}}^{\beta _2} (\gamma -1)^{-q}\gamma ^{-1}\beta ^{-3} d\beta \zeta \left(\frac{\epsilon _1}{\epsilon },\beta \right), \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} \zeta \left(\frac{\epsilon _1}{\epsilon },\beta \right)&=&\frac{\epsilon _1}{\epsilon } \left\lbrace 2\ln \frac{\epsilon _1}{\epsilon }\frac{1}{(1+\beta)^2 \gamma ^2}+\frac{2\beta }{1+\beta }\right.\nonumber\\ &&\left.+\frac{\epsilon }{\epsilon _1}(1+\beta)(1+\beta ^2)\gamma ^2-\frac{\epsilon _1}{\epsilon }\frac{1}{(1+\beta)\gamma ^2}\right\rbrace,\qquad \end{eqnarray} \tag{ 22 }$$

and ₀ is given by (₁/₀ − 1)(₁/₀ + 1)⁻¹ = β₂, e.g., ₀ = ₁(1 − β₂)/(1 + β₂). In fact, H(₁/, q) can be simplified as

$$\begin{eqnarray} H\left(\frac{\epsilon _1}{\epsilon },q\right)&=&\int _{\beta _1}^{\beta _2} (\gamma -1)^{-q}\gamma ^{-1}\beta ^{-3} d\beta \zeta \left(\frac{\epsilon _1}{\epsilon },\beta \right),\nonumber\\ && 1 \le \frac{\epsilon _1}{\epsilon } <\frac{1+\beta _1}{1-\beta _1},\nonumber \\ H\left(\frac{\epsilon _1}{\epsilon },q\right)&=&\int _{\beta _*}^{\beta _2} (\gamma -1)^{-q}\gamma ^{-1}\beta ^{-3} d\beta \zeta \left(\frac{\epsilon _1}{\epsilon },\beta \right),\nonumber\\ && \frac{1+\beta _1}{1-\beta _1}\le \frac{\epsilon _1}{\epsilon } \le \frac{1+\beta _2}{1-\beta _2},\nonumber \\ H\left(\frac{\epsilon _1}{\epsilon },q\right)&=&0, \frac{\epsilon _1}{\epsilon } > \frac{1+\beta _2}{1-\beta _2}. \end{eqnarray} \tag{ 23 }$$

The total Compton radiation spectrum is given by

$$\begin{eqnarray} L({\epsilon _1})&=& \int _{R_{{\rm in}}}^{R_{{\rm out}}} 2\pi R h_r \epsilon _1^2 \frac{dN}{dtd\epsilon _1}(R) dR \nonumber \\ &=&\frac{3}{64}H_1^{-1}(\beta _1,\beta _2)\epsilon _1^2 \int _{R_{{\rm in}}}^{R_{{\rm out}}} \frac{B^2 V_A}{ U_{{\rm ph}}}\nonumber\\ &&\times \int _{\epsilon _0}^{\epsilon _1}\frac{n(\epsilon,R)}{\epsilon }H\left(\frac{\epsilon _1}{\epsilon },q\right)d\epsilon R dR, \end{eqnarray} \tag{ 24 }$$

where we take R_in = 3 R_S and R_out = 20 R_S for the radiation-dominated region of the accretion disk, and B²V_A/U_ph is given by

$$\begin{eqnarray} \frac{B^2 V_A}{ U_{{\rm ph}}}&=&\frac{8\pi c}{3 \sigma } \xi ^{-\frac{3}{2}} \left(\frac{2a^3}{3m_p n_p}\right)^{\frac{1}{2}} \frac{T^6}{T_{{\rm eff}}^4} \nonumber \\ &=&3.6\times 10^9 \alpha ^{-1} \xi ^{-\frac{3}{2}} (1-r^{-\frac{1}{2}})\ {\rm cm\ s^{-1}}, \end{eqnarray} \tag{ 25 }$$

where we use the structure of the accretion disk as

$$\begin{eqnarray*} T(R)&=&2.3\times 10^5\, (\alpha m_8)^{-\frac{1}{4}} r^{-\frac{3}{8}}\ {\rm K}, \nonumber \\ n_p(R)&=&4.3\times 10^{9}\, (\alpha m_8)^{-1} \dot{m}^{-2} r^{\frac{3}{2}}(1-r^{-\frac{1}{2}})^{-2}\ {\rm cm^{-3}}, \nonumber \\ T_{{\rm eff}}(R)&=&6.2\times 10^5\, (\dot{m}^{-1} m_8)^{-\frac{1}{4}} r^{-\frac{3}{4}}\ {\rm K}, \nonumber \end{eqnarray*}$$

and m₈ = M/10⁸ M_☉.

Finally, we obtain

$$\begin{equation} L(\bar{\epsilon })=6.9\times 10^{48} \alpha ^{-1} \xi ^{-\frac{3}{2}} m_8^2 \bar{L}(\bar{\epsilon },q,\beta _1,\beta _2)\ {\rm erg\ s^{-1}}, \end{equation} \tag{ 26 }$$

where $\bar{\epsilon }=\epsilon _1/{\rm keV}$, and

$$\begin{eqnarray} \bar{L}(\bar{\epsilon },q,\beta _1,\beta _2,kT_0)&=&\bar{\epsilon }^2 H_1^{-1}(\beta _1,\beta _2)\int _{1}^{r_{{\rm out}}}(1-r^{-\frac{1}{2}})rdr\nonumber\\ &&\times \int _{\bar{\epsilon }_0}^{\bar{\epsilon }} \epsilon \left(e^{\frac{\epsilon }{kT_0}r^\frac{3}{8}}-1\right)^{-1} H\left(\frac{\bar{\epsilon }}{\epsilon },q\right) d\epsilon,\nonumber\\ \end{eqnarray} \tag{ 27 }$$

where kT₀ = 2.0 × 10⁻²(αm₈)^−1/4, which corresponds to the temperature of the inner stable radius of the accretion disk, and r_out = 20/3. The curves of $\bar{L}(\bar{\epsilon },q,\beta _1,\beta _2,kT_0)$ with $\bar{\epsilon }$ for the parameters of q, β₁, β₂, and kT₀ are shown in Figures 3–5.

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In this section, we apply the model to fit the X-ray spectra (EPIC-pn of XMM-Newton data) of AGNs (PG quasars and NLS1 galaxies) with high-mass accretion rates given by Crummy et al. (2006, see Table 1). The soft X-ray radiation of these objects includes thermal radiation and inverse Compton scattering. The Compton spectrum caused by magnetic reconnection is added in XSPEC v12.7.1 (Arnaud 1996) through the initpackage and lmod codes. As these samples have different black hole masses and accretion rates, different temperatures of the blackbody spectrum of the accretion disks appear. According to the standard disk model, for high accretion rate and low-mass black holes, the peak of the accretion disk blackbody spectrum can occur in an X-ray band and we need to add the blackbody model in spectral fitting. Because the radiation of the accretion disk is a multi-temperature blackbody spectrum, we use the diskbb model of XSPEC to fit the thermal spectrum. A power law with a spectral index Γ is used to fit the hard X-ray component because the hard X-ray radiation is considered to be from inverse Compton scattering in the accretion disk corona (Liu et al. 2002). For some sources with low black hole masses and high accretion rates, the thermal radiation of the accretion disk partly contributes to the soft X-rays. We use the mixing model absorption (diskbb + compton + powerlaw) to fit the soft excess and use the newpar code to link kT₀ and the temperature of the inner stable radius of diskbb. For other sources with high black hole masses and low accretion rates, the thermal radiation can be ignored due to the low blackbody temperature to radiate soft X-rays; we use the mixing model absorption (compton + powerlaw) to fit the soft excess. In the Compton model, there are four parameters that determine the spectrum: kT₀, q, β₁, and β₂. The fitting parameters are listed in Table 2 and the fitting spectra are shown in Figures 6 and 7. Then, we can estimate ξ by equating the observed soft X-ray luminosity with the Compton luminosity obtained by integrating Equation (26) from 0.1 keV to 5 keV. Combining Equation (4) which gives the maximum of the soft X-ray luminosity, we can estimate the maximum magnetic energy conversion rate. We take f = 0.1 because the mean growing speed of magnetic structures is 0.1c_s (Hirose et al. 2006). The results are listed in Table 3.

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The soft X-ray excess is a major component of the X-ray spectra in many AGNs and is usually well fit with a blackbody that has a roughly constant temperature of 0.1–0.2 keV over several decades in AGN mass. If the soft X-rays are thermal, their temperature is too high to be explained by the standard accretion disk model. Although a high temperature can occur in a slim accretion disk (Mineshige et al. 2000), its accretion rate is super-Eddington (Tanaka et al. 2005) and extreme ultraviolet photons are Comptonized (Porquet et al. 2004). Strong emission and absorption features are found in the soft X-ray spectra of most AGNs. There is a strong jump in opacity at 0.7 keV from partially ionized material, where O vii/O viii and the Fe M-shell unresolved transition array are combined to produce an apparent soft excess that is not intrinsic in two different geometries, either by reflection from optically thick material out of the line of sight, or through absorption by optically thin material in the line of sight. However, both geometries would show characteristic and sharp atomic features that can be smeared by strong relativistic effects in the reflection model, but the parameters required are quite extreme; the disk has to extend to the last stable orbit in the high spin black hole and the reflection has to be highly concentrated in the innermost regions (Crummy et al. 2006). Reflection off the surface of the accretion disk was proposed by Fabian et al. (2002) to explain the soft excess, in which the primary X-rays go through substantial reprocessing in the upper layers of the disk and emerge as a smooth spectrum due to relativistic blurring. Some sources are explained by disk reflection, such as Mrk 335 (Larsson et al. 2008), Mrk 478 (Zoghbi et al. 2008), Fairall 9 (Schmoll et al. 2009; Emmanoulopoulos et al. 2011), and Ark 120 (Nardini et al. 2011). A self-consistent, dual-reflector model is needed to explain the soft excess of TonS180 at different epochs (Nardini et al. 2012). Recently, intensity-correlated spectral analysis was developed to constrain the origin of the soft X-ray excess in several AGNs (Noda et al. 2011, 2013), in which multi-zone Comptonization is required to interpret a stable soft excess component that is independent of the dominant power-law emission. Mehdipour et al. (2011) and Petrucci et al. (2013) find that the soft X-ray excess emission varies in association with the UV emission and is independent of the X-ray power-law component based on the UV and X-ray observations of Mrk 509. A long-look Suzaku observation of Mrk 509 also supports the hypothesis that the soft excess is independent of the X-ray continuum (Rivers et al. 2012). It is indicated that the soft X-ray excess could be produced by the Comptonization of the thermal optical–UV photons from the accretion disk by a warm corona (Mehdipour et al. 2011) or with the disk photosphere itself (Done et al. 2012). Done et al. (2012) also find that the Compton upscattered soft X-ray excess decreases in importance with increasing L/L_Edd, in which the strongest soft excesses are associated with low-mass accretion rate AGNs rather than being tied to some change in the disk structure around Eddington.

Based on the standard disk model, we present a new scenario to explain the soft X-ray excess. Magnetic reconnection is assumed to happen in a thin layer on the surface of the accretion disk; its released energy accelerates electrons through a shock wave and it triggeres turbulence. These electrons then undergo inverse Compton scattering above the accretion disk; this process contributes the soft X-ray excess. We find that the magnetic field declines continuously with the radius of the accretion disk and that it decreases rapidly inside of the disk, but that it is relatively flat outside of the disk. We then calculate the upper limit on the energy released by the Poynting flux on the accretion disk. The distribution of the Poynting flux implies that the soft X-ray excess should arise from the inner region of the disk. In the inner region of the accretion disk, gas pressure can be ignored and we estimate the maximum luminosity by integrating the Poynting flux from the innermost steady radius to 20 R_S. In Section 3, we compared the electron energy losses from Bremsstrahlung and Compton radiation. The results show that the electron energy losses through Bremsstrahlung are far less than those through Compton radiation. This means that the Compton radiation of electrons dominates over Bremsstrahlung radiation. Then, we calculated the Compton spectrum of non-relativistic and non-thermal electrons and obtained that the Compton spectrum is determined by kT₀, q, β₁, β₂ and is independent of the accretion rate as found by Winter et al. (2009). From Equation (27), we know that the Compton spectrum is mainly determined by kT₀. According to the standard disk model, T₀∝M^−1/4 and the masses of the black holes in AGNs are in the range of 10⁶–10⁹ M_☉ and the Compton spectrum weakly depends on mass. Figure 3 shows that the Compton spectrum with different kT₀ values maintains a narrow range of 0.1–0.5 keV corresponding to the characteristic temperature of the soft X-ray excess. Therefore, the Compton spectrum can fit the soft X-ray excess well, as shown in Figures 6 and 7. Figure 7 includes the component with a multi-temperature blackbody spectrum. It is shown that the soft photons from the accretion disk are scattered to higher energies by non-thermal electrons generated by magnetic reconnection; these photons contribute to the soft excess. We also give the Compton spectrum depending on q, β₁, and β₂, shown in Figures 4 and 5. The spectrum maintains a similar shape and shifts to lower energy when q increases. The spectrum also shifts in energy with β₁ and β₂. Finally, we estimate ξ representing the magnetic strength and the magnetic energy conversion rate listed in Table 3, and find that the magnetic energy conversion rates are low for many sources.

We are grateful to the referee for useful comments that improved the paper. We acknowledge financial support from the National Basic Research Program of China (973 Program 2009CB824800); the National Natural Science Foundation of China 11133006, 11163006, 11173054; and the Policy Research Program of the Chinese Academy of Sciences (KJCX2-YW-T24).