Model of a ‘Warm Corona’ as the Origin of the Soft X-ray Excess of Active Galactic Nuclei

Some sort of ‘anomalous’ heating mechanism should work within a warm corona. By anomalous heating we mean that the heating rate is not simply proportional to local density nor pressure but that it is much more enhanced within the corona; i.e., $f_{\rm w}\gg\Sigma_{\rm w}/\Sigma_{\rm tot}(\ll 1)$.

This is in analogue with the hot corona model by Haardt & Maraschi (1991), who assumed that the energy dissipation occurs not entirely within the disk body, but its certain fraction ($f$) is transported to coronae and is dissipated there. The reason for this is simple (in the context of hot corona modeling); without such an anomalous heating the corona temperature will be too low to reproduce the observational values, $\sim 100$ keV, because of significant Compton cooling by soft photons emerging from the disk body. We, here, assume the same line for a warm corona (Petrucci et al., 2018, 2020). We will explicitly show in section 3 that anomalous heating is necessary to explain the warm corona temperature within our model.

What the observations indicate is that if the soft X-ray excess is produced by Comptonization in a warm corona, the cooling process in a warm corona should be dominated by Compton cooling over bound-free/free-free emissions. Naively, this condition can be easily satisfied, since we know that scattering opacity ($\kappa_{\rm es}$) dominates over free-free absorption opacity ($\kappa_{\rm ff}$) in the inner portions of the standard-type accretion disks around black holes (Shakura & Sunyaev, 1973); that is, the condition reads (e.g., Kato et al. (2008)),

not depending on the black hole mass.

We should note, however, that the situation is not so simple for two main reasons: One is that the bound-free opacity, $\kappa_{\rm bf}$, dominates in the temperature ranges well below $\sim 10^{7}$ K (i.e., $\kappa_{\rm abs}=\kappa_{\rm ff}+\kappa_{\rm bf}\sim\kappa_{\rm bf}$, where $\kappa_{\rm abs}$ is the absorption opacity; see Fig.2) whereas it is not considered in the classical paper by Shakura & Sunyaev (1973). Roughly speaking, this leads to underestimation of the absorption opacity by a factor of 30 or so (see, e.g., Cox & Daltabuit (1971)).

Second, the dominance of the Compton cooling rate (over absorption cooling rate) within the corona is not equivalent to the condition of $\kappa_{\rm es}>\kappa_{\rm abs}$, since we crudely estimate

where $Q^{-}_{\rm Comp}$ and $Q^{-}_{\rm bf}$ are the cooling rates due to Compton scattering and bound-free emission, respectively (note that bound-free emission is dominant in the temperature ranges of the warm corona), and $\tau_{\rm bf}=\kappa_{\rm bf}\Sigma_{\rm w}$ (note that $\Sigma_{\rm w}$ is defined as the surface density above the equatorial plane in the present paper). We assumed $\tau_{\rm bf}\ll 1$, since $\tau_{\rm eff}\sim 1$ and $\tau_{\rm es}\gg 1$ within the warm corona. Note that the expression for $Q_{\rm bf}^{-}$ in Eq.(15) is valid only when $\tau_{\rm bf}\ll 1$. From equation (15) the condition of the Compton domination is

We should make remark that these expressions, especially that for the absorption cooling rate in equation (15), is not so accurate, since the Kirchhoff’s law is applicable only to the absorption coefficient and emissivity as functions of frequency and is not exact when we use the Rosseland mean opacity and frequency-integrated emissivity. Nevertheless we adopt these in this study to grasp insight into the physics underlying warm corona formation.

One of the largest uncertainties in the simple model presented in the previous section is the crude evaluation of the bound-free cooling rate [equation (15)]. For reference, we show in figure 2 a schematic diagram of the bound-free cooling rate of plasma in collisional ionization equilibrium as a function of temperature (adapted from Fig.1 of Cox & Daltabuit (1971)). Contributions by heavy elements, such as O, Ne, Mg, Si, and S, are dominant in the temperature range around $\sim 10^{6}$ K (see also Cox & Tucker (1969)).

Remarkably, the bound-free cooling curve shows a rapid decrease with increasing temperature in the temperature range below several million degree K (which we are concerned with). Roughly, we find $q^{-}_{\rm bf}\propto T^{-2}$ (or $\propto T^{-0.5}$) below (above) $\sim 10^{6}$ K, where $q^{-}_{\rm bf}$ represents a bound-free cooling rate per unit volume, per electron, and per ion. This is due to the collisional ionization of metal elements.

In reality, however, we need non-grey calculations. Also needed are the considerations of photo-ionization effects, which are totally missing in Figure (2). More accurate evaluations are needed as future work in this point, but we wish to emphasize that our results will not be so sensitive to the precise treatments of the bound-free cooling rate, since it is Compton cooling, and not bound-free cooling, that is balanced with anomalous heating (see below).

In figure 2 we also schematically plot the Compton cooling rate, which is an increasing function of temperature, as long as we assume that $\Sigma_{\rm w}$ is kept constant. Note that its magnitude is taken arbitrarily, since it varies, depending on $\Sigma_{\rm w}$ (or $H_{\rm w}=\Sigma_{\rm w}/\rho_{\rm w}$, where $\rho_{\rm w}$ is the mass density of the warm corona) and $F_{\rm soft}$. We thus understand that the bound-free cooling rate (or the Compton cooling rate) exhibits negative (or positive) temperature dependence (see next subsection).

In our simple model, we introduced a parameter, $C_{\rm w}$ (ratio of the Compton cooling rate to the bound-free cooling rate in the warm corona), whose exact value cannot be determined within our model. This should be greater than unity in order to account for the observations, and this condition can be fulfilled by adjusting the height of the warm corona ($H_{\rm w}$), keeping its surface density $\Sigma_{\rm w}$ constant. Actually, the Compton cooling rate ($Q_{\rm Comp}^{-}\propto\tau_{\rm es,w}^{2}\propto\Sigma_{\rm w}^{2}$) remains unchanged, as long as $\Sigma_{\rm w}$ is kept constant, while the bound-free cooling rate is

is inversely proportional to $H_{\rm w}$. From the condition of $C_{\rm w}>1$, one can derive the condition for $H_{\rm w}$ as

The height of the warm corona, in turn, depends on how much anomalous heating takes place at which height, and it depends on the detailed magnetohydrodynamic (MHD) processes, such as magnetic reconnection, within the corona. This is very difficult to evaluate and we need await future radiation-MHD simulations with high numerical resolutions (see discussion in sec. 4.3.2).

The different temperature dependence of the radiative and Compton cooling rates may lead to an interesting consequence. Let us examine the stability of the warm corona exposed to constant soft radiation flux from the disk body ($F_{\rm soft}$) for given surface density ($\Sigma_{\rm w}$) and heating rate. Figure 3 is a schematic picture drawing the bound-free cooling rate, the Compton cooling rate, and the heating rate as functions of temperature (all the heating/cooling rates are measured per unit surface area). We understand that the bound-free cooling rate rapidly drops as temperature increases because of the ionization (see equation 24). By contrast, the Compton cooling rate is an increasing function of temperature, when $\Sigma_{\rm w}$ and thus $\tau_{\rm w}$ are fixed. By assumption, we fix the value of the heating rate ($Q^{+}$).²²2We here assume only one simple case, since we are not aware of the precise functional form of the anomalous heating rate. The precise shape of the heating curve will not affect the following discussion, unless the slope of the heating curve is greater than $\propto T$ or less than $Q^{+}\propto T^{-2}$.

We just point out that there could be two thermal equilibrium solutions, in which heating rate is equal to cooling rate. One is the bound-free branch (in which the bound-free emission dominates) found at lower temperatures and another is the Compton branch (in which Compton-cooling dominates) found at higher temperatures. From Fig. 3, we understand that not only the Compton cooling rate but also the heating rate should exceed the absorption cooling rate at $T\sim(1-3)\times 10^{6}$ K in order for a warm corona to appear. In other words, the occurrence of anomalous heating should be necessary, since otherwise the heating rate would be below the intersection of the two cooling curves. Since the cooling rate is the sum of the bound-free cooling rate and the Compton cooling rate, there would be no thermal equilibrium solution in such a case.

Suppose the situation in general that there are multiple thermal equilibrium solutions, which one will be realized? The answer is, the one(s) which is (are) thermally stable. We thus need to check the criterion for the thermal stability; that is

where $Q^{+}$ and $Q^{-}$ are the heating and cooling rate per unit surface area and differentiation is made at constant $\Sigma_{\rm w}$, since thermal timescale is much shorter than the viscous timescale, over which $\Sigma_{\rm w}$ varies (see, e.g., Shibazaki & Hōshi (1975); Shakura & Sunyaev (1976); Pringle (1981); see also Chapter 4 of Kato et al. (2008) for more generalized discussion).

Let us examine the two thermal equilibrium solutions found in Fig. 3. We can immediately understand that the Compton branch is thermally stable, while the bound-free branch is thermally unstable. In conclusion, it is natural that the observations of soft X-ray excess show the Compton-dominated spectra, which are very smooth without any atomic feature.

What exactly happens in the warm corona? Suppose that the disk surface layer (with $\tau_{\rm eff}\sim 1$) is on the bound-free branch. (The effective optical depth is assumed to be unity there.) Since such a layer is thermally unstable, its temperature should either decrease or increase. When the temperature decreases down to the surface temperature of the disk body; that is, the warm corona will disappear. When the temperature increases, conversely, a transition to the thermally stable Compton branch will be completed. We should note, however, the effective optical depth is no longer unity but should decrease as a result of decrease of $\kappa_{\rm bf}$ at higher temperatures. This may indicate that the warm corona may have a multiple-zone structure (see discussion in sec. 4.3.2).

We need to make a remark on the stability analysis made by Czerny et al. (2003) (see also similar but independent discussion by Nakamura & Osaki (1993)). They examined the thermal stability of the disk-corona system starting with the similar energy equation to our equations (5) and (6), but explicitly write down the expression for the energy transportation rate, and find the marginal stability condition as $f_{\rm w}=0.5$ ($w=0.5$ in their notation).

At first glance, our result seems an apparent contradiction with theirs, but this is not the case, since the situation is different. They examined the stability of the disk body, assuming that a certain fraction of the energy produced in the disk body is transported to a corona, and examined the condition how the original criterion is affected. [As is well known, the standard-type disk suffers thermal and viscous instabilities, when radiation pressure dominates over gas pressure (Lightman & Eardley, 1974; Shibazaki & Hōshi, 1975; Shakura & Sunyaev, 1976).] By contrast, we examined the stability of the corona, assuming that the disk body is stable (and hence the radiation flux from the disk body are unchanged). The case of thermally unstable disk body need to be examined as future work.

As shown in Sec. 2, the fraction of the accretion energy that is dissipated in a warm corona, $f_{\rm w}$, should be of order unity to reproduce the observational features of soft X-ray excess within our model, since otherwise the temperature and so the Compton-$y$ parameter of a warm corona would be much less, which means that the warm Comptonized emission would be too weak to observe. Note that the effects of internal Compton scattering on the structure and spectra of scattering-dominated accretion disk (with no anomalous heating) were intensively discussed in the 1980’s and 1990’s (see, e.g., Czerny & Elvis (1987); Ross et al. (1992); Shimura & Takahara (1995)). Although Compton up-scattering produces enhancements in the Wien part of the UV bump emission, the enhancements are not enough to explain the soft excess component (without anomalous heating). Some previous studies (e.g., Petrucci et al. (2013, 2020)) also introduced the parameter $f_{\rm w}$ in the models and constrained its value as being close to unity. The difference between them and our study is that we put the condition for a warm corona whose effective optical depth is unity, while the previous studies did not considered such a condition. This makes some quantitatively different results while the condition of $f_{\rm w}\simeq 1$ is common.

The magnetic energy dissipation in the upper layer of an accretion disk in AGNs or X-ray binaries has been discussed previously to reproduced their observational features (Merloni et al., 2000; Begelman et al., 2015; Gronkiewicz & Różańska, 2020; Gronkiewicz et al., 2023). Interestingly, the significant energy dissipation at the disk surface layer has been implied by several numerical simulations of an accretion disk (Turner et al., 2003; Hirose et al., 2006; Beckwith et al., 2009; Ohsuga & Mineshige, 2011; Zhu & Stone, 2018; Jiang et al., 2019; Mishra et al., 2020). They performed the multi-dimensional radiation-magnetohydrodynamical simulations of black hole accretion flows and outflows for various accretion rates, and presented their global structures. Especially they have shown the vertical dependence of the energy dissipation rate per unit volume and magnetic energy density becomes shallower as the distance from the equatorial plane increases. This means that it is not necessary for the accretion energy to dissipate mainly at the equatorial plane of the disk, and that a significant heating may occur at the disk surface layer. More accurate estimation regarding the heating rate distribution in a scattering-dominated layer is needed as future work.

The big assumption made in our simple analysis resides in (vertically) one-zone approximation. It is more likely that the anomalous heating occurs in a non-uniform fashion so that a warm corona may have a multi-zone structure. Suppose that the anomalous heating takes place predominantly only in an upper layer of the warm corona, while heating is much less in other parts. It then follows that the optical depth of the heated layer with a temperature of $\sim T_{\rm w}$ could be less than unity. ³³3We still keep the terminology of the warm corona to be a layer with $\tau_{\rm eff}\sim 1$ formed above the disk body. Note also that the optical depth of the unheated region cannot be negligibly small, since the absorption opacity is normally a decreasing function of temperature so that the value of $\kappa_{\rm bf}$ is larger in the unheated, high-density region than in the heated, low-density one.

Then, we may have an inequality,

[instead of equation (17)], leading to

[instead of equation (23)]. We can then find a higher temperature solution for $\tau_{\rm eff}<1$. If we set $(y,f_{\rm bf},C_{\rm w},\tau_{\rm eff})=(2,1,1,0.1)$, for example, we find $T_{\rm w}=1.3\times 10^{6}$ K for $T_{\rm soft}=0.1T_{0}$ ($=10^{5}$ K). In conclusion, partial (surface) heating of an upper layer ($\tau_{\rm eff}<1$ but $\tau_{\rm es}\gg 1$) of the warm corona is probably a solution for making the warm corona temperature above $\sim 10^{6}$ K.

Both a warm corona and a hot corona are introduced to theoretical models of AGNs as Comptonizing plasma that lie above the body of an accretion disk, which provide thermal seed photons to be Compton up-scattered. Their origins are distinct, however.

A warm corona is introduced in our model as a natural extension of accretion disk models; that is a scattering-dominated layer at the surface. In this sense, a warm corona can exist whenever or wherever there is a scattering-dominated atmosphere above an optically-thick accretion disk. A noteworthy feature of the warm corona is that the temperature difference between the disk and corona is not so large ($T_{\rm w}\sim 10^{6}~{}{\rm K}$ and $T_{\rm SSdisk}\sim 10^{5}~{}{\rm K}$) that the thermal conduction flux between them cannot be effective.

In the case of the hot corona, by contrast, the temperature difference should be much larger ($T_{\rm hot}\gtrsim 10^{9}~{}{\rm K}$ in the corona), which is required to account for the hard power-law component in the X-ray spectra of AGNs. Therefore, the heat conduction from a hot corona to the underlying disk is expected (as in the case of solar corona), and it can drive the mass evaporation from a disk into a corona (Liu et al. (1995, 1999); Meyer et al. (2000); Liu et al. (2002)). This is a big distinction between a warm corona and a hot corona. How these two components coexist in the innermost part of an accretion flow is beyond the scope of this work.

Let us consider what will occur when anomalous heating takes place at scattering-dominated atmosphere. The atmospheric temperature should increase, when an anomalous heating sets in, but the temperature increase will be up to $\sim 10^{6}$ K, at which Thomson thick Compton cooling is balanced with anomalous heating, as we have so far discussed. The temperature increase is moderate (i.e., $T_{\rm w}\ll 10^{9}~{}{\rm K}$) to acheive the condition of Compton $y\sim 1$ when $\tau_{\rm es}\gg 1$ and $\tau_{\rm eff}=1$. Note that $T_{\rm w}$ can be higher if the condition of $\tau_{\rm eff}$ is relaxed to allow that it can be smaller than unity.

It is observationally indicated that the warm corona temperature $T_{\rm w}$ is rather insensitive to the black hole mass or luminosity (or equivalently, accretion rate). How can we explain these facts?

We first discuss the accretion-rate dependence on $T_{\rm w}$. Suppose that the warm corona lies on the Compton branch as is displayed in figure 3 and let mass accretion rate decrease for a fixed $M_{\rm BH}$ and fixed $r$. We immediately understand that the coronal heating rate decreases for a fixed $f_{\rm w}$, since it is proportional to $\dot{M}$ [see equations (2) and (3)]. We can also easily understand that the Compton cooling rate should also decrease in proportion to $\dot{M}$ for a fixed $\Sigma_{\rm w}$, since $F_{\rm soft}\sim F_{\rm SSdisk}\sim Q_{\rm tot}^{+}$, as long as $f_{\rm w}=O(1)$ [see equation (11)]. Since both rates decrease in the same way, the equilibrium temperature should be kept the same (see figure 4). This naturally explains why $T_{\rm w}$ is insensitive to $\dot{M}$.

Note that the bound-free cooling rate should also decrease, since $Q_{\rm bf}^{-}\propto\rho^{2}H_{\rm w}\approx\rho\Sigma_{\rm w}$ and $\rho$ decreases, as $\dot{M}$ decreases. Although the amount of decreases may not be the same as that of the heating (or Compton cooling) rate, this is not critical, since the equilibrium temperature is determined not by the bound-free rate but by the Compton rate.

Note also that there is a lower limit to the mass accretion rate, for which our model can apply. When the accretion rate is so small, sufficient amount of material can no longer be supplied to the surface layer to fulfill the condition of $\tau_{\rm eff}\sim 1$. The exact expression for this lower limit is difficult in the framework of the one-zone model. We leave this issue to future works.

The next issue is how to understand that $T_{\rm w}$ does not depend on $M_{\rm BH}$. In the framework of the simple model presented in Sec. 3, $T_{\rm w}$ should have $M_{\rm BH}$ dependence. One promising way to overcome this issue is to introduce a multi-zone structure for the warm corona (or, equivalently, to relax the condition of $\tau_{\rm eff}=1$). We again use figure 4. Suppose, this time, that $M_{\rm BH}$ increases, while keeping ${\dot{m}}(\equiv{\dot{M}}c^{2}/L_{\rm Edd})$ constant, at the same normalized radius ($r/r_{\rm S}$). Since $Q_{\rm tot}^{+}\propto M_{\rm BH}{\dot{M}}/r^{3}\propto{\dot{m}}(r/r_{\rm S})^{-3}M_{\rm BH}^{-1}$, we understand that the heating rate should decrease in inversely proportional to $M_{\rm BH}$. So does Compton cooling rate. Since the heating and Compton cooling rate decreases in the same fashion, the equilibrium temperature should not change, as in the case of changing $\dot{M}$. The bound-free cooling should also decrease, since it is proportional to $\rho$ and $\rho$ usually decreases with increase of $M_{\rm BH}$ for a fixed $\dot{m}$ and $r/r_{\rm S}$. In fact, recalling the relationship, ${\dot{M}}=-2\pi rv_{r}\rho H\propto r^{2}\rho$ (as long as $v_{r}$ and $H/r$ are not so sensitive to $M_{\rm BH}$), one can derive $\rho\propto M_{\rm BH}^{-1}$ for a fixed $\dot{m}$ and fixed $r/r_{\rm s}$. Its precise $M_{\rm BH}$ dependence does not matter, since it does not affect the equilibrium temperature. We thus understand why $T_{\rm w}$ is not scaled by $M_{\rm BH}$.

We should note, however, that there is a lower limit to the black hole mass, since $T_{\rm soft}$ increases with a decrease of $M_{\rm BH}$ and eventually exceeds the typical warm corona temperature of $\sim 10^{6}$ K. Thus, the warm corona solution disappears in small $M_{\rm BH}$ objects.

As for the strength of the soft X-ray excess, it has also been indicated to be insensitive to the black hole mass (Gierliński & Done, 2004; Mitchell et al., 2022). In our model, the soft X-ray excess strength is determined by the energy dissipation fraction, $f_{\rm w}$, which is assumed as a free parameter in this work. In order to interpret this observational fact within our warm corona scenario, it is necessary to model how $f_{\rm w}$ should be determined from the physical properties of an accretion disk such as $M$ or $\dot{M}$, which is also left as a future work.

The next issue is the radial dependence of $T_{\rm w}$. We have suggested in section 3 that $T_{\rm w}$ may be a strong function of radius $r$, since it is roughly scaled with $T_{\rm soft}\approx T_{\rm SSdisk}\propto r^{-3/4}$. In other words, a warm corona can exist only at small radii. This can be easily understood, since much cooler seed photons emerging at larger radii requires a large Compton amplification factor, or large $y$, in contradiction with the observation (which indicate similar photon indices and so similar $y$-values).