Accretion disk coronae

An accretion disk described exactly by the standard model produces a relatively soft, quasi-thermal spectrum (dominated by optical/UV radiation in AGN, and soft X-rays in GBHCs). However, accreting black hole systems (both stellar and supermassive) often exhibit power-law components to their spectra which extend to hard X-ray energies with exponential rollovers at high energy ($E\sim 300{\rm\thinspace keV}$). A promising mechanism for producing such a spectrum is unsaturated inverse Compton scattering [135,136]. Extensive descriptions of inverse Comptonization and its applications to astrophysical sources can be found in the literature [137,138]. Here we provide a brief summary of these descriptions.

Compton scattering is the scattering of a photon by an electron. In the rest frame of the electron, the differential scattering cross section for a photon of energy $E\ll m_{\rm e}c^2$ is given by

$$\frac{{\rm d} \sigma_T}{{\rm d}\Omega} = \frac{3}{8 \pi} \sigma_T \left ( \frac{1 + \cos^2 \theta}{2} \right ) ~~,$$ 20

where

$$\sigma_T = \frac{8 \pi}{3} \left ( \frac{e^2}{m_e c^2} \right )^2 = 6.67 \times 10^{-25} ~{\rm cm^{2}}$$ 21

is the Thomson cross section and $\theta$ is the angle between the photon's initial and final trajectories. In the initial rest frame of the electron, the photon imparts kinetic energy to the recoiling electron, and hence the photon loses energy,

$$\Delta E = - \frac{E^2}{m_e c^2} ( 1 - \cos \theta )~~.$$ 22

In the laboratory frame, however, the electron can impart energy to the photon, up to $\Delta E = (\gamma - 1) m_e c^2$, where $\gamma$ is the electron's relativistic Lorentz factor.

It can be shown [137,138] that for a non-relativistic thermal distribution of electrons with temperature $T_e$, the average photon energy change for a single scattering is given by

$$\langle \Delta E \rangle = (4 k T_e - E) \frac{E}{m_e c^2} ~~.$$ 23

If $E \gg 4 kT_e$, the electron's kinetic energy is negligible and we recover the angle average of the previous equation for photon energy loss. If, however, $E \ll 4 k T_e$, the photon gains energy, with the fractional energy gain being approximately proportional to $\gamma^2$. This proportionality can be understood as one factor of $\gamma$ coming from the boosting of the photon into the initial electron rest frame, and another factor of $\gamma$ coming from a boosting of the scattered photon back into the lab frame [137,138]. This gain of photon energy is referred to as inverse Compton scattering.

So long as $E \ll 4 k T_e$, the photon continues to gain energy after each scattering event with $\Delta E/E \approx 4 k T_e / m_e c^2$. Thus, after $N$ scattering events, the average final photon energy, $E_f$, compared to the initial photon energy, $E_i$, is given by

$$E_f \approx E_i \exp \left ( N \frac {4 k T_e}{m_e c^2} \right ) ~~.$$ 24

For a medium with an optical depth of $\tau_{\rm es}$, the average number of scatters is roughly max($\tau_{\rm es}$, $\tau_{\rm es}^2$). This leads one to define the Compton $y$ parameter,

$$y \equiv {\rm max}(\tau_{\rm es}, \tau_{\rm es}^2) (4 k T_e/m_e c^2),$$ 25

such that $E_f \approx E_i \exp(y)$ [137]. For $y > 1$, the average photon energy increases by an `amplification factor' $A(y) \approx \exp (y)$. For $y \gg 1$, the average photon energy reaches the thermal energy of the electrons. This latter case is referred to as saturated Compton scattering, while the former case is referred to as `unsaturated inverse Comptonization'. Unsaturated inverse Comptonization is of most interest in the study of black hole systems.

As discussed above, the `standard' optically-thick, geometrically-thin disk model produces a maximum temperature that scales with the compact object mass as $M^{-1/4}$, so one naturally expects lower disk temperatures in AGN, as opposed to stellar mass black hole systems. However, one can instead define a `virial temperature' which refers to the average accretion energy per particle. Specifically, the gravitational energy released per particle of mass $m$ scales as $GM m /R$, and therefore scales as $m c^2/[R/(GM/c^2)]$. Thus, in terms of geometrical radii, the virial temperature is independent of compact object mass and furthermore can reach a substantial fraction of the rest mass energy of the accreted particles. Electron virial temperatures of tens to hundreds of keV can be readily achieved in the innermost regions of compact object systems. (The proton virial temperature is a factor of $\approx 2000$ greater; therefore, it is conceivable that electrons can achieve even higher temperatures depending upon the temperature of the protons and the coupling between the electrons and protons.)

It is such a distribution of low-to-moderate optical depth, high temperature electrons that is suspected of creating the observed power law spectra in black hole systems. Although multiple photon scatterings in such a corona become exponentially unlikely, multiple scatterings lead to exponential photon energy gain. The two effects balance, to some degree, and produce a power law spectrum. Simple estimates [137] show that for a given Compton $y$ parameter, the photon index of the resulting power law is approximately given by

$$\Gamma = -\frac{1}{2} + \sqrt{ \frac{9}{4} + \frac{4}{y} } ~~,$$ 26

i.e., $\Gamma \approx 2$ for $y = 1$⁷. Of course, the exact function $\Gamma(y)$ is dependent upon geometry and other assumptions. A power law form only holds for photon energies somewhat less than the electron thermal energy. As photons approach the electron thermal energy, they no longer gain energy from scattering, and a sharp rollover is expected in the spectrum. Thus, in inverse Comptonization models, the observed high energy spectral cutoff yields information about the temperature of the underlying electron distribution.

There are two principle uncertainties in applying coronal models to observed black hole systems, and both are substantial. First, the mechanism for heating the electrons to near virial temperatures is currently unknown. Current hypotheses invoke magnetic processes, perhaps akin to solar flares on the Sun or heating of the solar corona. Contrary to solar models, however, black hole coronae may be energetically dominant. Second, the geometry of the corona is also completely unknown. Thus, models have taken to hypothesizing specific geometries (see Fig. 6), and parameterizing the energy input into the corona.

Figure 6: Suggested geometries for an accretion disk and Comptonizing corona for predominantly spectrally hard states. The top figure is referred to as a ``slab'' or ``sandwich'' geometry; however, it tends to predict spectra softer than observed. The remaining three show ``photon starved geometries'' wherein the corona is less effectively cooled by soft photons from the disk. The middle two geometries are often referred to as ``sphere+disk geometries'', while the bottom geometry is often referred to as a ``patchy corona'' or ``pill box'' model [140].

An early suggestion [141] for the geometry of coronal systems was that the accretion disk consisted of two zones-- a cool, outer disk, and a hot inner, disk. It was envisioned that energy release was actively occurring within the disk itself, but that a ``two-temperature'' disk was being formed within the inner regions. Ions would reach temperatures of $\approx 10^{11-12}$K, while electrons would achieve temperatures of only $\approx 10^9$K. Hard X-ray emission would be due to Comptonization of soft photons from the outer disk by the hot electrons in the inner disk. It was later shown [142,143] that this specific disk model was thermally unstable; however, the basic geometry, which we will refer to as the `sphere+disk' geometry, has been proposed in a variety of forms since that time. The attraction of such a geometry is that the hot, inner corona sees only a fraction of the soft flux from the outer, cool disk, and is therefore not strongly Compton cooled. This allows the corona to remain very hot and to produce hard spectra [144].

The above disk models postulate a radial separation between the geometrically thin, optically thick disk and the geometrically thick, small optical depth corona. Other sets of coronal models exist, however, wherein both `disk' and `corona' exist at the same radii, but the accretion energy is postulated to be dissipated predominantly within the corona [145,146,147,148]. It was thought early on that a promising source for the necessary coronal energy would be dissipation of magnetic flux tubes as they buoyantly rose above the disk into the corona [149]. The recent studies of MHD instabilities in accretion flows have revived interest in such mechanisms [118,108,150].

The simplest such model is one where the corona sandwiches the accretion disk [151,152,153,154]. For a uniform and plane parallel corona and disk, the basic energetics of this geometry can be obtained from conservation principles. Let us decompose the radiative flux into that from the corona $F_c$ and that from the optically-thick part of the disk $F_d$. The coronal flux is comprised of two parts, an outward component, $F_c^+$, and an inward component, $F_c^-$, which will be intercepted and reprocessed by the accretion disk. The total flux from the disk, $F_d^t$, is comprised of any intrinsic dissipation within the disk, $F_d^i$, plus the flux intercepted from the corona. Thus $F_d^t = F_d^i + F_c^-$. The total flux from the corona, $F_c^+ + F_c^-$, is comprised of any intrinsic dissipation within the corona, $F_c^i$, plus the total outward flux of the disk. By definition, this is set equal to the outward flux from the disk multiplied by the Compton amplification factor, $A_c$. Specifically,

$$F_c^+ + F_c^- = A_c F_d^t ~~.$$ 27

After multiple scatterings within the corona, the radiation field becomes nearly isotropic, therefore $F_c^+ \approx F_c^- + \Theta F_d^t$, where $0 \le \Theta \le 1$ is a measure of the anisotropy of the coronal flux. As the optical depth of the corona, $\tau_{\rm es} \rightarrow 0$, $\Theta \rightarrow 0$. If one now assumes that a fraction, $f$, of the total (local) accretion energy is dissipated in the corona, while $1-f$ is dissipated within the disk, one obtains $F_d^i/F_c^i = (1-f)/f$, and

$$A_c = \frac{1 + \Theta f / 2}{1 - f/2} ~~.$$ 28

The above highlights two important considerations of this slab geometry. First, even under the most extreme conditions of $\Theta \approx 1$ and $f \approx 1$ (i.e., all accretion energy is dissipated within the corona), the Compton amplification factor does not exceed 3. This is equivalent to saying that the Compton $y$ parameter does not exceed $\approx 1$. Second, 100% of the soft, seed photons from the disk pass through the corona. Thus, the corona is relatively easily Compton cooled and it is difficult for it to obtain very high temperatures [140,144]. Even if the optical depth of such a corona is made small, pair production within the corona will ensue until the rate of heating is balanced by Compton cooling. With a pure slab geometry, extremely hard spectra (i.e., with very high energy cutoffs) are impossible to achieve [140,155].

There have been several suggestions put forward that allow the above basic geometry to achieve higher coronal temperatures, and thus harder spectra. If the corona is `patchy', such that small coronal regions sit atop the disk (sometimes referred to as a `pill box' geometry), then a large fraction of the coronal flux can be intercepted by the disk, with a relatively small fraction of the (reprocessed) disk flux being intercepted by the corona. Thus, the coronal patches are not very effectively Compton cooled [140]. Another method for varying the amount of soft flux reprocessed by the corona is to invoke relativistic motion within the corona. Downward motion beams the radiation towards the disk and increases the local disk reprocessing, while motion away from the disk leads toward beaming away from the disk, thereby decreasing reprocessing [156,157]. Finally, as we will further discuss below, if the disk surface is highly ionized, then rather than being reprocessed by the disk into soft radiation, the hard radiation will merely reflect back through the corona, without substantially cooling it [158,159,160,161].