1 Introduction

The X-ray spectra of galactic black hole candidates (BH) and active galactic nuclei (AGN) are remarkably similar, in spite of their large differences in mass and length scales. The similarities of the observed spectra are evident in their power law shape of energy index $s \approx 1$ in the medium X-ray range, and their high energy cut-off at $E_{\rm c} \approx 200$ keV. It is widely accepted that the power law spectra are produced by inverse Compton scattering of soft photons on hot thermal electrons (Shapiro et al. 1976; Sunyaev & Titarchuk 1980; Pozdnyakov et al. 1983). Such spectra depend on the optical depth and the temperature of the Comptonizing region. In view of the similarities in the observed power laws these two parameters apparently vary only within narrow ranges for different sources.

Another common feature evident in many of the observed spectra is the signature of a fluorescent Fe emission line and a Compton-reflection component. These two components provide a strong indication that the X-ray production region is very close to cold matter in the central regions of BHs or AGNs (e.g.  Ross & Fabian 1993).

The classical accretion disk model (Shakura & Sunyaev 1973) can not explain the simultaneous presence of cold and hot matter near a compact object. Another mode of accretion was subsequently proposed by Shapiro et al. (1976). They showed that there is an optically thin hot disk solution where the plasma is in a two-temperature state, with the ions near their virial temperature in the inner region of the accretion disk. This model could in principle account for the observed Comptonized spectra. In this model the accretion flow is optically thin and radiatively inefficient. It was called an ion supported torus by Rees et al. (1982).

The solution described by Shapiro et al. (1976) turned out to be thermally unstable. Stable accretion is obtained when the advection of internal energy with the flow is taken into account (Narayan & Yi 1994; Narayan & Yi 1995a,b); these models are called advection dominated accretion flows (ADAF). Most of the viscously dissipated energy is advected radially with the flow. The protons are near their virial temperature, whereas the electrons are much cooler due to their strong interaction with the radiation field and the low rate at which they can exchange energy with the protons via Coulomb interactions. The flow may in principle coexist with an optically thick, cold accretion disk such that the cold disk partly extends into the hot torus (Esin et al. 1997). Another possibility for the simultaneous presence of cold and hot matter are accretion disk corona (ADC) models (e.g. Nakamura & Osaki 1993; Haardt & Maraschi 1991; Haardt & Maraschi 1993; Svensson & Zdziarski 1994; Dove et al. 1997). In these models the cold accretion disk is embedded in a hot corona in a plane-parallel slab configuration. A large fraction of the gravitational energy is assumed to be released in the corona by magnetic fields, although the details of the coronal heating process are still unclear. The protons in the corona are hot, whereas the disk beneath is cool ( $kT\simeq 1$ keV), optically thick, geometrically thin and gas pressure supported.

In both the ADAF and the ADC models there is an energy coupling between the cold disk material and the hot tenuous plasma above. Traditionally this interaction was seen in terms of an exchange of radiation (Haardt & Maraschi 1991, 1993; Haardt et al. 1994, 1997). Here we are interested in the energetic coupling via the exchange of matter. We investigate the penetration of hot protons from the ADC/ADAF into the cool disk, i.e. "ion illumination''. The hot protons are stopped in the cold disk due to Coulomb interactions with the ambient electrons, rather than by binary interactions with target ions.

Proton illumination is not restricted to accretion disks. It was applied early in X-ray astronomy to explain spectra of galactic X-ray sources as neutron star surfaces heated by radially infalling ions (Zel'dovich & Shakura 1969; Alme & Wilson 1973; Turolla et al. 1994). Recently Deufel et al. (2001) revisited the proton illumination of a neutron star in the context of accretion from an ADAF.

The importance of proton illumination for accretion disks was suggested by Spruit (1997) and Spruit & Haardt (2000). Detailed Monte Carlo calculations of the Comptonization were presented by Deufel & Spruit (2000) (henceforth Paper I). For the present work we have improved the treatment of the radiative processes by solving the radiative transfer equation including thermal emission due to bremsstrahlung. Pair production is also taken into account. We further allow for energy redistribution due to thermal electron conduction within the disk. The interaction between the impinging protons and the accretion disk is computed time dependently in a one-dimensional, plane-parallel approximation. The density distribution is found from hydrostatic equilibrium including the force exerted by the decelerating protons.

The cool disk acts as an effective thermalizer for energetic (Comptonized) photons as long as its optical depth is large enough. At lower optical depth, such as occurs near the inner edge of the disk, photon production is less efficient. Therefore we expect different solutions for disks of low optical depth. This is borne out by the results reported in Sect. 5.