6 Discussion and conclusions

We have investigated a model for the production of a hard spectral component ($h\nu\gg 1$ keV) from a cool disk around a compact object. The model assumes that a standard cool accretion disk is surrounded by a hot atmosphere. This atmosphere can either be due to an ion supported accretion flow (ADAF) which partially overlaps in radial extent with the cool disk or to a hot corona which is e.g. magnetically heated (e.g. di Matteo et al. 1999). For the actual calculations presented here, we have used parameters representative for ADAFs. Fast protons from this atmosphere interact with the cool disk material and produce a warm ($\sim$80 keV) surface layer, where soft photons from the cool disk are Comptonized. We have considered two cases in detail: a disk around an $8~M_\odot$ black hole (galactic BHC case), and an AGN case with an $8\times 10^6~M_\odot$ black hole.

We find that the response of the cool disk to proton illumination depends on its optical thickness. At sufficient thickness ( $\tau\gtrsim 1.5$) the cool disk is able to effectively reprocess incoming energy flux into soft photons. Here, a two-layer structure develops, consisting of a warm surface layer of depth $\tau\sim 1$ on top of a cool disk. At lower total optical depth, the soft photon emissivity is insufficient to cool the disk and the disk heats up, by the proton energy flux, to an approximately isothermal layer of intermediate (a few 100 keV) temperatures. We call these cases warm disks: their temperature is higher than an accretion disk radiating as a black-body, but still much lower than the virial temperature.

In the optically thick case, the energy balance between the warm surface layer and the cool disk is like that in the Haardt-Maraschi (1991, 1993) model of a corona over a cool disk. For geometrical reasons about half of the Comptonized photons is reprocessed by the cool disk, the other escapes. The Compton y-parameter adjusts such that the amplification factor matches this requirement. This balance regulates the temperature of the layer, while its optical depth is determined by the penetration depth of the protons. The balance is strongly self-regulating: the temperature and optical depth of the warm layer depend little (within limits) on temperature and energy flux of the incident protons, so that also the shape of the emergent X-ray spectrum varies little (see Haardt & Maraschi 1993, and Paper I for a more detailed discussion).

We find that the existence of these proton-heated surface layers is limited to the innermost regions $r\lesssim 50$ of the disk. At larger distances from the hole, the optical depth of the warm layer drops rapidly, and its contribution to the spectrum becomes small.

For our BHC case the temperature and optical depth of the warm layer are in the right range to produce spectra with the main features of black hole binaries in their hard states. In the AGN case, the Comptonization is weaker, and the hard spectral component somewhat steeper (photon index around 2.6) than in the BHC case (index around 2.2). In both the BHC and AGN cases the spectra cut off around 200 keV. The maximum distance from the hole where a significant warm layer is formed is somewhat smaller in the AGN case ( $R_{\rm max}/R_{\rm S}\approx 20$) than in the BHC case ( $R_{\rm max}/R_{\rm S}\approx 50$). This is due to the lower energy flux per unit of surface area, and the reduced penetration depths in the cooler AGN disks (see also Deufel et al. 2001).

Near the inner edge of a cool accretion disk (at the distance $R_{\rm i}$ from the hole), its optical depth becomes small (in the thin disk limit, it varies by the well known factor $f=[1-(R_{\rm i}/R)^{1/2}$]). Though the optical depth of a cool accretion disk is generally substantial at the accretion rates inferred for BHC and AGN, there is always a region of low optical depth near the inner edge. This is also where the disk would be exposed to the largest proton energy flux from an ADAF or corona. At optical depths $\tau_{1/2}\lesssim 1.5$ we find that an initially cool disk heats up to a new equilibrium with nearly uniform temperatures of a few 100 keV (Fig. 5). The transition takes place in a time of the order of the dynamical time scale $1/\Omega$. Initially, the protons are stopped in the cool layers before reaching the midplane. Due to the thermal expansion of the plasma the electron densities drop, and as a consequence the proton heating finally overcomes the cooling by bremsstrahlung ($\sim$ $n_{\rm e}^2$). A new equilibrium is found when pair production sets in. The pairs add to the optical depth of the layer and increase the radiative loss of the plasma (by Comptonization) until it matches the input by proton heating.

Our solutions are local equilibria for conditions at a given distance from the hole. In disks in which the optical depth is low ($\lesssim$1) near the inner edge but at least a few at larger distance, proton heating would produce our warm disk structure of approximately uniform temperature in a (possibly narrow) zone near the inner edge. Further out it would instead produce a warm surface layer overlying the cool disk. Both regions would contribute to the overall X-ray spectrum.

Since the energy flux by proton heating is highest near the inner edge, the warm disk component at that location could contribute significantly to the overall spectrum even if its radial extent is limited. In our solutions, it would add a very flat component (photon index around 1, up to about 1 MeV, Fig. 4), so the overall spectral index could be significantly less than 2, as observed in some of the hardest spectra. The warm surface layers over the cool disk do not have such flat spectra (Fig. 2), and alone can not explain the hardest observed spectra (with indices around 1.5). The combination has the potential to explain the entire range of observed hard spectra, but for a quantitative result the radial structure of a proton-illuminated disk needs to be treated in more detail.

In this context we note that our warm disk solutions ignore a potentially important cooling mechanism, namely Compton cooling by soft photons from a cool disk. Though there are no cool regions internally in the warm disk, the nearby cool disk at larger radii produces a soft photon flux. A fraction of this flux could travel radially, illuminating the warm disk region. The flux of such soft photons is probably small, since the disk is thin at all radii and the radial optical depths therefore large. For quantitative assessment a two-dimensional radiative transfer model has to be developed, which is outside the scope of the present treatment.

The warm disk solution has a combination of temperature and optical depth just in the region where a normal, internally heated disk cannot exist. In such disks, the ($T,\tau$) combination characterizing our warm disks would lie on the SLE (Shapiro-Lightman-Eardley) branch of accretion flows, which is thermal/viscously unstable. The combination is quite stable in our warm disk solution precisely because it is not heated internally but externally by the incident protons. In this context it is interesting that model fits of low/hard states in terms of stationary ADAF models sometimes yield parameter combinations on the unstable SLE branch (for example in XTE 1118 +480 as analyzed by Frontera et al. 2001). We interpret this as a strong indication for the existence of our warm disk solutions.

The physics of the warm disk solutions presented here has further interesting consequences. In anticipation of results to be given in more detail in a future paper, we note here that at the temperatures and densities in our warm disks, the time scale for establishing thermal equilibrium between the electrons and the protons in the disk (not the illuminating protons) is not short compared to the dynamical time scale any more. Any mechanism which now provides energy to the disk protons faster than they can exchange energy with the disk electrons would give rise to a heating of the disk protons and, depending on the temperature and density dependence of such a process, might lead to runaway heating of the protons. Two energy channels immediately come to mind: proton-proton interactions, i.e. heating of the disk protons by the penetrating external protons, and internal viscous heating according to standard disk theory. We have neglected these here, as their energetic contribution is negligible in our model (the main energy channel is from the incident protons to the radiation field, via the disk electrons). If such an instability exists, it might heat the disk protons sufficiently for the disk to expand and feed its mass into the surrounding ADAF.

Acknowledgements

This work was done in the research network "Accretion onto black holes, compact stars and proto stars'' funded by the European Commission under contract number ERBFMRX-CT98-0195.