1 Introduction

Accreting galactic black holes (BHC) and active galactic nuclei (AGN) are often observed with two different spectral components: a soft component which is probably due to a multi-color blackbody from an optically thick, geometrically thin standard disk (SSD Shakura & Sunyaev 1973), and a hard component which is linked to an optically thin and geometrically thick flow. The hard component (an approximate power law with high energy cut off at $E_{\rm c}\approx 100$ keV) is most likely produced by inverse Compton scattering of soft photons on a hot thermal plasma (Sunyaev & Titarchuk 1980). Shapiro et al. 1976 showed that accretion can take place in the form of a two-temperature plasma, with the properties needed to produce such Comptonized radiation. Stable, optically thin, two-temperature flows were studied by Ichimaru (1977) and Rees et al. (1982). Extensive theoretical work on these accretion flows, stressing the role of the advection of internal energy was done by Narayan & Yi (1994, 1995a,b), for a review see Narayan et al. (1998), see also Esin et al. (1997) and references therein.

The effects of advection are the same for the optically thin two-temperature flows and optically thick, radiation supported flows, and both types of flow are now customarily called ADAF's. The distinction between these types, which was explicit in the older labels "ion supported'' and "radiation supported'', is still needed in many applications, however. For this reason, we propose here the name "ion supported accretion flow'' or ISAF as a means of distinguishing the optically thin type of ADAF from the radiation supported ("RSAF'') type.

Figure 1: Energy channels in a cool accretion disk embedded in a hot region. The energy dissipation in the cool disk is assumed small compared the atmosphere (a corona or an ion supported ADAF, here called ISAF). Squared boxes show physical processes, round boxes the particles involved. Heavy arrows and boxes show the main energy channel: viscous dissipation in the ISAF heats the protons there, which illuminate the cool disk below. By Coulomb interactions the ISAF protons loose their energy mainly to electrons, producing a warm ($\sim$80 keV) layer which radiates this energy by Compton-upscattering of soft photons from the cool disk below. The protons in the warm surface layer are largely outside this main energy channel. Near the transition radius  $R_{\rm tr}$ the cool part disappears and the warm part heats up to several hundred keV. Then the energy exchange of disk protons with the electrons is slow (shown by the saw tooth line) and viscous heating of protons becomes important.

ISAFs have attracted attention because of their potential to explain the spectra of X-ray transients (Esin et al. 1997 and references therein). The observations are consistent with an accretion flow that consists of two zones: an interior ISAF that extends from the black hole horizon to a transition radius  $R_{\rm tr}$, followed by an optically thick, geometrically thin and cool standard disk outside  $R_{\rm tr}$. A partial overlap between the two regions is probable since observations show evidence for the close vicinity of hot and cold matter in the central regions of BHCs and AGNs. This is indicated by a K$_\alpha$ iron fluorescence line at 6.4 keV and a Compton reflection component between $\approx$10-30 keV.

A critical element of such an accretion geometry is the change from the geometrically thin SSD to the hot ISAF flow at the transition radius. An alternative interpretation of the spectra is given by the "magnetic flare'' or disk corona model (Haardt & Maraschi 1991; Maraschi & Haardt 1997 and references therein; di Matteo et al. 1999; Merloni et al. 2000 and references therein), in which this transition is absent and the X-ray emitting plasma is a layer on top of a disk, heated by mechanical energy transfer from the disk.

Signatures of a disk transition radius have recently been found in the power density spectra of Cyg X-1 (Churazov et al. 2001; Gilfanov et al. 2000). From an analysis of the reflection component Di Salvo et al. (2001) showed that in Cyg X-1 the transition radius $R_{\rm tr}$is located between $10~R_{\rm S}$ and $70~R_{\rm S}$, if the observed reflected spectrum is due to a smeared component, or $6~R_{\rm S}<R_{\rm tr}<20~R_{\rm S}$ if the reflection is unsmeared, e.g. from the companion star or the outer disk.

We take these observations as reasonable indications that a transition from a cool optically thick to a hot optically thin accretion flow does in fact take place. But how the SSD-ISAF transition works is still under debate (cf. Manmoto et al. 2000 and references therein). Meyer et al. (2000) propose that the transition from the cold disk to the optically thin flow is due to a heat flow by electron conduction from a hot, friction-heated corona to the cold disk below (see also Meyer & Meyer-Hofmeister 1994). This model has its maximum evaporation efficiency at a large distance from the hole (a few 100 Schwarzschild radii). If, due to a high accretion rate in the cool disk, not all material is evaporated until that distance, the cool disk will survive until the last stable orbit. A transition radius further in than 100 Schwarzschild radii is inconsistent with this picture. Rózanska & Czerny (2000) investigate conductive and radiative coupling of an accretion powered corona with an underlying cool disk. For low accretion rates they find that the disk completely evaporates whereas high accretion rates prevent the SSD-ISAF transition as in Meyer et al. (2000). From a mathematical point of view Abramowicz et al. (1998) show that, if the transition region is not too wide, the region must rotate with super Keplerian orbital speed. Based on this property Kato & Manmoto (2000) demonstrate that trapped low-frequency oscillations are possible in the transition region.

In this paper we show how the inner disk regions, where the coronal evaporation process does not work, can evaporate into an ISAF. Only a few, well-known ingredients need to be invoked: the coupling of protons and electrons by the Coulomb interaction in a fully ionized plasma, standard radiation processes, and viscous heating.

The starting point is the view, supported by the observations mentioned, that an ISAF and a cool disk can coexist. That is, there is a partial overlap between the cold disk and the ISAF. In the overlap region, there is a very strong interaction between the two, since the ISAF consists of energetic ions (10-100 MeV) that penetrate the cool disk to a significant depth. The goal in the next sections is to determine the nature of the energy and mass exchange in this interaction region, and to show that it will lead to evaporation at the inner edge of the cool disk. The argument is then closed by determining the conditions under which these processes can consistently lead to the coexistence of the disk and the ISAF that was assumed at the outset.

1.1 Interaction between ISAF and cool disk

The essence of the processes described below is given by the energetic interaction between ISAF and disk. In Fig. 1 we have sketched the main energy channels involved in this interaction. The ultimate source of energy is the release of gravitational binding energy. We assume here that a significant fraction (at least a few tens of per cent) of this energy goes into the protons (on account of their much higher mass compared to the electrons). This viscous energy release predominantly takes place in the hot region (the ISAF). The protons and electrons in the ISAF form a two-temperature plasma, where the protons have temperatures near their virial temperature. The electrons are much cooler due to their strong interaction with the radiation field and the slow rate at which they exchange energy with the ISAF protons.

Due to their low temperature, the conductive energy flux carried from the ISAF to the disk by the electrons is negligible. This is one of the reasons why the coronal evaporation mechanism that functions well at larger distances from the central mass fails for a two-temperature plasma. The energy flux to the cool disk is carried almost entirely by the ions. This ion energy flux can not take over the role played by electron conduction in the coronal evaporation process, for two reasons. One is that the mean free path of the ions is not negligible as it is in the case of electrons. The stopping length of 10-100 MeV ions penetrating into a cool disk corresponds to a Thomson optical depth of order unity. This prevents the development of the thin energy deposition layer that is needed to heat the plasma to high temperatures. More important is the fact that the energy carried by these ions primarily heats the electrons of the disk. These in turn radiate it very effectively by inverse Compton scattering and limit the plasma temperature in the interaction region to values of $\sim$100 keV, well below the virial temperature. Instead of evaporation, the loss of the ISAF ions to the cool disk is a very effective condensation process. This condensation is an important sink to an ISAF flow generated by evaporation at larger distances, and has to be overcome by a sufficiently powerful evaporation process at some location in the disk. This location will turn out to be a region near the inner edge of the disk.

   
1.2 A "warm'' surface layer on the cool disk

The ISAF and the cool disk are separated by a "warm'' surface skin of temperature $\sim$100 keV, intermediate between the ion temperature of the hot ISAF and that of the cool disk. This heated surface layer is produced by the energy flux from the ISAF to the cool disk. This flux can be in the form of radiation (photons) or particles (ions). Radiative coupling was investigated by Haardt & Maraschi (1991, 1993) in the context of their "two-phase model'' for the hard X-ray spectra of accreting black holes. Explanation of these spectra as resulting from the coupling between an ISAF and a cool disk via hot the protons was proposed by Spruit (1997) and Spruit & Haardt (2000), and studied in greater detail by Deufel & Spruit (2000) (henceforth Paper I) and Deufel et al. (2002) (henceforth Paper II). The physics of this process has been studied before in the context of accretion onto a neutron star surface by Zel'dovich & Shakura (1969), Alme & Wilson (1973), Deufel et al. (2001).

The surface layer radiates its energy by Compton-upscattering of soft photons from the cool disk below. The energy of the ISAF is thus carried to the disk by the hot ISAF protons, and radiated away by the disk electrons. The protons in the warm surface layer are largely outside this main energy channel.

In the present analysis we concentrate on ion heating, which is necessarily very strong in the overlap region between an ion supported flow and a cool disk. A radiative flux from the ISAF can exist additionally, but in order to separate these contributions we assume here that the ISAF itself is radiatively inefficient, so that its direct contribution to the radiation from the accretion flow negligible. Finally, the viscous energy release in the warm layer turns out to be small compared with incident energy flux from the ISAF. Viscous heating is thus energetically unimportant in the surface layer, but it will turn out to be crucial to the evaporation process.

Finally there is the cool disk ($\sim$1 keV) below the warm layer. This region is outside of the reach of the impinging hot protons. The cool disk serves as a thermalizer for the downward directed radiation of the warm layer. It is the source of the soft photons which are Comptonized in the warm region and keep it at moderate temperatures. The viscous energy release in the cool disk is assumed to be negligible as a source of soft photons.

In Paper II we have shown that close to the inner edge of an accretion disk, where the surface density (and the optical depth) of the disk gets small, the penetration of virialized protons heats the entire vertical disk structure to temperatures of several 100 keV (which is equivalent to a disappearance of the cold part in Fig. 1). This corresponds to the first step in our model and is at the same time the starting point of the present investigation. At high temperatures the time scale for establishing thermal equilibrium between the disk protons and electrons is not short compared to the dynamical time scale any more. The viscous energy channel (due to internal heating of protons by friction) gets important now because coupling to the electrons is weak.

It will turn out that the physics described depends almost only on the dimensionless radius from the hole $r=R/R_{\rm S}$, the dimensionless accretion rate $\dot M/\dot M_{\rm Edd}$, and the dimensionless viscosity parameter $\alpha$. The results are thus scaleable between AGN and BHC cases. Where explicit values of the physical parameters are needed, we take those of a typical BHC case.

1.3 Coulomb interactions in an ionized plasma

 In the next three subsections, we briefly review the physics associated with the penetration of the protons into the disk. This has been discussed before in detail in the references given in the previous subsection. Here, we address a few conceptual issues, such as the validity of the approximation that the energy of the incident protons is transferred mainly to the electrons in the disk, the accuracy of a nonrelativistic treatment, and the charge balance between ISAF and disk.

At the typical energies of the protons incident on the cool disk, the energy loss is mostly by long-range Coulomb interactions with the electrons in the disk (small-angle scattering on the large number of electrons in a Debye sphere). This is opposite to the case of protons with a temperature near that of the plasma in which they move. In the latter case, the equilibration among the protons is faster than between electrons and protons, by a factor of order $(m_{\rm p}/m_{\rm e})^{1/2}$. To see how this apparent contradiction is resolved, consider the basic result for the energy loss of a charged particle moving in a fully ionized, charge-neutral plasma. This was derived by Spitzer (1962) (making use of Chandrasekhar's (1942) earlier results on dynamical friction). Introduce as a measure of distance in the plasma the Thomson optical depth $\tau$, i.e. ${\rm d}\tau=\sigma_{\rm T}n {\rm d}l$, where $\sigma_{\rm T}=8\pi e^4/(3m_{\rm e}^2c^4)$ is the Thomson cross section, n the electron density and l the distance. The rate of change of energy $E=m_{\rm p}v^2/2$ of a proton moving with velocity v in a field of particles with charge e and mass $m_{\rm f}$ (the "field particles'' in Spitzer's nomenclature) is then given by Spitzer's Eqs. (5)-(15). In our notation, this can be written in terms of the energy loss length $\tau_{\rm f}$ for interaction with the field particles f,

$$\tau_{\rm f}^{-1}\!=\!{1\over E}\!\left({{\rm d}E\over {\rm d... ...4\!\!\left(1+{m_{\rm p}\over m_{\rm f}}\right)\!F(x_{\rm f}),$$ (1)

where $\ln\Lambda$ is the Coulomb logarithm (which is determined by the size of the Debye sphere). Here

$$F(x)=\psi(x)-x\psi^\prime (x),$$ (2)

where $\psi$ is the error function and $\psi^\prime$ its derivative, and $x_{\rm f}= v[{m_{\rm f}}/(2kT)]^{1/2}$ is (up to a numerical factor) the ratio of the incident proton's velocity to the thermal velocity of the field particles. The limiting forms of F are

$$F(x)\rightarrow x^3/3\quad (x\rightarrow 0),\qquad F\rightarrow 1\quad (x\rightarrow\infty).$$ (3)

We can evaluate (1) under the assumption that the field particles that are most relevant for the energy loss are the protons or the electrons, respectively, and compare the loss lengths. If the incident proton has velocity comparable with the thermal velocity of the field protons, we have $x_{\rm p} \sim 1$, and $x_{\rm e}=(m_{\rm e}/m_{\rm p})^{1/2}x_{\rm p}\ll 1$. $F(x_{\rm p})$ is then of order unity and $F(x_{\rm e})\approx x^3_{\rm e}/3$. Setting ${\rm f}={\rm e}$ respectively ${\rm f=p}$ in (1) and taking the ratio, we have

$${\tau_{\rm e}\over\tau_{\rm p}}\approx \left({m_{\rm p}\over m_{\rm e}}\right)^{1/2}\cdot$$ (4)

The loss length for interaction with the electrons is thus much longer than for interaction with the protons, and the interaction with electrons can be neglected. This is the well known result for the relaxation of a proton distribution in a plasma that is not too far from its thermal equilibrium.

For incoming protons of high energy, however, the result is different because $x_{\rm e}$ is not sufficiently small any more. In the high-v limit, $F(x_{\rm e})= F(x_{\rm p})=1$, and one has

$${\tau_{\rm e}\over\tau_{\rm p}}\approx 2{m_{\rm e}\over m_{\rm p}}=10^{-3}.$$ (5)

In this limit, the energy loss is thus predominantly to the electrons. A related case is that of the ionization losses of fast particles in neutral matter (for references see Ryter et al. 1970). The case of an ionized plasma is simpler, since the electrons are not bound in atoms. The change from proton-dominated loss to electron-dominated loss takes place at an intermediate velocity $v_{\rm c}$, at which $F(x_{\rm p})\approx 1$ but $x_{\rm e}$ still small, so that $F(x_{\rm e})\approx x^3_{\rm e}/3$. Equating $\tau_{\rm e}$ and  $\tau_{\rm p}$ then yields

$$x_{\rm e,c}\approx \left({m_{\rm e}\over m_{\rm p}}\right)^{1/3},$$ (6)

or

$${E_{\rm c}\over kT}\approx \left({m_{\rm p}\over m_{\rm e}}\right)^{1/3}\approx 12.$$ (7)

For the electron temperatures we encounter in our models, $T\sim 100$ keV, energy loss to the field protons can thus be neglected for incoming protons with energy $E\ga 1$ MeV. This is the case in all calculations presented here.

1.4 Corrections at high and low energies

Spitzer's treatment is non-relativistic, while virialized ISAF protons near the hole can reach sub-relativistic temperatures. A fully relativistic treatment of the Coulomb interactions in a plasma has been given by Stepney & Guilbert (1983). We have compared the classical treatment according to Spitzer's theory with this relativistic result in Deufel et al. (2001, 2002), and found it to be accurate to better than 5% for proton temperatures <100 MeV. The classical approximation in Spitzer's analysis therefore does not introduce a significant error for the problem considered here.

For high energies, the Coulomb energy loss becomes so small that loss by direct nuclear collisions becomes competitive. This happens (cf. Stepney & Guilbert 1983) at $E\ga 300$ MeV, an energy that is not reached by virialized protons except in the tail of their distribution. We ignore these direct nuclear collisions. Note, however, that a gradual nuclear processing by such collisions can be important (Aharonian & Sunyaev 1984), in particular for the production of the Lithium. The Lithium overabundances seen in the companions of LMXB (Martín et al. 1994a), may in fact be a characteristic signature of the interaction of an ADAF and a disk described here (Martín et al. 1994b; Spruit 1997).

As the protons slow down, they eventually equilibrate with the field protons. This last part of the process is not accurately described by the energy loss formula (1). In addition to the simple energy loss of a particle moving on a straight path through the plasma, one has to take into account the random drift in direction and energy resulting from the interaction with the fluctuating electric field in the plasma. This drift can be ignored to first order (end of Sect. 5.2 in Spitzer 1962), but takes over in the final process of equilibration with the plasma. This last phase involves negligible energy transfer compared with the initial energy of the protons in the present calculations, and can be ignored here.

1.5 Charge balance

The protons penetrating into the disk imply a current that has to be balanced by a "return current''. As in all such situations, this return current results from the electric field that builds up due to the proton current. As this electric field develops, it drives a flow of electrons from the ADAF to the disk which maintains the charge balance. Since the electron density in the disk is high, the return current does not involve a high field strength.