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Astron. Astrophys. 362, L41-L44 (2000)

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3. The effects of conduction and enhanced Coulomb coupling

In a previous paper (HCb) we have shown that, in the absence of conduction, the disk truncates close to the last stable orbit for a variety of accretion rates, forming a hot ion torus around the central BH. When conduction is included, three different flow regions have been identified. There is an outer region, characterized by a Keplerian velocity, [FORMULA] which increases inwards slightly, where radiation is the dominant cooling process, the Compton-Y parameter is large, conduction is negligible and the cooling time equals the heating time (Fig. 3, 4 and 5). There is a torus region, characterized by a hot ion plasma which rotates at sub-keplerian velocities, is optically thin, has a small Compton-Y, [FORMULA] and the heating time decreases inwards becoming considerably smaller than the cooling time close to the inner radius. The ions here cool via advection and conduction (Fig. 3, Fig. 4 and Fig. 5).

[FIGURE] Fig. 3. The profiles of the radial velocity U (in units of [FORMULA]/left axis), density [FORMULA] in [FORMULA], modified optical depth [FORMULA] [FORMULA] and the Compton [FORMULA]parameter ([FORMULA]), right axis) along the equator. Note the rapid acceleration and the strong decrease of the variables in the torus region.

[FIGURE] Fig. 4. Radial profiles of the ion- and electron-conduction [FORMULA], radiative [FORMULA], Compton and the synchrotron cooling [FORMULA] (top). The bottom figure shows the centrifugal force [FORMULA] (right axis/dashed line) and gravitational force [FORMULA] (dotted line) normalized to [FORMULA] Note that [FORMULA] almost everywhere save the torus region where the ions rotate sub-keplerian. The solid line in the bottom figure shows the radial distribution of the ratio of the ion cooling time to the heating time. Note the relatively long cooling time of the ions in the torus region justifying the use of 2T description.

[FIGURE] Fig. 5. The quasi-stationary profiles of the ion, electron and radiative temperatures along the equator for different models. In panel A (top) we show the initial distribution of [FORMULA] and [FORMULA] obtained without conduction and which are used to initiate the subsequent models. The maximum of [FORMULA] in units of [FORMULA] and their corresponding ratios are assigned as well. Panel B corresponds to the model with conduction. In C, the three profiles are shown when the Coulomb coupling is enhanced by one order of magnitude. In D the Coulomb coupling is enhanced by two orders of magnitude, and E corresponds to the 1T accretion flows. Note the increase of [FORMULA], decrease of [FORMULA] and their convergence to 1T flows with the Coulomb enhancement.

The third region is within [FORMULA]. The flow here is super-thin, isothermal and the gradient of the rotational velocity decreases strongly inwards, thereby slowing the heating- relative to the cooling-rate (Fig. 4). Both fluids cool via advection and conduction, but the electrons continue to cool efficiently via Comptonization also.

The ratio of ion- to electron-conduction time scale is [FORMULA]. Thus, for [FORMULA], which can be easily maintained in 2T flows, the ion-conduction largely exceeds that of the electrons. In the torus region the radial velocity [FORMULA], and since conduction increases [FORMULA], it follows that [FORMULA], hence the Coulomb interaction term [FORMULA]. This implies that the weakest coupling starts from the innermost region and propagates outwards.

Consequently, a considerable heat flux is carried out from this region via conduction which goes to heat the ions in the innermost part of the disk, and subsequently increasing [FORMULA] via Coulomb interaction. Since [FORMULA] increases with increasing [FORMULA], the matter starts to accelerate, decreasing thereby the density, and subsequently weakening the radiative cooling and the Coulomb coupling ([FORMULA]) (Fig. 4). Therefore, the ion-torus expands and the truncation radius moves outwards and settles at approximately [FORMULA].

Across [FORMULA], the Compton-Y parameter is close to unity, ion- and electron-conduction and the modified Compton cooling are efficient. Soft photons are compton-upscattered by hot electrons with temperatures ranging between [FORMULA].

In the 1T case, conduction, as expected, lowers the maximum temperature in the hot torus and heats up the incoming matter from the disk. Therefore, the effect of advection becomes significant inducing an inwards shift of the truncation radius from 7 to [FORMULA]. In this case unsaturated Comptonization region penetrates deeper. The soft photons here are copious and efficiently compton-upscattered by more energetic electrons than in the 2T case [FORMULA] in this region ranges between [FORMULA].

Furthermore, the pressure gradient is two orders of magnitude smaller and therefore gives rise to less powerful outflows than in the 2T case. Across the inner boundary, the gas speed is subsonic relative to [FORMULA] whereas it is supersonic in the 1T case. This may explain why the accretion rate decreases inwards much more strongly in the 2T than in the 1T case.

We note that our 1T results disagree principally with the `unusual' flow structure obtained by Esin et al. (1996). Here, the 1T is as good as the 2T description in forming a truncated disk-advective torus and providing a temperature range that can be well accommodated within the observational data of Cyg X-1 in its high state. Fig. 5 shows that the stronger the ion-electron coupling, the lower is [FORMULA], the higher [FORMULA], and the closer they become to the 1T accretion flows. However, we have assumed that the electron and ion mean molecular weight remain unchanged as the coupling strength is enhanced.

We note that in both cases the temperature-increase in the inner region is much stronger than in the disk (more than [FORMULA]). This implies that conduction can easily supply the required energy to maintain a smooth transition from an optically thick disk to an optically thin advective torus (Fig. 5).

The observational relevance of our results are twofold. We suggest that 1T descriptions are more appropriate for accretion flows at higher rates. We expect outflows and winds to be less powerful, a pronounced modified BB spectrum and that the hard photons are well represented by a power law profile with a cutoff at [FORMULA]. On the other hand, powerful outflows and winds, [FORMULA] between [FORMULA], a less pronounced modified BB spectrum and a power law spectrum describing the hard photons with a cutoff at [FORMULA] are signatures for 2T accretion flows. This implies that the overall bolometric luminosity here is considerably lower than in the former case.

When the accretion rate is enhanced due to external mechanisms, the density increases, thereby enhancing the ion-electron thermal coupling, and allowing a transition from a 2T to 1T flow. In this case, the torus shrinks in volume only (but it continues to survive within [FORMULA]), and the disk moves inwards, truncating between [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000