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Astron. Astrophys. 344, 639-646 (1999)

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3. The thermal equilibrium condition

Consider the thermal equilibrium equation (9) at a point [FORMULA] above the disc where the optical depth from infinity to [FORMULA] is sufficiently small. Note the appearance of J in the equation. If


and there is no significant positive source function gradient, J may be neglected in condition (9). In general, the value of J depends on the structure of the disc and the radiation inside it. We may write that [FORMULA] where [FORMULA] is related to the effective temperature in the disc, which is given by the local mass transfer rate in the disc [FORMULA] and [FORMULA] is the Stefan Boltzmann constant.

In this paper, we consider the important regime of low [FORMULA] and restrict the discussion mainly to regimes where absorption is due to hydrogen free-free. This is regime (b) in the SS models where gas pressure dominates over radiation pressure, and is expected to encompass most of the inner regions of the discs in compact star binaries and in AGN in the optically thin, namely the low [FORMULA] regime.

We approximate the absorption coefficient by a power law


In the present numerical calculations we use free free opacity i.e. [FORMULA] and [FORMULA]. The inclusion of more accurate expressions for the free free opacity as well as other sources of opacity and consideration of Compton cooling, is left for future investigation. We note however, that Compton cooling is not expected to be a significant coolant in the body of the disc in the regime that we presently consider. Here, we contend ourselves with an approximation that serves to demonstrate the fundamental physical effect that we wish to highlight in this paper.

For the adopted power law opacity, the thermal equilibrium condition (9) leads to


This relation is shown in Fig. 1 for two cases. An accretion disc around a black hole of [FORMULA] at [FORMULA] cm (10 Schwarzschild radii) for accretion rates of [FORMULA],[FORMULA],[FORMULA],[FORMULA], [FORMULA] [FORMULA] and around a black hole soft x-ray transient (BHSXT of mass [FORMULA] at [FORMULA] cm ([FORMULA] 300 Schwarzschild radii) for [FORMULA],[FORMULA],[FORMULA],[FORMULA]and [FORMULA] [FORMULA].

[FIGURE] Fig. 1. The pressure - temperature relationship corresponding to thermal equilibrium in the upper atmosphere of a disc around a [FORMULA] black hole soft x-ray transient at [FORMULA] cm and a [FORMULA] AGN disc at [FORMULA] cm. The different curves correspond to mass transfer rates of [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] [FORMULA] for the AGN disc and [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] [FORMULA] for the BHSXT disc.

Along the sharply declining vertical portions of these curves, the mean intensity J plays an important role in determining the thermal structure, whereas the rising parts of the curves are dominated by optically thin free-free cooling. We note that for every J or equivalently [FORMULA], there is a minimal pressure below which there is no solution, and above which there are two solutions for a given pressure. In fact, such a minimum exists for all opacity laws with [FORMULA] and [FORMULA]. A mass element which finds itself on the left hand part of these curves could be in thermal and hydrostatic equilibrium, whereas if it is on the rising part (on the right of the minimum), no hydrostatic state is possible with pressure rising outward. We note that for the adopted absorption the structure line of a classical disc should have mass elements which lie along the thermally stable sharply declining parts of these curves. We have used the above results to represent schematically an example of a possible disc structure in thermal and hydrostatic equilibrium. The example is shown in Fig. 2, where [FORMULA] corresponds to the symmetry plane while [FORMULA] corresponds to the surface of the disc, [FORMULA] where eventually the vertical structure is truncated.

[FIGURE] Fig. 2. The P-T plane, the thermal equilibrium lines [FORMULA] and the structure lines in two cases. At each point in the disc there is a corresponding P-T curve with the appropriate J. The thermal equilibrium lines correspond to [FORMULA], where [FORMULA]. The heavy line marked [FORMULA] denotes the structure line of a disc for which all points lie on a corresponding P-T equilibrium line. [FORMULA] is the [FORMULA] plane and [FORMULA] the outermost point. In the second structure line [FORMULA], which is more realistic, point [FORMULA] does not lie any longer on an energy equilibrium line.

For a physical model, the point [FORMULA] must be close to where the pressure tends to zero (in the sense that the matter above [FORMULA] is sufficiently optically thin and produces only a negligible luminosity), or more realistically at some prescribed low value (say [FORMULA]) of the order of the environment pressure in the ISM (Wehrse, Wickramasinghe & Shaviv 1998). The shape of the cooling curve, however, precludes such a solution in general, as can be seen from Fig. 1. In this case the structure is better described in Fig. 2 by point [FORMULA] which does not lie on any thermal equilibrium line but is in a region where the heating dominates over the cooling. The point [FORMULA] cannot lie on the rising part of the curve since this configuration is thermally unstable. So far we have seen that there is a critical pressure [FORMULA], and corresponding to this pressure we can define a critical temperature [FORMULA]. The quantities [FORMULA] and [FORMULA] will play an important role in the subsequent discussion. It is obvious that the temperature is always higher than [FORMULA]!

The minimum values of pressure and temperature for mass transfer rates of and for the AGN and black hole transient cases are shown in Fig. 4.

[FIGURE] Fig. 4. The values of [FORMULA] and [FORMULA] as a function of the radial coordinate for a typical accreting WD with [FORMULA] and a BH with [FORMULA]. The units of the radial distance are [FORMULA] and [FORMULA].

The next question is what price the disc pays for not obeying the thermal equilibrium relationship. Said differently, the part [FORMULA] is clearly not also in hydrostatic equilibrium. So what happens to it?

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999