## 3. The thermal equilibrium conditionConsider the thermal equilibrium equation (9) at a point
above the disc where the optical
depth from infinity to is
sufficiently small. Note the appearance of and there is no significant positive source function gradient,
In this paper, we consider the important regime of low 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 regime. We approximate the absorption coefficient by a power law In the present numerical calculations we use free free opacity i.e. and . 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 at cm (10 Schwarzschild radii) for accretion rates of ,,,, and around a black hole soft x-ray transient (BHSXT of mass at cm ( 300 Schwarzschild radii) for ,,,and .
Along the sharply declining vertical portions of these curves, the
mean intensity
For a physical model, the point must be close to where the pressure tends to zero (in the sense that the matter above is sufficiently optically thin and produces only a negligible luminosity), or more realistically at some prescribed low value (say ) 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 which does not lie on any thermal equilibrium line but is in a region where the heating dominates over the cooling. The point 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 , and corresponding to this pressure we can define a critical temperature . The quantities and will play an important role in the subsequent discussion. It is obvious that the temperature is always higher than ! 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.
The next question is what price the disc pays for not obeying the
thermal equilibrium relationship. Said differently, the part
is clearly © European Southern Observatory (ESO) 1999 Online publication: March 18, 1999 |