## 4. Understanding the P-T planeTo understand the behavior of the solution in the P-T plane described in Figs. (1, 3) we define the following time scales: The viscous heating time scale defined by: where is the adiabatic speed of sound and is the local viscous heating per unit volume. The kinematic viscosity is given by The cooling time is given by: Finally the hydrodynamic time scale is It is easy to see that and Thus and are constants in this approximation. Let us draw now the following curves in the P-T plane: and It is instructive to examine the behavior of a mass element in the
P-T plane from the point of view of the ratio between the various time
scales. For this reason we redraw Fig. 1 and incorporate in it the
relevant time scales. The curves marked with Consider a mass element at point A in Fig. 3. Since the
hydrodynamic time is the shortest, the element wants to expand.
However, the cooling time is shorter than the viscous heating time,
and hence the element will move to lower pressures and lower
temperatures. The motion will continue until the element will reach
the thermal equilibrium line (lines marked by
At this point we can return and examine the disc structure which
penetrates into the region of point C. The part of the structure from
point A (cf. Fig. 3) to point cannot
be in hydrostatic equilibrium and will heat up and expand. Thus, this
part of the disc represents a slowly expanding wind moving away from
the plane. Whenever the structure
point representing the pressure decreases below
(on the thermal equilibrium line
corresponding to the proper The minimal pressure is a function of and . Since all discs must lose mass via a wind at some level, cannot be constant along . So . The total mass loss rate from the disc in the form of a wind is where is the outer radius of the disc and is the inner one, or to a good approximation the radius of the accreting object. For sufficiently high we have The disc models calculated by SW were calculated in this limit. When decreases the above condition is not satisfied any more and the change of throughout the disc must be taken into account. At a certain accretion rate which we denote by , will vanish for some . In this case the disc structure disappears before reaching the central object and the disc has a hole inside. We can use our calculations to roughly estimate the critical mass transfer rate below which we expect a hole to develop at some radius as follows: We assume for this estimate an isothermal atmosphere and the free-free opacity law (Allen 1976). For a thermally driven wind, the specific mass loss rate will be given approximately by the product of the density and the adiabatic sound speed (cf. Czerny & King, 1989a,b) evaluated at the structure point where hydrostatic equilibrium breaks down for the first time, namely at and . Hence: and can be obtained from a complete solution to our equations, or approximately as follows. Using the condition Eq. 12, we find that the minimum in the pressure-temperature relationship occurs at or, for sufficiently high point in the disc, where is to a good approximation equal to Hence we can write that: where The pressure at the critical point is given by For the particular case of free-free opacity we obtain the following expressions. and hence The total wind mass loss rate outside a given radius is obtained by integrating the above expression, and is given approximately by where we have assumed that the outer radius of the disc is much larger than the radius at which the inner boundary conditions should be imposed. (Note that so the above estimate is an upper limit for ) The specific mass loss rate (and of course also the integrated mass loss rate) increases as the inner radius decreases. For certain conditions it is possible that the wind mass loss rate will equal the mass transfer rate into the disc before the stellar surface is reached. Under these circumstances, a hole may develop in the inner regions of the disc. The radius of this hole can be obtained from the condition , and is given by The above estimates indicate that for a typical BHSXT of mass , a hole of radius of approximately will develop for a mass transfer rate of . We note the strong dependence of the radius of the hole on the uncertain viscosity parameter . Nevertheless it is interesting to note that the numerical value given above is close to the value deduced for the hole in the disc in BHSXT GRO1655-40 (Hameury et al. 1997). We can make similar estimates for the white dwarf and the AGN black hole cases, but here more careful attention will need to be paid to the nature and details of the opacity. The estimates based on free-free opacity are likely to be appropriate only for restricted regions of CV and AGN discs. The above results imply that in general, for very low , there is a regime where no static solutions will exist, and the matter will expand as soon as it tries to lose angular momentum. Matter that tries to accrete at extremely low accretion rates is blown away from the central object by the very process that is supposed to remove the angular momentum. The extreme sensitivity of the radius of the hole to the value of may in principle be used to infer a good estimate of when such a hole and it radius are observed. How far will the material go? From observation of Fig. 2 one can
conclude that the material will expand in the P-T plane for ever.
However, the matter loses angular momentum via the dissipation that
heats it. At the same time, as while the energy dissipated in this ring is A necessary but not sufficient condition for a wind is that the dissipation inside the ring be greater than the gravitational energy of the matter at this point, in other words: a condition which is satisfied only for . Hence, for sufficiently large radii, there is a possibility for a strict mass loss from the system via a wind. The extra energy needed to lift the matter to infinity comes from the dissipation energy at lower radii which was convected outward. But at lower radii there is apparently not enough energy and the consequence is a hot corona and eventually radial inflow into the accretor. Our estimates of the wind expansion ignores many effects like radiation pressure, radiation pressure in lines etc. It seems therefore that the present estimate is a lower limit to the actual phenomenon. © European Southern Observatory (ESO) 1999 Online publication: March 18, 1999 |