4. Understanding the P-T plane
To 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
Thus and are constants in this approximation.
Let us draw now the following curves in the P-T plane:
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 Jn are the thermal balance condition for the relevant J.
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 Jn in Fig. 2) appropriate for its conditions. The movement of an element at point B is in the opposite direction, and it will reach the thermal equilibrium line from the other side. This behavior can exist as long as and However, viscous heating of an element at C will cause it to heat and expand in such a way that it will never reach a thermal equilibrium line. We see also that the equilibrium line to the right of the minimal pressure is also unstable and for the same reason. An element below it does not approach the line, but moves away from it.
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 J) an expanding wind appears. The expanding wind is an integral part of any disc with a proper opacity law. The degree of mass loss through the wind depends how high is the point above . For very optically thick discs, we expect to be very much less than the total external mass input rate into the disc.
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 f depends on the nature of the inner boundary condition. Assuming , which is valid for a point far from the inner boundary
The pressure at the critical point is given by
For the particular case of free-free opacity we obtain the following expressions.
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 z increases the effective gravitational acceleration increases for a while and then decreases. It is not clear a priori which one wins. If the loss of angular momentum is the larger, then as the matter move up it moves inward. When the matter will reach radial motion there will be no more viscous heating and the thermal instability will die away. On the other hand, if the loss of angular momentum is minimal the matter will move outward. It is instructive to consider the energy balance. The gravitational energy released by the matter as it moves a distance is:
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