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Astron. Astrophys. 328, 247-252 (1997)

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4. Computational results of disk evolution

For a typical dwarf nova system we take the mass of the central white dwarf as [FORMULA], the initial inner boundary after an outburst at [FORMULA] cm, the outer boundary at [FORMULA] cm. The viscous coefficient for the cold state is taken as [FORMULA].

We start our computation with the surface density distribution left over from the preceding outburst. The diffusion Eq. (7) is integrated in similar way as in earlier work (compare e.g. Meyer-Hofmeister & Meyer 1988). Results of a first rough investigation of the effect of evaporation were presented earlier (Meyer & Meyer-Hofmeister 1994). Now the evolution of the cool disk is computed in detail using the coronal structure and evaporation rate by Liu et al. (1995) and the physical relations deduced in Sect. 2.

4.1. Computations including the relations derived in Sect. 2

During quiescence the matter flowing over from the secondary star is accumulated in the disk, only a small part is accreted onto the white dwarf. How the mass distribution changes depends on the assumed parameter [FORMULA]. The smaller [FORMULA] is the longer it takes for the gas to diffuse inward and the more matter is piled up in the outer disk. For outburst modelling the parameter [FORMULA] is chosen to fit the duration of the quiescence. Both values, [FORMULA] and the overflow rate (derived from observations), determine the evolving surface density distribution. A problem of dwarf nova outburst modelling was the fact that for adequately chosen [FORMULA] and [FORMULA] the onset of instability occurred dominantly in the innermost disk region (Cannizzo 1993, Ludwig et al. 1994, Ludwig & Meyer 1997), which is different from the observations.

For comparison we show in Fig. 1 the evolution of the disk without evaporation, in Fig. 2 including evaporation. Without evaporation the mass overflow from the secondary leads to a continuous increase of surface density everywhere in the disk. With evaporation the situation is different. From Fig. 2 one can see that the innermost disk is evaporated immediately after the preceding outburst. The inner boundary shifts outwards very fast at first and then more slowly. After 30 days a large hole is created and the inner boundary is at [FORMULA]. At this distance from the white dwarf the evaporation has become very low and just about balances the low local quiescent mass flow there. This means that mass flow via the corona onto the white dwarf continues at a low rate. In connection with the mass accumulation outside the hole the disk mass flow rate can increase and the boundary of the hole can then slightly be pushed inwards. Thus the observed X-rays and UV radiation in this later quiescence is expected to be low, maybe slightly varying. The consequence of the hole in the inner disk is accumulation of mass in the remaining outer disk. This leads to the onset of instability much farther out in the disk than without evaporation. Our new computations show that with the evaporation the onset of the outburst is at the distance of about [FORMULA] cm from the white dwarf, in much better agreement with the observations. The higher amount of matter needed to reach the critical surface density [FORMULA] at a larger radius demands a longer time of accumulation. In our example the onset of instability is found after 123 days with evaporation, after 73 days without evaporation. These are essential differences in the outburst behavior. Taking this into account one might then choose a larger value of [FORMULA] for the quiescent state.

[FIGURE] Fig. 1. Evolution of the disk during quiescence without evaporation. [FORMULA] surface density, r distance from central object. The two straight lines are critical surface densities. The diffusion time after outburst is indicated in days
[FIGURE] Fig. 2. Evolution of the disk during quiescence with evaporation. Parameters are same as in Fig. 1. Dashed lines show the preceding evaporation of the inner disk during early quiescence. For later quiescence the inner boundary stays at about the same distance from the white dwarf. The full lines show the evaporation after a large hole has been created until the next outburst is triggered

4.2. Simplified computations

We compare our results with those of a simplified computation, where we take only the angular momentum away from the cool disk that is lost with the wind but neglect the angular momentum transport within the corona. The mass flow density [FORMULA] and the wind loss fraction [FORMULA] are taken as derived by Liu et al. (1995), [FORMULA] g [FORMULA], [FORMULA]. The equations for conservation of mass and angular momentum in the cool disk then are the same as Eqs. (3) and (4), except that at the right side of Eq. (4) the factor [FORMULA] appears. Instead of relations (5) and (6) we have




This procedure conserves mass and angular momentum properly, it only restores the angular momentum released from coronal accretion locally to the disk.

The definition of the boundary is the same as the former (Sect. 3.3), i.e. [FORMULA]. From Eq. (16) follows [FORMULA], or


We now obtain the location and the motion of the inner boundary under the influence of disk evolution and evaporation through [FORMULA] and Eq. (18) as


where [FORMULA] is the grid point closest to the inner boundary. This procedure does not require the solution of a transcendental equation (like Eq. (15)).

The results of this simplified calculation are shown in Fig. 3. One sees that the size of the hole, the duration of quiescence, the location where the critical surface density is reached are very similar to the former results. However, the profile of the surface density distribution near the inner boundary is much steeper than the former analytical "tail", obviously it is caused by the rough approximation of the inner boundary motion, such a steep profile also occurs when interactions between disk and corona are taken into account but no analytical "tail" is included. In fact, the profile of the [FORMULA] distribution of the boundary just affects the time needed to create the hole during the early evolution, it hardly affects the final state before the outburst is triggered.

[FIGURE] Fig. 3. Evolution of the disk during quiescence with evaporation in a simplified computation. Parameters are same as in Fig. 1. It shows similar evolution to Fig. 2 except for a very steep profile near the inner boundary

The fact that the two methods lead to so similar results illustrates that redistribution of angular momentum by the corona is not important. The changes of angular momentum only concern the region near the inner boundary and the amount of angular momentum there is small compared to that in the outer region.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998