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Astron. Astrophys. 329, 559-570 (1998)

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5. Comparison with other investigations

5.1. Computations by Mineshige

Mineshige (1987) has carried out fine mesh calculations which can almost fully resolve the transition fronts. He has found that the effects of lateral heat diffusion become very important only when the heating wave passes by. This is in full agreement with our results. From his results Mineshige concluded that lateral heat diffusion may modify more or less the transition wave propagation, but that drastic changes cannot be expected. From our calculations however, we have found that lateral heat diffusion leads to drastic changes of the long-term behavior of the disk. Regarding this point one has to note, that Mineshige used a somewhat different cooling function, which may of course influence the results. The main difference to his results concerns the profile of the heating wave. In his heating waves the peak of the spike exceeds the upper critical line [FORMULA]. Also the width of the heating waves in his calculations is clearly smaller than in our calculations. Nevertheless, most of his results are qualitatively in good agreement with ours:
(1) The surface density distribution of a heating wave shows a profile with sharp spikes.
(2) The cooling wave starts on the hot side with the value [FORMULA]. The value of [FORMULA] at each point does never drop below [FORMULA].
(3) The [FORMULA] distribution around the transition waves (Fig. 2, 7, 12 and 13) propagating in the disk are almost the same in both computations. The shape of the heating front is different from that of the cooling front.

Mineshige also discussed the localized front approximation. He concluded that the localized front approximation is relatively good for heating waves but it is marginal for cooling waves. This conclusion was drawn from the fact that only the width of a heating front is very small compared to its distance from the white dwarf, the width of a cooling wave is not. The validity of the localized front approximation will be discussed further down.

5.2. Computations by Cannizzo

Cannizzo (1993) examined how secular changes in the input parameters of the model affect the outbursts. He also examined the dependence of the outburst behavior on terms in the energy equation. His results agree with our present analysis:
(1) The advective term strongly influences the outburst behavior and is thus important.
(2) The pressure term can be neglected.
(3) The radial heat flux is only important in the region of a heating wave.
In his calculations Cannizzo considered only the radial energy flux due to viscous diffusion. The radial energy flux due to radiative diffusion was neglected because of numerical problems. But he notes, that from test calculations he has found the two terms to be comparable in magnitude. We have found here that only the radial energy flux due to viscous diffusion need be considered, the other flux can be neglected.

His results for cooling and heating waves and our results are qualitatively in good agreement. A noticeable difference however, concerns the maximum value of [FORMULA] inside a heating wave. In Cannizzo's calculations the spike of the heating wave is always close to the [FORMULA] -line (see Fig. 3 in his paper).

Cannizzo's and our findings concerning a multi-modal outburst behavior are the same: The sequence of alternating long outbursts separated by one or several short outbursts is a natural consequence of the model. The mass present in the disk at the onset of the instability determines whether an outburst will be long or short. Cannizzo also notes, that the observed sequencing in the SU UMa systems can be explained in a natural way by the thermal instability alone and that the fact that the number of short outburst between two long outbursts is larger for systems below the period gap could be a consequence of the lower mass transfer rates in the SU UMa systems.

Another (at least ocassionally) observed feature in outbursts of SU UMa systems is that the recurrence time increases during the sequence of short normal outbursts between two superoutbursts. In Cannizzo's calculations such an outburst behavior was not obtained. In our calculations however, we find indeed such an increase of the quiescence between two short outbursts within the supercycle.

5.3. Earlier computations by Ludwig et al. (1994)

Finally, we compare our new calculations with our earlier work (Ludwig et al. 1994). In that paper results for simulations of VW Hydri, U Gem and SS Cygni were shown. For the simulations we used the localized front approximation (see Meyer 1984) for which the computing time required is only about 1 [FORMULA] of that required here. The front velocities obtained in the present investigation are in good agreement with those obtained earlier. In particular, this agreement holds also for cooling wave velocities, despite the fact that the width of our cooling fronts are relatively broad. But, if one compares the calculated long-term evolution of the accretion disk in the simulation for VW Hydri, strong differences can be seen. Fig. 11 of Ludwig et al. (1994) shows the calculated light curve for VW Hydri: the outbursts repeat strongly periodically. This is a direct consequence of the boundary condition used at the front: it was assumed that [FORMULA] is the same on both sides of the front. By this, the outward moving heating wave has always transformed the entire disk into the hot state. A reflection of a heating wave as a cooling wave was not possible. In addition, the localized front approximation does not yield the redistribution of mass by a heating wave which is, however, very important for obtaining a multi-modal outburst behavior. Also the development of a transition front and the effects of lateral heat diffusion (broadening of a heating wave) are neglected within the localized front approximation.

Thus, for the detailed structure of transition waves and its consequences for the long-term behavior it is necessary to solve the diffusive evolution and the thermal adjustment simultaneously.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997