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

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4. Discussion

4.1. The relevance of the terms in the energy equation

We have investigated the dependence of the outburst behavior on terms appearing in the energy equation similar to Cannizzo (1993). For this we have turned off different terms to judge from the resulting changes their importance.

From our calculations we have found, that lateral heat diffusion can be neglected outside of the transition front. Lateral heat diffusion is also unimportant in a cooling wave. However, the situation is different inside a heating wave. The radial energy flux carried by viscous diffusion leads to a significant broadening of the radial extension of the front. Further more we find that the radial energy flux carried by radiative diffusion can be neglected everywhere. Fig. 16 shows the evolution of the surface density during the outward propagation of a heating wave calculated without the lateral heat diffusion terms in the energy equation. If we compare this evolution with the one shown in Fig. 2 we see, that by turning off the lateral heat diffusion the heating wave becomes much narrower. The peak of the spike of the heating wave exceeds the upper critical surface density [FORMULA]. The resulting differences in the distribution of [FORMULA] inside the heating wave lead to a different redistribution of the mass inside the disk. Test calculations have shown, that the long-term behavior of a disk calculated without lateral heat diffusion is quite different from that calculated including it.

[FIGURE] Fig. 16. The evolution of the surface density with time during the outward propagation of a heating wave. This calculation was carried out omitting the lateral heat diffusion terms in the energy equation, otherwise the same as Fig. 2.

Neglecting the pressure term [FORMULA] leads to only very small changes. The contribution of the frictional stresses [FORMULA] and [FORMULA] to the viscous heating is small compared to that of [FORMULA] as expected. Even inside a heating wave, these contributions are three orders of magnitude smaller than the contribution from [FORMULA]. Nevertheless, these terms can not be neglected (in Eqs. 4and 10) because they are responsible for the damping of sound waves. The advective term [FORMULA] should not be neglected in calculation of dwarf nova outbursts, its neglect would make the width of the heating wave much smaller compared to that obtained including this term. Thus, the long-term behavior of the disk is strongly affected by the advective term.

4.2. The outburst pattern

The long-term light curve of SS Cyg, which includes observations since 1896 (see Cannizzo & Mattei 1992), shows a strong tendency for a bimodal outburst behavior. The alternation between one short and one long outburst is the most common behavior. The next most frequent behavior consists of stretches during which two short outbursts lie between two long ones. This behavior is reproduced in our calculations (see Fig. 15). The long outbursts are those in which the entire disk is transformed to the hot state, while in the short outbursts the outward moving heating wave is reflected as a cooling wave. A similar explanation is given by Cannizzo (1993). In our calculations both kinds of outburst start near the inner disk edge. This is a direct consequence of the two-alpha description used for the viscosity, where the parameters [FORMULA] and [FORMULA] are chosen to fit the durations of outburst and quiescence. But from observations of SS Cygni we know (see Mauche 1996), that at least the long outbursts can start near the inner disk edge (inside-out outbursts) as well as near the outer disk edge (outside-in outbursts). Within the two-alpha description outside-in outbursts can not be obtained by allowing for example small variations of the mass-transfer rate or by taking slightly different disk radii. What one needs for getting outside-in outbursts is a viscosity descriptions which produces a steep radial gradient in viscosity. This, for example, can be achieved by taking [FORMULA] proportional to [FORMULA] with [FORMULA], where [FORMULA] is the tidal radius which is about [FORMULA] of the binary separation. Ichikawa & Osaki (1992) used such an [FORMULA] parametrization to obtain an outside-in outburst. Also an ([FORMULA])-dependent [FORMULA] can lead to outside-in outbursts (Meyer & Meyer-Hofmeister 1984). For the modeling of the system SS Cyg one needs a viscosity description which allows both inside-out and outside-in outbursts. For the multi-modal outburst behavior this would mean that a long outburst can start from either edge of the disk. For example, it could be possible that due to small variations in the mass-transfer rate from the secondary sometimes the critical surface density for the long outburst is first reached in the inner disk region sometimes first in the outer region. However, within the framework of this model short outbursts cannot start at the outer edge.

4.3. The superoutburst phenomenon

Finally, in this section we show a connection to the superoutburst phenomenon of SU UMa stars. Three different models have so far been proposed to explain this phenomenon (for a review see Osaki 1996): (1) the enhanced mass-transfer model, (2) the thermal limit-cycle model, and (3) the thermal-tidal instability model. Here, we consider in particular the second model. In this model the superoutburst and the normal outburst of SU UMa stars are identical to the long and short outbursts seen in dwarf novae above the period gap (van Paradijs 1983). Howell et al. (1995) adopted this model arguing that the thermal limit-cycle instability is complex enough to produce several short outbursts sandwiched between two successive long outbursts. The supercycle of SU UMa stars would then be understood just by the thermal limit-cycle instability model of dwarf novae. Our calculations show (see Fig. 15), that indeed a supercycle light curve of SU UMa stars (here VW Hydri) can be produced with only the thermal limit-cycle instability operating inside the disk. The "superhump" phenomenon observed after the superoutburst has begun and which is explained by the 3:1 resonance between Kepler frequency at the outer disk and orbital frequency would then be a later consequence as the outbursting disk expands to the resonance radius, and not the cause for the superoutburst.

We note however, that observationally the superoutburst light curve seems to merge to a variable degree with the light curve of a preceding standart short outburst (Marino & Walker 1979, Warner 1995). The apparent variable superposition of the two light curves might indicate that the preceding short outburst does not directly trigger the superoutburst and might even have occasionally decayed significantly before the superoutburst takes its course. This would probably mean that the superoutburst occurs as an outward-in event whose beginning is not directly affected by the interior short outburst.

It will require further numerical investigation to see whether this course of events does require a release of the superoutburst by the 3:1 resonance as suggested by the Osaki model, or whether the phenomenon of alternating short and long outbursts will cover this case.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997