3.1. Initial distribution
We take as model parameters those of the system VW Hydri. The orbital parameters are (see Ritter & Kolb 1993): period P = 1.78 h, mass of the white dwarf , mass of the secondary . For the outer disk boundary we take (= 70% of primary's Roche radius). The disk's inner edge is set equal to the radius of the white dwarf (determined from according to Nauenberg 1972). For the mass inflow rate at the outer radius we take a value derived from observations: (Cannizzo et al. 1988). The total number of mesh points used in this calculation is 200. This is enough to resolve the detailed structure of transition fronts as confirmed by test calculations with 500 grid points. Cannizzo (1993) also had found that at least 100 grid points are necessary.
We start our computations of the outburst cycles with a stationary hot mass distribution. With arbitrary initial distribution of , the stationary state is obtained by taking a factor 100 times higher than the observed value. This rate is higher than the critical rate at the outer disk boundary below which dwarf nova behavior would set in. Once the stationary state is numerically established, we lower to the observed value and start a long-term calculation of the accretion disk.
Our code generates a strictly regular outburst behavior: two short outbursts alternate with one long outburst. In the following we describe the results for the outburst cycles starting with the onset of a long outburst.
3.2. Rise to a long outburst
First we show results for the change to the hot state. Figs. 2, 3 and 4 show the evolution of , T, and during the onset of a long outburst. An outward propagating heating wave, which has started near the inner disk edge, transforms the entire disk into the hot state. It is interesting to see in Fig. 2, that a narrow spike in builts up at the heating front. The peak of the spike does not exceed the upper critical line . Through radial energy flux carried by viscous processes, the critical temperature inside the heating wave is exceeded although stays below the critical surface density .
The double peaked distribution in the heating wave arises if the material on the cool side of the front is optically thin. In the transition to the hot state two unstable regimes of the - -relation are involved: the unstable intermediate branch, producing the first maximum, and an unstable part of the cool optically thick branch (compare Sect. 2.2.4), producing the second maximum. The latter effect is due to dissociation of molecules on the transition from an optically thin to an optically thick cool state. Dissociation leads to an unstable regime of the equilibrium - -relation: in the lower part of the cool optically thick branch there is . This part of the branch is, like the unstable intermidiate branch, thermally and diffusively unstable.
We further see, that the surface density on the hot side of the front is lower than on the cool side. We will later see (Sect. 3.5), that this is especially important for short outbursts, where heating waves are reflected as cooling waves before the entire disk is transformed to the hot state.
Fig. 3 shows the evolution of the temperature distribution. In that part of the disk, which has been transformed to the hot state, the temperature distribution is similar to that of a hot stationary disk. We see the high temperature region expanding to the outer disk for the consecutive time steps. The steep decrease marks the border of the hot structure.
When the heating wave reaches the outer boundary, all the material interior to the heating wave has been transformed to the hot state. For a short time after the transition this leads to relatively high temperatures and high inward directed radial flow velocities in the outer region of the disk (see Fig. 4). In each of the Figs. 2, 3 and 4 this distribution of the corresponding quantity is shown at a time shortly after the entire disk was transformed to the hot state as the last of the 7 consecutive states.
As for the surface density, a narrow spike in the distribution is characteristic for the heating wave regime. One notes that, for the entire hot part of the disk, the radial flow velocity is directed inward. This is possible because the heating wave is able to store mass and angular momentum.
In addition to the main results for disk evolution in Figs. 2, 3 and 4 we show in Fig. 5 the azimuthal velocity relative to the Kepler velocity . We see, that even inside the heating wave, the deviation from the Keplerian value is only a few percent. The gradient of relative to the Keplerian gradient is shown in Fig. 6, since this is the quantity relevant for the transition fronts. Inside the regime of the heating wave deviations up to 20% from the Keplerian value can occur. However, a test calculation, where we have assumed the disk to be Keplerian, has shown that these deviations, though they affect the detailed transition front structure, do not lead to a significant change of the outburst behavior. Thus it may be stated that deviations from Keplerian flow velocity lead only to small effects on the outburst behavior.
3.3. Outburst maximum
Soon, after the entire disk is transformed into the hot state, a quasi-stationary state is established. In this state the surface density decreases in a self-similar way. This quasi-stationary state ends, as soon as the critical temperature is reached at the outer disk rim. A cooling wave starts to develop, which transforms the entire disk back to the cool state.
3.4. Decline from outburst
The evolution of the basic quantities , T, and is shown for three successive moments during the inward propagation of the cooling wave in Fig. 7, 8and 9.
The -distribution of the cooling wave (see Fig. 7) shows the following features: the front starts at the hot side with the value . One notes, that the value of the surface density never decreases below this value . Then increases steeply inside the front, and finally turns into the relatively flat -distribution of the cool disk. During the inward propagation of the cooliung wave the decrease of is essential in the inner disk. The extend of the hot region from the inner edge to the location of the transition front (), becomes smaller and smaller. Further out stays practically at the same value during this phase. The temperature decrease during this evolution is shown in Fig. 8.
At the hot side of the front (), the outward directed radial flow velocity reaches a maxima (see Fig. 9). It is interesting to note that for a wide range interior to the inward propagating cooling wave the mass flow is directed outwards (positive values of the flow velocity). While in the cool part of the disk nearly no transport of angular momentum occurs, it is an essential process in the hot part. At the hot side of the front the value of decreases to the flow velocity (now directed again inward) of a cool disk. The radial flow velocities of a cool disk are two orders of magnitude smaller than those of the hot disk.
In Fig. 10 and 11 the corresponding deviations of and of from the Kepler values are shown. They remain small through the whole cooling wave. On the formation of the cooling wave sound waves are generated at the outer disk boundary. Their damping is caused by the and components of the viscous stress tensor. At the beginning of the cooling phase, these sound waves cause additional deviations from the Keplerian flow velocity. However, also these deviations are small and can be neglected.
3.5. Rise to a short outburst
As mentioned already the resulting outbursts are alternatingly long and short. In the following we discuss the differences. Fig. 12 shows the evolution of the surface density with time during the heating phase for a short outburst. The important phenomenon is that the outward moving heating wave can not propagate to the outer disk edge. Instead the outward moving heating wave is reflected as a cooling wave before it can reach the outer disk. The reflection occures, when the surface density on the hot side of the front has decreased to the value . No hot state is possible for . The subsequent evolution, i.e. the cooling phase, is shown in Fig. 13. So the essential difference between this evolution and that in a long outburst, where the heating wave reaches the outer disk edge, is the fact that the outer part of the disk remaines in the cool state.
In Fig. 14 the velocity of the propagation of the instability during the short outburst is shown. The part of the curve with positive values of shows the heating wave velocity, the part with negative values the cooling wave velocity. We obtain values for the transition front velocities which are comparable to those obtained by other authors (e.g. Meyer 1984, Mineshige 1987, Cannizzo 1993, Ludwig et al. 1994).
3.6. Long-term behavior
The long-term behavior of the accretion disk is shown in Fig. 15. Shown, from top to bottom, is the evolution of the visual luminosity, mass, and angular momentum of the disk, and the rate of mass accretion onto the central white dwarf. The visual luminosity of optically thin parts of the disk is set to zero. Parts of the quiescence, where the entire disk is in the optically thin state, shows up as gaps in the light curve. The small variations of the luminosity during the quiescence are caused by "tiny" transition waves running through a very small disk region. They are caused by the unstable part of the cool optically thick branch of the - -relation (compare Sect. 2.2.4).
The light curve shows the two types of outbursts, long ones and short ones. The two types results from differences in the surface density distribution at the time of onset of the instability. For both types of outbursts, the instability starts near the inner edge of the disk. For outbursts which end up being long, the entire disk is transformed to the hot state. During such long outbursts about 30% of the disk mass and angular momentum are drained from the disk, mostly during the quasi-stationary hot state. Short outbursts result when the heating wave is not able to transform the entire disk to the hot state because the critical surface density is reached on the hot side of the front before the front reaches the disk's outer edge. In such a case during the entire outburst, the outer part of the disk remains cool, and the disk loses less then 5% of its mass and angular momentum.
Our calculations show, that by an outward moving heating wave, mass is partially redistributed from inner parts to outer parts of the disk. Thus, from short outburst to short outburst, the surface density in the outer part of the disk increases by both, the redistribution of mass by a heating wave and the mass transfer from the secondary. If finally after a sufficient number of short outbursts (here two), the surface density is high enough in the entire disk, the outward moving heating wave transforms the entire disk to the hot state. With this long outburst then a significant depletion occurs, so that the next two outburst are short again.
How many short outbursts lie between two long ones depends on the values of following parameters: the outer disk radius , the inner disk radius , the mass transfer rate , the viscosity parameters and , as investigated by Cannizzo (1993).
© European Southern Observatory (ESO) 1998
Online publication: December 8, 1997