4 Possible solutions

4.1 ER UMa supercycles

The standard TTI model cannot reproduce the very short supercycles observed in ultra-low mass-ratio systems. Osaki (1995a) reproduced RZ LMI's supercycle by ending prematurely the superoutburst assuming $r_{\rm crit0} = 0.42 a$ instead of the usual value of 0.35 a (note that in the TTI framework, one assumes $r_{\rm crit0} = 0.35 a$ when modelling WZ Sge, even though its mass ratio is presumably smaller than in most SU UMa stars). In addition, the presence of superhumps after the end of the superoutburst implies that the eccentricity stops much later than the end of a superoutburst. The only ways for the disc to remain eccentric after the end of a superoutburst are then: (i) $t_{\rm f}$ is very long, or (ii) the accretion disc is always eccentric in ER UMa systems.

In the first case, the transition time $t_{\rm f}$ must be much larger than the one used here. However, as shown in Fig. 3, the interval between a superoutburst and the next normal outburst is far too long compared to observations. The second solution implies that the tidal torque is no longer responsible for superoutbursts, since c(r) = c₁ is a constant. However, the presence of large outbursts is still possible, as the (modified) DIM predicts the alternation of narrow and wide outbursts (see Paper I) for large enough mass transfer rates. If the mass transfer rate is constant, the light curve consists of one or two small outbursts surrounded by large ones, but small variations of the mass transfer rate can easily lead to ER UMa type light curves, provided that these variations show some regularity. Irradiation of the disc and of the secondary can also account for the presence of long and short outbursts; Hameury et al. (2000) have included these effects in the standard disc instability model and produced light curves typical of systems such as RZ LMi.

We therefore conclude that ER UMa stars should be dwarf novae with a permanently eccentric accretion disc, thereby accounting for superhumps, and where the illumination of the disc and the secondary star plays an important role. We thus predict that superhumps should exist at all phases of the supercycle of ER UMa stars; this apparently agrees with observations (Gao et al. 1999).

  
4.2 Echo outbursts in WZ Sge stars

Figure 5: Response of a steady accretion disc to a mass transfer rate variation by a factor 20. We use the parameters of EG Cnc, include irradiation, the presence of a hole in the disc, and the existence of a tidal instability. Initially $\dot{M}_2 = 1.5\times 10^{15}$ g s-1; at $t=10,~\dot{M}_2$ increases up to $3.0\times 10^{16}$ g s-1, and at t=30 returns to its quiescent value. The top panel shows the mass accretion rate onto the white dwarf (solid line) and the mass transfer rate from the secondary (dashed line); the other panels show the outer disc radius, its mass and visual magnitude.

WZ Sge stars have very long supercycles and superoutbursts. In addition, no normal outbursts between two consecutive superoutbursts are observed. Some WZ Sge stars also show echo outbursts at the end of the superoutburst: several small outbursts, spaced every tens of days, during which superhumps are still present.

Smak (1993) deduces the mass transfer rate of WZ Sge from the luminosity of the hot spot. It is observed to increase at least by a factor 10 during superoutburst, and it decreases afterwards, remaining larger than the quiescent value during several tens of days. Such mass transfer rate fluctuation could result from irradiation of the secondary star by the white dwarf. Hameury et al. (2000) have shown that irradiation (including disc irradiation) could indeed account for some peculiarities of WZ Sge stars: for example, they reproduced the echo outbursts phenomenon without including the tidal instability in the DIM; however, they did not reproduce a full cycle with long recurrence times, and the echo outbursts they obtained were slightly too dim.

The long recurrence times can be due to low alpha value (Smak 1993; Osaki 1995b), possibly due to a decay of the MHD turbulence that would lead to a time-dependant $\alpha-$prescription (Osaki et al. 2001), thereby explaining the echo outbursts. A low viscosity could result from the secondary being a brown dwarf (Meyer & Meyer-Hofmeister 1999), but one would have to explain why the viscosity is so much lower in these systems as compared to other SU UMa systems which have comparable or even shorter orbital periods. Another possibility (Lasota et al. 1995, 1999; Hameury et al. 1997) is that WZ Sge stars are in a stable low state between superoutbursts, thus explaining the absence of normal outbursts. This requires a hole in the central regions of the disc, as a result of either a moderate magnetic field, or of evaporation. The superoutbursts would then be triggered by an externally imposed increase of the mass transfer rate, the long duration of the outburst and the large mass accreted onto the white dwarf being due to the irradiation of the secondary. They did not include the tidal instability and, as in the case of ER UMa stars, the presence of superhumps and late superhumps is explained if the accretion disc is always eccentric. If one combines these results with those of Hameury et al. (2000) on the echo outburst, one should be able to reproduce a WZ Sge star light curve with echo outbursts, which would however be slightly different than the observed ones.

We have used our TTI model including irradiation and the presence of an inner hole for the accretion disc as determined by Eq. (7) of Hameury et al. (2000). We use the parameters of EG Cnc. The mass transfer rate from the secondary is assumed to be affected by irradiation according to:

$$\dot{M}_{\rm tr} = \max(\dot{M}_2,\gamma <\dot{M}_{\rm acc}>)$$ (4)

where $\dot{M}_2$ is the mass transfer rate that would be obtained in the absence of illumination, $\gamma$ is a constant in the range [0-1], and $<\dot{M}_{\rm acc}>$ is some average of the mass accretion onto the white dwarf, defined as:

$$<\dot{M}_{\rm acc}>~ = \int_{-\infty}^{t_0}\dot{M}_{\rm acc} {\rm e}^{-(t_0-t)/\Delta t}{\rm d}t$$ (5)

t₀ being the actual time. This prescription differs slightly from that of Hameury et al. (1997), in that the average of the mass accretion rate is used instead of its value at time t₀; this accounts for the fact that a fraction of the accretion luminosity is released at later times when the white dwarf cools. In the following, we have taken $\Delta t = 14$ days. We start with a stable system in the low state with a mass transfer rate of $1.5\times 10^{15}$ g s^-1, in agreement with Smak's (1993) estimate for WZ Sge ( $2\times 10^{15}$ g s^-1). At t=10 days, the mass transfer rate $\dot{M}_2$ is increased to $3 \times 10^{16}$ g s^-1; 20 days later, $\dot{M}_2$ is reduced to its initial value. Our results are insensitive on how $\dot{M}_2$returns to its normal value, because illumination dominates in Eq. (4).

This model predicts that the luminosity should increase just before the triggering of a superoutburst. The increase need not be large - one simply requires the mass transfer rate to exceed the critical rate for stability; this therefore does not contradict the observation of Ishioka et al. (2002) that no strong orbital hump due to a high mass transfer rate was detected during the early phase of WZ Sge 2001 outburst.

We obtain a light curve with a very long superoutburst followed by 3 normal outbursts that last $\sim$3 days and occur within two months of the main outburst (Fig. 5). The disc radius remains always larger than 0.35a, and is always in an eccentric state; permanent superhumps are thus expected. However, the interval between normal outbursts is twice longer than for EG Cnc echo outbursts, which are brighter than observed. We did not obtain a better fit with ER UMa light curves; in view of the very crude assumptions made to derive the mass transfer rate from the secondary under the influence of illumination, this is not really a surprise.