3 Decoupling the thermal and tidal instabilities

  
3.1 A property of ultra-low mass ratio systems?

Hellier (2001) suggested that the ultra-low mass ratio of ER UMa and WZ Sge stars is responsible for their peculiarities. In their case, the Roche lobe extends far beyond the 3:1 resonance radius, and the disc could in principle be larger than r_3:1. Hellier proposed that regions outside the resonance radius suffer a lower tidal dissipation than the portions in the 3:1 resonance zone. Then, during a superoutburst, heating by the tidal torque will not be as efficient as in other SU UMa stars, and a cooling wave could start while the disc is still eccentric, and will remain so several days after the end of the superoutburst.

This would then account for late superhumps. In addition, Hellier suggested that this model could also explain the echo outbursts in WZ Sge stars and the short ER UMa supercycle: at the end of a superoutburst, the disc would be still eccentric, and the tidal dissipation sufficiently large to trigger short outbursts during which the heating wave does not propagate to the outer edge (one should add that such outbursts are not only short but also have amplitudes lower than outbursts in which the heating front reaches the outer disc's edge). ER UMa stars would then be systems always in an eccentric state, while WZ Sge stars would finally return to complete quiescence with circular discs. The difference between the supercycle duration of ER UMa and WZ Sge stars would be due to higher mass-transfer rates in ER UMa's.

Figure 1 shows the simulated light curve for the parameters of SU UMa, assuming a factor 20 between c₁ and c₀. The superoutburst ends when the disc radius becomes lower than $r_{\rm crit0}$, i.e. the end of the tidal instability causes the end of the superoutburst. Figure 2 shows the simulated light curve for the parameters of ER UMa, with the same c₁ / c₀. The mass transfer rate of 10¹⁶ g s^-1 chosen here is close to the critical value above which the system is stable (within a factor 1.3). In this case, as predicted by Hellier, the superoutburst ends because the thermal instability stops, while the tidal instability is still effective. However, contrary to Hellier's hypothesis (2001), this phase is very short because of the rapid shrinking of the disc. Therefore, the duration of the supercycle is much larger than observed (there is practically no quiescence phase in real ER UMa stars): short outbursts are due to the usual thermal-viscous instability. As a consequence, contrary to Hellier's assumption, the heating front in short outbursts always reaches the outer disc's edge. Mass transfer rates closer to the stability limit do not lead to shorter supercycles, but rather to longer superoutbursts; this is actually the reason why it is impossible to obtain very short supercycles in the standard TTI model without changing the tidal instability condition (Osaki 1995a).

Figure 2: Same as Fig. 1 for ER UMa parameters.

We have investigated the influence of the ratio c₁ / c₀ and of the Roche size on our results. Figure 3 compares the superoutburst behavior for c₁ / c₀ factors of 20, 50 and 80 for ULMR stars with prescription I. As can be seen, the width of the superoutburst strongly depends on c₁. Very small and very large values of c₁ both give short superoutburst, either because the tidal heating is too small (small c₁), or because the disc contraction under the effect of the tidal torque is too large (large c₁). Typical c₁/c₀ should be in the range $\sim$20-50 in order to reproduce the observed superoutburst durations.

The decoupling of the tidal and thermal instability was found for values of c₁/c₀ lower than 80, for both prescription I and II for ER UMa parameters, thereby confirming the first part of Hellier (2001) proposal. On the other hand, for the parameters of SU UMa, the instability stopped because the disc shrinks below 0.35 a, except for c₁/c₀< 10; this limit is increased to $\sim$20 when one uses prescription II.

Figure 3: Radius behavior during superoutburst for ultra-low mass ratio stars with $\dot{M}_{\rm tr} = 10^{16}$ g s-1. The dot-dashed lines represents the 0.46 a (upper) and 0.35 a(lower) radius. For c1 / c0 factors of 20 (dashed line) and 50 (solid line), the superoutburst lasts $\sim$21 days ($\sim$17 days for c1 / c0 = 50) and is terminated by a cooling wave. For a factor 80 (solid line) it lasts $\sim$9 days and ends by the disc becoming circular.

These results strongly support the idea that the decoupling of thermal and tidal instability is a property of the ultra low mass ratio systems. One should however keep in mind that this conclusion depends somewhat on the assumed radial dependence of c, and on the assumed c₁/c₀; extreme values (less than 10 or more than 50) cause too short superoutburst to be triggered but ER UMa stars sometimes have short superoutbursts.

3.2 The superoutburst cooling wave

As shown in the previous section, decoupling of tidal and thermal instabilities is expected in ultra low mass ratio SU UMa stars. However, this is not sufficient to explain the late superhumps or echo outbursts observed in ER UMa and WZ Sge systems, as suggested by Hellier (2001). When a cooling front starts to propagate, the radius decreases rapidly (in less than a few days) to its minimum value in the supercycle. If this minimum is less than  $r_{\rm crit0}$, then the disc will return to a circular shape in a time scale $t_{\rm f}$, and one does not expect to observe late superhumps. If on the other hand the minimum radius is larger than  $r_{\rm crit0}$, the disc always remains eccentric, the superhumps are permanent, but one then loses the very mechanism that was supposed to trigger superoutbursts.

The disc shrinkage at the end of an outburst or a superoutburst is therefore a key question. The radius evolves under the influence of three main effects, as shown in Eq. (1): (i) the addition of mass with low specific angular momentum at the outer edge of the disc and (ii) the tidal torque that both act to contract the disc, and (iii) the viscous outward transport of angular momentum that tends to make the disc larger. These effects are parameterized by $\dot{M}_{\rm tr}$, c and $\alpha$respectively. In order to determine their influence on the disc radius variations, we have changed their values by a factor of 2 (linear variation on a time scale of 10⁴ s) for a steady disc in a hot state (Fig. 4). The disc radius in the new steady state is unchanged, except in the case where c is modified; the short term influence of these variations are summarized in Table 2 which shows the value of ${\rm d} \log r / {\rm d} \log x$, 10⁴ s after the beginning of the variation, x being one of the three parameters. As can be seen, ${\rm d} \log r / {\rm d} \log c$ is 5 to 10 times smaller than the two other quantities. This explains why c₁/ c₀ has to be larger than 10 in order to obtain a superoutburst, and why this ratio needs to be larger than 20 in order for the radius to decrease fast enough during a superoutburst and reach  $r_{\rm crit0}$ before the cooling wave starts.

Figure 4: Responses of the outer radius of a steady accretion disc in the hot state to variation by a factor 2 of $\dot{M}_{\rm tr}$(solid line), c (dashed line) and $\alpha$ (dot-dashed line). The parameters vary linearly in 104 s. Note that the perturbation starts at $t \sim 0.3$ d.

 

 

Table 2: Response of the outer disc radius $r_{\rm d}$ to a decrease of $\vert\dot{M}\vert _{\rm tr}$, c and $\alpha$ by a factor 2.

x	${\rm d}\log r_{\rm d} / {\rm d}\log x$ (at 104 s)
$\vert\dot{M}\vert _{\rm tr}$	$2.5 \times 10^{-2}$
c	$2.67 \times 10^{-3}$
$\alpha$	$-4.37 \times 10^{-2}$

In principle, the disc shrinkage could be slower, or possibly delayed, if the mass transfer rate from the secondary were reduced, possibly as a result of a decrease in the illumination of the secondary. However, when the cooling wave starts, $\alpha$decreases from $\alpha_{\rm hot}$ to $\alpha_{\rm low}$ in the outer disc regions, i.e. by a factor 5. c has to be kept fixed, since one requires the disc to remain eccentric. Because of the large change in viscosity when a cooling front starts to propagate, $\dot{M}_{\rm tr}$ would then have to be reduced by an extremely large factor; we made several numerical experiments with various prescriptions for a change in $\dot{M}_{\rm tr}$(amplitude and possible delay with respect to the formation of a cooling front), and we were unable to find a case in which the minimum disc radius was not obtained shortly after the end of a superoutburst.