ABSTRACT
I present a variation on Osaki's tidal thermal instability model for SU UMa behavior. I suggest that in systems with the lowest mass ratios, the angular momentum dissipation in an eccentric disk is unable to sustain the disk on the hot side of the thermal instability. This decoupling of the tidal and thermal instabilities in systems with q≲0.07 allows a better explanation of the "echo" outbursts of EG Cnc and the short supercycles of RZ LMi and DI UMa. The idea might also apply to the soft X‐ray transients.
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1. INTRODUCTION
The SU UMa subclass of cataclysmic variable is characterized by two distinct accretion disk instabilities. The first is a thermal instability caused by partially ionized hydrogen, which results in dwarf nova outbursts (e.g., Cannizzo 1993). The second is a tidal instability occurring when orbits in the disk resonate with the secondary‐star orbit, driving the disk elliptical and causing it to precess with a period of a few days (e.g., Whitehurst & King 1991). The 3:1 resonance that is responsible occurs at a disk radius of ≈0.46a, where a is the binary separation. In the elliptical disk the tidal stresses vary with the interaction of the orbital and precessional periods, to produce excess light recurring with a period a few percent longer than the orbital period, called a superhump.
2. SU UMa SUPERCYCLES
Observationally, SU UMa stars show a succession of normal outbursts, caused by the thermal instability alone, followed by a longer lasting superoutburst, during which the presence of superhumps implies that the tidal instability is also excited. This pattern (called a supercycle) then repeats.
Osaki (1989, 1996) has presented a model (the tidal thermal instability, or TTI, model) which combines the instabilities to explain the supercycle. He suggests that a normal outburst depletes the disk by less than the matter accumulated since the last outburst. The disk therefore grows over a succession of outbursts until, at the onset of another outburst, it expands beyond 0.46a and becomes tidally unstable. The tidal dissipation of the eccentric state enhances the mass flow through the disk, sustaining the disk in its hot state for a longer superoutburst, and so draining the disk of the majority of its material. Once the disk has shrunk to ≈0.35a, neither the eccentricity nor the hot state can be sustained, and the disk reverts to a cold, circular quiescence, to begin the cycle anew.
The TTI model is successful in explaining typical SU UMa stars, which have supercycles of 100–1000 days. However, some SU UMa stars (those with abnormally short or abnormally long supercycles) have outburst patterns that do not fit the standard model. This has led Hameury, Lasota, & Warner (2000) to present alternative models involving evaporation of the inner disk and irradiation‐induced increases in the mass transfer rate. The purpose of this paper is to suggest that the TTI model can, after all, explain the anomalous systems, given one alteration.
2.1. The Problem of "Echo" Outbursts in WZ Sagittae Systems
The term "WZ Sagittae star" is a shorthand for SU UMa systems with supercycles lasting decades (see Fig. 1). Part of the explanation for this is that the mass transfer rates,M˙, are very low, typically 1015 g s−1, compared to 1016 g s−1 for normal SU UMa stars (Osaki 1995a). Also, such stars rarely or never show normal outbursts between the superoutbursts. This implies that material does not accumulate in the inner disk, which requires an abnormally low viscosity, inner‐disk evaporation, or a magnetic propeller (e.g., Osaki 1995a; Hameury et al. 2000; Wynn, Leach, & King 2001). However, I do not address this issue here.
Fig. 1.— Supercycle lengths of SU UMa stars. Those with abnormally long supercycles are the WZ Sagittae stars, while those with abnormally short supercycles are the ER UMa stars. Data are from Warner (1995), supplemented by papers mentioned in the text.
Instead, my concern is the "echo" outbursts seen following superoutbursts in WZ Sagittae stars. These were most striking in EG Cnc, which, after the first superoutburst for 19 years, showed a rapid succession of six shorter outbursts, spanning 1 month, before dropping by a further magnitude to full quiescence (Fig. 2; Patterson et al. 1998). Similar, although less pronounced, echoes have been seen in other WZ Sagittae systems (e.g., AL Com: Patterson et al. 1996; UZ Boo: Kuulkers, Howell & van Paradijs 1996: and WZ Sge itself: Patterson et al. 1981).
The first problem raised by EG Cnc is that superhumps were seen throughout the period of echo outbursts, even during quiescence (Patterson et al. 1998). This contradicts the standard TTI model, in which the superoutburst ends with both the thermal and tidal instabilities shutting down.
The second problem is that the mean accretion rate is clearly higher over the echo period than in full quiescence. In the TTI model this requires an ad hoc increase of the quiescent viscosity to an α‐parameter,αcold, of 0.1 (Osaki, Shimuza, & Tsugawa 1997), a value more typical of a disk in outburst. Alternatively, Hameury et al. (2000) have suggested that irradiation of the disk by a hot white dwarf produces the echoes.
2.2. The Problem of ER UMa Supercycles
The ER UMa stars have short supercycles lasting 40–50 days (V1159 Ori and ER UMa itself), 25–35 days (DI UMa), and 19 days (RZ LMi) (Fig. 1; Robertson, Honeycutt, & Turner 1995; Kato et al. 1999).
Osaki (1995b) has shown that, in the TTI model, increasing the M˙ reduces the interoutburst intervals and thus leads to a shorter supercycle. However, this effect can reduce the supercycle only to ≈40 days (corresponding to M˙ ≈ 4 × 1016 g s−1); increasing M˙ further results in longer superoutbursts, and thus the supercycle lengthens again. To achieve a supercycle of only 19 days, the superoutburst must be truncated prematurely. Osaki (1995c) modeled RZ LMi's supercycle by artificially ending the superoutburst when the disk had shrunk from 0.46a to only 0.42a, whereas an end value of 0.35a is used for other SU UMa stars.
A second problem with ER UMa stars is that they show superhumps beyond the end of the superoutburst, including in quiescence (e.g., Patterson et al. 1995). This again violates the standard TTI model since the disk radius is then at a minimum (0.26a in the ER UMa model of Osaki 1995b, well inside the resonance zone at 0.46a) and so should not show superhumps.
3. A TTI VARIATION FOR SYSTEMS WITH ULTRALOW MASS RATIOS
In the standard TTI model, the extra drain of angular momentum in an eccentric disk provides a sufficient mass flow to ensure that the disk remains in the hot state, and thus the end of superoutburst requires the end of the eccentricity. However, the above difficulties are solved if the tidal and thermal instabilities are decoupled, allowing the superoutburst to end when a disk that is still eccentric drops out of the hot state. This, in any case, is virtually required by the observations of superhumps beyond the end of the superoutburst in ER UMa and WZ Sagittae systems.
I suggest that the key factor causing this behavior is an ultralow mass ratio. Figure 3 shows the radius at which the disk is truncated by tidal effects and also the radius of the 3:1 superhump resonance, both as a function of mass ratio, q (see Whitehurst & King 1991). For systems with q≳0.3, the disk cannot extend to the resonance radius, and the binary is not an SU UMa star. As the mass ratio decreases, there is increasing room in the disk outside the 3:1 resonance radius. In this region, other, weaker, higher order resonances operate, and these eventually combine to produce the tidal limit. However, the lower the mass ratio, the more likely that matter can accumulate in regions of the disk outside 0.46a, which, avoiding the main superhump resonance, interact less strongly with the secondary star and so suffer a lower tidal dissipation of angular momentum, J˙.
Fig. 3.— The variation in the 3:1 resonance radius and the tidal‐limit radius (in units of the separation a), as a function of mass ratio. Based on Whitehurst & King (1991).
Another factor is that the strength of the tidal interaction decreases for lower mass ratios (assuming a fixed orbital period and white dwarf mass), and this might assist in decoupling the tidal and thermal instabilities at ultralow mass ratios. However, it should be noted that the tidal interaction is generally stronger at shorter orbital periods, owing to the smaller size of the binary, which would counteract this effect.
In SU UMa stars we can investigate q without having to measure masses, since q correlates with the amount by which the superhump period exceeds the orbital period, 
, defined by = (Psh - Porb)/Porb. The exact relation between and q is not certain (see, e.g., Patterson 1998; Murray 2000), but for current purposes we require only that increases monotonically with q.
Note that the stars giving difficulty to the TTI model have exceptionally low values of and thus q (Fig. 4). EG Cnc has the lowest value of all, followed by two other stars that have shown echoes (WZ Sge and AL Com). DI UMa is one of two stars with supercycles shorter than 40 days, while for the other, RZ LMi, has not yet been measured.
Fig. 4.— The superhump period excess, , vs. orbital period, for SU UMa stars. The data, and the translation to mass ratio, q, are taken from Patterson (1998).
3.1. Application to Echo Outbursts
Applying the idea to EG Cnc, I suggest that the disk was larger than 0.46a and eccentric for the entire ≈75 day period of superoutburst and echoes (again, this is required by the observation of superhumps throughout this period by Patterson et al. 1998). However, after 15 days of superoutburst the disk outside 0.46a was depleted sufficiently to revert to the cool state—the lower tidal J˙ in this region could not drive the disk edge inward quickly enough to prevent this. Thus a cooling wave, originating outside 0.46a, ended the outburst. However, the enhanced J˙ from the 3:1 resonance did ensure an enhanced flow into the inner disk and this triggered a new outburst, which originated in a heating wave from the inner disk. The fact that this outburst was short suggests that the heating wave petered out before reaching the edge of the disk, and thus a cooling wave quickly followed. Thus a succession of short outbursts ensued, until the disk had depleted sufficiently to revert to the circular state. Then, when J˙ declined, the mass accretion rate dropped, and as the last superhumps died away the system faded to full quiescence.
Turning now to WZ Sge and AL Com, both of these systems, near the end of outbursts in 1978 and 1995, respectively, showed a brief drop in magnitude followed by the resumption of the superoutburst (Patterson et al. 1981, 1996). This again suggests that the eccentric J˙ was insufficient to sustain the outer disk in the hot state, resulting in a cooling wave. But, in these systems, the heating wave did run right to the edge of the disk, and thus reestablished the superoutburst. Note that the first cooling waves in WZ Sge and AL Com occurred 25–30 days after the start of superoutburst, whereas that in EG Cnc occurred much earlier, at 15 days. Thus WZ Sge and AL Com appear to be systems on the borderline of showing the echoes that are fully fledged in EG Cnc.
3.2. Application to ER UMa Supercycles
As noted above, supercycles shorter than ≈40 days (those of DI UMa and RZ LMi) require a premature end to the superoutburst. As in EG Cnc, I suggest that a cooling wave from the lower J˙ region outside 0.46a causes the disk to revert to the cold state while still eccentric. The early end to the superoutburst means that the disk retains a larger fraction of mass than is usual. This, and the enhanced J˙ from the 3:1 resonance region, means that the flow of material into the inner disk soon triggers another outburst. Again, the heating wave does not propagate to the outer edge, and so the new outburst is short.
Such stars therefore exhibit "echo" outbursts identical to those in EG Cnc—short, but with superhumps. The difference is that in EG Cnc the mass transfer rate,M˙, was much lower than the mean mass accretion rate,M˙acc, during the echoes. Thus the disk shrank to quiescence. In RZ LMi and DI UMa M˙ is higher than M˙acc, and the disk gains material until a heating wave does propagate to the outer edge of the disk, resulting in another superoutburst.
The two other ER UMa stars, V1159 Ori and ER UMa itself, have longer supercycles (43 and 48 days) that can be explained in the TTI model solely by a high M˙ (Osaki 1995b). Further, their values imply mass ratios that are typical of SU UMa stars (Fig. 4). Thus, the TTI variant discussed above might be required only for RZ LMi and DI UMa. However, the observation of superhumps in V1159 Ori beyond the end of the superoutburst (Patterson et al. 1995) suggests that the above ideas might come into play in all ER UMa stars.
3.3. Application to Soft X‐Ray Transients
Kuulkers et al. (1996) have pointed out the similarities between soft X‐ray transients (SXTs) and WZ Sagittae stars. SXTs are, for many intents and purposes, dwarf novae with a black hole replacing the white dwarf. The greater mass of the black hole in SXTs (∼10 M⊙ compared to a white dwarf's ∼1 M⊙) ensures that they have low mass ratios. Thus it is notable that they can show echo outbursts (most obviously in GRO J0422+32: Fig. 2; Kuulkers et al. 1996) along with quiescent superhumps around the time of the echo outbursts. From superhumps of GRO J0422+32, O'Donoghue & Charles (1996) measure = 0.016, which is lower than in most SU UMa stars (Fig. 4). I propose that the echoes arise for the same reason as they do in EG Cnc: the ultralow mass ratio allows the disk to cycle over the thermal instability while still in the eccentric state. Note, though, that this requires that the greater irradiation of an SXT does not prevent the outer disk from cooling. Perhaps the disk is convex, and the outer disk is shielded from the X‐ray flux.
4. CONCLUSIONS AND PREDICTIONS
Osaki (1996) has outlined a unification model for cataclysmic variables in which the different outburst properties are controlled by the two parameters q and M˙. I propose that echo outbursts and ultrashort supercycles (≲ 40 days) can be brought into the same scheme and that they occur in binaries with mass ratios of q≲0.07. In such systems the superoutbursts end prematurely because the lower tidal J˙ outside the 3:1 resonance radius allows cooling and heating waves to run through the disk while it is in the eccentric state. This produces "echo" outbursts characterized by short durations and the presence of superhumps. If M˙<M˙acc over the echo period, the system subsides to quiescence (EG Cnc behavior); if M˙>M˙acc, another superoutburst results (RZ LMi behavior).
This model predicts that superhumps should be present (even in quiescence) thoughout echo periods and perpetually in stars with ultrashort supercycles. Also, any system showing these characteristics should have an exceptionally low value of the superhump excess and mass ratio q. Of the systems discussed in this paper, we know for all except RZ LMi, the star with the shortest supercycle; I predict that it has a value of even lower than the 0.013 found for DI UMa.
I thank Joe Patterson, Andrew King, and Brian Warner for valuable comments on this work.