Multi-wavelength photometry during the 2018 superoutburst of the WZ Sge-type dwarf nova EG Cancri

We show the overall light curve of the 2018 superoutburst of EG Cnc in Figure 1 with the $V$-band light curve of its 1996–1997 superoutburst for comparison, which is derived from Fig. 1 of Patterson et al. (1998). Although it is unclear when the 2018 superoutburst began, the luminosity increase would occur at least after BJD 2458387.7. After the increase, the luminosity suddenly dropped by $\sim$2 mag around BJD 2458399 at the plateau stage. After the dip, the plateau continued during a week. Multiple rebrightenings (type-B rebrightening) were confirmed soon after the main superoutburst. The rebrightening behavior including the timing of each outburst in the 2018 superoutburst seems to be completely the same as that in the previous superoutburst. However, the depth of the luminosity dip in the plateau stage is different between these two outbursts. The duration of the plateau stage might be different as well.

We have combined our $R$-band, $V$-band, and unfiltered data, and have made the $O-C$ curve of times of superhump maxima to search for superhumps. The $O-C$ curve before the rebrightening stage is given in the top panel of Figure 2 with the amplitude of superhumps and the light curve in its middle and bottom panels. The $O-C$ value is summarized in Table E2 in the supplementary information. Some points with large errors are removed from this figure. According to the past studies on some WZ Sge stars entering superoutbursts with a luminosity dip in the middle of the plateau, there is a possibility that EG Cnc exhibited early superhumps before the dip (Kimura et al., 2018); however, we do not find them. It is unknown whether no detection of early superhumps is attributed to no observations or the fact that early superhumps did not develop.

Small-amplitude light variations seem to have appeared on BJD 2458398 and the onset of superhumps would be delayed with respect to the first appearance of the small-amplitude variations. The linear trend starting from the cycle 0 in the $O-C$ curve and the increasing amplitude of humps suggest that superhumps began developing on the cycle 0 and that the interval during BJD 2458399.7–2458400.8 ($0\leq E\leq 12$) is stage A. The $O-C$ curve bends around the cycle 12, which means that the superhump period suddenly decreased. This indicates the onset of stage B superhumps. The period variation after this (see the texts in the next paragraph) and the decreasing amplitude of humps suggest that the interval during BJD 2458400.8–2458408.7 ($16\leq E\leq 144$) is stage B. However, the fluctuations in the $O-C$ curve before the cycle 0 is not consistent with the observational properties of superhumps and early superhumps, which have been well investigated (see e.g., Kato et al. (2009) and Kato (2015)).

The PDM results and the phase-averaged profiles of stage A superhumps, stage B superhumps, and light modulations before stage A are displayed in the upper left, upper middle, and upper right panels of Figure 3. These modulations are clearly single-peaked. The stage A superhump period ($P_{\rm shA}$) estimated by PDM is 0.06103(2) d (see the upper left panel of Figure 3), and the averaged period at stage B ($P_{\rm shB}$) estimated by PDM is 0.060324(3) d, respectively (see the upper middle panel of Figure 3). The derivative of the superhump period during stage B ($P_{\rm dot}(\equiv\dot{P}_{\rm sh}/P_{\rm sh})$) is $-1.5(4.0)\times 10^{-6}~{}{\rm s}~{}{\rm s}^{-1}$. The averaged superhump amplitude during the plateau stage is 0.08 mag. On the other hand, the humps before the onset of stage A superhumps have a $\sim$0.0604-d period, which is shorter than the stage A superhump period and longer than the orbital period 0.05997(9) d (Patterson et al., 1998) (see the upper right panel of Figure 3). They do not seem to be either superhumps or early superhumps from their periodicity.

The superhumps seem to have continued for a long time after the main superoutburst. Long-lasting superhumps have been observed previously in WZ Sge stars including EG Cnc itself in the past (Patterson et al., 1998, 2002; Kato et al., 2008; Neustroev et al., 2017; Tampo et al., 2020). Figure 4 shows the $O-C$ curve of times of superhump maxima extending to the cycle $\sim$1200. Long-lasting late-stage superhumps seem to have begun at the cycle 215 in the 2018 superoutburst of EG Cnc. The superhumps in the cycles 181–200 may be of the end of stage C. As seen in some other WZ Sge-type superoutbursts, $\sim$0.5 phase shift in superhumps seem to have occurred when this system entered the post-stage of the main superoutburst (see also Kato (2015)) and this phenomenon was also confirmed in the 1996–1997 superoutburst of this object (Kato et al., 2004). This evolution of late-stage superhumps in the $O-C$ curve would be similar to that in the 1996–1997 superoutburst in EG Cnc (see also Patterson et al. (1998)). Both of the $O-C$ curves seem to smoothly bend around the cycle 600. We can see that the superhump period fluctuated in the middle of the late-stage and that the period became constant again after that. We thus divide the late-stage superhumps into the three intervals according to their period variations: during BJD 2458412.9–2458431.7 ($215\leq E\leq 526$) as stage La, during BJD 2458435.5–2458453.7 ($590\leq E\leq 891$) as stage Lb, and during BJD 2458457.5–2458471.7 ($956\leq E\leq 1189$) as stage Lc. It is difficult to distinguish stage Lb and stage Lc, and we determine the time interval of stage Lc to minimize $P_{\rm dot}$. We give the PDM results and the phase-averaged profiles of stage La, stage Lb, and stage Lc superhumps in the lower left, lower middle, and lower right panels of Figure 3. The superhump periods during stage La and stage Lc are 0.06050(1) d and 0.06042(2) d, respectively. The averaged superhump period in stage Lb is 0.060383(7) d, and the derivative of the superhump period during stage B ($P_{\rm dot}$) is $4.8(5.6)\times 10^{-5}~{}{\rm s}~{}{\rm s}^{-1}$.

We have explored the daily variation of optical and NIR colors during the rebrightening stage of the 2018 superoutburst in EG Cnc. In Figure 5, we show daily averages of the colors and the $V$-band light curve. The $B-V$ and $J-K_{\rm S}$ colors are extracted from the data obtained by KU1 and HHO, respectively. During quiescence between rebrightenings, the averaged $B-I$ color was 0.8, which is redder than that in quiescence in typical DNe (Bailey, 1980). This phenomenon is consistent with the unusually red $V-I$ color in the rebrightening phase of the previous superoutburst of this object (Patterson et al., 1998). Also, the $J-K_{\rm S}$ color became bluer. On the other hand, the $B-I$ color became bluer during each rebrightening.

Let us consider the continuum emission from a WZ Sge star. In WZ Sge stars, the radiation from the accretion disk is dominant over UV, optical, and NIR wavelengths, since the white dwarf and the secondary star are tiny and since the boundary layer emits X-rays. If the disk is optically thick and geometrically thin, the emission from the disk surface is approximately blackbody radiation. Generally, the temperature of the inner part is high and that of the outer part is low in the accretion disk since the gravitational potential of the white dwarf becomes deeper inwards in the disk. Simply, each annulus in the disk has each different temperature and multi-temperature blackbody emission synthesizes the radiation from the disk. In fact, the observed continuum spectra of the disk around the outburst maximum agree well with multi-color blackbody emission expected from the standard-disk model (Horne and Cook, 1985; Shakura, Sunyaev, 1973). It is considered that UV photons come mainly from the inner part of the disk. A Rayleigh-Jeans slope of the multi-color blackbody typically exists around optical and NIR wavelengths in the spectrum, which depends on the disk radius and the temperature of the outermost part of the disk. The $B-I$ color is sensitive to the temperature distribution of the outer disk and the $J-K_{\rm S}$ color is affected by the activity at the lower-temperature outermost region of the disk and the time variation of the disk radius. The $B-I$ color of 0.8 in our observations corresponds to the black-body radiation of $\sim$7000 K. Also, the $J-K_{\rm S}$ color of 0.2–0.6 corresponds to the black-body radiation of $\sim$4000–7000 K.

According to the picture described above, the red $B-I$ and $J-K_{\rm S}$ colors at the beginning of the rebrightening stage suggests the presence of the cool component at the outermost disk. If the low-temperature gas at the outermost disk is depleted with accretion during rebrightenings, i.e., the disk shrinks with time, the $J-K_{\rm S}$ color in the quiescent state between rebrightenings should become bluer. The bluer $J-K_{\rm S}$ color in our observations agrees with this idea. On the other hand, the bluer $B-I$ color during each rebrightening implies that the temperature of the outer disk increased at that time, because this region goes up to the outburst state. The $B-I$ color of 0.0 corresponds to the black-body radiation of $\sim$18000 K, which is higher than the minimum temperature of the outburst state (see e.g., Mineshige, Osaki (1983)). For the solid confirmation, we need to analyze the spectral energy distribution of the disk, which is beyond the scope of this paper.

We also have investigated the color variation of the phase-averaged profile of stage B and late-stage superhumps. The phase-averaged profiles and the $V-R$ colors are exhibited in Figure 6. The color became redder at the maximum of superhumps during stage B, while it became bluer after the main superoutburst.

Moreover, we have obtained multi-wavelength light curves during rebrightening by using the Swift satellite. The optical, $UV$, and X-ray light curves are exhibited in Figure 7. This system was faint at X-ray wavelengths during the rebrightening stage, which suggests that the boundary layer between the inner disk rim and the white dwarf became optically thick during multiple rebrightenings because of the high accretion rate to the white dwarf (e.g., Pringle, Savonije (1979); Patterson, Raymond (1985)). On the other hand, the $UV$ light curve is sensitive to multiple rebrightenings and seems to well trace the optical one. The daily $U-B$ color is also displayed in the top panel of Figure 7. Although the $UV$ observations were performed by 4 different bands, we here treat them as the same $U$ band for simplicity. The $U-B$ color became redder at least during quiescence between rebrightenings. As described in the second paragraph of this section, the inner part of the accretion disk is generally hotter than its outer part and $UV$ photons are regarded to be released from the innermost region of the disk. The $U-B$ color is thus considered to be sensitive to the activity in the inner disk as expected by numerical simulations (Cannizzo and Kenyon, 1987). The redder $U-B$ color during the rebrightening phase may represent that the inner region of the disk becomes cooler. The $U-B$ color seems to be at a similar level in comparison with the observed $U-B$ colors in other dwarf novae (Bailey, 1980; Echevarria and Jones, 1983; Shugarov et al., 2018).

We also show a diagram of the daily $U-B$ and $B-I$ colors during the rebrightening phase in Figure 8. We separate the colors into three intervals: the quiescence between rebrightenings, the rising part, and the fading part. The $U-B$ color during the rising part and quiescence has a large dispersion, because that color became redder during the rebrightening phase as described above. Despite the large error bars, we see the trend that the $B-I$ colors are reddest at the rising part, and are bluest at the fading part. The $B-I$ color was slightly redder at the rising part of each rebrightening than that in quiescence and would represent the change in the temperature distribution of the outer disk and the disk radius. In the disk-instability model, the disk rapidly expands at the onset of outbursts and the cool outer region temporarily expands before the outer disk edge completely goes up to the hot state (Ichikawa and Osaki, 1992) and the color should become redder only at the very early stage of the outbursts. Our observations are consistent with this picture. On the other hand, the $B-I$ color around the light maximum and/or at the fading part became much bluer, although there is one exception observed at the small brightening between the first and the second rebrightenings. This behavior agrees with the interpretation that the outer disk became hotter during each rebrightening, as described in the third paragraph of this subsection. The small rebrighetening between the first and the second rebrightenings is likely to have been limited in the narrow region of the disk at that time. Then the temperature of the outer disk does not rise very much, which keeps the $B-I$ color red.

We firstly estimate the mass ratio of EG Cnc by using the stage A superhump period that we have derived (see Sec. 3.2) and the orbital period (0.05997(9) d) reported by Patterson et al. (1998), complying with the method proposed by Kato, Osaki (2013). The superhump period is basically determined by the dynamical precession of the eccentric disk and the pressure effect decreasing the precession rate (Lubow, 1992; Hirose and Osaki, 1993), and hence, the apsidal precession rate of the eccentric disk, $\omega_{\rm pr}$, is approximately formulated as

where $\omega_{\rm dyn}$ and $\omega_{\rm press}$ are the dynamical precession rate and the precession rate by the pressure effect, respectively. Stage A superhumps are the developing superhumps and are believed to represent the dynamical precession at the 3:1 resonance radius. According to Hirose and Osaki (1990), the dynamical precession rate, is expressed as:

where $Q(q)$ and $R(r)$ are the functions of the mass ratio and that of a given radius in the accretion disk, respectively (see equations (1) and (2) in Kato, Osaki (2013) for details), and $r$ is expressed as $(3^{2}(1+q))^{-1/3}a$, which is the 3:1 resonance radius in the case of stage A superhumps. Here $a$ is the binary separation. By using equation (2), the mass ratio of EG Cnc is estimated to be 0.045(5) which is consistent with the $P_{\rm dot}$ value derived in Sec. 3.2 and the type-B rebrightening (see also Fig. 17 in Kato (2015)).

However, the period of stage A superhumps might be diluted by other light modulations before stage A (see Sec. 3.2). For confirmation, we have constrained the mass ratio by using late-stage superhumps. The constant period of late-stage superhumps observed a long time after the main superoutburst would represent the dynamical precession at the outer rim of the disk as suggested by Kato, Osaki (2013). If we know the mass ratio, the disk radius can be estimated from equation (2). We thus can constrain the mass ratio by using the period of stage Lc superhumps, if we determine the range of the disk radius. According to Kato, Osaki (2013), the disk radius after the rebrightening phase in WZ Sge stars ranges between the Lubow-Shu radius (Lubow, Shu, 1975) and the 3:1 resonance radius. The derived limit of the mass ratio of EG Cnc is 0.019–0.057 with 95% confidence intervals. We take the median value as 0.048 since the disk radius long after the superoutburst accompanied by type-B rebrightening is empirically 0.30–0.32$a$ (Kato, Osaki, 2013), which is consistent with the estimated mass ratio in the previous paragraph within 1$\sigma$ errors.

Our results thus allow a larger mass ratio for EG Cnc in comparison with the estimates in Patterson et al. (1998) and Patterson (2011). The mass ratio of this object would be underestimated in the past since the period of stage B superhumps was used for the estimation, as pointed out by Nakata et al. (2013). The evolutionary stage of EG Cnc is exhibited in the $q-P_{\rm orb}$ plane as Figure 9 based on our estimates of the mass ratio. This object still seems one of the good candidates for the period bouncer, since the mass ratio and the orbital period are similar to those of other period-bouncer candidates (the diamonds in Figure 9), even if we consider the wide range of the possible mass ratio.

Also, we have found some characteristic observational features which period-bouncer candidates share in the 2018 superoutburst of EG Cnc. The mean amplitude of superhumps in the main superoutburst was smaller than that of ordinary WZ Sge-type stars (more than 0.1 mag) (see also Sec. 3.2). The fading rate of the middle of the plateau stage in which stage B superhumps were observed was small, 0.098 mag d^-1, which indicates a slow decline (see also Kimura et al. (2018)). These properties are considered to originate from weak tidal dissipations due to extremely low-mass secondary stars. This idea was already described in detail by Kimura et al. (2018).

However, the recurrence time of superoutbursts and the duration of stage A superhumps in EG Cnc are shorter than those of the other candidates (see e.g., Nakata et al. (2014); Kimura et al. (2018)). Although the duration of stage A superhumps was possibly longer than that we have presented in Sec. 3.2, it is still less than a few days even if we consider the zero point of the superhump amplitude expected from the time evolution of the amplitude given in the middle panel of Figure 2. Since the long duration of stage A superhumps is considered to originate from the very small tidal effect by the low-mass star, and since the long recurrence time of superoutbursts would be due to the very low mass-transfer rate from the low-mass secondary, the short duration of stage A superhumps and the short recurrence time of superoutbursts in EG Cnc seem to be inconsistent with its low binary mass ratio. Although it is unclear what this inconsistency comes from, as for the short recurrence time, the sustained eccentricity in the disk expected from the long-lasting superhumps in this object might contribute because the tidal dissipation of the disk by the secondary star increases the viscosity in the disk (Ichikawa and Osaki, 1994). The temporal increase of the viscosity in the elliptical disk may be long-lasting even after superoutbursts, although the viscosity would gradually become lower with time.

Kimura et al. (2016a) proposed that the slow development of the 3:1 resonance tidal instability cannot keep the efficient removal of the angular momentum from the outer disk after the fading of the 2:1 resonance due to weak tidal torques by the small secondary star in extremely low-$q$ objects, which makes a luminosity dip in the middle of the plateau stage in the main superoutburst. However, the dip in the 2018 superoutburst in EG Cnc may be caused by a different reason, since the disk mass accumulated before that superoutburst seems to be small. EG Cnc entered one normal outburst between the 1996–1997 superoutburst and the 2018 superoutburst and some of the disk mass seems to have been depleted at that time. Although the disk is believed to expand far from the 2:1 resonance radius at the onset of superoutbursts in extremely low-$q$ objects, period-bouncer candidates, there may be a possibility that the disk radius did not exceed beyond the 2:1 resonance in the 2018 superoutburst of EG Cnc because of the small amount of mass stored in the disk.

We thus consider that the brightening before the dip in the 2018 outburst of EG Cnc is probably a precursor, i.e., a normal outburst triggering a superoutburst, which is usually observed before the superoutbursts in SU UMa-type stars (see e.g., Osaki, Kato (2013)), as confirmed in the 2015 superoutburst of AL Com, another WZ Sge star, which was triggered after only 1.5 yr quiescence (Kimura et al., 2016b).

The light curve in the rebrightening stage of the 2018 superoutburst in EG Cnc was completely the same as that of its previous superoutburst in 1996–1997 despite the difference in the light curve of the main superoutburst (see Figure 1). This is the second example following the 2015 superoutburst in AL Com (Kimura et al., 2016b). In the case of AL Com, the duration of the type-A (long duration) rebrightening was a little shorter in its 2015 superoutburst than that in its 2013 superoutburst (see also Kato et al. (2014a)). However, not only the rebrightening type but also the timing of each rebrightening were the same between the two superoutbursts in EG Cnc. This phenomenon strengthens the interpretation proposed by Kimura et al. (2016b), that is, the rebrightening pattern is inherent to each object. The rebrightening type would be independent of the amount of the initial disk mass at the beginning of superoutbursts. This idea is consistent with the picture that the rebrightening type is related to the evolutionary stage of DNe, which was suggested by Kato (2015). Kato (2015) also noticed that WZ Sge showed the same type of rebrightening in its superoutbursts despite a little decrease in the duration of the plateau stage of the main superoutburst, and other WZ Sge-type DNe entered superoutbursts more than twice exhibited the same type of rebrightening every superoutburst (Kato et al., 2009; Pavlenko et al., 2012; Kato et al., 2015). This agrees with the above interpretation.

The physical mechanism of rebrightenings observed in WZ Sge stars is unclear. As mentioned in the introduction, two possibilities were proposed as the trigger of rebrightening up to date: the enhancement of the mass transfer by irradiation (Hameury, 2000; Patterson et al., 2002) and the mass reservoir outside the 3:1 resonance radius (Kato et al., 1998; Hellier, 2001). As for the multiple rebrightenings observed in EG Cnc, we have detected superhumps during the rebrightening phase of the 2018 outburst of EG Cnc, and the $B-I$ and $J-K_{\rm S}$ colors were redder than those in the typical quiescence (see Sec. 3.3). This may suggest that substantial mass was left at the outermost region in the disk, which supplies some materials to the inner disk, as many other optical and NIR observations suggested in the past (Kato et al., 1997; Patterson et al., 1998; Uemura et al., 2008; Isogai et al., 2015; Nakata et al., 2014; Tampo et al., 2020). Also, the redder color of $B-I$ at the beginning of each rebrightening may indicate the disk expansion (see also Figure 8). This contradicts the enhanced mass transfer hypothesis since in that scenario the influx of low specific angular momentum material from the secondary makes the disk shrink at the onset of outbursts, which is reproduced by numerical simulations (Ichikawa and Osaki, 1992). The bluer $i-z$ and $J-K_{\rm S}$ colors during the rebrightening phase may indicate the depletion of the mass reservoir by accretion. We thus have found positive evidence for the mass reservoir model, while we have not for the enhanced mass-transfer model.

If the disk possesses a lot of cool material even after the main superoutburst, the small increase of the viscosity could reproduce sustained rebrightenings as observed in EG Cnc, which was proposed by Osaki et al. (2001). We have confirmed that the optical flux in the quiescence between rebrightenings in the 2018 superoutburst of EG Cnc was higher than that in the quiescence before the main outburst and after the end of the rebrightening phase (see Figure 1). Since more energy should be dissipated due to more efficient angular-momentum transfer, if the viscosity increases in the disk, the increase of the optical flux may represent the temporary enhancement of the viscosity of the disk. Osaki et al. (2001) considered that the temporary enhancement of the viscosity is caused by MHD turbulence. We also consider that the strong tidal dissipation lasting even after the main superoutburst could contribute to the enhancement as mentioned in Sec. 4.1.

Recent theoretical work by Meyer, Meyer-Hofmeister (2015) suggested that the enhancement of the viscosity halts the cooling-wave propagation over the whole disk at the decline from the plateau stage of the main superoutburst, and repetitive reflections of the heating and cooling waves in the disk could be the source of the multiple rebrightenings. The heating and cooling waves are the transition waves propagating over the disk. The former occurs when a region goes up to the hot state and the latter occurs when a region goes down to the cool state. They considered that the inner disk is permanently hot and that the entire disk easily jumps between the hot state and the cool state by the repetitive reflections.

However, we have found that the $U-B$ color became redder during the rebrightening phase in the case of 2018 superoutburst in EG Cnc (see the top panel of Figure 7). This would mean that the inner disk became cooler during the rebrightening stage (see Sec. 3.3). Also, the $U$-band light curve is similar to the optical light curve. These imply that the heating and cooling waves would go back and forth not only at the outermost disk but also at the inner disk at some point in the rebrightening phase. On the other hand, the $B-I$ color keeps a similar level during quiescence between multiple rebrightenings (see also Figure 5). Also, the rapid rise of the rebrightenings except for the last one suggests that most of the rebrightenings were triggered at the outer disk, i.e., outside-in outbursts. If the heating wave propagates outwards from the inner disk, we should always observe a slow rise at the beginning of each rebrightening, i.e., inside-out outbursts (see e.g., Mineshige, Osaki (1985) for the difference between outside-in and inside-out outbursts). Taking into account our results, the inner portion of the disk is unlikely permanently hot in EG Cnc and the heating wave would be newly triggered at the beginning of each rebrightening at the outer disk rather than repetitive reflections.

We have investigated the time evolution of the color variation of superhumps in Sec. 3.3. The redder color at the superhump maximum during stage B was also confirmed in the other two WZ Sge stars (Isogai et al., 2015; Neustroev et al., 2017), although the different types of color variations were found in V455 And (Matsui et al., 2009). Imada et al. (2018) showed the color variations of stage B superhumps change with time in the two WZ Sge stars HV Vir and OT J012059.6$+$325545, and interpreted that the time-varying pressure effect working on the precessing disk (Lubow, 1992; Hirose and Osaki, 1993) makes the variety. The difference in the color variation of superhumps among WZ Sge-type stars maybe because of the time variation of the pressure effect and/or the difference of the color bands used for the analyses. We have explored the $V-R$ color, while Matsui et al. (2009) investigated the $V-J$ color, and the data did not cover the entire period of the stage B superhumps. On the other hand, we have confirmed the bluer color around the superhump maximum in the post-superoutburst stage. This trend was confirmed also in SSS J122221.7$-$311525 (Neustroev et al., 2017).

The color variation of superhumps seems to be weaker after the main superoutburst (see also Figure 6). Superhumps are regarded to be produced by periodic tidal stressing working on the eccentric disk (Whitehurst, 1988; Hirose and Osaki, 1990; Lubow, 1991a). The difference in the amplitude of the color variation is likely attributed to the difference in the size of the region in which the eccentricity works. The wider region the eccentricity propagates over, the larger the change in the temperature, geometry, and/or density of the disk during each cycle of superhumps would be. The superhump period is the synodic period between the period of the prograde precession of the disk and the orbital period (Whitehurst, 1988; Hirose and Osaki, 1990; Lubow, 1991a), and the period change during stage B is believed to represent that the eccentricity propagates over the entire disk (Kato, Osaki, 2013). On the other hand, the period variation during the rebrightening phase would be moderate in comparison with that during stage B, since the late-stage superhumps included two constant-period stages.