title: Multi-wavelength photometry during the 2018 superoutburst of the WZ Sge-type dwarf nova EG Cancri
authors: Mariko Kimura Keisuke Isogai Taichi Kato Naoto Kojiguchi Yasuyuki Wakamatsu Ryuhei Ohnishi Yuki Sugiura Hanami Matsumoto Sho Sumiya Daiki Ito Kengo Nikai Katsura Matsumoto Sergey Yu. Shugarov Natalia Kathysheva Hiroshi Itoh Pavol A. Dubovsky Igor Kudzej Hiroshi Akitaya Kohei Oide Takahiro Kanai Chihiro Ishioka Yumiko Oasa Tonny Vanmunster Arto Oksanen Tamás Tordai Katsuhiro L. Murata Kazuki Shiraishi Ryo Adachi Motoki Oeda Yutaro Tachibana Seiichiro Kiyota Elena P. Pavlenko Kirill Antonyuk Oksana Antonyuk Nikolai Pit Aleksei Sosnovskij Julia Babina Alex Baklanov Koji S. Kawabata Miho Kawabata Tatsuya Nakaoka Masayuki Yamanaka Kiyoshi Kasai Ian Miller Stephen M. Brincat Wei Liu Mahito Sasada Daisaku Nogami
text: 
  
  
  mariko.kimura@riken.jp
  
  1Department of Astronomy, Graduate School of Science, Kyoto University, Oiwakecho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502
  
  2Extreme Natural Phenomena RIKEN Hakubi Research Team, Cluster for Pioneering Research, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198
  
  3Okayama Observatory, Kyoto University, 3037-5 Honjo, Kamogatacho, Asakuchi, Okayama 719-0232
  
  4Osaka Kyoiku University, 4-698-1 Asahigaoka, Kashiwara, Osaka 582-8582
  
  5Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetsky Ave., 13, Moscow 119992, Russia
  
  6Astronomical Institute of the Slovak Academy of Sciences, 05960 Tatranska Lomnica, Slovakia
  
  7Variable Star Observers League in Japan (VSOLJ), 1001-105 Nishiterakata, Hachioji, Tokyo 192-0153
  
  8Vihorlat Observatory, Mierova 4, Humenne, Slovakia
  
  9Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570
  
  10Graduate school of Education, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570
  
  11Faculty of Education, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570
  
  12Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-
  Hiroshima, Hiroshima 739-8526
  
  13Center for Backyard Astrophysics (Belgium), Walhostraat 1A, B-3401, Landen, Belgium
  
  14Hankasalmi observatory, Jyvaskylan Sirius ry, Verkkoniementie 30, FI-40950 Muurame, Finland
  
  15Polaris Observatory, Hungarian Astronomical Association, Laborc utca 2/c, 1037 Budapest, Hungary
  
  16Department of Physics, Tokyo Institute of Technology, 2-12-1
  Ookayama, Meguro-ku, Tokyo 152-8551
  
  17VSOLJ, 7-1 Kitahatsutomi, Kamagaya, Chiba 273-0126
  
  18Federal State Budget Scientific Institution “Crimean Astrophysical Observatory of RAS”, Nauchny, 298409, Republic of Crimea
  
  19Baselstrasse 133D, CH-4132 Muttenz, Switzerland
  
  20Furzehill House, Ilston, Swansea, SA2 7LE, UK
  
  21Flarestar Observatory, San Gwann SGN 3160, Malta
  
  22Purple Mountain Observatory, Chinese Academy of Sciences, No.10 Yuanhua Road, Qixia District, Nanjing, 210023, China
  
  
  
  
   
  
  accretion, accretion disks - novae, cataclysmic 
  variables - stars: dwarf novae - stars: individual 
  (EG Cancri)
  
  Kimura et al.The 2018 superoutburst of EG Cnc
  
  Multi-wavelength photometry during the 2018 superoutburst of the WZ Sge-type dwarf nova EG Cancri
      Mariko Kimura,1,2,*
          Keisuke Isogai,1,3
          Taichi Kato,1
          Naoto Kojiguchi,1
          Yasuyuki Wakamatsu, 1
          Ryuhei Ohnishi,1
          Yuki Sugiura,4
          Hanami Matsumoto,4
          Sho Sumiya,4
          Daiki Ito,4
          Kengo Nikai,4
          Katsura Matsumoto,4
          Sergey Yu. Shugarov,5,6
          Natalia Katysheva,5
          Hiroshi Itoh,7
          Pavol A. Dubovsky,8
          Igor Kudzej,8
          Hiroshi Akitaya,9,12
          Kohei Oide,10
          Takahiro Kanai,10
          Chihiro Ishioka,10
  
          Yumiko Oasa,9,10,11
          Tonny Vanmunster,13
          Arto Oksanen,14
          Tamás Tordai,15
          Katsuhiro L. Murata,16
          Kazuki Shiraishi,16
          Ryo Adachi,16
          Motoki Oeda,16
          Yutaro Tachibana,16
          Seiichiro Kiyota,17
          Elena P. Pavlenko,18
          Kirill Antonyuk,18
          Oksana Antonyuk,18 
          Nikolai Pit,18
          Aleksei Sosnovskij,18
          Julia Babina,18
          Alex Baklanov,18
          Koji S. Kawabata,12
          Miho Kawabata,12
          Tatsuya Nakaoka,12
          Masayuki Yamanaka,12
          Kiyoshi Kasai,19
          
          Ian Miller,20
          Stephen M. Brincat,21
          Wei Liu,22
          Mahito Sasada,12
          Daisaku Nogami1
          
      December 30, 2023
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  We report on the multi-wavelength photometry of the 2018 
  superoutburst in EG Cnc.  
  We have detected stage A superhumps and long-lasting late-stage 
  superhumps via the optical photometry and have constrained 
  the binary mass ratio and its possible range.  
  The median value of the mass ratio is 0.048 and the upper limit 
  is 0.057, which still implies that EG Cnc is one of the possible 
  candidates for the period bouncer.  
  This object also showed multiple rebrightenings in this superoutburst, 
  which are the same as those in its previous superoutburst 
  in 1996–1997 despite the difference in the main superoutburst.  
  This would represent that the rebrightening type is inherent to 
  each object and is independent of the initial disk mass 
  at the beginning of superoutbursts.  
  We also found that B-I and J-K_ S colors were 
  unusually red just before the rebrightening phase and became bluer 
  during the quiescence between rebrightenings, which would mean 
  that the low-temperature mass reservoir at the outermost disk 
  accreted with time after the main superoutburst.  
  Also, the ultraviolet flux was sensitive to rebrightenings 
  as well as the optical flux, and the U-B color became redder 
  during the rebrightening phase, which would indicate that 
  the inner disk became cooler when this object repeated 
  rebrightenings.  
  Our results thus basically support the idea that the cool 
  mass reservoir in the outermost disk is responsible for 
  rebrightenings.  
  
  
  
  
  
  § INTRODUCTION
  
  
  Cataclysmic variables (CVs) are close binary systems consisting 
  of a white dwarf (the primary star) and a low-mass star 
  (the secondary star).  An accretion disk is formed around 
  the primary via Roche-lobe overflow from the secondary.  
  Dwarf novae (DNe) are a subclass of CVs and show outbursts 
  representing a sudden release of gravitational energy.  
  The mass accumulated at the disk in the quiescent state 
  accretes onto the primary in the outburst state and 
  a huge amount of gravitational energy is released during 
  a short duration.  
  This sudden accretion is triggered by the thermal-viscous instability 
  in the disk (see <cit.> and <cit.> for 
  comprehensive reviews).   
  
  SU UMa-type stars, one of the subclasses of DNe, have short orbital 
  periods (∼1 hr < P_ orb < ∼ 2 hr) and show 
  occasional superoutbursts which are defined as large-amplitude 
  (∼6–8 mag) outbursts with superhumps.  
  The superhumps are small-amplitude periodic light variations 
  with periods slightly longer than the orbital period and 
  are believed to be induced by the 3:1 resonance tidal instability 
  <cit.>.
  <cit.> investigated the time evolution of superhumps and 
  proved that the superhumps have three stages: 
  stage A superhumps with a long and constant period and increasing 
  amplitudes, stage B superhumps with a systematically varying period 
  and decreasing amplitudes, and stage C superhumps with a short and 
  constant period.  
  
  WZ Sge-type stars, a subclass of SU UMa-type stars, have 
  extremely small binary mass ratios and predominantly show 
  superoutbursts.  
  Here the mass ratio is defined as the ratio of the secondary mass 
  with respect to the primary mass (q ≡ M_2/M_1).  
  Double-peaked modulations called early superhumps and rebrightening 
  are their two representative observational properties 
  (see <cit.> and references therein).  
  Early superhumps have a period almost equal to the orbital one 
  and are observed at the early stage of superoutbursts 
  <cit.>.
  They are considered to be triggered by the 2:1 resonance tidal 
  instability <cit.>.  
  Rebrightening is usually observed soon after the main superoutburst 
  with a long plateau stage and is classified into five types 
  according to the morphology of light curves: 
  type-A (long duration rebrightening), type-B (multiple rebrightenings), 
  type-C (single rebrightening), type-D (no rebrightening), and 
  type-E (double plateaus) <cit.>.  
  
  
  Rebrightenings in WZ Sge stars are not yet explained well 
  by the current model of the disk instability.  
  
  Some people argued that the mass transfer from 
  the secondary star is temporally enhanced via 
  the irradiation by the hot accretion disk 
  <cit.>.  
  However, it would be difficult for the irradiation to raise 
  the mass transfer rate <cit.>, and no observational 
  evidence of the increase of the mass-transfer rate has been 
  detected.  
  On the other hand, there is a possibility that 
  substantial mass remains outside the 3:1 resonance radius, 
  which accretes on the white dwarf delayed against 
  the main superoutburst <cit.>.  
  A large amount of mass seems to be accumulated in the disk 
  before superoutbursts because of long quiescence in WZ Sge stars, 
  and once a superoutburst is triggered, the disk rapidly expands 
  due to the angular-momentum transfer in the disk.  
  Then the substantial mass is possibly conveyed beyond the 3:1 
  resonance radius <cit.>.  
  The observational features like the presence of superhumps, 
  the strong NaD absorption, the unusually red color, and 
  the infrared activities, which were confirmed after 
  the main superoutburst in some WZ Sge stars may support 
  this idea 
  <cit.>.  
  Moreover, TCP J21040470+4631129, a recently observed 
  WZ Sge-type star, triggered superoutbursts twice 
  as a part of multiple rebrightenings after the main 
  superoutburst <cit.>.  
  This implies that there was a large amount of mass reservoir 
  in the outermost part of the disk and that the enhanced 
  mass transfer making the disk shrink unlikely occurred.  
  
  WZ Sge stars are interesting targets also in the study of 
  the CV evolution because some of them are believed to be 
  the best candidates for period bouncers.  
  Period bouncers are defined as the post-period-minimum CVs 
  containing degenerate secondaries.  
  CVs evolve as the orbital period becomes shorter for a long time 
  due to angular momentum losses by magnetic breaking and 
  gravitational-wave radiation.  
  Once the secondary star becomes degenerate, the system evolves as 
  the orbital period becomes longer, since the thermal timescale of 
  the secondary becomes longer than the mass-transfer timescale 
  (see <cit.> and references therein).  
  Although the population of period bouncers is predicted to be 
  ∼70% of CVs by theoretical works, the number of 
  the good candidates identified by observations is only around 
  a few dozen (e.g., <cit.>).  
  The gap between observational and theoretical populations of 
  period bouncers is one of the big problems in the CV evolution.  
  
  <cit.> proposed a new dynamical method for 
  estimating the binary mass ratio by using the period of 
  stage A superhumps.  The new method was applied to many 
  WZ Sge-type stars, resulting in the identification of 
  some period-bouncer candidates.  
  Recent observations found that some of WZ Sge-type stars with 
  type-B or type-E rebrightening are good candidates for the period 
  bouncer and that the candidates share the following common 
  observational features: (1) repeating rebrightenings or dips 
  in brightness at the main superoutburst stage, (2) long-lasting 
  stage A superhumps, (3) large decreases in the superhump period 
  at the stage A to B transition in the objects with repeating 
  rebrightenings, (4) small superhump amplitudes (≲ 0.1 mag), 
  (5) long delays of the appearance of ordinary superhumps, 
  (6) slow fading rates at the plateau stage with ordinary superhumps 
  in the superoutburst, and (7) large outburst amplitudes 
  at the time of appearance of ordinary superhumps (e.g., 
  <cit.> and <cit.>).  
  
  In this paper, we report on our multi-wavelength photometric 
  observations of the 2018 superoutburst of 
  the famous WZ Sge-type star EG Cnc which was originally 
  discovered by <cit.>.  
  This system underwent a superoutburst in 1996–1997 
  <cit.> and is supposed to have entered a normal 
  outburst in 2009 <cit.>.  
  We started a photometric campaign of the 2018 superoutburst 
  of this system soon after the detection of the luminosity 
  increase by Patrick Schmeer.  
  The mass ratio of EG Cnc was estimated to be 0.027–0.035 
  by using the superhump period in its 1996–1997 superoutburst, 
  which is the smallest among the identified WZ Sge stars 
  <cit.>, 
  and hence, EG Cnc is considered to be the best candidate 
  for the period bouncer.  
  However, the estimate was derived from the empirical relation 
  between the excess of superhumps and the mass ratio 
  <cit.>.  
  Also, EG Cnc showed 6 consecutive rebrightenings 
  just after the main superoutburst in the past.  
  Each rebrightening is similar to a normal outburst with 
  an interval of about a week, and hence, this target is 
  suitable for the detailed observations of rebrightenings.  
  Our purpose in this paper is to more accurately estimate 
  the mass ratio of EG Cnc to examine if it is a possible 
  period-bouncer candidate or not and to constrain 
  the origin of multiple rebrightenings.  
  This paper is structured as follows.  We describe the methods of 
  our observations and data analyses in Sec. 2 and present 
  the results in Sec. 3.  
  In Section 4, we discuss the results, and a summary and 
  conclusions are given in Sec. 5.  
  
  
  
  
  § OBSERVATIONS AND ANALYSES
  
  
  Time-resolved optical and near-infrared CCD photometry was 
  carried out by the Variable Star Network (VSNET) collaboration 
  team and the Optical and Infrared Synergistic Telescopes 
  for Education and Research (Oister) at 21 sites.  
  Table E1 shows the log of photometric observations.  
  All of the observation times are converted to barycentric 
  Julian date (BJD).  
  Each observer performed differential photometry.  
  We have applied zero-point corrections to each observer 
  by adding constants before analyses.  
  The constancy of the comparison star that each observer used 
  was checked by nearby stars within the same images.  
  The observational log is summarized in Table E1 in 
  the supplementary information.  
  
  The X-ray and ultraviolet (UV) observations were 
  performed by Swift/X-ray Telescope (XRT) and 
  Swift/UltraViolet and Optical Telescope (UVOT), 
  respectively.  The ObsIDs are 00035446003–00035446042.  
  The X-ray data are processed through xrtpipeline.  
  The UV flux is obtained via the standard tool uvot2pha 
  provided by the swift team.  
  All of the observation times are converted to barycentric 
  Julian date (BJD) as did in the optical and NIR data.  
  
  We adopt the phase dispersion minimization (PDM) method <cit.>
  for period analyses.  Before applying PDM, we subtract 
  the long-term trend of the light curve by locally weighted 
  polynomial regression (LOWESS: <cit.>).  
  The 1σ error of the best estimated period is determined 
  by the same method as that described by <cit.> and 
  <cit.>.  
  A variety of bootstraps was used for evaluating the robustness of 
  the PDM result.  We make 100 samples which randomly contained 
  50% of observations, and perform PDM analyses for these 
  samples.  
  The result of the bootstraps is expressed as a form of 90% 
  confidence intervals in the resultant θ statistics.  
  Also, we determine the times of maxima of 
  ordinary superhumps in the same way as in <cit.>, 
  when drawing the O-C curve.  
  
  
  
  
  § RESULTS
  
  
  
  
   §.§ Overall light curves
  
  
  
  
  We show the overall light curve of the 2018 superoutburst of 
  EG Cnc in Figure <ref> with the V-band light curve 
  of its 1996–1997 superoutburst for comparison, which is derived 
  from Fig. 1 of <cit.>.  
  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 ∼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.  
  
  
  
  
   §.§ Time evolution of superhumps
  
  
  
  
  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 <ref> 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 <cit.>; 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 ≤ E ≤ 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 ≤ E ≤ 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., <cit.> and 
  <cit.>).  
  
  
  
  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 <ref>.  These modulations are clearly single-peaked.  
  The stage A superhump period (P_ shA) estimated by PDM is 
  0.06103(2) d (see the upper left panel of 
  Figure <ref>), and the averaged period at stage B 
  (P_ shB) estimated by PDM is 0.060324(3) d, respectively 
  (see the upper middle panel of Figure <ref>).  
  The derivative of the superhump period during stage B 
  (P_ dot (≡Ṗ_ sh/P_ sh)) is 
  -1.5 (4.0) × 10^-6  s  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 ∼0.0604-d period, which is shorter than the stage A 
  superhump period and longer than the orbital period 
  0.05997(9) d <cit.> 
  (see the upper right panel of Figure <ref>).  
  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 
  <cit.>.  
  Figure <ref> shows the O-C curve of times of 
  superhump maxima extending to the cycle ∼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, 
  ∼0.5 phase shift in superhumps seem to have 
  occurred when this system entered the post-stage of 
  the main superoutburst (see also <cit.>) 
  and this phenomenon was also confirmed in the 1996–1997 
  superoutburst of this object <cit.>.  
  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 <cit.>).  
  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 ≤ E ≤ 526) 
  as stage La, during BJD 2458435.5–2458453.7 (590 ≤ E ≤ 891) 
  as stage Lb, and during BJD 2458457.5–2458471.7 (956 ≤ E 
  ≤ 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_ 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 <ref>.  
  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_ dot) is 4.8 (5.6) × 10^-5  s  s^-1.  
  
  
  
  
   §.§ Multi-wavelength behavior and color variations
  
  
  
  
  We have explored the daily variation of optical and NIR colors 
  during the rebrightening stage of the 2018 superoutburst in EG Cnc.  
  
  In Figure <ref>, we show daily averages of 
  the colors and the V-band light curve.  
  The B-V and J-K_ 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 <cit.>.  
  This phenomenon is consistent with the unusually red V-I color 
  in the rebrightening phase of the previous superoutburst of 
  this object <cit.>.  
  Also, the J-K_ 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 
  <cit.>.  
  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_ 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 ∼7000 K.  
  Also, the J-K_ S color of 0.2–0.6 corresponds to 
  the black-body radiation of ∼4000–7000 K.  
  
  According to the picture described above, the red B-I 
  and J-K_ 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_ S color in the quiescent state 
  between rebrightenings should become bluer.  
  The bluer J-K_ 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 ∼18000 K, which is higher than the minimum temperature 
  of the outburst state (see e.g., <cit.>).  
  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 <ref>.  
  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 <ref>.  
  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., <cit.>).  
  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 <ref>.  
  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 
  <cit.>.  
  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 
  <cit.>.  
  
  We also show a diagram of the daily U-B and B-I colors 
  during the rebrightening phase in Figure <ref>.  
  
  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 <cit.> 
  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.  
  
  
  
  
  § DISCUSSION
  
  
  
  
   §.§ Is EG Cnc one of the best candidates for the period bouncer ?
  
  
  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 <cit.>, complying with the method 
  proposed by <cit.>.  
  The superhump period is basically determined by the dynamical 
  precession of the eccentric disk and the pressure effect 
  decreasing the precession rate <cit.>, 
  and hence, the apsidal precession rate of the eccentric disk, 
  ω_ pr, is approximately formulated as
  
      ω_ pr = ω_ dyn + ω_ press,
  
  where ω_ dyn and ω_ 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 <cit.>, the dynamical precession 
  rate, is expressed as: 
  
      ω_ dyn / ω_ orb = Q(q)R(r),
  
  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 <cit.> for 
  details), and r is expressed as (3^2(1+q))^-1/3a, which is 
  the 3:1 resonance radius in the case of stage A superhumps.  
  Here a is the binary separation.  
  By using equation (<ref>), the mass ratio of 
  EG Cnc is estimated to be 0.045(5) which is consistent with 
  the P_ dot value derived in Sec. 3.2 and 
  the type-B rebrightening (see also Fig. 17 in 
  <cit.>).  
  
  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 <cit.>.  
  If we know the mass ratio, the disk radius can be estimated 
  from equation (<ref>).  
  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 <cit.>, the disk radius after 
  the rebrightening phase in WZ Sge stars ranges between 
  the Lubow-Shu radius <cit.> 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 0.30–0.32a <cit.>, which 
  is consistent with the estimated mass ratio in the previous 
  paragraph within 1σ errors.  
  
  
  
  Our results thus allow a larger mass ratio for EG Cnc 
  in comparison with the estimates in <cit.> and 
  <cit.>.  
  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 
  <cit.>.  
  The evolutionary stage of EG Cnc is exhibited in 
  the q - P_ orb plane as Figure <ref> 
  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 <ref>), 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 <cit.>).  
  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 <cit.>.  
  
  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., 
  <cit.>).  
  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 <ref>.  
  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 <cit.>.  
  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.  
  
  
  
  
   §.§ Luminosity dip in the plateau stage
  
  
  <cit.> 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., <cit.>), as confirmed 
  in the 2015 superoutburst of AL Com, another WZ Sge star, 
  which was triggered after only 1.5 yr quiescence <cit.>.  
  
  
  
  
   §.§ Implications to the mechanism of rebrightenings
  
  
  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 <ref>).  
  This is the second example following 
  the 2015 superoutburst in AL Com <cit.>.  
  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 <cit.>).  
  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 <cit.>, 
  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 <cit.>.  
  <cit.> 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 
  <cit.>.  
  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 
  <cit.> 
  and the mass reservoir outside the 3:1 resonance radius 
  <cit.>.  
  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_ 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 
  <cit.>.  
  Also, the redder color of B-I at the beginning of 
  each rebrightening may indicate the disk expansion 
  (see also Figure <ref>).  
  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 <cit.>.  
  The bluer i-z and J-K_ 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 <cit.>.  
  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 <ref>).  
  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.  
  <cit.> 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 <cit.> 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 <ref>).  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 <ref>).  
  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., <cit.> 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.  
  
  
  
  
   §.§ Color variations of superhumps
  
  
  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 
  <cit.>, although the different types of 
  color variations were found in V455 And <cit.>.  
  <cit.> 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 
  <cit.> 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 <cit.> 
  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 
  <cit.>.  
  
  The color variation of superhumps seems to be weaker 
  after the main superoutburst (see also Figure <ref>).  
  Superhumps are regarded to be produced by 
  periodic tidal stressing working on the eccentric disk 
  <cit.>.  
  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 <cit.>, 
  and the period change during stage B is believed to 
  represent that the eccentricity propagates over 
  the entire disk <cit.>.  
  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.  
  
  
  
  
  § SUMMARY
  
  
  We have analyzed our multi-wavelength photometric data 
  during the 2018 superoutburst of the WZ Sge-type DN EG Cnc 
  to investigate whether EG Cnc is a possible candidate 
  for the period bouncer and the origin of its multiple 
  rebrightenings.  
  The main results are as follows.  
  
  
    * The rebrightenings in the 2018 superoutburst of EG Cnc 
  almost completely reproduced those in its previous superoutburst 
  in 1996–1997.  EG Cnc is the second object following AL Com, 
  in which this kind of phenomenon was confirmed.  This phenomenon 
  implies that some parameters inherent to each object are 
  related to the rebrightening type in WZ Sge-type stars.  
  
    * The binary mass ratio of EG Cnc is estimated to be 0.045(5) 
  and the possible range is estimated to be 0.019–0.057  
  by using the period of the stage A superhumps and the late-stage 
  superhumps that we have detected.  
  Also, the amplitude of superhumps in the main superoutburst and 
  the fading rate at the plateau stage where stage B superhumps 
  were observed was small, which is common in period-bouncer 
  candidates among WZ Sge stars.  
  Although the interval between superoutbursts and the duration of 
  stage A superhumps are short in this object in comparison with 
  other candidates, the increase of the viscosity in the disk, 
  which might be caused by the long-lasting elliptical disk 
  suggested by the sustained late-stage superhumps, could affect 
  at least the short recurrence time of superoutbursts.  
  Taking into account the estimated mass ratio and some observational 
  features common in period-bouncer candidates, we consider that 
  EG Cnc is a possible candidate for the period bouncer.  
  
    * The B-I and J-K_ S colors were unusually red 
  at the beginning of the rebrightening phase, and gradually became 
  bluer in quiescence between rebrightenings with time.  
  This would mean that cool material existed at the outermost disk 
  before multiple rebrightenings and was depleted during 
  the rebrightening phase.  
  The slightly redder B-I color at the rising part of each rebrightening 
  suggests the disk expansion.  
  These results support the idea of the mass reservoir as the mechanism 
  of rebrightenings rather than the enhancement of the mass-transfer rate.  
  
    * The UV light curve well traced the optical light variations 
  of multiple rebrightenings and the U-B color became redder during 
  the rebrightening phase, which means that the inner disk became cooler.  
  The thermal-viscous instability may have newly occurred at each rebrightening.  
  
  
  
  
  
  § ACKNOWLEDGEMENTS
  
  
  This work was financially supported by Grants-in-Aid for JSPS Fellows
  for young researchers (M. Kimura).  
  M. Kimura acknowledges support by the Special Postdoctoral Researchers
  Program at RIKEN.  
  We are thankful to many amateur observers for providing a lot of 
  the data used in this research.  
  This work is also supported by the Optical and Near-infrared Astronomy 
  Inter-University Cooperation Program.  
  M. Kimura is grateful to K. Takagi, I. Otsubo, Y. Yamazaki, and H. Kimura 
  at Higashi-Hiroshima Observatory and Y. Uzawa at Saitama University 
  for engaging in the observation of this star.  
  The work of N. Katysheva and S. Shugarov was supported by the Program 
  of Development of Lomonosov Moscow State University `Leading 
  Scientific Schools', project `Physics of Stars, Relativistic 
  Objects and Galaxies'. 
  The work by S. Shugarov was supported by the Slovak Research and 
  Development Agency under the contract No. APVV-15-0458 and by 
  the Slovak Academy of Sciences grant VEGA No. 2/0008/17.
  The work by P. A. Dubovsky, I. Kudzej, and S. Shugarov was supported 
  by the Slovak Research and Development Agency under the contract 
  No. APVV-15-0458.  
  We thank the anonymous referee for helpful comments.  
  
  
  
  
  § SUPPORTING INFORMATION
  
  
  Additional supporting information can be found in the online version 
  of this article:
  Supplementary tables 1 and 2.
  Supplementary data is available at PASJ Journal online.   
  
  
  
  
  
  
  
  
  
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