Analysis of Three SU UMa-Type Dwarf Novae in the Kepler Field

Publications of the Astronomical Society of Japan, Oct 2013

We studied Kepler light curves of three SU UMa-type dwarf novae: a background dwarf nova of KIC 4378554, V585 Lyr, and V516 Lyr. Both the background dwarf nova and V516 Lyr showed a combination of a precursor and a main superoutburst, during which superhumps always developed in the fading branch of the precursor. This finding supports that the thermal-tidal instability theory explains the origin of superoutburst. A superoutburst of V585 Lyr recorded by Kepler did not show a precursor outburst, and the superhumps developed only after the maximum light: namely, the first-ever example in the Kepler data. Such a superoutburst is understood based on the thermaltidal instability model to be a “case B” superoutburst, discussed by Osaki and Meyer (2003, A&A, 401, 325). From the observation of V585 Lyr, Kepler first clearly revealed the positive period derivative commonly seen in the “stage B” superhumps of dwarf novae with a short orbital period. In all three objects, there was no strong signature of a transition to the dominating stream impact-type component of superhumps. This finding suggests that there is no strong indication of an enhanced mass-transfer following the superoutburst. In V585 Lyr, there were “mini-rebrightenings” with an amplitude of 0.2–0.4 mag and its period of 0.4–0.6d during the period between the superoutburst and the rebrightening. We have determined that the orbital period of V516 Lyr is 0.083999(8)d. In V516 Lyr, some of outbursts were double outbursts with varying degrees. The preceding outburst in the double was of the inside-out nature, while the following one was of the outside-in nature. One of the superoutbursts in V516 Lyr was preceded by a double precursor. The preceding precursor failed to trigger a superoutburst, and the following precursor triggered a superoutburst by developing positive superhumps. We have also developed new methods of reconstructing the light curve of superhumps, and of measuring the times of maxima from poorly sampled Kepler LC data.

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Analysis of Three SU UMa-Type Dwarf Novae in the Kepler Field

PASJ: Publ. Astron. Soc. Japan Analysis of Three SU UMa-Type Dwarf Novae in the Kepler Field Taichi KATO 0 0 Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502 and Yoji OSAKI Department of Astronomy, School of Science, The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 We studied Kepler light curves of three SU UMa-type dwarf novae: a background dwarf nova of KIC 4378554, V585 Lyr, and V516 Lyr. Both the background dwarf nova and V516 Lyr showed a combination of a precursor and a main superoutburst, during which superhumps always developed in the fading branch of the precursor. This finding supports that the thermal-tidal instability theory explains the origin of superoutburst. A superoutburst of V585 Lyr recorded by Kepler did not show a precursor outburst, and the superhumps developed only after the maximum light: namely, the first-ever example in the Kepler data. Such a superoutburst is understood based on the thermaltidal instability model to be a ?case B? superoutburst, discussed by Osaki and Meyer (2003, A&A, 401, 325). From the observation of V585 Lyr, Kepler first clearly revealed the positive period derivative commonly seen in the ?stage B? superhumps of dwarf novae with a short orbital period. In all three objects, there was no strong signature of a transition to the dominating stream impact-type component of superhumps. This finding suggests that there is no strong indication of an enhanced mass-transfer following the superoutburst. In V585 Lyr, there were ?mini-rebrightenings? with an amplitude of 0.2-0.4 mag and its period of 0.4-0.6 d during the period between the superoutburst and the rebrightening. We have determined that the orbital period of V516 Lyr is 0.083999(8) d. In V516 Lyr, some of outbursts were double outbursts with varying degrees. The preceding outburst in the double was of the inside-out nature, while the following one was of the outside-in nature. One of the superoutbursts in V516 Lyr was preceded by a double precursor. The preceding precursor failed to trigger a superoutburst, and the following precursor triggered a superoutburst by developing positive superhumps. We have also developed new methods of reconstructing the light curve of superhumps, and of measuring the times of maxima from poorly sampled Kepler LC data. - Cataclysmic variables (CVs) are close binary systems consisting of a white dwarf and a mass-transferring red dwarf star. Among CVs, there are dwarf novae (DNe) that show semiperiodic outbursts with an amplitude of 2?8 mag [for a review of CVs and DNe, see, e.g., Warner (1995) ]. Some of DNe, called the SU UMa subtype, show longer and brighter outbursts, called superoutbursts, during which the period of semiperiodic modulations (superhumps) is one-to-several percent longer than the orbital period (Porb). The superhump period (PSH) is believed to represent the synodic period of Porb and the precession period of the non-axisymmetric (also referred to as eccentric or flexing) accretion disk. This non-axisymmetric deformation of the disk is believed to be caused by the 3 : 1 resonance tidal instability (Whitehurst 1988; Hirose & Osaki 1990; Lubow 1991) . It is generally believed that the SU UMa-type phenomenon is due to a combination of the thermal instability (Osaki 1974; Meyer & Meyer-Hofmeister 1981) and the tidal instability; this is called the thermal-tidal instability (TTI) model (Osaki 1989; for a review, see Osaki 1996) . There have, however, been long-lasting debates as to whether enhanced (or modulated) mass-transfer plays a role in producing either superoutbursts or superhumps [generally called the enhanced mass-transfer (EMT) model; e.g., Smak 1991 , 2004, 2008], or whether the enhanced mass-transfer caused by an outburst may modify the behavior of the outburst (cf. Lasota 2001; Patterson et al. 2002) , while it has been argued that an enhanced mass-transfer hardly occurs in an irradiated secondary star in DNe (cf. Osaki & Meyer 2004; Viallet & Hameury 2007, 2008) . By analyzing the Kepler data of V1504 Cyg, Osaki and Kato (2013a) demonstrated that the TTI model fully explains the general behavior of SU UMa-type dwarf novae, particularly the disk-radius variation during the supercycle (the cycle of the successive superoutbursts) by employing the frequency of negative superhumps as a probe. The data presented by Osaki and Kato (2013a) also indicated that there is no pronounced enhanced mass-transfer before or at around the start of the superoutburst, which indicates that the EMT model is disqualified from being a dominant mechanism to produce superoutbursts. Furthermore, Osaki and Kato (2013a) demonstrated that the appearance of superhumps (manifestation of the tidal instability) is closely related to the development of the superoutburst, namely, the development from the precursor outburst to the main superoutburst; they concluded that the superhump and superoutburst are so much entwined with one another that it is very difficult to find some interpretation other than the TTI model. Osaki and Kato (2013b) further studied the Kepler observations of V1504 Cyg and V344 Lyr by using a newly developed, two-dimensional, least absolute shrinkage and selection operator (Lasso) power spectra to verify their conclusion. The situation concerning enhanced mass-transfer caused by the outburst is unclear. There have been the Kepler observations of V344 Lyr (Wood et al. 2011; Kato et al. 2012) , which showed prominent secondary superhump maxima in the late stage of the superoutburst, which smoothly developed into ?late superhumps,? having an 0.5 phase shift from main superhumps (Vogt 1983). The strength of these secondary superhump maxima might suggest that the enhanced mass-transfer is their origin. Since the strength of the secondary superhump was reported to be much weaker in V1504 Cyg (Kato et al. 2012) , it is necessary to study whether or not the secondary superhump or the late superhump with an 0.5 phase offset is generally present in many SU UMa-type dwarf novae. The ground-based study seems to suggest that these late superhumps with an 0.5 phase offset are relatively rare, except for objects with a high mass-transfer rate (MP ) (Kato et al. 2012) . This needs to be tested by high-quality observations. The presence of late superhumps with an 0.5 phase shift in low-MP systems is also a point for debate, since a close examination of a compilation from large ground-based observations shows that most of the previously reported late superhumps with an 0.5 phase shift can be explained by a combination of a discontinuous decrease of the superhump period and a gap (traditional observations in one observing location usually had gaps in the daytime) in the observation (Kato et al. 2009) . This interpretation also requires confirmation from uninterrupted, high-quality space observations. NASA?s Kepler satellite (Borucki et al. 2010; Koch et al. 2010) , which was designed to detect extrasolar planets, provides the best opportunity to answer such questions. The Kepler field contains several CVs, and so far observations of four SU UMa-type dwarf novae and an SS Cyg-type dwarf nova have been reported using the Kepler data. They are as follows: V344 Lyr (Still et al. 2010; Cannizzo et al. 2010, 2012; Wood et al. 2011; Kato et al. 2012; Osaki & Kato 2013b) ; V1504 Cyg (Kato et al. 2012; Cannizzo et al. 2012; Coyne et al. 2012; Osaki & Kato 2013a, 2013b) ; V516 Lyr (Garnavich et al. 2011) ; a background dwarf nova of KIC 4378554 (Barclay et al. 2012) ; an SS Cyg-type dwarf nova with a shallow eclipse?V447 Lyr (Ramsay et al. 2012) . In addition to these dwarf novae, other classes of CVs have also been studied or discovered: MV Lyr (Scaringi et al. 2012b, 2012a) (novalike variable), KIC 8751494 (Kato & Maehara 2013) (novalike variable), KIC 4547333 (Fontaine et al. 2011) (AM CVn-type object). Kepler observations were performed either in a high time-resolution ( 1 min) mode on short cadence (SC) runs or a low time-resolution ( 29.4 min) mode on long cadence (LC) runs. Since the number of targets observable in the SC mode is severely limited, most of Kepler targets were recorded in the LC mode. Some of these CVs were also recorded in the SC mode. Although these objects are expected to play an important role in studying superhumps in detail, there are several obstacles that were not met in V1504 Cyg and V344 Lyr: faintness of the objects (V516 Lyr; the background dwarf nova of KIC 4378554), the presence of a close contaminating star (the background dwarf nova of KIC 4378554), and the lack of suitable SC runs (many objects). The lack of suitable SC runs makes the analysis particularly difficult, because the timeresolution of LC runs is typically a third or a fourth of the superhump period of a typical SU UMa-type dwarf nova. There is a need to develop methods of conquering these difficulties. Here, we present the results of the three SU UMa-type dwarf novae obtained from new techniques. 2. A Background Dwarf Nova of KIC 4378554 2.1. Introduction and Data Analysis This SU UMa-type dwarf nova was discovered by Barclay et al. (2012) to be a background object of the Kepler target star, KIC 4378554. We analyzed the public LC Kepler data to understand the characteristics of this object. Since the light of this object is mostly outside the aperture of the original target star, KIC 4378554 (SAP FLUX), we applied a custom aperture centered on this object, the same as in Barclay et al. (2012) . Since we are mainly interested in its short-term variation (i.e., superhumps), rather than the outburst amplitude, we did not adopt a sophisticated method described in Barclay et al. (2012) . We added the observed count rates in the custom aperture, and subtracted the ?sky? values estimated from the surrounding pixels, the same as in ordinary aperture photometry (see appendix 1 for details). Since the background is highly contaminated by the light of KIC 4378554 and the systematic background level is slowly variable (Barclay et al. 2012) , we subtracted linear fits obtained from the data outside the outburst. In this procedure we determine the zero count for the averaged quiescent brightness of this object. This is considered to be a good first-order approximation because this object was undetectable in a Kepler full-frame image in quiescence (Barclay et al. 2012) . The numbers of pixels for the custom aperture were three for Q3 and Q4 and two for Q5 and Q7. For quarters in which this object was not detected in an outburst, we used the apertures used in the nearest quarters (three for Q1 and Q2 and two for Q8). Since the superhump period is relatively long ( 4 LC exposures), we simply applied the fitting method described in Kato et al. (2009) to determine the times of the superhump maxima and the amplitudes in the following. We extracted the times of the superhump maxima by numerically fitting a template superhump light curve at around the times of the observed maxima. We used a template-averaged (spline interpolated) superhump light curve in GW Lib, which shows a typical asymmetric profile of superhumps in SU UMa-type dwarf novae. We used two superhump cycles (typically containing 6?8 data points) at around the maxima. This method worked well, and it has become evident that we do not have to pay special attention to the poor time-resolution of LC measurements in characterizing superhumps in long-PSH systems. The result for the superoutburst and its associated normal outbursts in Q3?Q4 is shown in figure 1. In order to better reproduce the brightness variation during the outbursts, we added a constant to the flux (which was originally adjusted to be zero outside the outbursts), so that the outburst amplitude would match a typical amplitude ( 5 mag) of superoutbursts in SU UMa-type dwarf novae with similar PSH. Although this treatment is artificial, and the global trend of variation in the faintest part is not reliable, this correction is expected to produce a better representation of the light curve when the system was in its low brightness, especially in the postsuperoutburst stage. We could not detect an orbital period in the quiescent data. In calculating the superhump period (figure 1, third panel), we applied a linear regression to 20 adjacent times of the maxima measured by template fitting. We should note that the variation of the period was smoothed by this time scale ( 1.5 d). The 1 error is a formal error of linear regression. 2.2. System Characteristics With respect to this object, Barclay et al. (2012) quoted VW Hyi which is a prototype SU UMa-type dwarf nova with frequent outbursts. However, we suggest that this object belongs to a group of objects with a much low MP , because a typical interval of normal outbursts (OB3 and OB4, OB4 and OB5 in Barclay et al. 2012) is on an order of a 200 d quiescent interval, and more than 200 d was recorded before OB1. These are much longer than a typical recurrence time of 28 d in VW Hyi (e.g., van der Woerd & van Paradijs 1987). The long recurrence times recorded in this object are highly characteristic of SU UMa-type dwarf novae with a lower MP . If we assume that the cycle length of normal outbursts (Tn) is proportional to MP 2 (cf. Ichikawa & Osaki 1994; Osaki 1996) , MP in this object is expected to be 2.7 times smaller than that in VW Hyi, and given this MP the superoutburst recurrence time (i.e., the supercycle length) is expected to be 500?700 d. The sequence of outbursts shown in figure 1 is very characteristic of such low-MP dwarf novae, i.e., precursor outburst? main superoutburst?post-superoutburst rebrightening. Similar examples are found in many objects: the 2003 superoutburst of V699 Oph (Kato et al. 2009) and the 2007 and 2009 superoutbursts of QZ Vir (Ohshima et al. 2010) . Even when precursor outbursts were absent (this may have been partly due to lack of observations before the superoutbursts), a post-superoutburst rebrightening is frequently observed in many SU UMa-type dwarf novae (see examples in Kato et al. 2009) , and the precursor outburst is also a commonly found feature in many SU UMa-type dwarf novae (see, e.g., Kato 1997; Uemura et al. 2005; Imada et al. 2009; Shears et al. 2011; Wood et al. 2011) . We therefore identified the sequence of outbursts (OB1?OB3 in Barclay et al. 2012) as a complex of precursor outburst (OB1)? main superoutburst (OB2)?post-superoutburst rebrightening (OB3). The short interval between the end of OB2 and OB3, as compared with intervals between other normal outbursts, is then naturally understood. Post-superoutburst rebrightenings are, however, more frequently observed in shorter-Porb systems (cf. Kato et al. 1998). There was a deep dip between the precursor and the main superoutburst. Such a deep dip is difficult to explain in the pure thermal instability model proposed by Cannizzo et al. (2010) , because the precursor in their model just looks like a shoulder in the superoutburst light curve (cf. subsection 3.2 in Osaki & Kato 2013a) . SU UMa-type dwarf novae with PSH similar to this object tend to show more frequent outbursts. There are only a few moderately studied systems having PSH and outburst frequencies that are comparable to those of this object. We give an example of EG Aqr [PSH = 0.078828(6) d: Imada et al. 2008 ]. EG Aqr showed a superoutburst in 2006 with a small precursor and the intervals of its superoutbursts were 800?1000 d (Kato et al. 2013) . The supercycle length of EG Aqr is similar to what is expected ( 500?700 d) from this object with a cycle length of normal outbursts. QY Per [PSH = 0.07861(2) d: Kato et al. 2009] may be another similar object, although the frequency of outbursts is lower in QY Per. The representative objects having PSH similar to this object but showing less frequent outbursts, RZ Leo (Ishioka et al. 2001) and V1251 Cyg (Kato 1995; Kato et al. 2010) , do not resemble this object in showing intervals of several days (with double-wave modulations) before superhumps appear, and in showing no precursor outbursts. 2.3. Development of Superhumps The second and bottom panels of figure 1 show the development of superhumps in this system. As can be more clearly seen in the enlarged O C diagram of the second panel, it is clearly that there are three distinct stages, as discussed in Kato et al. (2009) : a stage with a relatively constant, long superhump period (E . 30; stage A), a stage with a shorter superhump period having a zero to slightly positive derivative (Pdot = PP =P ) (30 . E . 70; stage B), and a stage with a shorter superhump period (E & 70; stage C). As shown in figure 4 of Kato et al. (2009) , objects with a longer PSH tend to have shorter duration of stage B. This tendency well matches the behavior in this object. The transition from stage A to stage B, however, took place later than the time when the amplitude of superhumps reached a maximum. Such a finding was not recorded in the groundbased observation, and it may be one of the advantages in analyzing the Kepler data. Using the epochs of the superhump maxima during 0 E 17 before the amplitude of the superhumps suddenly grew, we obtained a mean superhump period of 0.07781(10) d. We regarded this period as a representative superhump period when the superhumps were growing (here we call it stage A1). The latter part of stage A (we call it stage A2) has a mean period of 0.07727(5) d, slightly shorter than the period during stage A1. Using the segment of 36 E 70, when the O C diagram showed a positive Pdot, we determined that the mean period of stage-B superhumps is 0.07690(2) d and Pdot is +13(4) 10 5. Using the segment of 70 E 105, we determined that the period of stage-C superhumps is 0.07650(2) d. The period of stage-C superhumps is 0.5% shorter than the mean period of stage-B superhumps, in agreement with the general behavior described in Kato et al. (2009) . During stage B, the amplitude of superhumps monotonically decreased, while it grew again slightly after the transition to stage C. This behavior is also similar to that in other systems 175 180 185 ?0.5 0.0 0.5 1.0 (Kato et al. 2012) . Individual mean profiles of superhumps for a 1 d segment are shown in figure 2. Although there was an unavoidable gap (a gap between Kepler quarters) after E = 105, the later (E 144) development in the O C diagram after Kepler restarted observations appears to show a smooth extension of stageC superhumps. This suggests that the period and phase of superhumps have remained nearly unchanged during the gap in the observation. Using all E 70 times of the maxima, we obtained a period of 0.07643(1) d. The object immediately entered the rapidly fading stage of the superoutburst when Kepler resumed the observation. The O C diagram of superhamp maxima did not show a strong variation, despite a rapid decline in the brightness. This feature was also commonly seen in many SU UMa-type dwarf novae (e.g., Kato et al. 2009) . The amplitudes of superhumps were also decreasing as the system was fading (see also figure 3), and this feature makes the difference between persisting superhumps in V344 Lyr (Wood et al. 2011; Osaki & Kato 2013b) clear, which are supposed to arise from the streamdisk interaction of accretion, corresponding to the traditional picture of late superhumps (Vogt 1983) . We should note that the enhancement of the secondary hump in figure 9 of Barclay et al. (2012) corresponds to the late post-superoutburst stage (corresponding to BJD 2455189.5 of the fifth profile in figure 3). There are some hints about the enhancement of the superhump amplitudes (measured in electrons s 1) during the post-superoutburst rebrightening. As already discussed in Kato et al. (2009) and a subsequent series of papers, the late stage (post-superoutburst) superhump in low-MP SU UMa-type dwarf novae bears more characteristics of a continuation of stage-C superhumps than the traditional, stream-disk interaction-type late superhumps. This finding in highly sensitive Kepler data provides another support for the result that the mass-transfer rate is not greatly enhanced during the superoutburst (see also a discussion in Osaki & Meyer 2003) . before the Post-Superoutburst 2.4. Oscillations Rebrightening In the light curve, there appears to be some signs of oscillations between the superoutburst and the post-superoutburst rebrightening. There are small bumplike structures on around BJD 2455188.3 and BJD 2455189.3 (the latter becomes more evident after the subtraction of a rising trend from the rebrightening). A trace of these features can also be seen in figure 5 of Barclay et al. (2012) , and we consider that these features are real. Since a similar feature can be better seen in V585 Lyr, we describe the characteristics in section 3 concerning V585 Lyr. 3. V585 Lyrae 3.1. Introduction and Kepler Data V585 Lyr is a dwarf nova discovered by Kryachko (2001) . Kryachko (2001) reported on a long outburst and two short ones, and suggested that the object is an SU UMa-type dwarf nova. It is interesting that the long outburst showed a temporary fading near the start of the outburst. It was most likely accompanied by a precursor. Kato et al. (2009) studied the 2003 superoutburst, and Kato et al. (2013) also reported on the less observed 2012 superoutburst. Howell et al. (2013) reported on an unsuccessful attempt to obtain a spectrum. The Kepler public data for V585 Lyr available in this work are those of Q2?Q10 for LC and Q9 and Q14 for SC. Since V585 Lyr is sufficiently isolated from nearby stars, we used SAP FLUX for the SC and LC data. In these data there was only one outburst event recorded by Kepler; it is a superoutburst and its post-superoutburst rebrightening, starting at the end of 2010 January and ending at the end of February. Only the LC data were available during this superoutburst, and there was a longer gap in the data than 4 d immediately after the peak of the superoutburst. Because of these restrictions, the early development of the superhumps was not sufficiently recorded. We can, however, say that there was no precursor outburst, and that low-amplitude modulations already appeared 6 hr after its peak brightness. These modulations likely had long periods (several hours) at around the appearance, but there was an indication of emerging signals (BJD 2455229.4) having a period close to the superhump period, which we discuss later. Unfortunately, the Kepler observation was stopped soon after these signals appeared. We therefore mainly focus on the analysis after the resumption of the observation at BJD 2455233.8. The overall light curve of this superoutburst is shown in figure 4. The superoutburst started on around BJD 2455227, and ended on around BJD 2455248; thus, its duration was 21 d. It was followed by the post-superoutburst stage with semiperiodic variations (?mini-rebrightenings? during BJD 2455247.5?2455252; figure 5) and a distinct post-outburst rebrightening peaking at around BJD 2455252.7. The object then faded slowly with semiperiodic fluctuations with a period of 2 d and an amplitude of 0.1?0.2 mag. These fluctuations almost ceased after the following 20 d (BJD 2455276). The SC runs (Q9 and Q14) did not record any outbursts. Although the object was closely observed in quiescence in the two SC runs, we were unable to detect any significant periodic signal on each SC run. And we were not able to detect the signature of orbital variation in the LC data, either. 3.2. Variations of Superhump Profile from LC Data In order to obtain an O C diagram of superhump maxima or a period variation (subsection 3.3), we need a mean profile of the superhump beforehand. We therefore first obtain the mean superhump profile and then discuss the O C variation. As discussed below, we can see a periodic light variation of superhump origin during the superoutburst of V585 Lyr with a period of 0.060 d (or 87 min or 16.5 c=d), which is very close to three times the LC sampling interval of 29.42 min. It is particularly difficult to obtain times of the maxima or a superhump profile by conventional methods due to such a low 14 16 18 20 sampling rate. We therefore reconstructed a template light curve for each 0.5 d bin using the Markov-chain Monte Carlo (MCMC) modeling of the Kepler data (see appendix 2). The result is shown in figure 6. The profile looked like being double-peaked at the end of stage A (BJD 2455234.0). This feature may have been spurious, since the profile was not double-peaked in the incomplete (shorter than 0.5 d) preceding bin (BJD 2455233.5). The period increased during stage B (BJD 2455234.5?2455242.5), and secondary humps only appeared near the end of B stage (BJD 2455241.5?2455242.5). The amplitudes of the superhumps grew again. Stage C with ?0.5 0 0.0 0.5 1.0 1.5 a shorter superhump period was recorded during the later stage of the superoutburst, and during the subsequent fading stage. BJD 2455246.0 corresponds to the start of the rapid fading. Despite this fading, no alternation between the main and secondary humps was observed as in V344 Lyr (Wood et al. 2011) . After BJD 2455247.5, the object entered the post-superoutburst stage (before the rebrightening). During this stage, mini-rebrightenings with a period of 0.5?0.7 d and an amplitude of 0.3 mag were recorded. This phenomenon is discussed later. The superhumps became undetectable at around the peak of the rebrightening, and the waveform became difficult to trace after this rebrightening (not shown in the figure). This result well reproduces ground-based observations of short-Porb systems, such as SW UMa (Soejima et al. 2009; Kato et al. 2009) and V585 Lyr itself (Kato et al. 2009) , although the ground-based observation only recorded a part of the superoutburst of V585 Lyr with limited accuracy. 3.3. O C Analysis of LC Data Using the method described in appendix 3, we determined the times of the superhump maxima (figure 7). We here define the superhump maximum for BJD 2455233.8536 as E = 0. The result very well demonstrates a familiar stage A? B?C variation in the O C diagrams in short-Porb systems [cf. Kato et al. (2009) , particularly figure 3 for SW UMa (Porb = 0.056815 d)]; there were stage A with a long superhump period and evolving superhumps (E . 10), stage B with a increasing superhump period (10 . E . 140), and stage C with an almost constant, shorter superhump period (E & 140). Individual mean periods for these three stages were 0.06128(4) d (stage A), 0.06041(1) d (stage B), and 0.06013(1) d (stage C, limited to 140 E 220). The Pdot for stage B was +9.6(5) 10 5, which is especially characteristic of this PSH (Kato et al. 2009, 2010, 2012, 2013) . These values very well agree with the ground-based value for the 2003 superoutburst; 0.06113(8) d (stage A), 0.06036(2) d (stage B), and Pdot = +10.7(12) 10 5 for stage B. The period of the stage-C superhump was not well determined in 2003 due to the short observational coverage (Kato et al. 2009) . The value of 0.06035(4) d for the stage-B superhumps was obtained from a more sparse data set in the 2012 superoutburst (Kato et al. 2013) . Although there was a possible rapid increase in the superhump period in around the rising phase of the rebrightening (E 290), it was difficult to confirm the reality of this phenomenon on account of the signal close to the detection limit. An independent confirmation of this feature of period variation was provided by the two-dimensional Lasso spectrum (figure 8). This figure corresponds to figure 8 in Barclay et al. (2012) , who used Fourier analysis. The frequency of the strongest (superhump) signal initially decreased (up to BJD 2455241; corresponding to stage B) and then increased (stage B?C transition) in both the fundamental and the first harmonics. This trend (i.e., initial increase and then decrease in period) confirmed the result from the O C analysis. Note that the frequency of the first harmonics is higher than the Nyquist frequency. The Lasso analysis can handle such a case by using appropriate windows, and by suppressing the 1 2 3 4 5 230 235 240 245 250 14 16 18 20 0.02 0.00 ?0.02 ?0.04 0.0615 0.0610 0.0605 0.0600 4000 3000 2000 1000 0 A B C ?50 0 50 100 150 200 250 300 350 14 15 16 17 18 19 33.6 33.4 reflected signal at the Nyquist frequency, which was seen in Barclay et al. (2012) . The O C analysis of the Kepler LC data thus perfectly confirmed the trend in short-Porb systems recorded in groundbased observations. This object is the first Kepler CV showing the very distinct stages A?B?C, particularly the clear presence of stage B with an unambiguously positive Pdot. The Kepler observation, however, recorded the details of stage C with unprecedented high accuracy. This observation unambiguously demonstrated a lack of 0.5 phase jump (corresponding to the ?traditional? late superhumps). The variation of the superhump amplitude was similar to those in short-Porb systems recorded in ground-based observations; the superhump amplitude reaches a peak near the stage A?B transition, decreases during stage B, reaches a minimum before the stage B?C transition, and again increases near the stage B?C transition (Kato et al. 2012) . In conjunction with the increase in amplitude near the stage B? C transition, the brightness of the system also shows an upward (brighter) deviation from the linear fading. This brightening trend is commonly seen in short-Porb systems (Kato et al. 2003). 3.4. Mini-Rebrightenings and the distinct post(BJD 2455247.5?2455252; figure 5), there were ?mini-rebrightenings? with an amplitude of 0.2?0.4 mag and a period of 0.4?0.6 d. So far as we know, such variations have never been documented before. The amplitude appears to be too large to be explained by a beat between two different periods (e.g., between a superhump period and another period), and the period of 0.4?0.6 d appears to be too short to be explained by a beat phenomenon. We thus regard the variation in brightness on this time-scale as real. There was a hint as to the same phenomenon on the declining branch in the superoutburst (BJD 2455247.4). When the object exhibited the distinct rebrightening that peaked on BJD 2455252.7, this phenomenon disappeared and has never again appeared after the fading of the rebrightening. Although this phenomenon appears to be physically related to the appearance of the post-superoutburst rebrightenings, and to be potentially important for understanding the origin of the rebrightening, we still do not give a physical explanation of this phenomenon. Not all SU UMa-type dwarf novae show the post-superoutburst rebrightening (Kato et al. 2009) , and a systematic search for such variations might bring about a solution. As we have already seen, the background dwarf nova of KIC 4378554 also possibly showed the same phenomenon (subsection 2.4). This phenomenon may be more prevalent and will deserve a further study. 4. V516 Lyrae 4.1. Introduction V516 Lyr was originally identified as a blue object (NGC 6791 B8) in the region of the old open cluster NGC 6791 (Kaluzny & Udalski 1992) . Kaluzny and Rucinski (1995) suspected that the object is likely a cataclysmic variable based on the color and the variability. The dwarf novatype nature was confirmed with photometry and spectroscopy by Kaluzny et al. (1997) . Ground-based observations indicated that this object showed 2 mag outbursts outside the V = 21 quiescence and it was suggested that this object is an SS Cyg-type dwarf nova (Mochejska et al. 2003) . Garnavich, Still, and Barclay (2011) reported from the Kepler SC data that this object underwent a superoutburst on 2011 October 13 and determined that the superhump period is 2.097(3) hr (in average) or 2.109(3) hr (initial two days) (Garnavich et al. 2011) . Howell et al. (2013) presented a spectrum that is in agreement with that of the dwarf nova-type classification, and also presented Kepler light curves showing normal outbursts with a recurrence time of 18 d. Howell et al. (2013) 1 also reported on a possible detection of a period of 0.087478 d based on the LC data in the quiescence on around BJD 2455468. 4.2. Global Light Curves We first examine the Kepler LC light curves of V516 Lyr available for public at the present work. They are those of Q6? Q10 and Q14. Figure 9 illustrates the Kepler light curve of the LC data of V516 Lyr during the periods of Q6?10 and Q14. Since there is no long-term stable zero-point in the Kepler light 97-10 Q6?2 Q6?3 55380 55390 Q6?3 Q6?4 55400 55410 Q6?5 Starting date and ending one (BJD 2455000). Outburst duration (d). Abbreviations: precursor (PC), rebrightening (RB), and superoutburst (SO). Quiescence duration prior to the outburst (d). k Comment: 1, data gap; 2, too noisy. curve, we artificially added constants to adjust the quiescent level to be 21 mag, as recorded in Mochejska, Stanek, and Kaluzny (2003). As can be seen in figure 9, V516 Lyr exhibited 40 outbursts including two superoutbursts on a time scale of about one and a half years. We summarize the main characteristics of these outbursts in table 1. The first column of the table is the sequential number of outbursts within a given quarter. The second and the third are the dates of the start and the end of an outburst (estimated by eye), respectively, where dates are counted from BJD 2455000. The fourth gives the outburst duration and the fifth the outburst type. The sixth gives the duration of quiescence preceding to a given outburst and the last is just comments, if there are any. As discussed by Smak (1984) , two types of normal outbursts are recognized: Type-A outburst or the ?outside-in? outburst in which the heating transition of the thermal instability starts from the outer-part of the disk and the heating front propagates toward the inside, and Type-B outburst or the ?inside-out? outburst in which the heating transition starts in the inner-part of the disk and the heating wave propagates toward the outside. The light curve of Type-A outburst is characterized by a rapid rise to its maximum as compared to its slow decay from the maximum while Type-B outburst is characterized by its more or less symmetrical rise and fall at around the maximum. A judgement of outburst types is made by eye, and we must admit that there are not a few cases in which the judgement is ambiguous. Two superoutbursts are seen in our data, which occurred on BJD 2455600 (SO No. 1) and BJD 2455785 (SO No. 2). Thus, the length of the supercycle between these two superoutbursts is 185 d. However, since the starting date of observations in Q6 was BJD 2455372, and since no superoutburst occurred until BJD 2455600 (SO No. 1), the length of the preceding supercycle must be longer than at least 220 d. The next superoutburst is expected to have likely occurred during the period of Q12, but so far we have not found a way to confirm or disprove it. A possible orbital period of V516 Lyr was found to be 0.0840 d (2.016 hr or 11.9 c=d), which is very near to that of V344 Lyr (figure 11 in subsection 4.3). However, the recurrence cycles of both the normal outburst and the superoutburst suggest that V516 Lyr has a slightly lower mass-transfer rate than V344 Lyr. One of the most interesting characteristics in the light curve of V516 Cyg was the occurrence of several double outbursts. They are as follows: Q6-1, Q6-2, Q6-3, Q9-6, Q14-4, and Q14-8. The two outbursts, Q14-6 and Q14-7, are likely a double outburst with an interval of one day in quiescence. The degree of doubling varies from outburst to outburst, such as an outburst with a shoulder (e.g., Q6-3) and an outburst with such a deep dip as reaches nearly to the quiescence level (e.g., Q14-4). As shown in figure 9, the preceding outburst in the double was always Type-B (?inside-out?) outburst while the following one was Type-A (?outside-in?) outburst, and there has been no exception to this rule so far. The double outburst tended to occur as a group (e.g., Q6, Q14). This phenomenon is understood in the following way. As mentioned above, the preceding outburst is the inside-out outburst in which the heating transition first occurs in the inner part and the heating front propagates outward. But the heating front fails to reach the outer edge of the disk, and is reflected at the middle of the disk (i.e., Type-Bb outburst in Smak?s classification). This brings about a special situation in which a thermal instability is easily triggered in the outer part of the disk because a large amount of matter outwardly pushed by the heating front is left behind in the outer-part by reflection (see Cannizzo 1993) . The second thermal instability in the outer part of the disk is then triggered (Type-A outburst) before the cooling wave of the preceding thermal instability completely reached the inner edge of the disk. The behavior of the double outburst can be well explained within the framework of the thermal disk instability. 4.3. SC Data V516 Lyr was observed in the Kepler SC in Q8?Q10 and Q14. Since the object is faint in quiescence ( 30 electrons s 1) and the baseline shows a systematic trend, we adjusted the quiescent level to Kp = 21 (same as in the LC data) by subtracting low-order (up to second) polynomials from the observed count rates in each continuous quarter (67?97 d in duration). The resultant quiescent magnitudes are not real, but this treatment is sufficient, since one of our interests is the residual pulsed flux. The mean superhump waveform during one of its superoutbursts (Q10-5) is shown in figure 10. We also analyzed the SC data after excluding two superoutbursts and the post-superoutburst phase before the next normal outburst (but not excluding other normal outbursts), i.e., a combination of three data (before BJD 2455592, BJD 2455612?2455777, and after BJD 2455805). The only detected signal was 0.083999(8) d, which is likely the orbital period (figure 11). We were not able to confirm a period of 0.087478 d reported on by Howell et al. (2013) . There was no hint of persisting negative superhumps or positive superhumps in quiescence in this interval. The fractional superhump excess is 4.0%, typical of objects having this orbital period. The same as in V1504 Cyg and V344 Lyr (Osaki & Kato 2013b) , we calculated Fourier (figure 12) and Lasso (figure 13) two-dimensional power spectra (Kato & Uemura 2012; Kato & Maehara 2013) . Due to faintness of the object, these power spectra are not as clear as those of V1504 Cyg and V344 Lyr, particularly in the quiescence. During Q8?Q10, V516 Lyr underwent a relatively regular feature of normal outburst and two superoutbursts separated in time by 185 d. The characteristic signals of positive superhumps during the superoutbursts showed an increase of the frequency (shortening of the superhump period) as reported on by Garnavich, Still, and Barclay (2011). The orbital signal was also sometimes detected in Lasso power spectra, but it was not clearly seen in Fourier. Although there were no clear, long-standing negative superhumps such as were observed in V344 Lyr and V1504 Cyg, impulsive (failed) negative superhumps occurred on several occasions; they are negative superhumps that are transiently seen during some normal outbursts (terminology introduced in Osaki & Kato 2013b; see also the description of the phenomenon in Wood et al. 2011) . These impulsive negative superhumps occurred on around BJD 2455735 (outburst Q9-8) and BJD 2455750 (Q10-1); these dates are four and three cycles ahead of the superoutburst, respectively. 4.4. Superoutburst with Double Precursor The most notable feature in the light curve of V516 Lyr is 55778.0 40 20 0 a ?double precursor? in SO No. 2 (i.e., Q10-4). Here, we discuss two possible origins of this double precursor. One of possible explanations is the same as that of the double outburst presented in subsection 4.2. As already discussed there, the preceding outburst is an inside-out one in which the heating wave is turned over at the middle of the disk. As discussed in subsection 3.2 of Osaki and Kato (2013a), this type of outburst is not accompanied with the disk expansion, and so it does not trigger the 3 : 1 resonance tidal instability. On the other hand, since the following outburst is an outside-in one, it is accompanied with the disk expansion, and thus it can trigger the 3 : 1 eccentric tidal instability and start superhumps and a superoutburst. Another possibility is that related to the failed precursor due to an appearance of impulsive negative superhumps. During the preceding precursor (BJD 2455778), negative superhumps seem to have emerged [their period between BJD 2455778?2445780 was 0.0828(3) d by the PDM method; figure 14]. However, this signal diminished during the following precursor, but instead positive superhumps appeared the same as in ordinary precursor outbursts of V344 Lyr and V1504 Cyg. In this interpretation, the preceding-precursor outburst triggered the development of impulsive negative superhumps which were unable to sustain the disk in a hot state, while the following precursor successfully triggered the development of the eccentric instability to produce positive superhumps, and succeeded in starting a superoutburst. This kind of ?failed superoutburst? due to the impulsive negative superhump was also seen in V1504 Cyg (sub-subsection 4.5.2 and figure 77 in Kato et al. 2012) . It is interesting to note that this failed superoutburst in V1504 Cyg occurred in the second last normal outburst prior to the following superoutburst (cf. subsection 2.7 in Osaki & Kato 2013b) . The two explanations presented above seem to conflict with each other, because the inside-out type precursor with Smak?s type Bb in the first explanation is not expected to excite impulsive negative superhumps. This is because the disk does not expand with outbursts of this type. Although this may contradict observed impulsive negative superhumps, we would like to mark both explanations as possibilities at the moment, because the amplitudes of seeming impulsive negative superhumps were close to the detection limit, and it is difficult to make any definite conclusion. We, however, note there have been at least two objects that showed an inside-out type rise in the superoutburst: ER UMa (Ohshima et al. 2012) and BZ UMa (figure 150 in Kato et al. 2009) . In figure 2 of Ohshima et al. (2012) for ER UMa, the transition from negative to positive superhumps took place 2 d after the start of the superoutburst with a kink in the light curve. This case does not cause any problems because an expansion of the disk occurs if the inside-out outburst is Smak?s type Ba, in which the outwardly propagating heating front reaches the outer edge of the disk. In BZ UMa, there appears to be a weak second precursor at around BJD 2454203, 0.0 0.5 Fig. 16. Variation of superhump profiles of V516 Lyr. The phase-averaged profiles were calculated for 2 d segments. The phases were calculated by using the same ephemeris as in figure 15. after which superhumps developed, although the double nature of precursor is not as evident as that in V516 Lyr. It appeared to be understood that this superoutburst is an inside-out type precursor followed by an outside-in type second precursor, which were not well separated from the preceding precursor. 4.5. O C Analysis of Superhumps We calculated the O C diagram of the superoutburst during BJD 2455780?2455797 (the superoutburst with double precursor). The O C diagram of the times of superhump maxima was roughly composed of stages A?B?C (figure 15), as described in Kato et al. (2009) . The transitions from stage A to stage B and from stage B to stage C were rather smooth, as compared with short-Porb systems described in Kato et al. (2009) , and more resembles those of the long-Porb, high-MP system V344 Lyr (Kato et al. 2012) . Although it is not very clear to find the most adequate segments in E for the determination of the period in each stage, we adopted periods of 0.08732(3) d for stage B (38 E 70) and 0.08725(3) d for stage C (70 E 120). During stage B, an increase in period, Pdot = +27(7) 10 5, was recorded. There was a break in the O C diagram at around the time when the amplitude of the superhump reached a maximum. The same as in the background dwarf nova of KIC 4378554, we refer to these stages as A1 (E 15) and A2 (16 E 38). The mean periods of superhumps during stage A1 and stage A2 were 0.0894(4) d and 0.08814(3) d, and 6.4(5)% and 4.93(4)% longer than the orbital period, respectively. The profile of the superhumps (figure 16) did not show a strong secondary peak, as seen in V344 Lyr (Wood et al. 2011; Kato et al. 2012) . There was no clear transition to traditional late superhumps having an 0.5 phase jump. During the late stage of the superoutburst, the amplitudes of the superhumps (including the secondary maxima, if they exist) decreased to an undetectable limit (amplitude . 1 electrons s 1). These amplitudes were not larger than that of the orbital variation in quiescence (subsection 4.3). 4.6. LC Data We also made a two-dimensional Lasso power spectrum of the LC data (figure 17). Since the number of data points was much smaller than that in the SC data, we were able to detect relatively persisting signals only. The faintness of V516 Lyr also made it more difficult than in V344 Lyr or V1504 Cyg (Osaki & Kato 2013b) to analyze the data. We adopted a window of 30 d, which was shown to give the best signal-to-noise ratio to the orbital modulation, and hence is expected to detect signals with a similar strength. This length of the window was too long to resolve any variation of positive superhumps during the superoutburst, and these positive superhumps were only found to be broad bands. In figure 17, however, there seem to be transient weak signals of possible negative superhumps at around the frequencies 12.32?12.35 c=d during BJD 2455500?2455560 (this may also extend to a possible frequency 12.30 c=d during BJD 2455420?2455450), BJD 2455730?2455760, and BJD 2456120?2456150. Although it is difficult to confirm the reality of these signals, we suspect that they were likely present because the initial segment corresponds to a period of decreased frequency of outbursts (type ?L? in Osaki & Kato 2013a) , and the second segment corresponds to a period of impulsive negative superhumps discussed in subsection 4.3. The fractional deficiency of superhump frequencies over orbital frequencies, (see sub-subsection 5.2.1), for these possible negative superhumps was 3.5%?3.7%. Although the result is not very conclusive, the tendency that negative superhumps suppress the occurrence of outbursts (Ohshima et al. 2012; Osaki & Kato 2013a; Zemko et al. 2013) seems to be valid for V516 Lyr. Indeed, if the signals during BJD 2455420?2455450 were also negative superhumps, there was an increase in the frequency of the negative superhumps as the phase of the supercycle progressed, which has been confirmed in V1504 Cyg and V344 Lyr (Osaki & Kato 2013a, 2013b) . We expect that future Kepler SC runs (if available) in a state with negative superhumps will provide us a better characterization of phenomena. The Kepler results up to now appear to suggest that negative superhumps in quiescent SU UMa-type dwarf novae are more prevalent than those which have been considered. An additional feature in the two-dimensional power spectrum analysis is that the orbital signal was more strongly detected in Q14 than in other quarters. Such a variation of strength of the orbital signal was also seen in V1504 Cyg and V344 Lyr (Osaki & Kato 2013a, 2013b) . 4.7. Interpretation of Historical Data Although Mochejska, Stanek, and Kaluzny (2003) suggested 12.4 12.2 to classify V516 Lyr as an SS Cyg-type dwarf nova, it has now become evident that this object is an SU UMa-type dwarf nova. We reexamined the material in Mochejska, Stanek, and Kaluzny (2003). The phased light curve (figure 6) in Mochejska, Stanek, and Kaluzny (2003) indicates a mean period of 17.7298 d, which appears to be consistent with the Kepler observations. Their long-term light curve (figure 6, object B8) showed at least six major outbursts, most of which reached a maximum of V = 19. One of them (in their window No. 12) lasted at least 5 d, and this outburst must have been a superoutburst. On one occasion (in their window No. 3), the object was discovered to be so faint as to have a magnitude of V = 22?23, more than 1 mag fainter than its ordinary one in quiescence. Since this object is not eclipsing, this phenomenon may have been caused by a temporary reduction of the masstransfer rate. It is difficult to check whether or not a similar phenomenon was recorded in the Kepler data due to faintness of the object and their highly variable zero-point. 5. Discussion 5.1. Appearance of Superhumps and its Implication in the Thermal-Tidal Instability Theory In the background dwarf nova of KIC 4378554 and V516 Lyr, all superoutbursts took the form of a combination of precursor and superoutburst, the same as reported in V1504 Cyg and V344 Lyr (Still et al. 2010; Wood et al. 2011; Cannizzo et al. 2010, 2012; Osaki & Kato 2013a, 2013b) . In both systems, superhumps always started to appear in the fading branch of precursor and reached the maximum of amplitude at around the peak of the main superoutburst. This sequence of phenomena?a precursor, development of superhumps, and main superoutburst?very well fits with the picture of the TTI model (Osaki 1989; Osaki & Kato 2013a, 2013b) , and justifies the universal application of the TTI theory to two more SU UMa-type dwarf novae. In the case of V585 Lyr, the superhump took 7 d to reach the maximum of amplitude. When compared with the background dwarf nova of KIC 4378554 and V516 Lyr, it can be naturally understood that this long waiting time is an effect of a lower q (mass ratio of the binary) in V585 Lyr. Since the growth rate of the eccentric mode is proportional to q2 (Lubow 1991) , a typical q = 0.10 for the short period V585 Lyr requires 3 times the growth time longer than a case of q = 0.18 in V516 Lyr. Since it took 2?3 d to reach the maximum of amplitude in the cases of V516 Lyr and the background dwarf nova of KIC 4378554, this estimated time closely agrees with 7 d of V585 Lyr, and strengthens the TTI theory. 5.2.1. Global variation We studied the period variation of superhumps in all three SU UMa stars using the O C data. We did not use periodfinding methods, such as Fourier transform and PDM, because the O C data are more sensitive to the period variation and our method gives a more stable result when the sampling rate is very low. Our results are shown in the third panels of figures 1 [which is an improvement of Barclay et al. (2012) ], 7, and 15. We, however, note that there appeared some wiggles in V585 Lyr (figure 7), which should not be considered to be real, since the LC sampling rate in V585 Lyr was almost exactly one third of the superhump period, which resulted in artificial signals. All three objects show a global decrease in superhump period (shown by a broken line in the figures): from 0.0778 d to 0.0765 d in the background dwarf nova of KIC 4378554, from 0.0613 d to 0.0600 d in V585 Lyr, and from 0.0895 d to 0.0870 d in V516 Lyr. The global decrease in superhump period during the superoutburst of these three stars is thought to be produced by monotonic decrease in disk radius (Osaki & Kato 2013b) . One of them, V516 Lyr, has a photometric orbital period (Porb), and we can determine the precession rate of the eccentric mode (the superhump mode) in the accretion disk. As discussed in Osaki and Kato (2013b), it is very convenient to introduce the fractional deficiency of superhump frequencies over orbital frequencies, ( orb SH)= orb = 1 Porb=PSH, where orb and SH are the orbital and superhump frequencies, respectively. The ordinary excess of superhump periods, (PSH Porb)=Porb, is related to the former quantity by = =(1 + ). The quantity of positive superhumps, +, is then related to the apsidal precession rate of the eccentric disk, prec, by + = prec= orb. The quantity + in the growing stage of positive superhumps is thought to reflect the precession rate at the 3 : 1 resonance, because the superhump wave is restricted to the resonance region (Osaki & Kato 2013b) . The precession rate in this condition can be expressed as (Hirose & Osaki 1990, 1993) + = prec = p orb q 1 1 d 1 + q 2 pr d r r 2 dB0 d r ; where r is the radius from the white dwarf primary (in units of the binary?s separation). B0 is written as (1) (2) effects, where A is the binary?s separation. 5.2.2. Pressure effects As discussed in Osaki and Kato (2013b), we consider that the temporary deviation of period variation from its general trend is caused by pressure effects. The pressure effects bring about a decrease in superhump eigenfrequency (Hirose & Osaki 1993) and thus a decrease in the (positive) superhump period (Osaki & Kato 2013b) , besides the effect of disk-radius variation. We interpret that the temporary deviation from the general trend (temporary decrease in period) commonly seen in the three systems was caused by the pressure effect. Although strengths of this effect in the systems seem to be different from object to object, the effect is the strongest in the early phase of the superoutburst, particularly in the stage A?B transition. This is because the disk temperature is the highest near the start of the superoutburst. It may be understood that a much stronger downward deviation seen in V585 Lyr during the period of BJD 2455235? 2455239 is a stronger pressure effect. As discussed in Osaki and Kato (2013b), the precession rate of the eccentric mode is given by two different terms: the dynamical prograde precession by the tidal perturbation of the secondary star and the retrograde precession by pressure effects. V585 Lyr is thought to have the lowest q among the mass ratios of three SU UMa stars, judging from their superhump periods. The pressure effects may affect the precession frequency of the eccentric mode (and the superhump period) more strongly in a low-q system such as V585 Lyr. This is because the prograde precession rate due to the tidal effect is lower in a lower-q system while the pressure effects work for retrograde precession similarly regardless of q. Thus the relative importance of pressure effects may increase if we consider a system of lower q. 5.2.3. Period after superoutburst After the superoutburst, the period tends to remain at a short value, probably reflecting a smaller disk radius. It looks like that the period increased at around the rebrightening in the background dwarf nova of KIC 4378554 and V585 Lyr which may reflect the increase in the disk radius due to the thermal instability (cf. Osaki & Kato 2013b) . The periods of the superhump at that time, however, may not be very reliable because the amplitudes of superhumps were very low. The period variation at the time of ?mini-rebrightenings? of V585 Lyr was not very informative due to the noisy O C diagram, and the time resolution was insufficient to resolve ?mini-rebrightenings.? 5.3. Superoutburst without a Precursor The outburst of V585 Lyr recorded with Kepler did not show an outburst with a precursor. This is currently the only known superoutburst without a precursor in the Kepler data. In the framework of the thermal-tidal instability model (Osaki 2005) , it can be understood that this superoutburst is as follows. If the mass sufficiently accumulates on the accretion disk during a long quiescence, the next normal outburst could make the disk expand up to the tidal truncation radius, by passing the 3 : 1 resonance. The expansion of the disk is stopped at this radius, and the disk is then in fully hot state, and the viscous depletion of matter ensues even when the eccentric tidal instability has not yet developed. This makes the disk develop the B0.r / = 2 b1.0=/2 = 2 Z 0 =2 q1 d r 2sin2 ; which is the Laplace coefficient of the order 0 in celestial mechanics (Smart 1953) . Using + = 0.060(4) for stageA1 superhumps, we obtained a mass ratio of q = 0.18(2) for V516 Lyr where we use the radius of the 3 : 1 resonance, r3W1 = 3. 2=3/.1 + q/ 1=3. The global decrease in superhump period and hence in the quantity indicates that the disk radius decreased during the superoutbursts in these three SU UMa stars. The at around the end of the plateau phase of V516 Lyr was 3.4%. This value corresponds to a radius of 0.36 A if we ignore the pressure growth of an eccentric structure (and thus superhumps) a few days after the outburst maximum. This situation corresponds to the ?case B? superoutburst discussed in Osaki and Meyer (2003) and Osaki (2005) . As shown by Kryachko (2001) , V585 Lyr also showed a superoutburst with a precursor [?case A? in Osaki & Meyer (2003) ]. This indicates that the properties of V585 Lyr are not so extreme as WZ Sge-type dwarf novae, which predominantly show superoutbursts without a precursor and very rarely2 show a precursor-superoutburst type outburst. [For a recent review of WZ Sge-type dwarf novae, see Kato et al. 2001 ; Kato et al. 2009]. The Kepler data clearly demonstrate that the same object can show two types of superoutbursts, with a precursor and without a precursor. 5.4. Late-Stage Superhumps It has been well known that the prototypical Kepler SU UMa-type dwarf novae, V344 Lyr and V1504 Cyg, showed the development of secondary humps that evolved into the strongest signal after the fading from the superoutburst (Wood et al. 2011) . This picture agrees with the traditional interpretation of the late superhumps (Vogt 1983), namely, the varying-energy release on an eccentric disk around the stream impact point. It is, however, reported that there is a difference in the degree of the development of the secondary hump between V344 Lyr and V1504 Cyg, and the latter has much less prominent secondary humps (Kato et al. 2012) . In the present study, three objects showed neither a strong sign of the secondary hump nor an 0.5 phase jump expected for traditional late superhumps. This result seems to confirm the ground-based results (Kato et al. 2009, 2010, 2012, 2013) that most of SU UMa-type dwarf novae, particularly low-MP ones, do not show an 0.5 phase jump characteristic of traditional late superhumps. From the low amplitudes of post-superoutburst superhumps in the three objects, we may conclude that there was no evidence of enhanced mass transfer during the superoutburst. It appears that V344 Lyr is rather exceptional in strength of the secondary hump. This may be related to its high MP , as inferred from its short supercycle. The present result of V585 Lyr clearly confirms that ?stage-C superhumps? survive after the fading from the superoutburst without showing an 0.5 phase jump. This has demonstrated our suggestion it can be understood that at least some ?late superhumps? (supposing an 0.5 phase jump) are due to a mistaken cycle count for stage-C superhumps with a shorter period (Kato et al. 2009) . 2 Richter (1992) reported on a possible dip in the early stage of the outburst of DV Dra in 1991. The object underwent a superoutburst in 2005 (Kato et al. 2009) and its property was that of a WZ Sge-type dwarf nova (Porb = 0.0588 d). V585 Lyr and DV Dra may be similar objects. Recently, a dwarf nova with a very large outburst amplitude ( 8 mag), OT J075418.7+381225 = CSS130131:075419+381225, showed a precursor outburst (vsnet-alert 15438). This object has a long superhump period of 0.07194 d (vsnet-alert 15438) and may be different from an ordinary WZ Sge-type dwarf nova (C. Nakata in preparation). Conclusion We studied the Kepler SC and LC light curves of three SU UMa-type dwarf novae: a background dwarf nova of KIC 4378554, V585 Lyr, and V516 Lyr. The background dwarf nova and V516 Lyr showed a similar combination of precursor and main superoutburst, during which their superhumps always started to appear in the fading branch of the precursor (the ?type A? superoutburst in Osaki & Meyer 2003 and Osaki 2005) . This behavior is common to V344 Lyr and V1504 Cyg and strongly supports the TTI theory in regard to the origin of the superoutburst. V585 Lyr is on the borderline between SU UMa-type dwarf novae and WZ Sge-type dwarf novae. The Kepler LC data recorded one superoutburst without a precursor. It can be understood based on the TTI model that the development of this outburst is a case where the disk with mass sufficiently accumulated first so expanded as to reach the tidal truncation radius by triggering a normal outburst, and then both the tidal instability and the superhump developed after the maximum (the ?case B? in Osaki & Meyer 2003 and Osaki 2005) . The subsequent development of the O C diagram of the times of superhump maxima perfectly confirmed the existence of three distinct stages (A?B?C) for short-Porb SU UMa-type dwarf novae inferred from ground-based observations. This observation made the first clear Kepler detection of the positive period derivative commonly seen in the stage-B superhump in such dwarf novae. There were recurring ?mini-rebrightenings? between the superoutburst and the rebrightening. V516 Lyr is an SU UMa star similar to V344 Lyr but frequencies of normal outburst and superoutburst are both lower than those of V344 Lyr, which suggests a lower masstransfer rate of V516 Lyr than that of V344 Lyr. One of the most interesting aspects in the Kepler light curves of V516 Lyr is an appearance of double outbursts. In the double outburst, the preceding outburst is of an inside-out nature while the following outburst is of an outside-in nature. The double outburst tends to occur as a group. We have determined that the orbital period of V516 Lyr is 0.083999(8) d. One of superoutbursts of V516 Lyr was preceded by a double precursor. The preceding precursor failed to trigger a superoutburst and the following precursor triggered a superoutburst by developing positive superhumps. There were possible negative superhumps in the LC data of V516 Lyr. An occasion of an appearance of possible negative superhumps coincided the time when a reduction of normal outbursts occurred, which has strengthened the conclusion derived from V1504 Cyg and V344 Lyr (Osaki & Kato 2013a, 2013b) that negative superhumps tend to suppress normal outbursts. None of these three SU UMa-type dwarf novae showed a strong sign of transition to a dominating stream impact-type component of superhumps. This finding suggests that a case of V344 Lyr is rather exceptional; that is, the transition is probably associated with a high MP . There was no strong indication of enhanced mass-transfer following the superoutburst. We have also developed a method for analyzing superhumps of short-PSH SU UMa-type dwarf novae in the Kepler LC data by modeling the observation and applying the MCMC method. Since most of the new dwarf novae included in the Kepler field by chance (cf. Barclay et al. 2012) are expected to be observed only on LC runs and a large part of dwarf novae is expected to be short-PSH systems (Ga?nsicke et al. 2009) , our method will be effective in characterizing these dwarf novae. This work was supported by a Grant-in-Aid for Scientific Research on Innovation Area ?Initiative for High-Dimensional Data-Driven Science through Deepening of Sparse Modeling? from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. We thank the Kepler Mission team and the data calibration engineers for making Kepler data available to the public. Appendix 1. with R Method to Handle Kepler Pixel Image Data Barclay et al. (2012) introduced PyKE routines to extract custom apertures. We introduce a different convenient way: by using FITSio package of R,3 and loading the target pixel FITS file in a variable d, one can easily obtain a time-series of data for pixel and draw pixel light curves as shown in figure 2 in Barclay et al. (2012) , and make custom-aperture photometry without a special library. To load FITS pixel data, we can use the following command (the file name was split due to its long length; it should be written in a single line upon execution): library(FITSio) d <- readFITS("kplr004378554-2009350155506 _lpd-targ.fits"). The resultant d$col contains the table values in the FITS file. The counts for individual pixel number n are stored in d$col[[5]][,n], where 5 means the fifth member of the FITS data (pixel counts). For example, we can draw a light curve for pixel n=3 by plot(d$col[[5]][,3]). If one wishes to add three pixels n=3, 4, and 8, one can get the summed values by f <- d$col[[5]][,3] + d$col[[5]][,4] + d$col[[5]][,8], and plot(f), to draw a light curve for the custom aperture. Appendix 2. Reconstruction of the Superhump Profile Using Kepler LC Data Since the Kepler LC data were too sparse for such a shortperiod object as V585 Lyr (only three measurements in one superhump), we modeled the observation from the template superhump light curve and obtained the best model referring to the actual observation (subsection 3.2). In reconstructing the superhump profile, we subdivided observations into 24 segments (a segments of 0.5 d). We first subtracted longer trends caused by outbursts from these 3 The R Foundation for Statistical Computing hhttp://cran.r-project.org/i. ?=1 ?=0.1 ?=0.01 Fig. 18. Dependence on ? in reconstructing the superhump profile from the Kepler LC data. The data are for V585 Lyr, 0.5 d segment starting on BJD 2455235 (near the peak amplitude). segments, and adjusted the mean residuals to zero. We assumed a template light curve consisting of 20 phase bins. We can then model the observation data by numerically integrating the spline-interpolated template for the phases corresponding to each Kepler LC exposure (29.4 min), making assumptions about epoch and period. From an assumed template, we can get a set of [ymodel.i /], where i represents the i -th Kepler LC exposure. We then compare [ymodel(i )] with the actual observation [yobs(i )]. In principle, we can minimize the difference between [ymodel(i )] and [yobs(i )] by changing the assumed template. The best-fit model, however, gives a highly complicated light curve, but does not resemble a light curve of superhump. This is probably caused by the ?overexpression? of random or systematic variation other than the superhump variation. We therefore estimated the best superhump profile by using both the residual and the smoothness used in the Bayesian framework. The value (j ) (j = 1,: : :,N ), where N is the number of phase bins, represents the template light curve. The notation follows the Appendix in Kato et al. (2010) . Bayes? theorem gives P r . jD/ / P r .Dj / . /; where is the model parameter (the template light curve in the present case), D the observed data, P r (Dj ) the likelihood, and ( ) the prior probability. L0 = log[P r (Dj ) ( )] can be expressed as follows: L0 = Lres + Ls0m; 0 (A1) (A2) where the subscripts ?res? and ?sm? mean the residual and the smoothness, respectively, and is the parameter. Here, ?yobs.i / ymodel.i / 2 2 2 where is the observation error. Following the standard technique in Bayesian analysis, we then express the condition in smoothness in by introducing a prior function, assuming that the second order difference of [ (j )] follows a normal distribution. Then, Ls0m = X j 1/ 2 .j / + .j + 1/ 2 2 s2 where (0) = (N 1) and (N + 1) = (1) reflecting the cyclic condition; s is the standard deviation of the normal distribution. Since only the difference of L0 is important in the MCMC steps, we can omit the constant terms. Since and s are constants, we can now write the likehood in the form of 1 L = 2 2 .L1 + L2/; L1 = L2 = X?yobs.i / i 0 X? .j j ymodel.i / 2; 1/ 2 .j / + .j + 1/ 2; and 0 = ( = s)2. In the process of calculation, we estimated contributions of L1 and L2 to L, and expressed ? = L2=L1 as the ratio of L2 to L1. We changed ? and selected the most likely template light curve. After making a comparison between the actual Kepler data and the superhump light curve derived from the ground-based observation, we adopted ? = 0.1 (for example, ? = 1 gives too smooth light curve, and ? = 0.01 gives too complicated, as compared with the ground-based observation; figure 18). The result, however, does not very strongly depend on ?. Since all the LC measurements of V585 Lyr have more than 7 105 electrons, the photon noise is not the dominant source of the error. We therefore used = 0.001 mag for analyzing this computation. The resultant profile, however, is not sensitive to this value. The probability density function (PDF) of model parameters from the observed data can be determined by the MCMC method (cf. Kato et al. 2010) , and we adopted the mean value ; (A3) (A7) and the standard deviation for each phase of the PDF to obtain its template light curve. Note that the standard deviation is not the actual error in estimation, but only a measure of residuals which is strongly affected of the condition set of the prior. A length of 20000 MCMC runs was sufficient to obtain a reasonable PDF and the initial burn-in period was 5000. We should note that this kind of reverse estimation may be prone to artificial features, but in the present case solutions were mostly stable. Appendix 3. Determination of Superhump Maxima in Kepler LC Data In subsection 3.3, we determined the times of superhump maxima by using the MCMC method. As described in appendix 2, we modeled the observation by using a template with 20 phase bins. In this case, we used the template superhump light curve obtained in appendix 2. Because the superhump profiles varied with the time, we averaged the template superhump profiles ?1 d of observations. Since the superhump profile was not well determined during stage A, we used the template on BJD 2455236 before this epoch. For the same reason, we used the template on BJD 2455245 after this epoch. The template was normalized to have full amplitudes of 0.2 mag and the zero mean value. The phase of the maximum (determined by spline interpolation) was shifted to phase zero. The definition of the superhump maximum in the present case refers to the light maximum, and is slightly different from that in Kato et al. 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Kato, Taichi, Osaki, Yoji. Analysis of Three SU UMa-Type Dwarf Novae in the Kepler Field, Publications of the Astronomical Society of Japan, 2013, DOI: 10.1093/pasj/65.5.97