1
No.90

Feb. 2022

ISSN 0917-2211

Variable Star Bulletin
Emerging ordinary superhumps as the standard candle for WZ Sge stars
Taichi Kato1
1

Department of Astronomy, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japantkato@kusastro.kyoto-u.ac.jp

Received 2022 Feb. 05

Abstract
In Kato (2015, arXiv:1507.07659), I suggested that the magnitude when ordinary superhumps appear can
be a standard candle for WZ Sge stars. Using Gaia EDR3 parallaxes, I studied 53 WZ Sge stars to examine
this suggestion. The analysis indicated that the absolute magnitudes when ordinary superhumps appear are
strongly dependent on orbital inclinations, which is consistent with what is expected for the projected area
for optical thick accretion disks. I showed that there is a linear relation between these absolute magnitudes
and logarithmic amplitudes of early superhumps, which are also dependent on the inclinations. I conﬁrmedthat the magnitude when ordinary superhumps appear can be used as the standard candle for WZ Sge stars
particularly when the amplitude of early superhumps is observationally known. The resultant median absolutemagnitude when ordinary superhumps appear was +5.4 (for an average inclination of 1 radian). A few objects
with multiple rebrightenings, which are good candidates for period bouncers, showed slightly fainter absolutemagnitudes, although the majority of the same class of objects follows the relation. Using the relation, I derived
an empirical relation between the inclination and the amplitude of early superhumps. I applied the relation
to MASTER OT J030227.28+191754.5, which was initially considered as a possible optical counterpart of thehigh-energy neutrino event IceCube-211125A, and concluded that this WZ Sge star was one of the faintest
before the outburst and one of the brightest around the optical peak in terms of absolute magnitudes.

1

Introduction

WZ Sge stars are a subclass of dwarf novae and they usually show rare (typically once in a decade) and largeamplitude (typically 8 mag) superoutbursts [for general information of cataclysmic variables and dwarf novae,
see e.g. Warner (1995)]. Although WZ Sge stars were originally deﬁned as dwarf novae showing rare, largeamplitude superoutbursts and almost lacking normal outbursts (see e.g. Bailey 1979; Downes and Margon 1981;
Downes 1990), this deﬁnition remained somewhat ambiguous. Following the dramatic superoutburst of WZ Sge
in 2001 (Patterson et al. 2002; Ishioka et al. 2002; Baba et al. 2002), our understanding of WZ Sge stars has
been refreshed. It was known that periodic variations having the orbital period were observed in WZ Sge stars
during the early phase of their superoutbursts. Patterson et al. (1981) considered that they are orbital humps
and and that they reﬂect a greatly enhanced mass-transfer from the secondary. At that time, the only example
was the 1978–1979 superoutburst of WZ Sge (Patterson et al. 1981). The number of objects showing this feature
gradually increased: HV Vir in 1992 (Kato et al. 2001) [although Barwig et al. (1992); Mendelson et al. (1992);
Leibowitz et al. (1994) reported the same feature, they considered the variations to be usual superhumps] andAL Com in 1995 (Kato et al. 1996; Patterson et al. 1996; Howell et al. 1996; Nogami et al. 1997). Following theoutburst of AL Com, the presence of double-wave modulations during the early stage of the superoutburst wasestablished. These variations are currently referred to as early superhumps. Osaki and Meyer (2002) identiﬁed
early superhumps as the manifestation of the 2:1 resonance, in addition to the 3:1 resonance which causes
superhumps and superoutbursts in SU UMa stars (Whitehurst 1988; Osaki 1989; Hirose and Osaki 1990; Lubow2∆v(i )

0

2

4

0

20

40

60

80

Inclination (i )Figure 1: Dependence of the apparent brightness of an optically thick disk on the orbital inclination using equation(1). Fainter magnitudes are displayed lower.
1991) and in WZ Sge stars later during superoutbursts. WZ Sge stars are currently deﬁned as dwarf novae inwhich the 2:1 resonance plays a role during their superoutbursts (Kato 2015).
In Kato (2015), I suggested in its subsection 7.10 that the brightness of WZ Sge stars when ordinary
superhumps appear could be used as the standard candle since the disk is expected to have a size close to theradius of the 3:1 resonance. This could be equally applied to SU UMa stars. There are, however, unavoidable
large uncertainties arising from inclinations. Paczyński and Schwarzenberg-Czerny (1980) derived a formula for
an optically thick disk (as in dwarf-nova outbursts) considering the projected area and limb darkening of the disk[]3
∆Mv (i) = −2.5 log10 (1 + cos i) cos i ,(1)2
where ∆Mv and i are the correction to produce the visual absolute magnitude and the inclination, respectively.
These corrections can become large (Warner 1987; Patterson 2011). This relation is shown in ﬁgure 1. Pole-on
systems (i=0) are observed 1.0 mag brighter than the average and systems with i=80◦ are nearly 2 mag fainter.
The use of the accretion disk as the standard candle is somewhat limited since i is relatively diﬃcult to measureother than in eclipsing systems.
In WZ Sge stars, however, the amplitude of early superhumps is considered to depend on the inclination since
early superhumps arise from a geometric eﬀect (to the observer) of vertical structures of the disk (Osaki and Meyer
2002). Uemura et al. (2012) succeeded in reproducing the proﬁle of early superhumps by considering self-eclipseof a vertically extended disk. Kato (2015) used the code by Uemura et al. (2012) and reasonably reproduced the
distribution of the observed amplitudes of early superhumps except objects with very large amplitudes (in its
subsection 5.5). The amplitudes of early superhumps could thus be used instead of i.
At the time of Kato (2015), WZ Sge stars with known parallaxes were only three (WZ Sge, GW Lib andV455 And). Now that Gaia parallaxes are available (GaiaEDR3: Gaia Collaboration et al. 2021), I calibrated
this “standard candle” and studied the dependence on the amplitude of early superhumps.

2

The Data

2.1

Data source

The data were mostly taken from table 5 in Kato (2015). The quiescent magnitudes were replaced by average
G magnitudes in Gaia Collaboration et al. (2021). The amplitudes of early superhumps were taken from table3
2 in Kato (2015). The full amplitudes of early superhumps (AESH ) correspond to A2 (mean amplitude) intable 2 in Kato (2015). Some objects in table 5 in Kato (2015) had more recent superoutbursts. If the values
were improved, I supplied the data from these superoutbursts. As stated in Kato (2015), the accuracy of themagnitudes at which ordinary superhumps appear is ∼0.1 mag. In some cases, zero-point calibrations of unﬁltered
CCD magnitudes caused more uncertainties (particularly in the past). In such cases, I used All-Sky AutomatedSurvey (ASAS-3: Pojmański 2002) V data and All-Sky Automated Survey for Supernovae (ASAS-SN) Sky Patrol
V data (Shappee et al. 2014; Kochanek et al. 2017) to obtain well-calibrated magnitudes. Observations from theAAVSO International Database1 were sometimes used.
Additional objects were taken from Kato et al. (2017, 2020). Some objects (mainly since 2017) reported to
VSNET (Kato et al. 2004) were also included to increase the sample. The detailed source of the data of eachobject is given later in this section. The objects having 1σ errors in Gaia parallaxes typically less than 20% were
included in the table. When drawing ﬁgures and making statistical analysis, I limited objects with parallax errors
less than 10%, corresponding to errors of 0.2 mag. Dwarf novae whose parallaxes are measured to this accuracy
are nearby objects and I ignored interstellar extinction. The lack of reddening was conﬁrmed by the blue colors(BP − RP ) in Gaia magnitudes. In examining the light curves, I also used Public Data Releases of the Zwicky
Transient Facility (Masci et al. 2019) observations2 .

2.2

Notes on individual objects

Simply referring to “magnitude”, here I mean the magnitude when ordinary superhumps appeared.
AL Com: updated using the 2019 April data (Tampo et al. 2021).
EG Cnc: updated using the 2018 October data (Kimura et al. 2021).
HV Vir: the same magnitude was obtained using the 2016 March data (Imada et al. 2018, and also VSNET
data).
RZ Leo: updated using the 2022 January data (Outburst detection by Tadashi Kojima. H. Maehara, vsnetalert 265223 ; T. Kato, vsnet-alert 265294 and Y. Tampo, vsnet-alert 265325 ). The main observers were Kyoto U.
team, Hiroshi Itoh, Seiichiro Kiyota, Osaka Kyoiku U. team, Shawn Dvorak, Tamás Tordai, Stephen M. Brincat,
Vihorlat Observatory team and Filipp Romanov.
UZ Boo: magnitude updated using snapshot V measurements.
ASAS J102522−1542.4: value updated using ASAS-3 V magnitudes. The amplitude of early superhumps
was taken from Kato et al. (2009a).
EZ Lyn: magnitude updated based on V data used in Kato et al. (2009a).
V455 And: magnitude updated based on V data used in Kato et al. (2009a).
OT J111217.4−353829: the epoch of the appearance of ordinary superhumps was somewhat uncertain.
V624 Peg: the epoch of the appearance of ordinary superhumps was somewhat uncertain. The magnitude
was from AAVSO V observations.
V1838 Aql: The amplitude of early superhumps was taken from Kato et al. (2014). See also Echevarría et al.
(2019).
ASASSN-14cv: early superhumps were reanalyzed in this work. The magnitude was conﬁrmed during the2020 July–August superoutburst.
V529 Dra: magnitude updated using snapshot V data. First case of double superoutburst (Kato et al.
2013a) and is one of the best candidates for period bouncers.
GS Cet: data from Kato et al. (2017). The magnitude was from ASAS-SN V data.
ASASSN-16eg: data from Wakamatsu et al. (2017). WZ Sge star with an unusually long orbital period.
ASASSN-16js: data from Kato et al. (2017). The magnitude was from ASAS-SN V data.
ASASSN-17el: data from Kato et al. (2020). The amplitude of early superhump was corrected in thiswork. The magnitude was from ASAS-SN V data.
PNV J20205397+2508145: data from Kato et al. (2020).
1 <http://www.aavso.org/data-download>.
2 The ZTF data can be obtained from IRSA <https://irsa.ipac.caltech.edu/Missions/ztf.html> using the interface <https://irsa.ipac.caltech.edu/docs/program_interface/ztf_api.html> or using a wrapper of the above IRSA API<https://github.com/MickaelRigault/ztfquery>.
3 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26522>.
4 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26529>.
5 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26532>.

4
HO Cet: data from Kato et al. (2009a). The magnitude was from ASAS-SN V data.
V627 Peg: data from Kato et al. (2010).
PNV J17144255−2943481: data from C. Nakata et al. in preparation; reanalyzed in this paper. Thisobject showed ﬁve post-superoutburst rebrightenings (Kato 2015).
OV Boo: data from Ohnishi et al. (2019). Population II object (Patterson et al. 2017).
TCP J18154219+3515598: superoutburst in 2017 June6 (H. Maehara, vsnet-alert 210987 ; K. Isogai,
vsnet-alert 211018 ; T. Kato, vsnet-alert 211059 ; spectrum by P. Berardi10 and T. Kato, vsnet-alert 2110911 ).
This object showed 10 post-superoutburst rebrightenings (June–September), nine of which were announced on
VSNET: R. J. Modic, vsnet-alert 2117512 ; R. J. Modic, vsnet-alert 2122413 ; E. de Miguel, vsnet-alert 2124414 ; H.
Maehara, vsnet-alert 2126815 ; T. Kato, vsnet-alert 2127816 ; K. Isogai, vsnet-alert 2130217 ; H. Maehara, vsnetalert 2132218 ; R. J. Modic, vsnet-alert 2133419 and H. Maehara, vsnet-alert 2134720 . See also Zubareva et al.
(2018). The amplitude of early superhumps is based on re-analysis of the data reported to VSNET. The main
observers were Geoﬀ Stone, Tamás Tordai, Enrique de Miguel, Tonny Vanmunster, Lisnyky Observatory team,Kyoto U. team, Vihorlat Observatory team, Stephen M. Brincat, Hiroshi Itoh, Rudolf Novák, Terskol Observatory
team, Seiichiro Kiyota, Kiyoshi Kasai, Roger D. Pickard, Natalia Katysheva, Ian Miller, Alexandra M. Zubareva,
Javier Ruiz, William Stein, Sergey Yu. Shugarov, Lewis M. Cook, TSHAO Observatory team and Crimean
Astrophysical Observatory team.
ASASSN-17pm: also known as PNV J05580574−001115521 . Superoutburst in 2017 November–December(T. Vanmunster, vsnet-alert 2162422 ; T. Kato, vsnet-alert 2162723 and T. Kato, vsnet-alert 2164624 ). The
amplitude of early superhump was re-analyzed in this work. There was a short rebrightening, a shallow dip anda long rebrightening. The main observers were Hiroshi Itoh, Crimean Astrophysical Observatory team, FranzJosef Hambsch, Seiichiro Kiyota, Tonny Vanmunster, Kyoto U. team, Tamás Tordai, Ian Miller and Domenico
Licchelli.
ASASSN-18do: the data were not very good around the appearance of ordinary superhumps. Superoutburst in 2018 February–March (T. Vanmunster, vsnet-alert 2190625 ; T. Kato, vsnet-alert 2192126 ). This object
is eclipsing (T. Kato, vsnet-alert 2193327 ). The main observers were Tonny Vanmunster, Hiroshi Itoh, Jochen
Pietz, Tamás Tordai, Crimean Astrophysical Observatory team, Sergey Yu. Shugarov and Kyoto U. team.
ASASSN-18wa: superoutburst in 2018 September–October (T. Vanmunster, vsnet-alert 2256028 ; Y. Wakamatsu, vsnet-alert 2256529 ; Y. Wakamatsu, vsnet-alert 2259130 ). There was a short gap in the observations after
the superoutburst and it was unclear whether there was a post-superoutburst rebrightening. The ZTF data
showed a smooth fading tail characteristic to a WZ Sge star. The main observers were Tamás Tordai, Osaka
Kyoiku U. team, Tonny Vanmunster, Hiroshi Itoh and Geoﬀ Stone.
ASASSN-19ag: superoutburst in 2019 January (T. Vanmunster, vsnet-alert 2292731 ). Although brightening and growth of ordinary superhumps were recorded, the proﬁle before the growth of ordinary superhumps was
6 <http://www.cbat.eps.harvard.edu/unconf/followups/J18154219+3515598.html>.
7 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21098>.
8 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21101>.
9 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21105>.
10 <http://quasar.teoth.it/html/spectra/tcpj18154219+3515598_PB.png>.
11 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21109>.
12 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21175>.
13 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21224>.
14 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21244>.
15 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21268>.
16 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21278>.
17 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21302>.
18 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21322>.
19 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21334>.
20 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21347>.
21 <http://www.cbat.eps.harvard.edu/unconf/followups/J05580574−0011155.html>.
22 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21624>.
23 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21627>.
24 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21646>.
25 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21906>.
26 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21921>.
27 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21933>.
28 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/22560>.
29 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/22565>.
30 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/22591>.
31 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/22927>.

5
not similar to that of early superhumps (T. Kato, vsnet-alert 2293732 ). I therefore did not give the amplitude ofearly superhumps. The object showed at least one post-superoutburst rebrightening in the ZTF data.
TCP J06373299−0935420 = ASASSN-19de: superoutburst in 2019 February–March33 (spectrum by
R. Leadbeater, vsnet-alert 2301434 and T. Kato, vsnet-alert 2303635 ). There was one post-superoutburst rebrightening (T. Kato, vsnet-alert 2308336 ). The main observers were Osaka Kyoiku U. team, Hiroshi Itoh,
Stephen M. Brincat, Berto Monard, Franz-Josef Hambsch, Seiichiro Kiyota, Arto Oksanen, Masanori Mizutani,
Tonny Vanmunster and Yasui Sano. According to vsnet-alert postings, Gianluca Masi also reported observations
but I could not receive the data and are not included in this analysis.
TCP J05390410+4748030 = ASASSN-19hh: superoutburst in 2019 March–April37 (T. Vanmunster,
vsnet-alert 2306838 ; T. Kato, vsnet-alert 2307239 ; T. Kato, vsnet-alert 2312040 and T. Kato, vsnet-alert 2312441 ).
The main observers were Tonny Vanmunster, Seiichiro Kiyota, Hiroshi Itoh, Crimean Astrophysical Observatory
team, Vihorlat Observatory team, Tamás Tordai, Stephen M. Brincat, Ian Miller and Lewis M. Cook.
ASASSN-19hl: superoutburst in 2019 March–April (D. Denisenko, vsnet-alert 2309042 ; T. Kato, vsnetalert 2310343 and T. Kato, vsnet-alert 2314444 ). The main observers were Berto Monard, Franz-Josef Hambschand Hiroshi Itoh.
V3101 Cyg: data from Tampo et al. (2020). The amplitude of early superhumps was re-analyzed in thiswork.
EQ Lyn: data from Tampo et al. (2021).
GY Cet: superoutburst in 2020 July–August (P. Schmeer, vsnet-alert 2444645 ; T. Kato, vsnet-alert 2448646
and T. Kato, vsnet-alert 2449847 ). No post-superoutburst rebrightening was present. The main observers wereFranz-Josef Hambsch and Berto Monard.
VX For: superoutburst in 2021 January–February (Y. Maeda, vsnet-alert 2526448 ; P. Schmeer, vsnet-alert
2526549 and Y. Tampo, vsnet-alert 2528850 ). There were ﬁve post-superoutburst rebrightenings (P. Schmeer,
vsnet-alert 2535851 ; P. Schmeer, vsnet-alert 2538652 ; R. Stubbings, vsnet-outburst 2665153 ; P. Schmeer, vsnetalert 2541454 and ASAS-SN g=14.3 on 2021 February 28). This object also showed ﬁve post-superoutburst
rebrightenings in 2009 (Kato et al. 2010) and has been suggested as a period bouncer (Kato 2022). The main
observers were Hiroshi Itoh, Berto Monard, Peter Nelson and Franz-Josef Hambsch.
ASASSN-21et = TCP J06154200−2756220: superoutburst in 2021 April-May55 (P. Schmeer, vsnet-alert
2563756 ; T. Kato, vsnet-alert 2565857 and T. Kato, vsnet-alert 2578058 ). There was no post-superoutburst
rebrightening in the ASAS-SN data. The main observers were Berto Monard and Franz-Josef Hambsch.
I only deal with statistics in this paper and the full analysis and ﬁgures for these unpublished objects areplanned to appear in separate paper(s).
32 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/22937>.
33 <http://www.cbat.eps.harvard.edu/unconf/followups/J06373299−0935420.html>.
34 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23014> and<https://britastro.org/specdb/data_graph.php?obs_id=3925>.
35 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23036>.
36 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23083>.
37 <http://www.cbat.eps.harvard.edu/unconf/followups/J05390410+4748030.html>.
38 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23068>.
39 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23072>.
40 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23120>.
41 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23124>.
42 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23090>.
43 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23103>.
44 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/23144>.
45 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/24446>.
46 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/24486>.
47 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/24498>.
48 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25264>.
49 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25265>.
50 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25288>.
51 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25358>.
52 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25386>.
53 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-outburst/26651>.
54 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25414>.
55 <http://www.cbat.eps.harvard.edu/unconf/followups/J06154200−2756220.html>.
56 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25637>.
57 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25658>.
58 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/25780>.

6

3

Result and Discussion

3.1

Absolute magnitudes when ordinary superhumps appear

The absolute magnitudes when ordinary superhumps appear [hereafter MV (SH)] are listed in table 1. Note that
this table includes objects with relative large errors in parallax. For example, the bright MV (SH) in HO Cet
is likely a result of the uncertainty in the parallax. In this table, I gave 1σ errors estimated from the errors inparallax. They are the main cause of the overall errors of MV (SH).
Table 1: Brightness when superhumps appear
ObjectYear
P∗
Mag1† Mag2‡ AESH
ϖ§ϖerrorWZ Sge2011 0.05669
15.2
9.9
0.14 22.104
0.030
AL Com2019 0.05667
19.7
13.7
0.04
1.903
0.565
EG Cnc2018 0.05997
18.8
12.7
0.018 5.361
0.199
HV Vir1992 0.05707
19.0
13.3
0.044 3.152
0.261
RZ Leo2022 0.07603
18.2
13.1
0.05
3.564
0.149
QZ Lib2004 0.06460s
18.8
12.2
–
5.023
0.254
UZ Boo2013 0.0620s
19.9
12.8
–
3.086
0.511
ASAS J102522−1542.4
2006 0.06136
19.3
12.5
0.025 3.675
0.342
EZ Lyn2010 0.05901
17.8
12.7
0.067 7.004
0.110
GW Lib2007 0.05332
16.5
10.2
0.00
8.846
0.061
V455 And2007 0.05631
16.0
11.0
0.22 13.189
0.043
OT J111217.4−353829
2007 0.05847
20.2
14.4
0.14
2.003
0.531
SDSS J161027.61+090738.4
2009 0.05687
19.8
14.6
0.05
2.599
0.430
CRTS J104411.4+211307
2010 0.05909
19.4
13.7
0.030 2.425
0.342
OT J012059.6+325545
2010 0.05716
19.8
14.2
0.045 3.030
0.427
V355 UMa
2011 0.05729
17.4
10.8
0.01
6.619
0.064
V624 Peg
2011 0.05865
18.4
12.2
0.00
4.553
0.193
BW Scl2011 0.05432
16.3
10.7
0.10 10.679
0.052
MASTER OT J211258.65+242145.4 2012 0.05973
20.0
15.2
0.050 1.754
0.493
V1838 Aql
2013 0.05706
17.9
11.7
0.011 4.940
0.142
ASASSN-14cl
2014 0.05838
18.2
11.9
0.018 3.825
0.188
ASASSN-14cv
2014 0.05992
19.1
12.9
0.03
3.407
0.171
PNV J23052314−0225455
2014 0.05456
19.2
13.4
0.035 2.103
0.299
PNV J03093063+2638031
2014 0.05615
18.7
12.2
0.018 4.132
0.188
ASASSN-14jv
2014 0.05442
18.8
12.5
0.017 3.333
0.147
ASASSN-15bp
2014 0.05563
19.8
13.7
0.014 1.792
0.325
V1251 Cyg
2008 0.07433
20.2
13.4
0.018 1.965
0.517
BC UMa2003 0.06261
18.3
12.6
0.04
3.392
0.126
V529 Dra
2011 0.07168
20.4
13.7
0.005 1.661
0.488
CRTS J122221.6−311524
2013 0.07649s
18.9
12.3
–
4.136
0.222
PNV J06000985+1426152
2014 0.06331s
20.0
12.9
–
1.833
0.706
GS Cet2016 0.05597
20.7
14.2
0.065 3.846
1.388
ASASSN-16eg
2016 0.07548
19.4
13.4
0.05
2.260
0.204
ASASSN-16js
2016 0.06034
20.5
15.3
0.18
2.475
0.350
ASASSN-17el
2017 0.05434
18.7
12.6
0.025 2.889
0.120
PNV J20205397+2508145
2017 0.05651
20.1
14.2
0.06
1.561
0.526
HO Cet2006 0.05490
19.3
13.5
0.035 1.134
0.392
V627 Peg
2010 0.05452
15.6
10.3
0.045 10.079
0.037∗
Orbital or superhump (with a suﬃx ‘s’) period (d).
†Quiescent brightness.
‡
Brightness when ordinary superhumps appear.
§
Gaia EDR3 parallax (mas).
∥
Absolute magnitude when ordinary superhumps appear. The 1σ error estimated from ϖerror is given.

MV (SH)∥
6.6(0)
5.1(8)
6.3(1)
5.8(2)
5.9(1)
5.7(1)
5.2(4)
5.3(2)
6.9(0)
4.9(0)
6.6(0)
5.9(7)
6.7(4)
5.6(3)
6.6(3)
4.9(0)
5.5(1)
5.8(0)
6.4(7)
5.2(1)
4.8(1)
5.6(1)
5.0(3)
5.3(1)
5.1(1)
5.0(4)
4.9(7)
5.3(1)
4.8(8)
5.4(1)
4.2(11)7.1(10)5.2(2)
7.3(3)
4.9(1)
5.2(9)
3.8(9)
5.3(0)

7
Table 1: Brightness when superhumps appear (continued).
ObjectYear
P∗
Mag1† Mag2‡ AESH
ϖ§ϖerrorPNV J17144255−2943481
2014 0.05956
17.2
12.3
0.05
5.665
0.098
OV Boo2017 0.04626
18.2
13.5
0.27
4.715
0.091
TCP J18154219+3515598
2017 0.06168s
19.1
12.4
0.015 4.600
0.171
ASASSN-17pm
2017 0.05973
17.8
14.2
0.14
3.363
0.120
ASASSN-18do
2018 0.05498
20.5
15.9
0.6
1.593
0.671
ASASSN-18wa
2018 0.05441
20.2
14.4
0.02
2.068
0.949
ASASSN-19ag
2019 0.06616
20.0
13.0
–
2.512
0.635
TCP J06373299−0935420
2019 0.06642s
19.2
12.1
0.01
3.432
0.275
TCP J05390410+4748030
2019 0.05534
19.6
13.7
0.10
2.957
0.318
ASASSN-19hl
2019 0.05415
19.5
12.9
0.03
2.889
0.274
V3101 Cyg2019 0.05352
17.7
10.9
0.045 9.226
0.072
EQ Lyn2019
0.0528
18.9
12.0
–
3.223
0.222
GY Cet2020 0.05663
18.5
11.9
0.015 3.712
0.181
VX For2021 0.06133s
20.4
13.2
–
2.060
0.574
ASASSN-21et
2021 0.05820
20.2
14.5
0.12
2.816
0.508
∗
Orbital or superhump (with a suﬃx ‘s’) period (d).
†Quiescent brightness.
‡
Brightness when ordinary superhumps appear.
§
Gaia EDR3 parallax (mas).
∥
Absolute magnitude when ordinary superhumps appear. The 1σ error estimated from ϖerror is given.

MV (SH)∥
6.1(0)
6.9(0)
5.7(1)
6.8(1)
6.9(12)6.0(13)5.0(6)
4.8(2)
6.1(2)
5.2(2)
5.7(0)
4.5(2)
4.7(1)
4.8(7)
6.7(4)The distribution of absolute magnitudes when ordinary superhumps appear is shown in ﬁgure 2. The meanvaule is ⟨MV (SH)⟩=+5.57 and the standard deviation is 0.67 mag. This standard deviation is too large to be
directly used as the standard candle. Furthermore, the mean vaule is not adequate since MV (SH) does not follow
a Gaussian distribution as explained later. The dispersion in MV (SH) is largely due to the inclination eﬀect as
seen in equation (1).

3.2

Correlation with the amplitude of early superhumps

Considering that AESH is expected to be a function of i, the relation between AESH and MV (SH) is shown inﬁgure 3. Systems with large AESH have fainter MV (SH) as expected. The relation becomes linear using log
AESH (ﬁgure 4). The solid line in the ﬁgure corresponds to
MV (SH) = 7.5(3) + 1.28(20) log10 AESH .

(2)

In drawing the ﬁgure and making a regression, MV (SH)=0.01 was given for objects with MV (SH) <0.01 (Thevalue of 0.01 mag is a realistic upper limit of the detection of early superhumps; use the same value in applying
the formula to WZ Sge stars without detectable early superhumps). The standard deviation from this relation is0.42 mag.

3.3

Special objects and the updated relation

The two most deviating objects were EG Cnc (1.1 mag fainter than this relation) and EZ Lyn (0.9 mag fainterthan this relation). One of the possible reason for EG Cnc is that the observations of EG Cnc did not start earlyenough (Kimura et al. 2021) and AESH might have been underestimated. EZ Lyn is an eclipsing system and issuggested to be a period bouncer (Zharikov et al. 2008; Pavlenko et al. 2007; Kato et al. 2009b; Kato and Osaki
2013; Amantayeva et al. 2021).
Although TCP J18154219+3515598 with 10 rebrightenings is also 0.5 mag fainter than this relation, ASASSN14cv with 5–8 rebrightenings [8 in 2014 (Sklyanov et al. 2016; Kato 2015); 5 in 2020, VSNET data] does
not show a strong deviation. QZ Lib with four rebrightenings (Kato et al. 2009a; Pala et al. 2018) and PNV

4
3
2
0

1

Frequency

5

6

8

4.5

5.0

5.5

6.0

6.5

7.0

M V(SH)
Figure 2: Distribution of absolute magnitudes when ordinary superhumps appear [MV (SH)]. Objects withparallax errors less than 10% were selected.

M V(SH)

5

6

7

0.00

0.05

0.10

0.15

0.20

0.25

Figure 3: Dependence of absolute magnitudes when ordinary superhumps appear [MV (SH)] on amplitudes of
early superhumps. Objects with parallax errors less than 10% were selected. The error bars reﬂect the errors
from parallax measurements.

95

M V(SH)ASASSN−16eg

6

ASASSN−14cv
V3101 Cyg
TCP J18154219+3515598
PNV J17144255−2943481
EG Cnc7

EZ LynOV Boo−2.2 −2.0 −1.8 −1.6 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4

log 10 amplitude of early superhumps
Figure 4: Dependence of absolute magnitudes when ordinary superhumps appear [MV (SH)] on amplitudes of earlysuperhumps in logarithmic scale. Objects with parallax errors less than 10% were selected. Several unusual objects
are plotted with diﬀerent colors. Red marks represent objects with multiple post-superoutburst rebrightenings.
(See text for the details). The solid and dashed lines represent equations (2) and (3), respectively. The error
bars reﬂect the errors from parallax measurements.

10
J17144255−2943481 with ﬁve rebrightenings are also on this relation. The very unusual object V3101 Cyg,
which showed multiple rebrightenings and superoutbursts following the initial superoutburst (Tampo et al. 2020;
Hameury and Lasota 2021), is also on the relation. The most extreme period bouncer in this sample CRTS
J122221.6−311524 (Kato et al. 2013b; Neustroev et al. 2017, 2018) is not in ﬁgure 4 since AESH is unknown, but
has an ordinary MV (SH)=5.4.
There is a possible reason of the deviation for (some) period bouncers if the deviation is real. They are likely
to have small mass ratios (Kato 2022, and references therein) and the tidal torque on the disk is expected to be
weak. This may cause a weaker eﬀect of the 2:1 resonance and AESH may be smaller than in other WZ Sge stars.
Since a number of WZ Sge stars with multiple rebrightenings are on the relation, this explanation does not seemto apply to all period bouncers. This possibility needs to be studied further both from the observational andtheoretical sides.
The only population II object below the period minimum OV Boo (Littlefair et al. 2007; Patterson et al.
2008; Uthas et al. 2011; Patterson et al. 2017; Ohnishi et al. 2019) is on the relation. This object is located near
the period minimum of population II cataclysmic variables are it may not be as unusual as more evolved period
bouncers. ASASSN-16eg is a WZ Sge star with a very long orbital period (Wakamatsu et al. 2017). The deviation
to the brighter side may reﬂect the large disk size, but the error in the parallax is still large to draw a deﬁnite
conclusion.
Disregarding the objects deviating by more than 0.6 mag from equation (2), I obtained
MV (SH) = 7.5(2) + 1.31(15) log10 AESH .

(3)

The standard deviation from this regression is 0.31 mag. This equation can currently be regarded as the best
relation for WZ Sge stars with potential exceptions of evolved period bouncers and systems with very long orbital
periods.

3.4

Standard candle for the average inclination

The median value of MV (SH) for all the objects is +5.38, which can be considered as a value for the average
inclination (=1 radian = 57◦ ). This value can be used as the standard candle if AESH is unknown, but with a 1σ
error of 0.67 mag. Although, errors may be even larger in high-inclination systems, i can probably be estimatedusing eclipses in such systems and there would be no need for the use of the median MV (SH).

3.5

Estimation of the inclination from the amplitude of early superhumps

By equating the equation (1) and the equation (3) and using the value of the standard candle of MV (SH)=+5.38
for i=1 radian, one can obtain an experimental relation between the amplitude of early superhumps and theinclination (ﬁgure 5). Note that this ﬁgure is based on an assumption that the relation between logAESH and
MV (SH) is linear. This ﬁgure suggests that AESH is 0.02 mag for the average inclination (=1 radian). It is likelythat early superhumps are not detectable for i <40◦ . There is only one object (OV Boo) with i >80◦ in our
sample. It is possible that the amplitude of early superhumps is saturated for very large i and the linear relation
would break. The relation in ﬁgure 5 would be helpful for estimating i for objects with moderate AESH .

4

Application to MASTER OT J030227.28+191754.5

MASTER OT J030227.28+191754.5 = PNV J03022732+1917552 was reported as a possible counterpart (Zhirkov
et al. 2021) of IceCube-211125A high-energy neutrino event (IceCube Collaboration 2021). This object had alarge outburst amplitude of 10 mag (Zhirkov et al. 2021). It was independently discovered by Y. Nakamura59 .
Due to the large outburst amplitude and the possible association with the neutrino event, it was suspected to bea nova. Early spectroscopy indeed showed narrow emission lines and it was suggested to be either a narrow-lined
He(/N) nova or a dwarf nova with an exceptionally large amplitude (Taguchi et al. 2021). The initial spectrum
did not resemble that of a dwarf nova and the object was initially favored to be a nova (Paliya 2021). Follow-up
observations did not detect very-high-energy gamma-ray ﬂux (Quinn et al. 2021; Ayala 2021a,b). The optical
spectrum on the second night clariﬁed that the object is a very high-amplitude dwarf nova (Isogai et al. 2021).
CCD images taken 8.5 hr before the neutrino event indicated that the object was already 1 mag above the
59 <http://www.cbat.eps.harvard.edu/unconf/followups/J03022732+1917552.html>.

11Inclination (i )

80

60

40

20

0
0.00

0.05

0.10

0.15

0.20

0.25

Figure 5: Relation between the amplitude of early superhumps and the inclination.
preoutburst level (Sarneczky et al. 2021). Although this observation suggested the association with the neutrino
event less likely, the magnitude might have been too faint to be considered as the ignition of the outburst.
Early superhumps with an amplitude of 0.03 mag were detected (T. Kato, vsnet-alert 2647760 ). The object
eventually started to show ordinary superhumps on 2021 December 30, 30–32 d after the outburst detection (T.
Kato, vsnet-alert 2650161 ). This delay of the appearance of ordinary superhumps was the longest ever observed(Kato 2015).
Due to the unusual nature and the extremely slow development of ordinary superhumps in this object,
I applied the present method to estimate the distance and luminosity. Ordinary superhumps appeared at amagnitude of 14.9 (VSNET data; Y. Tampo et al. in preparation). Using the equation (3), MV (SH) is expected
to be +5.6 and the distance modulus is estimated to be 9.3, which corresponds to ∼720 pc. The SDSS magnitude
g=22.0 (Abazajian et al. 2009) corresponds to the quiescent absolute magnitude of +12.7, which is one of the
faintest recorded in WZ Sge stars (Tampo et al. 2020). The peak V magnitude 11.8 corresponds to MV =+2.6,
which is also among the brightest (Tampo et al. 2020).
To examine the possibility if any nuclear reaction was involved in the outburst of this object, I made acomparison with the faintest measurement of an outbursting nova. The recurrent nova T Pyx was recorded on
the rise at a magnitude of 13.0 during the 2011 eruption by M. Linnolt (Waagan et al. 2011; Schaefer et al. 2013)[the initial spectrum was taken several hours later by Arai et al. (2015), conﬁrming the nova eruption]. This
corresponds to MV =+0.7 and the maximum MV of MASTER OT J030227.28+191754.5 is still 6 times below
this value. The faint brightness of T Pyx during the rise was probably due to the large bolometric correction andthe actual upper limit of the luminosity of the white dwarf due to nuclear reaction should be much lower than
this estimate (compared to T Pyx) based on the V -band data.

Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 21K03616. This research has made use of the
AAVSO Variable Star Index (VSX, Watson et al. 2006) and NASA’s Astrophysics Data System. I am deeply
indebted to world-wide observers who reported observations, outburst detection and information to VSNET,VSOLJ and AAVSO. The author is grateful to the ASAS-3 and ASAS-SN teams for making their data available tothe public. We acknowledge with thanks the variable star observations from the AAVSO International Database
60 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26477>.
61 <http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26501>.

12
contributed by observers worldwide and used in this research. The author is grateful to Naoto Kojiguchi for
helping downloading the ZTF data.
Based on observations obtained with the Samuel Oschin 48-inch Telescope at the Palomar Observatory as
part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant
No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar
Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches
Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortiumof Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations
are conducted by COO, IPAC, and UW.
The ztfquery code was funded by the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement n◦ 759194 – USNAC, PI: Rigault).

List of objects in this paper
V455 And, V1838 Aql, UZ Boo, OV Boo, GS Cet, GY Cet, HO Cet, EG Cnc, AL Com, V1251 Cyg, V3101 Cyg,
V529 Dra, VX For, RZ Leo, GW Lib, QZ Lib, EQ Lyn, EZ Lyn, V624 Peg, V627 Peg, T Pyx, BW Scl, WZ Sge,
SU UMa, BC UMa, V355 UMa, HV Vir, ASAS J102522−1542.4, ASASSN-14cl, ASASSN-14cv, ASASSN-14jv,
ASASSN-15bp, ASASSN-16eg, ASASSN-16js, ASASSN-17el, ASASSN-17pm, ASASSN-18do, ASASSN-18wa,
ASASSN-19ag, ASASSN-19hl, ASASSN-21et, CRTS J104411.4+211307, CRTS J122221.6−311524, IceCube211125A, MASTER OT J030227.28+191754.5, MASTER OT J211258.65+242145.4, OT J012059.6+325545,
OT J111217.4−353829, PNV J03093063+2638031, PNV J06000985+1426152, PNV J17144255−2943481, PNV
J20205397+2508145, PNV J23052314−0225455, SDSS J161027.61+090738.4, TCP J05390410+4748030, TCP
J06373299−0935420, TCP J18154219+3515598

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c/o Keiichi Saijo National Science Museum, Ueno-Park, Tokyo Japan
Editor Seiichiro Kiyota
e-mail: skiyotax@gmail.com