1
No.89

Jan. 2022

ISSN 0917-2211

Variable Star Bulletin
Evolution of short-period cataclysmic variables: implications from eclipse
modeling and stage A superhump method (with New Year’s gift)q=M 2/M 1

0.15

J194419

0.10

ASASSN−15po

0.05

0.050

0.055

0.060

0.065

0.070

Figure 1: Mass ratios (q) versus orbital periods (Porb ) determined by the eclipse modeling method and the stage
A superhump method, enlargement around the period minimum. See text for the detail; the answer for thesymbols is shown in ﬁgure 10. The dashed and solid curves represent the standard and optimal evolutionary
tracks in Knigge et al. (2011), respectively.
Taichi Kato1
1

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

Received 2022 Jan. 07

2(Abstract is given at the end of the paper).

Prologue“Look to the skies, and you will feel it — a deep universal fascination ...” — stargazers will completely agreewith the phrase, but this is the beginning of the narration of the TV documentary “Extraordinary Birds” directedby Tom Simon in 2000. I refrain from talking about my favorite birds for a while; now, just look at ﬁgure 1!
Although this may not be as fascinating as what you see in the skies, this is a ﬁgure summarizing our current
best knowledge about the terminal evolution of cataclysmic variables (CVs). CVs are close binaries consisting ofa white dwarf and a mass-transferring low-mass dwarf star. CVs evolve from upper right on this ﬁgure from longorbital periods (Porb ) to shorter Porb and to lower mass ratios (q) by transferring the matter from the secondary.
CVs then reach the “period minimum”, after which Porb lengthens while q is still decreasing. Such objects arecalled “period bouncers”.
This ﬁgure contains two symbols representing measurements by two state-of-the-art methods to determine q.
One method employs 3.5–8.2 m telescopes equipped with a specially designed multicolor high-speed camera(there were even Nature and Science papers among them: Littlefair et al. 2006; Hernández Santisteban et al.
2016). The other employs 20–50 cm telescopes, which are often owned by amateur astronomers equippedwith oﬀ-the-shelf CCD cameras. Can you tell which symbol is which?
This question would be diﬃcult to answer: these two methods give almost the same results and mutually
reinforce the reliability each other. In other words, the second method (stage A superhump method) is as reliable
as the ﬁrst method (eclipse modeling). I will explain the reason in the following sections. If you are interested tocontribute to observations by the second method, a book “Cataclysmic Variable Stars: How and why they vary”
by Hellier (2001) will be helpful. Our Variable Star Network team (VSNET Collaboration: Kato et al. 2004)
regularly receive observations of superhumps from amateurs and professionals worldwide and your observations
will surely contribute to reveal the secrets of CVs.
It is a pity, however, that the measurements of q by the stage A superhump method tend to be neglected byresearchers of the CV evolution, probably due to a persistent misunderstanding that the reliability of superhumps
for determining q is limited partly because it is dependent on experimental calibration based on old knowledge
before the 2010s. In this paper, I review the history of the misunderstanding, the current reliable method anda comparison with the results of the eclipse modeling method using a high-speed photometer, which is usuallyconsidered to be most accurate.

1

Historical Development

Superhumps in SU UMa-type dwarf novae have periods (superhump period, PSH ) a few percent longer than theorbital period Porb [for general information of cataclysmic variables and dwarf novae, see e.g. Warner (1995)].
Superhumps are widely accepted to be caused by a precessing eccentric accretion disk which arises from the 3:1
resonance (Whitehurst 1988; Hirose and Osaki 1990; Lubow 1991). The presence of the gravity of the secondary
star causes the deviation of the gravitational ﬁeld from the inverse square law by the white dwarf primary [see e.g.
Hirose and Osaki (1990) for a mathematical treatment], it is natural to consider that the precession rate (ωpr )can be used as a measure of the binary mass ratio q = M2 /M1 , where M1 and M2 are the masses of the primary(white dwarf) and the secondary which transfers matter to the primary, respectively. The relation between Porb ,PSH and ωpr is:ϵ∗ ≡ ωpr /ωorb = 1 − Porb /PSH ,(1)
where ωorb is the orbital angular frequency of the binary. In actual observations, the fractional superhump excess(ϵ) is widely used:ϵ ≡ PSH /Porb − 1.
(2)
The relation between ϵ and ϵ∗ is:ϵ∗ = ϵ/(1 + ϵ).

(3)

The fractional superhump excesses were, however, historically only used to derive approximate values of q
mainly due to the two reasons:

3

Figure 2: VSNET page telling the excitement of the unexpected outburst of WZ Sge in 2001.
(1) The precession rate depends on the radius or the mass distribution of the disk, which is usually diﬃcult todetermine by observations.
(2) The precession rate usually does not reﬂect the purely dynamical precession. The pressure eﬀect slows down
the precession rate. (Lubow 1992; Hirose and Osaki 1993; Murray 1998; Montgomery 2001; Pearson 2006),
while the pressure eﬀect is diﬃcult to formulate (see e.g. Montgomery 2001; Pearson 2006) or measure byobservations.
Before the identiﬁcation of the nature of superhumps, Stolz and Schoembs (1984) made a pioneering work
illustrating that there is a linear relation between ϵ and Porb . This relation was updated by Robinson et al. (1987).
This Stolz-Schoembs relation has widely been used to estimate Porb from PSH (e.g. Ritter 1984). This relation
also implied that ϵ should be a strong function of q following the evolution of cataclysmic variables (Warner 1976;
Patterson 1984). Molnar and Kobulnicky (1992) ﬁrst systematically studied the relation between ϵ and Porb , andthen q. They found very good linear relations between ϵ and Porb , and between ϵ and q assuming main-sequence
secondaries. This Molnar-Kobulnicky relation was used to estimate q from PSH or ϵ (such as Howell et al. 1993;
Kato et al. 2001). Lemm et al. (1993); Skillman and Patterson (1993); Leibowitz et al. (1994) presented updated
ﬁgures of the Molnar-Kobulnicky relation. The work by Mineshige et al. (1992) was one of the ﬁrst to directly
estimate q from ϵ. They introduced η, the ratio between the disk radius and the radius of the 3:1 resonance, andestimated η from measurements of SU UMa stars and applied to superhumps in black-hole X-ray transients toestimate the masses of the black holes. This empirical calibration, however, was not widely used by researchersof cataclysmic variables [I could only ﬁnd Retter et al. (1997)].
An independent eﬀort to calibrate the ϵ-q relation became necessary following an burst of new detectionsof superhumps in WZ Sge stars [the examples being HV Vir in 1992: Leibowitz et al. (1994); Kato et al.
(2001), AL Com in 1995: Pych and Olech (1995); Kato et al. (1996); Howell et al. (1996); Patterson et al.
(1996); Nogami et al. (1997), EG Cnc in 1996: Patterson et al. (1998); Kato et al. (2004), V2176 Cyg in
1997: Novák et al. (2001); Kwast and Semeniuk (1998), V592 Her in 1998: Duerbeck and Mennickent (1998);

4
Kato et al. (2002), WZ Sge in 2001: Patterson et al. (2002); Ishioka et al. (2002); Baba et al. (2002), ﬁgure 2;
see Kato (2015) for a modern review of WZ Sge stars]. These WZ Sge stars drew attention of researchers since
they have very low-mass secondaries which may be brown dwarfs. Patterson (1998) derived an equation
ϵ=

0.23q
1 + 0.27q(4)

combined with the analytical formula of the precession rate. Patterson (2001) further calibrated the relation
using observed superhump excesses and obtained
ϵ = 0.216(±0.018)q.

(5)

This relation was derived from various classes of objects ranging from dwarf novae to novalike variables and X-ray
binaries. Patterson et al. (2005a) published a reﬁnement of the empirical relation
ϵ = 0.18q + 0.29q 2(6)

based on a broader sample of objects ranging from dwarf novae to novalike variables. In Patterson (2001), only
one X-ray binary (KV UMa) was used in contrast to Patterson (2001). These three formulae are still most widelyused. These “Patterson” relations, however, have intrinsic diﬃculties. In addition to the reasons listed earlier inthis section, there are diﬃculties:(3) The pressure eﬀect is expected to be diﬀerent depending on the state of the accretion disk (see e.g. Pearson
2006). Patterson’s calibration relied on diﬀerent states: non-stationary outbursting dwarf novae, steadystate novalike disks and an X-ray binaries, which can have disk sizes and temperatures diﬀerent from CVs
and contributions of the pressure eﬀect may be diﬀerent.
(4) Patterson’s formulae assumed ϵ=0 at q=0, which is incorrect if the pressure eﬀect is taken into account.
According to Lubow (1992), the (apsidal) precession rate can be written as a form:ωpr = ωdyn + ωpressure + ωstress ,(7)

where the ﬁrst term, ωdyn , represents a contribution to disk precession due to the gravitational potential of thesecondary, giving rise to prograde precession, the second term, ωpressure (negative value), the pressure eﬀect giving
rise to retrograde precession, and the last term, ωstress , the minor wave-wave interaction. The functional form ofωdyn is given in the next section. As one could naturally see, ωdyn is small when q is small. ωpressure is, however,not very dependent on q and the eﬀect of this term can become larger than that of the ﬁrst term for very small q.
This is the reason why the item (4) is important. The same issue was also pointed out by Goodchild and Ogilvie(2006).

2

Modern Method Using Stage A Superhumps

2.1

Superhumps and dynamical precession

The situation has changed since the identiﬁcation of superhump stages (stages A, B and C: Kato et al. 2009a).
Kato et al. (2009a) showed that the periods of superhumps systematically vary. Stage A superhumps appear ﬁrst
when superhumps grow. It was not certain at that time which stage is suitable for estimating q. In Kato and Osaki(2013), however, stage A superhumps were identiﬁed to reﬂect the dynamical precession rate of the disk at theradius of the 3:1 resonance. Stage B superhumps with smaller ϵ is most strongly aﬀected by the pressure eﬀect
and was found to be inadequate to derive ϵ [Patterson’s formulae used stage B superhumps for dwarf novae; seefootnote 11 in Patterson (2011)]. Following the treatment in Kato and Osaki (2013),1
ωdyn= Q(q)R(r),
ωorb

(8)

1 The equations in this part are not absolutely necessary to understand the stage A superhump method and its applications. One
can skip this part and proceed to the next section if necessary.

5r is the dimensionless radius measured in units of the binary separation A. The dependence on q and r can be
described as (cf. Hirose and Osaki 1990)
1q
Q(q) = √
,
(9)
2 1+q
and
R(r) =1 √ (1)
rb3/2 (r),
2

(10)

(j)

where 12 bs/2 is the Laplace coeﬃcient2
1
1 (j)bs/2 (r) =
2
2π

Z 2π
0

cos(jϕ)dϕ.
(1 + r2 − 2r cos ϕ)s/2(11)

There is also a polynomial expression (Pearson 2003, 2006):

where

∞
X
ωdyn
3
q
= √
r3/2
cn r2(n−1) ,
ωorb
4 1+q
n=1

(12)

2

n
Y
2
2m − 1cn = (2n)(2n + 1)
.
3
2m
m=1

(13)

The full polynomial formula for the dynamical precession rate can be written down as


ωdyn
q
3
15 2 175 4 3675 6 72765 8 693693 10 6441435 12
3/2
r +
r +
r +
r +
r + ... .
= √r
1+ r +
ωorb
4 1+q
8
64
1024
16384
1310721048576(14)

The convergence of this formula is rather slow and it requires 12–13 terms (i.e. up to r22 or r24 ) to obtain aprecision of 10−6 around the radius the 3:1 resonance.

2.2

Stage A superhumps and mass ratio

During stage A, it is considered that the superhump wave is conﬁned to the 3:1 resonance region and the thepressure eﬀect can be neglected [see ﬁgure 13 in Osaki and Kato (2013) or ﬁgure 4 in Niijima et al. (2021) for
schematic representations of the pressure eﬀect and the regions of the superhump wave]. One can substitute r
by the radius of the 3:1 resonance.
r3:1 = 3(−2/3) (1 + q)−1/3 ,(15)Then Q(q)R(r3:1 ) becomes a function of q and we can directly estimate q from ϵ∗ of stage A superhumps.
Originally in Kato and Osaki (2013), the equations (8) to (10) were combined and described asωdyn
q h 1 1 d  2 db1/2 iq h 1 1 (1) i
√
√ b
=√
r
=√
ωorb
dr
1 + q 4 r dr
1 + q 4 r 3/2(0)⇐ incorrect!

(16)

There was a typo introduced while writing down an equation in LATEX in equation (16) = equation (1) in
Kato and Osaki (2013), and the correction was made in Kato et al. (2016c). The correct equation isωdynq h 1 √ (1) iq h 1 1 d  2 db1/2 i
√
r
=√
rb3/2
=√
ωorb
dr
1 + q 4 r dr
1+q 4
(0)

⇐ correct.

(17)

The same incorrect equation was written in Kato et al. (2013b), Nakata et al. (2013) and Kato (2015). The ﬁgureand table dealing with this equation in Kato and Osaki (2013) were correct. No published q values by the stage
A superhump method were aﬀected by this typo.
The stage A superhump method is a dynamical method to determine q in that it relies only on celestialmechanics. The equation is analytical and no experimental calibration is needed. These features are clearlyadvantageous over the classical superhump methods such as the Patterson relations. For users’ convenience for
2 Please don’t be discouraged by a formula with an integral. Modern computer languages have functions for numerical integrations
and this integral can be very quickly computed.

6
Table 1: Relation between ϵ∗ of stage A superhumps and q.
ϵ∗
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
0.020
0.021
0.022
0.023

q
0.00000.00240.00490.00740.00990.01240.01490.01740.02000.02260.02520.02780.03040.03310.03580.03850.04120.04390.04660.04940.05220.05500.05780.0607

ϵ∗
0.024
0.025
0.026
0.027
0.028
0.029
0.030
0.031
0.032
0.033
0.034
0.035
0.036
0.037
0.038
0.039
0.040
0.041
0.042
0.043
0.044
0.045
0.046
0.047

q
0.0636
0.06650.06940.07230.07530.07820.08130.08430.08730.0904
0.0935
0.0966
0.09980.10290.10610.10940.11260.11590.11920.12250.12580.12920.13260.1361

ϵ∗
0.048
0.049
0.050
0.051
0.052
0.053
0.054
0.055
0.056
0.057
0.058
0.059
0.060
0.061
0.062
0.063
0.064
0.065
0.066
0.067
0.068
0.069
0.070
0.071

q
0.13950.14300.14650.15010.15370.15730.16090.16460.16830.17200.17580.17960.18340.18730.1912
0.19510.19910.20310.20720.21120.21540.21950.22370.2279

ϵ∗
0.072
0.073
0.074
0.075
0.076
0.077
0.078
0.079
0.080
0.081
0.082
0.083
0.084
0.085
0.086
0.087
0.088
0.089
0.090
0.091
0.092
0.093
0.094
0.095

q
0.2322
0.23650.24090.24530.24970.25420.25870.26330.26790.27250.27720.28200.28680.2916
0.29650.30140.30640.31140.31650.32170.32690.33210.33740.3428

ϵ∗
0.096
0.097
0.098
0.099
0.100
0.101
0.102
0.103
0.104
0.105
0.106
0.107
0.108
0.109
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
0.118
0.119

q
0.34820.35370.35920.36480.37050.37620.38200.38790.39380.39980.40580.41190.41810.42440.43070.43710.44360.45020.45680.46350.47030.47720.4842
0.4912interpolation, I provide an extended version of table 1 and ﬁgure 2 in Kato and Osaki (2013) in table 1 and ﬁgure
3. Note that the values are given for (probably) unrealistic values of ϵ∗ . There is also a polynomial expression ofthis relation [equation (7) in Warner (1995) = equation (3.41b) in Warner (1995)]:


ωdyn
0.146
0.044
1 q
0.433
0.013
+
+
+
...
.
(18)
=
1+
+
ωorb r=r3:1
41+q(1 + q)2(1 + q)2/3(1 + q)4/3(1 + q)8/3
I have conﬁrmed that this equation (up to this term) gives the same value in table 1 to a precision of 0.001 inϵ∗ . Although there is also a polynomial equation (4) in Kato and Osaki (2013), please do not cite this equation
since it is a regression, not an analytical formula.
This stage A superhump method has been applied to many SU UMa and WZ Sge stars, such as in a seriesof papers Kato et al. (2014b)–Kato et al. (2020). At the time of writing of the present paper, q values have been
determined by the stage A superhump method for more than 100 objects. There have been indirect estimationsof q values using Kato and Osaki (2013), such as a combination of the periods of stage A superhumps andpost-superoutburst superhumps (Kato et al. 2013b, 2019).

2.3

Relation between stage B superhump period and mass ratio

Using the empirical relation between fractional superhump excesses of stage A and stage B superhumps [equation(9) in Kato and Osaki 2013]:ϵ∗ (3 : 1) = 0.016(3) + 0.94(12)ϵ(stage B),(19)
where ϵ∗ (3 : 1) is the expected precession rate at the 3:1 resonance. This indirect method is helpful when the
periods of stage A superhumps are unknown but uncertainties resulting from an experimental calibration remain(the problem of ϵ=0 at q=0 in Patterson’s formulae is, however, avoided). Examples of applications of thismethod can be found in Pavlenko et al. (2021); Shugarov et al. (2021). These results are not included in theanalysis in this paper.

7

0.12

ε*=1−P orb/P SH(stageA)

0.10

0.08

0.06

0.04

0.02

0.00
0.0

0.1

0.2

0.3

q=M 2/M 1
Figure 3: Relation between ϵ∗ of stage A superhumps and q.

0.4

8
In this paper, I provide a new calibration since the number of calibrators has dramatically increased since
Kato et al. (2013b). Most of them are from a series of papers Kato et al. (2009a)–Kato et al. (2020) (hereafter“Pdot” papers; see section 3 for the detail). I also used ϵ∗ instead of ϵ since the former reﬂects the precession rate.
The calibrators are given in tables 2 (I removed one doubtful measurement of stage B superhumps ASASSN-16os)
for the q values obtained by the stage A superhump method and table 3 the q values obtained by the eclipse
modeling method. The PSH values in these tables refer to the mean period during stage B. When there wereseveral measurements of superhumps, I selected the best one (longest baseline or smallest error). I sometimes
averaged two measurements when the quality of the two were comparable (see individual reference for details of
the selection of the data).
Table 2: Calibrators for ϵ∗ (stage B)-q relation. The q values were measured by the stage A superhump method.
Object
V455 And
V466 And
V1838 Aql
VY AqrV342 CamHT CasBO CetWX CetGS Cet
HO CetPU CMaYZ CncGZ CncAL Com
V503 Cyg
V632 Cyg
V1006 Cyg
V1504 Cyg
V3101 Cyg
MN DraV529 Dra
IR Gem
V592 HerVW Hyi
GW Lib
QZ Lib
BR LupEZ LynV344 Lyr
V453 NorDT OctV681 Peg
UV PerTY PsABW Scl
V493 SerWZ SgeKK TelEK TrASW UMaER UMaKS UMaPorb (d)0.056310.05637(1)0.05706(2)0.06309(4)0.07531(8)0.07365
0.13984
0.05826
0.05597(3)0.05490(2)0.05669(4)0.08680.08825(28)0.05667
0.07776
0.06377(8)0.09903(9)0.06955
0.05347
0.0995(1)0.07168(1)0.06840.056100.07427
0.05332(2)0.06436(2)0.07948(2)0.059010.087900.06338(4)0.07271(1)0.05277(2)0.06489(1)0.08423(1)0.054320.08001(1)0.056690.0845(8)0.06288(5)0.056810.06366(3)0.06796(10)q (stage A)
0.080(4)0.083(4)0.095(4)0.106(12)0.164(4)0.167(2)0.323(13)0.094(1)0.078(2)0.090(4)0.110(11)0.168(5)0.30(2)0.090(5)0.218(5)0.106(3)0.34(2)0.172(2)0.0880(9)0.29(5)0.042(3)0.22(4)0.054(14)0.126(5)0.069(3)0.048(10)0.142(4)0.078(3)0.174(2)0.137(6)0.147(7)0.082(2)0.138(4)0.142(4)0.067(6)0.132(4)0.078(3)0.188(9)0.118(21)0.077(1)0.100(15)0.120(4)

PSH (stage B)
0.057133(10)0.057203(15)0.058382(10)0.064657(14)0.078399(36)0.076333(5)0.15069(3)0.059529(14)0.056645(14)0.055987(17)0.058033(33)0.090648(96)0.092699(56)0.057323(22)0.081446(96)0.065833(27)0.1076(1)0.072221(9)0.054182(4)0.105299(61)0.072342(18)0.071098(20)0.056607(16)0.076916(14)0.054095(10)0.064602(24)0.082241(37)0.059541(9)0.091594(8)0.064970(17)0.074755(19)0.053438(12)0.066671(10)0.087809(19)0.055000(8)0.082961(22)0.057204(5)0.087606(23)0.064832(6)0.058208(34)0.065747(24)0.070179(9)PSH reference
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2014a)
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2017)
Kato et al. (2021)
Kato et al. (2009a)
Kato et al. (2017)
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2014b)
Kato et al. (2014a)
Kato et al. (2014a)
Kato et al. (2014b)
Kato et al. (2009a)
Kato et al. (2016d)
Kato and Osaki (2013)
Tampo et al. (2020)
Kato et al. (2014a)
Kato et al. (2013a)
Kato et al. (2017)
Kato et al. (2010)
Kato et al. (2013a)
Kato et al. (2009a)
Pala et al. (2018)
Kato et al. (2015)
Kato et al. (2012a)
Kato et al. (2012a)
Kato et al. (2009a)
Kato et al. (2014a)
Kato et al. (2014b)
Kato et al. (2009a)
Kato et al. (2014b)
Kato et al. (2013a)
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2016b)
Kato et al. (2012a)
Kato et al. (2012a)
Ohshima et al. (2014)
Kato et al. (2009a)

9
Table 2: Calibrators for ϵ∗ (stage B)-q (by stage A superhump method) relation (continued).
ObjectV355 UMa
HV Vir
QZ Vir
ASAS J102522−1542.4
ASASSN-14cv
ASASSN-14jf
ASASSN-14jv
ASASSN-15bp
ASASSN-15gq
ASASSN-15hd
ASASSN-15na
ASASSN-15ni
ASASSN-15po
ASASSN-15pu
ASASSN-15uj
ASASSN-16bh
ASASSN-16bu
ASASSN-16da
ASASSN-16dt
ASASSN-16eg
ASASSN-16hj
ASASSN-16iw
ASASSN-16jb
ASASSN-16js
ASASSN-16oi
ASASSN-17bl
ASASSN-17ei
ASASSN-17el
ASASSN-17fn
ASASSN-17hw
ASASSN-18aan
CRTS J035905.9+175034
CRTS J112619.4+084650
CRTS J122221.6−311524
CRTS J174033.4+414756
CRTS J200331.3−284941
CRTS J214738.4+244554
Cze V404
GALEX J194419.33+491257.0
KSN:BS-C11a
LSPM J03338+3320
MASTER OT J005740.99+443101.5
MASTER OT J094759.83+061044.4
MASTER OT J181953.76+361356.5
MASTER OT J203749.39+552210.3
MASTER OT J211258.65+242145.4
OT J012059.6+325545
PNV J03093063+2638031
PNV J17144255−2943481
PNV J17292916+0054043
PNV J20205397+2508145
PNV J23052314−0225455

Porb (d)0.057290.057070.05882
0.06136(6)0.05992
0.05539(1)0.05442(1)0.05563(2)0.06490(3)0.05541(1)0.06297(2)0.05517(4)0.050450.05757(1)0.05527(1)0.05346(2)0.05934(13)0.05610(3)0.06420(2)0.07548(1)0.05499(6)0.06495(5)0.06305(2)0.06034(5)0.05548(7)0.05466(5)0.05646(1)0.05434(3)0.06096(1)0.05886(2)0.149450.07956
0.05423(3)0.07625(5)0.04503
0.05870
0.09273(3)0.098020.05282
0.05703(2)0.06663(7)0.056190.05588(9)0.05684(2)0.06051
0.05973
0.05715(2)0.05615(2)0.05956
0.05973(3)0.05651(1)0.05456(1)q (stage A)
0.066(1)0.093(3)0.108(3)0.120(5)0.077(1)0.070(5)0.074(3)0.079(2)0.107(8)0.076(12)0.081(5)0.074(2)0.0669(8)0.074(16)0.064(4)0.076(1)0.10(1)0.12(1)0.036(2)0.166(2)0.09(2)0.079(2)0.088(1)0.056(5)0.091(7)0.062(3)0.074(3)0.071(3)0.097(1)0.078(1)0.278(1)0.281(15)0.086(2)0.032(2)0.077(5)0.084(1)0.207(8)0.247(5)0.142(2)0.070(5)0.172(4)0.076(16)0.059(8)0.069(1)0.097(8)0.081(2)0.073(7)0.078(1)0.076(1)0.073(2)0.090(3)0.102(2)

PSH (stage B)
0.058094(7)0.058244(9)0.060442(15)0.063365(16)0.060413(7)0.055949(5)0.055102(13)0.056702(9)0.066726(34)0.056105(7)0.063720(27)0.055854(9)0.050913(2)0.058254(24)0.055805(12)0.054027(6)0.060513(71)0.057344(24)0.06451(1)0.077880(3)0.055644(41)0.065462(39)0.064397(21)0.060934(15)0.056241(17)0.055367(10)0.057257(11)0.055183(13)0.061584(14)0.059717(13)0.15821(4)0.08346(4)0.054886(10)0.076486(13)0.045515(29)0.059720(88)0.097147(21)0.10421(3)0.05479(1)0.05861(4)0.06902(2)0.057067(11)0.056121(20)0.057519(10)0.061307(9)0.060291(4)0.057729(8)0.057437(15)0.060092(9)0.060282(15)0.057392(10)0.055595(23)PSH reference
Kato et al. (2012a)
Kato et al. (2017)
Imada et al. (2017)
Kato et al. (2009a)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Namekata et al. (2017)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2017)
Kimura et al. (2018)
Wakamatsu et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2020)
Kato et al. (2020)
Kato et al. (2020)
Kato et al. (2020)
Wakamatsu et al. (2021)
Littleﬁeld et al. (2018)
Kato et al. (2014b)
Kato et al. (2020)
Imada et al. (2018b)
Kato et al. (2016b)
Kato et al. (2013a)
Kára et al. (2021)
Kato and Osaki (2014)
Ridden-Harper et al. (2019)
Kato et al. (2016a)
Kato et al. (2014a)
Kato et al. (2014b)
Kato et al. (2014b)
Nakata et al. (2013)
Nakata et al. (2013)
Imada et al. (2018a)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2020)
Kato et al. (2015)

10
Table 2: Calibrators for ϵ∗ (stage B)-q (by stage A superhump method) relation (continued).
ObjectSDSS J161027.61+090738.4
SDSS J162520.29+120308.7
TCP J23382254−2049518

Porb (d)0.05687(1)0.091430.05726(1)q (stage A)
0.090(5)0.23(1)
0.061(4)

PSH (stage B)
0.057820(19)0.096054(47)0.057868(14)PSH reference
Kato et al. (2010)
Kato et al. (2014b)
Kato et al. (2014a)

Table 3: Calibrators for ϵ∗ (stage B)-q relation. The q values were measured by the eclipse modeling method.
ObjectNZ BooOY CarHT CasZ Cha
GP CVnXZ Eri
V1239 Her
V2051 Oph
V4140 Sgr
DV UMaIY UMaPU UMaOU VirASASSN-14ag
CRTS J043112.4−031452
CRTS J200331.3−284941
SDSS J090350.73+330036.1
SDSS J115207.00+404947.8
SDSS J152419.33+220920.0

Porb (d)0.05891
0.063120.073650.07450
0.06295
0.061160.10008
0.06243
0.061430.08585
0.073910.07788
0.072710.060310.06605
0.058700.05907
0.06775
0.06532q (eclipse)
0.10990.10650.15
0.189
0.11150.1098
0.248
0.19
0.125
0.172
0.146
0.182
0.1641
0.149
0.105
0.095
0.113
0.153
0.17

Error
0.0007+0.0009/−0.0028
0.03
0.004
0.00160.0017
0.005
0.03
0.015
+0.002/−0.007+0.009/−0.001+0.009/−0.004
0.0013
0.016
0.006
0.004
0.004
+0.015/−0.011
0.03

PSH (stage B)
0.060463(13)0.064653(28)0.076333(5)0.077360(82)0.064604(29)0.062807(18)0.105005(56)0.064367(29)0.063510(43)0.088800(30)0.076210(25)0.081090(48)0.074962(132)0.062059(55)0.067583(26)0.059720(88)0.060364(50)0.070362(44)0.067136(23)

PSH reference
Kato et al. (2010)
Kato et al. (2009a)
Kato et al. (2017)
Kato et al. (2015)
Kato et al. (2012a)
Kato et al. (2009a)
Kato et al. (2013a)
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2010)
Kato et al. (2013a)
Kato et al. (2009a)
Kato et al. (2015)
Kato et al. (2012a)
Kato et al. (2016b)
Kato et al. (2010)
Kato et al. (2017)
Kato et al. (2009a)

The updated relation is shown in ﬁgure 4 (updated version of ﬁgure 9 in Kato and Osaki 2013; ϵ∗ is usedinstead of ϵ for stage B superhumps). The linear relation is now very clear (note that the measurements of stage
A and stage B superhumps are independent). The updated relation isϵ∗ (3 : 1) = 0.0140(11) + 1.11(4)ϵ∗ (stage B).

(20)

This relation can be safely used in all SU UMa-type dwarf novae. For user’s convenience I provide table 4 bothin reference to ϵ and ϵ∗ . However, when stage A superhumps are well observed, do net rely on stage
B and use table 1. The coeﬃcient for ϵ∗ (stage B) in equation (20) is close to unity, and if I assume it to be 1,
the equation becomesϵ∗ (3 : 1) = 0.0169(6) + ϵ∗ (stage B).
(21)
This equation is based on the assumption that ωpressure for stage B superhumps is constant regardless of q or ϵ.
The small deviation of the coeﬃcient from unity in equation (20) suggests that this is not a bad assumption. Twoﬁgures (5, 6) show this relation. The result looks very diﬀerent from ﬁgure 1 in Smak (2020). This was because
Smak (2020) used diﬀerent classes objects including (nearly) steady state AM CVn stars and novalike stars, andobjects with large errors. It is now apparent that the mass ratio of AM CVn (Roelofs et al. 2006) used in Smak(2020) had a large uncertainty. This is one of the reasons why I basically did not use the q values from Doppler
tomography for comparing the stage A superhump method and the eclipse modeling method (sections 3, 4).
Equation (20) would be meaningful only when stage A superhump are not well observed. Although thisformula depends on an experimental calibration, it is expected to be still better than Patterson’s formulae in thatit deals with the pressure eﬀect properly. The calibration was done only for hydrogen-rich dwarf novae; it is not11

ε*=1−P orb/P SH(stageB)

0.08

stage A superhumps
eclipse analysis

0.06

0.04

0.02

0.00
0.00

0.02

0.04

0.06

0.08

0.10

ε*(3:1) observed or estimated from q

Figure 4: Relation between ϵ∗ of stage B superhumps and dynamical precession rate = ϵ∗ (3:1) observed or
estimated from q. The dashed line indicates a linear ﬁt to the data.

12

ε*(stage B) − ε*(3:1)

0.06

0.04

0.02

0.00

stage A superhumps
eclipse analysis
−0.02
0.04

0.06

0.08

0.10

0.12

0.14

0.16

Figure 5: Relation between Porb and pressure eﬀect working on stage B superhumps. The dashed line indicatesthe relation equation (21).

13

ε*(stage B) − ε*(3:1)

0.06

0.04

0.02

0.00

stage A superhumps
eclipse analysis
−0.02
0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

q=M 2/M 1
Figure 6: Relation between q and pressure eﬀect working on stage B superhumps. The dashed line indicates therelation equation (21).

14

Table 4: Relation between ϵ of stage B superhumps and q. Note that this relation was experimentally calibrated
and not analytically derived. q < 0 is not realistic, but is given to estimate the ϵ and ϵ∗ values for q = 0.
ϵ−0.0130−0.0120−0.0110−0.0100−0.0090−0.0080−0.0070−0.0060−0.0050−0.0040−0.0030−0.0020−0.00100.00000.00100.00200.00300.00400.00500.00600.00700.00800.00900.01000.01100.01200.01300.01400.0150ϵ∗−0.0132−0.0121−0.0111−0.0101−0.0091−0.0081−0.0070−0.0060−0.0050−0.0040−0.0030−0.0020−0.00100.00000.00100.00200.00300.00400.00500.00600.00700.00790.00890.00990.01090.01190.01280.01380.0148q−0.002
0.001
0.004
0.007
0.010
0.012
0.015
0.018
0.021
0.024
0.027
0.030
0.033
0.036
0.039
0.042
0.045
0.048
0.051
0.054
0.057
0.060
0.063
0.066
0.070
0.073
0.076
0.079
0.082

ϵ
0.01600.01700.01800.01900.02000.02100.02200.02300.02400.02500.02600.02700.02800.02900.03000.03100.03200.03300.03400.03500.03600.03700.03800.03900.04000.04100.04200.04300.0440ϵ∗
0.01570.01670.01770.01860.01960.02060.02150.02250.02340.02440.02530.02630.02720.02820.02910.03010.03100.03190.03290.03380.03470.0357
0.0366
0.03750.03850.03940.04030.04120.0421

q
0.086
0.089
0.092
0.096
0.099
0.102
0.106
0.109
0.113
0.116
0.120
0.123
0.127
0.130
0.134
0.137
0.141
0.145
0.148
0.152
0.156
0.159
0.163
0.167
0.171
0.175
0.179
0.183
0.186

ϵ
0.04500.04600.04700.04800.04900.05000.05100.05200.05300.05400.05500.05600.05700.05800.05900.06000.06100.06200.06300.06400.06500.06600.06700.06800.06900.07000.07100.07200.0730ϵ∗
0.0431
0.04400.04490.04580.04670.0476
0.04850.04940.05030.05120.0521
0.05300.05390.05480.0557
0.0566
0.05750.05840.05930.0602
0.06100.06190.06280.06370.06450.06540.06630.06720.0680

q
0.190
0.194
0.198
0.202
0.207
0.211
0.215
0.219
0.223
0.227
0.232
0.236
0.240
0.245
0.249
0.253
0.258
0.262
0.267
0.271
0.276
0.281
0.285
0.290
0.295
0.300
0.304
0.309
0.314

ϵ
0.07400.07500.07600.07700.07800.07900.08000.08100.08200.08300.08400.08500.08600.08700.08800.08900.09000.09100.09200.09300.09400.09500.09600.09700.09800.09900.10000.10100.1020ϵ∗
0.06890.06980.0706
0.07150.07240.07320.07410.07490.07580.07660.07750.07830.07920.08000.08090.08170.08260.08340.08420.08510.08590.08680.08760.08840.08930.09010.09090.0917
0.0926

q
0.319
0.324
0.329
0.334
0.339
0.344
0.349
0.355
0.360
0.365
0.371
0.376
0.381
0.387
0.392
0.398
0.404
0.409
0.415
0.421
0.427
0.433
0.439
0.445
0.451
0.457
0.463
0.469
0.475

known how the strength the pressure eﬀect aﬀects the relation other than in hydrogen-rich dwarf novae (see e.g.
Pearson 2007). Please note that ϵ∗ (stage B) or ϵ(stage B) is not zero for q=0. ϵ(stage B) can be even negative(i.e. PSH can be shorter than Porb ; these superhumps are “negative superhumps” by deﬁnition!?) in systems withq < 0.04.
Since Knigge’s group published a slightly diﬀerent form of calibration (using ϵ instead; see section 6 for moredetails) between stage B superhumps and q, I provide a comparison of this type of formula using the same dataset (ﬁgure 7). There is a clear tendency of deviation in the region of q > 0.20. For systems with q < 0.20, therelation (dashed line in ﬁgure 7) is
q = 0.036(4) + 3.25(17)ϵ∗ (stage B).

(22)

Note that the calibration is valid only in the range of 0.003 < ϵ(stage B) < 0.045. The reason for the deviation forlarge q is evident: the precession rate at the radius of the 3:1 resonance, ϵ∗ (3 : 1), is not a linear function of q (seeﬁgure 3) and a regression should not be done between ϵ (or ϵ∗ ) and q. The relation between this calibration withPatterson’s formulae is shown in ﬁgure 8. Patterson’s formulae systematically give smaller q values for q < 0.1
because pressure eﬀect was not properly considered. The advantage of the treatment in equation (20) or ﬁgure
4 over Knigge-type treatment is now also obvious.

15

ε*=1−P orb/P SH(stageB)

0.08

stage A superhumps
eclipse analysis

0.06

0.04

0.02

0.00
0.00

Eq. (20), this work
Linear fit to q<0.20
McAllister et al. (2019)

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

q (stage A superhump/eclipse)
Figure 7: Relation between ϵ∗ of stage B superhumps and q. The dotted line indicates a linear ﬁt to the data for
q < 0.20 [corresponding to linear regressions used in Knigge (2006), but limited in the range]. The dashed curve
indicates a linear relation using ϵ in equation (26) = equation (2) in McAllister et al. (2019). The solid curve
represents the relation in equation (20).

16

ε(stageB)=P SH(stageB)/P orb − 1

0.10

0.08

)

on

al.
t
e

0.06

(

s
er

t

t
Pa

05
20

)

01

20
n(

rso

tte
Pa

0.04

0.02

0.00
0.00

0.05

0.10

0.15

0.20

0.25

0.30

q=M 2/M 1
Figure 8: Estimated relation between ϵ of stage B superhumps and q. The blue curve (note that it has a curvature)
represents equation (20) on the ϵ-q plane. Patterson’s formulae systematically give smaller q values for q < 0.1.

17

2.4

Application to other classes of binaries

There have been applications of the stage A superhump method to AM CVn stars (Kato et al. 2014c; Isogai et al.
2016, 2019; Han et al. 2021), but they are not treated in this paper. There has been the ﬁrst application of thestage A superhump method to the classical black-hole X-ray transient V3721 Oph = ASASSN-18ey = MAXI
J1820+070 (Niijima et al. 2021). This stage A superhump method is expected to be widely used in black-hole
X-ray transients in future in determining the black hole masses (this is complimentary to radial-velocity studies
combined with ellipsoidal modeling in quiescence), since the stage A superhump method is not aﬀected byinclinations, which usually have large uncertainties.

3

The Data for Comparison of the Eclipse Modeling Method and StageA Superhump Method

The q values determined by the stage A superhump method are listed in table 5. V627 Peg (Kato et al. 2015)
is not included in this list since Porb was not apparently determined reliably. The q values determined by themodern eclipse method are listed in table 6. Only hydrogen-rich dwarf novae are treated in this table.
By “modern eclipse method”, I mean detailed modeling of quiescent eclipses using modern equipment such as
Ultracam (Dhillon and Marsh 2001; Dhillon et al. 2007). Such a model was described as a form of decompositionof the eclipse light curve in Wood et al. (1986). Modern treatments can be found such as in Feline et al. (2004a);
Savoury et al. (2011). The model lcurve, which was a generalization of the code used in Horne et al. (1994), wasdescribed in Copperwheat et al. (2010) (see also Pyrzas et al. 2009). Feline et al. (2004a) used the classical leastχ2 ﬁnding algorithm Amoeba (downhill simplex, Press et al. 1986) in determining the parameters. In the later
series of analysis by the same group, such as Southworth et al. (2009); Copperwheat et al. (2010); Savoury et al.
(2011), this was taken over by the modern tool Markov-Chain Monte Carlo (MCMC) method (Ford 2006; Gregory2007) and the results by the MCMC method are probably more reliable (Savoury et al. 2011). McAllister et al.
(2017a) employed a parallel-tempered MCMC sampler (Earl and Deem 2005; Foreman-Mackey et al. 2013). Thisdetailed modeling of quiescent eclipses is now considered as the golden standard and q values derived by old
methods or by radial-velocity studies, which have large uncertainties, are excluded from the analysis in thispaper. I included HT Cas (Horne et al. 1991) despite that the q value was measured by the traditional method
since there is no reﬁned modern measurement (the hot spot is not always present in HT Cas; see Feline et al.
2005). I also included EX Dra (Fiedler et al. 1997) which had a relatively well-resolved eclipse light curve. Theq value of 0.59(17) for SDSS J075059.97+141150.1 (Southworth et al. 2010a) is too large for Porb =0.09317 d andis not used.
I apologize if there are omissions, although I extensively used NASA’s Astrophysics Data System (ADS). I
used only already published results in these tables.
Table 5: Mass ratios determined by stage A superhump method.
Object
V455 And
V466 And
V1838 Aql
VY AqrV342 CamHT CasBO CetWX CetGS Cet
HO CetZ Cha
PU CMaYZ CncGZ CncAL ComPorb (d)0.056310.05637(1)0.05706(2)0.06309(4)0.07531(8)0.07365
0.13984
0.05826
0.05597(3)0.05490(2)0.07450
0.05669(4)0.08680.08825(28)0.05667

q
0.080
0.083
0.095
0.106
0.164
0.167
0.323
0.094
0.078
0.090
0.22
0.110
0.168
0.30
0.090

Error
0.004
0.004
0.004
0.012
0.004
0.002
0.013
0.001
0.002
0.004
0.01
0.011
0.005
0.02
0.005

References
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2014a)
Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2017)
Kato et al. (2021)
Kato et al. (2009a)
Kato et al. (2017)
Kato et al. (2009a)
Kato et al. (2015)
Kato et al. (2009a)
Kato et al. (2014b)
Kato et al. (2014a)
Kato et al. (2014a)

18
Table 5: Mass ratios determined by stage A superhump method (continued).
Object
V503 Cyg
V632 Cyg
V1006 Cyg
V1504 Cyg
V3101 Cyg
MN DraV529 Dra
IR Gem
V592 HerVW Hyi
GW Lib
QZ Lib
BR LupEZ LynV344 Lyr
V453 NorDT OctV681 Peg
UV PerTY PsABW Scl
V493 SerWZ SgeKK TelEK TrASW UMaER UMaIY UMaKS UMaV355 UMa
HV Vir
QZ Vir
ASAS J102522−1542.4
ASASSN-14cv
ASASSN-14jf
ASASSN-14jv
ASASSN-15bp
ASASSN-15gq
ASASSN-15hd
ASASSN-15na
ASASSN-15ni
ASASSN-15po
ASASSN-15pu
ASASSN-15uj
ASASSN-16bh
ASASSN-16bu
ASASSN-16da
ASASSN-16dt
ASASSN-16eg
ASASSN-16hj
ASASSN-16iw
ASASSN-16jb

Porb (d)0.07776
0.06377(8)0.09903(9)0.06955
0.05347
0.0995(1)0.07168(1)0.06840.056100.07427
0.05332(2)0.06436(2)0.07948(2)0.059010.087900.06338(4)0.07271(1)0.05277(2)0.06489(1)0.08423(1)0.054320.08001(1)0.056690.0845(8)0.06288(5)0.056810.06366(3)0.07391
0.06796(10)0.057290.057070.05882
0.06136(6)0.05992
0.05539(1)0.05442(1)0.05563(2)0.06490(3)0.05541(1)0.06297(2)0.05517(4)0.050450.05757(1)0.05527(1)0.05346(2)0.05934(13)0.05610(3)0.06420(2)0.07548(1)0.05499(6)0.06495(5)0.06305(2)

q
0.218
0.106
0.34
0.172
0.0880
0.29
0.042
0.22
0.054
0.126
0.069
0.048
0.142
0.078
0.174
0.137
0.147
0.082
0.138
0.142
0.067
0.132
0.078
0.188
0.118
0.077
0.100
0.120
0.120
0.066
0.093
0.108
0.120
0.077
0.070
0.074
0.079
0.107
0.076
0.081
0.074
0.0669
0.074
0.064
0.076
0.10
0.12
0.036
0.166
0.09
0.079
0.088

Error
0.005
0.003
0.02
0.002
0.0009
0.05
0.003
0.04
0.014
0.005
0.003
0.010
0.004
0.003
0.002
0.006
0.007
0.002
0.004
0.004
0.006
0.004
0.003
0.009
0.021
0.001
0.015
0.006
0.004
0.001
0.003
0.003
0.005
0.001
0.005
0.003
0.002
0.008
0.012
0.005
0.002
0.0008
0.016
0.004
0.001
0.01
0.01
0.002
0.002
0.02
0.002
0.001

References
Kato et al. (2014b)
Kato et al. (2009a)
Kato et al. (2016d)
Kato and Osaki (2013)
Tampo et al. (2020)
Kato et al. (2014a); Ba̧kowska et al. (2017)
Kato et al. (2013a)
Kato et al. (2017)
Kato et al. (2010)
Kato et al. (2013a)
Kato et al. (2009a)
Pala et al. (2018); Kato et al. (2009a)
Kato et al. (2015)
Kato et al. (2012a)
Kato et al. (2012a)
Kato et al. (2009a)
Kato et al. (2009a, 2014a)
Kato et al. (2014b)
Kato et al. (2009a)
Kato et al. (2014b)
Kato et al. (2013a)
Kato et al. (2009a, 2016b)
Kato et al. (2009a)
Kato et al. (2016b)
Kato et al. (2010)
Kato et al. (2012a)
Ohshima et al. (2014); Kato et al. (2009a)
Kato et al. (2010)
Kato et al. (2009a)
Kato et al. (2012a)
Kato et al. (2017)
Imada et al. (2017); Kato et al. (2009a)
Kato et al. (2009a)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Namekata et al. (2017)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2016b)
Kato et al. (2017)
Kimura et al. (2018)
Wakamatsu et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)

19
Table 5: Mass ratios determined by stage A superhump method (continued).
ObjectASASSN-16js
ASASSN-16oi
ASASSN-16os
ASASSN-17bl
ASASSN-17ei
ASASSN-17el
ASASSN-17fn
ASASSN-17hw
ASASSN-18aan
CRTS J035905.9+175034
CRTS J104411.4+211307
CRTS J112619.4+084650
CRTS J122221.6−311524
CRTS J174033.4+414756
CRTS J200331.3−284941
CRTS J214738.4+244554
Cze V404
GALEX J194419.33+491257.0
KSN:BS-C11a
LSPM J03338+3320
MASTER OT J005740.99+443101.5
MASTER OT J094759.83+061044.4
MASTER OT J181953.76+361356.5
MASTER OT J203749.39+552210.3
MASTER OT J211258.65+242145.4
OT J012059.6+325545
PNV J03093063+2638031
PNV J17144255−2943481
PNV J17292916+0054043
PNV J20205397+2508145
PNV J23052314−0225455
SDSS J161027.61+090738.4
SDSS J162520.29+120308.7
TCP J00590972+3438357
TCP J20034647+1335125
TCP J23382254−2049518

Porb (d)0.06034(5)0.05548(7)0.05494(6)0.05466(5)0.05646(1)0.05434(3)0.06096(1)0.05886(2)0.149450.07956
0.05909(1)0.05423(3)0.07625(5)0.04503
0.05870
0.09273(3)0.098020.05282
0.05703(2)0.06663(7)0.056190.05588(9)0.05684(2)0.06051
0.05973
0.05715(2)0.05615(2)0.05956
0.05973(3)0.05651(1)0.05456(1)0.05687(1)0.091430.0543(1)0.05526(4)0.05726(1)

q
0.056
0.091
0.047
0.062
0.074
0.071
0.097
0.078
0.278
0.281
0.077
0.086
0.032
0.077
0.084
0.207
0.247
0.142
0.070
0.172
0.076
0.059
0.069
0.097
0.081
0.073
0.078
0.076
0.073
0.090
0.102
0.090
0.23
0.103
0.109
0.061

Error
0.005
0.007
0.003
0.003
0.003
0.003
0.001
0.001
0.001
0.015
0.001
0.002
0.002
0.005
0.001
0.008
0.005
0.002
0.005
0.004
0.016
0.008
0.001
0.008
0.002
0.007
0.001
0.001
0.002
0.003
0.002
0.005
0.01
0.006
0.005
0.004

References
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2017)
Kato et al. (2020)
Kato et al. (2020)
Kato et al. (2020)
Kato et al. (2020)
Wakamatsu et al. (2021)
Littleﬁeld et al. (2018)
Kato et al. (2010)
Kato et al. (2014b)
Kato et al. (2020); Neustroev et al. (2017b)
Imada et al. (2018b)
Kato et al. (2016b)
Kato et al. (2013a)
Kato (2021b); Kára et al. (2021)
Kato and Osaki (2014)
Ridden-Harper et al. (2019)
Kato et al. (2016a)
Kato et al. (2014a)
Kato et al. (2014b)
Kato et al. (2014b)
Nakata et al. (2013)
Nakata et al. (2013)
Imada et al. (2018a)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2015)
Kato et al. (2020)
Kato et al. (2015)
Kato et al. (2010)
Kato et al. (2010, 2014b)
Tampo et al. (2021)
Tampo et al. (2021)
Kato et al. (2014a)

Table 6: Mass ratios determined by modern eclipse modeling.
ObjectNZ BooOV BooOY CarHT CasV1258 Cen
V713 CepKN CetZ Cha
GY CncGP CVnPorb (d)0.058910
0.046258
0.063121
0.073647
0.088941
0.085419
0.052985
0.074499
0.175442
0.062950q0.10990.06470.1065
0.15
0.233
0.246
0.087
0.189
0.448
0.1115Error
0.00070.0018+0.0009/−0.0028
0.03
0.004
+0.006/−0.014
0.006
0.004
+0.014/−0.021
0.0016

References
Savoury et al. (2011)
Savoury et al. (2011)
McAllister et al. (2019)
Horne et al. (1991)
McAllister et al. (2019)
McAllister et al. (2019)
McAllister et al. (2015)
McAllister et al. (2019)
McAllister et al. (2019)
Savoury et al. (2011)

20
Table 6: Mass ratios determined by modern eclipse modeling (continued).
Object
EX DraXZ Eri
V1239 Her
V2051 Oph
V2779 Oph
IP PegDI PheV4140 Sgr
DV UMaIY UMaPU UMaOU VirNSV 4618
1RXS J180834.7+101041
ASASSN-14ag
ASASSN-16kr
ASASSN-17jf
CRTS J043112.4−031452
CRTS J052209.7−350530
CRTS J200331.3−284941
KIS J192748.53+444724.5
PN Te11SDSS J090350.73+330036.1
SDSS J100658.40+233724.4
SDSS J103533.02+055158.3
SDSS J105754.25+275947.5
SDSS J115207.00+404947.8
SDSS J143317.78+101123.3
SDSS J150137.22+550123.4
SDSS J152419.33+220920.0
SEKBO 106646.2532

4

Porb (d)0.209937
0.061159
0.100082
0.062428
0.070037
0.158206
0.065550
0.061430
0.085853
0.073909
0.077881
0.072706
0.065769
0.070037
0.060311
0.061286
0.056790
0.066051
0.062193
0.058704
0.165308
0.120972
0.059074
0.185913
0.057007
0.062792
0.067750
0.054241
0.056841
0.065319
0.059579q
0.75
0.1098
0.248
0.19
0.168
0.48
0.1097
0.125
0.172
0.146
0.182
0.1641
0.169
0.168
0.149
0.044
0.085
0.105
0.055
0.095
0.570
0.236
0.113
0.46
0.0571
0.05460.153
0.0661
0.084
0.17
0.114

Error
0.03
0.0017
0.005
0.03
0.016
0.01
0.00080.015+0.002/−0.007+0.009/−0.001+0.009/−0.004
0.0013+0.011/−0.006
0.016
0.016
0.002
0.006
0.006
0.003
0.004
0.011
0.006
0.004
0.03
0.00100.0020+0.015/−0.011
0.0007
0.004
0.03
0.005

References
Fiedler et al. (1997)
Feline et al. (2004a)
Savoury et al. (2011)
Baptista et al. (1998)
Southworth and Copperwheat (2011)
Copperwheat et al. (2010)
Savoury et al. (2011)
Borges and Baptista (2005)
McAllister et al. (2019)
McAllister et al. (2019)
McAllister et al. (2019)
Savoury et al. (2011)
McAllister et al. (2019)
Southworth and Copperwheat (2011)
McAllister et al. (2017a)
Wild et al. (2022)
Wild et al. (2022)
McAllister et al. (2019)
Wild et al. (2022)
McAllister et al. (2019)Littlefair et al. (2014)
Miszalski et al. (2016)
Savoury et al. (2011)
McAllister et al. (2019)
Savoury et al. (2011)
McAllister et al. (2017b)
McAllister et al. (2019)
Savoury et al. (2011)
McAllister et al. (2019)
Southworth et al. (2010a)
McAllister et al. (2019)

Comparison between Eclipse Modeling Method and Stage A Superhump Method

Only three objects (HT Cas, Z Cha and IY UMa) are common to tables 5 and 6. XZ Eri, V1239 Her and DV
UMa, which were used in Kato and Osaki (2013), have been excluded from this list since it has become apparent
that stage A superhumps were observed only for a short segment [see ﬁgure 87 in Kato et al. (2009a) and ﬁgures
52 and 153 in Kato et al. (2013a); the values given in Kato et al. (2009a, 2013a) should be regarded as the lower
limit]. The adopted objects have Porb longer than 0.07 d and they are not very favorable targets for the stage
A superhump method, except HT Cas, which was very extensively observed during the 2017 superoutburst (seeﬁgure 9 in Kato et al. 2012a). This is because the accuracy of q by the stage A superhump method depends on the
duration of stage A, which is short in long-Porb systems. The presence of eclipses also complicates determinationof the superhump period (especially when eclipses overlap superhump maxima, the superhump period cannot be
reliably determined). Due to these factors, this direct comparison between the results of the eclipse modelingmethod and the stage A superhump method is not very fruitful as of now. This situation is expected to be
improved by an accumulation of continuous short-cadence observations such as by the Transiting Exoplanet
Survey Satellite (TESS).
The lack of short-period objects common to these two methods was due to the low frequency of outbursts

21

0.25

Z Cha

q (from stage A superhumps)

0.20

HT Cas0.15

IY UMa0.10

WZ Sge

0.05

0.00
0.00

OV Boo

0.05

0.10

0.15

0.20

0.25

q (from eclipses etc.)
Figure 9: Comparison of q measured by the eclipse modeling method (abscissa) and q estimated from the stage
A superhump method (ordinate). The q value for WZ Sge was determined spectroscopically by Steeghs et al.
(2007). We also used unpublished data for OV Boo (Ohnishi et al. 2019).
in the objects analyzed by the eclipse modeling method. This is because eclipse modeling is suitable for lowmass transfer systems (such as WZ Sge stars) which is less disturbed by ﬂickering and by the light from theaccretion disk (and probably because hunting brown-dwarf secondaries was one of the primary targets by the
eclipse modeling method). The stage A superhump method requires observations during superoutbursts. Suchoutbursts in WZ Sge stars usually occur once in decades and time will tell whether the stage A superhump
method gives the same q values for individual WZ Sge stars measured by the eclipse modeling method.
I nevertheless provide table 7 and ﬁgure 9 for comparison. In order to increase the sample size, I used the
dynamical constraints for WZ Sge by spectroscopic observations by Steeghs et al. (2007). Steeghs et al. (2007)
implied q=0.092(8) based on the gravitational redshift, which would give a smaller error bar shown in this ﬁgure.
Steeghs et al. (2007) also stated that q=0.050(15) from superhump observations by Patterson et al. (2005a) is too
small and that this cannot be a reliable independent determination. This is exactly what I described in section
2 (Patterson’s formulae give systematically small q values for low-q systems). I also included still unpublishedsuperhump observations of OV Boo (Ohnishi et al. 2019), which recorded the entire superoutburst with ideal
coverage. Let’s hope that this OV Boo paper will come into the world soon.

22

Table 7: Comparison of q values determined by the eclipse modeling method and the stage A superhump method.
ObjectOV BooWZ SgeHT CasIY UMaZ Cha

5

Porb (d)0.046258
0.056688
0.073647
0.073909
0.074499

q (eclipse)
0.0647
0.092
0.15
0.146
0.189

Error
0.0018+0.009/−0.017
0.03+0.009/−0.001
0.004

q (stage A)
0.066
0.078
0.167
0.120
0.22

Error
0.001
0.003
0.002
0.006
0.01

Distributions on the Porb -q Plane

In ﬁgures 1 and 10, I show the distribution of the measurements on the Porb -q plane. The stage A superhump
method and the eclipse modeling method give the same distribution. There are relatively large scatters for Porb
longer than 0.07 d. This is because long-Porb systems are not ideal targets both for the stage A superhump
method and the eclipse modeling method. The reason for the stage A superhump method was already describedin section 4. The duration of stage A is short for these systems and stage A tends to be contaminated by stage B(giving smaller q values), if observations are not made suﬃciently early (this problem can be avoided if continuous
short-cadence observations such as by TESS become available). The reason for the eclipse modeling method isthat these objects have high mass-transfer rates and the proﬁles of eclipses tend to be blurred or the light curveis dominated by ﬂickering, although this is alleviated by using Gaussian process modeling (McAllister et al.
2017a). This eclipse modeling also requires the presence of the hot spot. In Feline et al. (2005), q values were
not determined for IR Com and HT Cas due to the lack of the hot spot.
The two unusual objects plotted on ﬁgure 1 are ASASSN-15po and GALEX J194419.33+491257.0. The
former is an object having Porb below the period minimum and may not be on the standard evolutionary path ofCVs (Namekata et al. 2017). The latter is an unusually active SU UMa star with very short recurrence times andis suspected to be a CV with a stripped core evolved secondary (Kato and Osaki 2014). Additional two unusual
object plotted on ﬁgure 10 are CRTS J174033.4+414756, which is also considered to be a CV with a stripped core
evolved secondary (Chochol et al. 2015; Imada et al. 2018b) and OV Boo, which is considered to be a population
II CV (Patterson et al. 2017).

Conclusion of this section:
Both ﬁgures now clearly show that the eclipse modeling method and the stage A superhump method
give the same distribution. This means that the stage A superhump method is as reliable as the
eclipse modeling method.

6

CV Evolution and Period Minimum

It is well known that the distribution of CV has a period minimum (Paczyński and Sienkiewicz 1981). Aroundthis point during the course of the CV evolution, the Porb increases mainly due to two reasons: the thermal
time-scale (Kelvin-Helmholtz time) of the secondary exceeds the mass-transfer time-scale and the mass-radiusrelation is reversed for degenerate dwarfs [modern works have shown that the picture is a bit more complex,
see e.g. Kolb and Baraﬀe (1999); Araujo-Betancor et al. (2005b); Knigge et al. (2011)]. For hydrogen-rich CVs,
Paczyński and Sienkiewicz (1981) suggested a period of 74 min (0.051 d) for M1 =0.5M⊙ and 87 min (0.060 d)
M1 =1.0M⊙ . Paczyński and Sienkiewicz (1981) argued that these periods are remarkably close to the observed
period of 81 min (0.056 d) for WZ Sge. Although early models indicated the period between 60 and 80 min(0.042 and 0.060 d) (Paczyński and Sienkiewicz 1981; Rappaport et al. 1982; Paczyński and Sienkiewicz 1983),
later reﬁnement of the model yielded signiﬁcantly shorter values of 65–70 min (0.045–0.049 d) (Kolb and Baraﬀe
1999; Howell et al. 2001). This was apparently not in agreement with the observation (e.g. Kolb 1993), andthis problem was called the “period minimum problem”. Population synthesis studies expected that most CVs
have already reached the period minimum (Kolb 1993; Howell et al. 1997) and that here should be a heavy
accumulation of systems around the period minimum (period spike or period minimum spike), since the dropin the mass-transfer rate slows down the CV evolution. At that time, such a spike was not apparent and this23

q=M 2/M 1

0.3

0.2
J1944190.1
J174033

stage A superhumps
eclipse analysis

OV Boo

0.0
0.05

0.06

0.07

0.08

0.09

0.10

0.11

Figure 10: Mass ratios (q) versus orbital periods (Porb ) determined by the eclipse modeling method and thestage A superhump method. The dashed and solid curves represent the standard and optimal evolutionary tracks
in Knigge et al. (2011), respectively. Three unusual objects are also plotted: OV Boo (Patterson et al. 2017),
which is a population II CV, ASASSN-15po (Namekata et al. 2017) and GALEX J194419.33+491257.0 (J194419)(Kato and Osaki 2014).

24
disagreement was called the “period spike problem” (Kolb and Baraﬀe 1999; Renvoizé et al. 2002). There was an
idea that the mass-transfer rates rapidly decrease as CVs approach the period minimum and that they become
increasingly diﬃcult to ﬁnd by detections of outbursts. Uemura et al. (2010) assumed that the recurrence time ofoutbursts near the period minimum follows an exponential law and explored whether there could be a “missing”
population of CVs below the shortest known orbital period.
Systematic spectroscopic and photometric follow-up observations of the CV candidates selected by the Sloan
Digital Sky Survey (SDSS: York et al. 2000) (Szkody et al. 2002, 2003a,b, 2004, 2005, 2006, 2007, 2009, 2011, 2018;
Southworth et al. 2006, 2007, 2008a,b, 2009, 2010a,b, 2015; Wolfe et al. 2003; Homer et al. 2006; Dillon et al.
2008; Schmidt et al. 2008; Hilton et al. 2009; Skinner et al. 2011; Woudt and Warner 2004, 2010; Woudt et al.
2004, 2012; Rebassa-Mansergas et al. 2014; Thorstensen et al. 2015; this list includes other types of CVs nottreated in this paper) answered these issues. Gänsicke et al. (2009) collected SDSS-selected and other CVs andclariﬁed the presence of the period spike. Selection of CVs using the SDSS was advantageous in that it did not
require outburst detections and it was considered that this sample was less biased than the traditional one. Thereremained, however, a possibility that systems with very low-mass transfer rates can have colors indistinguishable
from white dwarfs and their number might have been underestimated (subsubsection 5.2.1 in Gänsicke et al.
2009; Kato et al. 2012b). Gänsicke et al. (2009) selected GW Lib [Porb =0.05332(2) d or 76.78 min] as theshortest-period “standard” hydrogen-rich CV. Gänsicke et al. (2009) reported the period spike of 82.4(7) min= 0.0572(5) d with a FWHM of 5.7±1.7 min = 0.0040(12) d. Considering the width of the period spike, aperiod of 80 min = 0.0556 d was considered to be an estimate for the period minimum. Gänsicke et al. (2009)
stated that the theoretically calculated period minimum is too short by about 10 min. This diﬀerence could bereconciled if either the orbital braking is about four times the value provided by gravitational wave radiation(Kolb and Baraﬀe 1999), or if the theoretical models underestimate the stellar radius for a given mass by about
20% (Barker and Kolb 2003).
Knigge (2006) used superhumping CVs (SU UMa stars and permanent superhumpers) and eclipsing CVs to
explore the mass-radius relation for the secondary stars in CVs. Knigge (2006) reported a period minimum of
76.2±1.0 min = 0.0529(7) d. It might be worth noting that Knigge (2006) used the originally calibrated formula
between q and ϵ (stage B for SU UMa stars):q(ϵ) = (0.114 ± 0.005) + (3.97 ± 0.41) × (ϵ − 0.025),(23)

using the same set of calibrators as in Patterson et al. (2005a). This formula is better than equation (6), therelation by Patterson et al. (2005a), in that it allows q ̸= 0 at ϵ = 0. For example, q=0.015 is obtained for ϵ = 0
by this equation (compare with ﬁgure 8).
Knigge et al. (2011) studied the evolution of CVs in detail considering various eﬀects. Using the material
in Knigge (2006), Knigge et al. (2011) concluded that the best-ﬁt model for short-Porb systems requires angularmomentum losses (AML) 2.47±0.22 larger than what is expected by gravitational wave radiation (optimal evolutionary track). This optimal evolutionary track and standard evolutionary track (angular momentum losses
exactly by gravitational wave radiation) are plotted on ﬁgures 10 and 1. Knigge et al. (2011) derived the theoretical period minimum of 73.2 min = 0.0508 d and 81.8 min = 0.0568 d for the standard and optimal evolutionary
tracks, respectively.
The value of the period minimum depends on the deﬁnition. Kato et al. (2017) used Porb values estimatedfrom PSH for all SU UMa/WZ Sge stars and derived a sharp cut-oﬀ at 0.052897(16). The used relations between
Porb and PSH (stage B or C) were in Kato et al. (2012a), but repeated here for easier reference:ϵ(stage B) = 0.000346(36)/(0.043 − Porb ) + 0.0443(21)(24)

ϵ(stage C) = 0.000273(24)/(0.044 − Porb ) + 0.0381(13).

(25)

and
These equations have smaller residuals (1-σ error of 0.0003 d) compared to older compared to the one inStolz and Schoembs (1984), which had a systematic error.
McAllister et al. (2019) used all SU UMa/WZ Sge stars as in Kato et al. (2017) and eclipsing CVs and derived
the period minimum (deﬁned as the period spike as in Gänsicke et al. 2009) of 79.6(2) min = 0.0553(1) d with a
FWHM of 4.0 min = 0.003 d. McAllister et al. (2019) claimed that they derived new Knigge-type calibrationsof the relationship between ϵ and q [they were probably unaware of Kato and Osaki (2013)], namely,q(ϵB ) = (0.118 ± 0.003) + (4.45 ± 0.28) × (ϵB − 0.025)

(26)

25
and
q(ϵC ) = (0.135 ± 0.004) + (5.0 ± 0.7) × (ϵC − 0.025),(27)

where ϵB and qB represent stage B superhumps and ϵC and qC represent stage C superhumps. Although therelation for stage B was greatly improved compared to Patterson (2001) or Patterson et al. (2005a), this relation
has is inferior to equation (20) since the ϵ-q relation is not linear (see subsection 2.3). McAllister et al. (2019)
also wrote: “While there is good coverage for systems with 0.1 < q < 0.2, more calibration systems with q outside
this range are required in order to further constrain the gradient.” — This is because systems with q < 0.1 areusually WZ Sge stars (or borderline SU UMa/WZ Sge stars) and they usually do not show prominent stage C
superhumps; hence the equation for stage C superhumps is not necessary for q < 0.1. In any case, the departureof the period minimum from the theoretically calculated one was also apparent.
Wild et al. (2022) added three examples and evaluated the mass-radius relation in Knigge et al. (2011). They
also used superhumps probably as in the way as in McAllister et al. (2019), but the details and references were
not shown. Wild et al. (2022) used our superhump periods of MASTER OT J220559.40−341434.9 = ASASSN16kr (Kato et al. 2017) and obtained q=0.059(7) using the relation in McAllister et al. (2019). Wild et al. (2022)
suggested that this may be preliminary evidence that the ϵ-q relation may overestimate q for CVs at shortperiods. I looked at the data of this object again and found that the data were not adequate for making such a
comparison. There were observations by Monard on four nights and Hambsch on four nights, the latter with a
low time resolution, and these observations covered the ﬁnal part of the superoutburst plateau (the observations
started 12 d after the initial detection of the superoutburst). As stated in Kato et al. (2017), most superhumps
were aﬀected by overlapping eclipses and orbital humps and the period should be regarded as on approximate.
The unusually [written as “usual” in Kato et al. (2017), which was a typo] large period derivative also indicatedthat the period was not well determined. This object should not be used as a ϵ-q calibrator until we have bettermeasurements of superhumps in the early part of a superoutburst in future. Wild et al. (2022) found a relation
between Pex = Pobs,orb − Pmodel,orb and M2 , which is a measure of excess AML against the optimal evolutionary
sequence in Knigge et al. (2011). Based on the complex behavior, Wild et al. (2022) wrote: “The ‘optimal’ tracks
add an extra source of AML that takes the form of 1.5 times the gravitational wave breaking. By examining theperiod excess between the growing set of observed CV donor radii and models, we demonstrate that this does notproperly describe the missing AML.” Although I did not examine the matter in detail, a systematic departure ofthe (linear) ϵ-q relations from the non-linear (correct) one might have complicated the problem [ﬁgure 7 suggeststhat the relation in McAllister et al. (2019) tends to overestimate q for 0.15 < q < 0.25 and underestimate q for
q < 0.10].
Upon a closer look at ﬁgure 1 or ﬁgure 10 in McAllister et al. (2019), it is apparent that the majority ofobjects on the Porb -q (or Porb -M2 ) plane near the period minimum have shorter Porb than the optimal evolutionary
track in Knigge et al. (2011), but still longer than the standard evolutionary track. It is also apparent that Porb
values around the period minimum are widespread and some of the objects apparently have Porb longer than the
optimal evolutionary track. Although the presence of objects with Porb longer than the optimal evolutionary track
is more apparent in q values measured from superhumps, there are indeed some systems measured by the eclipse
modeling method and they look like to be real. This suggests that the “period minimum” is more widespread (inPorb ) than has been thought. This implies that a given object passes a broad range (such as 0.052–0.060 d) of
the period minimum rather than evolving through a narrow, ﬁxed period. The reason of the spread is unknown.
The errors in q estimates are unlikely the main source, since the spread in Porb is least aﬀected by the errors of q
near the period minimum. Diﬀerent chemical compositions of the secondaries would be a cause. Diﬀerent degrees
of AML other than gravitational wave radiation may be the cause, although consequential angular momentum
loss (CAML) such as caused by nova eruptions (Schreiber et al. 2016) is expected to be less important aroundthe period minimum (McAllister et al. 2019).
In order to obtain the new “optimal” evolutionary track around the period minimum, I linearly interpolated
the standard and optimal evolutionary tracks in Knigge et al. (2011). I write the two tracks with functional formsofPorb (standard, optimal) = fstandard,optimal (q),(28)
using tables 3 and 4 in Knigge et al. (2011). I then minimizedi2Xh
Porb,i − cfoptimal (qi ) + (1 − c)fstandard (qi )

(29)

i

by changing c, where Porb,i and qi represents values for individual objects. c = 0 corresponds to the standard
evolutionary track and c = 1 corresponds to the optimal evolutionary track in Knigge et al. (2011). I have ex-

26

Table 8: Parameters for new optimal evolutionary track.
MethodStage A superhumps
Eclipse modeling
Combined

Number of objects
70
19
89

Optimal c
0.91
0.91
0.91

cluded ASASSN-15po, CRTS J174033.4+414756 and GALEX J194419.33+491257.0 (for the stage A superhummp
method) and OV Boo (for the eclipse modeling method) and the samples were limited to q > 0.05334 (lower
limit of the evolutionary track), q < 0.16 and Porb < 0.07 (d). The results are shown in table 8. The optimalvalue of c is in excellent agreement between the stage A superhummp method and the eclipse modeling method(again conﬁrming the reliability of the stage A superhummp method). I adopted c=0.91 using the combined
data. The means that the “new” optimal evolutionary track is closer to the original optimal evolutionary track
than to the standard evolutionary track. Considering that Porb at the period minimum is roughly proportional
to AML1/3 [equation (16) in Paczyński (1981) or equation (2) in Paczyński and Sienkiewicz (1983)], the angularmomentum loss is estimated to be 1.9 times larger than what is expected by gravitational wave radiation. The
period minimum on this track is 0.0562 d = 81.0 min. The can be compared to 81.8 min = 0.0568 d for the
optimal evolutionary track by Knigge et al. (2011). The new optimal evolutionary track is shown as a red curve
in ﬁgure 11.

7

Period Bouncers

There have been mainly two sources of (candidate) period bouncers. One is by eclipse modeling starting from thediscovery of SDSS J103533.02+055158.3 (Littlefair et al. 2006). The other is by superhump methods calibrated
by various authors. Patterson (1998) was probably the ﬁrst to provide a list of candidate period bouncers. There
have been a number of candidate since then and Patterson (2001) provided a list of candidates from various aspects(including those discovered by the eclipse modeling method). Since the establishment of the stage A superhump
method in 2013 and the discovery of double-superoutburst object with q=0.04 (Kato and Osaki 2013), there havebeen a growing number of candidate period bouncers by this method. It became evident that the type of WZ
Sge-type rebrightening is related to the evolutionary stage (see the next section for more details), and outburst
parameters became useful in selecting period bouncers. A list of candidate period bouncers from superhump
observations and outburst characteristics was given in Kimura et al. (2016) and updated in Kimura et al. (2018).
I list these two references since researchers using the eclipse modeling method did not appear to be aware of theseworks, but was properly referenced by Thorstensen (2020).
Patterson (2001) estimated that the period bounce occurs at a mass of 0.058(8)M⊙ , which was likely anunderestimate considering the systematic error in the ϵ-q relation (section 2). McAllister et al. (2019) estimatedthat 30% of their sample (including superhumpers) appear to be post-period minimum systems. They derived
the mass of the secondary at the period bounce to be Mbounce =0.063+0.005−0.002 M⊙ . This value can be comparedto the lower limit of hydrogen burning (Kumar’s limit: Kumar 1963) of 0.076(5)M⊙ (Henry et al. 1999) (fromobservations) and slightly below 0.072M⊙ (Chabrier and Baraﬀe 1997) (from theoretical calculation) for isolated
population I objects. Using their average mass (0.81±0.02M⊙ ) of white dwarfs below the period gap, thiscorresponds to q=0.078 (with an error of an order of 0.003). Using this value and ignoring all errors, 33 out of
103 objects determined by the stage A superhump method are expected to be post-period minimum systems.
The fraction is in very good agreement with McAllister et al. (2019).
The distribution of M2 determined by the stage B superhump method, ignoring all errors, is shown inﬁgure 12. It is worth noting the existence of a sharp peak around the period minimum (and perhaps near thehydrogen-burning limit). The number density is expected to be higher below this limit since the evolutionary
time scale becomes signiﬁcantly longer. The sharp drop of the number density below the period minimum may
be a reﬂection of the sharply decreasing mass-transfer rate, which makes outbursts infrequent and diﬃcult toﬁnd by outburst detections. Consequently, it would be natural to think that there are many “dormant” period
bouncers which still await our detections.
The chronology of period bouncers and brown-dwarf secondaries, together with the development of thetechniques, is listed in table 9. This table would be useful for searching which period bouncers were reported

27

q=M 2/M 1

0.15

J194419

0.10

ASASSN−15po

New optimal
evolutionary
track

0.05

Symbols:

stage A superhump method

0.050

0.055

eclipse modeling method

0.060

0.065

0.070

Figure 11: New optimal evolutionary track (red solid curve) plotted on the relation between mass ratios (q)and orbital periods (Porb ) determined by the eclipse modeling method and the stage A superhump method,
enlargement around the period minimum. The other symbols are the same as in ﬁgures 1 and 10.

28

15
10
5
0

Frequency

periodbouncers

0.05

0.10

0.15

0.20

0.25

M 2 (M solar)
Figure 12: Distribution of M2 determined by the stage B superhump method. An average white dwarf mass of
0.82M⊙ was assumed. A vertical dashed line represents the location of the period bounce (M2 = 0.063M⊙ ).
ﬁrst and what method was used for them. This table includes polars with brown-dwarf secondaries. SED stands
for spectral energy distribution in the table.
Table 9: Chronology of period bouncers and brown-dwarf secondaries.
Year

Superhump method, Porb , others
Boldface indicates stage A superhump method

1986
1993
1994
1996

1997

Eclipse modeling
Boldface indicates period bouncers
Decomposition of eclipse light curve of Z Cha
(Wood et al. 1986).

Smak (1993) suggested M2 =0.06M⊙ for WZ Sge.
Eclipse modeling of OY Car (Horne et al. 1994).
AL Com (Porb =0.05667 d, 2:1 resonance signal?
in quiescence, M2 < 0.04M⊙ ) (Patterson et al.
1996); q=0.090(5) from stage A superhumps(Kato and Osaki 2013; Kato et al. 2014a), not verysmall as suggested in Patterson et al. (1996).
Howell et al. (1997) suggested that EF Peg could
be the longest Porb (0.084 d) period bouncer, giving
a tentative lower limit to the age of the Galaxyof 8.6×109 yr, rejected by himself in Howell et al.
(2002).

29
Table 9: Chronology of period bouncers and brown-dwarf secondaries (continued).
Year
1998

1999

2000
2001

2004

Superhump method, Porb , others
Boldface indicates stage A superhump method
EG Cnc (Porb =0.0599 d, very small ϵ=0.0066(13),multiple rebrightenings) (Patterson et al. 1998);
SED: Neustroev et al. (2017a); Kimura et al.
(2021) obtained most likely q ∼ 0.048 and also constraint q < 0.057 from superhumps.
..................................................
“Late Evolution of Cataclysmic Variables” listed DI
UMa, AL Com, WZ Sge, EG Cnc as candidates
having small ϵ (Patterson 1998); but gave systematically small q, such as q=0.02–0.04 for WZ Sge
due to the ϵ-q calibration problem described in subsection 2.3 in this paper.
DI UMa (ϵ=0.013) (Fried et al. 1999), see also
Rutkowski et al. (2009) (doubtful consideringlarge superhump amplitudes; photometric Porb
might not be true Porb in ER UMa star).
..................................................
V592 Her (very faint absolute magnitude)(van Teeseling et al. 1999); Small ϵ consistent
with a brown dwarf secondary (Mennickent et al.
2002); could be a lower main-sequence secondary?
(Kato et al. 2002); q=0.054(12) based on stage
A superhumps (Kato et al. 2010; Kato and Osaki2013), conﬁrming the brown dwarf nature.
EF Eri (polar, spectrum) (Beuermann et al. 2000).
EF
Eri(polar,infrared
spectrum)(Howell and Ciardi 2001); also claimed thedetection of a brown dwarf-like secondary in LL
And, refuted by Littlefair et al. (2003).
..................................................
1RXS J105010.3−140431 (Porb =0.0615 d, verysmall mass function) (Mennickent et al. 2001).
..................................................
“Accretion-Disk Precession and Substellar Secondaries in Cataclysmic Variables”: collection of
data; no new objects; also included AM CVn stars(Patterson 2001).
IX Dra [ϵ=0.076(3)] (Olech et al. 2004) [probably
incorrect; confusion with stage C superhumps?
Kato et al. (2009a)]; spectroscopic Porb =0.0648 dﬁnally excluded the possibility of a period bouncer(Thorstensen 2020).
..................................................
V379
Vir(polar,spectrum)2004),
seealso(Farihi and Christopher
Burleigh et al.
(2006);
Farihi et al.
(2008);
Breedt et al. (2012).

Eclipse modeling
Boldface indicates period bouncers

Ultracam:(2001).

description by Dhillon and MarshParameter determination of XZ Eri, DV UMa andOU Vir with ultracam and lcurve (Feline et al.
2004a,b).

30
Table 9: Chronology of period bouncers and brown-dwarf secondaries (continued).
Year
2005

2006

Superhump method, Porb , others
Boldface indicates stage A superhump method
RX
J0502.8+1624(polar,spectrum)(Littlefair et al. 2005).
..................................................
V455 And (Porb =0.05631 d, infrared spectrum,
secondary later than L2 by SED modeling)(Araujo-Betancor et al. 2005a).
..................................................
PQ
And(Porb =0.0559
d,
faint
MK )
(Patterson et al. 2005b); ϵ not yet measured.
..................................................
MT Com (Porb =0.0829 d?, proposed as counterpart of EUV transient RE J1255+266, faintMV =14.6±1.3) (Patterson et al. 2005c); the exact
nature still unknown.
SED: Neustroev et al.
(2017a).
Gaia parallax gave MV =+11.5(Gaia Collaboration et al. 2021).
V406 Vir (Porb =0.0559 d, suggested by resemblance to other WZ Sge stars) (Zharikov et al.
2006); infrared data suggested an L4-type browndwarf (Aviles et al. 2010); large superhump amplitudes during the 2017 superoutburst made verysmall q unlikely (Kato et al. 2020).
..................................................
SDSS J121607.03+052013.9 (Porb =0.0686 d or its
one-day alias, small mass function, 0.012 < M2 <0.040M⊙ ) (Southworth et al. 2006).
..................................................
“The donor stars of cataclysmic variables” (Knigge2006): used superhumps listed in Patterson et al.
(2005a); linear calibration on the ϵ-q plane; showeda jump in mass-radius relation at period gap.

2007

2008

GD 552 = NSV 25966 (long Porb )=0.07134 d,no sign of M-type secondary (Unda-Sanzana et al.
2008). SED: Neustroev et al. (2017a).
..................................................
EZ Lyn (q=0.05 from superhumps) (Zharikov et al.
2008); multiple rebrightenings (Pavlenko et al.
2007; Kato et al. 2009b); q=0.078(3) from stage A
superhumps (Kato and Osaki 2013).
..................................................
OV Boo (population II from space velocity)(Patterson et al. 2008)

Eclipse modeling
Boldface indicates period bouncers
Ultracam and lcurve analysis of GY Cnc, IR
Com and HT Cas (Feline et al. 2005).

Science paper “A Brown Dwarf Mass Donor in an
Accreting Binary” (Littlefair et al. 2006): SDSS
J103533.02+055158.3 [M2 =0.052(2)M⊙ ].

OV Boo [below period minimum, Porb =0.04626 d,q=0.0623(7), formed directly from a detached
white dwarf/brown dwarf binary?] (Littlefair et al.
2007).
“On the evolutionary status of short-period
cataclysmic variables” (Littlefair et al. 2008):
SDSS J143317.78+101123.3 [Porb =0.05424 d,q=0.0661(7); not a bouncer? by Sirotkin and Kim(2010); bouncer Savoury et al. (2011)], SDSS
J150137.22+550123.4[Porb =0.05684
d,q=0.067(3); not a bouncer? by Sirotkin and Kim(2010); later not a bouncer by Savoury et al.
(2011)].

31
Table 9: Chronology of period bouncers and brown-dwarf secondaries (continued).
Year
2009

Superhump method, Porb , others
Boldface indicates stage A superhump method
1RXS J023238.8−371812 (PSH =0.06619 d, multiple rebrightenings) (Kato et al. 2009a).
..................................................
“SDSS unveils a population of intrinsically faintcataclysmic variables at the minimum orbital period ” (Gänsicke et al. 2009): mainly Porb statistics.
..................................................
“Survey of Period Variations of Superhumps in SU
UMa-Type Dwarf Novae” (Kato et al. 2009a): establishment of superhump stages A, B and C; bigcompilation of SU UMa/WZ Sge stars.

2010
2011

2012

2013

GW Lib (Porb =0.05332 d, M2 =0.06M⊙ from ϵ)(Vican et al. 2011). q=0.069(3) from stage A superhumps (Kato and Osaki 2013).
..................................................
“Distances and absolute magnitudes of dwarf novae: murmurs of period ” listed 22 candidates
selected from the literature (including two objects from eclipse modeling). The period bounce
was estimated to occur at a mass of 0.058(8)M⊙(Patterson 2011). Also selected from Kato et al.
(2009a) CT Tri, OT J111217.4−353829 (small ϵ)and UZ Boo (long PSH =0.06192 d and multiple rebrightenings).
..................................................
OV Boo (UV spectroscopy, consistent with population II) (Uthas et al. 2011).
..................................................
“The Evolution of Cataclysmic Variables as Revealed by Their Donor Stars” (Knigge et al. 2011):
reﬁnement of (Knigge 2006); basically same
method; derivation of optimal evolutionary track.
SDSS
J125044.42+154957.3,
SDSS
J151415.65+074446.5(polars,
spectrum)(Breedt et al. 2012).
..................................................
BW Scl (Porb =0.05432 d, 2:1 resonance signal? inquiescence) (Uthas et al. 2012); q=0.067(6) fromstage A superhumps (Kato et al. 2013a).
“New Method to Estimate Binary Mass Ratios byUsing Superhumps” : establishment of stage A
superhump method (Kato and Osaki 2013).
..................................................
V529 Dra (ﬁrst case of double superoutburst,
Porb =0.0717 d, q=0.04) (Kato et al. 2013a).
..................................................
CRTS J122221.6−311524 (double superoutburst,
q <0.05 from stage A and post-superoutburst superhumps) (Kato et al. 2013b).

Eclipse modeling
Boldface indicates period bouncers
Light-curve modeling code applied to detached
white dwarf-main sequence binaries (Pyrzas et al.
2009).

Light-curve modeling code lcurve; error determination using MCMC (Copperwheat et al. 2010).
“Cataclysmic variables below the period gap:
mass determinations of 14 eclipsing systems”(Savoury et al. 2011): full introduction of MCMC;
only ultracam data were plotted on the evolutionary track.

32
Table 9: Chronology of period bouncers and brown-dwarf secondaries (continued).
Year
2014
2015

2016

2017

2018

Superhump method, Porb , others
Boldface indicates stage A superhump method
CRTS J075418.7+381224, KX Psc (long-period,
long-lasting stage A) (Nakata et al. 2014).
ASASSN-14cv[Porb =0.05992
d,q=0.077(1), multiple rebrightenings], PNV
J17144255−2943481[Porb =0.05956
d,q=0.076(1), multiple rebrightenings], PNV
J17292916+0054043[Porb =0.05973
d,q=0.073(2)],PNV
J06000985+1426152(PSH =0.06331 d, multiple rebrightenings) (Nakata
et al. in prep; Kato et al. 2015).
ASASSN-15jd (nearly double superoutburst andlong superhump PSH =0.06498 d) (Kimura et al.
2016); also provided a list of bouncer candidates
from photometric observations.
..................................................
ASASSN-15na [Porb =0.06297 d, q=0.081(5)],ASASSN-16bu [Porb =0.0593 d, q=0.10(1),slow fading], ASASSN-15gn (PSH =0.06364 d),
ASASSN-15hn (PSH =0.06183 d, slow superhump
evolution, low superhump amplitude), ASASSN15kh (PSH =0.06048 d, slow superhump evolution,
low superhump amplitude) (Kato et al. 2016b).
CRTS J122221.6−311524, originally from stage B
superhumps (Neustroev et al. 2017b), reﬁned tobe q=0.032 using stage A superhumps (Kato et al.
2020).
..................................................
ASASSN-16js [Porb =0.06034 d, q=0.056(5)](Kato et al. 2017).
..................................................
WZ Sge, claimed not to be a period bouncer basedon secular Porb decrease (Han et al. 2017); refuted
by Patterson et al. (2018).
..................................................
HO
Cet[Porb =0.05490
d]
from
SED
(Neustroev et al. 2017a); q=0.090(4) from stage A
superhumps (Kato et al. 2009a).
ASASSN-16dt[doublesuperoutburst,
q=0.036(2)] and ASASSN-16hg (long-period,
long-lasting stage A) (Kimura et al. 2018); a compiled list of bouncer candidates using superhumps
and light curves was also given.
..................................................
QZ Lib, originally from stage B superhumps(Pala et al. 2018); updated to q=0.05 in this work
using stage A superhumps.

Eclipse modeling
Boldface indicates period bouncers

KN Cet (substellar donor M2 =0.064(5)M⊙ ,below period minimum at Porb =0.05298 d)(McAllister et al. 2015); only ultracam data wereplotted on the evolutionary track.

Nature paper “An irradiated brown-dwarf companion to an accreting white dwarf ”: SDSS
J143317.78+101123.3 [M2 =0.0571(7)M⊙ , spectrum L1±1] (Hernández Santisteban et al. 2016).

SDSS
J105754.25+275947.5 [lowest-mass
donor M2 =0.0436(20)M⊙ ,Porb =0.06279 d](McAllister et al. 2017b); only ultracam data
were plotted on the evolutionary track.
..................................................
Introduction of Gaussian process modeling andparallel-tempered
MCMC(McAllister et al.
2017a).

33
Table 9: Chronology of period bouncers and brown-dwarf secondaries (continued).
Year
2019

2020

2021

2022

8

Superhump method, Porb , others
Boldface indicates stage A superhump method
KSN:BS-C11a [Porb =0.05703 d, q=0.070(5)](Ridden-Harper et al. 2019).

ASASSN-17ei [Porb =0.05646 d, q=0.074(3), long
rebrightening], ASASSN-17el [Porb =0.05434 d,q=0.071(3), multiple rebrightenings] (Kato et al.
2020)..................................................
New candidates for period bouncers from the literature and QZ Lib (Thorstensen 2020): mostly fromstage A and other superhump data from “Pdot” papers.
SMSS J160639.78−100010.7 (magnetic systemwith Porb =0.06391 d, low accretion rate and spectrum) (Kawka et al. 2021).
Let’s add DY CMi (PSH =0.06074 d), VX For(PSH =0.06133 d) (Kato et al. 2009a), MASTER OT J085854.16−274030.7 (PSH =0.05556 d,
Kato et al. 2015), TCP J23382254−2049518(PSH =0.05726 d, Kato et al. 2014a) — all withmultiple rebrightenings, Kato (2015) — to thecandidate list; comparison between stage A superhump method and eclipse modeling (this work).

Eclipse modeling
Boldface indicates period bouncers“The evolutionary status of Cataclysmic Variables:
eclipse modelling of 15 systems” (McAllister et al.
2019): reﬁnement of Knigge et al. (2011), using
both eclipsing CVs and stage B superhumps (referring to “Pdot” papers; Knigge-type ϵ-q calibration,
not using stage A method).

MASTEROT
J220559.40−341434.9[Porb =0.06129
d,=
ASASSN-16kr
M2 =0.042(1)M⊙ ], CRTS J052209.7−350530[Porb =0.06219 d, M2 =0.042(4)M⊙ ], ASASSN17jf[near borderline;
Porb =0.05679 d,
M2 =0.060(8)M⊙ ] (Wild et al. 2022);
periodexcess to the “optimal” evolutionary sequenceincreases as M2 decreases.

Porb , q and WZ Sge-type Rebrightenings

Most of short-Porb and low-q dwarf novae are WZ Sge stars. WZ Sge stars often show post-superoutburst rebrightenings. The rebrightenings have a variety of morphology and they are classiﬁed into type-A (long rebrightening),
type-B (multiple rebrightenings), type-C (single rebrightening), type-D (no rebrightening) and type-E (double
superoutburst) (Kato 2015). In WZ Sge stars, there is an empirical relation between the period derivative ofstage B superhumps (Pdot = Ṗ /P ) and q (ﬁgure 21 in Kato 2015). The empirical relation [equation (6) in Kato
2015] isq = 0.0043(9)Pdot × 105 + 0.060(5).
(30)
Although this relation is experimental and is calibrated only in the range 0.04 < q < 0.12, it is regarded asreliable since a Porb -Pdot diagram very clearly depicts the expected evolutionary track (see ﬁgure 17 in Kato 2015).
Systems with known Pdot can be plotted on this diagram with diﬀerent marks for diﬀerent types of rebrightenings.
It has become evident that the types of rebrightenings indicate the following evolutionary sequence: type C → D→ A → B → E (Kato 2015). The same type of ﬁgure is shown in ﬁgure 13.3 Objects with q values obtained by
the eclipse modeling method are also plotted on this ﬁgure using equation (30). It is apparent that these objects
are on the same evolutionary sequence depicted by Pdot .
The most interesting test from this ﬁgure is whether the lowest-q systems determined by the eclipse modelingmethod show rebrightenings expected by the stage A superhump method (or by the Pdot -q relation). These
objects below q=0.06 are ASASSN-16kr = MASTER OT J220559.40−341434.9, SDSS J105754.25+275947.5,
3 During the preparation of this ﬁgure, I noticed that the corresponding ﬁgure 61 in Kato et al. (2017) has incorrect labels for q.
The q labels are corrected in this ﬁgure.

34

20
0.14
15

⋅
P /P × 105

10

0.12

0.10

5

0.08

0

0.06

−5

0.04
0.085

0.050

0.055

0.060

0.065

0.070

0.075

0.080

q=M 2/M 1

type A
type B
type C
type D
type E
unknown
eclipseFigure 13: Pdot versus Porb for WZ Sge-type dwarf novae. Symbols represent the type of outburst: type-A (longrebrightening, ﬁlled circles), type-B (multiple rebrightenings, ﬁlled squares), type-C (single rebrightening, ﬁlled
triangles), type-D (no rebrightening, open circles) and type-E (double superoutburst, ﬁlled stars). On the rightside, I show mass ratios estimated using equation (30). We can regard this ﬁgure as to represent an evolutionary
diagram. The objects with q values obtained by the eclipse modeling method are also plotted. This ﬁgure is
reﬁnement of ﬁgure 61 in Kato et al. (2017).

35

Table 10: Very light (< 0.5M⊙ ) white dwarfs in CVs.
ObjectBC UMaCU VelSDSS J100515.39+191108.0
V754 Lyr

MWD
(M⊙ )
+0.08
0.48−0.09+0.04
0.47−0.05+0.15
0.44−0.09
0.46(2)HY Eri+0.10
0.43−0.07

Method, comment
Revised ultraviolet spectral ﬁt to Gänsicke et al. (2005)
Ultraviolet spectral ﬁt (Pala et al. 2017)
Ultraviolet spectral ﬁt (Pala et al. 2017)
Modeling ellipsoidal light variations,
low mass-transfer rate for Porb =0.3641 d (Yu et al. 2019)
Burwitz et al. (1999), polar

CRTS J052209.7−350530 and SDSS J103533.02+055158.3. They are expected to show either type-B (multiple
rebrightenings) or type-E (double superoutburst) superoutbursts. A superoutburst was observed only one of thesefour (ASASSN-16kr, but the observational coverage was insuﬃcient). Observations reported to VSNET did notcover the post-superoutburst phase. The All-Sky Automated Survey for Supernovae (ASAS-SN: Shappee et al.
2014; Kochanek et al. 2017) data4 showed at least one post-superoutburst rebrightening on 2016 October 7 at
V =15.5. There were gaps of observations and I could not conﬁrm whether there were more rebrightening(s).
Future observations of these lowest-q systems during superoutbursts (stage A superhumps, of course, to obtain
independent q estimates) and post-superoutburst states will be a very interesting topic to test the evolutionary
picture derived from superhump observations.

9

Do Some CVs Have Very Light White Dwarfs?

Pala et al. (2021) published a compilation of masses of white dwarfs (MWD ) in 89 CVs. While the mean mass
⟨MWD ⟩ = 0.081M⊙ was obtained, they identiﬁed ﬁve systems with MWD < 0.5M⊙ (table 10). Since MWD values
for eclipsing SU UMa/WZ Sge stars were known to be in a narrow region (table 11), I examined whether the initial
two SU UMa stars have anomalously large ϵ (it is a pity that Porb is now known for SDSS J100515.39+191108.0,
superhump observations were present). For an 0.5M⊙ white dwarf, ϵ is expected to be 1.6 times larger than
average SU UMa stars with the same Porb . The result is shown in ﬁgure 14. BC UMa and CU Vel do not haveunusually large ϵ, and this ﬁgure suggests that they have white dwarf masses typical for CVs. The masses ofwhite dwarfs in these two objects were measured by ﬁtting the ultraviolet spectra obtained by the Hubble Space
Telescope (HST). As judged from table 6 in Pala et al. (2021), it looks like that this method is responsible for alarge number of small-mass white dwarfs.
It might be worth noting that QZ Vir was reported to have a very light (< 0.4M⊙ ) white dwarf in the past
(Shafter and Szkody 1984), which has a completely normal q=0.108(3) from the stage A superhump method.
This instance also suggests that very large errors in the masses of white dwarfs are expected by traditional
methods. Zorotovic et al. (2011) wrote: “Only 7±3% of the 104 CVs with available WD-mass estimates anderrors have M1 ≤ 0.5M⊙ , and none of the systems in the sub-sample of 32 with presumably more reliable mass
determinations. We therefore conclude that the fraction of He-core WDs in the observed sample of CVs is ≤10%”.
It may be that such low-mass white dwarfs are totally missing in dwarf novae below the period gap considering
the absence of objects largely deviating from the main distribution on the Porb -ϵ plane.

10

Advantages and Disadvantages of Eclipse Modeling Method andStage A Superhump Method

I summarize the advantages and disadvantages of the two methods.

Eclipse modeling method
Advantages:
4 <https://asas-sn.osu.edu/sky-patrol/coordinate/4822659d-5613-4692-beb0-5724f2401d9b>.

36

0.08

ε (stage B)

0.06

0.04

CU VelBC UMa

0.02

0.00

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Figure 14: Relation between Porb and ϵ of stage B superhumps. Although BC UMa and CU Vel have beenreported have low-mass white dwarfs, they do not have unusually large ϵ, suggesting that they have white dwarf
masses typical for CVs.

Table 11: Masses of white dwarfs in CVs.
⟨MWD ⟩ below period gap(M⊙ )0.83±0.02
0.80
0.82±0.02
0.81

Intrinsic scatter
(M⊙ )
0.07
0.12
0.10
−0.16/+0.17

Reference
Savoury et al. (2011)
Zorotovic et al. (2011)
McAllister et al. (2019)
Pala et al. (2021)

37• Basically applicable to eclipsing systems regardless of the dwarf nova type.
• Considered to be the most reliable.
• Both the mass and radius of the secondary can be measured to directly determine the evolutionary stateof the secondary.
Disadvantages:• Only applicable to eclipsing systems. Inclinations need to be high enough and cannot be directly applied
to grazing eclipsers.
• Requires a large telescope and a high-speed imaging device.
• Requires relatively bright (∼20 mag) objects in quiescence.
• Systems without orbital humps are not good targets.
• Aﬀected by ﬂickering in high mass-transfer systems.
• Requires complicated modeling of the light curve and an advanced numerical method such as (paralleltempered) MCMC.

Stage A superhump method
Advantages:• Can be applied to any inclination.
• Small telescopes (20–60 cm) are usually suﬃcient.
• Very high time resolutions are not needed. Time resolutions around 30 s are usually suﬃcient.
• Relatively faint (∼16 mag using the above equipment) objects can be observed.
• Less disturbed by ﬂickering.
• Period determination is numerically very easy.
• A considerable fraction of WZ Sge stars show early superhumps and orbital periods and q values can be
determined only by outburst observations.
• The sample is expected to grow steadily as new transients are discovered. The number now exceeded 100
and is 2.5 times those determined by the eclipse modeling.
Disadvantages:• Only applicable to superhumping systems (SU UMa/WZ Sge stars). Cannot be applied to SS Cyg stars.
• Observations of the growing phase (stage A) of superhumps are necessary. Timely alert and intensive
observations (not always easy) during these superhumps are essential.
• Systems with relative large q have short durations of stage A and the accuracy of the q estimate is limitedby the short baseline of the observations.
• Not very adequate for eclipsing systems.

38

Epilogue
Although I wrote this paper myself, this works has been impossible without contributions of professional andamateur astronomers who wish to catch and conﬁrm the nature of suddenly appearing new objects. It is sometimes said that birding (birdwatching) invokes our primeval hunting instincts,5 and our activity with “hunting
superhumps” may be invoked by the same type of our primeval hunting instincts. Since my childhood, I had beenacquainted with media reports telling a certain bird species ﬁrst recorded in Japan or spotted after absence oftens of years (these birds are called “vagrants” or “rarities”). When I became enchanted by the ﬁeld of variable
stars, particularly “UG” (=U Gem) stars (currently known as dwarf novae; subclasses were not speciﬁed at thattime), I noticed the striking similarity between birding and hunting rarely outbursting stars (recurrent novae andrarely outbursting UG stars; WZ Sge was considered as a recurrent nova at that time). There were even parallel
terminology (jargon) both in birding and UG hunting, at least in the Japanese language. We were informing eachother via telephone which stars were in outburst; this is exactly what present-day birders do with cell phones.
Some present-day observers, particularly such as the renowned CV enthusiast Patrick Schmeer, still know how I
was devoted to hunting UG outbursts (I have a vivid recollection of his international calls when I was observing
at the university observatory as a beginner professional: “Hello, this is Patrick Schmeer from Germany...", asking
conﬁrmation of his suspected detection of outbursts).
When I chose the carrier as a professional astronomer, I switched my main hobby from variable star observations to birding — this was initially from curiosity about what the world on the other side, connected by asimilar set of jargon, would look like. I got, of course, enchanted by the world of birds: not only because birding
itself was interesting (as interesting and deep as astronomy as a hobby) but also due to the fascination of thebirds themselves.
Looking back the history of studies of superhumps, I feel that the present accomplishment, just like thispaper, can be regarded as one of the most remarkable achievements of citizen science in astronomy — itrivals, or may even surpasses, the achievements by large telescopes. Many amateur and professional observers
have devoted themselves to detect outbursts and observe superhumps. I greatly owe this accomplishment tothem (and of course, to the theoretician Prof. Yoji Osaki for interpreting superoutbursts, superhumps and theprecession rate). The existence of international groups played an important role: our VSNET [and, of course,
the Variable Star Observers League in Japan (VSOLJ)] had a main coverage in the East Asia and Europe, while
the Center for Backyard Astrophysics (CBA)6 had a main coverage in the North America and Europe, and somesouthern hemisphere countries. These two parties were the driving force of studies of CVs in professional-amateurcollaborations, or citizen science in modern, more fashionable terminology. I sometimes recall competitions
between these groups in the late 1990s and 2000s when both groups observed SU UMa/WZ Sge stars. I feelsomewhat a pity that CBA observers do not observe SU UMa/WZ Sge stars as before. Because of this, data
above the skies over the Americas currently tend to be lacking.
Why the longitudinal coverage matters? — This comes from the nature of the transient objects. Stage A
superhumps in SU UMa stars usually last 1–2 d in long-Porb systems and a few days in short-Porb systems. A
good temporal coverage is necessary to obtain a reliable period. The durations are an order of a day, not a few
hours or several days, and this is why a longitudinal coverage is so important since a given object can be observed
only for several hours (up to an entire night in extreme cases) from a given location on the globe. This situationis very diﬀerent from objects like intermediate polars, the current main targets of the CBA. Intermediate polars
do not require continuous coverage and a longer baseline in time is more important, which can be achieved byobservations from a single location. Considering these characteristics, dwarf novae are ideal objects for uniting
world-wide observers particularly in the longitudinal direction.
Looking at broader ﬁelds of variable stars, the American Association of Variable Stars (AAVSO)7 plays
the central role, providing observational database and the world most comprehensive variable star catalog, theAAVSO Variable Star Index (VSX). Researchers engaged in variable objects should have already consulted thecatalog or database provided by the AAVSO. The AAVSO is just celebrating the 110 years of citizen science.8
Wouldn’t it be one of the most renowned citizen science programs with a long and distinguished history?
Let’s look at the other side of the world — I mean the world of birds. The online database of bird reports
eBird9 has a history of 20 years, although much less than the AAVSO, provides the function similar to the AAVSO.
5 See e.g. <http://www.birdwatching.com/birdingfaq.html>.
6 <https://cbastro.org>.
7 <https://www.aavso.org/>.
8 <https://www.aavso.org/110>.
9 <https://ebird.org/>.

39
Birders among the readers of this paper should certainly be aware of eBird and some may have already submitted
observations to it. Again, the analogy between variable star observing and birding is here. I recommend readers
who are not familiar with eBird to visit there and explore the functions of eBird, and you will ﬁnd the similarityof these ﬁelds. I even wonder if Sebastián Otero, the chief supervisor of VSX, is well familiar with the other sideof the world and his work in astronomy is also inspired by activities in the world of birding. His page on the
AAVSO website10 tells “When he’s not doing something related to astronomy (and he’s not sleeping) you may ﬁnd
him in some natural reserve taking pictures of every bird he is able to detect, because he is an active bird-watcher.
To ﬁnd several weird species, he likes travelling all around his country when he has time, so it seems everythingthat is up in the sky deserves his attention! ” I’m suspecting that he continuously gains a momentum from his
own birding activity or inspirations from activities of birding organizations, including eBird, to his enthusiastic
tasks in variable stars.
I have written that dwarf novae are ideal objects for uniting world-wide observers particularly in the longitudinal direction. What about the latitudinal direction? We indeed collaborate with southern hemisphereobservers to study southern object which we cannot see from the northern hemisphere, but there are far better“ambassadors of friendship” in the world of birds — migratory birds. Many migratory birds annually move fromthe north to the south and the reverse, and international collaboration particularly in the latitudinal direction
is indispensable for studies and conservation of these birds. I would propose to express that variable star observations and studies of migratory birds are the best examples of warp and woof (longitudinal and latitudinal
threads) of global scientiﬁc collaboration and friendship.
Just to name a few, shorebirds (waders) migrate from the arctic tundra to Australia and many people along
this East Asian-Australasian ﬂyway11 are engaged in observation and conservation. Similar kinds of ﬂyways
are present in the Americas (Paciﬁc Americas ﬂyway, Atlantic Americas ﬂyway), central Eurasia (Central Asian
ﬂyway), Europe-Africa (East Atlantic Flyway) and others.12 Some migratory birds are “visible” on migration.
One of the best examples is diurnal raptors (eagles, hawks and allies) [See e.g. Bildstein (2006); Panuccio et al.
(2021)]. There are hot spots of migration corridors around the world, such as Batumi in Georgia famous for thenumber of passing raptors (Verhelst et al. 2011).13 There was even a record of more than 200000 raptors passing
in a single day. During the seasons of migration, raptor watchers along the migratory route observe the sky all
the day just as meteor observers watch the sky. Among migratory raptors, I pick up the Crested (or Oriental)
Honey Buzzard (Pernis ptilorhynchus; ﬁgure 15). They breed in Russia, Japan, Korea and northeastern partof China and migrate to the Southeast Asia to winter. It has been demonstrated by satellite tracking that thisspecies encompasses all the East Asian and Southeast Asian countries during migration! (Higuchi et al. 2005).
Enthusiastic watchers observe them along the migration route, but the information is still limited to a relatively
small number of countries. Birds sometimes travel beyond our imagination. I was surprised to read the news
that a Crested Honey Buzzard was recorded ﬁrst time in the South African Republic in 2021 (Cohen 2021).14 I
had never imagined to make a collaboration with a South African researcher about this oriental species, although
I had a number of joint papers with a variable star observer in South Africa in the ﬁeld of dwarf novae. You cansee from these pages and images that the enthusiasm is the same between these two ﬁelds, and feel the wondersof the nature.
Scientiﬁc activities in astronomy and ornithology are not totally independent. Since migratory birds are
known to use the Sun, stars [this has been conﬁrmed by experiments using a planetarium as early as Emlen(1967a,b)] and perhaps the Moon in addition to geomagnetic cues for orientation. Light-level geolocators(Afanasyev and Prince 1993), tracking devices that use daylight to estimate location, are now widely used instudying movements of various types of birds (Croxall et al. 2005; Stutchbury et al. 2009). This technique isdependent on the solar ephemeris (astronomical almanac). I have even been invited to join a research programby providing a handy code for calculating solar and lunar ephemerides along the trajectory of moving birds byusing knowledge in astronomy (Holdaway et al. 2022). Movements of birds at night in relation to the moonlight(e.g. Norevik et al. 2019) are likely a ﬁeld in which astronomers can contribute to ornithology. We should not
10 <https://www.aavso.org/sebastián-otero>.
11 East Asian-Australasian Flyway Partnership (EAAFP) <https://www.eaaﬂyway.net/>.
12 The names of ﬂyways depend on the literature.
13 <https://www.batumiraptorcount.org/>.
14 See, for example, <http://www.alexaitkenhead.co.za/2021/02/crested-honey-buzzard-pernis.html> (There is a photograph ofpeople waiting for the bird to appear. It was written: “Oﬃcial Conﬁrmation registering the CHB (=Crested Honey Buzzard) as
the ﬁrst record for South Africa, making this raptor from the orient the highest priority amongst the birding fraternity. Saturday
6th February was our ﬁrst opportunity to get onto the view site and hopefully get a glimpse of this famous buzzard”. There wasalso an interesting story at <https://www.iol.co.za/capetimes/news/rare-sighting-of-a-crested-honey-buzzard-in-somerset-west-hasbirders-abuzz-172fd2d5-7afd-42a2-a0e6-43821a5ca4b3> from the Cape Times.

40

Figure 15: Migratory raptors. (Upper:) Crested Honey Buzzards (Pernis ptilorhynchus) on migration. Female(left) and male (right). (Lower:) Thermally soaring group (“kettle”) of Grey-faced Buzzards (Butastur indicus)
on migration (photos by author).

41
forget that counting migratory birds on the disk of the Moon has long been used (e.g. Very 1897) — many (particularly small) birds migrate at night to avoid overheating in daytime ﬂight or to avoid predation. The scene
was also beautifully described in Rachel Carson’s famous book “The Sense of Wonder” (published posthumously:
Carson 1965). Migrating birds often issue calls (nocturnal ﬂight calls or NFC, ﬁrst described in Ball 1952) atnight, probably communicating each other. Recording of these calls (nocmig)15 is also an important tool to study
bird migration (e.g. Evans and Mellinger 1999; Larkin et al. 2002; Farnsworth et al. 2004). The nights suitable
for astronomical observations are also suitable for migratory birds to move. Won’t someone working in suitable
astronomical observatories regularly make recordings of these nocturnal ﬂight calls?
Modern space science technologies also help biologists to understand the behavior of moving animals and takepart in their conservation. In particular, the “Space for Birds” project from the International Space Station (ISS)maps the routes taken by seven endangered or threatened bird species, highlighting along those routes habitatchanges caused mainly by human activities.16 There is also the International Cooperation for Animal Research
Using Space (ICARUS) project using the ISS by collaboration of the Max Planck Institute for Ornithology,
Germany and the Russian Space Agency Roscosmos.17 This page explains: “To achieve the ICARUS goals,
the following prerequisites have to be fulﬁlled: Global tag coverage to record long distance migration patterns;
Simultaneous communication with a multitude of animal tags; Extremely low tag mass and size to allow tracking
of small animals; Long, maintenance-free tag life in order to cover complete migration cycles; and logging of theinternal (physiological) and the external (environmental) state of animals”. The system went into operation in2020 March, and the data are stored in Movebank (Fiedler and Davidson 2012).18 The data are made publicly
open and scientists around the world can make their own analyses, just as we do with observations of major
astronomical observatories and astronomical satellites. Movebank also hosts other sets of tracking data obtainedby various researchers worldwide.
During the preparation of this paper, I heard the passing of the renowned, and sometimes debated, biologist
Edward Osborne Wilson (1929-2021), who is not only renowned for inventing the concept “sociobiology” in his
book “Sociobiology: The New Synthesis” (Wilson 1975) but also publishing “The Diversity of Life” (Wilson 1992)
in conjunction with the 1992 United Nations Conference on Environment and Development, Rio de Janeiro, Brazil(commonly known as the “Earth Summit”). This summit resulted “Rio Convention” including “The Convention on
Biological Diversity” and “The United Nations Framework Convention on Climate Change”, both of which are thebiggest issues facing our planet and its inhabitants, including humans — this will be also true for extraterrestrial
civilization, if it exists, since any civilization must have evolved from rich biological diversity. The popularity ofWilson’s book (Wilson 1992) was one of the motives to spread the concept of biological diversity, or biodiversity.
When I met the former book (Wilson 1975), I was surprised to see that biologists, including Wilson himself, applied Shannon entropy (information entropy, Shannon 1948) in describing and discussing the evolution ofcommunication in biology [in his chapter 8; biologists wanted to explain why the number of types of signals ishighly conserved (10–40) between vastly diﬀerent species; this chapter is a very good introduction to the theoryof communication in biology; if the book is in your library, it is certainly worth looking at] so early in the history.
Both Haldane and Spurway (1954) and Wilson (1962) used the expression of entropy for continuous variables(Shannon and Weaver 1949) to describe animal behavior. Hazlett and Bossert (1965) measured transition probabilities of Markov chains dealing with three types of behavior of crabs and estimated the information entropy
associated with communication. We astronomers tend to think (sorry if it is my misunderstanding or prejudice!)that biologists are less mathematically inclined, but this is not always true. The maximum entropy method(MEM) is the most frequently used form of Shannon entropy in astronomy. By searching using ADS, the ﬁrst
applications of MEM in astronomy appears to be Richer et al. (1973); Richer and Ulrych (1974) [These papers
cited applications to geophysics: Ulrych (1972)]. The ﬁrst application of MEM to inversion problems, which wemeet more frequently now, appeared in Bryan and Skilling (1980). Compared to them, you can see how early
the applications to biology were.
E. O. Wilson left “Advice to young scientists” in his TED talk in 2012.19 Although the main audience of thistalk was apparently young biologists or medical students and the talk as a whole may not be directly applicableto young astronomers of physicists, he stated: “In time, all of science will come to be a continuum of description,
an explanation of networks, of principles and laws. That’s why you need not just be training in one specialty,
but also acquire breadth in other ﬁelds, related to and even distant from your own initial choice. Keep your eyes
15 <https://nocmig.com/>.
16 <https://www.nasa.gov/mission_pages/station/research/news/amass-ceo>.
17 <https://directory.eoportal.org/web/eoportal/satellite-missions/i/iss-icarus>.
18 <https://www.movebank.org/>.
19 <https://www.ted.com/talks/e_o_wilson_advice_to_a_young_scientist>.

42
lifted and your head turning. The search for knowledge is in our genes. It was put there by our distant ancestors
who spread across the world, and it’s never going to be quenched ”. This part of the talk probably referred tohis concept of “Consilience” (Wilson 1998), although humans have not yet reached a point to “unite the sciences
and might in the future unite them with the humanities” as proposed by Wilson. This talk, however, conveys
an important message for us, too. Keeping the eyes open to diﬀerent ﬁelds of science is undoubtedly important.
For example, by applying the existing methods to other ﬁelds of science, one may be able to obtain a deeper
insight and a positive feedback to the original ﬁeld, just as I attempted in Kato (2021a). Such multidisciplinary
applications are always appreciated for development of science. As I have shown, two ﬁelds of science, variable
star observing (astronomy) and birding (ornithology), which are usually considered to be distant in terms ofscience, are two sides of the same coin in human activity invoked by our primeval hunting instincts. It would beeasy to understand the fun of the other side from a diﬀerent side and join or respect the other activity. More
importantly, these two activities share the same character in that they are essentially international collaborations
and use the similar types of information such as public databases. These activities have a common aspect thatthey contribute to uniting people worldwide via exchanges of observations. If experience, knowledge and methodsof the two ﬁelds are mutually exchanged, these ﬁelds would be expected to be even more powerful.
Astronomers can contribute to ornithology by introducing techniques (including mathematical modeling orcomputer science) or ideas used in astronomy (astronomy is a bit ahead of ornithology in the use of the literature database ADS, the preprint server arXiv and open databases, and I hope that ornithological community
assimilates the usefulness of these services and introduces some of functions in the astronomical services into
their activity). Although the migration routes of the Crested Honey Buzzard are very diﬀerent between spring
and fall migrations (Higuchi et al. 2005; Yamaguchi et al. 2008), the reason of the complexity of the routes is
not yet well understood. Biologists have considered the weather (particularly assisting winds) during migration and foraging ecology at stopover sites (Yamaguchi et al. 2012; Mardiyanto et al. 2015; Nourani et al. 2017;
Sugasawa and Higuchi 2019) to be main reasons. Agostini and Mellone (2007), however, claimed that food diversity and abundance along the migration route should not be of critical importance and suggested that there might
not yet have been suﬃcient time to evolve the more direct route through the southwest islands. Astronomers may
provide ideas to solve the issue by using mathematical modeling or by the help of data-driven science. Although
the tracking data of the Crested Honey Buzzard are unfortunately not yet available on Movebank, there are
public data for the European Honey Buzzard (Pernis apivorus), the species sister to the Crested Honey Buzzard,
and other migratory raptors. They can be used for modeling the migration routes. This is only an example, andI feel that astronomers can contribute to ornithology in various aspects using public data.
In order to make an actual contribution, basic knowledge and experience are required to some extent. Birding
would be one of ideal tools to start with to become familiar with biology (if one is not yet), since we can directly
learn from our winged neighbors. It may be needless to say, but I am tempted to add a bit (a lot?) about“birdbrain” for readers who are not familiar with modern biological development. “Birdbrain” literally means astupid person, or a person lacking intelligence. It was indeed written in old textbooks that birds lack the partof the brain (cerebral cortex), which is present in mammals, and the prefrontal cortex in the cerebral cortex is
known to be particularly relevant for intelligence. I indeed learned that behavior of birds is simply a complex
collection of reﬂexes and instinctual behavior governed by the basal ganglia (old part of the brain). It has
become, however, apparent that hyperpallium (in modern terminology) in birds is equivalent to the mammaliancerebral cortex and a consortium of neuroscientists has proposed renaming of the structures of the bird brain(Jarvis et al. 2005; Herculano-Houzel 2020), which can be directly compared to the mammalian brain (following
the nomenclature common to birds and mammals, the mammalian neocortex is also called neopallium). It has
been even shown that birds have numbers of neurons in the forebrain (advanced part of the brain) comparable tothe primates (Olkowicza et al. 2016). These results have already been reﬂected on the wikipedia page20 and thecomparison is amazing. The Rook (Corvus frugilegus), a relatively small species of a crow, has a number of neurons
comparable to a Beagle dog and exceeds that of the Lion (Panthera leo)! Some bird species have cognitive ability
rivalling great apes [such as tool use, mirror recognition, plan for future needs and vocal learning; for a review
see Olkowicza et al. (2016)]. It has been shown that the working memory has the same neuronal mechanism
between crows and monkeys (Hahn et al. 2021). The nidopallium caudolaterale in birds is considered to beequivalent to the prefrontal cortex in mammals. It is also known to work when pigeons discriminate the abstract
features of paintings (Anderson et al. 2020). Avian nidopallium caudolaterale and mammalian prefrontal cortex
are considered to be an example of convergent evolution and it has been suggested that there may be limited
degrees of freedom in developing intelligence (Güntürkün 2005). The intelligence of birds is probably a part of the
20 <https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons>.

43
reason why we understand birds easily and birds attract many people. I can list at least three highly intelligent
groups of birds: parrots (very well-known), crows (annoy and sometimes amuse us21 everywhere) and the StriatedCaracara (Phalcoboenus australis). A Grey Parrot (Psittacus erithacus) named Alex (1976-2007)22 could evenmake elementary vocal conversations with humans. The New Caledonian Crow (Corvus moneduloides)23 could
use, store and even manufacture tools (Weir et al. 2002; von Bayern et al. 2018). The Striated Caracara is related
to falcons and would be less known (even to ornithologists), but there are a number of YouTube videos of behavior
in the wild and laboratory experiments. Just have a look at “Flying Devils” provided by National Geographic(2007).24 There is even a book “A Most Remarkable Creature – The Hidden Life and Epic Journey of the World’s
Smartest Birds of Prey” (Meiburg 2021). The important point is that intelligence evolved multiple times in
diﬀerent lineages (this would be also true for mammals considering intelligent dolphins or elephants). We tend
to think that intelligence evolved on a single path within the primates and we humans are on the top of it, but
this picture may be an oversimpliﬁcation of the evolutionary process of intelligence. The diversiﬁcation time
between birds and mammals is estimated to be 297–326 million years.25 Considering that all the three groupsof very intelligent birds belong the modern clade Eufalconimorphae (diversiﬁcation time of 73–87 million years
from other clades of birds), the appearance of the clades hosting intelligent species independently from mammals
required ∼230 million years since the diversiﬁcation between birds and mammals. This is nearly 40% of time of
600 million years of evolution of multicellular organisms (Chen et al. 2014). This value could be a measure toconsider the evolutionary time-scales of intelligence. This lesson learned from very intelligent birds may provide
previously neglected insights into the life helpful for dreaming of or searching for extraterrestrial intelligence.
Yes, learning about birds in modern perspective could contribute to understanding of ourselves and possibly(distantly?) to astrobiology.
We can also learn from the ﬁeld of ornithology such as eBird and other databases like xeno-canto26 , which
deals with vocalizations of birds. The xeno-canto service is tolerant to multiple languages, and data can beposted or discussed sometimes using non-English languages (even in languages distant from English, such as
Chinese and Russian). Such an aspect may be missing in USA-based systems like the AAVSO and eBird, andxeno-canto could be a model for multilingual international citizen science programs. We can also learn more
broadly from ecology or biology. There are methods in ecology with which astronomers are not usually familiar.
For example, the open source software “MaxEnt” 27 (Phillips et al. 2004, 2006; Elith et al. 2011) is widely used inlandscape ecology for modeling species’ distributions and selection of explanatory variables (e.g. choosing factors
which determine the presence of a certain species and a prediction of the distribution of a certain species whenobservations are fragmentary). This method is an extension of the well-known logistic regression. We might
be urged to apply Least Absolute Shrinkage and Selection Operator (Lasso: Tibshirani 1996) to such a class ofproblems, but MaxEnt can handle the data in which non-detection data points are largely missing and is likely
a better solution under such circumstances. Such a method may inspire applications in astronomy under similar
circumstances.
Returning to E. O. Wilson, there is a concept of the Biophilia hypothesis, which suggests that humans
possess an innate tendency to seek connections with nature and other forms of life (from wikipedia). Wilson(1984) deﬁned it in his book “Biophilia” as “the urge to aﬃliate with other forms of life”. The current popularity
of astrobiology and related ﬁelds may be a manifestation of the “Biophilia gene” (Wilson might have alreadynoticed this, but he would have been happy to know this idea if he were still alive). In some future, humans maybuild space colonies and live within them. This is quite understandable since “the search for knowledge is in our
genes. It was put there by our distant ancestors who spread across the world, and it’s never going to be quenched ”(Wilson) and spreading outside the Earth would be a natural outcome of expression of these genes. In such spacecolonies, we can eventually test the Biophilia hypothesis. In isolated artiﬁcial environments, people with strong
expression of the Biophilia gene may miss the lack of biodiversity, or may miss the absence of migratory birds
seasonally passing over or visiting us. I hope that our distant descendants at that time still have a healthy, but
fragile, living planet to return and can enjoy biodiversity and sometimes complain of the weather when observing
stars or waiting for meteors.

21 See e.g. “Crowboarding: Russian roof-surﬁn’ bird caught on tape”: <https://www.youtube.com/watch?v=3dWw9GLcOeA>.
22 <https://en.wikipedia.org/wiki/Alex_(parrot)>.
23 <https://en.wikipedia.org/wiki/New_Caledonian_crow>.
24 <https://www.youtube.com/watch?v=Y7qcNiJTfVU>.
25 <http://www.timetree.org/>.
26 <http://www.xeno-canto.org/>.
27 <https://biodiversityinformatics.amnh.org/open_source/maxent/>.

44
Abstract
In 2013, the method of determining mass ratios (q) of dwarf novae using stage A superhumps (growing
superhumps) was established. This method is a dynamical one in that it relies only on celestial mechanics.
It is not dependent on an experimental calibration. Since then, more than 100 objects have been measured
by this method and this method have also been applied to AM CVn stars and a black-hole X-ray binary,
but these achievements have often been neglected. In this paper, I provide an updated description of themethod. Comparisons with the results of the modern eclipse modeling method, which is considered to be thegolden standard, have shown that these two methods agree very well and I have conﬁrmed that the stage
A superhump method is as accurate and as reliable as the modern eclipse modeling method. The stage
A superhump method has many points advantageous over the eclipse modeling method in that the former
does not require large telescopes and can be applicable to non-eclipsing systems. The number of objects
determined by the stage A superhump method is now a few times of that by the modern eclipse modeling andthey are now indispensable to study the terminal evolution of cataclysmic variables. I also showed that paststudies by the other groups assumed incorrect fractional superhump excess (ϵ)-q relations, causing biases in
discussing the evolution. In particular, formulae given in Patterson (2001) and Patterson et al. (2005a) should
be avoided. I derived a new experimental ϵ-q relation of stage B superhumps and showed that the pressure
eﬀect during stage B of superhumps has an only weak dependence on q. I derived a reﬁned evolutionary
track around the period minimum suggesting that the angular momentum loss is 1.9 times larger than what is
expected by gravitational wave radiation. The period minimum on this track occurs at 0.0562 d = 81.0 min.
There is a sharp peak in the distribution of the mass of the secondaries around the period minimum. The
ϵ values rule out the claimed existence of very low-mass white dwarfs (< 0.5M⊙ ) among dwarf novae below
the period gap. The measurements of stage A superhumps greatly owe to international collaborations withamateurs and professionals. I describe a brief summary of these collaborations and highlight the similarity
with the world of birding (or ornithology) in that they play a role in uniting the world via international
exchanges of observations. I also describe my thoughts about the similarity and relation between astronomy
and ornithology, and give prospects how multidisciplinary works can be made possible between these seemingly
distant ﬁelds of science.

Key words: accretion, accretion disks — astronomical data bases — catalogs — methods: data analysis —stars: binaries: eclipsing — stars: dwarf novae — stars: novae, cataclysmic variables — stars: variables — SU
UMa stars — WZ Sge stars — superhumps — mass ratios — period bouncers — brown dwarfs — citizen science— methods in biology — bird migration — raptor migration — avian biology — evolution of intelligence

Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 21K03616. This research has made use of NASA’s
Astrophysics Data System. I am deeply indebted to world-wide observers to study superhumps. Complete listsof collaborators have been given in our Pdot papers.

List of objects in this paper
LL And, PQ And, V455 And, V466 And, V1838 Aql, VY Aqr, UZ Boo, NZ Boo, OV Boo, V342 Cam, OY
Car, HT Cas, V1258 Cen, V713 Cep, WX Cet, BO Cet, GS Cet, HO Cet, KN Cet, Z Cha, PU CMa, DY CMi,YZ Cnc, EG Cnc, GY Cnc, GZ Cnc, AL Com, IR Com, MT Com, GP CVn, V503 Cyg, V632 Cyg, V1006
Cyg, V1504 Cyg, V2176 Cyg, V3101 Cyg, EX Dra, IX Dra, MN Dra, V529 Dra, XZ Eri, EF Eri, HY Eri,
VX For, IR Gem, V592 Her, V1239 Her, VW Hyi, GW Lib, QZ Lib, BR Lup, EZ Lyn, V344 Lyr, V754 Lyr,
V453 Nor, DT Oct, V2051 Oph, V2779 Oph, V3721 Oph, EF Peg, IP Peg, V681 Peg, UV Per, DI Phe, TY
PsA, KX Psc, BW Scl, V493 Ser, WZ Sge, V4140 Sgr, KK Tel, EK TrA, CT Tri, SU UMa, SW UMa, BC
UMa, DI UMa, DV UMa, ER UMa, IY UMa, KS UMa, KV UMa, PU UMa, V355 UMa, CU Vel, HV Vir, OU
Vir, QZ Vir, V379 Vir, V406 Vir, 1RXS J023238.8−371812, 1RXS J105010.3−140431, 1RXS J180834.7+101041,
ASAS J102522−1542.4, ASASSN-14ag, ASASSN-14cv, ASASSN-14jf, ASASSN-14jv, ASASSN-15bp, ASASSN15gn, ASASSN-15gq, ASASSN-15hd, ASASSN-15hn, ASASSN-15jd, ASASSN-15kh, ASASSN-15na, ASASSN15ni, ASASSN-15po, ASASSN-15pu, ASASSN-15uj, ASASSN-16bh, ASASSN-16bu, ASASSN-16da, ASASSN16dt, ASASSN-16eg, ASASSN-16hg, ASASSN-16hj, ASASSN-16iw, ASASSN-16jb, ASASSN-16js, ASASSN16kr, ASASSN-16oi, ASASSN-16os, ASASSN-17bl, ASASSN-17ei, ASASSN-17el, ASASSN-17fn, ASASSN-17hw,
ASASSN-17jf, ASASSN-18aan, CRTS J035905.9+175034, CRTS J043112.4−031452, CRTS J052209.7−350530,
CRTS J075418.7+381224, CRTS J104411.4+211307, CRTS J112619.4+084650, CRTS J122221.6−311524, CRTS

45
J174033.4+414756, CRTS J200331.3−284941, CRTS J214738.4+244554, Cze V404, GALEX J194419.33+491257.0,
GD 552, KIS J192748.53+444724.5, KSN:BS-C11a, LSPM J03338+3320, MASTER OT J005740.99+443101.5,
MASTER OT J085854.16−274030.7, MASTER OT J094759.83+061044.4, MASTER OT J181953.76+361356.5,
MASTER OT J203749.39+552210.3, MASTER OT J211258.65+242145.4, NSV 4618, OT J012059.6+325545, OT
J111217.4−353829, PN Te11, PNV J03093063+2638031, PNV J06000985+1426152, PNV J17144255−2943481,
PNV J17292916+0054043, PNV J20205397+2508145, PNV J23052314−0225455, RX J0502.8+1624,
SDSS J075059.97+141150.1, SDSS J090350.73+330036.1, SDSS J100515.39+191108.0, SDSS J100658.40+233724.4,
SDSS J103533.02+055158.3, SDSS J105754.25+275947.5, SDSS J115207.00+404947.8, SDSS J121607.03+052013.9,
SDSS J125044.42+154957.3, SDSS J143317.78+101123.3, SDSS J150137.22+550123.4, SDSS J151415.65+074446.5,
SDSS J152419.33+220920.0, SDSS J161027.61+090738.4, SDSS J162520.29+120308.7, SEKBO 106646.2532,
SMSS J160639.78−100010.7, TCP J00590972+3438357, TCP J20034647+1335125, TCP J23382254−2049518

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