ISSN 0917-2211

Variable Star Bulletin
No.25

Dec. 1997

Least-Squares Strategy for Analyzing Pogson’s Step Method
Taichi Kato (Kyoto, Japan)
Received 1997 Nov. 27
Pogson’s step method has been widely used in (especially visual) observations of variable stars.
The “textbook” reduction procedure for this method is often rather empirical: 1) first determine thediﬀerence in steps between pairs of comparison stars of nearest magnitudes, 2) second determine thebrightness of the variable star by using the step values of comparison stars. This usual procedureintroduces some ambiguity in selecting pairs from the whole sample of comparison stars, and somebiases caused by inhomogeneous usage of individual comparison stars. I present a more robust,
least-squares strategy for analyzing Pogson’s step method, which simultaneously solves the step
values of comparison stars and the variable star.
The usual expression of Pogson’s method (1) tells that the variable was observed two steps
fainter than comparison a, one step brighter than comparison b and three steps brighter thancomparison c.
a2v v1b v3c(1)In this strategy, we set step values of comparisons a, b, etc. as ma , mb , etc. The step value ofthe variable star at time ti is expressed as vi . In the sample (1) we can get a set of equations:v1 − ma = 2 + ²1 ,v1 − mb = −1 + ²2 ,v1 − mc = −3 + ²3

(2)

Similar formalism can be applied to a set of observations giving a set of equations:vi − mj = sij + ²ij(3)for all set of observation (t = ti ) and comparison star j.
By applying the least-squares regime, we define the square sum of residuals
S=X²ij 2(4)i,jwhere the summation represents the sum over all set of individual step estimates.
To minimize S, we then have to solve the following equations:
∂S=0,
∂vi
∂S=0
∂mj
1(5)

Eq. 5 yields a set of normal equations with unknowns vi and mj . In order to reduce the∂Sequations, the first set of equations ( ∂v= 0) are used to represent vi by mj :ivi =1 X(mi + sij )Ni j(6)

where Ni represents number of comparison stars used at t = ti . By replacing vi by eq. 6, eq. 5yields a reduced set of normal equations having unknowns mj .
Xjaij mj =X
isij −

X 1 XiNisik(7)kwhere aij = N1i − 1(i = j), N1i (i 6= j).
As is evident from the definition, all mj are not independent. So one of these normal equations
in eq. 7 can be replaced by defining the zero point:
m0 = 0(8)A C source program to convert observations to these normal equations to solve them, together
with a DOS executable is available at the URL:
ftp://ftp.kusastro.kyoto-u.ac.jp/pub/vsnet/others/prog/step.

Reanalysis of Photographic Observations of CY UMa in 1988
Taichi Kato (Kyoto, Japan)
Received 1997 Nov 27
1.

Introduction

CY UMa was discovered as a dwarf nova by Goranskij (1977). CY UMa was first claimed to be
an SU UMa-type, ultrashort orbital-period dwarf nova by photographic and visual observations(Kato et al. 1988). Their suggested superhump period of 0.0593 d, however, was disproved by thelater CCD observations giving superhump periods of 0.0719 0.0005 d (Kato 1995) during two
superoutbursts in December, 1991 — January, 1992 and in March, 1993, and later 0.07210 0.00003
d (Harvey, Patterson 1995) during the March, 1995 superoutburst. The author tested whether thissuperhump period was observed on the 1988 January occasion.
Among the data from the VSOLJ database, I used Fujino’s time-resolved photographic photometry for the following analysis. The data were taken on 1988 November 11, 13 and 14. The
total number of estimates was 284. Fujino used his own sequence which he had calibrated photographically using Huruhata’s variable star charts.
2.

Analysis

The resultant light curve is shown in Figure 1, which shows a characteristic slow decline seen in
the later stage of a superoutburst.
After applying heliocentric corrections and removing a general trend of decline, the data wereanalyzed using the Phase Dispersion Minimization (PDM) method (Stellingwerf 1978). The resultant theta diagram (Figure 2) clearly shows the existence of superhump variation near the frequencyof 13.8 d−1 , and rather disproves our earlier analysis (Kato et al. 1988). Adjacent signals near
12.8 d−1 and 14.8 d−1 are one-day aliases resulting from the short nightly coverage. The best
determined frequency (disregarding the possibilities of aliasing) was 13.76 d−1 , corresponding to
the superhump period of 0.0727 d.
2

Figure 1. Light curve of CY UMa

Figure 2. Period analysis of CY UMa

Figure 3. Folded light curve of CY UMa
This result confirmed superhumps were actually detected by Fujino’s photographic
photometry, and retrospectively demonstrates the efficacy of skilled photographic
photometry of detecting small amplitude variation like superhumps. The folded light curve(Figure 3) shows a typical superhump light curve having a steeper rise and a slower decline.
The total amplitude of superhumps was 0.17mag.
Acknowledgments. This work has been based on the VSOLJ Database. The author is
grateful to the VSOLJ staffs, and to S. Fujino, without his extraordinarily skillfulobservations, this star would not have received that timely attention as early as in 1988.References
Goranskij V. P. 1977, Astron. Tsirk. No. 942
Harvey D., Patterson J. 1995, PASP 107, 1055
Kato T. 1995, IBVS No. 4236
Kato T., Fujino S., Iida M., Makiguchi N., Koshiro M. 1988, VSOLJ Variable Star Bulletin 5, 18
Stellingwerf R. F. 1978, Astrophys. J. 224, 953

Prediscovery Photographic Observation of V4334 Sgr (Sakurai’s Object)
Kesao Takamizawa(65-1 Oohinata, Saku-machi, Nagano, 384-0502 Japan)
E-mail: GHA07243@niftyserve.or.jp
Received 1997 Dec. 05
S. Sakurai discovered the final helium flash candidate, V4334 Sgr on Feb. 20.806 UT, 1996
(IAUC 6322). I examined my patrol films and measured prediscovry photographic magnitude ofthis star. The observations done at Saku Observatory (Saku-machi, Nagano, Japan), using twin
10-cm patrol cameras (PENTAX 100SDUF, fl=400mm) and T-Max400 films. The magnitudes weredetermined against neighboring GSC stars.
Table 1. Observations of V4334 Sgr before discovery
JD-2400000
49457.1618
49775.2854
49831.2076
49870.1306
49943.9618
50135.3028

mag
<15.5
12.412.3
12.0
11.3
11.3

JD-2400000
49647.9028
49783.2618
49834.2479
49888.1118
50003.9347

mag
<15.0
12.2
12.2
11.9
11.4

JD-2400000
49759.3118
49788.2528
49860.1424
49922.0660
50015.8910

mag
12.4
12.3
11.8
11.4
11.2

Note: These data have been reported as Vsnet-alert 341 (http://kusastro.kyoto-u.ac.jp/vsnet/Mail/vsnet-alert/msg00341.html), previously.

Variable Star Observers League in Japan
National Science Museum
Ueno Park Taito-ku Tokyo 110 Japan4